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CONTINUUM MECHANICS
(Lecture Notes)
Zden�ek Martinec
Department of Geophysics
Faculty of Mathematics and Physics
Charles University in Prague
V Hole�sovi�ck�ach 2, 180 00 Prague 8
Czech Republice-mail: [email protected]�.cuni.cz
May 9, 2003
Preface
This text is suitable for a two-semester course on continuum mechanics. It is based on notesfrom undergraduate courses that I have taught over the last few years. The material is intendedfor use by undergraduate students of physics with a year or more of college calculus behind them.
I would like to thank Erik Grafarend, Ctirad Matyska, Detlef Wolf and Ji�r�� Zahradn��k, whoseinterest encouraged me to write this text. I would also like to thank my oldest son Zden�ek whoplotted most of �gures embedded in the text.
Readers of this text are encouraged to contact me with their comments, suggestions, andquestions. I would be very happy to hear what you think I did well and I could do better. Mye-mail address is [email protected]�.cuni.cz and a full mailing address is found on the titlepage.
Zden�ek Martinec
ii
Contents
Preface
1. GEOMETRY OF DEFORMATION
1.1 Body, con�gurations, and motion1.2 Description of motion1.3 Lagrangian and Eulerian coordinates1.4 Lagrangian and Eulerian variables1.5 Deformation gradient1.6 Polar decomposition of the deformation gradient1.7 Measures of deformation1.8 Length and angle changes1.9 Area and volume changes1.10 Strain invariants, principal strains1.11 Displacement vector1.12 Geometrical linearization
1.12.1 Linearized analysis of deformation1.12.2 Length and angle changes1.12.3 Area and volume changes
2. KINEMATICS
2.1 Material and spatial time derivatives2.2 Time changes of some geometric objects2.2 Reynolds's transport theorem2.3 Modi�ed Reynolds's transport theorem
3. MEASURES OF STRESS
3.1 Body and surface forces, mass density3.2 Cauchy traction principle3.3 Other measures of stress
4. FUNDAMENTAL BALANCE LAWS
4.1 Global balance laws4.2 Local balance laws in the spatial description4.2.1 Continuity equation4.2.2 Equation of motion4.2.3 Symmetry of the Cauchy stress tensor4.2.4 Energy equation4.2.5 Entropy inequality4.2.6 R�esum�e of local balance laws
4.3 Jump conditions in special cases
iii
4.4 Local balance laws in the referential description4.4.1 Continuity equation4.4.2 Equation of motion4.4.3 Symmetries of the Cauchy stress tensors4.4.4 Energy equation4.4.5 Entropy inequality
5. CONSTITUTIVE EQUATIONS
5.1 The need for constitutive equations5.2 Formulation of thermomechanical constitutive equations5.3 Simple materials5.4 Frame indi�erence5.5 Reduction by polar decomposition5.6 Kinematic constraints5.7 Material symmetry5.8 Noll's rule5.9 Material symmetry of frame indi�erent functionals5.10 Isotropic materials5.11 Solids5.12 Fluids5.13 Current con�guration as reference5.14 Isotropic constitutive functionals in relative representation5.15 A general constitutive equation of a uid5.16 The principle of bounded memory5.17 Representation theorems for isotropic functions
5.17.1 Representation theorem for a scalar isotropic function5.17.2 Representation theorem for a vector isotropic function5.17.3 Representation theorem for a symmetric tensor isotropic function
5.18 Examples of isotropic materials with bounded memory5.18.1 Elastic solid5.18.2 Thermoelastic solid5.18.3 Kelvin-Voigt viscoelastic solid5.18.4 Maxwell viscoelastic solid5.18.5 Elastic uid5.18.6 Thermoelastic uid5.18.7 Viscous uid5.18.8 Incompressible viscous uid5.18.9 Viscous heat-conducting uid
6. ENTROPY PRINCIPLE
6.1 The Clausius-Duhem inequality6.2 Application of the Clausius-Duhem inequality to a classical viscous heat-conducting uid6.3 Application of the Clausius-Duhem inequality to a classical viscous heat-conducting in-
compressible uid6.4 Application of the Clausius-Duhem inequality to a classical thermoelastic solid6.5 The M�uller entropy principle6.6 Application of the M�uller entropy principle to a classical thermoelastic uid
iv
7. CLASSICAL LINEAR ELASTICITY
7.1 Linear elastic solid7.2 The elastic tensor7.3 Isotropic linear elastic solid7.4 Restrictions on elastic coeÆcients7.5 Field equations7.5.1 Isotropic linear elastic solid7.5.2 Incompressible isotropic linear elastic solid7.5.3 Isotropic linear thermoelastic solid
8. SMALL MOTIONS IN A MEDIUM WITH A FINITE PRE-STRESS
8.1 Equations for the initial state8.2 Application of an in�nitesimal deformation8.3 Lagrangian and Eulerian increments8.4 Linearized continuity equation8.5 Increments in stress8.6 Linearized equation of motion8.7 Linearized interface conditions8.7.1 Kinematic interface conditions8.7.2 Dynamic interface conditions
8.8 Linearized elastic constitutive equation8.9 Gravitational potential theory8.9.1 Equations for the initial state8.9.2 Increments in gravitation8.9.3 Linearized Poisson's equation8.9.4 Linearized interface condition for potential increments8.9.5 Linearized interface condition for increments in gravitation8.9.6 Boundary-value problem for the Eulerian potential increment8.9.7 Boundary-value problem for the Lagrangian potential increment8.9.8 Linearized integral relations
8.10 Equation of motion for a self-gravitating body
Appendix A. Vector di�erential identities
Appendix B. Fundamental formulae for surfaces
B.1 Tangent vectors and tensorsB.2 Surface gradientB.3 IdentitiesB.4 Curvature tensorB.5 Divergence of vector, Laplacian of scalarB.6 Gradient of vectorB.7 Divergence of tensorB.8 Vector ~n � grad grad�
Appendix C. Orthogonal curvilinear coordinates
C.1 Coordinate transformationC.2 Base vectorsC.3 Derivatives of unit base vectors
v
C.4 Derivatives of vectors and tensorsC.5 Invariant di�erential operators
C.5.1 Gradient of a scalarC.5.2 Divergence of a vectorC.5.3 Curl of a vectorC.5.4 Gradient of a vectorC.5.5 Divergence of a tensorC.5.6 Laplacian of a scalar and a vector
Literature
vi
Notation(not �nished yet)
b entropy source per unit mass~f body force per unit mass~F Lagrangian description of ~fh heat source per unit mass~n normal at �~N normal at �0~q heat ux at �~Q heat ux at �0s bounding surface at �S bounding surface at �0~s ux of entropy at �~S ux of entropy at �0t time~t(~n) stress vector on a surface with the external normal ~n
t(~x; t) Eulerian Cauchy stress tensor
t( ~X; t) Lagrangian Cauchy stress tensor
T (1) �rst Piola-Kirchho� stress tensor
T (2) second Piola-Kirchho� stress tensorv volume at �V volume at �0~v velocity at �~V Lagrangian description of ~v~x Eulerian Cartesian coordinates~X Lagrangian Cartesian coordinatesE total internal energyH total entropyK total kinetic energyW total mechanical power" internal energy density per unit mass� entropy density per unit mass% mass density at �% Lagrangian description of %%0 mass density at �0� present con�guration�0 reference con�guration� singular surface at �� singular surface at �0� temperature~� the speed of singular surface �� total entropy production
vii
1. GEOMETRY OF DEFORMATION
1.1 Body, con�gurations, and motion
The object of all consideration in continuum mechanics and the domain of all physical quantitiesis the material body. A material body B = fXg is a compact measurable set of elements X ,called the material particles or material points, which can be put into one-to-one correspondencewith triplets of real numbers. Such triplets are sometimes called the intrinsic coordinates ofthe particles. Note that whereas a "particle" of classical mechanics has an assigned mass, a"continuum particle" is essentially a material point for which a density is de�ned.
A body is available to us only in its con�guration. The con�guration � of B is the speci�cationof the position of all particles of B in the physical space E3 (usually the Euclidean space). Oftenit is convenient to select one particular con�guration, the reference con�guration �0, and refereverything concerning the body to that con�guration. Mathematically, the de�nition of thereference con�guration �0 is expressed by mapping
~ 0 : B ! E3
X ! ~X = ~ 0(X ) ; (1.1)
where ~X is the position occupied by the particle X in the reference con�guration �0, as indicatedin Figure 1.1.
The choice of reference con�guration is arbitrary. It may be any smooth image of the bodyB, need not even be a con�guration ever occupied by the body. For some choice of �0, we mayget a particularly simple description, just as in geometry one choice of coordinates may lead toa simple equation for a particular �gure, but the reference con�guration itself has nothing to dowith such motions as it may be used to describe, just as the coordinate system has nothing to dowith geometrical �gures themselves. A reference con�guration is introduced so as to allow us toemploy the mathematical apparatus of Euclidean geometry.
Under the in uence of external loads, the body B deforms, moves and changes its con�gura-tion. The con�guration of body B at the present time t is called the present con�guration �t andis de�ned by mapping
~ t : B ! E3
X ! ~x = ~ t(X ; t) ; (1.2)
where ~x is the position occupied by the particle X in the present con�guration �t.A motion of body B is a sequence of mappings ~� between the reference con�guration �0 and
the present con�guration �t:~� : E3 ! E3
~X ! ~x = ~�( ~X; t) :(1.3)
This equation states that the motion takes a material point X from its position ~X in the referencecon�guration �0 to a position ~x in the present con�guration �t. The functional form for a givenmotion as described by (1.3) depends on the choice of the reference con�guration. If morethan one reference con�guration is used in the discussion, it is necessary to label the functionaccordingly. For example, relative to two reference con�gurations �1 and �2 the same motioncould be represented symbolically by the two equations ~x = ~��1(
~X; t) = ~��2(~X; t).
1
Æ~X
~x
X
~ 0(X )~ t(X ; t)
~�( ~X; t)
�0
�t
B
Figure 1.1. Body, reference con�guration versus present con�guration.
It is often convenient to change the reference con�guration in the description of motion. To seehow the motion is described in a new reference con�guration, consider two di�erent con�gurations�� and �t of body B at two di�erent times � and t:
~� = ~�( ~X; �) ; ~x = ~�( ~X; t) ; (1.4)
that is, ~� is the place occupied at time � by the particle that occupies ~x at time t. Since thefunction ~� is invertible, that is,
~X = ~��1(~�; �) = ~��1(~x; t) ; (1.5)
we have either~� = ~�(~��1(~x; t); �) =: ~�t(~x; �) ; (1.6)
or~x = ~�(~��1(~�; �); t) =: ~�� (~�; t) : (1.7)
The map ~�t(~x; �) de�nes the deformation of the new con�guration �� of the body B relativeto the present con�guration �t, which is considered as reference. On the other hand, the map~�� (~�; t) de�nes the deformation of the new reference con�guration �� of the body B onto thecon�guration �t. Evidently, it holds
~�t(~x; �) = ~���1(~x; t) : (1.8)
The functions ~�t(~x; �) and ~�� (~�; t) are called the relative motion functions. The subscripts t and� at functions � are used to recall which con�guration is taken as reference.
We assume that the functions ~ 0, ~ t, ~�, ~�t and ~�� are single-valued and possess continuouspartial derivatives with respect to their arguments for whatever order is desired, except possibly
2
at some singular points, curves, and surfaces. Moreover, each of these function can uniquelly beinverted. This assumption is known as the axiom of continuity. It expresses the fact that thematter is indestructible, that is, no region of positive, �nite volume of matter is deformed intoa zero or in�nite volume. Another implication of this axiom is that the matter is impenetrable,that is, the motion caries every region into a region, every surface into a surface, and everycurve into a curve. One portion of matter never penetrates into another. In practice, there arecases in which this axiom is violated. We cannot describe such processes as the creation of newmaterial surfaces, that is, we cannot describe the cutting, tearing, or the propagation of cracks,etc. Continuum theories dealing with such processes must weaken the continuity assumption inthe neighbourhood of those parts of the body, where the map (1.3) becomes discontinuous. Theaxiom of continuity is secured through the well-known implicit function theorem.
It is common practice in continuum mechanics to write the mapping (1.3) in the alternativeform
~x = ~x( ~X; t) (1.9)
with understanding that the symbol ~x on the right-hand side of the equation represents thefunction whose arguments are ~X and t while the same symbol on the left-hand side representsthe value of the function, that is a point in the space. We shall use this notation frequently inthe text that follows.
1.2 Description of motion
The motion can be described by four types of description. Under the assumption that thefunctions ~�0, ~�t, ~�, ~�t and ~�� are di�erentiable and invertible, all descriptions are equivalent. Wecall them the:
� Material description, given by the mapping (1.2), whose independent variables are theabstract particle X and the time t.
� Referential description, given by the mapping (1.3), whose independent variables are theposition ~X of the particle X in an arbitrarily chosen reference con�guration, and the time t.When the reference con�guration is chosen to be the actual initial con�guration at t = 0, thereferential description is often called the Lagrangian description, although many authorscall it the material description, using the particle position ~X in the reference con�guration asa label for the material particle X in the material description.
� Spatial description, whose independent variables are the present position ~x accupied by theparticle at the time t and the present time t. It is the description most used in uid mechanics,often called the Eulerian description.
� Relative description, given by the mapping (1.6), whose independent variables are thepresent position ~x of the particle and a variable time � . The variable time � is the timewhen the particle occupied another position ~� and the motion is described with ~� as dependentvariable. Alternatively, the motion can be described by the mapping (1.7), whose independentvariables are the position ~� at time � and the present time t. These two relative descriptions areactually special cases of the referential description, di�ering from the Lagrangian descriptionin that the reference positions are now denoted by ~x at time t and ~� at time � , respectively,instead of ~X at time t = 0.
3
�O P~P
~I1~I2
~I3
X1
X2
X3
S
�0
V
t = 0
�o~i1
~i2
~i3
x1
x2
x3
p~p
s
�t
v
t
Figure 1.2. Lagrangian and Eulerian coordinates for the reference con�guration �0 and the presentcon�guration �t.
1.3 Lagrangian and Eulerian coordinates
Consider now that the position ~X of the particle X in the reference con�guration �0 may beassigned by three rectangular Cartesian coordinates XK , K = 1; 2; 3, or by a position vector ~Pthat extends from an origin O of the coordinates to the point P , the place of the particle X in thereference con�guration, as indicated in Figure 1.2. The notations ~X and ~P are interchangeable ifwe use only one reference origin for the position vector ~P , but both "place" and "particle" havemeaning independent of the choice of origin. The same position ~X (or place P ) may have manydi�erent position vectors ~P corresponding to di�erent choices of origin, but the relative positionvectors d ~X = d~P of neighboring positions will be the same for all origins.
At the present con�guration �t, a material particle X occupies the position ~x or a spatialplace p. We may locate place p by a position vector ~p extending from the origin o of a newset of rectangular coordinates xk, k = 1; 2; 3. Following the current terminology, we shall callXK the Lagrangian or material coordinates and xk the Eulerian or spatial coordinates. In nextconsiderations we assume that these two coordinate systems, one for the reference con�guration�0 and one for the present con�guration �t, are nonidentical.
The reference position ~X of a point P in �0 and the present position ~x of p in �t, respectively,referred to Cartesian coordinates XK and xk are given by
~X = XK~IK ; ~x = xk~ik : (1.10)
where ~IK and ~ik are the respective unit base vectors in Figure 1.2. and the usual summationconvention over repeated indices is employed. Since the rectangular Cartesian coordinates areemployed, the base vectors are mutually orthogonal,
~IK � ~IL = ÆKL ; ~ik �~il = Ækl ; (1.11)
4
where ÆKL and Ækl are the Kronecker symbols, which are equal to 1 when the two indices are equaland zero otherwise; the dot `�' stands for the scalar product of vectors. 1
When the two Cartesian coordinates are not identical, we shall also express a unit base vectorin one coordinates in terms of its projection in another coordinates. It is readily to show that
~ik = ÆkK~IK ; ~IK = ÆKk~ik ; (1.12)
whereÆkK :=~ik � ~IK ; ÆKk := ~IK �~ik (1.13)
are called shifters. They are not a Kronecker symbol except when the two coordinate systemsare identical. It is clear that (1.13) is none other than the cosine directors of the two coordinatesxk and XK . From the identity
Ækl~il =~ik = ÆkK~IK = ÆkKÆKl~il ;
we �nd thatÆkKÆKl = Ækl ; ÆKkÆkL = ÆKL : (1.14)
The notation convention will be such that the quantities associated with the reference con-�guration �0 will be denoted by capital letters, and the quantities associated with the presentcon�guration �t by lower case letters. When these quantities are referred to coordinates XK , theirindices will be majuscules; and when they are referred to xk, their indices will be minuscules. Forexample, a vector ~V in �0 referred to XK will have components VK , and referred to xk will havethe components Vk:
VK = ~V � ~IK ; Vk = ~V �~ik : (1.15)
Using (1.12), the components VK and Vk can be related by
VK = VkÆkK ; Vk = VKÆKk : (1.16)
Conversely, we consider vector ~v in �t that, in general, di�ers from ~V . Its components vK andvk referred to XK and xk, respectively, are
vK = ~v � ~IK ; vk = ~v �~ik : (1.17)
Again, using (1.12), the components vK and vk can be related by
vK = vkÆkK ; vk = vKÆKk : (1.18)
1.4 Lagrangian and Eulerian variables
The coordinate form of the motion (1.3) is
xk = �k(X1;X2;X3; t) � xk(X1;X2;X3; t) ; k = 1; 2; 3 ; (1.19)
1If various curvilinear coordinate systems are employed, the appropriate form of equations in these systems canbe obtained by means of the well-known transformation rules. For instance, the partial derivatives in the Cartesiancoordinates must be replaced by the partial covariant derivatives. However, in general considerations, we shallalways rely on the above introduced Cartesian systems.
5
or, conversely,
XK = ��1K (x1; x2; x3; t) � XK(x1; x2; x3; t) ; K = 1; 2; 3 : (1.20)
According to the implicit function theorem, the mathematical condition that guarantees theexistence of such an unique inversion is the non-vanishing of the jacobian determinant J , that is,
J( ~X; t) := det
�@�k@XK
�� det
�@xk@XK
�6= 0 : (1.21)
Every scalar, vector or tensor physical quantity Q de�ned for the body B such as the den-sity, the temperature, or the velocity has both an Eulerian description q(~x; t) and a Lagrangiandescription Q( ~X; t). In the Lagrangian description, attention is focused on what is happening tothe individual particles during the motion, whereas in the Eulerian description the emphasis is di-rected to the events taking place at speci�c points in space. If, for instance, Q is the temperature,then Q( ~X; t) gives the temperature recorded by a thermometer attached to a moving particle ~X,whereas q(~x; t) gives the temperature recorded at a �xed point ~x in space. The relation betweenthese two descriptions is
Q( ~X; t) = q(~x( ~X; t); t) ; q(~x; t) = Q( ~X(~x; t); t) ; (1.22)
where the small and capital letters emphasize di�erent functional forms resulting from the changein variables. We can see that the Lagrangian and Eulerian variablesQ( ~X; t) and q(~x; t) are referredto the reference con�guration �0 and the present con�guration �t of the body B, respectively.
As an example, we demonstrate the de�nition of the Lagrangian and Eulerian variables for avector quantity ~V . Let us assume that ~V is given in the Eulerian description,
~V � ~v(~x; t) : (1.23)
Vector ~v may be expressed in the Lagrangian or Eulerian components vK(~x; t) and vk(~x; t) as
~v(~x; t) = vK(~x; t)~IK = vk(~x; t)~ik : (1.24)
The Lagrangian description of ~V, the vector ~V ( ~X; t), is de�ned by (1.22)1:
~V ( ~X; t) := ~v(~x( ~X; t); t) : (1.25)
Representing ~V ( ~X; t) in the Lagrangian or Eulerian components VK( ~X; t) and Vk( ~X; t),
~V ( ~X; t) = VK( ~X; t)~IK = Vk( ~X; t)~ik ; (1.26)
the de�nition (1.25) can be interpreted in two possible component forms:
VK( ~X; t) := vK(~x( ~X; t); t) ; Vk( ~X; t) := vk(~x( ~X; t); t) : (1.27)
Expressing the Lagrangian components vK in terms of the Eulerian components vk according to(1.18), we also have
VK( ~X; t) = vk(~x( ~X; t); t)ÆkK ; Vk( ~X; t) = vK(~x( ~X; t); t)ÆKk : (1.28)
6
An analogous consideration can be carried out for the Eulerian variables vK(~x; t) and vk(~x; t) inthe case that ~V is represented by the Lagrangian variables VK( ~X; t) and Vk( ~X; t).
1.5 Deformation gradient
From (1.19) and (1.20), for �xed time, we have
dxk = xk;KdXK ; dXK = XK;kdxk ; (1.29)
where indices following a comma represent partial di�erentiation with respect to XK , when theyare majuscules, and with respect to xk when they are minuscules, that is,
xk;K :=@xk@XK
; XK;k :=@XK
@xk: (1.30)
The two sets of quantities de�ned by (1.30) are components of thematerial and spatial deformationgradient tensors F and F�1, respectively,
F ( ~X; t) := xk;K( ~X; t)(~ik ~IK) ; F�1(~x; t) := XK;k(~x; t)(~IK ~ik) ; (1.31)
where symbol denotes the dyadic product of vectors. In symbolic notation, we have 2
F ( ~X; t) := (Grad ~x)T ; F�1(~x; t) := (grad ~X)T : (1.32)
The deformation gradients F and F�1 are two-point tensor �elds because they relate a vectord~x in the present con�guration to a vector d ~X in the reference con�guration. Their componentstransform like those of a vector under rotations of only one of two reference axes and like a two-point tensor when the two sets of axes are rotated independently. In symbolic notation, equation(1.29) appears in the form
d~x = dxk~ik = F � d ~X ; d ~X = dXK~IK = F�1 � d~x : (1.33)
The material deformation gradient F can thus be thought of as a mapping of the in�nitesimal vec-tor d ~X of the reference con�guration into the in�nitesimal vector d~x of the current con�guration;the inverse mapping is performed by the spatial deformation gradient F�1.
Through the chain rule of partial di�erentiation it is clear that
xk;KXK;l = Ækl ; XK;kxk;L = ÆKL ; (1.34)
or written it in symbolic notation:
F � F�1 = F�1 � F = I ; (1.35)
where I is the identity tensor. Strictly speaking, there are two identity tensors I, one in theLagrangian coordinates and one in the Eulerian coordinates; we shall, however, disregard thissubtlety. Equation (1.35) shows that the spatial deformation gradient F�1 is the inverse tensorof the material deformation gradient F . Each of the two sets of equations (1.34) consists of nine
2"Grad" denotes the gradient operator applied to functions of ~X , while "grad" referes to functions of ~x, thetime t being held constant in each case. The gradient of a vector function ~� is de�ned by the left dyadic productof the nabla operator ~r with ~�, that is, grad ~� := ~r~�. Hence, F = (~r~x)T .
7
linear equations for the nine unknown xk;K or XK;k. Since the jacobian is assumed not to vanish,a unique solution exists and, according to Cramer's rule of determinants, the solution for XK;k
may be obtained in terms of xk;K as
XK;k =cofactor (xk;K)
J=
1
2J�KLM�klmxl;Lxm;M ; (1.36)
where �KLM and �klm are the permutation symbols, and
J := det (xk;K) =1
3!�KLM�klmxk;Kxl;Lxm;M = detF : (1.37)
Note that the jacobian J is identical with the determinant of tensor F only in the case that boththe Lagrangian and Eulerian coordinates are of the same type, as for instance, in our case whenboth are the Cartesian coordinates. If the Lagrangian coordinates are of di�erent type than theEulerian coordinates, the volume element in these coordinates is not the same and, consequently,the jacobian J di�ers from the determinant of F .
By di�erentiating (1.36) and (1.37), we get the following Jacobi identities:
d J
d xk;K= JXK;k ;
(JXK;k);K = 0 ; or (J�1xk;K);k = 0 :
(1.38)
The �rst identity can be proved as follows:
d J
d xr;R=
1
3!�KLM�klm
"@xk;K@xr;R
xl;Lxm;M + xk;K@xl;L@xr;R
xm;M + xk;Kxl;L@xm;M
@xr;R
#
=1
3!
��RLM�rlmxl;Lxm;M + �KRM�krmxk;Kxm;M + �KLR�klrxk;Kxl;L
�
=1
2�RLM�rlmxl;Lxm;M = cofactor (xr;R) = JXR;r :
Furthermore, multiplying (1.36) by J and di�erentiating the result with respect to XK yields
(JXK;k);K =1
2�KLM�klm(xl;LKxm;M + xl;Lxm;MK)
= �KLM�klmxl;LKxm;M :
For k = 1 we have
(JXK;1);K = �KLM�1lmxl;LKxm;M
= �KLM�123x2;LKx3;M + �KLM�132x3;LKx2;M
= �KLM(x2;LKx3;M � x3;LKx2;M )
= x2;21x3;3 � x3;21x2;3 � x2;31x3;2 + x3;31x2;2 + x2;32x3;1 � x3;32x2;1
�x2;12x3;3 + x3;21x2;3 + x2;13x3;2 � x3;13x2;2 � x2;23x3;1 + x3;23x2;1
= 0 :
Likewise, the above relation vanishes for k = 2; 3, which proves (1.38)2.
8
Making use of the identity
d
dF(detF ) = (detF )F�T ; (1.39)
which is valid for all invertible second-order tensors F , the above Jacobi identities can be writtenin symbolic notation:
d J
dF= JF�T ;
Div (JF�1) = ~0 ; or div (J�1F ) = ~0 :
(1.40)
Furthermore, the two identities 3
(detF )[( ~A� ~B) � ~C] = [(F � ~A)� (F � ~B)] � (F � ~C) ;(detF )F�T � ( ~A� ~B) = (F � ~A)� (F � ~B) ;
(1.41)
which are valid for all vectors ~A, ~B and ~C, enable to write (1.36) and (1.37) in the form
J�KLMXK;k = �klmxl;Lxm;M ; and J�KLM = �klmxk;Kxl;Lxm;M : (1.42)
The di�erential operations in the Eulerian coordinates applied to Eulerian variables can beconverted to the di�erential operations in the Lagrangian coordinates applied to Lagrangianvariables by means of the chain rule of di�erentiation. For instance, the following identities canbe proved:
grad � = F�T �Grad � ;div � = F�T :: Grad � ;
div grad � = F�1 � F�T :: GradGrad �+divF�T �Grad � ;(1.43)
or, conversely,
Grad � = F T � grad � ;Div � = F T :: grad � ;
DivGrad � = F � F T :: grad grad �+DivF T � grad � ;(1.44)
where the symbol :: denotes the double-dot product of tensors.
1.6 Polar decomposition of the deformation gradient
The basic properties of the local behavior of deformation emerge from the possibility to decomposethe deformation into a rotation and stretch which, roughly speaking, is a change of the shapeof the volume element. This decomposition is called the polar decomposition of the deformationgradient, and it is summarized in the following theorem.
A non-singular tensor F (detF 6= 0) admits the polar decompositions such that
F = R �U = V �R ; (1.45)
where the particular factors have the following properties:3Note that F�T �
�FT��1��F�1�T
.
9
1. The tensors U and V are symmetric and positive de�nite.
2. The tensor R is orthogonal, R �RT = RT �R = I , i.e., R is a rotation tensor.
3. U , V and R are uniquely determined.
4. The eigenvalues of U and V are identical; if ~e is an eigenvector of U , thenR�~e is an eigenvectorof V .
As a preliminary to proving these statements, we note that an arbitrary tensor T is positivede�nite if ~v � T � ~v > 0 for all vectors ~v 6= ~0. A necessary and suÆcient condition for T to bepositive de�nite is that all its eigenvalues be positive. In this regard, consider the tensor C,C := F T �F . Since F is assumed to be non-singular (detF 6= 0) and F �~v 6= ~0 if ~v 6= ~0, it followsthat (F � ~v) � (F � ~v) is a sum of squares and hence greater than zero. Thus
0 < (F � ~v) � (F � ~v) = ~v � F T � F � ~v = ~v �C � ~v ;
and C is positive de�nite. By the same arguments, we may show that the tensor b, b := F �F T ,is also positive de�nite.
The positive roots of C and b de�ne two tensors U and V ,
U :=pC =
pF T � F ; V :=
pb =
pF � F T : (1.46)
The tensors U and V , called the right and left stretch tensors, are symmetric, positive de�niteand are uniquely determined. Next, two tensors R and ~R are de�ned by
R := F �U�1 ; ~R := V �1 � F : (1.47)
We recognize that both are orthogonal since by de�nition we have
R �RT = (F �U�1) � (F �U�1)T = F �U�1 �U�1 � F T = F �U�2 � F T =
= F � (F T � F )�1 � F T = F � F�1 � F�T � F T = I :
A similar proof holds for ~R. So far we have demonstrated two decompositions F = R �U = V � ~R,where U and V are symmetric, positive de�nite and R and ~R are orthogonal. From
F = V � ~R = ( ~R � ~RT) � V � ~R = ~R � ( ~RT � V � ~R) = ~R � ~U ;
it may be concluded that there might be two decompositions of F , namely F = R � U and
F = ~R � ~U . However, if this were true we were forced to conclude that C = F T � F = ~U2= U2,
whence follows that ~U = U , because of the uniqueness of the positive root. This implies ~R = R;consequently, U , V and R are unique. Finally, we assume ~e and � to be an eigenvector andeigenvalue of U . Then, we have �~e = U �~e, as well as �R �~e = (R �U) �~e = (V �R) �~e = V � (R �~e).Thus � is also eigenvalue of V and R � ~e is an eigenvector. This completes the proof of thetheorem.
It is instructive to write the relation (1.45) in componental form. We have
FkK = RkLULK = VklRlK ; (1.48)
which means that R is a two-point tensor while U and V are ordinary (one-point) tensors.
10
Equation d~x = F �d ~X shows that the deformation gradient F can be thought of as a mappingof the in�nitesimal vector d ~X of the reference con�guration into the in�nitesimal vector d~x ofthe current con�guration. The theorem of polar decomposition replaces the linear transformationd~x = F � d ~X by two sequential transformations, by rotation and stretching, where the sequenceof these two steps may be interchanged, as illustrated in Figure 1.3. The combination of rotationand stretching corresponds to the multiplication of two tensors, namely, R and U or V and R,
d~x = (R �U) � d ~X = (V �R) � d ~X : (1.49)
However, R should not be understood as a rigid body rotation since, in general case, it varies frompoint to point. Thus the polar decomposition theorem re ects only a local property of motion.
~I3 ~I2
~I1
~i3
~i2~i1
~X
U
R
F
R
V~x
d ~X d~x
Figure 1.3. Polar decomposition of deformation gradient.
1.7 Measures of deformation
Local changes in the geometry of continuous bodies can be described, as usual in di�erentialgeometry, by the changes in the metric tensor. In the Euclidean space, it is particularly simple toaccomplish. Consider a material point ~X and an in�nitesimal material vector d ~X . The changes inthe length of three such linearly independent vectors describe the local changes in the geometry.
The in�nitesimal vector d ~X in �0 is mapped onto the in�nitesimal vector d~x in �t. The metricproperties of the present con�guration �t can be described by the square of the length of d~x:
ds2 = d~x � d~x = (F � d ~X) � (F � d ~X) = d ~X �C � d ~X = C :: (d ~X d ~X) ; (1.50)
where the Green deformation tensor C de�ned by
C( ~X; t) := F T � F ; (1.51)
11
has already been used in the proof of the polar decomposition theorem. Equation (1.50) describesthe local geometry property of the present con�guration �t with respect to that of the referencecon�guration �0. Alternatively, we can use the inverse tensor expressing the geometry of thereference con�guration �0 relative to that of the present con�guration �t. We have
dS2 = d ~X � d ~X = (F�1 � d~x) � (F�1 � d~x) = d~x � c � d~x = c :: (d~x d~x) ; (1.52)
where c is the Cauchy deformation tensor de�ned by
c(~x; t) := F�T � F�1 : (1.53)
Both tensors c and C are symmetric, that is, c = cT and C = CT . We see that, in contrast tothe deformation gradient tensor F with generally nine independent components, the changes inmetric properties, following from the changes of con�guration, are described by six independentcomponents of the deformation tensors c or C.
Apart from the Cauchy deformation tensor c and the Green deformation tensor C, we canintroduce other equivalent geometrical measures of deformation. Equation (1.50) and (1.52)yield two di�erent expressions of the squares of element of length, ds2 and dS2. The di�erenceds2 � dS2 for the same material points is a relative measure of the change of length. When thisdi�erence vanishes for any two neighboring points, the deformation has not changed the distancebetween the pair. When it is zero for all points in the body, the body has undergone only a rigiddisplacement. From (1.50) and (1.52) for this di�erence we obtain
ds2 � dS2 = dX � 2E � dX = dx � 2e � dx ; (1.54)
where we have introduced the Lagrangian and Eulerian strain tensors, respectively,
E( ~X; t) :=1
2(C � I) ; e(~x; t) :=
1
2(I � c) : (1.55)
Clearly, when either vanishes, ds2 = dS2. Both tensors e and E are symmetric, that is, e = eT
and E = ET . Therefore, in three dimensions there are only six independent components for eachof these tensors, for example, E11, E22, E33, E12 = E21, E13 = E31, and E23 = E32. The �rstthree components E11, E22, and E33 are called normal strains and the last three E12, E13, andE23 are called shear strains. The reason for this will be discussed later in this chapter.
Two other equivalent measures of deformation are the reciprocal tensors b and B (known asthe Finger and Piola deformation tensors, respectively) de�ned by
b(~x; t) := F � F T ; B( ~X; t) := F�1 � F�T : (1.56)
They satisfyb � c = c � b = I ; B �C = C �B = I ; (1.57)
which can be shown by mere substitution of (1.53) and (1.56).We have been using the word tensor for quantities such as C. This term referees to a set
of quantities that transform according to a certain de�nite law upon coordinate transformation.Suppose that Lagrangian Cartesian coordinates XK are transformed into X 0
K according to
XK = XK(X01;X
02;X
03) : (1.58)
12
The left-hand side of (1.50) is independent of the coordinate transformations. If we substitute
dXK =@XK
@X 0M
dX 0M ;
on the right-hand side of (1.50), we obtain
ds2 = CKLdXKdXL = CKL@XK
@X 0M
@XL
@X 0N
dX 0MdX
0N
!= C 0MNdX
0MdX
0N :
Hence
C 0MN ( ~X0; t) = CKL( ~X; t)
@XK
@X 0M
@XL
@X 0N
(1.59)
since dX 0M is arbitrary and CKL = CLK . Thus, knowing CKL in one set of coordinates XK , we
can �nd the corresponding quantities in another set X 0K once the relations (1.58) between XK
and X 0K are given. Quantities that transform according to the law of transformation (1.59) are
known as absolute tensors.
1.8 Length and angle changes
A geometrical meaning of the normal strains E11, E22 and E33 is provided by considering thelength and angle changes as a result of deformation. We consider material line element d ~X of thelength dS which deforms to the element d~x of the length ds. Let ~K be the unit vector along d ~X,
~K :=d ~X
dS: (1.60)
The relative change of length,
E( ~K) :=ds� dS
dS; (1.61)
is called the extension or elongation. Dividing (1.54)1 by dS2, we have
ds2 � dS2
dS2= ~K � 2E � ~K : (1.62)
Expressing the left-hand side by the extension E( ~K), we get the quadratic equation for E( ~K):
E( ~K)(E( ~K) + 2)� ~K � 2E � ~K = 0 : (1.63)
From the two possible solutions of this equation, we choose physically admissible one:
E( ~K) = �1 +q1 + ~K � 2E � ~K : (1.64)
Since ~K is a unit vector, ~K � 2E � ~K + 1 = ~K � C � ~K. As we have proved in Section 1.6, theGreen deformation tensor C is positive symmetric, that is, ~K �C � ~K > 0. Hence, the argumentof square root in (1.64) is positive. Particularly, when ~K is taken along the X1�axis, this gives
E(1) = �1 +p1 + 2E11 : (1.65)
13
The geometrical meaning of the shear strains E12, E13, and E23 is found by considering theangles between two directions ~K(1) and ~K(2),
~K(1) :=d ~X(1)
dS(1); ~K(2) :=
d ~X(2)
dS(2): (1.66)
The angle � between these vectors in the reference con�guration �0,
cos� =d ~X(1)
dS(1)� d
~X(2)
dS(2); (1.67)
is changed by deformation to
cos � =d~x(1)
ds(1)� d~x
(2)
ds(2)=F � d ~X(1)
ds(1)� F � d
~X(2)
ds(2)=d ~X(1) �C � d ~X(2)
ds(1) ds(2): (1.68)
By (1.60) and (1.61), we further have
cos � = ( ~K(1) �C � ~K(2))dS(1)
ds(1)dS(2)
ds(2)=
~K(1) �C � ~K(2)
(E( ~K1)
+ 1)(E( ~K2)
+ 1); (1.69)
which can be rewritten in terms of the Lagrangian strain tensor as
cos � =~K(1) � (I + 2E) � ~K(2)q
1 + ~K(1) � 2E � ~K(1)
q1 + ~K(2) � 2E � ~K(2)
: (1.70)
When ~K(1) is taken along X1�axis and ~K(2) along X2�axis, (1.70) reduces to
cos �(12) =2E12p
1 + 2E11
p1 + 2E22
: (1.71)
~I1 ~I2
~I3
~i1
~i2
~i3
Pp
~X
~u
~x
~b
d ~X(1)d ~X(2)
d ~X(3)
d~x(1)
d~x(2)
d~x(3)
Figure 1.4. Deformation of an in�nitesimal rectilinear parallelepiped.
14
1.9 Area and volume changes
Let us investigate the change of area and volume with deformation. The oriented element of areabuilt on the edge vectors d ~X(1) and d ~X(2) after deformation becomes to the oriented area withedge vectors d~x(1) and d~x(2). Thus the deformed area is given by
d~a := d~x(1) � d~x(2) = (F � d ~X(1))� (F � d ~X(2)) = JF�T � (d ~X(1) � d ~X(2)) ;
where we have used the identity (1.41)2. Substituting for the undeformed oriented area d ~A,
d ~A := d ~X(1) � d ~X(2) ; (1.72)
we haved~a = JF�T � d ~A ; or dak = JXK;kdAK : (1.73)
To express the interface conditions at discontinuity surfaces in the Lagrangian description, weneed to �nd the relation between the unit normals ~n and ~N to the deformed discontinuity � andthe undeformed discontinuity �, respectively. Considering the transformation (1.73) between thespatial and referential surface elements, and writting d~a = ~nda, d ~A = ~NdA, we readily �nd that
da = J
q~N �B � ~N dA ; (1.74)
where B is the Piola deformation tensor, B = F�1 � F�T . Combining (1.73) with (1.74) resultsin
~n =~N � F�1p~N �B � ~N
: (1.75)
To calculate the deformed volume element, let us consider the in�nitesimal rectilinear paral-lelepiped in the reference con�guration �0 spanned by the vectors d ~X(1), d ~X(2) and d ~X(3), seeFigure 1.4. Its volume is given by the scalar product of d ~X(3) with d ~X(1) � d ~X(2):
dV := (d ~X(1) � d ~X(2)) � d ~X(3) : (1.76)
In the present con�guration �t, the volume of the parallelepiped is
dv := (d~x(1) � d~x(2)) � d~x(3) = [(F � d ~X(1))� (F � d ~X(2))] � (F � d ~X(3)) = J(d ~X(1) � d ~X(2)) � d ~X(3) ;
where we have used the identity (1.41)1. In view of (1.76), we thus have
dv = JdV : (1.77)
Consequently, the determinant J of the deformation gradient F measures the volume changes ofin�nitesimal elements. For this reason, J must be positive for material media.
1.10 Strain invariants, principal strains
In this section, we give a brief account on the invariants for a second-order symmetric tensor. TheLagrangian strain tensors E will be consider as a typical tensor of this group. It is of interest todetermine, at a given point ~X, the directions ~V for which the expression ~V �E � ~V takes extremum
15
values. For this we must di�erentiate ~V �E �~V with respect to ~V subject to the condition ~V �~V = 1.Using Lagrange's method of multipliers, we set
@
@VK
h~V �E � ~V � �(~V � ~V � 1)
i= 0 ; (1.78)
where � is the unknown Lagrange multiplier. This gives
(EKL � �ÆKL) VL = 0 : (1.79)
A nontrivial solution of homogeneous equations (1.79) exists only if the characteristic determinantvanishes,
det (E � �I) = 0 : (1.80)
Upon expanding this determinant we obtain a cubic algebraic equation in �, known as the char-acteristic equation of the tensor E:
��3 + IE �2 � IIE �+ IIIE = 0 ; (1.81)
where
IE := E11 +E22 +E33 � trE ;
IIE := E11E22 +E11E33 +E22E33 �E212 �E2
13 �E223 �
1
2
h(trE)2 � tr (E2)
i; (1.82)
IIIE := detE :
The quantities IE , IIE and IIIE are known as the principal invariants of tensor E. Thesequantities remain invariant upon any orthogonal transformation of E, E� = Q �E �QT , where Qis an orthogonal tensor. This can be deduced from the equivalence of the characteristic equationsfor E� and E:
0 = det (E� � �I) = det (Q �E �QT � �I) = dethQ � (E � �I) �QT
i= detQ det (E � �I) detQT = (detQ)2 det (E � �I) = det (E � �I) :
A second-order tensor E in three dimensions possesses only three independent invariants,that is, all other invariants of E can be shown to be functions of the above three invariants. Forinstance, three other invariants are
~IE := trE ; ~IIE := trE2 ; ~IIIE := trE3 : (1.83)
The relations of these to IE, IIE and IIIE are
IE = ~IE ; ~IE = IE ;
IIE = 12
�~I2E � ~IIE
�; ~IIE = I2E � 2IIE ;
IIIE = 13
�~IIIE � 3
2~IE ~IIE + 1
2~I3E
�; ~IIIE = I3E � 3IE IIE + 3IIIE :
(1.84)
The roots ��, � = 1; 2; 3, of the characteristic equation (1.80) are called the characteristic
roots. If E is the Lagrangian or Eulerian strain tensor, �� are called the principal strains. With
16
each of the characteristic roots, we can determine a principal direction ~V�, � = 1; 2; 3, solving theequation
E � ~V� = ��~V� ; (1.85)
together with the normalizing condition ~V� � ~V� = 1. For a symmetric tensor E, it is not diÆcultto show that (i) all characteristic roots are real, and (ii) the principal directions correspondingto two distinct characteristic roots are unique and mutually orthogonal. If, however, there isa pair of equal roots, say �1 = �2, then only the direction associated with �3 will be unique.In this case, any other two directions which are orthogonal to ~V3, and to one another so as toform a right-handed system, may be taken as principal directions. If �1 = �2 = �3, every setof right-handed orthogonal axes quali�es as principal axes, and every directions is said to be aprincipal direction. Thus we can see that it is always possible to �nd at point ~X, at least threemutually orthogonal directions for which the expression ~V �E � ~V takes the stationary values.
The tensor E takes a particularly simple form when the reference coordinate system is selectedto coincide with the principal directions. Let the component of tensor E be given initially withrespect to arbitrary Cartesian axes XK with the base vectors ~IK , and let the principal axes ofE be designated by X� with the base vectors ~I� � ~V�. In invariant notation, the tensor E isrepresented in the form
E = EKL(~IK ~IL) = E��(~V� ~V�) ; (1.86)
where the diagonal elements E�� are equal to the principal values ��, while the o�-diagonalelements E�� = 0, � 6= �, are zero. The projection of the vector ~IK on the base of vectors ~V� is
~IK = (~IK � ~V�)~V� ; (1.87)
where ~IK � ~V� are the direction cosines between the two established sets of axes XK and X�. Bycarrying (1.87) into (1.86), we �nd that
E�� = (~IM � ~V�)(~IN � ~V�)EMN : (1.88)
Hence, the determination of principal directions ~V� and characteristic roots �� of a tensor E isequivalent to �nding a rectangular coordinate system of reference in which the matrix jjEKLjjtakes the diagonal form.
1.11 Displacement vector
The geometrical measures of deformation can also be expressed in terms of the displacement
vector ~u that extends from a material points ~X in the reference con�guration �0 to its spatialposition ~x in the present con�guration �t, as illustrated in Figure 1.5.:
~u := ~x� ~X +~b : (1.89)
We may interpret (1.89) in either the Lagrangian or Eulerian descriptions of ~u:
~U( ~X; t) = ~x( ~X; t)� ~X +~b ; ~u(~x; t) = ~x� ~X(~x; t) +~b : (1.90)
By taking the scalar product of both sides of (1.90)1 and (1.90)2 by ~ik and ~IK , respectively, weobtain
Uk( ~X; t) = xk � ÆkLXL + bk ; uK(~x; t) = ÆKlxl �XK +BK ; (1.91)
17
�~I1 ~I2
~I3
X3
X2
X1
P
~X
~u
�0
~i1
~i2
~i3
x1x2
x3
p
~x
�t
~b
Figure 1.5. Displacement vector.
where bk = ~b�~ik and BK = ~b�~IK . Here again we can see the appearance of shifters. Di�erentiating(1.91)1 with respect to XK and (1.91)2 with respect to xk, we have
Uk;K( ~X; t) = xk;K � ÆkLÆLK ; uK;k(~x; t) = ÆKlÆlk �XK;k :
In view of (1.14), this can be simpli�ed as
Uk;K( ~X; t) = xk;K � ÆkK ; uK;k(~x; t) = ÆKk �XK;k : (1.92)
In principle, all physical quantities representing the measures of deformation can be expressedin terms of the displacement vector and its gradient. For instance, from the �rst of (1.92) theLagrangian strain tensor can, in indicial notation, be written in the form
2EKL = CKL � ÆKL = xk;Kxk;L � ÆKL = (ÆkK + Uk;K)(ÆkL + Uk;L)� ÆKL ;
which reduces to2EKL = UK;L + UL;K + Uk;KUk;L : (1.93)
Likewise, from the second of (1.92) the Eulerian strain tensor can be written in the form
2ekl = Ækl � ckl = Ækl �XK;kXK;l = Ækl � (ÆKk � uK;k)(ÆKl � uK;l) ;
which reduces to2ekl = uk;l + ul;k � uK;kuK;l : (1.94)
It is often convenient not to distinguish between the Lagrangian coordinates XK and theEulerian coordinates xk. In such a case the shifter symbol ÆKl reduces to the Kronecker delta Ækland the displacement vector is de�ned by
~U( ~X; t) = ~x( ~X; t)� ~X : (1.95)
18
Consequently, equation (1.92)1 can be written in symbolic form:
F = I +HT ; (1.96)
where H is the displacement gradient tensor de�ned by
H( ~X; t) := Grad ~U( ~X; t) : (1.97)
Furthermore, the indicial notation (1.93) of the Lagrangian strain tensor is simpli�ed to symbolicnotation:
E =1
2(H +HT +H �HT ) : (1.98)
1.12 Geometrical linearization
If the numerical values of all components of the displacement gradient tensor are very smallcompared to one, we may neglect the squares and products of these quantities in comparison tothe gradients themselves. A convenient measure of smallness of deformations is the norm of thedisplacement gradient,
kHk :=pH :: H � 1 : (1.99)
In the following, the term small deformation will correspond to the case of small displacementgradients. Geometrical linearization is the process of the developing of all kinematic variablescorrect to �rst order in kHk and neglecting all terms of orders higher than O(kHk). In itsgeometrical interpretation, a small value of kHk implies small strains as well as small rotations.
1.12.1 Linearized analysis of deformation
The geometrical linearization results in the following relations:
F�1 = I �HT +O(kHk2) ; (1.100)
detF = 1 + trH +O(kHk2) ; (1.101)
C = I +H +HT +O(kHk2) ; (1.102)
B = I �H �HT +O(kHk2) ; (1.103)
U = I +1
2(H +HT ) +O(kHk2) ; (1.104)
V = I +1
2(H +HT ) +O(kHk2) ; (1.105)
R = I +1
2(HT �H) +O(kHk2) : (1.106)
From the last three relations we infer
R �U =hI +
1
2(HT �H) +O(kHk2)
i�hI +
1
2(H +HT ) +O(kHk2)
i= I +HT +O(kHk2) ; (1.107)
and
V �R =hI +
1
2(H +HT ) +O(kHk2)
i�hI +
1
2(HT �H) +O(kHk2)
i= I +HT +O(kHk2) : (1.108)
19
Thus we can see that for small deformations the multiplicative decomposition of the deformationgradient into orthogonal, symmetric and positive de�nite factors is approximated by the additivedecomposition of the displacement gradient into symmetric and skew-symmetric parts:
F = R �U = V �R = I +HT +O(kHk2) = I +1
2(H +HT ) +
1
2(HT �H) +O(kHk2) ;
which shows that the deformation gradient can be decomposed into the symmetric and skew-symmetric parts:
F = I + ~E + ~R+O(kHk2) ; (1.109)
where~E :=
1
2(H +HT ) ; ~R :=
1
2(HT �H) : (1.110)
The symmetric part ~E = ~ET is the linearized or in�nitesimal Lagrangian strain tensor, andthe skew-symmetric part ~R = � ~RT is the linearized or in�nitesimal Lagrangian rotation tensor.Summing up the last two equations results in
HT = ~E + ~R (exact) : (1.111)
Carrying this into (1.98), we obtain
E = ~E +1
2( ~E + ~R)T � ( ~E + ~R) (exact) : (1.112)
Now, it is clear that in order that E � ~E, not only strains ~E must be small, but rotations ~R mustalso be small so that products such as ~ET � ~E, ~ET � ~R, and ~RT � ~R will be negligible comparedto ~E.
The transformations (1.43) between the di�erential operations applied to the Eulerian vari-ables and those applied to the Lagrangian variables can be linearized in the following manner:
grad � = Grad � �H �Grad �+O(kHk2) ;div � = Div � �H :: Grad �+O(kHk2) ;
div grad � = DivGrad � �2H :: GradGrad � �DivH �Grad �+O(kHk2) ;(1.113)
or, conversely,
Grad � = grad �+H � grad �+O(kHk2) ;Div � = div �+H :: grad �+O(kHk2) ;
DivGrad � = div grad �+2H :: grad grad �+DivH � grad �+O(kHk2) :(1.114)
Equation (1.94) shows that the geometrical linearization of the Eulerian strain tensor e resultsin the linearized Eulerian strain tensor ~e:
e � ~e =1
2
hgrad ~u+ (grad ~u)T
i: (1.115)
Because of (1.113)1, the tensor ~e is, correct to �rst order of kHk, equal to the tensor ~E,
~e = ~E +O(kHk2) : (1.116)
20
Thus, in the linear theory the distinction between the Lagrangian and Eulerian strain tensorsdisappears.
1.12.2 Length and angle changes
When the deformation is small, E11 � 1, by expanding (1.65) and neglecting the square andhigher powers of E11, we get
E(1) � E11 � ~E11 : (1.117)
Similar results are of course valid for E22 and E33, which indicates that the in�nitesimal normalstrains are approximately the extensions of the �bers along the coordinate axes in the referencecon�guration.
The linearization of (1.71) yields
cos �(12) � 2E12 � 2 ~E12 : (1.118)
Hence, writing cos �(12) = sin�(12) � (12), we have
(12) � 2E12 � 2 ~E12 : (1.119)
Similar results are valid for E13 and E23. This provides geometrical meaning for shear strains. Thein�nitesimal shear strains are approximately one half of the angle change between the coordinateaxes in the reference con�guration.
1.12.3 Area and volume changes
Let us express the foregoing expressions in the linear theory in which the displacement gradientis suÆciently small that the linearization is justi�ed. Substituting the linearized forms (1.100)and (1.101) of the spatial deformation gradient F�1 and the jacobian J into (1.73), we obtainthe linearized relation between d~a and d ~A:
d~a = [I + (trH)I �H ] � d ~A+O(kHk2) : (1.120)
We may also linearize the separate contributions to d~a. Using the linearized form (1.103) of thePiola deformation tensor B, we can write
1p~N �B � ~N
=1q
1� 2 ~N �H � ~N= 1 + ~N �H � ~N +O(kHk2) : (1.121)
Employing this and the linearized forms (1.91) and (1.92) for the spatial deformation gradientF�1 and the jacobian J , the unit normal ~n to the deformed surface � and the surface elementda of � may, within the framework of linear approximation, be written as
da = (1 + trH � ~N �H � ~N)dA+O(kHk2) ; (1.122)
~n = (1 + ~N �H � ~N) ~N �H � ~N +O(kHk2) : (1.123)
where ~N is the unit normal to the undeformed surface �, the Lagrangian description of thedeformed surface �. The �rst equation accounts for the change in the elementary area, thesecond gives the de ection of the unit normal.
21
The linearized expressions for the deformed surface area element can also be expressed interms of the surface gradient operator Grad� and the surface divergence operator Div� de�nedin Appendix B. For example, the surface gradient and the surface divergence of the displacementvector ~U are given by (B.30) and (B.31):
Grad�~U = H � ~N ( ~N �H) ; (1.124)
Div�~U = trH � ~N �H � ~N ; (1.125)
where H = Grad ~U and trH = Div ~U . In terms of the surface displacement gradient anddivergence, equations (1.120), (1.122) and (1.123) have the form
d~a = [I + (Div�~U)I �Grad�
~U ] � d ~A+O(kHk2) ; (1.126)
da = (1 + Div�~U)dA+O(kHk2) ; (1.127)
~n = (I �Grad�~U) � ~N +O(kHk2) : (1.128)
Equation (1.127) is the areal analogue of the volumetric relation (1.77); the deformed and unde-formed volume elements dv and dV are related by
dv = (1 + Div ~U)dV +O(kHk2) : (1.129)
22
2. KINEMATICS
2.1 Material and spatial time derivatives
In the previous chapter we discussed the geometrical properties of the present con�guration �tunder the assumption that the parameter t describing time changes of the body is kept �xed,that is, the function ~� was considered as a deformation map ~�(�; t). Now we turn our attentionto problems arising in the case of motion, that is, the function ~� is discussed as the map ~�( ~X; �)for a chosen point ~X. We assume that ~�( ~X; �) is twice di�erentiable.
If we focus attention on a speci�c particle XP having the material position vector ~XP , (1.3)takes the form
~xP = ~�( ~XP ; t) (2.1)
and describes the path or trajectory of that particle as a function of time. The velocity ~vP of theparticle along this path is de�ned as the time rate of change of position, or
~V P :=d~xP
dt=
@~�
@t
!�����~X= ~XP
; (2.2)
where the subscript ~X accompanying a vertical bar indicates that ~X is held constant (equal to~XP ) in the di�erentiation of ~�. In an obvious generalization, we may de�ne the velocity of thetotal body as the derivative
~V ( ~X; t) :=d~x
dt=
@~�
@t
!�����~X
: (2.3)
This is the Lagrangian representation of velocity and the time rate of change with respect to amoving particle is called the material time derivative. Similarly, the material time derivative of~v de�nes the Lagrangian representation of acceleration,
~A( ~X; t) :=d~v
dt=
@~V
@t
!�����~X
=
@2~�( ~X; t)
@2t
!�����~X
: (2.4)
By using (1.90)1 we may also write
~V ( ~X; t) =
@ ~U
@t
!�����~X
; ~A( ~X; t) =
@2 ~U
@2t
!�����~X
: (2.5)
The material particle with a given velocity or acceleration is, in the Lagrangian representation,identi�able, so that, both the velocity and acceleration �elds are de�ned on the reference con�gu-ration �0. This is frequently not convenient, for instance, in the classical uid mechanics. Whenthe present con�guration �t is chosen as the reference con�guration, the function ~�( ~X; t) cannotbe speci�ed. Thus, in the Eulerian description, the velocity and acceleration at time t at a spatialpoint are known, but the particle occupying this point is not known.
Since the fundamental laws of continuum dynamics involve the acceleration of particles andsince the Lagrangian formulation of velocity may not be available, the acceleration must becalculated from the Eulerian formulation of velocity. To accomplish this, only the existence of
23
the unknown trajectories, ~x = ~�( ~X; t), must be assumed. By the substitution of (1.5) for ~X in(2.3), we have
~v(~x; t) = ~V (~��1(~x; t); t) ; (2.6)
which gives the velocity �eld at each spatial point ~x at time t with no speci�cation of its relationto the material point ~X. This is the Eulerian representation of velocity.
Based on the same assumption of the existence of the trajectories ~x = ~�( ~X; t), the Eulerianrepresentation of velocity (2.6) may be considered in the form ~v = ~v(~x( ~X; t); t). By chain rule ofcalculus, we get
d~v
dt
����~X=@~v
@t
����~x+ ~v � grad~v ; (2.7)
where di�erentiation in the grad-operator is taken with respect to the spatial variables. 4 This isthe desired equation for the Eulerian representation of acceleration which is expressed in termsof the Eulerian representation of velocity,
~a(~x; t) =@~v
@t
����~x+ ~v � grad~v : (2.8)
In this equation, the �rst term on the right-hand side gives the time rate of change of velocity ata �xed position ~x, known as the local rate of change or spatial time derivatives; the second termresults from the particles changing position in space and is referred to as the convective term.Note that the convective term can equivalently be represented in the form
~v � grad~v = gradv2
2� ~v � rot~v : (2.9)
The material time derivative of any other �eld quantity can be calculated in the same way ifits Lagrangian or Eulerian representation is known. This suggests to introduce the material timederivative operator
D
Dt�
:( ):=
d
dt
����~X=
8>><>>:
@@t
���~X
for a �eld in the Lagrangian representation ;
@@t
���~x+ ~v � grad for a �eld in the Eulerian representation ;
(2.10)which can be applied to any �eld quantity given in the Lagrangian or Eulerian representation.
2.2 Time changes of some geometric objects
All geometric objects, which we discussed in Chapter 1, can be calculated from the deformationgradient F . For this reason, we begin with the investigation of the time derivative of F . Inindicial notation, we can write
(xk;K)� =
�@xk@XK
��
=@:xk
@XK= vk;K = vk;lxl;K ; (2.11)
where we have used the fact that XK are kept �xed in:
( ) so that material time derivative:
( )and material gradient @=@XK commute. In symbolic notation, we have
:F= l � F ; or l =
:F �F�1 ; (2.12)
4The components of grad~v in the Cartesian coordinates xk are (grad~v)kl = vl;k.
24
where l is the transpose spatial velocity gradient,
l(~x; t) := gradT~v(~x; t); or lkl(~x; t) :=@vk(~x; t)
@xl: (2.13)
A corollary of this lemma is(F�1)� = �F�1 � l : (2.14)
To prove it, we take the material time derivative of F � F�1 = I . Hence,
:F �F�1 + F � (F�1)� = 0 :
Using (2.12), we obtain (2.14).The material time derivative of the jacobian is given by
:J= J div~v = J tr l : (2.15)
To show this, we have
:J= (detxk;K)
� =@J
@xk;K(xk;K)
� =@J
@xk;Kvk;lxl;K :
Using (1.38), this gives (2.15).The velocity gradient l determines the material time derivatives of the material line element
d~x, surface element d~a and volume element dv according to the formulae:
(d~x)� = l � d~x ; (2.16)
(d~a)� =h(div~v)I � lT
i� d~a ; (2.17)
(dv)� = div~v dv : (2.18)
The proof is immediate. We take the material time derivative of (1.33)1 and substitute from(2.12):
(d~x)� =:F � d ~X = l � F � d ~X :
Replacing d ~X by d~x proves (2.16). By di�erentiating (1.73) and using (2.14) and (2.15), we have
(d~a)� =h :J F
�T + J(F�T )�i� d ~A =
hJ div~vF�T � J lT � F�T
i� d ~A :
Replacing d ~A by d~a we obtain (2.17). The third statement can be veri�ed by di�erentiating (1.77)and using (2.15):
(dv)� =:J dV = J div~v dV = div~v dv ;
which completes the proofs of statements.As any 2nd order tensor, the spatial velocity gradient l can be decomposed into its symmetric
and skew-symmetric partsl = d+w ; (2.19)
where
d :=1
2(l+ lT ) ; w :=
1
2(l� lT ) : (2.20)
25
The symmetric tensor d, d = dT , is called the strain-rate tensor and the skew-symmetric tensorw, w = �wT , is the spin or vorticity tensor.
To highlight the meaning of the spin tensor w, we readily see that any skew-symmetric 2nd-order tensor w,
w =
0B@ 0 �w3 w2
w3 0 �w1
�w2 w1 0
1CA (~ik ~il) ;
can be represented in terms of the so-called vorticity vector ~w in the form:
wkl = "lkmwm ; or wk =1
2"klmwml :
Since wkl =12(vk;l � vl;k), we obtain that the vorticity vector ~w is equal to one-half of the curl of
the velocity vector ~v:
~w =1
2rot~v : (2.21)
Hence the name spin, or vorticity, given to the tensor w. The scalar product of w with a vector~a is given by
w � ~a = ~w � ~a:Equation (2.16) can now be written in the form
(d~x)� = d � d~x+ ~w � d~x : (2.22)
The physical signi�cance of the spin tensor may be seen by considering the case when the strain-rate tensor is zero, d = 0. Then (d~x)� = ~w�d~x, according to which the spin tensor w describes aninstantaneous local rigid-body rotation about an axis passing through a point ~x; the correspondingangular velocity, the direction and the sense of this rotation is given by the spin vector ~w. If,on the other hand, the spin tensor is zero in a region, w = 0, the velocity �eld is said to beirrotational in the region.
The material time derivative of the Lagrangian strain tensor is given by
:E=
1
2
:C= F T � d � F : (2.23)
To show this, we have
:C= (F T � F )� = (F T )� � F + F T �
:F= F T � lT � F + F T � l � F = F T � (l+ lT ) � F = 2F T � d � F :
The material time derivative of strain tensor E is determined by the tensor d, but not by thetensor w. This is why d is called the strain-rate tensor.
Likewise, it can readily be shown that the material time derivative of the Piola deformationtensor B = F�1 � F�T is :
B= �F�1 � 2d � F�T : (2.24)
In view of this, the material time of the unit normal ~n to surface � in the present con�guration is
(~n)� = �~n � l+ (~n � l � ~n)~n : (2.25)
26
To show this, we take the material time derivative of (1.75) and use (2.14) and (2.24):
(~n)� =~N � (F�1)�
( ~N �B � ~N)1=2�
~N � F�1
2( ~N �B � ~N)3=2( ~N �
:B � ~N)
= �~N � F�1 � l
( ~N �B � ~N)1=2+
~N � F�1
( ~N �B � ~N)3=2( ~N � F�1 � d � F�T � ~N) :
By substituting from (1.75), we obtain (2.25).The material time derivative of the square of the arc length is given by
(ds2)� = 2 d~x � d � d~x : (2.26)
The proof follows from (1.49) and (2.23):
(ds2)� = (d~x �d~x)� = (d ~X �C �d ~X)� = d ~X �:C � d ~X = 2 d ~X �F T �d �F � d ~X = 2F � d ~X �d �F � d ~X :
Using (1.33)2, this gives (2.26).Higher-order material time derivatives can be carried out by repeated di�erentiation. Thus,
for instance, the nth material time derivative of ds2 can be expressed as
(ds2)(n) = d~x � an � d~x ; (2.27)
where
an(~x; t) := F�T �(n)
C �F�1 (2.28)
are known as the Rivlin-Ericksen tensors of order n. The proof of (2.27) is similar to that of
(2.26). Note that(n)
C is the nth material time derivative of the Green deformation tensor. The�rst two Rivlin-Ericksen tensors are
a0 = I ; a1 = 2d : (2.29)
The Rivlin-Ericksen tensors were introduced in the construction of the non-linear viscoelasticity,and, in particular, in the description of non-Newtonian uids, see Chapter 5.
2.3 Reynolds's transport theorem
We aim to prove that the material time derivative of a volume integral of any scalar or vector�eld � over the spatial volume v(t) is given by
D
Dt
Zv(t)
�dv =
Zv(t)
�D�
Dt+ �div~v
�dv : (2.30)
To prove this, we �rstly transform the integral over the spatial volume to an integral over thematerial volume V . Under the assumption of existence of the mapping (1.3), and by using (1.76),we have
D
Dt
Zv(t)
�dv =D
Dt
ZV�JdV ;
27
where �( ~X; t) = �(~x( ~X; t); t). Since V is a �xed volume in the Lagrangian con�guration, thedi�erentiation D=Dt and the integration over V commute and the di�erentiation D=Dt can beperformed inside the integral sign,
D
Dt
ZV�JdV =
ZV
D
Dt(�J) dV =
ZV
�D�
DtJ +�
DJ
Dt
�dV =
ZV
�D�
Dt+�div~v
�JdV :
By converting this back to the spatial formulation by (1.76), we prove (2.30). Equation (2.30) isoften spoken of as the Reynolds transport theorem.
This theorem may be expressed in a di�erent form. To do it, we calculate the material timederivative of a scalar �eld �,
D�
Dt=@�
@t+ ~v � grad� ; (2.31)
and substitute this back into (2.30). With the product rule
�div~v + ~v � grad� = div (~v�) ; (2.32)
which is valid for a scalar �eld �, we then arrive
D
Dt
Zv(t)
�dv =
Zv(t)
@�
@t+ div (~v�)
!dv :
Arranging the second term on the right-hand side according to the Gauss's theoremZvdiv~u dv =
Is~n � ~u da ; (2.33)
where ~u is a vector-valued function continuously di�erentiable in v, s is the surface boundingvolume v and ~n is the outward unit normal to s, we get an equivalent form of the Reynoldstransport theorem
D
Dt
Zv(t)
�dv =
Zv(t)
@�
@tdv +
Is(t)
(~n � ~v)�da ; (2.34)
where both � and ~v are again required to be continuously di�erentiable in v.The same form (2.34) holds also for a vector-valued function �. The only di�erence is that
the product rule�div~v + ~v � grad� = div (~v �) ; (2.35)
must be applied instead of (2.32) and the Gauss theorem must be used for a 2nd-order tensor A:ZvdivA dv =
Is~n �A da : (2.36)
The material time derivative of a ux of a vector ~q across a surface s(t) is
D
Dt
Zs(t)
~q � d~a =Zs(t)
�D~q
Dt+ (div~v)~q � ~q � grad~v
�� d~a : (2.37)
To prove this transport theorem, transform the integral over spatial surface s(t) into the referen-tial, time-independent surface S, carry D=DT inside of the integral, use (1.72) and then proceedin a similar way as in the proof of (2.17)2.
28
2.4 Reynolds's transport theorem for a volume with moving di-continuity
The time rate of integral over a region containing a discontinuity surface is common occurrencein the study of geophysical phenomena. We give below an expression modifying the Reynoldstransport theorem for the case when a volume is intersected by a moving discontinuity surface.
The moving discontinuity �(t) can be de�ned implicitly by the equation
f�(~x; t) = 0 ~x 2 �(t): (2.38)
The unit normal ~n to �(t) is then given by the relation
~n(~x; t) =grad f�jgrad f�j ; (2.39)
where grad denotes the gradient with respect to ~x. The velocity of the points of �(t) in thedirection of the normal ~n has the form
wn = �@f�@t
jgrad f�j : (2.40)
By combining (2.39) and (2.40), we obtain
@f�@t
+ ~w � grad f� = 0 ; (2.41)
where ~w is the speed of propagation of �(t); wn is its normal component, wn = ~w � ~n. Equation(2.41) can also be written in the form
dwf�(~x; t)
dt= 0 ; (2.42)
saying, that the time derivative of function f�(~x; t) following the motion of moving discontinuity�(t) with velocity ~w must vanish. The condition (2.41) or (2.42) is called the kinematic conditionfor time evolution of �(t).
Consider a material volume v which is intersected by a discontinuity surface �(t) across whicha tensor-valued function A undergoes a jump (Figure 2.1). The surface �(t) divides the materialvolume v into two parts, namely v+ on the side of the normal ~n and v� on the other side. Thenthe Gauss theorem (2.36) is to be modi�ed asZ
v��divA dv =
Is��
~n �A da�Z�~n � [A]+� da ; (2.43)
where the unit normal ~n points toward the + side of �, as shown in Figure 2.1. The volumeintegral over v � � means the volume v of the body excluding the material points located onthe discontinuity surface �. Similarly, the integral over the surface s � � excludes the line ofintersection of � with s, that is,
v � � := v+ + v� ; s� � := s+ + s� : (2.44)
29
��+
�(t)
��
s�
s+
~n+
~n�~n
~n
~n
v+
v�
~w
Figure 2.1. Discontinuity surface.
The brackets indicate the jump of the enclosed quantity across �(t), e.g.,
[A]+� := A+ �A� : (2.45)
To prove (2.43), we apply the Gauss theorem (2.36) to the two volumes v+ and v� boundedby s+ + �+ and s� + ��, respectively. HenceZ
v+divA dv =
Zs+~n �A da+
Z�+~n+ �A+ da ;
Zv�
divA dv =
Zs�~n �A da+
Z��~n� �A� da ;
where ~n+ and ~n� are the exterior normals to �+ and ��, respectively. Upon adding these twoequations, we getZ
v++v�divA dv =
Is++s�
~n �A da+
Z�+~n+ �A+ da+
Z��~n� �A� da :
But we have~n+ = �~n� = �~n ;
and letting �+ and �� approach �, so thatZ�+~n+ �A+ da+
Z��~n� �A� da =
Z�~n � �A� �A+� da = �
Z�~n � [A]+� da
The Reynolds transport theorem (2.30) has also to be modi�ed once the discontinuity surface�(t) moves with velocity ~w which di�ers, in general, from the material velocity ~v. The modi�cationof (2.30) reads
D
Dt
Zv��
�dv =
Zv��
�D�
Dt+ �div~v
�dv +
Z�~n � [(~v � ~w) �]+� da : (2.46)
30
Both � and ~v are required to be continuously di�erentiable in v � �. To prove (2.46), we apply(2.34) to the two volumes v+ and v� bounded by s+ + �+ and s� + ��, respectively. Hence
D
Dt
Zv+�dv =
Zv+
@�
@tdv +
Zs+
(~n � ~v)�da+Z�+
�~n+ � ~w��+ da ;
D
Dt
Zv��dv =
Zv�
@�
@tdv +
Zs�
(~n � ~v)�da+Z��
�~n� � ~w��� da :
Upon adding these two equations, letting �+ and �� approach � and realizing that ~n+ = �~n� =�~n, we obtain
D
Dt
Zv++v�
�dv =
Zv++v�
@�
@tdv +
Is++s�
(~n � ~v)�da�Z�(~n � ~w) [�]+� da :
Using the Gauss theorem (2.43) for A = ~v� to replace the second term on the right-hand side,we get
D
Dt
Zv++v�
�dv =
Zv++v�
�@�
@t+ div (~v �)
�dv +
Z�~n � [(~v � ~w) �]+� da :
To complete the proof of modi�ed Reynolds's transport theorem (2.46), the �rst term on theright-hand side is to be arranged by making use of (2.31) and (2.35).
31
3. MEASURES OF STRESS
3.1 Body and surface forces, mass density
The forces acting on a continuum or between portions of it divide into long-range forces andshort-range forces.
Long-range forces comprise gravitational, electromagnetic and inertial forces. These forcesdecrease very gradually with an increase in distance between the interacting particles. As aresult, long-range forces act uniformly on all matter contained in a suÆciently small volume, sothat, they are proportional to its size. In continuum mechanics, long-range forces are thereforecalled body or volume forces.
Short-range forces comprise several types of molecular forces. Their characteristic featureis that they decrease extremely abruptly with an increase in distance between the interactingparticles. Hence, they are appreciable only when this distance does not exceed molecular dimen-sions. A consequence is that, if the matter inside some volume is acted on by short-range forcesoriginating from interactions with matter outside this volume, these forces can only act on a thinlayer immediately below its surface. In continuum mechanics, short-range forces are thereforecalled surface forces; they are speci�ed more closely by constitutive equations (Chapter 5).
In the following, we assume that volume and surface forces arise due to interactions that areequal, opposite and collinear (so-called the strong law of action and reaction). On this assumption,volume and surface couple stresses cannot arise.
In continuum mechanics, with each body there is associated a measure called mass. It is non-negative and additive, and it is invariant under the motion. If the mass is absolutely continuous
in the space variables, then there exists a density % called the mass density. The total mass ofthe body is then determined by
m =
Zv%dv : (3.1)
If the mass is not continuous throughout volume v, then instead of (3.1) we write
m =
Zv1%dv +
X�
m� ; (3.2)
where the summation is taken over all discrete masses contained in the body. We shall be dealingwith a continuous mass medium in which (3.1) is valid, which implies that m! 0 as v ! 0. Wetherefore have
0 � % <1 : (3.3)
Let ~f be the body force per unit mass. The resultant body force acting on the body currentlyoccupying some �nite volume v is then Z
v%~fdv :
3.2 Cauchy traction principle
We consider a material body b(t) which is subject to body forces ~f and surface forces ~g. Let pbe an interior point of b(t) and imagine a plane surface a� passing through point p (sometimesreferred to as a cutting plane) so as to partition the body into two portions, designated I and
32
II (Figure 3.1). Point p is lying in the small element of area �a� of the cutting plane, which isde�ned by the unit normal ~n pointing in the direction from Portion I into Portion II, as shownin Figure 3.1. The internal forces being transmitted across the cutting plane due to the actionof Portion II upon Portion I will give rise to a force distribution on �a� equivalent to a resul-tant surface force �~g as is also shown in Figure 3.1. (For simplicity, body forces and surface forces
acting on the body as a whole are not drawn inFigure 3.1.) Notice that �~g are not necessar-ily in the direction of the unit normal vector ~n.The Cauchy traction principle postulates that thelimit when the area �a� shrinks to zero, with premaining an interior point, exists and is
~t(~n) = lim�a�!0
�~g
�a�: (3.4)
Obviously, this limit is meaningful only if �a�
degenerates not into a curve but into a point p.The vector ~t(~n) is called the Cauchy stress vec-
tor or the Cauchy traction vector (force per unitarea). It is important to note that, in general, ~t(~n) depends not only on the position of p on�a� but also the orientation of surface �a�, i.e., on its external normal ~n. This dependence istherefore indicated by the subscript ~n. Thus, for the in�nity of cutting planes imaginable throughpoint p, each identi�ed by a speci�c ~n, there is also an in�nity of associated stress vectors ~t(~n) fora given loading of the body.
�p
�a�
a�
~n
�~g
I
II
b(t)
Figure 3.1.Surface force on surface element �a�.
We incidentally mention that a continuous distribution of surface forces acting across somesurface is, in general, equivalent to a resultant force and a resultant couple. In (3.4) we havemade the assumption that, in the limit at p, the couple per unit area vanishes and therefore thereis no remaining concentrated moment, or couple stress as it is called. For a discussion of couplestresses, the reader should referred to Eringen, 1967.
To determine the dependence of the stress vector on the exterior normal, we next apply theprinciple of balance of linear momentum to a small tetrahedron of volume �v having its vertexat p, three coordinate surfaces �ak, and the base �a on a with an oriented normal ~n (Figure3.2). The stress vector 5 on the coordinate surface xk = const: is denoted by �~tk.
We now apply the equation of balance of linear momentum (Sect.4.1) to this tetrahedron.Z�v%~fdv �
Z�ak
~tkdak +
Z�a
~t(~n)da =D
Dt
Z�v%~vdv :
An estimate of the surface and volume integrals may be made by use of the mean value theorem:
%� ~f��v � ~t�k�ak + ~t�(~n)�a =D
Dt(%�~v��v) ; (3.5)
where %�, ~f�, and ~v� are, respectively, the values of %, ~f , and ~v at some interior points of thetetrahedron and ~t�(~n) and
~t�k are the values of ~t(~n) and ~tk on the surface �a and on coordinate
5Since the exterior normal of a coordinate surface xk = const: is in the direction of �xk, without loss ingenerality, we denote the stress vector acting on this coordinate surface by �~tk rather than ~tk.
33
�p
x1
x2
x3
�~t3�a3
�~t2�a2
�~t1�a1
~t(~n)�a~n
%~f�v
h
Figure 3.2. Equilibrium of an in�nitesimal tetrahedron.
surfaces �ak. The volume of the tetrahedron is given by
�v =1
3h�a ; (3.6)
where h is the perpendicular distance from point p to the base �a. Moreover, the area vector�~a is equal to the sum of coordinate area vectors, that is,
�~a = ~n�a = �ak~ik : (3.7)
Thus�ak = nk�a : (3.8)
Inserting (3.6) and (3.8) into (3.5) and canceling the common factor �a, we obtain
1
3%� ~f�h� ~t�knk + ~t�(~n) =
1
3
D
Dt(%�~v�h) : (3.9)
Now, letting the tetrahedron shrink to point p by taking the limit h! 0 and noting that in thislimiting process the starred quantities take on the actual values of those same quantities at pointp, we have
~t(~n) = ~tknk ; (3.10)
which is the Cauchy stress formula. Equation (3.10) allows us to determine the Cauchy stressvector at some point acting across an arbitrarily inclined plane, if the Cauchy stress vectors actingacross the three coordinate surfaces through that point are known.
The stress vectors ~tk are, by de�nition, independent of ~n. From (3.10) it therefore followsthat
~t(�~n) = �~t(~n) : (3.11)
34
We now introduce the de�nition of the Cauchy stress tensor. The tkl component of the Cauchystress tensor t is given by the lth component of the stress vector ~tk acting on the positive side ofthe kth coordinate surface:
~tk = tkl~il or tkl = ~tk �~il : (3.12)
The �rst subscript in tkl indicates the coordinate surface xk = const: on which the stress vector~tk acts, and the second subscript the direction of the component of ~tk. For example, t23 is thex3-components of the stress vector ~t2 acting on the coordinate surface x2 = const:. Now, ifthe exterior normal of x2 = const: points in the positive direction of the x2-axis, t23 points inthe positive direction of the x3-axis. If the exterior normal of x2 = const: is in the negativedirection of the x2-axis, t23 is directed in the negative direction of the x3-axis. The positive stresscomponents on the faces of a parallelepiped built on coordinate surfaces are shown on Figure3.3. The components t11, t22 and t23 are called normal stresses and the mixed components t12,t13, etc. are called shear stresses. The nine components tkl of the Cauchy stress tensor t may bearranged in a matrix form
t =
0B@ t11 t12 t13t21 t22 t23t31 t32 t33
1CA : (3.13)
Considering (3.12), the Cauchy stress formula (3.10) reads
~t(~n) = ~n � t ; (3.14)
saying that the Cauchy stress vector acting on any plane through a point is fully characterizedas a linear function of the stress tensor at the point.
�0x1
x2
x3 ~t3
t31t32
t33
~t2
t21
t22
t23
�~t2
t21
t22
t23
�~t3
t31t32
t33
Figure 3.3. The components of the stress tensor.
35
3.3 Other measures of stress
So far, we have represented short-range intermolecular forces in terms of the Cauchy stress vector~tk or tensor t. There are, however, other two di�erent ways of measuring or de�ning the stress,each of which plays a certain role in the theory of continuum mechanics. The Eulerian Cauchystress tensor gives the surface force acting on the deformed elementary area da in the form
d~g = ~t(~n)da = (~n � t)da = d~a � t : (3.15)
The Cauchy stress tensor, like any other variable, has both an Eulerian and a Lagrangian de-scription; the corresponding Lagrangian Cauchy stress tensor is de�ned, as usual, by T ( ~X; t) :=t(~x( ~X; t); t). We make, however, an exception in the notation and use t( ~X; t) for the Lagrangiandescription of the Cauchy stress tensor. The Eulerian Cauchy stress tensor t(~x; t) arises naturallyin the Eulerian form of the balance of linear momentum; the corresponding Lagrangian form ofthis principle cannot, however, be readily expressed in terms of the Lagrangian Cauchy stresstensor t( ~X; t).
A simple Lagrangian form of the balance of linear momentum can be obtained if a stressmeasure is referred to a surface in the reference con�guration. This can be achieved by introducingthe so-called �rst Piola-Kirchho� stress tensor T (1) as a stress measure referred to the referentialarea element d ~A:
d~g = d~a � t =: d ~A � T (1) : (3.16)
Evidently, the tensor T (1) gives the surface force acting on the deformed area d~a at ~x in termsof the corresponding referential element d ~A at the point ~X . Thus, T (1) is a measure of the forceper unit referential area, whereas both the Eulerian and Lagrangian Cauchy stresses t(~x; t) andt( ~X; t) are measures of the force per unit spatial area. We can �nd the relation between T (1) andt by using the transformation (1.73) between the deformed and undeformed elementary areas.The result can be expressed in either of the two equivalent forms
T (1)( ~X; t) = JF�1 � t( ~X; t) ; t( ~X; t) = J�1F � T (1)( ~X; t) : (3.17)
The surface force vector d~g in (3.16) acts upon the displaced point ~x whereas the surface elementvector d ~A is referred to the reference point ~X . The �rst Piola-Kirchho� stress tensor T (1) istherefore a two-point tensor. This can also be observed from the componental form of (3.17):
T(1)Kl (
~X; t) = JXK;ktkl( ~X; t) ; tkl( ~X; t) = J�1xk;KT(1)Kl (
~X; t) : (3.18)
The constitutive equations for a simple materials (see equation (5.37) in Chapter 5) are mostconvenient to express in terms of another yet measure of stress, known as the second Piola-Kirchho� stress tensor. This quantity, denoted by T (2), gives, instead of the actual surface forced~g acting on the deformed area element d~a, a force d ~G related to d~g in the same way as thereferential di�erential d ~X is related to the spatial di�erential d~x, that is,
d ~G = F�1 � d~g ; (3.19)
just as d ~X = F�1 � d~x. De�ning T (2) by
d ~G =: d ~A � T (2) ; (3.20)
36
we �nd the �rst and second Piola-Kirchho� stresses are related by
T (2) = T (1) � F�T ; T (1) = T (2) � F T : (3.21)
By comparing with equation (3.17), we obtain the corresponding relation between the secondPiola-Kirchho� stress tensor and the Lagrangian Cauchy stress tensor:
T (2)( ~X; t) = JF�1 � t( ~X; t) � F�T ; t( ~X; t) = J�1F � T (2)( ~X; t) � F T : (3.22)
Since the transformed surface force d ~G may be considered to act at the referential position ~Xrather than at the spatial position ~x, the second Piola-Kirchho� stress tensor is an ordinary (aone-point) rather than a two-point tensor. This can also be seen from the componental form of(3.22):
T(2)KL(
~X; t) = JXK;kXL;ltkl( ~X; t) ; tkl( ~X; t) = J�1xk;Kxl;LT(2)KL(
~X; t) : (3.23)
The foregoing expressions may be used as a source for approximate theories in which displace-ment gradient H is much small as compared to unity that the linearization is justi�ed. To thisend, we carry the linearized forms (1.100) and (1.101) into (3.17)1 and (3.22)1 and we obtain
T (1) = (1 + trH) t�HT � t+O�Æ2�;
T (2) = (1 + trH) t�HT � t� t �H +O�Æ2�:
(3.24)
The last equation demonstrates that the symmetry of tensor T (2) has not been violated bylinearization process. Conversely,
t = (1� trH)T (1) +HT � T (1) +O�Æ2�
= (1� trH)T (2) +HT � T (2) + T (2) �H +O�Æ2�:
(3.25)
Supposing, in addition, that stresses are small as compared to unity (the in�nitesimal deformationand stress theory), then
T (1) �= T (2) �= t ; (3.26)
showing that, in the in�nitesimal deformation and stress theory, the distinction between theCauchy and the Piola-Kirchho� stresses is not necessary.
37
4. FUNDAMENTAL BALANCE LAWS
4.1 Global balance laws
The fundamental laws of continuum mechanics are principles of the conservation of some physicalquantity. These balance laws, as they are often called, are postulated to all material continua,irrespective of material constitution and geometry, and result in equations that must always besatis�ed. As usual in continuum mechanics, we consider the �ve conservations laws dealing withmass, linear and angular momentum, energy and entropy. They are valid for all bodies subjectto thermomechanical e�ects.
The balance laws are usually formulated in global (integral) form derived by a considerationof the conservation of some property of the body as a whole. The global equations then may beused to develop associated �eld equations which are valid at all points within the body and onits boundary.
Fundamental Principle 1 (Conservation of Mass). The total mass of a body is unchanged
with motion.
This principle assumes that the mass production and supply is zero. It thus states that thetotal mass of the body in the reference con�guration is the same as the total mass of the bodyat any other time: Z
V%0 dV =
Zv(t)
% dv ; (4.1)
where V and v(t) is the reference and the current volume of the body, %0( ~X) is the mass densityof the body in the reference con�guration, %(~x; t) is the mass density of the body in the presentcon�guration.
Fundamental Principle 2 (Balance of Linear Momentum). The time rate of change of the
total linear momentum of a body is equal to the resultant force acting on the body.
Let a body having a current volume v(t) and bounding surface s(t) with exterior unit normal~n be subject to surface traction ~t(~n) and body force ~f (body force per unit mass of the body).The resultant force acting on the body isI
s(t)
~t(~n) da+
Zv(t)
%~f dv :
In addition, let the body be in motion under the velocity �eld ~v(~x; t). The linear momentum ofthe body is de�ned by the vector Z
v(t)%~v dv :
Thus the balance of linear momentum states that
D
Dt
Zv(t)
%~v dv =
Is(t)
~t(~n) da+
Zv(t)
%~f dv : (4.2)
Fundamental Principle 3 (Balance of Angular Momentum). The time rate of change ofthe total angular momentum of a body is equal to the resultant moment of all forces acting on the
body.
38
Mathematically,
D
Dt
Zv(t)
%~x� ~v dv =Is(t)
~x� ~t(~n) da+Zv(t)
%~x� ~f dv ; (4.3)
where the left-hand side is the time rate of change of the total angular momentum about theorigin, which is also frequently called the moment of momentum. On the right-hand side thesurface integral is the moment of the surface tractions about the origin, and the volume integralis the total moment of body forces about the origin.
Fundamental Principle 4 (Conservation of Energy). The time rate of change of the sum
of kinetic energy K and internal energy E is equal to the sum of the rate of work W of the surfaceand body forces and all other energies U� that enter and leave body per unit time.
Mathematically,D
Dt(K + E) =W +
X�
U� : (4.4)
The total kinetic energy of the body is given by
K =1
2
Zv(t)
%~v � ~v dv : (4.5)
In continuum mechanics the existence of the internal energy density " is postulated:
E =
Zv(t)
%" dv : (4.6)
The mechanical power, or rate of work of the surface traction ~t(~n) and body forces ~f is given by
W =
Is(t)
~t(~n) � ~v da+Zv(t)
%~f � ~v dv : (4.7)
Other energies U� (� = 1; 2; :::; n) that enter and leave the body may be of thermal, electro-magnetic, chemical, or some other origin. In this text, we consider that the energy transfer incontinuum is thermo-mechanical and thus only due to work or heat. The heat energy consists ofthe heat ux per unit area ~q that enters or leaves through the surface of the body and the heatsource, or heat supply, h per unit mass produced by internal sources (possibly from a radiation�eld). Thus we set U� = 0 except for
U1 := �Is(t)
~q � ~n da+Zv(t)
%h dv ; (4.8)
where the unit normal ~n is directed outward from the surface of the body. The negative sign atthe surface integral is needed because the heat ux vector ~q is pointing from high temperaturetoward lower temperature, so that
Hs(t) ~q � ~n da is the total outward heat ux. Thus the principle
of conservation of energy states that
D
Dt
Zv(t)
(%"+1
2%~v � ~v) dv =
Is(t)
(~t(~n) � ~v � ~q � ~n) da+Zv(t)
(%~f � ~v + %h) dv ; (4.9)
which is the statement of the �rst law of thermodynamics.
39
Fundamental Principle 5 (Entropy inequality). The time rate of change of the total entropy
H is never less than the sum of the ux of entropy ~s through the surface of the body and the entropyB supplied by the body forces. This law is postulated to hold for all independent processes.
Mathematically,
� :=DH
Dt�B +
Is(t)
~s � ~n da � 0 ; (4.10)
where � so de�ned is the total entropy production. Note that ~s is the outward entropy ux vector.In classical continuum mechanics the entropy density � per unit mass and the entropy source bper unit mass and unit time are postulated to exist such that
H =
Zv(t)
%� dv ; B =
Zv(t)
%b dv :
The entropy inequality then becomes
D
Dt
Zv(t)
%� dv �Zv(t)
%b dv +
Is(t)
~s � ~n da � 0 : (4.11)
The inequality implies internal entropy production in an irreversible process; the equality holdsfor a reversible process.
The foregoing �ve laws are postulated to hold for all bodies irrespective of their geometriesand constitutions. To obtain local equations, further restrictions are necessary, which are madein the next section.
4.2 Local balance laws in the spatial description
4.2.1 Continuity equation
The law of conservation of mass asserts that the mass of a body is invariant under motion, thatis, it remains constant in every con�guration. Thus, the material derivative of (4.1) is zero,
D
Dt
Zv(t)
% dv = 0 ; (4.12)
which upon application of the Reynolds transport theorem (2.46) with � = % becomesZv(t)��(t)
�D%
Dt+ %div~v
�dv +
Z�(t)
~n � [%(~v � ~w)]+� da = 0 : (4.13)
We now assume that density %(~x; t), velocity ~v and [%(~v � ~w)]+� are continuously di�erentiablefunctions of the spatial variables xk and time t in the volume v(t)� �(t) and across the disconti-nuity �(t), respectively, which implies that the integrands of the volume and surface integrals in(4.13) are continuous in xk and t. Moreover, we postulate that all global balance laws are valid
for arbitrary part of the volume and of the discontinuity surface. 6 Applied to (4.13) this impliesthat integrands of each of the integral must vanish identically. Thus
D%
Dt+ %div~v = 0 in v(t)� �(t) ; (4.14)
~n � [%(~v � ~w)]+� = 0 on �(t) : (4.15)
6For nonlocal continuum theories this postulate is revoked, and only the global balance laws (valid for the entirebody) are considered to be valid.
40
These are the equations of local conservation of mass and the interface condition in spatial form.Equation (4.14) is often called the continuity equation. But the material time derivative of % canbe written as
D%
Dt=@%
@t+ ~v � grad % ;
so that (4.14) may be expressed in the alternative form
@%
@t+ div (%~v) = 0 in v(t)� �(t) : (4.16)
Equation (4.15) means that the mass ux (the amount of mass per unit surface and unittime) entering the discontinuity surface must leave it on the other side, that is, there is no masscummulation on the discontinuity. In other words, the quantity in square brackets in (4.15) is theamount of mass swept through a running discontinuity in relation to the motion of body behindand ahead of the surface. According to (4.15), they must be equal.
4.2.2 Equation of motion
By substituting for the Cauchy stress vector ~t(~n) from (3.14), the equation of global balance oflinear momentum (4.2) reads
D
Dt
Zv(t)��(t)
%~v dv =
Is(t)��(t)
~n � t da+Zv(t)��(t)
%~f dv : (4.17)
By applying the generalized Gauss's theorem (2.43) to the surface integral on the right-hand side,we obtain
D
Dt
Zv(t)��(t)
%~v dv =
Zv(t)��(t)
(div t+ %~f) dv +
Z�(t)
~n � [t]+� da ; (4.18)
which upon using Reynolds's transport theorem (2.46) with � = %~v yields
Zv(t)��(t)
�D(%~v)
Dt+ %~v div~v � div t� %~f
�dv +
Z�(t)
~n � [%(~v � ~w) ~v � t]+� da = ~0 : (4.19)
This is postulated to be valid for all parts of the body. Thus the integrands vanish separately.Upon using (4.14), this is simpli�ed to
div t+ %~f = %D~v
Dtin v(t)� �(t) ; (4.20)
~n � [%(~v � ~w) ~v � t]+� = ~0 on �(t) : (4.21)
Equation (4.20) is known as Cauchy's equation of motion expressing the local balance of linearmomentum in spatial form, and (4.21) is the associated interface condition on the singular surface�.
4.2.3 Symmetry of the Cauchy stress tensor
The angular momentum of the surface tractions about the origin occurring in the global law ofbalance of the angular momentum (4.3), can be rewritten by using the Cauchy stress formula(3.14), the tensor identity
~v � (~w �A) = �~w � (A� ~v) ; (4.22)
41
where ~v, ~w are vectors, A is a second-order tensor, and the generalized Gauss theorem (2.43), tothe form I
s(t)��(t)~x� ~t(~n) da = �
Zv(t)��(t)
div (t� ~x) dv �Z�(t)
~n � [t� ~x]+� da : (4.23)
By making use of the two di�erential identities,
div (t� ~v) = div t� ~v + tT:� grad~v ; (4.24)
grad ~x = I ; (4.25)
where the superscript T at tensor t stands for transposition,:� denotes the dot-cross product of
the 2nd order tensors, and I is the second-order identity tensor, we can writeZv(t)��(t)
div (t� ~x) dv =Zv(t)��(t)
(div t� ~x+ tT:� I) dv :
Upon carrying this and (4.23) into the equation of balance of the angular momentum (4.3) andusing the Reynolds transport theorem (2.46) with � = ~x� %~v, we obtain
Zv(t)��(t)
�D(%~x� ~v)
Dt+ (%~x� ~v) div~v + div t� ~x+ tT
:� I � %~x� ~f
�dv
+
Z�(t)
~n � [(~v � ~w) (~x� %~v)]+� da+
Z�(t)
~n � [t� ~x]+� da = ~0 ;
which can be arranged to the formZv(t)��(t)
�(~x� ~v)
�D%
Dt+ %div~v
�+ %
D~x
Dt� ~v + ~x�
�%D~v
Dt� div t� %~f
�+ tT
:� I
�dv
�Z�(t)
~n � [%(~v � ~w) ~v � t]+� � ~x da = ~0 : (4.26)
ConsideringD~x
Dt� ~v = ~v � ~v = ~0
along with the local laws of conservation of mass (4.14), the balance of linear momentum (4.20),and the associated interface condition (4.21), equation (4.26) reduces toZ
v(t)��(t)tT
:� I dv = ~0 : (4.27)
Again, postulating that this to be valid for all parts of v(t)� �(t), the integrand must vanish, sothat,
tT:� I = ~0 or tT = t in v(t)� �(t) : (4.28)
Thus, the necessary and suÆcient condition for the satisfaction of the local balance of angularmomentum is the symmetry of the Cauchy stress tensor t. We have seen that the associatedinterface condition for the angular momentum is satis�ed identically.
Note that in formulating the angular principle by (4.3) we have assumed that no body orsurface couples act on the body. If any such concentrated moments do act, the material is said
42
to be a polar material and the symmetry property of t no longer holds. But this is a ratherspecialized situation and we shall not consider it here.
4.2.4 Energy equation
The same program can be carried out for the equation of energy balance (4.9). The integrandof the surface integral on the right-hand side of (4.9) is expressed by the Cauchy stress formula(3.14) and then the surface integral is converted to the volume integral by the Gauss theorem:I
s(t)��(t)(~t(~n) � ~v � ~q � ~n) da =
Zv(t)��(t)
(div (t � ~v)� div ~q) dv +
Z�(t)
~n � [t � ~v � ~q]+� da : (4.29)
The divergence of vector t � ~v will be arranged by making use of the identity
div (t � ~v) = div t � ~v + tT :: grad~v ; (4.30)
here :: stands for the double-dot product of tensors. The left-hand side of the equation of energybalance (4.9) can be arranged by Reynolds's transport theorem (2.46) with � = %"+ 1
2%~v � ~v asD
Dt
Zv(t)��(t)
(%"+1
2%~v � ~v) dv
=
Zv(t)��(t)
�D
Dt(%"+
1
2%~v � ~v) + (%"+
1
2%~v � ~v) div~v
�dv +
Z�(t)
~n ��(~v � ~w)(%"+
1
2%~v � ~v)
�+�da
=
Zv(t)��(t)
�("+
1
2%~v � ~v)
�D%
Dt+ %div~v
�+ %
D"
Dt+ %
D~v
Dt� ~v�dv+
Z�(t)
~n��(~v � ~w)(%"+
1
2%~v � ~v)
�+�da
which, by the law of mass conservation (4.14), reduces to
D
Dt
Zv(t)��(t)
(%"+1
2%~v�~v)dv =
Zv(t)��(t)
%
�D"
Dt+D~v
Dt� ~v�dv+
Z�(t)
~n��(~v � ~w)(%"+
1
2%~v � ~v)
�+�da :
(4.31)In view of (4.29){(4.31), the equation of motion (4.20), the symmetry of the Cauchy stress tensor(4.28), and upon setting the integrand of the result equal to zero, we obtain
%D"
Dt= t :: l� div ~q + %h in v(t)� �(t) ; (4.32)
~n ��(~v � ~w)(%" +
1
2%~v � ~v)� t � ~v + ~q
�+�= 0 on �(t) ; (4.33)
where l is the transposed velocity gradient tensor introduced by (2.13). Because of the symmetryof the Cauchy stress tensor t, equation (4.32) can be written in the alternative form
%D"
Dt= t :: d� div~q + %h in v(t)� �(t) ; (4.34)
where d is the strain-rate tensor introduced by (2.20)1. Equation (4.34) is the energy equationfor a thermomechanical continuum in spatial form and (4.33) is the associated interface conditionon the discontinuity �.
43
4.2.5 Entropy inequality
The same program can be applied to the global law of entropy to carry out it to the local form.Using again the Reynolds transport theorem, the Cauchy stress formula and the Gauss theorem,and assuming that the global law of entropy (4.11) is valid for any part of the body, we get thelocal form of the entropy inequality
%D�
Dt+ div~s� %b � 0 in v(t)� �(t) ; (4.35)
~n � [%�(~v � ~w) + ~s]+� � 0 on �(t) : (4.36)
4.2.6 R�esum�e of local balance laws
For a quick reference, we collect all local balance laws in the spatial description.
(i) Conservation of mass
@%
@t+ div (%~v) = 0 in v(t)� �(t) ; (4.37)
~n � [%(~v � ~w)]+� = 0 on �(t) : (4.38)
(ii) Balance of linear momentum
div t+ %~f = %D~v
Dtin v(t)� �(t) ; (4.39)
~n � [%(~v � ~w) ~v � t]+� = ~0 on �(t) : (4.40)
(iii) Balance of angular momentum
t = tT in v(t)� �(t) : (4.41)
(iv) Conservation of energy
%D"
Dt= t :: d� div~q + %h in v(t)� �(t) ; (4.42)
~n ��(~v � ~w)(%" +
1
2%~v � ~v)� t � ~v + ~q
�+�= 0 on �(t) : (4.43)
(v) Entropy inequality
%D�
Dt+ div~s� %b � 0 in v(t)� �(t) ; (4.44)
~n � [%�(~v � ~w) + ~s]+� � 0 on �(t) : (4.45)
4.3 Interface conditions in special cases
If there is a moving discontinuity surface �(t) sweeping the body with a velocity ~w in the directionof the unit normal ~n of �(t), then the interface conditions (4.38), (4.40) (4.43) and (4.45) must
44
be satis�ed on the surface �(t). Some of these interface conditions will now be applied to twospecial cases:
(i) The discontinuity surface is amaterial surface, that is, the surface frozen in the body (i.e.always containing the same material points). In this case, ~w = ~v, (4.38) is satis�ed identically,(4.40) and (4.43) reduce to
~n � [t]+� = ~0 ; (4.46)
~n � [t � ~v � ~q]+� = 0 : (4.47)
With the help of (4.46), the interface condition (4.47) may also be written in the form
~n � t � [~v]+� � ~n � [~q]+� = 0 : (4.48)
Hence, on a material interface between two media the surface traction ~n � t is continuous, andthe jump on the energy of tractions across this interface is balanced with that of the normalcomponent of the heat vector.
The dynamic interface conditions (4.46) and (4.47) must be supplemented by the kinematicinterface conditions. At a welded interface, like on a discontinuity between two solids, there is notangential slip across the interface and the spatial velocity must be continuous:
[~v]+� = ~0 : (4.49)
Consequently, the interface condition (4.48) is further reduces to
~n � [~q]+� = 0 ; (4.50)
which states the continuity in the normal component of the heat vector across �. Tangential slipis allowed at an interface between a solid and an inviscid uid or it may also occur on an idealizedfault surface separating two solids. At such a slipping interface, equation (4.49) is replaced by
[~n � ~v]+� = 0 : (4.51)
The last condition guarantees that there is no separation or interpenetration of the two materialsat the interface.
A frictionless interface of two materials is the material discontinuity across which the motionfrom one of its side runs without friction. This means that the shear stresses of the Cauchy stresstensor t are equal to zero from one of discontinuity side (e.g., with superscript `�'),
~n� � t� � �I � ~n� ~n�� = ~0 or ~n� � t� =�~n� � t� � ~n��~n� : (4.52)
Carrying this into (4.46) and considering that ~n = ~n� = �~n+ across �, the stress vector ~n � tfrom both side of a frictionless discontinuity is of the form
~n � t = �p~n ; (4.53)
where p is the negative normal component of the stress vector, p = �(~n � t � ~n). The condition(4.46) reduces to the condition of the continuity p across �,
[p]+� = 0 : (4.54)
45
(ii) The discontinuity surface coincides with the surface of the body. In this case %+ = 0,~v� = ~w. Again (4.38) gives an identity and the others reduce to
~n � [t]+� = ~0 ; (4.55)
~n � [t � ~v � ~q]+� = 0 ; (4.56)
where ~n � t+ is interpreted as the external surface load and t+ � ~v+ as the energy of this load. Ifthe external surface load is equal to zero, t+ = 0, then ~n � t� = ~0, and the �rst term on the leftof (4.56) is equal to zero. Hence, we obtain the boundary condition
~n � [~q]+� = 0 (4.57)
involving the heat alone.
4.4 Local balance laws in the referential description
In the previous section, the local balance laws have been expressed in spatial form. These equa-tions may also be cast in the referential form, which we now introduce.
4.4.1 Continuity equation
The referential form of continuity equation can be derived from (4.1) by using the transformationlaw dv = JdV : Z
V
�%0 �%J
�dV = 0 ; (4.58)
where V is the volume of the body in the reference con�guration and %( ~X; t) is the Lagrangiandescription of the density,
%( ~X; t) := %(~x( ~X; t); t) : (4.59)
But V is arbitrary, so that,%0 = %J in V � � ; (4.60)
which is equivalent toD%0Dt
= 0 in V � � ; (4.61)
Equations (4.60) and (4.61) are called the referential form of the continuity equation.The relation (4.60) can be considered as the general solution to the continuity equation (4.14).
To show it, let a body occupies the reference con�guration �0 and the present con�guration �t attime t = 0 and at time t, respectively. Making use of the relation for the material time derivativeof the jacobian,
:J= J div~v, the continuity equation (4.14) can be written in the form
:%
%+
:J
J= 0 :
Integrating this equation with respect to time from t = 0 to t, we �nd that
ln%
%0+ lnJ = 0:
This gives (4.60) after short algebraic manipulation. Hence, in the case that the deformationgradient F is given as the solution to the other �eld equations, the mass density % can be
46
calculated from (4.60) after solving these �eld equations. It means that equation (4.60) doesnot have to be included in the set of the governing equations. Such a situation appears inthe Lagrangian description of solids. This is, however, not the case if we employ the Euleriandescription. Since the initial con�guration is not the reference con�guration, neither F nor Jfollow from the �eld equations, and the continuity equation (4.14) must be included in the set ofthe governing equations in order to specify %.
To express the spatial interface conditions (4.15) in the referential form, we must �rst modify(4.15) such that
[~nda � %(~v � ~w)]+� = 0 on �(t) ; (4.62)
where we have inserted the spatial surface element da into equation (4.15) since it appears as thesurface element in the surface integral in (4.13). Moreover, we have involved the unit normal ~ninto the square brackets assuming that ~n points towards the + side of the discontinuity �(t) onboth sides of �(t). Since the spatial surface element da of �(t) changes continuously across �(t),that is, da+ = da�, the interface condition (4.62) is equivalent to the form (4.15). Consideringthe transformation (1.73) between the spatial and referential surface elements, d~a = Jd ~A � F�1,(4.62) transforms to h
%0( ~NdA � ~W )i+�= 0 on � ; (4.63)
where � is the surface discontinuity in the reference con�guration and
~W ( ~X; t) := F�1 � (~v � ~w) : (4.64)
In contrast to the spatial surface element da, the referential surface element dA may changediscontinuously on �, dA+ 6= dA�, if there is a tangential slip on this discontinuity, like on a uid-solid discontinuity (see Figure 8.1). On the other hand, at a discontinuity with no tangentialslip, like on a solid-solid discontinuity, dA+ = dA�, and this factor can be dropped from theinterface condition (4.63).
4.4.2 Equation of motion
A simple form of the Lagrangian equation of motion can be obtained in terms of the �rst Piola-Kirchho� stress tensor that is related to the Lagrangian Cauchy stress tensor by (3.17). Applyingthe divergence operation on (3.17)2 and making use of the following di�erential identity
div (A �B) = (divA) �B +AT :: gradB ; (4.65)
which is valid for all di�erentiable tensors A and B, the divergence of the Cauchy stress tensorcan be arranged as follows
div t = div (J�1F � T (1)) = div (J�1F ) � T (1) + J�1F T :: gradT (1) :
The �rst term is equal to zero because of the Jacobi identity (1.40)2 and the second term can bearranged according to the di�erential identity (1.43)2:
div t = J�1DivT (1) : (4.66)
Using (4.60), the equation of motion (4.20) can be expressed in the referential form as
DivT (1) + %0 ~F = %0D~v
Dtin V � � ; (4.67)
47
where ~F ( ~X; t) is the Lagrangian description of the body force,
~F ( ~X; t) := ~f(~x( ~X; t); t) : (4.68)
Note that the divergence of the Eulerian Cauchy stress tensor div t( ~X; t) in (4.20) is tranformedinto the divergence of the �rst Piola-Kirchho� stress tensor DivT (1)( ~X; t) in (4.67). In fact, theoriginal de�nition (3.16) of T (1) was motivated by this simple transformation.
In similar way as for the conservation of mass, the interface condition (4.21) can be expressedin the referential form. Using (1.73), (4.64) and (3.17)2, we obtainh
%0( ~NdA � ~W )~v � ~NdA � T (1)i+�= ~0 on � : (4.69)
Note that the Lagrangian equation of motion and the interface conditions can also be written interms of the second Piola-Kirchho� stress tensor rather than in terms of the �rst Piola-Kirchho�stress tensor by substituting T (1) = T (2) � F T .
4.4.3 Symmetries of the Piola-Kirchho� stress tensors
Upon (3.17)2 and (3.22)2, the symmetry of the Cauchy stress tensor t = tT has two di�erentforms �
T (1)�T
= F � T (1) � F�T ;�T (2)
�T= T (2) : (4.70)
It shows that T (2) is symmetric whenever t is symmetric (nonpolar case), but T (1) is in generalnot symmetric.
4.4.4 Energy equation
Let ~Q be de�ned as the heat ux with respect to the surface element d ~A in the reference con�g-uration:
~q � d~a = ~Q � d ~A ; (4.71)
where ~q is the heat ux with respect to the surface element d~a in the present con�guration. By us-ing the transformation rule for surface elements between the present and reference con�gurations,d~a = Jd ~A � F�1, the referential and spatial heat uxes are related by
~q = J�1F � ~Q ; ~Q = JF�1 � ~q : (4.72)
The referential heat ux and the Piola-Kirchho� stress tensors will be employed to transformthe energy equation from the spatial form to the referential form. First, the divergence of heat ux ~q can be arranged by making use of the di�erential identity (4.30) as follows
div ~q = div (J�1F � ~Q) = div (J�1F ) � ~Q+ J�1F T :: grad ~Q :
The �rst term is equal to zero because of the Jacobi identity (1.40)2 and the second term can bearranged according to the di�erential identity (1.43)2:
div ~q = J�1Div ~Q : (4.73)
48
Next, the stress power per unit volume in the present con�guration, t :: d, can be referred backto the reference con�guration. Using two tensor identities
(A �B) :: C = A :: (B �C) ; (A �B) :: (C �D) = (D �A) :: (B �C) ; (4.74)
valid for the 2nd order tensors A, B, C and D, equations (2.12)2 and (3.17)2, we have
t :: d = t :: l = J�1(F � T (1)) :: (:F �F�1) = J�1(F�1 � F ) :: (T (1)�
:F ) = J�1I :: (T (1)�
:F )
= J�1(I � T (1)) :::F ;
or,
t :: d = J�1T (1) :::F ; (4.75)
where:F is the material time derivative of the deformation gradient tensor. Introducing the
second Piola-Kirchho� stress tensor T (2) instead of T (1), the stress power can alternatively beexpressed as
t :: d = J�1(T (2) � F T ) :::F= J�1T (2) :: (F T �
:F ) = J�1T (2) :: (F T � l � F ) = J�1T (2) :: (F T � d � F );
or,
t :: d = J�1T (2) :::E ; (4.76)
where:E is the material time derivative of the Lagrangian strain tensor E given by (2.23).
We are now ready to express the conservation of energy (4.34) in the referential form. Using(4.73) and (4.76), along with (4.60), we obtain
%0D"
Dt= T (2) ::
:E �Div ~Q+ %0h in V � � : (4.77)
The dissipative term is also equal to T (1) :::F .
The referential form of energy interface condition (4.33) can be derived by substituting (4.62)and (4.64) into (4.33) and considering (4.71):
�%0( ~NdA � ~W )(" +
1
2%~v � ~v)� ~NdA � T (1) � ~v + ~NdA � ~Q
�+�= 0 on � : (4.78)
4.4.5 Entropy inequality
The same program can be applied to the entropy inequality to carry out it to the referential form.Let ~S be de�ned as the entropy in ux with respect to the surface element d ~A in the referencecon�guration:
~s � d~a = ~S � d ~A ; (4.79)
where ~s is the heat ux with respect to the surface element d~a in the present con�guration.Considering the transformation between the spatial and referential surface elements, d~a = Jd ~A �F�1, the referential and spatial entropy in uxes are related by
~s = J�1F � ~S ; ~S = JF�1 � ~s : (4.80)
49
In an analogous way as in the preceding section, the entropy inequality (4.35) and the entropyinterface condition (4.36) may then be expressed in the referential form as
%0D�
Dt+Div ~S � %0b � 0 in V � � ; (4.81)h
%0�( ~NdA � ~W ) + ~NdA � ~Si+�
� 0 on � : (4.82)
50
5. CONSTITUTIVE EQUATIONS
5.1 The need for constitutive equations
Basic principles of continuum mechanics, namely, conservation of mass, balance of momenta, andconservation of energy, discussed in Chapter 4, lead to the fundamental equations:
@%
@t+ div (%~v) = 0 ; (5.1)
div t+ %~f = %D~v
Dt; t = tT ; (5.2)
%D"
Dt= t :: d� div ~q + %h : (5.3)
In total, they constitute �ve independent equations (one for mass, three for linear momentumand one for energy) for �fteen unknown �eld variables, namely,
� mass density %,
� velocity ~v,
� Cauchy's stress tensor t,
� internal energy ",
� heat ux ~q,
� temperature �
provided that body forces ~f and distribution of heat sources h are given. Clearly, the foregoingbasic equations are not adequate for the determination of these unknowns except for some triv-ial situations, for example, rigid body motions in the absence of heat conduction. Hence, tenadditional equations must be supplied to make the problem well-posed.
In the derivation of the equations (5.1) to (5.3) no di�erentiation has been made between var-ious types materials. It is therefore not surprising that the foregoing equations are not suÆcientto explain fully the motions of materials having various type of physical properties. The characterof the material is brought into the formulation through the so-called constitutive equations, whichspecify the mechanical and thermal properties of particular materials based upon their internalconstitution. Mathematically, the usefulness of these constitutive equations is to describe therelationships among the kinematic, mechanical, and thermal �eld variables and to permit theformulations of well-posed problems of continuum mechanics. Physically, the constitutive equa-tions de�ne various idealized materials which serve as models for the behavior of real materials.However, it is not possible to write one equation capable of representing a given material over itsentire range of application, since many materials behave quite di�erently under changing levels ofloading, such as elastic-plastic response due to increasing stress. Thus, in this sense it is perhapsbetter to think of constitutive equations as representative of a particular behavior rather than ofa particular material.
5.2 Formulation of thermomechanical constitutive equations
In this text we deal with the constitutive equations of thermomechanical materials. The studyof the chemical changes and electromagnetic e�ects are excluded. A large class of materials does
51
not undergo chemical transition or produce appreciable electromagnetic e�ects when deformed.However, the deformation and motion generally produce heat. Conversely, materials subjectedto thermal changes deform and ow. The e�ect of thermal changes on the material behaviordepends on the range and severity of such changes.
The thermomechanical constitutive equations are relations between a set of thermomechanicalvariables. They may be expressed as an implicit tensor-valued functional R of 15 unknown �eldvariables:
R~X0 2 B� � t
h%( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); t( ~X 0; �); ~q( ~X 0; �); "( ~X 0; �); ~X
i= 0 ; (5.4)
where � are all past times and t is the present time. The constraints ~X 0 2 B and � � t express theprinciple of determinism postulating that the present state of the thermomechanical variablesat a material point ~X of the body B at time t is uniquely determined by the past history of themotion and the temperature of all material points of the body B. The principle of determinismis a principle of exclusions. It excludes the dependence of the material behavior on any pointoutside the body and any future events.
We shall restrict the functional in (5.4) to be of a type that does not change with time, thatis, that does not depend on the present time t explicitly but only implicitly via thermomechanicalvariables. Such a functional is invariant with respect to translation in time. The materials de-scribed by (5.4) possess time-independent thermomechanical property. Note that the dependenceof functionalR on the position of material point ~X means that the material property changes withthe change of position within the body B; such a material is called heterogeneous. If functionalR is independent of ~X , material is homogeneous.
For a simple material (see the next section), the implicit functional equation (5.4) is supposedto be solved uniquely for the present values of thermomechanical variables. In this case, theimplicit functional equation (5.4) is replaced by a set of explicit functional equations:
t( ~X; t) = F~X0 2 B� � t
h%( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); ~X
i;
~q( ~X; t) = Q~X0 2 B� � t
h%( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); ~X
i;
"( ~X; t) = E~X0 2 B� � t
h%( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); ~X
i;
(5.5)
where F , Q and E are respectively tensor-valued, vector-valued and scalar-valued functionals.Note that all constitutive functionals F , Q and E are assumed to depend on the same set ofvariables %( ~X 0; �), ~x( ~X 0; �), �( ~X 0; �) and ~X. This is known as the principle of equipresence.
However, the implicit functional equation (5.4) need not be of such a nature as to determinet, ~q and " at ( ~X; t) explicitly. For instance, the stress at ( ~X; t) may depend not only on themotion and temperature at all other points of the body but also on the histories of the stress,the heat ux and the internal energy. Various types of approximations of (5.4) exist in which thedependence on t( ~X 0; �) is replaced by the history of various order of stress rates, heat rates, etc.For example, in a special case the constitutive equation (5.4) may be written explicitly for the
52
stress rates:t at ( ~X; t):
:t ( ~X; t) = F
~X0 2 B� � t
ht( ~X; t); %( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); ~X
i: (5.6)
More generally, we may have
t(p)( ~X; t) = F~X0 2 B� � t
ht(p�1)( ~X; t); t(p�2)( ~X; t); : : : ; t( ~X; t); %( ~X 0; �); ~x( ~X 0; �); �( ~X 0; �); ~X
i; (5.7)
which involves the stress rates up to the pth order at ( ~X; t). This type of generalization is, forinstance, needed to interpret creep data.
Except the Maxwell viscoelastic solid, we shall consider the explicit constitutive equations(5.5). Together with �ve basic balance equations (5.1){(5.3) they form 15 equations for 15 un-knowns. Since t, ~q, and " are expressed explicitly in (5.5), it is, in principle, possible to eliminatethese variables in (5.1){(5.3). Then we obtain �ve equations (the so called �eld equations of
thermomechanics) for �ve unknown �eld variables: mass density %, velocity ~v and temperature �.We now proceed to deduce the consequence of additional restrictions on the functionals F , Q
and E . Since the procedure is similar for all functionals, for the sake of brevity, we carry out theanalysis only for stress functional F . The results for Q and E are then written down immediatelyby analogy.
5.3 Simple materials
The constitutive functionals F , Q and E are subject to another fundamental principle, the prin-ciple of local action postulating that the motion and the temperature at distant material pointsfrom ~X does not a�ect appreciably the stress, the heat ux and the internal energy at ~X. Supposethat the functions %( ~X 0; �), ~x( ~X 0; �) and �( ~X 0; �) admit Taylor series expansion about ~X for all� < t. According to the principle of local action, a relative deformation of the neighborhood ofmaterial point ~X is permissible to approximate only by the �rst-order gradient
%( ~X 0; �) � %( ~X; �) + Grad %( ~X; �) � d ~X ;
~x( ~X 0; �) � ~x( ~X; �) + F ( ~X; �) � d ~X ; (5.8)
�( ~X 0; �) � �( ~X; �) + Grad �( ~X; �) � d ~X ;
where F ( ~X; �) is the deformation gradient tensor at ~X and time � , and d ~X = ~X 0 � ~X . Sincethe relative motion and temperature history of an in�nitesimal neighborhood of ~X is completelydetermined by the history of density, deformation and temperature gradients at ~X , the stresst( ~X; t) must be determined by the history of Grad %( ~X; �), F ( ~X; �) and Grad �( ~X; �) for � � t.Such materials are called simple materials. We also note that if we retain higher-order gradientsin (5.8), then we obtain nonsimple materials of various classes. For example, by including thesecond-order gradients into argument of F we get the theory of couple stress. In other words, thebehavior of the material point ~X within a simple material is not a�ected by the histories of thedistant points from ~X . To any desired degree of accuracy, the whole con�guration of a suÆcientlysmall neighborhood of the material point ~X is determined by the history of Grad %( ~X; �), F ( ~X; �)and Grad �( ~X; �), and we may say that the stress t( ~X; t), which was assumed to be determined
53
by the local con�guration, is completely determined by Grad %( ~X; �), F ( ~X; �) and Grad �( ~X; �).That is, the general constitutive equation (5.5)1 reduces to the form
t( ~X; t) = F��t
h%( ~X; �);Grad %( ~X; �); ~x( ~X; �);F ( ~X; �); �( ~X; �);Grad �( ~X; �); ~X
i: (5.9)
Given the deformation gradient F , the density in the reference con�guration is expressedthrough the continuity equation (4.60) as
%( ~X; �) =%0( ~X)
detF ( ~X; �): (5.10)
We can thus drop the argument %( ~X; �) in the functional F since %( ~X; �) is expressible in termsof F ( ~X; �) (the factor %0( ~X) is a �xed expression { not dependent on time { for a given referencecon�guration). Moreover, by applying the gradient operator on (5.10), the term Grad %( ~X; �) canbe expressed in term of GradF ( ~X; �). For a simple material, however, GradF can be neglectedwith respect to F . In summary, the principle of local action applied for a simple material leadsto the following constitutive equation:
t( ~X; t) = F��t
h~x( ~X; �);F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i; (5.11)
where ~G� stands for the temperature gradient, ~G� := Grad �.
5.4 Frame indi�erence
Mathematically, an observer is identi�ed with a frame. The term frame therefore means a systemequipped to measure position in the Euclidean space and time on the real line. If ~x and ~x� denotethe position vectors of the same observer P in two spatial frames (see Figure 5.1), then they are
related to each other by the relation
~x� = ~x+~b ; (5.12)
where ~b corresponds to the translation ofthe two spatial frames; in fact, ~b can betime-dependent, ~b = ~b(t). In componentform, (5.12) reads
x�k~i�k = xl~il + b�k~i
�k ;
where ~ik and ~i�k are the Cartesian basevectors associated with the two spatialframes. By a consideration of the scalarproduct ~i�k �~i�l = Ækl, we obtain
x�k = (~i�k �~il)xl + b�k : (5.13)
�~i1 ~i2
~i3
P
~x
~x�
~b
~i�1
~i�2
~i�3
Figure 5.1. Euclidean transformation.
In view of de�nition (1.13), we can see that the scalar products~i�k �~il are the shifters between thetwo spatial frames. Instead of the Kronecker symbol Æ, we denote them as
Qkl :=~i�k �~il :
54
In fact, Qkl can again be time-dependent, Qkl = Qkl(t). Equation (1.12) yields the relationbetween the Cartesian base vectors ~ik and ~i
�k:
~i�k = Qkl~il ; ~ik = (QT )kl~i
�l ; (5.14)
where(QT )kl :=~ik �~i�l :
The orthogonality relation (1.14) for the shifters has now the form
(~i�k �~il)(~il �~i�m) = Ækm ; (~ik �~i�l )(~i�l �~im) = Ækm ;
which can be expressed in terms of Qkl as
Qkl(QT )lm = Ækm ; (QT )klQlm = Ækm : (5.15)
Equation (5.13) can be written in symbolic notation as
~x� = Q(t) � ~x+~b(t) : (5.16)
Here ~x� and ~x are regarded algebraically as triples of the components, ~x = (x1; x2; x3) and~x� = (x�1; x
�2; x
�3), rather than as geometric objects ~x = xk~ik and ~x� = x�k
~i�k. Hence, the symbolicequation (5.16) has always to be interpreted in terms of vector components. 7 Likewise, thesymbolic notation of matrix multiplication (5.15) is
Q(t) �QT (t) = QT (t) �Q(t) = I : (5.17)
Equation (5.16) expresses a rigid motion of the spatial frame. In fact, ~b(t) corresponds to thetranslation, and Q(t) to the rotation of the spatial frame. The transformation (5.16) is oftenreferred to as the Euclidean transformation or the observer transformation (x; t)! (x�; t). 8
We say that any scalar-, vector- and tensor-valued quantity � is frame indi�erent, if it isinvariant under all changes of frame (5.16), that is, if �� = �. For instance, the components of aframe indi�erent second-order tensor a transform under rigid motion of spatial frame accordingto the relation
a�kl = QkmamnQln ; (5.18)
which can be obtained from the transformation rule (5.14). In symbolic notations, we can write
a� = Q � a �QT : (5.19)
7Vector notation is somewhat ambiguous. The same symbol is used for geometric quantity ~x = xk~ik and thealgebraic quantity ~x = (x1; x2; x3), which denotes triples of components of the geometric quantity ~x. We shallnot distinguish between geometric and algebraic notation by using di�erent symbols. For example, ~x� in (5.12) isgeometric notation while ~x� in (5.16) is algebraic notation.
8The most general change of frame (x; t)! (x�; t�) is, in addition, characterized by the shift of time:
t� = t� a ;
where a is a particular time. It can be shown that the consequence of frame indi�erence with respect to the shiftof time is that the constitutive functionals do not depend explicitly on the current time t. We considered this factas a starting assumption in formulating a general form (5.4) of constitutive equations.
55
The symbols a and a� represent the same tensor in two di�erent spatial frames as algebraic9-tuples of tensor components rather than as geometric objects. Hence, the symbolic equation(5.19) has always to be interpreted in terms of tensor components. In an analogous way, scalar-and vector-valued physical quantities � and ~v are called frame indi�erent if they transform undera rigid motion of spatial frame according to
�� = � ;v�k = Qklvl ; or; symbolically; ~v� = Q � ~v : (5.20)
In the sense of this de�nition, the mass of a material body is a frame indi�erent scalar, the forceis a frame-indi�erent vector and the Cauchy stress tensor is a frame indi�erent tensor. That is,for instance,
t�( ~X; t) = Q(t) � t( ~X; t) �QT (t): (5.21)
The transformation rule for the deformation gradient is
F �( ~X; t) = Q(t) � F ( ~X; t) : (5.22)
To show it, we express the deformation gradient in the starred frame according to (1.31)1 andsubstitute from (5.13):
F �kK =@x�k@XK
=@
@XK
�Qklxl + b�k
�= Qkl
@xl@XK
= QklFlK :
Written in symbolic notation, we obtain (5.22). Thus, the two-point deformation gradient tensorF is not a frame indi�erent tensor. However, three columns of F (for K = 1; 2; 3) are frame indif-ferent vectors. Other quantities which are not frame indi�erent are, among others, the positionvector and the velocity vector. The property of a particular quantity to be frame indi�erent ornot is either a priori postulated or derived from other de�nitions.
Let us verify that the jacobian J , the referential heat ux vector ~G�, the Green deformationtensor C, the right stretch tensor U and the second Piola-Kirchho� stress tensor T (2) are allframe indi�erent scalars:
J� = J; ~G�� = ~G�; C� = C; U � = U ; T (2)� = T (2): (5.23)
The proof is immediate by making use of (1.37), (1.46), (1.51), (3.22)1, (5.17), (5.21) and (5.22):
J� = detF � = det(Q � F ) = detQdetF = detF = J;
~G�� = Grad �� = Grad � = ~G�;
C� = (F �)T � F � = F T �QT �Q � F = F T � F = C;
U� =pC� =
pC = U ;
T (2)� = J�(F �)�1 � t� � (F �)�T = JF�1 �QT �Q � t �QT �Q � F�T = JF�1 � t � F�T = T (2):
The constitutive functionals are subject to yet another fundamental principle, the principleof frame indi�erence or (material objectivity). This principle states that a constitutiveequation must be form-invariant under rigid motions of the spatial frame or, in other words,
56
the material properties cannot depend on the motion of an observer. To express this principlemathematically, let us write the constitutive equation (5.11) in the unstarred and starred frames:
t( ~X; t) = F��t
h~x( ~X; �);F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i;
t�( ~X; t) = F���t
h~x�( ~X; �);F �( ~X; �); ��( ~X; �); ~G��(
~X; �); ~Xi;
(5.24)
where ~x( ~X; t) � �( ~X; t) and ~x�( ~X; t) � ��( ~X; t) are related by (5.16), that is, ~x�( ~X; t) =Q(t) � ~x( ~X; t) + ~b(t). In general, the starred and unstarred functionals F and F� may di�er,but the principle of frame indi�erence requires that the form of the constitutive functional Fmust be the same under any two rigid motions of spatial frame. Mathematically, no star isattached to the functional F :
F(�) = F�(�) ; (5.25)
where the arguments are those of (5.24).The frame indi�erence of the Cauchy stress tensor, see (5.21), places the restriction on con-
stitutive functional F :Q(t) � F
��t
h~x( ~X; �);F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�QT (t)
= F��t
h~x�( ~X; �);F �( ~X; �); ��( ~X; �); ~G��(
~X; �); ~Xi:
Taking into account the transformation relations (5.16), (5.22) and (5.23)2, we obtain
Q(t) � F��t
h~x( ~X; �);F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�QT (t)
= F��t
hQ(�) � ~x( ~X; �) +~b(�);Q(�) � F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i:
(5.26)
This relation must hold for any arbitrary orthogonal tensor-valued functions Q(t) and any arbi-trary vector-valued function ~b(t).
Now, let us examine the restrictions imposed on the form of the constitutive functional byconsidering separately two special rigid changes of frame, which taken successively in any ordercan represent the general rigid motion of the spatial frame of reference.
(a) Rigid translation of the spatial frame such that the moving origin moves with the material
point ~X :Q(�) = I ; ~b(�) = �~x( ~X; �) : (5.27)
This means that the spatial frame of reference is translated so that the material point ~X at anytime � remains at the origin. From (5.16) it follows that ~x�( ~X; �) = 0 and (5.26) becomes
F��t
h~x( ~X; �);F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i= F��t
h~0;F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i: (5.28)
Thus the stress at the material point ~X and time t cannot depend explicitly on the history ofmotion of this point. Consequently, the general constitutive equation (5.11) reduces to the form
t( ~X; t) = F��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i: (5.29)
57
(b) Time-dependent rigid rotations of the spatial frame of reference. We now consider the re-strictions imposed on F by the principle of frame indi�erence under arbitrary time-dependentrotation such that ~b(�) = 0 and Q(�) is arbitrary. Using (5.26) we can conclude that the func-tional F must satisfy a restrictive condition of the form
Q(t) � F��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�QT (t) = F
��t
hQ(�) � F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i(5.30)
for all orthogonal tensor-valued functions Q(t) and all invertible deformation processes F ( ~X; t).
5.5 Reduction by polar decomposition
The condition (5.30) will now be used to reduce the constitutive equation (5.29). Let us recall thepolar decomposition (1.45) of the deformation gradient F ( ~X; t) = R( ~X; t)�U( ~X; t) into a rotation
tensor R and the right stretch tensor U =pC =
pF T � F . We now make a special choice for
the orthogonal tensor Q(t) in (5.30). For any �xed reference place ~X , we put Q(t) = RT ( ~X; t)for all t. This special choice of Q(t) in (5.30) yields:
RT ( ~X; t) � F��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�R( ~X; t)
= F��t
hRT ( ~X; �) � F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i;
(5.31)
which with U = RT � F reduces to
F��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i= R( ~X; t) � F
��t
hU( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�RT ( ~X; t) :
(5.32)This reduced form has been obtained for a special choice of Q(t) in the principle of frame in-di�erence (5.30). This means that (5.32) is a necessary relation for satisfying the principle offrame indi�erence. Now, we shall prove that (5.32) is also a suÆcient relation for satisfying thisprinciple. Suppose, that F is of the form (5.32) and consider an arbitrary orthogonal tensorhistory Q(t). Since the polar decomposition of Q � F is (Q �R) �U , (5.32) for Q � F reads
F��t
hQ(�) � F ( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i
= Q(t) �R( ~X; t) � F��t
hU( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�hQ(t) �R( ~X; t)
iT;
which in view of (5.31) reduces to (5.30), so that (5.30) is satis�ed. Therefore the reduced form(5.32) is necessary and suÆcient to satisfy the principle of frame indi�erence.
We have proved that the constitutive equation of a simple material may be put into the form
t( ~X; t) = R( ~X; t) � F��t
hU( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�RT ( ~X; t) : (5.33)
A constitutive equation of this kind, in which the functionals are not subject to any furtherrestriction, is called a reduced form. The result (5.33) shows that while the stretch history of asimple material may a�ect its present stress, past rotations have no e�ect at all. The presentrotation enters (5.33) explicitly.
58
There are many other reduced forms for the constitutive equation of a simple material. Re-placing the stretch U by the Green deformation tensor C, U =
pC, and denoting the functional
F(pC; � � �) again as F (C; � � �), we obtain
t( ~X; t) = R( ~X; t) � F��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i�RT ( ~X; t) : (5.34)
Likewise, expressing rotation R through the deformation gradient, R = F � U�1, and replacingthe stretch tensor U by Green's deformation tensor C, equation (5.34), after introducing a newfunctional ~F of the deformation history C, can be put into another reduced form
t( ~X; t) = F ( ~X; t) � ~F��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i� F T ( ~X; t) : (5.35)
We should emphasize that the constitutive equations (5.34) and (5.35) automatically satisfy theprinciple of material indi�erence.
Another useful reduced form may be obtained if the second Piola-Kirchho� stress tensor T (2)
de�ned by (3.22)1,T (2) = (detF )F�1 � t � F�T ; (5.36)
is used in the constitutive equation instead of the Cauchy stress tensor t. With
J = detF = det (R �U) = detU =pdetC ; (5.37)
and de�ning a new functional G of the deformation history C, the constitutive equation (5.35)can be rewritten for the second Piola-Kirchho� stress tensor as
T (2)( ~X; t) = G��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i: (5.38)
According to this result, the second Piola-Kirchho� stress tensor depends only on the Greendeformation tensor and not on the rotation. In view of (5.23)5, the second Piola-Kirchho� tensoris frame indi�erent scalar. Therefore, the reduced form (5.38) satis�es the principle of frameindi�erence.
The principle of frame indi�erence applied to the constitutive equations (5.5) for the heat uxand the internal energy in analogous way as for the stress results in the following reduced forms:
~q( ~X; t) = R( ~X; t) � Q��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i;
"( ~X; t) = E��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �); ~X
i:
(5.39)
5.6 Kinematic constraints
A condition of kinematic constraint is a geometric restriction on the set of all motions which arepossible for a material body. A condition of constraint can be de�ned as a restriction on thedeformation gradients:
�(F (t)) = 0 ; (5.40)
59
where to lighten notations in this section we drop the place ~X from the notation as in F (t) �F ( ~X; t). The requirement that kinematic constraints shall be frame indi�erent implies thatequation (5.40) has to be replaced by
�(C(t)) = 0 : (5.41)
In the context of kinematic constraints, the principle of determinism must be modi�ed; it pos-tulates that only a part of the stress tensor is related to the history of the deformation. Thusinstead of (5.29), we write
t(t) = �(t)+ F��t
hF (�); �(�); ~G�(�)
i: (5.42)
The tensor t�� is called the determinate stress because it is uniquely determined by the motion.The tensor � is called indeterminate stress and represents the reaction stress produced by thekinematic constraint (5.41). It is not determined by the motion. In analogy with analyticalmechanics, it is assumed that the reaction stress does not perform any work, that is the stresspower vanishes for all motions compatible with the constraint condition (5.41):
�(t) : d(t) = 0 ; (5.43)
where d is the strain-rate tensor de�ned by (2.20)1.To derive an alternative form, we modify the constitutive equation (5.38) for the second
Piola-Kirchho� stress tensor as
T (2)(t) = �(t)+ G��t
hC(�); �(�); ~G�(�)
i: (5.44)
If we take into account (2.23) and (3.22)2, the double-dot product � : d can be expressed in termsof the reaction stress � as
� : d =1
J(F �� � F T ) :
1
2(F�T �
:C �F�1) =
1
2J(� :
:C) :
Hence, the constraint (5.43) takes the referential form
�(t) ::C (t) = 0 : (5.45)
This says that the reaction stress � has no power to work for all motions compatible with theconstraints (5.41). This compatibility can be evaluated in a more speci�c form. Di�erentiating(5.41) with respect to time, we obtain
d�(C(t))
dCKL
:CKL= 0 or; symbolically;
d�(C(t))
dC::C= 0 : (5.46)
This shows that the normal d�=dC to the surface �(C)=const. is orthogonal to all strain rates:C which are allowed by the constraint condition (5.41). Equation (5.45) suggests that just thesame orthogonality holds for the reaction stress �, whence follows that � must be parallel tod�=dC :
�(t) = �(t)d�(C(t))
dC: (5.47)
60
Here, the factor � is left undetermined and must be regarded as an independent �eld variable inthe balance law of linear momentum.
The same arguments apply if simultaneously more constraints are speci�ed. For
�i(C(t)) = 0 ; i = 1; 2; : : : ; (5.48)
we haveT (2)(t) =
Xi
�i(t)+ G��t
[C(�); �(�);Grad �(�)] (5.49)
with
�i(t) = �i(t)d�i(C(t))
dC: (5.50)
We note that at most six constraint conditions can be speci�ed. The maximum of six constraintscorresponds to a rigid body; in this case the state of stress is completely underdetermined.
As an example we investigate the constraint of incompressibility. If a body is assumed to beincompressible, only volume-preserving motions are possible. It also means that the mass densityremains unchanged, that is,
J =pdetC =
%o%
= 1 ; (5.51)
where the �rst equality follows from (1.37) and (1.51). This condition places a restriction onC, namely, the components of C are not all independent. In this particular case, the constraint(5.41) reads
�(C) = detC � 1 : (5.52)
Making use of the identityd
dA(detA) = (detA)A�T ; (5.53)
which is valid for all invertible second-order tensors A, we derive
d
dC(detC � 1) = (detC)C�1 = C�1 : (5.54)
Equation (5.47) then implies�(t) = �(t)C�1(t) : (5.55)
Instead of � we write �p in order to indicate that p is a pressure. Equation (5.44) now reads
T (2)(t) = �p(t)C�1(t)+ G��t
hC(�); �(�); ~G�(�)
i: (5.56)
Complementary, the Cauchy stress tensor is expressed as
t(t) = �p(t)I + F � F��t
hC(�); �(�); ~G�(�)
i� F T : (5.57)
This equation shows that for an incompressible simple material the stress is determined by themotion only to within a pressure p.
61
X1
X2
X3
X1
X2
X3
x1 x2
x3
~x = ~�R( ~X; t)
~x = ~�R(~X; t)
~X = ~�( ~X)
Ref. con�guration R
Ref. con�guration R
Present con�guration
Figure 5.2. The motions from two di�erent reference con�gurations.
5.7 Material symmetry
In this section, we will con�ne ourselves to the homogeneous material. We say that the materialis homogeneous if the constitutive equations do not depend on the translations of the origin thereference con�guration. In other words, there is at least one reference con�guration in which aconstitutive equation of a homogeneous material is the same for all particles. It means that theexplicit dependence of the constitutive functionals F , Q and E on the position ~X disappears. Forinstance, equation (5.29) for a homogeneous material reduces to
t( ~X; t) =FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i: (5.58)
Note that the functional form may, in general, be di�erent for di�erent particles. Then theexplicit dependence of constitutive functionals on the position ~X of a particle X cannot beomitted. The following discussion is, however, concerned with the constitutive functional at aparticle. Moreover, the functional F is labelled by the subscript R since F depends on the choiceof reference con�guration.
Material symmetry or isotropy, if it exists, can be characterized by invariance propertiesof the constitutive equations with respect to a change of reference con�guration. To make thisstatement more precise and to exploit its consequences, denote by R and R two di�erent referencecon�gurations (see Figure 5.2) related by the one-to-one mapping
~X = ~�( ~X) () ~X = ~��1( ~X) ; (5.59)
62
where ~X and ~X are the positions of the particle X in the reference con�gurations R and R,
respectively. Then the motion ~x = ~�R( ~X; t) takes the form ~x = ~�R(~X; t), and we obtain the
identity~�R( ~X; t) = ~�R(
~�( ~X); t) ; (5.60)
which holds for all ~X and t. Di�erentiating ~x = ~�R( ~X; t) and �( ~X; t) with respect to ~X leads tothe transformation rule for the deformation and temperature gradients:
FkK =@xk@XK
=@xk@�L
@�L@XK
= FkLPLK ;
( ~G�)K =@�
@XK=
@�
@�L
@�L@XK
= ( ~G�)LPLK ;
or, in symbolic notation,
F ( ~X; t) = F ( ~X; t) � P ( ~X) ;
�( ~X; t) = �( ~X; t) ; (5.61)
~G�( ~X; t) = ~G�(~X; t) � P ( ~X) :
Here,
F ( ~X; t) := (Grad�R( ~X; t))T (5.62)
denotes the transformed deformation gradient, and
P ( ~X) := (Grad ~�( ~X))T (5.63)
is the gradient of the transformation ~� that maps the con�guration R onto the con�gurationR. The deformation and temperature gradients therefore depend on the choice of con�guration.Likewise, the Green deformation tensor C, C = F T � F , transforms according to the rule:
C( ~X; t) = P T ( ~X) � C( ~X; t) � P ( ~X) : (5.64)
The constitutive equation (5.58) for the reference con�guration R is of the form
t( ~X; t) =FR��t
�F ( ~X; �); �( ~X; �); ~G�(
~X; �)
�: (5.65)
The relation between the di�erent functionals FR and FR is given by the identity
FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i=FR��t
�F ( ~X; �); �( ~X; �); ~G�(
~X; �)
�; (5.66)
which says that the stress at the present con�guration can be described by means of two di�erentdeformation and temperature histories of the same material with respect to two di�erent referencecon�gurations. By using (5.61), we express the arguments in con�guration R in terms of thosein con�guration R:
FR��t
�F ( ~X; �) � P ( ~X); �( ~X; �); ~G�(
~X; �) � P ( ~X)
�=FR��t
�F ( ~X; �); �( ~X; �); ~G�(
~X; �)
�: (5.67)
63
Since (5.67) holds for any motion, we obtain the identity
FR��t
hF ( ~X; �) � P ( ~X); �( ~X; �); ~G�( ~X; �) � P ( ~X)
i=FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i; (5.68)
which is valid for all deformation and temperature histories. Identity (5.68) can alternatively bearranged by expressing the arguments in con�guration R in terms of those in con�guration R:
FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i=FR��t
hF ( ~X; �) � P�1( ~X); �( ~X; �); ~G�( ~X; �) � P�1( ~X)
i: (5.69)
To introduce the concept of material symmetry, consider a particle which at time t = �1 wasin con�guration R and subsequently su�ered from certain histories of the deformation gradientF , the temperature � and the temperature gradient ~G�. At the present time t, the constitutiveequation for the stress is
t( ~X; t) =FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i: (5.70)
Let the same particle but now in the con�guration R is experienced the same histories of defor-mation gradient, temperature and temperature gradient as before. The resulting stress at presenttime t is
t( ~X; t) =FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i: (5.71)
Since FR and FR are not, in general, the same functionals, it follows that t 6= t. However, itmay happen that these values coincide, which expresses a certain symmetry of material. Thecondition for this case is
FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i=FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i: (5.72)
By expressing the right-hand side according to identity (5.68), we obtain
FR��t
hF ( ~X; �) � P ( ~X); �( ~X; �); ~G�( ~X; �) � P ( ~X)
i=FR��t
hF ( ~X; �); �( ~X; �); ~G�( ~X; �)
i: (5.73)
If this relation holds for all deformation and temperature histories, we say that the material atthe particle is symmetric with respect to the transformation P : R! R.
It shows that any such P as in (5.73) corresponds to a static local deformation (at a given par-ticle X ) from R to another reference con�guration R such that any deformation and temperaturehistories lead to the same stress in either R or R (at a given particle X ). Hence P corresponds toa change of reference con�guration which cannot be detected by any experiment at a given parti-cle X . The matrices P , for which (5.73) applies, are called the symmetry transformations. Oftenthe constitutive functionals remain invariant under volume-preserving transformations, that is,the transformations with detP = �1. A transformation with this property is called unimodu-
lar. From now on, we will consider changes of reference con�guration for which the associateddeformation gradient is unimodular. We point out that the unimodularity condition on the trans-formation of reference con�guration could, in principle, be omitted. However, no materials areknown that satisfy the symmetry condition (5.73) for a non-unimodular transformation.
64
The properties of material symmetry can be represented in terms of the symmetry group.The set of all unimodular transformations which leave the constitutive equation invariant withrespect to R, that is, for which (5.73) holds forms a group. 9 This group is called the symmetrygroup gR of the material at the particle X whose place in the reference con�guration R is ~X:
gR :=nH j detH = �1 and
FR��t
hF (�) �H ; �(�); ~G�(�) �H
i=FR��t
hF (�); �(�); ~G�(�)
i8F (�); 8�(�); 8 ~G�(�)
o;
(5.74)where we have dropped the place ~X from the notation as, for instance, in F ( ~X; �) � F (�). Theoperation de�ned on gR is the scalar product of tensors and the identity element is the identitytensor.
To show that gR is a group, let H1 and H2 be two elements of gR. Then we can replace Fin (5.74) with F �H1, ~G� with ~G� �H1 and take H =H2 to �nd
FR��t
hF (�) �H1 �H2; �(�); ~G�(�) �H1 �H2
i=FR��t
hF (�) �H1; �(�); ~G�(�) �H1
i:
Since H1 2 gR, we �nd that
FR��t
hF (�) �H1 �H2; �(�); ~G�(�) �H1 �H2
i=FR��t
hF (�); �(�); ~G�(�)
i:
This shows that the productH1 �H2 2 gR. Furthermore, the identity tensorH=I clearly satis�es(5.74). Finally, if H 2 gR and F is invertible, then F �H�1 is an invertible tensor. Hence (5.74)must hold with F replaced by F �H�1 (and ~G� with ~G� �H�1):
FR��t
hF (�) �H�1 �H ; �(�); ~G�(�) �H�1 �H
i=FR��t
hF (�) �H�1; �(�); ~G�(�) �H�1
i:
Hence we �nd that
FR��t
hF (�); �(�); ~G�(�)
i=FR��t
hF (�) �H�1; �(�); ~G�(�) �H�1
i;
which shows that H�1 2 gR. We have proved that gR has the structure of a group.
5.8 Noll's rule
The symmetry group gR generally depends upon the choice of reference con�guration, since thesymmetries a body enjoys with respect to one con�guration generally di�er from those it enjoyswith respect to another. However, the symmetry groups gR and gR (for the same particle) relative
to two di�erent reference con�gurations R and R are related by Noll's rule:
gR = P � gR � P�1 ; (5.75)
9A group is a set of abstract elements A, B, C, � � �, with a de�ned operation A �B (as, for instance, `multipli-
cation') such that: (1) the product A �B is de�ned for all elements A and B of the set, (2) this product A �B isitself an element of the set for all A and B (the set is closed under the operation), (3) the set contains an identityelement I such that I � A = A = A � I for all A, (4) every element has an inverse A�1 in the set such thatA �A�1 = A
�1 �A = I.
65
where P is the �xed local deformation tensor for the deformation from R to R given by (5.63).Noll's rule (5.75) means that every tensor H 2 gR must be of the form H = P �H �P�1 for some
tensor H 2 gR, and conversely every tensor H 2 gR is of the form H = P�1 � H � P for someH 2 gR. Note that P not necessarily be unimodular, because P represents a change of referencecon�guration, not a member of symmetry group. Noll's rule shows that if gR is known for onecon�guration, it is known for all. That is, the symmetries of a material in any one con�gurationdetermine its symmetries in every other.
To prove Noll's rule, let H 2 gR. Then (5.74) holds and it is permissible to replace F byF � P and ~G� by ~G� � P :
FR��t
hF (�) � P �H ; �(�); ~G�(�) � P �H
i=FR��t
hF (�) � P ; �(�); ~G�(�) � P
i:
By applying identity (5.69) to the both side of this equation, we obtain
FR��t
hF (�) � P �H � P�1; �(�); ~G�(�) � P �H � P�1
i=FR��t
hF (�); �(�); ~G�(�)
i;
which holds for all deformation and temperature histories. Thus, ifH 2 gR, then P �H �P�1 2 gR.The argument can be reversed which completes the proof of Noll's rule.
By de�nition, the symmetry group is a subgroup of the group of all unimodular transforma-tions,
gR � unim: (5.76)
The `size' of a speci�c symmetry group is the measure for the amount of material symmetry.The group of unimodular transformations contains the group of orthogonal transformations (as,for instance, rotations or re ections) but also the group of non-orthogonal volume-preservingtransformations (as, for instance, torsions), i.e.,
orth: � unim: (5.77)
The smallest symmetry group fI ;�Ig, fI ;�Ig � gR ; corresponds to a material that has no(nontrivial) symmetries. Such a material is called triclinic. Obviously,
P � fI ;�Ig � P�1 = fI ;�Ig : (5.78)
Hence a triclinic material has no symmetries in any reference con�guration: no deformation canbring this material into a con�guration that has a nontrivial symmetry.
The largest symmetry group is the group of all unimodular transformations. Noll's rule showsthat if P is an arbitrary invertible tensor, then P �H � P�1 is unimodular for all unimodularH , and if H is any unimodular tensor, the tensor P�1 � H � P is a unimodular tensor, andH = P �H � P�1. Therefore,
P � unim: � P�1 = unim: (5.79)
Hence, the maximum symmetry of such a material cannot be destroyed by deformation.
5.9 Material symmetry of frame indi�erent functionals
So far, we have been dealing with material symmetries of the stress constitutive functional FRwhich is not frame indi�erent. We now employ the de�nition of the symmetry group to �nd a
66
symmetry criterion for the functional G, see (5.38), that de�nes the constitutive equation for thesecond Piola-Kirchho� tensor. As we have seen stress functional G is frame indi�erent.
First, we derive the transformation rule for the components of the second Piola-Kirchho�
tensor in two reference coordinate systems. Let ~IK and ~IK be the Cartesian base vectors of tworectangular coordinate systems XK and XK that describe the reference con�gurations R and R.In view of (5.59), they are related as follows
~IK = ~IL PLK ; or; conversely; ~IK = ~IL P�1LK : (5.80)
The components of the second Piola-Kirchho� tensor T (2)( ~X; t) transform under the mapping(5.59) according to the relation
T(2)KL(
~X; t) = PKM ( ~X)T(2)MN (
~X; t)PLN ( ~X) ; (5.81)
where T(2)KL(
~X; t) and T(2)KL(
~X; t) are the components of tensor T (2)( ~X; t) with respect to XK and
XK coordinates, respectively. In symbolic notation, we can write
T(2)( ~X; t) = P ( ~X) � T (2)( ~X; t) � P T ( ~X) : (5.82)
Here, T (2)( ~X; t) and T(2)( ~X; t) represent the same tensor in two di�erent coordinate systems.
Hence, symbolic equation (5.82) has always to be interpreted in terms of tensor components.For a homogeneous material, constitutive equation (5.38) for the second Piola-Kirchho� tensor
in the reference con�gurations R and R has the form
T (2)( ~X; t) = GR��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �)
i;
T(2)( ~X; t) = GR
��t
�C( ~X; �); �( ~X; �); ~G�(
~X; �)
�;
(5.83)
where the subscripts R and R refer to the underlying reference con�guration. Making use oftransformation relation (5.82), the functional GR satis�es an analogous identity to that for thefunctional FR, see (5.66):
P ( ~X) � GR��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �)
i� P T ( ~X) =GR
��t
�C( ~X; �); �( ~X; �); ~G�(
~X; �)
�: (5.84)
With the help of the transformation relation for the Green deformation tensor, see (5.64), andthe temperature gradient, see (5.61), we obtain
P ( ~X) � GR��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �)
i� P T ( ~X)
=GR��t
hP�T ( ~X) �C( ~X; �) � P�1( ~X); �( ~X; �); ~G�( ~X; �) � P�1( ~X)
i:
(5.85)
If for two di�erent con�gurations R and R the constitutive functional happens to be unchanged,then GR = GR, or,
P ( ~X) � GR��t
hC( ~X; �); �( ~X; �); ~G�( ~X; �)
i� P T ( ~X)
=GR��t
hP�T ( ~X) �C( ~X; �) � P�1( ~X); �( ~X; �); ~G�( ~X; �) � P�1( ~X)
i:
(5.86)
67
We say that a material at the particle X (whose place in the reference con�guration is ~X) issymmetric with respect to the transformation P : R ! R if the last relation is valid for alldeformation and temperature histories. The symmetry group gR of a material characterized bythe constitutive functional GR is de�ned as
gR :=nHj detH = �1 and
H � GR��t
hC(�); �(�); ~G�(�)
i�HT =GR
��t
hH�T �C(�) �H�1; �(�); ~G�(�) �H�1
ifor 8C(�); 8�(�); 8 ~G�(�)
o;
(5.87)
where we have dropped the place ~X from the notation as, for instance, in C( ~X; �) � C(�).By an analogous way, it can be shown that the condition of material symmetry for the consti-
tutive functional ~FR in constitutive equation (5.35) which is frame indi�erent is similar to thatof GR:
H � ~FR��t
hC(�); �(�); ~G�(�)
i�HT = ~FR
��t
hH�T �C(�) �H�1; �(�); ~G�(�) �H�1
i: (5.88)
5.10 Isotropic materials
We say that a material is isotropic if it possesses a con�guration R, in which the symmetrycondition (5.74) or (5.87) applies for all orthogonal transformations. In other words, a materialis isotropic if there exists a reference con�guration R such that
gR � orth: (5.89)
Such a reference con�guration is called an undistorted con�guration of the isotropic material. Foran isotropic material, (5.76) and (5.89) can be combined to give
orth: � gR � unim: (5.90)
We further know that gR is also a group. The group theory states that the orthogonal group is amaximal subgroup of the unimodular group. Consequently, there are only two groups satisfying(5.90), either orthogonal or unimodular; no other group exists between them. Hence, there aretwo kinds of isotropic materials, either those with gR = orth: or those with gR = unim:
The concept of the symmetry group can be used to de�ne solids and uids according to Noll.
5.11 Solids
Solids have the property that any change of shape (as represented by a non-orthogonal trans-formation of the reference con�guration) brings the material into a new reference con�gurationfrom which its response is di�erent. Hence, according to Noll, a simple material is called a solid
if there is a reference con�guration such that every element of the symmetry group is a rotation,
gR � orth: (5.91)
We again call this preferred con�guration R as an undistorted con�guration of the solid.
68
Let R and R be two undistorted con�gurations of a solid, gR � orth: and gR � orth:, and let
P := (Grad ~�)T , where ~� : R! R. If the polar decomposition of P is P = R �U , then
gR = R � gR �R�1 : (5.92)
To prove (5.92), we �rst use Noll's rule: every element Q 2 gR � orth: must be of the form
Q = P �Q � P�1 for some Q 2 gR � orth:, that is Q � P = P �Q. The polar decomposition of Pthen implies Q �R �U = R �U �Q, or
(Q �R) �U = R �Q � (QT �U �Q) ;
where both Q �R and R �Q are orthogonal, while U and QT �U �Q are symmetric and positivede�nite. If we compare the respective polar decompositions of the left side and right side of thelast equation, it follows that it must hold
Q �R = R �Q and U = QT �U �Q ;
since the polar decomposition of a given tensor is unique. This �rst result proves (5.92), while thesecond shows that any elements of gR commutes with the right stretch tensor U , Q �U = U �Q.
An isotropic solid is described by two concepts of undistorted con�gurations. By hypothesis,there exist con�gurations R and R such that
gR � orth: ; gR � orth: ; (5.93)
where R is an undistorted con�guration of the isotropic material, while R is an undistortedcon�guration of the solid. Let us show that these concepts agree. By (5.90), either gR = orth:or gR = unim: If gR = unim:, then by (5.79) also gR = unim:, which contradicts (5.93)2. Hence,gR = orth:, and R is also an undistorted con�guration of the solid. Since both gR � orth:and gR � orth:, we can employ (5.92): gR = R � gR � R�1, or with gR = orth: we also havegR = R �orth: �R�1. Since R 2 orth:, it holds R �orth: �R�1 = orth: Hence gR = orth: In summary,we have proved that gR = gR = orth: and can conclude that the symmetry group of an isotropic
solid in any undistorted con�guration is equal to the full orthogonal group:
gR = orth: (5.94)
If the symmetry group of a solid is smaller than the full orthogonal group, the solid is calledanisotropic.
For an isotropic solid, the set of unimodular transformations is equal to the set of full orthog-onal transformations, H = Q 2 orth, so that the symmetry condition (5.87) (or (5.88)) for theconstitutive functional GR (or ~FR) is reduced to
Q � GR��t
hC(�); �(�); ~G�(�)
i�QT =GR
��t
hQ �C(�) �QT ; �(�);Q � ~G�(�)
i; (5.95)
where we have identi�ed Q � ~G�(�) � ~G�(�) �QT . Similar constraints can also be derived for theconstitutive functionals QR and ER of the heat ux and the internal energy, respectively:
Q � QR��t
hC(�); �(�); ~G�(�)
i= QR
��t
hQ �C(�) �QT ; �(�);Q � ~G�(�)
i;
ER��t
hC(�); �(�); ~G�(�)
i= ER
��t
hQ �C(�) �QT ; �(�);Q � ~G�(�)
i:
(5.96)
69
Functionals which satisfy constraints (5.95) and (5.96) are called tensorial, vectorial and scalar
isotropic functionals with respect to orthogonal transformations.
5.12 Fluids
Fluids have the property that can be poured from one container to another and, after some time,no evidence of the previous circumstances remains. Such a change of container can be thought asa change of reference con�guration, so that for uids all reference con�gurations with the samedensity are indistinguishable. According to Noll, a simple material is a uid if its symmetry grouphas a maximal symmetry, being identical with the set of all unimodular transformations,
gR = unim: (5.97)
Moreover, Noll's rule implies that the condition gR = unim: holds for every con�guration if itholds for any one reference con�guration R, so a uid has maximal symmetry relative to everyreference con�guration. A uid is thus a non-solid material with no preferred con�gurations. Interms of symmetry groups, gR = gR (= unim:). Moreover, a uid is isotropic, because of (5.77).
The simple uid and the simple solid do not exhaust the possible types of simple materials.There is, for instance, a simple material, called liquid crystal which is neither a simple uid nora simple solid.
5.13 Current con�guration as reference
So far, we have employed a reference con�guration �xed in time, but we can use a referencecon�guration varying in time. Thus one motion may be described in terms of any other. Theonly time-variable reference con�guration useful in this way is the present con�guration. If wetake it as reference, we describe past events as they seem to an observer �xed to the particle Xnow at the place ~x. The corresponding description is called relative and is has been introducedin section 1.2.
To see how such the relative description is constructed, consider the con�gurations of body Bat the two times � and t:
~� = ~�( ~X; �) ;
~x = ~�( ~X; t) ;(5.98)
that is, � is the place occupied at time � by the particle that occupies ~x at time t. Since (5.98)2is invertible,
~X = ~��1(~x; t) ; (5.99)
we have~� = ~�(~��1(~x; t); �) =: ~�t(~x; �) ; (5.100)
where the function ~�t(~x; �) is called the relative motion function. The subscript t at the function~� is used to recall that the reference con�guration is the con�guration at time t. This is actuallya special case of the referential description, di�ering from the Lagrangian description in that thereference position is now denoted by ~x at time t instead of ~X at time t = 0.
The relative deformation gradient F t is the gradient of the relative motion function:
F t(~x; �) := (grad ~�t(~x; �))T ; (F t)kl :=
@�k@xl
: (5.101)
70
It can be expressed in terms of the (absolute) deformation gradient F . Di�erentiating the fol-lowing identity for the motion function,
~� = ~�( ~X; �) = ~�t(~x; �) = ~�t(~�( ~X; t); �) ;
with respect to XK and using the chain rule of di�erentiation, we get
@�k( ~X; �)
@XK=@�k(~x; �)
@xl
@xl( ~X; t)
@XKor (F )kK( ~X; �) = (F t)kl(~x; �)(F )lK( ~X; t) ;
which represents the tensor equation
F ( ~X; �) = F t(~x; �) � F ( ~X; t) ; or F t(~x; �) = F ( ~X; �) � F�1( ~X; t) : (5.102)
In particular,F t(~x; t) = I : (5.103)
The Green deformation tensor C = F T � F may also be developed in terms of the relativedeformation gradient:
C( ~X; �) = F T ( ~X; �) � F ( ~X; �) = F T ( ~X; t) � F Tt (~x; �) � F t(~x; �) � F ( ~X; t) :
By introducing the relative Green deformation tensor
ct(~x; �) := F Tt (~x; �) � F t(~x; �) ; (5.104)
we haveC( ~X; �) = F T ( ~X; t) � ct(~x; �) � F ( ~X; t) : (5.105)
5.14 Isotropic constitutive functionals in relative representation
Let now the motion be considered in relative representation (5.100). Thus, the present con�gu-ration at the current time t serves as reference, while the con�guration at past time � , (� � t),is taken as present. The relative deformation gradient F t(~x; �) can be expressed in terms of theabsolute deformation gradient according to (5.102). Similarly, the relative temperature gradient,~g� := grad �, can be expressed in terms of the absolute temperatute gradient, ~G� := Grad �, as
~g�(~x; �) =@�(~x; �)
@xk~ik =
@�( ~X; �)
@XKXK;k
~ik = XK;k~ik~IK � @�(
~X; �)
@XM
~IM = F�T (~x; t) � ~G�( ~X; �) :
To abbreviate notations, ( ~X; t) and (~x; t) will be dropped from notations and ( ~X; �) and (~x; �)will be shortened as (�):
~G�(�) = F T � ~g�(�) : (5.106)
With the help of the polar decomposition F = R �U , (5.105) and (5.106) can also be written inthe form
C(�) = U �RT � ct(�) �R �U ;
~G�(�) = U �RT � ~g�(�) :(5.107)
71
We consider the reduced form (5.33) of the constitutive equation for the Cauchy stress tensor:
t = R � F��t
hC(�); �(�); ~G�(�)
i�RT : (5.108)
Making use of (5.107) together with C = U2, we obtain
t = R � F��t
hpC �RT � ct(�) �R �
pC; �(�);
pC �RT � ~g�(�)
i�RT : (5.109)
The information contained in the last constitutive equation can alternatively be expressed as
t = R � F��t
hRT � ct(�) �R; �(�);RT � ~g�(�);C
i�RT ; (5.110)
where the present strain C(t) appears in the functional F as a parameter. This shows that itis not possible to express the e�ect of deformation history on the stress entirely by measuringdeformation with respect to the present con�guration. While the e�ect of all the past historyis accounted for, a �xed reference con�guration is required, in general, to allow for the e�ect ofthe deformation at the present instant, as indicated by the appearance of C(t) as a parameter in(5.110).
We are searching for a form of functional F for isotropic materials only. In this case, thefunctional F must satisfy relation (5.95) for isotropic functionals, that is, if we replace
R! Q �R �QT ; C ! Q �C �QT ; ct(�)! Q � ct(�) �QT ; ~g�(�)! Q � ~g�(�) ;
in (5.110), where Q is any constant orthogonal tensor, it must hold:
(Q �R �QT ) � F��t
h(Q �RT �QT ) � (Q � ct(�) �QT ) � (Q �R �QT ); �(�);
(Q �RT �QT ) � (Q � ~g�(�)); (Q �C �QT )i� (Q �RT �QT ) (5.111)
= Q ��R � F
��t
hRT � ct(�) �R; �(�);RT � ~g�(�);C
i�RT
��QT :
Since ~X and t are �xed in the relative description of motion, we may choose Q = R( ~X; t). Makinguse of the orthogonality property of R, R �RT = RT �R = I, we obtain
R � F��t
[ct(�); �(�); ~g�(�); b] �RT = R2 � F��t
hRT � ct(�) �R; �(�);RT � ~g�(�);C
i� (RT )2 ; (5.112)
where b is the Finger deformation tensor, b = F � F T = R � C � RT . By multiplying the lastequation by RT from the left side and by R from the right side, we obtain
F��t
[ct(�); �(�); ~g�(�); b] = R � F��t
hRT � ct(�) �R; �(�);RT � ~g�(�);C
i�RT ; (5.113)
which in view of (5.110) takes the form
t = F��t
[ct(�); �(�); ~g�(�); b] : (5.114)
72
This is a general form of the constitutive equation for a simple homogeneous isotropic materialsexpressed in relative representation. It remains to show that this constitutive equation is invariantto any orthogonal transformation Q(t) of reference con�guration R, not only to our special choiceQ = R( ~X; t). Since none of the arguments in (5.114) does depend on the reference con�gurationR, this requirement is satis�ed trivially.
By construction, the constitutive functional F is not automatically frame indi�erent. Theframe indi�erence of the Cauchy stress tensor implies that the functional F must transformunder a rigid motion of spatial frame as
F��t
[c�t (�); ��(�); ~g���(�); b
�] = Q � F��t
[ct(�); �(�); ~g�(�); b] �QT (5.115)
for all orthogonal tensors Q. It can readily be veri�ed the transformation rules for the relativedeformation tensor, the relative temperature gradient and the Finger deformation tensor are
c�t (�) = Q � ct(�) �QT ;
~g���(�) = Q � ~g�(�) ; (5.116)
b� = Q � b �QT :
By this and realizing that the temperature �(�) as a scalar quantity is invariant to any transfor-mation of spatial frame, the constraint (5.115) reduces to
F��t
hQ � ct(�) �QT ; �(�);Q � ~g�(�);Q � b �QT
i= Q � F
��t[ct(�); �(�); ~g�(�); b] �QT : (5.117)
This shows that the functional F must be an isotropic functional of variables ct(�), �(�), ~g�(�)and an isotropic function of b; then F is frame indi�erent.
An analogous arrangement can be carried out for the constitutive equation (5.39) for the heat ux and the internal energy:
~q = Q��t
[ct(�); �(�); ~g�(�); b] ;
" = E��t
[ct(�); �(�); ~g�(�); b] :(5.118)
Note that the constitutive equations in relative representation will be used to describe materialproperties of a uid, while for a solid we will employ the constitutive equation (5.38) in referentialrepresentation.
5.15 A general constitutive equation of a uid
A uid is characterized through the largest symmetry group, being identical with the set of all uni-modular transformations, gR = unim: Mathematically, the condition for the material symmetryof functional F in (5.114) under unimodular transformation H is expressed as
F��t
hct(�); �(�); ~g�(�);F � F T
i= F��t
hct(�); �(�); ~g�(�);F �H �HT � F T
i; (5.119)
since the Finger deformation tensor b = F � F T transforms under unimodular transformationH as F �H �HT � F T . This general characterization of a uid must also hold for the specialunimodular transformation
H = (detF )1=3 F�1 =
�%0%
�1=3F�1 ; (5.120)
73
where %(~x; t) and %0( ~X) is the mass density of a body in the present and reference con�guration,respectively. The transformation (5.120) is indeed unimodular since
detH =h(detF )1=3
i3detF�1 = 1 :
For such H , F �H �HT � F T = (detF )2=3 I = (%0=%)2=3I , and the constitutive equation (5.114)
for an isotropic body can be rewritten with new denotation:
t =Ft��t
[ct(�); �(�); ~g�(�); %] : (5.121)
This is a general form of the constitutive equation for a simple uid in relative representation.It shows that the Cauchy stress tensor for uid depends on the relative deformation history, thetemperature and temperature gradient histories, and on the present mass density as a parameter.Since none of ct(~x; �), �(~x; �), ~g�(~x; �) and %(~x; t) does depend on the reference con�guration,constitutive equation (5.121) is invariant to any transformation of the reference con�guration.Hence, it satis�es trivially the requirement that gR = unim.
5.16 The principle of bounded memory
This principle states that deformation and temperature events at distant past from the presenthave a very small in uence on the present behaviour of material. In other words, the memoryof past motions and temperatures of any material point decays rapidly with time. This principleis the counterpart of the principle of smooth neighborhood in the time domain. No uniquemathematical formulation can be made of this principle. The following limited interpretationsuÆces for our purpose.
To express the boudedness of a memory, let h( ~X; �) be a function in the argument of constitu-tive functional. Suppose that h( ~X; �) is an analytical function such that it possesses continuouspartial derivatives with respect to � at � = t. For small t � � , it can be approximated by thetruncated Taylor series expansion at � = t:
h( ~X; �) =NXn=0
(�1)nn!
@nh( ~X; �)
@�n
������=t
(� � t)n : (5.122)
If a constitutive functional is suÆciently smooth so that the dependence on h( ~X; �) can bereplaced by the list of functions
(n)
h ( ~X; t) :=@nh( ~X; �)
@�n
������=t
for n = 0; 1; : : : ; N ; (5.123)
that is,(n)
h is the nth material time derivative of h,(n)
h = Dnh=D�n, we say that the material is ofthe rate type of degree N with respect to the variable h. If the material is of the rate type in allits variables, the constitutive equation (5.38) for a homogeneous solid can be approximated by
T (2) = T (2)(C;:C; : : : ;
(NC)
C ; �;:�; : : : ;
(N�)
� ; ~G�;:~G�; : : : ;
(NG)
~G� ) : (5.124)
74
The number of time derivatives for each of the gradients depends on the strength of the memoryof these gradients. We note that T (2) is no longer functional. It is a tensor-valued function of thearguments listed. They involve time rates of the deformation gradient up to the order NC andtemperature and its gradient up to order N� and NG, respectively. The degrees NC , N� and NG
need not be the same. 10
The general constitutive equation for a uid is given by (5.121). We again assume that therelative Green deformation tensor ct(�) can be approximated by the truncated Taylor seriesexpansion of the form (5.122):
ct(�) = a0(t)� a1(t)(� � t) + a2(t)(� � t)2
2: : : ; (5.125)
where the expansion coeÆcients
an(~x; t) :=@nct(~x; �)
@�n
�����=t
(5.126)
can be shown to be the Rivlin-Ericksen tensors of order n de�ned by (2.28).We again assume that all the functions in the argument of functional Ft in (5.121) can
be approximated by the truncated Taylor series expansion. Then the functional for a class ofmaterials with boundedmemory may be represented by functions involving various time derivativeof the argument functions:
t = t(a1; : : : ;aNC; �;
:�; : : : ;
(N�)
� ;~g�;:~g�; : : : ;
(Ng)
~g� ; %) ; (5.127)
where the dependence on a0 = I has been ommitted because of its redundancy. Note that t isagain no longer functional, but a tensor-valued function. This is an asymptotic approximation ofthe most general constitutive equation (5.121) for a uid. Coleman and Noll (1960) showed thatthis approximation is valid for suÆciently slow deformation processes.
Similar expressions hold for the heat ux vector ~q and the internal energy ". The numberof time derivatives for each of the gradients depends on the strength of the memory of thesegradients. According to the axiom of equipresence, however, all constitutive functionals shouldbe expressed in terms of the same list of independent constitutive variables.
5.17 Representation theorems for isotropic functions
We are now looking for the representation of a scalar-, vector- and tensor-valued function whichare isotropic, that is, form-invariant for all orthogonal transformations.
10The functions in the argument of constitutive functionals may not be so smooth as to admit truncated Taylorseries expansion. Nevertheless, the constitutive functionals may be such as to smooth out past discontinuities inthese argument functions and/or their derivatives. The principle of bounded memory, in this context also calledthe principle of fading memory, is then a requirement on smoothness of constitutive functionals.The principle of fading memory, mathematically formulated by Coleman and Noll (1960), starts with the as-
sumption that the so-called Frech�et derivatives of constitutive functionals up to an order n exist and are continuousin the neighborhood of histories at time t in the Hilbert space normed by an in uence function of order greaterthan n+ 1
2. Then, the constitutive functionals can be approximated by linear functionals for which explicit math-
ematical representations are known. The most important result of the fading memory theory is the possibility toapproximate asymptotically a suÆciently slow strain and temperature histories by the Taylor series expansion.
75
5.17.1 Representation theorem for a scalar isotropic function
A scalar-valued function a depending on symmetric tensor S and vector ~v is called isotropic if itsatis�es the identity
a(S; ~v) = a(Q � S �QT ;Q � ~v) (5.128)
for all orthogonal tensors Q. This means that a does not depend on components Skl and vkarbitrarily, but it can only be considered as a function of scalar invariants that are independentof any orthogonal transformation.
Hence, we have to determine the set of all independent invariants of variables S and ~v. Thereare three independent invariants of tensor S, namely,
IS = trS ; IIS =1
2
h(trS)2 � tr (S2)
i; IIIS = detS ; (5.129)
and one invariant of vector ~v, the scalar product ~v � ~v. Combining S and ~v, we can create anumber of other invariants,
~v � S � ~v ; ~v � S2 � ~v ; ~v � S3 � ~v ; : : : ; ~v � Sn � ~v ; : : : ; (5.130)
of course, not all of them are mutually independent. According to the Cayley-Hamilton theoremsaying that any matrix satis�es its characteristic polynom, we have
S3 � IS S2 + IIS S � IIIS I = 0 ; (5.131)
where I and 0 are the unit and zero tensors of the same order as S. Multiplying this equationfrom left and right by vector ~v, we obtain
~v � S3 � ~v � IS (~v � S2 � ~v) + IIS (~v � S � ~v)� IIIS (~v � ~v) = 0 : (5.132)
We see that ~v �Sn � ~v for n � 3 can be expressed in terms of ~v � S2 � ~v, ~v �S � ~v and ~v � ~v but thesecannot be used to express three invariants of S. Hence, the set of all independent invariants ofvariables S and ~v is
IS; IIS ; IIIS ; ~v � ~v; ~v � S � ~v; ~v � S2 � ~v : (5.133)
We can conclude that scalar isotropic function a can be represented in the form
a(S; ~v) = a(IS ; IIS ; IIIS ; ~v � ~v; ~v � S � ~v; ~v � S2 � ~v) : (5.134)
5.17.2 Representation theorem for a vector isotropic function
A vector-valued function ~b depending on symmetric tensor S and vector ~v is called isotropic if itsatis�es the identity
Q �~b(S; ~v) = ~b(Q � S �QT ;Q � ~v) (5.135)
for all orthogonal tensors Q. For a vector ~c, let us de�ne function F ,
F (S; ~v;~c) := ~c �~b(S; ~v) : (5.136)
76
This function must be a scalar isotropic function of variables S, ~v and ~c. In analogy to thepreceding section, the set of all independent invariants of these three quantities is
IS ; IIS ; IIIS ; ~v � ~v; ~v � S � ~v; ~v � S2 � ~v;~c � ~v; ~c � S � ~v; ~c � S2 � ~v;~c � ~c; ~c � S � ~c; ~c � S2 � ~c :
(5.137)
If function F was an arbitrary function of S, ~v and ~c, it could be represented in terms of these12 invariants in an arbitrary manner. Function F is, however, linearly dependent on ~c, so that itcan be represented in terms of the invariants that are linear functions of ~c only:
F = a0 ~c � ~v + a1 ~c � S � ~v + a2 ~c � S2 � ~v ; (5.138)
where the coeÆcients ai depend on the six invariant that are independent of ~c. Since ~c was anarbitrary vector in equations (5.136){(5.138), we can conclude that vector isotropic function ~bcan be represented in the form
~b = a0 ~v + a1 S � ~v + a2 S2 � ~v ; (5.139)
whereai = ai (IS ; IIS ; IIIS ; ~v � ~v; ~v � S � ~v; ~v � S2 � ~v) : (5.140)
5.17.3 Representation theorem for a symmetric tensor isotropic function
(a) Symmetric tensor isotropic function of symmetric tensor and vector. First, let us con-sider symmetric tensor-valued function T depending on symmetric tensor S and vector ~v. TensorT is called isotropic if it satis�es the identity
Q � T (S; ~v) �QT = T (Q � S �QT ;Q � ~v) (5.141)
for all orthogonal tensors Q. For a tensor A, let us de�ne function F ,
F (S; ~v;A) := tr(A � T (S; ~v)) : (5.142)
This function must be a scalar isotropic function of variables S, ~v and A. Under the sameargument as in the previous case, function F must be represented as a linear combination of allindependent invariants that are linear functions of A only, that is, in terms of
trA; tr(A � S); tr(A � S2); ~v �A � ~v; ~v �A � S � ~v; ~v �A � S2 � ~v: (5.143)
Note that it is rather diÆcult to prove that there are no more independent invariants linearlydependent on A. Hence, we can write
F = a0 trA+ a1 tr(A �S) + a2 tr(A �S2) + a3 ~v �A � ~v+ a4 ~v �A �S � ~v+ a5 ~v �A �S2 � ~v ; (5.144)
where the coeÆcients ai depend on the invariants that are independent of A, that is, on the sixinvariants of S and ~v. Because of the identity
~v �A � ~v = tr(A � ~v ~v) ; (5.145)
77
where ~v ~v is the dyadic product of vector ~v by itself, we also have
F = a0 trA+a1 tr(A �S)+a2 tr(A �S2)+a3 tr(A �~v~v)+a4 tr(A �S �~v~v)+a5 tr(A �S2 �~v~v) :(5.146)
The comparison of (5.146) with (5.142) results in the representation of a symmetric tensor-valuedisotropic function T in the form
T (S; ~v) = a0I + a1S + a2S2 + a3 ~v ~v + a4 sym(S � ~v ~v) + a5 sym(S
2 � ~v ~v) ; (5.147)
whereai = ai(IS ; IIS ; IIIS ; ~v � ~v; ~v � S � ~v; ~v � S2 � ~v) (5.148)
and the symbol `sym' stands for the symmetric part of tensor.(b) Symmetric tensor isotropic function of two symmetric tensors. Second, let us consider
symmetric tensor-valued function T depending on two symmetric tensors A and B. Tensor T iscalled isotropic if it satis�es the identity
Q � T (A;B) �QT = T (Q �A �QT ;Q �B �QT ) (5.149)
for all orthogonal tensors Q. For a tensor C, let us de�ne function F ,
F (A;B;C) := tr(C � T (A;B)) : (5.150)
This function must be a scalar isotropic function of variables A, B and C. The set of allindependent invariants of these three symmetric tensors that are linear functions of C is
trC; tr(C�A); tr(C�A2); tr(C�B); tr(C�B2); tr(C�A�B); tr(C�A�B2); tr(C�A2�B); tr(C�A2�B2) :(5.151)
Hence, function F can be represented in the form
F = a0 trC + a1 tr(C �A) + a2 tr(C �A2) + a3 tr(C �B) + a4 tr(C �B2) + a5 tr(C �A �B)
+ a6 tr(C �A �B2) + a7 tr(C �A2 �B) + a8 tr(C �A2 �B2) ; (5.152)
where the coeÆcients ai depend on the invariants that are independent of C, that is, on the teninvariants of A and B:
IA; IIA; IIIA; IB ; IIB ; IIIB ; tr(A �B); tr(A �B2); tr(A2 �B); tr(A2 �B2) : (5.153)
The comparison of (5.152) with (5.150) yields the representation of a symmetric tensor-valuedisotropic function T in the form
T (A;B) = a0 I + a1A+ a2A2 + a3B + a4B
2 + a5 sym(A �B) + a6 sym(A �B2)
+ a7 sym(A2 �B) + a8 sym(A
2 �B2) ; (5.154)
where
ai = ai(IA; IIA; IIIA; IB ; IIB ; IIIB ; tr(A �B); tr(A �B2); tr(A2 �B); tr(A2 �B2)) : (5.155)
5.18 Examples of isotropic materials with bounded memory
78
5.18.1 Elastic solid
A solid is called elastic if the stress tensor t depends only on the deformation gradient F at thepresent time, not on the temperature � and not on the entire past thermomechanical history:
t = t(F ) : (5.156)
Hence, the stress in an elastic material at each particle is uniquely determined by the presentdeformation from a �xed reference con�guration. The material objectivity requires that thedependence on F is not arbitrary, but it has the form (5.33):
t = R � t(C) �RT : (5.157)
For isotropic materials, t must be an isotropic function of C. Making use of the representationtheorem for a tensor isotropic function, we have
t = R � (a0I + a1C + a2C2) �RT ; (5.158)
where ai are scalar functions of three principal invariants of C,
ai = ai(IC ; IIC ; IIIC) : (5.159)
With the help of the Finger deformation tensor b and its second power,
b = F � F T = V 2 = V T � V = R � F T � F �RT = R �C �RT ;
b2 = V T �V �V T �V = R �F T �F �RT �R �F T �F �RT = R �F T �F �F T �F �RT = R �C2 �RT ;
we obtain a general form of the constitutive equation of an isotropic elastic material:
t = a0I + a1b+ a2b2 ; ai = ai(Ib; IIb; IIIb) : (5.160)
This does not imply that the constitutive equation can only be a quadratic in the components ofb, since scalar functions ai may be non-linear in terms of the components of b.
The constitutive equation (5.38) for the second Piola-Kirchho� stress tensor for an elasticmaterial (even without isotropy) is reduced to the simple form:
T (2) = T (2)(C) : (5.161)
The function T (2) for isotropic materials can be represented in the form:
T (2) = a0I + a1C + a2C2 ; ai = ai(IC ; IIC ; IIIC) ; (5.162)
which is another general form, equivalent to (5.160), of the constitutive equation for isotropicelastic materials.
We now have a look at the case of small deformation when the displacement gradient H issuÆciently small such that the geometric linearization can be applied and the di�erence betweenthe reference con�guration and the present con�guration is not necessary to consider. Within thelimit of geometric linearization, the Finger deformation tensor is equal to the Green deformationtensor. Consequently, the linearized constitutive equation for the second Piola-Kirchho� tensor
79
coincides with that for the Cauchy stress tensor. Let us, for instance, linearize the constitutiveequation (5.162). Because of (1.102), we have
C = U2 = I+2 ~E ; C2 = I+4 ~E ; IC = 3+2 tr ~E ; IIC = 3+4 tr ~E ; IIIC = 1+2 tr ~E ;(5.163)
where ~E is the in�nitesimal strain tensor. Then the constitutive equations (5.160) and (5.162)may be linearized as
T (2) �= t = �(tr ~E)I + 2� ~E : (5.164)
This is the constitutive equation of an isotropic, linear elastic material, known also as Hook's law.The both parameters � and � are called the Lam�e elastic constants.
5.18.2 Thermoelastic solid
The e�ect of heat conduction can be incorporated in the behaviour of elastic materials by includingtemperature and temperature gradients among the constitutive variables. From (5.124) we deducethat the classical (non-classical) thermoelastic solid is de�ned by NC = 0, N� = 0(1) and Ng = 0.Hence, the constitutive equation for a thermoelastic solid will have the form:
T (2) = T (2)(C; �;:�; ~G�) ; ~Q = ~Q(C; �;
:�; ~G�) ; " = "(C; �;
:�; ~G�) : (5.165)
For isotropic materials, functions T (2), ~q and " must be isotropic functions of their arguments:
T (2) = a0I + a1C + a2C2 + a3 ~G� ~G� + a4 sym(C � ~G� ~G�) + a5 sym(C
2 � ~G� ~G�) ;
~Q = �� ~G� + c1C � ~G� + c2C2 � ~G� ; (5.166)
" = "(�;:�; IC ; IIC ; IIIC ; ~G� � ~G�; ~G� �C � ~G�; ~G� �C2 � ~G�) ;
where the scalar function ai, � and ci depend on the same set of arguments as the function " ofthe internal energy. We can see that the heat ux vector ~Q depends linearly on the temperaturegradient ~G� but contains two non-linear terms as the Green deformation tensor arises togetherwith ~G�. The coeÆcient � is called the heat conductivity, the remaining coeÆcients c1 and c2 donot have separate names.
5.18.3 Kelvin-Voigt viscoelastic solid
In general, under the viscoelastic solid we understand the material for which the stress is dependenton the strain, strain rate and stress rate tensors and not on the temperature. From (5.124) wededuce that the simplest viscoelastic solid is de�ned by NC = 1:
T (2) = T (2)(C;:C ) : (5.167)
This simple strain-rate dependent solid is called the Kelvin-Voigt viscoelastic solid. Since thePiola-Kirchho� stress tensor depends on two symmetric tensors, for isotropic materials it may berepresented according to (5.154):
T (2) = a0 I + a1C + a2C2 + a3
:C +a4
:C
2+a5 sym(C�
:C) + a6 sym(C�
:C
2)
+ a7 sym(C2�:C) + a8 sym(C
2�:C
2) ; (5.168)
80
where
ai = ai�trC; trC2; trC3; tr
:C; tr
:C
2; tr
:C
3tr(C�
:C); tr(C�
:C
2); tr(C2�
:C); tr(C2�
:C
2)�:
(5.169)
5.18.4 Maxwell viscoelastic solid
A special case of constitutive equation (5.6) expressed in terms of the second Piola-Kirchho�stress tensor is
(T (2)):= T (2)(T (2);
:C ) : (5.170)
This simple stress-rate dependent solid is called the Maxwell viscoelastic solid. Since T (2) is aframe indiferent scalar, see (5.23)5, and the material time derivative of a frame indiferent scalaris again a frame indiferent scalar, the constitutive equation (5.170) satis�es the principle of frameindi�erence. If T (2) is an isotropic function, we can proceed similarly as for the Kelvin-Voigtviscoelastic solid.
5.18.5 Elastic uid
An elastic material may be a solid or a uid. For an elastic uid, the functions in the argumentof the constitutive function t, see (5.127), are reduced to the dependence on the current density%:
t = t(%) : (5.171)
According to the assumption of material frame indi�erence, we have
t� = t(%�) (5.172)
under a change of frame. Because %� = % and t� = Q � t �QT , this implies that
Q � t(%) �QT = t(%) (5.173)
for all orthogonal tensors Q. The only tensor satisfying this identity is a spherical tensor:
t = �p(%)I : (5.174)
Therefore the stress in an elastic uid is always a hydrostatic pressure depending on the densityalone.
5.18.6 Thermoelastic uid
The e�ect of heat conduction can be incorporated in the behaviour of elastic uid by includingtemperature and temperature gradients among the constitutive variables. From (5.127) we deducethat the classical (non-classical) thermoelastic (heat-conducting) uid is de�ned by NC = 0,N� = 0(1) and Ng = 0. Hence, the constitutive equation for a thermoelastic uid will have theform:
t = t(�;:�;~g�; %) ; ~q = ~q(�;
:�;~g�; %) ; " = "(�;
:�;~g�; %) : (5.175)
Making use of the representation theorems for isotropic functions, we obtain
t = �I + � ~g� ~g� ;~q = ��~g� ; (5.176)
" = "(%; �;:�;~g� � ~g�) ;
81
where�; �; � = �; �; �(%; �;
:�;~g� � ~g�) : (5.177)
5.18.7 Viscous uid
Taking NC = 1 in (5.127), we obtain a viscous uid, an analog to a simple viscoelastic solid:
t = t(a1; %) = t(d; %) ; (5.178)
since a1 = 2d. This constitutive equation can further be simpli�ed by applying the representationtheorem for a symmetric isotropic tensor:
t = a0I + 2�vd+ a2d2 ; ai; �v = ai; �v(%; Id; IId; IIId) ; (5.179)
where �v is the so-called (shear) viscosity.Now, we wish to show that the constitutive equation (5.179) is frame indi�erent. Multiplying
(5.179) from the left by Q and from the right by QT , we obtain
Q �t�QT = a0Q�I �QT +2�vQ �d �QT +a2Q�d �QT �Q �d �QT ; ai; �v = ai; �v(%; Id; IId; IIId) :(5.180)
Since t and d are frame indi�erent tensors and % as well as the invariants of d are frame indi�erentscalars, it holds
t� = a0I + 2�vd� + a2(d
�)2 ; ai; �v = ai; �v(%�; I�d ; II
�d ; III
�d ) ; (5.181)
which is of the same form as that in the unstarred frame. This completes the proof.
5.18.8 Incompressible viscous uid
For an incompressible uid we have the kinematic constraint that the density is constant, equalto a known value: % = %0. The mass-conservation principle then yields: Id = trd = div~v = 0.Combining (5.57) and (5.127), the constitutive equation of an incompressible viscous uid is
t = �pI + tD : (5.182)
The pressure p is an additional unknown independent �eld variable and it would be determinedthrough solutions of �eld equations under a given set of boundary conditions. On contrary, thedeviatoric part tD of the stress tensor t 11 is given by a constitutive equation analogous to (5.179):
tD = 2�vd+ a2d2 ; ai; �v = ai; �v(IId; IIId) : (5.183)
The dependencies on % and Id are dropped since % is a known constant and Id = 0. Notice thatconstitutive relation (5.182) has been postulated in such a way that any spherical part of stresstensor t may, without loss of generality, be absorbed in the pressure.
11The stress tensor t may, in general, be decomposed into isotropic spherical tensor �I and the deviatoric stress
tensor tD:t = �I + t
D :
Under the choice of � = tr t=3, the trace of the deviatoric stress vanishes, tr tD = 0. The scalar � is thus the meanof the normal-stress components and is called mechanical pressure. A characteristic feature of a uid at rest is thatit cannot support shear stresses. Consequently, the deviatoric stress identically vanishes. Choosing p := ��, weobtain t = �pI. Therefore, the stress in a uid at rest is the so-called hydrostatic pressure.
82
5.18.9 Viscous heat-conducting uid
A classical (non-classical) viscous heat-conducting uid is de�ned by NC = 1, N� = 0(1) andNg = 0, so that (5.127) reduces to
t = t(d; �;:�;~g�; %) ; ~q = ~q(d; �;
:�;~g�; %) ; " = "(d; �;
:�;~g�; %) : (5.184)
Making use of the representation theorems for isotropic functions, we obtain
t = a0I + 2�vd+ a2d2 + a3 ~g� ~g� + a4 sym(d � ~g� ~g�) + a5 sym(d
2 � ~g� ~g�) ;~q = ��~g� + c1 d � ~g� + c2 d
2 � ~g� ; (5.185)
" = "(%; �;:�; Id; IId; IIId; ~g� � ~g�; ~g� � d � ~g�; ~g� � d2 � ~g�; ) ;
where the scalars ai, �v, � and ci depend on the same set of arguments as the function " of theinternal energy. The proof of the frame indi�erence of these constitutive equations is similar tothat for the viscous uid.
Fluids characterized by the above non-linearity are called Stokesian or non-Newtonian uids.On the other hand, linear behaviour is generally referred to as Newtonian behaviour. If, moreover,the constitutive equations are also linear with respect to the temperature gradient, the emerging�eld equations are called the Navier-Stokes equations.
83
6. ENTROPY PRINCIPLE
The preceding analysis makes no use of thermodynamic considerations. The treatment ofthe second law of thermodynamics is a part of the material theory since the entropy principleplaces restrictions on material properties. Thus, the entropy principle would belong to Chapter 5.Thermodynamic requirements are, however, so important that we prefer to deal with them in aseparate chapter.
6.1 The Clausius-Duhem inequality
The second law of thermodynamics states that the entropy production cannot be negative. Equa-tion (4.35) expresses this law in spatial form,
% := %:� +div~s� %b � 0 ; (6.1)
where is the local entropy production. This law does not, however, de�ne internal dissipationmechanisms uniquely. We must bring additional information to be able to draw conclusions onmaterial properties. In continuum thermodynamics, there are many other dissipation postulates,some of which do not even involve entropy as a �eld variable. In sections 6.1{6.4, we will con�neourselves to considerations that lead to the Clausius-Duhem inequality.
� We will postulate the existence of a positive-valued absolute temperature T as a measure ofhotness and assume that T is a frame indi�erent scalar. The absolute temperature may di�erfrom the empirical temperature � that can be measured in the Celsius scale and may, in contrastto T , take negative values. Later on we will present one possible way how to relate the absoluteand empirical temperatures.
� We shall be dealing with simple thermodynamic processes for which the entropy ux ~s andentropy source b are taken as
~s =~q
T; b =
h
T: (6.2)
In general, these two postulates are not provable, but they have been adopted as certain gener-alizations of results of special problems, such as the kinetic theory of gases. Their validity canonly be proved by physical experiments.
The entropy inequality (6.1) for a simple thermodynamic process then becomes
%:� +div
�~q
T
�� %h
T� 0 ; (6.3)
which is known as the Clausius-Duhem inequality. This inequality need not be satis�ed for alldensities %, motions ~x and temperatures T , but for all possible thermodynamic processes, thatis for all solutions of the balance laws for mass, linear momentum and energy, that is for allsolutions of the �eld equations of thermomechanics. To satisfy these additional constraints whenthe Clausius-Duhem inequality is applied, we can follow two di�erent points of view.
� The balance laws for linear momentum and energy contain two free �eld variables, namely, bodyforce ~f and heat supply h, which may be assigned arbitrary values. It implies that arbitraryhistories of density %, motion ~x and temperature T can be chosen and still appropriate bodyforce ~f and heat supply h can be found to balance the linear momentum and energy equations.
84
Hence, the linear momentum and energy equations do not raise any restrictions when theClausius-Duhem inequality is applied. It remains to satis�ed the continuity equation whichmay be considered as an additional constraint to the Clausius-Duhem inequality. For instance,:% cannot be chosen arbitrarily, but such that
:%= �%div~v.
� For a given physical problem, body force ~f and heat supply h are speci�ed as the inputinformation that cannot be altered during the solution of the problem. The linear momentumand energy equations must then be considered as additional constraints to the Clausius-Duheminequality.
Though the second criterion is more acceptable from physical point of view, the �rst criterion isusually applied because of its simplicity. Here, we also start with this criterion in the applicationof the Clausius-Duhem inequality, but later on we introduce the entropy principle in modernunderstanding that employs the second criterion.
With the help of the energy equation (4.34), the heat source h can be eliminated from (6.3).The entropy inequality then takes the form
%(T:� � :
") + t :: d� ~q � grad TT
� 0 ; (6.4)
which is known as the reduced Clausius-Duhem inequality in spatial form. The reduced Clausius-Duhem inequality referred to the reference con�guration can be obtained by combining the ref-erential form of the energy equation (4.80) and the entropy inequality (4.84):
%0(T:� � :
") + T (2) :::E �
~Q �GradTT
� 0 : (6.5)
6.2 Application of the Clausius-Duhem inequality to a classicalviscous heat-conducting uid
The constitutive equations for a classical viscous heat-conducting uid are introduced in section5.18.9:
t = t(d; T; grad T; %) ; ~q = ~q(d; T; grad T; %) ; " = "(d; T; grad T; %) ; (6.6)
where, instead of the empirical temperature �, the absolute temperature T is employed as a mea-sure of hotness. The Clausius-Duhem inequality introduces a new variable, the entropy density�. Since � is not determined by the �eld equations, a constitutive relation must be established forit. According to the principle of equipresence, we choose the same set of independent variables:
� = �(d; T; grad T; %) : (6.7)
Because of frame indi�erence and isotropy, the constitutive functions have the form (5.184). Thisform will be applied later.
We carry out time di�erentiation of " and � according to the chain rule and substitute theresult into the reduced Clausius-Duhem inequality (6.4). This yields
%
"�T@�
@%� @"
@%
�:% +
�T@�
@T� @"
@T
� :T +
�T@�
@d� @"
@d
�:::d
+
�T
@�
@(grad T )� @"
@(grad T )
�� (grad T )�
#+ t :: d� ~q � gradT
T� 0 :
(6.8)
85
The inequality must be satis�ed for all thermodynamic processes, that is for all solutions of thebalance laws for mass, linear momentum and energy. As we explained before, the linear mo-mentum and energy equations do not raise any restrictions when the Clausius-Duhem inequalityis applied. On contrary, the balance law of mass imposes an additional constraint to possiblethermodynamic process in such a way such that, for instance,
:% cannot be chosen arbitrarily, but
in accordance with the continuity equation:
:%= �%div~v = �% trd = �% I :: d :
Substituting this value for:% into inequality (6.8), the �rst term and the term t :: d may be put
together:
%
"�T@�
@T� @"
@T
� :T +
�T@�
@d� @"
@d
�:::d+
�T
@�
@(grad T )� @"
@(grad T )
�� (grad T )�
#
+
"t� %2
�T@�
@%� @"
@%
�I
#:: d� ~q � gradT
T� 0 :
(6.9)
This inequality must hold for all �elds %, ~x and T without any additional restrictions.
The quantities:T ,
:d and (grad T )� are not considered as independent variables in the con-
stitutive equations for a classical viscous heat-conducting uid, see (6.6) and (6.7), so that theydo not occur implicitly in (6.9) but only explicitly such that the �rst three constituents in (6.9)
are linear in:T ,
:d and (grad T )�. Since
:T ,
:d and (gradT )� may take any arbitrary values, the
inequality (6.9) would be violated unless the factors standing at:T ,
:d and (gradT )� vanish. This
argument gives the following constraints:
@�
@T=
1
T
@"
@T;
@�
@d=
1
T
@"
@d;
@�
@(grad T )=
1
T
@"
@(grad T ): (6.10)
By this, inequality (6.9) reduces to
�(%; T;d; grad T ) � 0 ; (6.11)
where
�(%; T;d; grad T ) :=
"t� %2
�T@�
@%� @"
@%
�I
#:: d� ~q � gradT
T: (6.12)
The di�erentiation of (6.10)1 with respect to d and (6.10)2 with respect to T yields
@2�
@d @T=
1
T
@2"
@d @T;
@2�
@T @d=
1
T
@2"
@T @d� 1
T 2
@"
@d:
By assuming the exchange of the order of partial derivatives of � and " with respect to d and T ,we obtain
1
T
@2"
@d @T=
1
T
@2"
@d @T� 1
T 2
@"
@d;
which shows that " cannot be a function of d,
@"
@d= 0 : (6.13)
86
Because of (6.10)2, it also holds@�
@d= 0 : (6.14)
Likewise, the cross-di�erentiation of (6.10)1 and (6.10)3 with respect to grad T and T , respectively,provides an analogous result:
@"
@(grad T )= 0 ;
@�
@(gradT )= 0 : (6.15)
This is the �rst result of the entropy principle. The constitutive functions " and � cannot dependon d and gradT , and, consequently, the constitutive equations (6.6)3 and (6.7) reduces to
" = "(%; T ) ; � = �(%; T ) : (6.16)
Functions " and � are still subject to constraints (6.10). Since (6.10)2;3 are satis�ed identically,it remains to satis�ed (6.10)1:
@�
@T=
1
T
@"
@T: (6.17)
We continue with the exploitation of the residual inequality (6.11). First, we de�ne thethermostatic equilibrium as a time-independent thermodynamic process with uniform (constantin space) and stationary (constant in time) velocity and temperature �elds. Mathematically, thethermostatic equilibrium is de�ned through equations
~v( ~X; t) = ~v(~x; t) = const ; T ( ~X; t) = T (~x; t) = const ; (6.18)
which implies thatd = 0 ; gradT = ~0 : (6.19)
Then (6.12) shows that�(%; T;0;~0) = 0 (6.20)
in thermostatic equilibrium. Together with inequality (6.11) we can see that � is minimal inthermostatic equilibrium. According to the theory extrema of functions of several variables, thenecessary condition that � reaches a minimum is that the �rst derivatives of � with respect to dand gradT vanish, and the matrix of the second partial derivatives is positive semi-de�nite:
@�
@d
����E= 0 ;
@�
@(grad T )
����E
= ~0 ; (6.21)
and the matrix
A :=
0BBBB@
@2�
@d @d
�����E
@2�
@d @(grad T )
�����E
@2�
@d @(grad T )
�����E
@2�
@(grad T ) @(grad T )
�����E
1CCCCA (6.22)
is positive semi-de�nite. The subscript E denotes the state of thermostatic equilibrium, that is,the state with d = 0 and gradT = ~0.
87
Let us �rst have a look at the constraints in (6.21). Di�erentiating (6.12) with respect to dand gradT , respectively, and taking the result in thermostatic equilibrium yields
@�
@d
����E= tjE � %2
�T@�
@%� @"
@%
�I ;
@�
@(grad T )
����E
= � ~qjET
:
Using this in (6.21), we obtain
tjE = �p(%; T )I ; ~qjE = ~0 ; (6.23)
where p(%; T ) is the thermodynamic pressure,
p(%; T ) := �%2�T@�
@%� @"
@%
�: (6.24)
The result (6.23) shows that in thermostatic equilibrium the stress is pure hydrostatic and theheat ux vanishes. The latter result does not bring any new information since the reducedconstitutive equation (5.184)2 involves this feature.
The constitutive equation (5.184)1 in thermostatic equilibrium reduces to
tjE = a0jE (%; T )I : (6.25)
Comparing this with (6.23)1 shows that
a0jE (%; T ) = �p(%; T ) ; (6.26)
which motivates to decompose function a0 in (5.184)1 as
a0(%; T; Id; IId; IIId; : : :) = �p(%; T ) + �0(%; T; Id; IId; IIId; : : :) : (6.27)
Consequently, the non-equilibrium part �0 of stress vanishes in thermostatic equilibrium,
�0jE = 0 : (6.28)
In view of (6.16)2, the total di�erential d� of � is
d� =@�
@%d%+
@�
@TdT : (6.29)
Let us arrange (6.24) for @�=@%,@�
@%=
1
T
�@"
@%� p
%2
�;
substitute it for the �rst term on the right-hand side of (6.29), and use (6.17) for the second term.We end up with
d� =1
T
�@"
@%� p
%2
�d%+
1
T
@"
@TdT ; (6.30)
which is known as a Gibbs relation for a classical viscous heat-conducting uid. Alternatively, inview of (6.16)1, we may write the total di�erential d" of ":
d" =@"
@%d%+
@"
@TdT : (6.31)
88
By this, the total di�erential d� in (6.30) can be simpli�ed as
d� =1
Td"� p
T
d%
%2: (6.32)
Introducing the speci�c volume v := 1=% and dv = �d%=%2, we have
d� =1
Td"+
p
Tdv ; (6.33)
which is equivalent tod" = T d� � p dv : (6.34)
This is another form of the Gibbs relation, in which " is considered as a function of � and v.The necessary and suÆcient condition that d� in (6.30) is an exact di�erential is
@
@%
�1
T
@"
@T
�=
@
@T
�1
T
�@"
@%� p
%2
��;
or1
T
@2"
@%@T=
1
T
@2"
@T@%� 1
%2@p
@T
!� 1
T 2
�@"
@%� p
%2
�:
After some manipulations, we obtain
@
@T
�p(%; T )
T
�= �
�%
T
�2 @"(%; T )@%
: (6.35)
The function p(%; T ) is usually called the thermal equation of state and the function "(%; T ) thecaloric equation of state. Equation (6.35) shows that the derivative @"=@% need not be sought bycaloric measurements if the thermal equation of state is known. The measurements of the latterare much simpler than the caloric measurements of @"=@%.
Let us summarize the results following from the application of the entropy principle.
� The constitutive equations (5.184) for a classical viscous heat-conducting uid are reduced to
t = (�p+ �0)I + 2�vd+ a2d2 + : : : ;
~q = �� gradT + c1 d � gradT + c2 d2 � grad T ;
" = "(%; T ) ;
� = �(%; T ) ;
(6.36)
wherep = p(%; T ) ;
�0; �v; ai; �; ci = �0; �v; ai; �; ci(%; T; Id; IId; IIId; : : :) ;
�0jE = 0 :
(6.37)
� Functions p, " and � are mutually related,
@�
@T=
1
T
@"
@T;
@�
@%=
1
T
�@"
@%� p
%2
�;
@
@T
�p
T
�= �
�%
T
�2 @"@%
: (6.38)
89
� Further constraints can be drawn from the positive de�niteness of matrix A de�ned by (6.22).
We will now con�ne ourselves to a special type of classical viscous heat-conducting uid whichis linear with respect to d and gradT . In this case, the constitutive equations (6.36) reduce to
t = (�p+ �v trd)I + 2�vd ;
~q = �� gradT ;
" = "(%; T ) ;
� = �(%; T ) ;
(6.39)
wherep; �v; �v; � = p; �v; �v; �(%; T ) : (6.40)
The uid characterized by the constitutive equation (6.39)1 is known as the Newton viscous uid,equation (6.39)2 is known as Fourier's law of heat conduction. The parameters �v and �v arecalled, respectively, the dilatational and shear viscosities. The constitutive equation (6.39)1 canbe written in an alternative form if the strain-rate tensor d is decomposed into spherical anddeviatoric parts, d = (trd) I=3 + dD:
t = (�p+ kvtrd)I + 2�vdD ; (6.41)
where
kv := �v +2
3�v (6.42)
is the bulk viscosity.To draw conclusions from (6.22), we substitute the constitutive equations (6.41) and (6.39)2
into (6.12) and consider (6.24):
�(%; T;d; gradT ) = (t+ pI) :: d� ~q � grad TT
= kvtrd (I :: d) + 2�v(dD :: d) + �
(gradT )2
T
= kv(trd)2 + 2�v
hdD ::
�13(trd) I + dD
�i+ �
(grad T )2
T
= kv(trd)2 + 2�v(d
D :: dD) + �(grad T )2
T: (6.43)
We proceed to calculate the second derivatives of � occurring in (6.22). The �rst derivatives of� are
@�
@(diag d)= 2kv(trd) diag I ;
@�
@dD= 4�vd
D ;@�
@(grad T )= 2�
grad T
T; (6.44)
where diagd is the vector composed from the diagonal elements of tensor d. Then the secondderivatives of � are
@2�
@(diagd) @(diag d)= 2kvI ;
@2�
@dD @dD= 4�vI I ; @2�
@(grad T ) @(grad T )=
2�
TI ;
@2�
@(diagd) @dD= ~0 ;
@2�
@dD @(gradT )= ~0 ;
@2�
@(diag d) @(grad T )= 0 ;
(6.45)
90
where ~0 is the 3rd order zero tensor. The matrix of the second derivatives of �, de�ned by (6.22),is then block-diagonal:
A =
0BB@
2kvI 0 00 4�vI I 0
0 02�
TI
1CCA : (6.46)
The necessary and suÆcient conditions that A is positive semi-de�nite is
kv � 0 ; �v � 0 ; � � 0 : (6.47)
We can conclude that to satisfy the Clausius-Duhem inequality for a linear classical viscous heat-conducting uid, the bulk viscosity kv, the shear viscosity �v and the heat conductivity � mustbe non-negative functions of % and T .
Moreover, the functions ", � and p must satisfy constraints (6.38). If we introduce theHelmholtz free energy
:= "� T� = (%; T ) ; (6.48)
we can then deduce that (6.38) are satis�ed if
� = �@ @T
; p = %2@
@%= �@
@v: (6.49)
This demonstrates that serves as a thermodynamic potential for the entropy and the pressure.The internal energy is also derivable from potential by (6.48).
6.3 Application of the Clausius-Duhem inequality to a classicalviscous heat-conducting incompressible uid
For an incompressible material, the density % remains unchanged, equal to a known value, % = %0.It implies that, compared to the preceding case, the density is excluded from a list of independentconstitutive variables. The stress constitutive equation of an incompressible uid is given by(5.181),
t = �pI + tD : (6.50)
Note that, in contrast to the thermodynamic pressure p(%; T ), pressure p is now an unknown �eldvariable. The constitutive equations for tD, ~q, " and � have the same form as (6.6) and (6.7)except that all functions are now independent of %,
tD = tD(d; T; grad T ) ; ~q = ~q(d; T; grad T ) ; " = "(d; T; grad T ) ; � = �(d; T; gradT ) ;(6.51)
and are subject to the additional constraint
div~v = trd = 0 : (6.52)
Under this constraint, the term t :: d occurring in the Clausius-Duhem inequality can be arrangedas follows:
t :: d = (�pI + tD) :: d = �p trd+ tD :: d = tD :: d :
91
The entropy inequality (6.8) then takes the form
%
"�T@�
@T� @"
@T
� :T +
�T@�
@d� @"
@d
�:::d
+
�T
@�
@(grad T )� @"
@(grad T )
�� (grad T )�
#+ tD :: d� ~q � gradT
T� 0 :
(6.53)
This inequality is linear in the variables:T ,
:d and (gradT )� which may have arbitrarily assigned
values. Thus the factors at these variables must vanish, which yields
@�
@T=
1
T
@"
@T;
@�
@d=
1
T
@"
@d;
@�
@(grad T )=
1
T
@"
@(grad T ): (6.54)
By cross-di�erentiation of (6.54), we again obtain that
@"
@d= 0 ;
@�
@d= 0 ;
@"
@(grad T )= 0 ;
@�
@(grad T )= 0 ; (6.55)
which shows that the internal energy and entropy are functions of temperature only,
" = "(T ) ; � = �(T ) : (6.56)
Functions " and � are still subject to the constraint
@�
@T=
1
T
@"
@T: (6.57)
By (6.54), inequality (6.53) reduces to
�(T;d; gradT ) := tD :: d� ~q � gradTT
� 0 : (6.58)
In thermostatic equilibrium, it again holds
�(T;0;~0) = 0 ; (6.59)
which shows that the conditions (6.21) for the minimum of � are also valid here. From (6.21)1 it
follows that tD���E= 0 which implies that
tjE = �pI : (6.60)
The second constraint in (6.21) yields~qjE = ~0 : (6.61)
We can conclude that also in this case the stress is hydrostatic and the heat ux vanishes inthermostatic equilibrium.
6.4 Application of the Clausius-Duhem inequality to a classicalthermoelastic solid
92
The constitutive equations for a classical thermoelastic solid are introduced in section 5.18.2:
T (2) = T (2)(E; T;GradT ) ; " = "(E; T;GradT ) ;
~Q = ~Q(E; T;Grad T ) ; � = �(E; T; gradT ) :(6.62)
Since C = 2E + I , we replaced the dependence on the Green deformation tensor C in (5.164)by an equivalent representation in terms of the Lagrangian strain tensor E. Note that thecontinuity equation does not play any role here, since the balance of mass results in the statement%0 = %0( ~X).
We carry out time di�erentiation of " and � according to the chain rule of di�erentiation andsubstitute the result into the reduced Clausius-Duhem inequality (6.5). This yields
%0
"�T@�
@T� @"
@T
� :T +
�T@�
@E� @"
@E+
1
%0T (2)
�:::E
+
�T
@�
@(GradT )� @"
@(Grad T )
�� (GradT )�
#�~Q �GradT
T� 0 :
(6.63)
Since:T ,
:E and (GradT )� are not considered as independent variables in the constitutive equation
(6.62), the coeÆcients standing at:T ,
:E and (GradT )� are independent of these quantities.
Hence,:T ,
:E and (GradT )� occur only linearly in the inequality (6.63). This inequality cannot
be maintained for all:T ,
:E and (Grad T )� unless the coeÆcients at these terms vanish. Hence,
@�
@T=
1
T
@"
@T;
1
%0T (2) =
@"
@E� T
@�
@E;
@�
@(Grad T )=
1
T
@"
@(Grad T ): (6.64)
By this and since T > 0, inequality (6.63) reduces to
� ~Q �GradT � 0 : (6.65)
The di�erentiation of (6.64)2 with respect to GradT and (6.64)3 with respect to E and assumingthe exchange of the order of di�erentiation of " and � with respect to E and GradT results in
@T (2)
@(Grad T )= 0 ;
which shows that the second Piola-Kirchho� stress tensor is independent of GradT . By ananalogous procedure, it may be shown that " and � are functions of E and T only. Except theheat ux, the constitutive equations (6.62) reduces to
T (2) = T (2)(E; T ) ; " = "(E; T ) ; � = �(E; T ) : (6.66)
The total di�erential d� of � is
d� =@�
@E:: dE +
@�
@TdT : (6.67)
Substituting from (6.64), we obtain
d� =1
T
�@"
@TdT +
�@"
@E� 1
%0T (2)
�:: dE
�: (6.68)
93
This is the Gibbs relation for a classical thermoelastic solid. If the constitutive equations for "and T (2) are known, it allows to determine the entropy �.
Moreover, the functions T (2), " and � must satisfy constraints in (6.64). Since (6.64)3 issatis�ed identically, it remains to satisfy (6.64)1;2. If we introduce the Helmholtz free energy
:= "� T� = (E; T ) ; (6.69)
we can then deduce that (6.64)1;2 are satis�ed if
� = �@ @T
; T (2) = %0@
@E: (6.70)
This demonstrates that serves as a thermodynamic potential for the entropy and the secondPiola-Kirchho� stress tensor. The internal energy is also derivable from potential by (6.69),
" = � T@
@T: (6.71)
In thermostatic equilibrium, which is de�ned as a process with GradT=0, it is reasonableto assume that there can be no heat conduction, that is, ~QjE = 0. This condition is satis�edprovided that
~Q = �� �GradT ; (6.72)
where � is the so-called tensor of heat conductivity, which depends, in general, on E, T andGradT . The residual inequality (6.65) then becomes GradT � � � GradT � 0 : Decomposingtensor � into the symmetric and skew-symmetric parts, this inequality reduces to
GradT � sym(�) �GradT � 0 ; (6.73)
which states that tensor sym(�) must be positive semi-de�nite. As far the skew-symmetric partof � is concerned, nothing can be concluded from the entropy inequality. It is the statementof the so-called Onsager reciprocity relations that the skew-symmetric part of � is equal to zeromaking � symmetric. This assumption is extensively adopted.
We will now con�ne ourselves to a special type of classical thermoelastic solid for which theconstitutive functions are isotropic and linear with respect to E, T and GradT . The isotropyrequirement means that thermodynamic potential is a function of invariants of E (and tem-perature T ),
= (~IE ; ~IIE ; ~IIIE ; T ) ; (6.74)
where~IE = trE ; ~IIE = trE2 ; ~IIIE = trE3 : (6.75)
To obtain the stress and entropy constitutive equations which are linear and uncoupled in E andT , function can, at most, include polynomials of second degree in E and T , that is,
%0 = %0 0 + � ~IE + �~I2E + 2� ~IIE � �T ~IE � %0�0T � 1
2 T 2 ; (6.76)
where 0, �, �, �, �, �0 and are constants independent of E and T . The third-order terms,
T 3; T 2 ~IE ; T ~I2E ; T~IIE; ~IIIE ;
94
and higher-order terms would generate non-linear terms in the constitutive equations and aretherefore omitted from in linear case. Substituting (6.76) into (6.70) and using the relations
@ ~IE@E
= I ;@ ~IIE@E
= E ; (6.77)
we obtain the constitutive equations for T (2) and �:
T (2) = (� + �trE � �T )I + 2�E ; � = �0 +�
%0trE +
%0T : (6.78)
The constitutive equation for the internal energy can be derived by using (6.71):
%0" = %0 0 + � ~IE + �~I2E + 2� ~IIE +1
2 T 2 : (6.79)
Moreover, comparing (5.165)2 with (6.72) we can deduce that for a linear, isotropic thermoe-lastic solid the tensor of heat conduction must be a spherical tensor, that is, � = �I . Hence,(6.72) reduces to
~Q = ��GradT ; (6.80)
where the non-negative heat conductivity �,
� � 0 ; (6.81)
does not depend on E, T and GradT . We can conclude that equations (6.78){(6.79) form theconstitutive equations for a linear isotropic classical thermoelastic solid. The inequality (6.81)expresses the fact that heat ows from hot to cold in such a solid.
6.5 The M�uller entropy principle
Even of a broad use of the Clausius-Duhem inequality, this contains certain limitations that mightbe violated in some physical situations. Recall that it comes out from
� special choice (6.2) of the entropy ux ~s and the entropy source b,
� the necessity to postulate the existence of the absolute temperature,
� simpli�ed treating of the balance law for mass, linear momentum and energy as additionalconstraints when the entropy inequality is applied (see the discussion after equation (6.3)).
We will be now dealing with a modern concept of the entropy inequality, formulated by M�uller(1983), that imposes a di�erent type of additional information to the entropy inequality. TheM�uller entropy principle also comes out from the second law of thermodynamics (6.1) and, inaddition, assumes:
� The entropy source b vanishes if there are no internal body forces, ~f = ~0, and no internal energysources, h = 0. This requirement is, in particular, satis�ed if the entropy source is linearlyproportional to the body force and the heat source,
b = ~�f � ~f + �hh : (6.82)
95
� There are impenetrable thin walls across which the empirical temperature and the normalcomponent of the entropy ux are continuous,
[�]+� = 0 ; [~n � ~s ]+� = 0 : (6.83)
In view of (6.82), the entropy inequality (6.1) reads
%:� +div~s� %(~�f � ~f + �hh) � 0 : (6.84)
This inequality must be satis�ed for all possible thermodynamic processes that are solutions ofthe balance laws for mass, linear momentum and energy. In contrast to the Clausius-Duheminequality, the M�uller entropy principle considers all these balance laws as additional restrictionsto the entropy inequality (6.84). To satisfy inequality (6.84) subject to the constraints (4.37),(4.39) and (4.42), we create a new unconstrained inequality
%:� +div~s�%(~�f � ~f+�hh)��%( :% +%div~v)�~�v �(%
:~v �div t�%~f)��"(% :" �t :: d+div ~q�%h) � 0 ;
(6.85)which must be satis�ed for all densities %, motions ~x and temperatures �, and also for all Lagrangemultipliers ~�f , �h, �%, ~�v and �" without any additional restrictions.
6.6 Application of the M�uller entropy principle to a classical ther-moelastic uid
In section 5.18.6 we introduced a classical thermoelastic uid as the material with the constitutivevariables %, � and grad �:
t = t(%; �; grad �) ; ~q = ~q(%; �; grad �) ; " = "(%; �; grad �) : (6.86)
According to the principle of equipresence, we choose the same set of constitutive variables forthe entropy � and the entropy ux ~s:
� = �(%; �; grad �) ; ~s = ~s(%; �; grad �) : (6.87)
Note that, in contrast to the Clausius-Duhem inequality, the existence of the absolute temperatureT as a measure of the empirical temperature � is not postulated in the M�uller entropy principle.The reduced form of the constitutive equations (6.86) and (6.87) can be obtained from (5.175):
t = �I + � grad � grad � ; � = �(%; �; g) ;
~q = �� grad � ; ~s = � grad � ;" = "(%; �; g) ; �; �; �; = �; �; �; (%; �; g) ;
(6.88)
where, for the sake of brevity, we introduced g := grad � grad �. The reduced constitutiveequations for � and ~s which are not included in (5.175) are set up analogously to that of " and ~qsince we assume that � and ~s is a frame indi�erent scalar and vector, respectively.
When the constitutive equations (6.86) and (6.87) are brought into (6.85) and the time-di�erentiation is carried out according to the chain rule of di�erentiation, we have
%
@�
@%
:% +
@�
@�
:� +
@�
@�;i
:�;i
!+@si@%
%;i +@si@�
�;i +@si@�;j
�;ji � %(�fi fi + �hh)� �%(:% +%vi;i)
96
��vi %:vi +�
vi
@tij@%
%;j +@tij@�
�;j +@tij@�;k
�;kj
!+ �vi %fi (6.89)
��"% @"
@%
:% +
@"
@�
:� +
@"
@�;i
:�;i
!+ �"tijvj;i � �"
@qi@%
%;i +@qi@�
�;i +@qi@�;j
�;ji
!+ �"%h � 0 ;
which can be rearranged to the form
�%@�
@%� �% � %�"
@"
@%
�:% +
�%@�
@�� %�"
@"
@�
� :� +
%@�
@�;i� %�" @"
@�;i
!:�;i � (%�vi )
:vi
+
@sj@%
+ �vi@tij@%
� �"@qj@%
!%;j +
@sj@�;k
+ �vi@tij@�;k
� �"@qj@�;k
!�;kj (6.90)
+��%�%Æij + �"tij
�vj;i +
��vi � �fi
�%fi +
��" � �h
�%h+
@sj@�
+ �vi@tij@�
� �"@qj@�
!�;j � 0 :
This inequality is linear in:%,:�,:�;i,
:vi, %;j, �;kj, vj;i, fi and h, but not in �;j, since �;j is the
constitutive variable and the factor at the last term in (6.90) implicitly contains �;j. To maintainthis inequality for all values of the variables which enter (6.90) linearly, the factors at these termsmust vanish. This yields
�vi = 0 ; sym
�@sj@�;k
� �"@qj@�;k
�= 0 ;
�% = %
�@�@% � �
" @"@%
�; �"tij = %�%Æij ;
@�@�
= �" @"@�
; �h = �" ;
@�@�;i
= �" @"@�;i
; �fi = �vi = 0 :
@sj@% = �"
@qj@% ;
(6.91)
Consequently, the inequality (6.90) reduces to
�@sj@�
� �"@qj@�
��;j � 0 : (6.92)
Equations (6.91)8;9 imply that the entropy source in a classical thermoelastic uid is of the form
b = �"h ; (6.93)
which also means that the body force ~f in linear model (6.82) does not contribute to the internalentropy sources.
The relation (6.91)6 together with (6.88)2;5, represented in indicial notation, sj = � �;j andqj = ���;j, can be arranged as follows:
(�"�� )Æjk + sym
�"
@�
@�;k� @
@�;k
!�;j = 0 :
97
This, along with@�
@�;k=@�
@g
@g
@�;k= 2
@�
@g�;k ;
results in
(�"�� )Æjk + 2
��"@�
@g� @
@g
��;j�;k = 0 :
For an arbitrary temperature gradient, this equation can only be satis�ed if the expressions inboth brackets vanish,
= �"� ;@
@g= �"
@�
@g: (6.94)
Upon di�erentiating (6.94)1 with respect to g, we obtain
@
@g= �"
@�
@g+@�"
@g� : (6.95)
Since we assume that � 6= 0 (otherwise our uid would not conduct heat), we equate (6.95) with(6.94)2 and obtain
@�"
@g= 0 : (6.96)
Substituting (6.94)1 into the reduced constitutive equations (6.88)2;5 for ~q and ~s, we have
~s = �"~q : (6.97)
Considering this result in (6.91)5 yields
@�"
@%= 0 : (6.98)
Equation (6.91)3 can formally be understood as the determining equation for �". Since the factors
occurring in this equation are functions of the three constitutive variables %, � and g, also �" canonly depend on these variables, �" = �"(%; �; g). Moreover, because of (6.96) and (6.98), �" isonly �-dependent and we employ �" for the de�nition of absolute temperature T (�) as follows
�" = �"(�) =:1
T (�): (6.99)
Equation (6.91)7 yields the representation of the stress tensor in the form
t = �p(%; �; g)I ; (6.100)
where p is referred to as the thermodynamic pressure,
p(%; �; g) := �%2�T (�)
@�
@%� @"
@%
�: (6.101)
This equation can be solved for @�=@%:
@�
@%=
1
T (�)
�@"
@%� p
%2
�: (6.102)
98
With the help of (6.91)3;4 and (6.102), we can express the total di�erential of the entropy� = �(%; �; grad �) in the form
d� =1
T (�)
�@"
@�d� +
�@"@%� p
%2
�d%+
@"
@(grad �)� d(grad �)
�: (6.103)
The integrability condition for the di�erential d� requires that the cross di�erentiation of the �rstconstituent with respect to grad � and the third constituent with respect to � are equal to eachother:
1
T (�)
@2"
@(grad �) @�=
1
T (�)
@2"
@� @(grad �)� 1
T 2(�)
dT (�)
d�
@"
@(grad �);
which shows that@"
@(grad �)= 0 : (6.104)
In view of (6.91)4, we also have@�
@(grad �)= 0 : (6.105)
Consequently, the internal energy and the entropy are functions of % and � only,
" = "(%; �) ; � = �(%; �) ; (6.106)
and (6.103) reduces to the Gibbs relation
d� =1
T (�)
�@"
@�d� +
�@"@%� p
%2
�d%
�: (6.107)
In view of (6.106), the thermodynamic pressure p also depends on % and � only,
p(%; �) = �%2�T (�)
@�(%; �)
@%� @"(%; �)
@%
�; (6.108)
and (6.100) for the stress tensor reduces to
t = �p(%; �)I : (6.109)
By equating this with the constitutive equation (6.88)1, we obtain
�(%; �; g) = �p(%; �) ; �(%; �; g) = 0 : (6.110)
The integrability condition for the Gibbs relation (6.107) is
1
T (�)
@2"
@% @�=
1
T (�)
@2"
@� @%� 1
%2@p
@�
!� 1
T 2(�)
dT (�)
d�
�@"
@%� p
%2
�:
This equation can be solved for the derivative of T (�):
d lnT (�)
d�=
�@p@�
%2@"
@%� p
:
99
Finally, the integration with respect to � yields
T (�) = T (�0) exp
8>><>>:
�Z�0
�@p@�
%2@"
@%� p
d�
9>>=>>; : (6.111)
Choosing integration constant T (�0) positive, function T (�) is a positive-valued function, asexpected for an absolute temperature. To show that T (�) is a meaningful absolute temperature,we need to prove that (i) T (�) is a universal function of �, that is, it is not di�erent for twodi�erent materials, and (ii) T (�) is a strict monotonic function of �, that is, TA > TB means that"A is warmer than B".
Universality
We now employ the postulate of the existence of impenetrable thin walls with the property (6.83).Let us consider two di�erent classical thermoelastic uids that are separated by an impenetrablethin wall. Let uid I be placed on the positive side and uid II on the negative side of the wall.In view of (6.97), the continuity condition (6.84)2 of the normal component of the entropy uxacross the wall becomes �
1
T (�)~n � ~q
�+�
= 0 : (6.112)
By an impenetrable wall we mean that no portion of matter from one side can penetrate intothe matter on the other side, that is, ~n � ( [~v]+� � ~�) = 0. In the terminology of section 4.3, animpenetrable thin wall is a material discontinuity surface. Moreover, we assume that there is no
tangential slip across the wall, so that,h~vki+�= ~0, where ~vk := (I � ~n ~n) � ~v. Consequently,
~n � [~q]+� = 0 and the interface condition (6.112) can be rewritten as
�1
T (�)
�+�
~n � ~q = 0 : (6.113)
Since, in general, ~n � ~q 6= 0, we �nally obtain�1
T (�)
�+�
= 0 ; or T I(�) = T II(�) : (6.114)
We can conclude that the function T (�) is the same for all classical thermoelastic uids. Thesame property of the function T (�) can be proved for other materials. Hence, T (�) is a universalfunction of the empirical temperature �.
Monotony
First, we de�ne the ideal gas as a classical thermoelastic uid with the following thermal andcaloric equations of state:
p(%; �) =%
�f(�) ; "(�) =
�1�f(�) + �2 ; (6.115)
where a positive-valued, strict monotonic function f(�), which is universal for all ideal gases,only depends on the choice of a measure of empirical temperature; � is the moll-mass, and �1
100
and �2 are characteristic constants of a particular ideal gas. For instance, in the Celsius scale oftemperature, we have
f(�) = R(� + �m) ; (6.116)
where the universal gas constant R=8.314 J/(mol oC), and �m = 273:15oC. In this case, theequations of state are of the form
p = %R
�(� + �m) ; " = �1
R
�(� + �m) + �2 : (6.117)
With the help of (6.111), we calculate function T (�) for the ideal gas:
T (�) = T (�0) exp
8>>><>>>:
�Z�0
�%�
df(�)
d�
�%�f(�)
d�
9>>>=>>>;= T (�0) exp
8><>:
�Z�0
d ln f(�)
d�d�
9>=>; = T (�0) exp
�ln
f(�)
f(�0)
�:
Hence,
T (�) =T (�0)
f(�0)f(�) : (6.118)
Choosing integration constant T (�0) positive, T (�) has the same property as function f(�), thatis, T (�) is a positive-valued, strict monotonic function. Because of the universality of T (�),this result holds not only for ideal gases but in general. Thus the function T (�0) satis�es bothrequirements of a meaningfully de�ned absolute temperature.
Let us have a look at the Celsius scale as a measure of empirical temperature. Putting�0 = 0oC and choosing the integration constant as
T (�0) = T (0oC) = 273:15K ; (6.119)
equation (6.118) along with (6.116) yields
T (�) =T (�0)
R�mR(� + �m) =
273:15K
273:15oC(� + 273:15oC) ;
that is,
T (�) = 273:15K + 1KoC
� : (6.120)
This measure of the absolute temperature is called the Kelvin scale.To complete the exploitation of the M�uller entropy principle for a classical thermoelastic uid,
it remains to evaluate the reduced inequality (6.92). With ~s = ~q=T (�), see (6.97) and (6.99), wehave �
1
T (�)
@~q
@�� 1
T 2(�)
dT (�)
d�~q � 1
T (�)
@~q
@�
�� grad � � 0 ;
which reduces to�~q � grad � � 0 :
Substituting for ~q from the constitutive equation (6.88)2, we can see that the heat conductivityis non-negative,
� � 0 : (6.121)
Let us summarize the results following from the application of the M�uller entropy principleto a classical thermoelastic uid.
101
� For the empirical temperature �, it is possible to construct a positive-valued, strict monotonicand universal function T (�),
T (�) =T (�0)
f(�0)f(�) ; (6.122)
which may serve as a measure of the absolute temperature.
� The constitutive equations (6.88) reduce to
t = �pI ; � = �(%; �) ;
~q = �� grad � ; ~s = ~qT (�)
;
" = "(%; �) ; p = p(%; �) ; � = �(%; �; g) :
(6.123)
� Functions p, ", � and T are mutually related:
@�
@�=
1
T (�)
@"
@�;
@�
@%=
1
T (�)
�@"
@%� p
%2
�; (6.124)
d lnT (�)
d�=
�@p@�
%2@"
@%� p
:
� The entropy source b and the heat source h are related as
b =h
T (�): (6.125)
Besides the possibility (6.122) to construct the universal absolute temperature T for a givenempirical temperature �, there is a particular choice to put
T = � : (6.126)
Under this choice, the results of the M�uller entropy principle for a classical thermoelastic uidcoincide with those of the Clausius-Duhem inequality. In particular, the relations ~s = ~q=T andb = h=T , postulated by the Clausius-Duhem inequality, follow from the M�uller entropy principleas the results (6.123)5 and (6.125). However, these two principles do not, in general, provideequivalent restrictions on material properties. For instance, for a non-classical heat-conducting
uid (for which the constitutive equations also depend on:�), the Clausius-Duhem inequality yields
that �(%; �;:�; g) � 0 in the stress constitutive equation, an analog to the classical case, while the
M�uller entropy principle allows non-vanishing � of the stress tensor. Hence, the Clausius-Duheminequality imposes stronger restrictions on constitutive equations than those resulting from theapplication of the M�uller entropy principle.
102
7. CLASSICAL LINEAR ELASTICITY
7.1 Linear elastic solid
In section 6.4 we found that the most general constitutive equations of a classical thermoelasticsolid have the forms expressed by equations (6.69){(6.73). If thermal e�ects are not considered,the Helmholtz free energy and the second Piola-Kirchho� stress tensor are functions of strainalone. According to section 5.18.1, this solid is called elastic. For an elastic solid, the constitutiveequation (6.70)2 reduces to
T (2) =@W
@E; (7.1)
where the so-called elastic strain energy density (strain energy per unit undeformed volume) isde�ned by
W := %0 = W (E) : (7.2)
It is worthwhile noting that elastic behavior is sometimes de�ned on the basis of the existenceof a strain energy function from which the stress may be determined by the di�erentiation in(7.1). A material de�ned in this way is called a hyperelastic material. The stress is still a uniquefunction of strain so that this \energy approach" is compatible with our earlier de�nition of elasticbehavior in section 5.18.1.
By using the constitutive equation (7.1) of nonlinear elastic solid, we can derive variousapproximation theories. Expanding W about the con�guration �0 from which the strain E isreckoned, we have
W (E) =W (0) +@W (0)
@E:: E +
1
2E ::
@2W (0)
@E @E:: E +O(jEj3) ; (7.3)
and, from (7.1),
T (2) =@W (0)
@E+@2W (0)
@E @E:: E +O(jEj2) : (7.4)
Using di�erent notation, we can shortly write
T (2) = T 0 +C :: E +O(jEj2) ; (7.5)
where T 0 is the stress in the con�guration �0 from which the strain E is reckoned,
T 0 :=@W (0)
@E: (7.6)
In the classical linear theory, the con�guration �0 is used as reference and the stress T 0 isconsidered as a tensor-valued function of the Lagrangian coordinates, T 0 = T 0( ~X). The fourth-order tensor C, introduced in (7.5), is de�ned by
C :=@2W (0)
@E @E; CKLMN :=
@2W (0)
@EKL @EMN: (7.7)
Due to the symmetry of both the stress and strain tensors, it is clear that
CKLMN = CLKMN = CKLNM ; (7.8)
103
which reduces the 34 = 81 components of C to 36 distinct coeÆcients CKLMN at most. Moreover,as a consequence of the equality of the mixed partial derivatives of W , the coeÆcients CKLMN
satisfy the further symmetry relation:
CKLMN = CMNKL ; (7.9)
which is known as the Maxwell relation. Thus, the existence of a strain energy function reducesthe number of distinct coeÆcients CKLMN from 36 to 21. Further reduction for special typesof elastic behavior are obtained from the material symmetry properties. Note that the samematerial may, in general, have di�erent coeÆcients CKLMN for di�erent con�gurations �0.
In the classical linear theory, we consider only in�nitesimal deformations from the referencecon�guration �0, and then the last term on the right-hand side of (7.5) is dropped. Moreover, thestrain tensor E is replaced by the in�nitesimal strain tensor ~E, so that, the constitutive equation(7.5) reduces to
T (2) = T 0 +C :: ~E : (7.10)
Within the framework of the classical in�nitesimal model, we may study the e�ects of an in-�nitesimal deformation superimposed upon a reference con�guration �0 with a �nite pre-stressT 0. Then (7.10) is taken as the exact constitutive equation de�ning the classical linear (in-�nitesimal) model of elasticity. However, within the framework of the general theory of elasticsimple materials, we see that (7.10) is an approximation, for small deformations, of the exactstress relation (7.1).
In fact, we have considerable extent in the choice of tensor C. The only requirement is thatwe satisfy the symmetries (7.8) and (7.9). It is easily veri�ed that these relations are satis�ed bya fourth-order tensor � of the form
�KLMN = CKLMN + a [T0;KLÆMN + T0;MNÆKL]
+ b [T0;KMÆLN + T0;LNÆKM ]
+ c [T0;KNÆML + T0;MLÆKN ] ;
(7.11)
or, in symbolic notation, 12
� = C + a [(T 0 I)1234 + (I T 0)1234]
+ b [(T 0 I)1324 + (I T 0)1324]
+ c [(T 0 I)1432 + (I T 0)1432] ;
(7.12)
where a, b and c are arbitrary scalars. The expressions in parentheses multiplying a, b and c arethe only three linear combinations of permutations of T 0 I satisfying the symmetries (7.8) and(7.9). Every choice of the scalars a, b and c de�nes the behaviour of a linear elastic solid. Wemay thus replace the tensor C in the constitutive equation (7.10) by the tensor �:
T (2) = T 0 +� :: ~E
= T 0 +C :: ~E + a[(tr ~E)T 0 + (T 0:: ~E)I ] + (b+ c)(T 0 � ~E + ~E � T 0) :
(7.13)
12The symbols ( )1324, ( )1432, etc., denote the transpose of a quadric, e.g. (~a~b ~c ~d)1324 = ~a ~c~b ~d.
104
The linearized constitutive equations for the Cauchy stress tensor and the �rst Piola-Kirchho�stress tensor can be obtained from (7.13)2 using the relation (3.24)1 and (3.25)2:
T (1) = T 0 + T 0 �H +C :: ~E + a[(tr ~E)T 0 + (T 0:: ~E)I ] + (b+ c)(T 0 � ~E + ~E � T 0) ; (7.14)
t = T 0 � (trH)T 0 +HT � T 0 + T 0 �H +C :: ~E + a[(tr ~E)T 0 + (T 0:: ~E)I ]
+ (b+ c)(T 0 � ~E + ~E � T 0) : (7.15)
By the decomposition HT = ~E + ~R, where ~R is the in�nitesimal rotation tensor, the represen-tations (7.14) and (7.15) become
T (1) = T 0 � T 0 � ~R+C :: ~E + a[(tr ~E)T 0 + (T 0:: ~E)I ] + (b+ c)( ~E � T 0)
+ (b+ c+ 1)(T 0 � ~E) ; (7.16)
t = T 0 + ~R � T 0 � T 0 � ~R+C :: ~E + (a� 1)(tr ~E)T 0 + a(T 0:: ~E)I
+ (b+ c+ 1)(T 0 � ~E + ~E � T 0) : (7.17)
The �rst terms in (7.16) and (7.17) can be identi�ed as the rotated initial stress, whereas theremaining terms represent the perturbation in stress due to the in�nitesimal strain ~E.
Upon substituting equations (7.6) and (7.7) into (7.3) and replacing the tensor C by thetensor �, the strain energy density can be rewritten, correct to second order in ~E, in the form
W ( ~E) = W (0) + T 0:: ~E + 1
2~E :: � :: ~E
= W (0) + T 0:: ~E + 1
2~E :: C :: ~E + a(tr ~E)(T 0
:: ~E) + (b+ c)tr ( ~E � T 0 � ~E) :(7.18)
The terms on the right-hands side of (7.18) proportional to T 0 represent the work done againstthe initial stress, whereas the term 1
2~E :: C :: ~E is the classical elastic energy density in the
absence of any initial stress.
7.2 The elastic tensor
A stress-free con�guration is called a natural state. If such a con�guration is used as reference,then T 0 = 0, and (7.13){(7.15) reduce to
� = C :: " ; (7.19)
where � can be interpreted as either t, T (1) or T (2) and " either as the in�nitesimal Lagrangianor Eulerian strain tensor, ~E or ~e. This classical linear elastic stress{strain relation is knownas generalized Hooke's law and the fourth-order tensor C is called the elastic tensor. The linearstress-strain constitutive relation for in�nitesimal deformations of an anisotropic solid in its stress{free natural state are speci�ed by 21 independent components of C. In the more general caseof an incremental stress superimposed upon the zeroth-order initial stress T 0, the stress-strainrelation is speci�ed by 27 coeÆcients: the 21 components of C and 6 components of the initialstress tensor T 0. Equations (7.16) and (7.17) constitute the generalization of Hooke's law to thecase of a pre-stressed elastic medium.
In the particular case T 0 = 0, the strain energy density reduces to
W (") =W (0) +1
2� :: " : (7.20)
105
7.3 Isotropic linear elastic solid
The highest symmetry of a solid is reached if the solid possesses no preferred direction withrespect to its elastic property. This also means that the linear elasticity C is invariant under anyorthogonal transformation of the coordinate system. We called such a solid isotropic. In section5.18.1, we showed the number of independent elastic constants reduces to 2 and the constitutiveequation of an isotropic linear elastic solid is of the form
� = �#I + 2�" ; (7.21)
where � and � are the Lam�e elastic constant,
# := tr " = div ~u ; (7.22)
and ~u is the displacement vector. Now, we will show, independently of considerations in section5.18.1, that the generalized Hooke's law (7.19) reduces to the form (7.21) for an isotropic linearelastic solid.
In the linear theory, it is not necessary to distinguish between the Lagrangian and Eulerian co-ordinate systems. We will follow this concept and use Cartesian coordinates xk for the coincidingcoordinate systems. Any orthogonal transformation of the coordinate system may be expressedby the Euclidean transformation (5.12) and (5.13) with ~b = ~0, that is, by the transformationequation
x0k = Qklxl (7.23)
subject toQklQml = QlkQlm = Ækm ; detQkl = �1 : (7.24)
The components of a second-order Cartesian tensor � transform under the orthogonal transfor-mation of the coordinate system according to (5.15):
� 0kl = QkmQln�mn ; (7.25)
which may readily be inverted with the help of the orthogonality conditions (7.24) to yield
�kl = QmkQnl�0mn : (7.26)
Note carefully the location of the summed indices m and n in (7.25) and (7.26).Proof of Hooke's law (7.21) for an isotropic linear elastic solid will be accomplished in four
steps. First, we show that for an isotropic linear elastic solid the principal axes of the stress
and in�nitesimal strain tensors coincide. To proof it, we take, without loss of generality, thecoordinate axes xk in the principal directions of strain tensor ". Then "12 = "13 = "23 = 0. Weshall now show that �23 = 0. We �rst have
�23 = A"11 +B"22 + C"33
with A := C2311, B := C2322, and C := C2333. We now rotate the coordinate system through anangle of 180o about the x3-axis. Then x
01 = �x1, x02 = �x2, and x03 = x3, and the matrix of this
transformation is
Q =
0B@ �1 0 0
0 �1 00 0 1
1CA :
106
In view of (7.25), we therefore have
� 023 = Q2mQ3n�mn = ��23 ;"011 = Q1mQ1n"mn = "11 ;
"022 = Q2mQ2n"mn = "22 ;
"033 = "33 :
The relation� 023 = A"011 +B"022 + C"033 = A"11 +B"22 + C"33 = �23
is now the consequence of isotropy since the constants A, B, and C do not depend on the referencecon�guration of a solid. Thus
��23 = � 023 = �23 ;
which implies that �23 = 0. Likewise, it can be shown that �12 = �13 = 0. We have proved thatthe principal axes of stress and strain coincide. Note that the principal stresses and strains are,in general, di�erent.
Second, consider the component �11. Taking the coordinate axes in principal directions ofstrain, we obtain
�11 = a1"11 + b1"22 + c1"33 ;
with a1 := C1111, b1 := C1122, and c1 := C1133. We now rotate the coordinate system through anangle 90o about the x1-axis in such a way that
x01 = x1 ; x02 = x3 ; x03 = �x2 :The matrix of this transformation is
Q =
0B@ 1 0 0
0 0 10 �1 0
1CA ;
and we have
� 011 = Q1mQ1n�mn = �11 ;
"011 = "11 ;
"022 = Q2mQ2n"mn = "33 ;
"033 = "22 :
In view of isotropy, the constants a1, b1, and c1 do not depend on the reference con�guration, sothat
� 011 = a1"011 + b1"
022 + c1"
033 ;
and substituting for "0kk, we get
� 011 = a1"11 + b1"33 + c1"22 :
This implies that b1 = c1 since �11 = a1"11 + b1"22 + c1"33. We can thus write �11 as
�11 = a1"11 + b1 ("22 + "33) = �1#+ 2�1"11 ;
107
where�1 := b1 = C1122 ; 2�1 := a1 � b1 = C1111 �C1122 :
The same relations can be obtained for subscripts 2 and 3:
�22 = �2#+ 2�2"22 ;
�33 = �3#+ 2�3"33 ;
where �2 := C2222, 2�2 := C2211 � C2222, �3 := C3322, 2�3 := C3311 � C3322.Third, rotate the coordinate system through an angle 90o about the x3-axis in such a way
thatx01 = x2 ; x02 = �x1 ; x03 = x3 :
The matrix of this transformation is
Q =
0B@ 0 1 0�1 0 00 0 1
1CA ;
and we have
� 011 = �22 ; "011 = "22 ;
� 022 = �11 ; "022 = "11 ;
� 033 = �33 ; "033 = "33 :
Combining the results of previous steps, we can write
� 022 = �2#0 + 2�2"
022
= �2#+ 2�2"022
= �2#+ 2�2"11!= �11
= �1#+ 2�1"11 :
This implies that �2 = �1 and �2 = �1. Likewise, it can be shown that �3 = �1 and �3 = �1. Insummary, we have
�11 = �#+ 2�"11 ;
�22 = �#+ 2�"22 ;
�33 = �#+ 2�"33 ;
�kl = 0 for k 6= j ;
which is the generalized Hooke's law for an isotropic body in principal directions. Shortly written,
�kl = �#Ækl + 2�"kl :
Fourth, we now rotate the coordinate system with the axes xk coinciding with principaldirections of strain to arbitrary coordinate system with the axes x0k and show that Hooke's
108
law for an isotropic solid also holds in a rotated coordinate system x0k. Denoting by Qkl thetransformation matrix of this rotation, the stress and strain tensor transform according to thetransformation law (7.25) for second-order tensors. Multiplying the above equation by QmkQnl,using the transformation law (7.25) and the orthogonality property (7.24) of Qkl, we have
� 0mn = �#Æmn + 2�"0mn :
Moreover,#0 = "0mm = QmkQml"kl = Ækl"kl = "kk = # :
We �nally have� 0mn = �#0Æmn + 2�"0mn ;
which proves Hooke's law (7.21) for an isotropic linear elastic solid.
We have seen that the Lam�e coeÆcients are expressed in terms of the elastic coeÆcients as
� = C1122 ; �+ 2� = C1111 : (7.27)
We note without proof that the most general form of a fourth-order isotropic tensor A is
Aklmn = �ÆklÆmn + � (ÆkmÆln + ÆknÆlm) + � (ÆkmÆln � ÆknÆlm) ; (7.28)
where �, �, and � are scalars. If A is replaced by the linear elasticity C, the symmetry relationsCklmn = Clkmn = Cklnm imply that � must be zero since by interchanging k and l in the expression
� (ÆkmÆln � ÆknÆlm) = � (ÆlmÆkn � ÆlnÆkm)
we see that � = �� and, consequently, � = 0. Thus, the linear elasticity for an isotropic solid hasthe form
Cklmn = �ÆklÆmn + � (ÆkmÆln + ÆknÆlm) : (7.29)
The Hooke's law for an isotropic solid takes a particularly simple form in spherical and devi-atoric parts of " and � . Let us decompose the in�nitesimal strain tensor " into the spherical anddeviatoric parts:
" =1
3#I + "D ; (7.30)
where the mean normal strain # is de�ned by (7.22), and the deviatoric strain tensor "D is atrace-free, symmetric, second-order tensor:
"D := "� 1
3#I : (7.31)
The same decomposition of the stress tensor � reads
� = �pI + �D ; (7.32)
where the scalar p is the negative of the mean normal stress and called mechanical pressure,
p := �1
3tr � ; (7.33)
109
and the deviatoric stress tensor �D is a trace-free, symmetric, second-order tensor, de�ned by
�D := � + pI : (7.34)
By substituting (7.30) and (7.32) into (7.21), Hooke's law for an isotropic linear elastic solidmay be written separately for the spherical and deviatoric parts of the stress and strain:
�p = k# ; �D = 2�"D ; (7.35)
where
k := �+2
3� (7.36)
is referred to as the elastic bulk modulus. Finally, we derive the strain energy density function foran isotropic linear elastic solid. For this purpose, we successively substitute (7.21) into (7.20),giving
W (") =W (0) +1
2�#2 + �(" :: ") ; (7.37)
which, using (7.31) and (7.36), can be rewritten as
W (") =W (0) +1
2k#2 + �("D :: "D) : (7.38)
7.4 Restrictions on elastic coeÆcients
In this section, we analyze several hypothetical experiments and consequent restrictions that mustbe placed upon elastic moduli in order that they may represent a real material adequately.
We �rst assume that Hooke's law (7.21) for an isotropic linear elastic solid is invertible for ".Applying the trace operator on this equation gives
tr � = (3�+ 2�)tr " : (7.39)
Now, by solving (7.21) for " and substituting from (7.39) for tr ", we obtain the inverse form ofHooke's law for an isotropic solid,
" =�
2�� �
2�(3�+ 2�)tr � I : (7.40)
We observe that for " to be uniquely determined by � we must have
� 6= 0 ; 3�+ 2� 6= 0 : (7.41)
(i) Hydrostatic pressure. Experimental observations indicate that under hydrostatic pres-sure the volume of an elastic solid diminishes. For hydrostatic compression, the stress tensor hasthe form
� = �pI ; p > 0 : (7.42)
From (7.35)1 we see thatp = �k# ; (7.43)
110
where # := tr " = div ~u = J � 1 = (dv � dV )=dV is the cubical dilatation. In the case ofhydrostatic pressure, clearly dv < dV and hence # < 0. We must thus have k > 0 which impliesthat
3�+ 2� > 0 : (7.44)
(ii) Simple shear. Consider a simple constant shear in which
�12 6= 0 ; �kl = 0 otherwise: (7.45)
In this case (7.40) gives�12 = 2�"12 : (7.46)
Experimental observations of small deformations of elastic solids subjected to simple shear indi-cate that �12 and "12 have the same direction. Consequently,
� > 0 : (7.47)
Under conditions (7.44) and (7.47) and provided that the strain energy density functionvanishes at the unstrained natural state, W (0) = 0, the strain energy density function W (") foran isotropic linear elastic solid is a positive de�nite, quadratic function of the in�nitesimal strain". This fact is basic for the proof of the uniqueness of solution of boundary-value problems inelastic equilibrium.
The conditions (7.44) and (7.47) can be shown to be suÆcient but not necessary for anisotropic linear isotropic solid to get plausible results from the in�nitesimal �eld theory. Thenecessary and suÆcient conditions for an isotropic linear elastic solid are weaker than (7.44) and(7.47), namely,
�+ 2� > 0 ; � > 0 : (7.48)
It can be shown that these conditions give positive wave speeds in isotropic linear elastic solids. Anumber of special a priori inequalities have been proposed based on certain \reasonable" physicalexpectations when an isotropic elastic solid is subjected to pressure, tensions, and shears. For adiscussion of these we refer the reader to Truesdell and Noll (1965).
In the classical in�nitesimal theory of elasticity other material constants are often used inplace of the Lam�e constants � and �. Following are relations among some of these constants:
E := �(3�+ 2�)=(�+ �) ; � := �=2(�+ �) ;
� = E�=(1 + �)(1� 2�) = 2G�=(1 � 2�) ; � � G = E=2(1 + �) ;
k = �+ 23� = E=3(1 � 2�) ;
(7.49)
where E is called Young's modulus, G is the elastic shear modulus, or modulus of rigidity, which,as noted, is identical to the Lam�e constant �, and � is Poisson's ratio. The inequalities
G > 0 ; �1 < � <1
2; (7.50)
are equivalent to (7.44) and (7.47).
7.5 Field equations
111
The �eld equations in continuum mechanics consist of the balance laws, valid for all continua,and a particular constitutive equation. Here we shall consider isotropic linear elastic or linearthermoelastic solids. The �eld equations may be expressed in Lagrangian or Eulerian form butfor the linearized theory of elasticity the two forms are identical.
7.5.1 Isotropic linear elastic solid
Within the frame of the in�nitesimal strain theory, the continuity equation (4.58) in the referentialform can be arranged as follows
% =%0J
=%0
1 + tr ~E+O(j ~Ej2) = %0 +O(j ~Ej) : (7.51)
Making use of this approximation and provided, in addition, that the volume force ~f is alsoin�nitesimally small, the equation of motion (4.39) in the present con�guration coincides withthe equation of motion (4.69) in the referential con�guration:
div � + %0 ~f = %0@2~u
@t2; (7.52)
where we have substituted for ~v = @~u=@t since ~u = ~u( ~X; t) and the partial derivative with respectto the material coordinates held constant. We can see that in the in�nitesimal strain and stresstheory the distinction between the reference and present con�guration of a material disappears.Moreover, the continuity equation does not play any role here, since the balance of mass resultsin the statement %0 = %0( ~X).
We substitute Hooke's law (7.21) for an isotropic linear elastic solid into the equation of motion(7.52). With the help of vector di�erential identities introduced in Appendix A, the divergenceof the stress tensor � can be arranged as follows
div � = div (�#I + 2�")
= grad (�#) + 2�div "+ 2grad� � "= � grad div~u+ div~u grad�+ �div(grad ~u+ grad T~u) + grad� � (grad ~u+ grad T~u)
= (�+ �) grad div~u+ �r2~u+ div~u grad�+ grad� � (grad ~u+ grad T~u)
= (�+ 2�) grad div~u� � rot rot ~u+ div~u grad�+ grad� � (grad ~u+ grad T~u) :
The linearized equation of motion (7.52) can now be expressed in the form
(�+2�) grad div~u�� rot rot ~u+div~u grad�+grad� � (grad ~u+grad T~u)+%0 ~f = %0@2~u
@t2; (7.53)
which is known as the Navier{Cauchy equation of an isotropic linear elastic solid. It consists ofthree second-order di�erential equations for the three displacement components. Note that thevector form (7.53) is independent of the coordinate system and may be expressed in curvilinearcoordinates by the methods of Appendix C. In particular, for a homogeneous solid, � and � areconstants (independent of ~X) and the Navier-Cauchy equation reduces to
(�+ 2�) grad div~u� � rot rot ~u+ %0 ~f = %0@2~u
@t2; (7.54)
112
or, alternatively,
(�+ �) grad div~u+ �r2~u+ %0 ~f = %0@2~u
@t2: (7.55)
Note that elastostatics is restricted to those situations in which inertia forces %0@2~u=@t2 may be
neglected. Since (7.52) does not involve displacements, an alternative formulation of elastostaticproblems in terms of stresses alone is possible.
7.5.2 Incompressible isotropic linear elastic solid
For incompressible solids we havediv~u = 0 ; (7.56)
and the density remains unchanged, equal to a known value, % = %0. Hooke's law (7.21) need tobe modi�ed such that
� = �pI + 2�" ; (7.57)
where p is a new unknown �eld variable. The Navier-Cauchy equation of motion (7.53) transformsto the form
�grad p+ �r2~u+ grad� � (grad ~u+ grad T~u) + %0 ~f = %0@2~u
@t2: (7.58)
Hence, (7.56) and (7.58) represents four di�erential equations for four unknown �eld variables:pressure p and three components of displacement vector ~u.
7.5.3 Isotropic linear thermoelastic solid
Equation (6.78){(6.81) form the constitutive equations for an isotropic linear thermoelastic solid.We now substitute these constitutive equations to the energy equation in the referential form(4.80). We obtain
�(~IE)�+�(~I2E)
�+2�( ~IIE)�+ T
:T= (�+�~IE��T )(~IE)�+2�E ::
:E +Div (�Grad T )+%0h : (7.59)
The material time derivatives of principal invariants of strain can be arranged as follows:
(~I ~E)� = Div
:~u ;
(~I2E)� = 2~IE(~IE)
� ;
( ~IIE)� = (trE2)� = tr (E2)� = 2tr (E �
:E ) :
(7.60)
In view of these expressions, equation (7.59) is non-linear with respect to E. Within the frame-work of linear theory, we consider only in�nitesimal deformations from the reference con�guration.Then the non-linear terms in (7.59) can be omitted and the strain tensor is replaced by the in-�nitesimal strain tensor ~E. The linearization with respect to strain brings (7.59) into the form
T:T= ��T Div
:~u +Div (�Grad T ) + %0h : (7.61)
This equation is still non-linear with respect to temperature. Further linearization can be madeby assuming that the instantaneous absolute temperature T in the present con�guration di�ersfrom the temperature T0( ~X) (> 0) in the reference con�guration by a small quantity, that is,
T = T0 + T1 ; jT1j � T0 : (7.62)
113
The linearization of (7.62) with respect to T yields
T0:T1= ��T0Div
:~u +Div (�Grad T1) + Div (�Grad T0) + %0h : (7.63)
If we introduce the speci�c heat cv at constant deformation Div:~u= 0 and no heat supply h = 0
through the equation
Div ~Q = %0cv:T1 ; (7.64)
then T0 = %0cv and the energy equation becomes
%0cv:T1= ��T0Div
:~u +Div (�Grad T1) + Div (�Grad T0) + %0h : (7.65)
This equation, known as the coupled heat conduction equation, gives the relationship between therate of change of the temperature and the strain with the heat conduction.
Under the decomposition (7.62)1, the linearized constitutive equation (6.78)1 for the secondPiola-Kirchho� stress tensor becomes
T (2) = (� � � T0)I + �(trE)I + 2�E � � T1I : (7.66)
Again, in the in�nitesimal theory T (2) and E can be replaced by the in�nitesimal stress tensor �and the in�nitesimal strain tensor ", respectively. Note that when the reference con�guration isstress free then ��� T0 = 0. The constitutive equation (7.66) is known as the Duhamel-Neumannform of Hooke's law. It may be inverted to give
E = � 1
3�+ 2�(� � � T0)I +
1
2�T (2) � �
2�(3�+ 2�)trT (2) I +
�
3�+ 2�T1I ; (7.67)
where� := �=(3�+ 2�) (7.68)
is the linear coeÆcient of thermal expansion.The coupled heat conduction equation, along with the equation of motion and thermoelas-
tic stress{strain equation constitute the basic set of �eld equations for coupled thermoelasticproblems. There are, however, problems in which the heat conduction equation can further besimpli�ed. For instance, if the reference con�guration of the thermoelastic solid coincides withthe thermostatic equilibrium, then GradT0 = ~0 and the third term on the right-hand side of(7.65) vanishes. Another simpli�cation arises when the isotropic linear thermoelastic solid is also
incompressible. Then Div ~u = 0, and also Div:~u= 0, so that, the �rst term on the right-hand side
of (7.65) vanishes. In this case, the heat conduction equation is decoupled from the equation ofmotion and the thermoelastic problem is decomposed into two separate problems, which must besolved consecutively, but independently.
7.5.4 Example: The deformation of a plate under its own weight
We consider a two-dimensional inclined plate of constant thickness H (slab) in�nitely extendedin the x- and y-directions and tightly connected to the bed (Figure 7.1). The material of slab isisotropic and linear elastic. We aim to determine the deformation of the slab by its own weight.
Solution. We assume that the inclination angle � of the slab is constant. Because of this, in�nite x-and y-dimensions and constant thickness of the slab, the �eld variables can only depend on the ver-tical coordinate z; all derivatives with respect to x and y must vanish, (@=@x)(�) = (@=@y)(�) = 0.Moreover, the y-component of the displacement vector is taken as zero, and the remaining
114
��; �
H
�
xy
z
~g
Figure 7.1. The deformation of a plate under its own weight.
components are considered as functions of z only,
ux = ux(z) ; uz = uz(z) : (7.69)
Consequently, the strain components "xx = "xy = "yz = "yy = 0, while "zz and "xz are functionsof z only,
"zz =duzdz
; "xz =1
2
duxdz
: (7.70)
Moreover, we neglect the inertia forces and treat the problem as elastostatic.As far the boundary conditions are concerned, the displacement vector vanishes at the bed,
and the surface traction vanishes at the upper free surface, that is,
ux = uz = 0 for z = 0 (7.71)
�xz = �zz = 0 for z = H: (7.72)
Since div~u = duz=dz, the stress components have the form
�xz = 2�"xz = �duxdz
;
�zz = �div~u+ 2�"zz = (�+ 2�)duzdz
:
In view of this, the boundary conditions (7.72) reduce to
duxdz
����z=H
=duzdz
����z=H
= 0 : (7.73)
Under the above simpli�cations, the x� and z-component of the Navier-Cauchy equation(7.55), that is,
(�+ �)@
@x(div~u) + �r2ux + %0fx = %0
@2ux@t2
;
(�+ �)@
@z(div~u) + �r2uz + %0fz = %0
@2uz@t2
;
115
where fx and fz are components of the constant gravity acceleration g,
fx = g sin� ; fz = �g cos� ; (7.74)
reduce to the form
d2uxdz2
+%0g sin�
�= 0 ;
d2uzdz2
� %0g cos�
�+ 2�= 0 : (7.75)
The double integration with respect to z yields a general solution of these equations:
ux(z) = �%0g sin�2�
z2 + C1z + C2 ; uz(z) =%0g cos�
2(�+ 2�)z2 + C3z + C4 ; (7.76)
where Ci, i = 1; : : : ; 4, are constants. The boundary condition (7.71) yields C2 = C4 = 0, while(7.73) implies that C1 = %0gH sin�=� and C3 = �%0gH cos�=(� + 2�). The solution for thedisplacement components can �nally be written in the form
ux(z) =%0gH sin�
�
�z � z2
2H
�; uz(z) = �%0gH cos�
�+ 2�
�z � z2
2H
�: (7.77)
We can see that the displacement components increase monotonically from zero values at thebase to their maximum values at the upper free surface. The parabolic displacement ux(z) withinthe slab is sketched in Figure 7.2. Note that the y-component of the Navier-Cauchy equation isidentically satis�ed. Hence, the problem is fully solved by (7.77).
��
xy
z
~g
ux
�xz
Figure 7.2. The displacement ux and stress �xz in the elastic plate deformed by its own weight.
We further determine the components of the stress tensor by Hooke's law (7.21) for a linearelastic, isotropic body:
�xx = �@uz@z
= � �
�+ 2�%0gH cos�
�1� z
H
�;
�yy = �xx ;
116
�zz = (�+ 2�)@uz@z
= �%0gH cos��1� z
H
�; (7.78)
�xy = �yz = 0 ;
�xz = �@ux@z
= %0gH sin��1� z
H
�:
The stress components are linear functions of the height of the elastic slab and vanish at theupper free surface. Figure 7.2 shows schematically the linear behaviour of stress �xz within theslab. It is interesting to note that the normal component �yy does not vanish even all the straincomponents vanish in y-direction.
In the particular case when the plate is not inclined (� = 0), equation (7.77) reduces to
ux(z) = 0 ; uz(z) = � %0gH
�+ 2�
�z � z2
2H
�; (7.79)
the upper free surface deforms due to the own weight of the plate by
uz(H) = � %0gH2
2(�+ 2�): (7.80)
For the Earth's lithosphere of 100 km thickness with material parameters � = 1:5 � 1011 N/m2,� = 0:67 � 1011 N/m2 and %0 = 3300 kg/m3, we have
uz(H) = �3300 kg=m3 � 9:81m=s2 � 1010m2
5:68 � 1011 N=m2 � �570m :
117
8. SMALL MOTIONS IN A MEDIUM WITH A FINITEPRE-STRESS
8.1 Equations for the initial state
We consider a body B to be composed of a number of solid and uid regions. We denote thevolume of solid regions by VS, and the volume of uid regions by VF. The entire volume of the bodywill be denoted by V = VS [ VF. The solid and uid regions are separated by non-intersecting,smooth and closed surfaces, called internal discontinuities. We denote all the internal weldedsolid-solid discontinuities between solid regions by �SS, and all the internal slipping uid-soliddiscontinuities between uid and solid regions by �FS. The union of all internal discontinuitieswill be denoted by � = �SS [ �FS. The exterior surface of B will be denoted by @V .
Let the body B occupy at time t = 0 the volume V in the con�guration �0, which we hereafteruse as the reference con�guration of the body. We suppose that this con�guration does notcorrespond to the natural, stress-free state of the body B but the body in the con�guration �0 ispre-stressed by a �nite stress in such a way that B is in a static equilibrium. In this con�guration,the Cauchy stress and the two Piola-Kirchho� stresses coincide; we denote this initial static stressby t0( ~X), where ~X denotes the position of a material particle in the con�guration �0. The staticequilibrium is guaranteed by the static linear momentum equation:
Div t0 + %0 ~f0 = 0 in V � � ; (8.1)
where %0( ~X) is the density of the body in the con�guration �0 and ~f0( ~X) is the body force perunit mass in �0. Since the body B contains uid regions, it is convenient to decompose the initialstatic stress t0 into the isotropic and deviatoric parts,
t0 = �p0I + tD0 ; (8.2)
where the pressure p0 = �13tr t0 and the trace of the deviatoric part vanishes, tr t
D0 = 0. Equation
of static equilibrium then transforms to the form
�Grad p0 +Div tD0 + %0 ~f0 = 0 in V �� : (8.3)
Since a uid is unable to support shear stresses when it is in the static equilibrium, the staticstress deviatoric tD0 vanishes in uid regions and the static linear momentum equation reducesto:
�Grad p0 + %0 ~f0 = 0 in VF � �FS : (8.4)
The internal discontinuities � within the body are assumed to be material surfaces. Theinterface condition at welded discontinuities between two solids and at slipping discontinuitiesbetween a solid and a uid is given by (4.46):
h~N � t0
i+�= ~0 on � ; (8.5)
where ~N is the unit outward normal to the discontinuity �. In addition, at a slipping discontinuitybetween a solid and a uid, there can be no shear stresses in the static equilibrium on the uid
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side and the stress vector ~N � t0 must be in the direction of the normal ~N ; the stress vector canbe expressed on the both sides of the discontinuity in the form (4.53):
~N � t0 = �p0 ~N on �FS ; (8.6)
where the initial pressure p0, p0 = �( ~N � t0 � ~N), passes through �FS continuously,
[p0]+� = 0 on �FS : (8.7)
On the outer free surface of B, the stress vector must vanish,~N � t0 = ~0 on @V : (8.8)
8.2 Application of an in�nitesimal deformation
We now consider a time and space dependent in�nitesimal deformation �eld, characterized bythe displacement ~u( ~X; t), superimposed upon the initial con�guration �0. The superimposeddisplacement �eld deforms the body into another time-dependent con�guration �t, which occu-pies volume Vt. Because of the smallness of the superimposed displacement �eld, the presentcon�guration �t is very closed to the reference con�guration �0. This motivates us to use thesame coordinates to describe the position of particles in both the con�gurations �0 and �t. Weadopt the Lagrangian description of motion and write the position of a material particle in thecon�guration �t in the form
~x = ~X + ~u( ~X; t) : (8.9)
We will apply the principle of geometrical linearization and develop �eld equations and interfaceconditions correct to �rst order in kHk, H = Grad ~u, and neglect all terms of higher order thanO(kHk). From mathematical point of view, the subsequent linearization process is self-consistent,and correct to �rst order in kHk, if the ratio of an incremental stress caused by the in�nitesimaldeformation (8.9) compared to the initial static stress is also of the order of O(kHk), see Section8.5.
8.3 Lagrangian and Eulerian increments
Time changes of any physical quantity Q that has a non-zero initial static value can be describedby the Eulerian (local) or Lagrangian (material) increments qE and qL, de�ned by
qE(~x; t) := q(~x; t)� q(~x; 0) ;
qL( ~X; t) := Q( ~X; t)� q( ~X; 0) ;(8.10)
where q(~x; t) and Q( ~X; t) are the Eulerian and Lagrangian descriptions of quantity Q at thepresent con�guration �t, respectively, and q(~x; 0) and q( ~X; 0) are the initial static values of Q atthe con�gurations �t and �0, respectively. The former are related by (1.22) and the latter by
q(~x; 0) = q( ~X + ~u; 0)
= q( ~X; 0) + ~u �Grad q( ~X; 0) +O(j~uj2) :(8.11)
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Note that the initial static values of Q at the con�guration �0 can be denoted by either q( ~X; 0)or Q( ~X; 0), that is q( ~X; 0) � Q( ~X; 0). Inserting (8.9){(8.11) into (1.22)1 yields
Q( ~X; t) = q(~x( ~X; t); t)
= q( ~X + ~u; 0) + qE( ~X + ~u; t)
= q( ~X; 0) + ~u �Grad q( ~X; 0) + qE( ~X + ~u; t) +O(j~uj2)= q( ~X; 0) + ~u �Grad q( ~X; 0) + qE( ~X; t) +O(j~uj2)!= q( ~X; 0) + qL( ~X; t) :
Hence, correct to �rst order in j~uj, the Lagrangian and Eulerian increments are related by
qL = qE + ~u �Grad q0( ~X) ; (8.12)
where q0( ~X) � q( ~X; 0). Note that we have dropped the dependence of qL and qE on the positions~X and ~x, respectively, since, in �rst-order theory, it is immaterial whether the increments qL andqE are regarded as functions of ~X or ~x.
8.4 Linearized continuity equation
Adopting the above concept, the Eulerian and Lagrangian increments in density are de�ned by
%E := %(~x; t)� %0(~x) ;
%L := %( ~X; t)� %0( ~X) ;(8.13)
where %(~x; t) and %( ~X; t) is the Eulerian and Lagrangian description of the density, respectively;they can be converted to each other by (1.22). The increments satisfy the �rst-order relation
%L = %E + ~u �Grad %0( ~X) : (8.14)
These two increments can be expressed in terms of displacement ~u by linearizing the Eulerianand Lagrangian conservation of mass laws. Inserting the decomposition (8.13)1 into the Euleriancontinuity equation (4.16) and integrating it with respect to time, we �nd that
%E = �Div [%0( ~X)~u] ; (8.15)
correct to �rst order in kHk. Inserting (1.101) together with (8.13)2 into the Lagrangian conser-vation of mass equation (4.60) yields
%L = �%0( ~X)Div ~u ; (8.16)
correct to the same order. It is easy to validate that the last two expressions for the Eulerian andLagrangian increments in density are consistent with the general expression (8.14).
8.5 Increments in stress
We now consider the state of stress in the present con�guration �t. The stress in �t referred tothe area element in the reference con�guration �0 is characterized by the �rst Piola-Kirchho�
120
stress tensor T (1) (see section 3.3). Since the initial static stress for T (1) (and also for the Cauchystress tensor t) is t0( ~X), we express T (1) in the form analogous to (8.10)2
T (1)( ~X; t) = t0( ~X) + T (1);L ; (8.17)
where T (1);L is the Lagrangian increment of the �rst Piola-Kirchho� stress tensor caused bythe in�nitesimal displacement ~u( ~X; t). We assume that the increment in stresses caused by thein�nitesimal deformation is also small, of the order of O(kHk:
kT (1);Lkkt0k � O(kHk) : (8.18)
The Lagrangian Cauchy stress tensor t is related to T (1) by (3.17)2:
t( ~X; t) = J�1F � T (1)( ~X; t) ; (8.19)
where J and F is the Jacobian and the deformation gradient of the in�nitesimal deformation(8.9), respectively. Correct to the �rst order in kHk, relation (8.19) can be expressed by (3.25)1:
t = (1� trH)T (1) +HT � T (1) +O(kHk2) : (8.20)
Substituting from (8.17) into (8.20) yields the decomposition:
t( ~X; t) = t0( ~X) + tL ; (8.21)
where tL is the Lagrangian increment of the Lagrangian Cauchy stress tensor t( ~X; t), which isrelated to the Lagrangian increment of the �rst Piola-Kirchho� stress tensor by the relation
tL = T (1);L � (trH)t0( ~X) +HT � t0( ~X) +O(kHk2) : (8.22)
Equations (8.17) and (8.22) can be combined to give
T (1)( ~X; t) = t0( ~X) + (trH)t0( ~X)�HT � t0( ~X) + tL +O(kHk2) : (8.23)
Therefore, the knowledge of displacement gradients, the initial stress distribution and the incre-ment in the Cauchy stress tensor determine the �rst Piola-Kirchho� stress tensor.
Beside the Lagrangian increment tL of the Cauchy stress tensor we can introduce the Eulerianincrement tE of the Cauchy stress tensor by de�nition
tE := t(~x; t)� t0(~x) ; (8.24)
where t(~x; t) is the Eulerian description of the Cauchy stress tensor. At a �xed point in space,the Lagrangian and Eulerian increments are related by the �rst-order relation (8.12):
tL = tE + ~u �Grad t0( ~X) : (8.25)
8.6 Linearized equation of motion
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The exact form of the equation of motion in the Lagrangian description is given by (4.67):
DivT (1) + %0 ~F = %0@2~u
@t2in V � � ; (8.26)
where ~F ( ~X; t) is the Lagrangian description of the body force per unit mass. Time changes of thebody force can be described by either the Eulerian increment ~fE or the Lagrangian increments~fL, de�ned by
~fE := ~f(~x; t)� ~f0(~x) ;
~fL := ~F ( ~X; t)� ~f0( ~X) ;(8.27)
where ~f(~x; t) is the Eulerian description of the body force, ~f(~x; t) = ~F ( ~X(~x; t); t). The incrementssatisfy the usual �rst-order relation:
~fL = ~fE + ~u �Grad ~f0( ~X) : (8.28)
To obtain the equation of motion in the Lagrangian increments, we substitute the representation(8.17) and (8.27)2 into the exact relation (8.26). Subtracting the static equilibrium equation(8.1), we obtain
DivT (1);L + %0 ~fL = %0
@2~u
@t2in V � � : (8.29)
This equation of motion is exactly valid in the initial con�guration �0 of the body, that is,everywhere in the volume V � �.
We shall now use this form for linearization in order to express the equation of motion interms of the Lagrangian increment of the Cauchy stress tensor, which is more convenient to bede�ned by the constitutive equation than the Lagrangian increment of the �rst Piola-Kirchho�stress tensor. We substitute (8.22) into (8.29), neglect the terms of second order in kHk, andobtain
Div tL +Div [(trH)t0]�Div (HT � t0) + %0 ~fL = %0
@2~u
@t2in V � � : (8.30)
By making use of the di�erential identities (A.14), (A.22) and (A.24), and recalling that trH =Div ~u, the second and the third term on the left-hand side of (8.30) can be simpli�ed as
Div [(trH)t0]�Div (HT � t0) = (trH)Div t0 �H :: Grad t0 :
In view of this and the static linear momentum equation (8.1), the equation of motion (8.30)becomes
Div tL + %0 ~fL + %L ~f0 �H :: Grad t0 = %0
@2~u
@t2in V � � : (8.31)
Decomposing the initial static stress t0 into the isotropic and deviatoric parts according to (8.2),the second term on the left-hand side of (8.31) can be arranged as
H :: Grad t0 = �H �Grad p0 +H :: Grad tD0 :
Eliminating Grad p0 by means of (8.3), the linearized form of the Lagrangian equation of motioncan �nally be written in the form
Div tL + %0(~fL +H � ~f0) + %L ~f0 �H :: Grad tD0 +H �Div tD0 = %0
@2~u
@t2in V � � : (8.32)
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In the uid regions tD0 = 0, and the equation of motion reduces to
Div tL + %0(~fL +H � ~f0) + %L ~f0 = %0
@2~u
@t2in VF � �FS : (8.33)
The equation of motion (8.31) can be rewritten explicitly in terms of the Eulerian incrementin the Cauchy stress. Applying operator Div on (8.25), using the di�erential identity (A.23) andsubstituting for Div t0 from the static momentum equation (8.1), we obtain
Div tL = Div tE +H :: Grad t0 � ~u �Grad (%0 ~f0) :
In view of this, the di�erential identity (A.2) and the relation (8.28) between the Lagrangian andEulerian increments in body force, the equation of motion (8.31) transforms to
Div tE + %0 ~fE + %E ~f0 = %0
@2~u
@t2; (8.34)
where %E is the Eulerian increment in density. Strictly speaking, this equation is valid at a point~x within the volume v(t)��(t) of the present con�guration �t of the body rather than at a point~X within the volume V � � of the initial con�guration �0. However, correct to �rst order inkHk, this distinction is immaterial.
The �nal interesting form of the equation of motion in this context is
Div tL +Grad (%0~u � ~f0) + %0 ~fE + %E ~f0 +Grad (~u �Div tD0 )�Div (~u �Grad tD0 ) = %0
@2~u
@t2; (8.35)
which can be derived from (8.32) by making use of the identity
�H :: Grad tD0 +H �Div tD0 = Grad (~u�Div tD0 )�Div (~u�Grad tD0 )+Grad (%0 ~f0)�~u�~u�Grad (%0 ~f0):(8.36)
The most suitable forms of the equation of motion valid within the volume V �� are summarizedfor convenience in Table 8.1.
Field variables Linearized equation of motion
T (1);L, ~fL, ~u DivT (1);L + %0 ~fL = %0
@2~u@t2
(exact)
tL, ~fL, ~u Div tL + %0 ~fL + %L ~f0 �Grad ~u :: Grad t0 = %0
@2~u@t2
Div tL + %0(~fL +Grad ~u � ~f0) + %L ~f0
�Grad~u :: Grad tD0 +Grad ~u � Div tD0 = %0@2~u@t2
tL, ~fE, ~u Div tL + %0(~fE + ~u �Grad ~f0) + %L ~f0 �Grad ~u :: Grad t0 = %0
@2~u@t2
Div tL +Grad (%0~u � ~f0) + %0 ~fE + %E ~f0
+Grad (~u � Div tD0 )�Div (~u �Grad tD0 ) = %0@2~u@t2
tE , ~fE, ~u Div tE + %0 ~fE + %E ~f0 = %0
@2~u@t2
Table 8.1. Summary of various forms of the equation of motion.
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8.7 Linearized interface conditions
The linearized equation of motion must be supplemented by the kinematic and dynamic interfaceconditions at the internal discontinuities.
8.7.1 Kinematic interface conditions
The kinematic interface condition on the welded solid-solid discontinuities is given by (4.49). Wetransform the Eulerian form (4.49) to the Lagrangian form and obtain
[~V ]+� = ~0 on �SS ; (8.37)
where ~V ( ~X; t) := ~v(~x( ~X; t); t) is the Lagrangian description of the velocity. We substitute for~V ( ~X; t) from (2.5)1 and integrate the result with respect to time. Choosing ~u( ~X; 0) = ~0, theinterface condition (8.37) transforms to
[~u]+� = ~0 on �SS : (8.38)
Note that the interface condition (8.38) is exact, like the incremental Lagrangian linear momentumequation (8.29).
On the uid-solid discontinuities, tangential slip is allowed and, hence, the interface condition(4.49) must be replaced by (4.51). Considering the transformation (1.123) between the unitnormal ~n to the deformed discontinuity � and the unit normal ~N to the undeformed discontinuity� and integrating (4.51) with respect to time, we obtain
[ ~N � ~u]+� = 0 on �FS ; (8.39)
correct to �rst order in kHk. This equation is the �rst-order condition that guarantees that thereis no separation or interpenetration of the material on either side of the uid-solid discontinuity�FS.
8.7.2 Dynamic interface conditions
The dynamic interface condition on the welded boundaries �SS can readily be obtained fromthe interface condition (4.69). Realizing that �SS are material non-slipping discontinuities acrosswhich the initial surface element dA changes continuously, scalar factor dA can be dropped from(4.69) and we have
[ ~N � T (1)]+� = ~0 on �SS : (8.40)
Substituting for T (1) from (8.17) and subtracting the static interface condition (8.5), we obtain
[ ~N � T (1);L]+� = ~0 on �SS : (8.41)
The corresponding condition on the outer free surface @V is simply
~N � T (1);L = ~0 on @V : (8.42)
Both (8.41) and (8.42) are exact, like the kinematic interface condition (8.38).The dynamic interface condition on the slipping uid-solid discontinuities �FS requires more
124
consideration. Figure 8.1 shows a small portion of a slipping boundary; ~N+dA+ and ~N�dA�
are two initial elements of surface areacentered on particles ~X+ and ~X� lyingon the upper and lower side of an un-deformed discontinuity �FS. Upon de-formation, both particles ~X+ and ~X�
move to the same point ~x and the sur-face elements ~N+dA+ and ~N�dA�
merge to form the continuous element~nda on the deformed discontinuity �.Since �FS are material discontinuities,the interface condition (4.69) reducestoh
~NdA � T (1)i+�= ~0 on �FS :
(8.43)Substituting for T (1) from (8.17) and for dA from (1.127), this exact condition can be written,correct to �rst order in kHk, in the form
�~X+
dA+
~N+
~X�
dA�
~N�
~x da
~n
[~u]+�
~u+ ~u�
�FS
�FS
Figure 8.1. A small portion of a uid-solid disconti-nuity before (�FS) and after (�FS) deformation.
h~N � (t0 + T (1);L)(1�Div�~u)da
i+�= ~0 on �FS : (8.44)
where da = da( ~X; t) is the Lagrangian description of the deformed surface element of the dis-continuity �. The continuity of the Eulerian surface element da(~x; t) on �, that is, the condition[da(~x; t)]+� = 0 on �, implies the continuity of the Lagrangian surface element da( ~X; t) on �, that
is, the conditionhda( ~X; t)
i+�= 0 on �. Consequently, the factor da can be dropped from the
condition (8.44). In addition, since the initial static stress vector ~N � t0 is continuous, the o�setquantities ~N+ � t+0 and ~N� � t�0 are related by
~N+ � t+0 = ~N� � t�0 � [~u]+� �Grad�( ~N � t0) on �FS ; (8.45)
where the superscripts � denote evaluation at ~X�. Correct to �rst order in j~uj, it is immaterialwhether the tensor Grad�(
~N � t0) is taken at ~X�. Using (8.45) and the di�erential identity(B.18)5, equation (8.44) can be put into the formh
~N � T (1);L �Div� [~u ( ~N � t0)]i+�= ~0 on �FS : (8.46)
This linearized interface condition guarantees the continuity of stress vector across a slippingboundary. At a welded discontinuity, which experiences zero slip, the displacement ~u is continuousand condition (8.46) reduces to (8.41).
In addition, on the uid-solid boundaries, the initial stress vector is of the form (8.6). Usingthis, (8.46) reduces toh
~N � T (1);L +Div�(p0~u ~N)i+�= ~0 on �FS ; (8.47)
which, by making use of the di�erential identity (B.18)5, can also be written ash~N � T (1);L + ~N Div�(p0~u) + p0~u �Grad� ~N
i+�= ~0 on �FS : (8.48)
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The kinematic condition (8.39) requires the normal component of displacement ~u to be continuousacross �FS. This means that the changes in ~N � ~u must be equally large on both sides of thediscontinuity �FS, that is, h
Grad�(~N � ~u)
i+�= ~0 on �FS : (8.49)
Employing the identity (B.18)2 and the symmetry of surface curvature tensor, (Grad�~N)T =
Grad�~N , equation (8.49) can be considered in the form
hGrad�~u � ~N + ~u �Grad� ~N
i+�= ~0 on �FS : (8.50)
If we substitute this form for the last term in the condition (8.48) and use the continuity of thepressure p0, the condition (8.48) can be written in the alternative formh
~N � T (1);L + ~N Div�(p0~u)� p0Grad�~u � ~Ni+�= ~0 on �FS : (8.51)
This is the most useful form of the linearized dynamical continuity condition on �FS.If the uid-solid boundary is, in addition, frictionless, the Cauchy stress vector ~n � t must be
in the direction of normal ~n to the deformed uid-solid boundary �FS , that is, its projection on�FS must vanish:
~n � t � (I � ~n ~n) = ~0 on �FS : (8.52)
Making use of (1.123), (8.21) and (8.22), the Eulerian Cauchy stress vector ~n � t at the deformeddiscontinuity �FS can be expressed in terms of the �rst Piola-Kirchho� stress tensor on theundeformed discontinuity �FS. Correct to �rst order in kHk, it holds~n � t = [ ~N + ( ~N �H � ~N) ~N �H � ~N ] � [t0 + T (1);L � (trH)t0 +HT � t0]
= ~N � t0 + ~N � T (1);L � (trH)( ~N � t0)+ ~N � (HT � t0)+( ~N �H � ~N)( ~N � t0)�(H � ~N) � t0= ~N � t0 + ~N � T (1);L � (trH � ~N �H � ~N)( ~N � t0) ;
where we have abbreviated t0 � t0( ~X) and used the identity ~N � (HT � t0) = (H � ~N) � t0. Thelast term in the above equation can further be expressed in terms of the surface divergence of thedisplacement ~u, see (1.125):
~n � t = ~N � t0 + ~N � T (1);L � (Div�~u)(~N � t0) on �FS : (8.53)
The dyadic product ~n ~n occurring in the interface condition (8.52) can be written, correct to�rst order in kHk, in the form
~n ~n = ~N ~N � (Grad�~u � ~N) ~N � ~N (Grad�~u � ~N) : (8.54)
Substituting (8.53) and (8.54) into (8.52) yields
[ ~N �t0+ ~N �T (1);L�(Div�~u)( ~N �t0)]�(I� ~N ~N)+( ~N �t0)�[(Grad�~u� ~N) ~N+ ~N(Grad�~u� ~N)] = ~0 ;
which is valid on �FS. At this discontinuity, the initial stress vector ~N � t0 has the form (8.6),which helps to reduce the last equation:
( ~N � T (1);L) � (I � ~N ~N)� p0(Grad�~u � ~N) = ~0 :
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Using the identity (B.33), we can also write
[ ~N � T (1);L � p0(Grad�~u � ~N)] � (I � ~N ~N) = ~0 :
Finally, we can add the term ~N Div�(p0~u) � (I � ~N ~N) = ~0 for convenience:
[ ~N � T (1);L + ~N Div�(p0~u)� p0(Grad�~u � ~N)] � (I � ~N ~N) = ~0 : (8.55)
Correct to �rst order in kHk, this equation guarantees that there is no shear stress vector on the uid-solid boundary �FS.
By comparing condition (8.51) and (8.55), we see that the quantity
~� (1);L := ~N � T (1);L + ~N Div�(p0~u)� p0(Grad�~u � ~N) (8.56)
is a continuous, normal vector on �FS:h~� (1);L
i+�= ~0 ; (8.57)
~� (1);L = ( ~N � ~� (1);L) ~N : (8.58)
At a solid-solid discontinuity �SS, the last two term in the de�nition (8.56) of ~� (1);L are continuousand the condition (8.54) coincides with the condition (8.41). Therefore, frictionless uid-solid andwelded solid-solid discontinuities are distinguished by the normality condition (8.58) only.
All above derived condition can be expressed in terms of the incremental Lagrangian Cauchystress vector ~N � tL rather than ~N � T (1);L using the relation (8.22). The left scalar product of(8.22) with the normal ~N yields
~N � T (1);L = ~N � tL + (trH)( ~N � t0)� (t0 �H) � ~N ;
where we have used the identity ~N � (HT � t0) = (t0 �H) � ~N which follows from the symmetryof t0. Decomposing the initial stress into the isotropic and deviatoric parts, see (8.2), and using(B.30) and (B.31) results in
~N � T (1);L = ~N � tL + p0(Grad�~u � ~N � ~N Div�~u) + (trH)( ~N � tD0 )� (tD0 �H) � ~N ; (8.59)
which is valid for both the solid-solid interface �SS and the uid-solid interface �FS. Moreover,the auxiliary vector ~� (1);L, which is de�ned at the uid-solid interface �FS, can be expressed inthe form
~� (1);L = ~N � tL + (Grad�p0 � ~u) ~N : (8.60)
The complete set of the interface conditions on � is summarized for convenience in Table 8.2.
127
Discontinuity type Exact and linearized interface condition
@V : free surface ~N � T (1);L = ~0 (exact)
�SS: solid-solid [~u]+� = ~0 (exact)
[ ~N � T (1);L]+� = ~0 (exact)
�FS: uid-solid [ ~N � ~u]+� = 0
[~� (1);L]+� = [ ~N � ~� (1);L]+� ~N = ~0
~� (1);L := ~N � T (1);L + ~N Div�(p0~u)� p0(Grad�~u � ~N)
Table 8.2. Summary of exact and linearized interface conditions.
8.8 Linearized elastic constitutive equation
All of the equations we have derived so far can be regarded as linearized balance laws of ei-ther geometry or physics. They are valid regardless of the constitution of the material. Tocomplete these equations, we specify the constitutive relation between the incremental stressesand displacement gradient. Here we will assume that the material behavior for the in�nitesimalsuperimposed deformation is linear and elastic, as discussed in section 7.1.
The linearized constitutive equations for the incremental �rst Piola-Kirchho� stress T (1);L
the incremental Lagrangian Cauchy stress tL can be obtained from (7.16) and (7.17) using thedecompositions (8.17) and (8.21):
T (1);L = C :: ~E � t0 � ~R+ a[(tr ~E)t0 + (t0 :: ~E)I ] + (b+ c)( ~E � t0)+ (b+ c+ 1)(t0 � ~E) ; (8.61)
tL = C :: ~E + ~R � t0 � t0 � ~R+ (a� 1)(tr ~E)t0 + a(t0 :: ~E)I
+ (b+ c+ 1)(t0 � ~E + ~E � t0) : (8.62)
Every choice of the scalars a, b and c de�nes the behavior of a linear elastic solid. The mostconvenient alternative, adopted by Dahlen and Tromp (1998), is
a = �b = �c = 1
2: (8.63)
Under this choice, the equation (8.61) and (8.62) reduce to
T (1);L = C :: ~E � t0 � ~R� ~E � t0 + 1
2(tr ~E)t0 +
1
2(t0 :: ~E)I ; (8.64)
tL = C :: ~E + ~R � t0 � t0 � ~R� 1
2(tr ~E)t0 +
1
2(t0 :: ~E)I : (8.65)
By the decomposition (8.2) of the initial static stress into the isotropic and deviatoric parts, wealso have
T (1);L = C :: ~E + p0[ ~E + ~R� (tr ~E)I ]� tD0 � ~R� ~E � tD0 +1
2(tr ~E)tD0 +
1
2(tD0
:: ~E)I ; (8.66)
tL = C :: ~E + ~R � tD0 � tD0 � ~R� 1
2(tr ~E)tD0 +
1
2(tD0
:: ~E)I : (8.67)
128
It deserves to note that the incremental Lagrangian Cauchy stress tL does not depend explicitlyon the initial pressure p0, but only on the elastic tensor C and the initial deviatoric stress tD0 . Infact, this is what motivated us to set a = �b = �c = 1=2 in equations (8.61) and (8.62). This isthe only choice that completely eliminates the explicit dependence of tL on p0.
8.9 Gravitational potential theory
So far we have not speci�ed a body force acting in body B. We will now assume that the Newtongravitational attraction is the body force acting in the body B; we call B a self-gravitating body.Since the Newton gravitation is a conservative force, the Eulerian gravitational force per unitmass ~g(~x; t) is deriveable from an Eulerian gravitational potential �(~x; t):
~g = �grad� : (8.68)
Given an instantaneous Eulerian density distribution %(~x; t), we can �nd �(~x; t) by the Newtonintegral:
�(~x; t) = �GZv(t)
%(~x0; t)
k~x� ~x0k dv0 ; (8.69)
whereG is Newton's gravitational constant and the primes denote the dummy point of integration.The integration in (8.69) is carried out over the volume v(t) which is �lled in by a material bodyat time t, that is over all point ~x0 where %(~x0; t) > 0. By combining (8.68) and (8.69), we obtainthe gravitational attraction ~g(~x; t):
~g(~x; t) = �GZv(t)
%(~x0; t)(~x� ~x0)k~x� ~x0k3 dv0 : (8.70)
It can be veri�ed that the gravitational potential satis�es Poisson's equation:
div grad� = 4�G% in v(t)� �(t) : (8.71)
At the internal discontinuities across which the density % has a jump, the Poisson equation (8.71)must be supplemented by the interface conditions
[�]+� = 0 on �(t) ; (8.72)
[~n � grad�]+� = 0 on �(t) : (8.73)
Note that the continuity of the scalars � and (~n�grad�) implies the continuity of the gravitationalattraction vector ~g = �grad�.
Classical gravitational potential theory, as summarized above, is an inherently Eulerian theory.The quantities � and ~g are the gravitational potential and gravitational attraction at a �xedpoint ~x in space. The corresponding Lagrangian variables can, however, readily be found fromthe relation
�( ~X; t) := �(~x( ~X; t); t) ~G( ~X; t) := ~g(~x( ~X; t); t) : (8.74)
Using (1.77) and (4.60), we transform (8.69) and (8.70) into integrals over the correspondingreference volume V,
�( ~X; t) =�GZV
%0( ~X0)
k~x( ~X; t)� ~x( ~X 0; t)k dV0 ; (8.75)
~G( ~X; t) =�GZV
%0( ~X0)[~x( ~X; t)� ~x( ~X 0; t)]
k~x( ~X; t)� ~x( ~X 0; t)k3 dV 0 : (8.76)
129
Making use of (1.43)3, it is straightforward to rewrite the Poisson equation (8.71) in the La-grangian form:
F�1 � F�T :: GradGrad� + divF�T �Grad� = 4�G% in V �� ; (8.77)
where %( ~X; t) is the Lagrangian description of the density %(~x; t). To transform the interfaceconditions (8.72) and (8.73) to the undeformed discontinuity �, we apply (1.43)1 and (1.75) andobtain:
[�]+� = 0 on � ; (8.78)"~N � F�1 � F�Tp
~N �B � ~N�Grad�
#+�
= 0 on � : (8.79)
8.9.1 Equations for the initial state
Let the density %0( ~X) in the initial con�guration �0 generates the initial gravitational potential�0( ~X), and let
~g0 = �Grad�0 (8.80)
denote the corresponding initial gravitational attraction. The two �elds �0 and ~g0 are given interms of %0 by
�0( ~X) = �GZV
%0( ~X0)
k ~X � ~X 0k dV0 ; (8.81)
and
~g0( ~X) = �GZV
%0( ~X0)( ~X � ~X 0)
k ~X � ~X 0k3 dV 0 : (8.82)
The density %0 is assumed to vanish outside the body, but �0 and ~g0 are both non-zero everywhereoutside and inside the body. The potential �0 satis�es Poisson's equation
13
r2�0 = 4�G%0 in V � � ; (8.83)
together with the interface conditions at the internal discontinuities across which the density %0has a jump:
[�0]+� = 0 on � ; (8.84)h
~N �Grad�0i+�
= 0 on � : (8.85)
In the region outside the body, the potential is harmonic,
r2�0 = 0 outside V : (8.86)
8.9.2 Increments in gravitation
13r2 � DivGrad ; we do not introduce a nabla operator for the Eulerian operator div grad .
130
The Eulerian and Lagrangian increments in gravitational potential are de�ned by
�(~x; t) =: �0(~x) + �E ;
�( ~X; t) =: �0( ~X) + �L ;(8.87)
where �(~x; t) and �( ~X; t) is the Eulerian and Lagrangian description of the gravitational potential,respectively; they are related to each other by (8.74)1. The increments satisfy the usual �rst-orderrelation
�L = �E + ~u �Grad�0( ~X) : (8.88)
Applying the operator �grad on (8.87)1, the Eulerian gravitation attraction can be arranged asfollows:
~g(~x; t) = �grad�(~x; t) = �grad�0(~x)� grad�E = ~g0(~x)� grad�E = ~g0(~x)�Grad�E ;
where, in the last step, we have neglected the term H � Grad�E, as can be seen from (1.114)1.De�ning the Eulerian increment in gravitation in accordance with (8.10)1,
~gE := ~g(~x; t)� ~g0(~x) ; (8.89)
the above result shows that ~gE is expressible as the gradient of the corresponding potentialincrement:
~gE = �Grad�E : (8.90)
The corresponding result for ~gL is more complicated. Writing the Lagrangian description ofthe gravitational attraction in the form
~G( ~X; t) = ~g(~x( ~X; t); t) = ~g( ~X + ~u; t) = ~g0( ~X + ~u; t) + ~gE = ~g0( ~X) + ~u �Grad~g0( ~X) + ~gE ;!= ~g0( ~X) + ~gL ;
it shows that the two increments in gravitation are related by
~gL = ~gE + ~u �Grad~g0( ~X) (8.91)
in agreement with the general relation (8.12). Inserting for ~gE from (8.90) and for ~g0( ~X) from(8.80), we obtain
~gL = �Grad�E � ~u �GradGrad�0( ~X) : (8.92)
On the other hand, applying the operator Grad on (8.88) and using the identity (A.3), we obtain
Grad�L = Grad�E +Grad~u �Grad�0( ~X) + GradGrad�0( ~X) � ~u :
Combining the last two relations and using the symmetry of tensor GradGrad�0, the Lagrangianincrement in gravitation can be expressed in an alternative form
~gL = �Grad�L +Grad ~u �Grad�0( ~X) : (8.93)
The di�erence between the two relations (8.90) and (8.93) re ects the inherently Eulerian natureof classical gravitational �eld theory.
131
8.9.3 Linearized Poisson's equation
To obtain the linearized form of Poisson equation, we apply geometrical linearization (1.113)3 ofthe Laplace operator to the exact relation (8.71) and �nd that
r2�� 2H :: GradGrad��DivH �Grad� = 4�G% in V � � ; (8.94)
correct to the �rst order in kHk. Substituting for � from (8.87)2 and for % from (8.13)1,subtracting Poisson's equation (8.83) for the initial con�guration, we obtain
r2�L � 2H :: GradGrad�0 �DivH �Grad�0 = 4�G%L in V � � ; (8.95)
where %L is the Lagrangian increment in density. Making use of the di�erential identities (A.37)and (A.38), and the symmetry of tensor GradGrad�0, we can derive the identity
r2(~u �Grad�0) = 2H :: GradGrad�0 +DivH �Grad�0 + ~u �Grad (r2�0) : (8.96)
Substituting for the expression 2H :: GradGrad�0 + DivH � Grad�0 from (8.96) into (8.95),using Poisson's equation (8.83) for the initial potential, and considering (8.15) for the Lagrangianincrement in density, Poisson's equation for �L can be put into an alternative form
r2�L +r2(~u � ~g0) = �4�GDiv (%0~u) in V � � : (8.97)
The linearized Poisson equation can also be written in terms of the Eulerian increments inpotential and density. By inspection of (8.14), the right-hand side of (8.97) is proportional to theEulerian increment in density %E . Moreover, taking into account (8.89), �L = �E �~u �~g0, and weobtain
r2�E = 4�G%E in V � � : (8.98)
8.9.4 Linearized interface condition for potential increments
The exact continuity condition (8.78) for the potential on the undeformed discontinuity � can beexpressed in terms of the Lagrangian increment in potential ash
�0( ~X) + �Li+�= 0 on � : (8.99)
At a welded discontinuity �SS, which experiences zero slip, the initial potential �0 is continuousand the condition (8.99) reduces toh
�Li+�= 0 on �SS : (8.100)
At a uid-solid discontinuity �FS, which may exhibit a tangential slip, the o�set quantities �+0and ��0 are related by an expansion analogous to (8.45). Correct to �rst order in k~uk, it holds
�+0 = ��0 � [~u]+� �Grad��0 on �FS ; (8.101)
where the superscripts � denote evaluation at ~X�. Within the same accuracy, it is immaterialwhether Grad��0 is taken at ~X+ or ~X�. Substituting (8.101) into (8.99) yieldsh
�L � ~u �Grad��0i+�= 0 on �FS : (8.102)
132
Since the normal component of displacement and the normal component of the initial gravitationare continuous across �FS, we can subtract the term ( ~N � ~u)( ~N �Grad�0) from the left-hand sideof (8.102) for convenience and obtain
h�L � ~u �Grad�0
i+�= 0 on �FS : (8.103)
Note that equation (8.100) is exact whereas (8.103) is correct to �rst order in k~uk.The corresponding results for �E can be obtained from the relation (8.88) between the incre-
ments �E and �L. At a welded discontinuity �SS, we haveh�E + ~u �Grad�0
i+�= 0 :
Since the displacement and the initial gravitation are continuous on �SS, the interface conditionsimpli�es to h
�Ei+�= 0 on �SS : (8.104)
At a uid-solid discontinuity �FS, the interface condition (8.103) transforms to a condition ofthe same form as (8.104). Hence, for the both types of discontinuities, the Eulerian increment inpotential satis�es the interface condition of the form
h�Ei+�= 0 on � ; (8.105)
where � = �SS [ �FS.
8.9.5 Linearized interface condition for increments in gravitation
To obtain the linearized form of the interface condition for gravitation, we apply the geometricallinearization (1.100) of F�1 and the linearized equation (1.121) in the exact interface condition(8.79) and �nd that
h~N �Grad�� ~N �H �Grad��Grad� �H � ~N + ( ~N �H � ~N)( ~N �Grad�)
i+�= 0 on �
(8.106)correct to the �rst order in kHk. Upon inserting the decomposition (8.87)2 of the Lagrangiangravitational potential into (8.106), we �nd, correct to �rst order in k~uk that
h� ~N � ~g0 + ~N �Grad�L + ~N �H � ~g0 + ~g0 �H � ~N � ( ~N �H � ~N)( ~N � ~g0)
i+�= 0 ; (8.107)
where the initial gravitation ~g0 is introduced for convenience. On the other hand, equations (B5)and (B.28) can be combined to give
~g0 �H � ~N � ( ~N �H � ~N)( ~N � ~g0) = Grad�(~N � ~u) � (~g0)� ; (8.108)
where (~g0)� is the tangential part of ~g0. Since both the vectors on the right-hand side of (8.108)are continuous on �, we obtain
h~g0 �H � ~N � ( ~N �H � ~N)( ~N � ~g0)
i+�= 0 on � : (8.109)
133
In view of this, the interface condition (8.107) reduces to
h� ~N � ~g0 + ~N �Grad�L + ~N �H � ~g0
i+�= 0 on � : (8.110)
Note that the interface condition (8.110) is valid at all discontinuities �. At a welded discontinuity�SS, the normal component of the initial gravitation is continuous and the condition (8.110)further reduces to h
~N �Grad�L + ~N �H � ~g0i+�= 0 on �SS : (8.111)
At a uid-solid discontinuity �FS which admits a tangential slip, the quantities ~N� � ~g�0 arerelated by an expansion analogous to (8.101):
~N+ � ~g+0 = ~N� � ~g�0 � [~u]+� �Grad�( ~N � ~g0) on �FS ; (8.112)
correct to �rst order in k~uk. Substituting (8.112) into (8.110) yieldsh~N �Grad�L + ~N �H � ~g0 + ~u �Grad�( ~N � ~g0)
i+�= 0 on �FS : (8.113)
The last term in the brackets can be developed by means of the di�erential identity (B.18)2resulting in:
h~N �Grad�L + ~N �H � ~g0 + ~u �Grad�~g0 � ~N + ~u �Grad� ~N � ~g0
i+�= 0 on �FS : (8.114)
In view of (8.50) and the continuity of the initial gravitation, the last term in the brackets canbe rewritten in a slightly di�erent form
h~N �Grad�L + ~N �H � ~g0 + ~u �Grad�~g0 � ~N � ~g0 �Grad�~u � ~N
i+�= 0 on �FS : (8.115)
By combining (B.32) and the interface condition (8.111), it is readily to show that the last termin the brackets is continuous on � and can, thus, be dropped from the brackets, leaving
h~N �Grad�L + ~N �H � ~g0 + ~u �Grad�~g0 � ~N
i+�= 0 on �FS : (8.116)
The corresponding results for �E can be obtained from the relation (8.88) between the incre-ments �E and �L. Substituting (8.88) into (8.111) and employing the di�erential identity (A.3),we obtain h
~N �Grad�E � ~N �Grad~g0 � ~ui+�= 0 on �SS : (8.117)
Making use of the identity (B.37), the continuity of ~u, �0 and the normal component of ~g0 on�SS, and the Poisson's equation (8.83), we �nd that
h~N �Grad~g0 � ~u
i+�=h( ~N � ~u)Div~g0
i+�= �4�G[%0]+�( ~N � ~u) on �SS : (8.118)
The interface condition (8.118) can then be put into the form
h~N �Grad�E + 4�G%0( ~N � ~u)
i+�= 0 on �SS : (8.119)
134
At a uid-solid discontinuity �FS, condition (8.119), expressed in terms of �E , becomes
h~N �Grad�E � ~N �Grad~g0 � ~u+ ~u �Grad�~g0 � ~N
i+�= 0 on �FS : (8.120)
Because of the symmetry of the second-order tensor Grad~g0 = �GradGrad�0, we can write~N �Grad~g0 � ~u = ~u �Grad~g0 � ~N . Moreover, the identity (B.32), applied to the initial gravitationvector ~g0, yields
Grad~g0 � ~N �Grad�~g0 � ~N = ~N( ~N �Grad~g0 � ~N) : (8.121)
The interface condition for the expression ~N � Grad~g0 � ~N follows from the identity (B.37), thecontinuity of the initial potential and normal component of initial gravitation, and Poisson'sequation (8.83):
h~N �Grad~g0 � ~N
i+�= [Div~g0]
+� = �4�G[%0]+� on �FS : (8.122)
A combination of (8.123) and (8.124) results in
h~N �Grad~g0 � ~u� ~u �Grad�~g0 � ~N
i+�= �4�G[%0]+�( ~N � ~u) on �FS : (8.123)
In view of this, the interface condition (8.122) for gravitation at a discontinuity �FS takes the form(8.121), that is valid at a discontinuity �SS. In summary, for the both types of discontinuities,the linearized form of the interface condition for gravitation is
h~N �Grad�E + 4�G%0( ~N � ~u)
i+�= 0 on � : (8.124)
The complete set of linearized interface conditions at a density discontinuity is summarized forconvenience in Table 8.3.
135
Discontinuity type Linearized interface condition
@V : free surfaceh�Li+�= 0 (exact)h
�Ei+�= 0h
~N �Grad�Li+�� ~N �Grad~u� � ~g�0 = 0h
~N �Grad�Ei+�� 4�G%�0 (
~N � ~u�) = 0
�SS: solid-solidh�Li+�= 0 (exact)h
�Ei+�= 0h
~N �Grad�L + ~N �Grad ~u � ~g0i+�= 0h
~N �Grad�E + 4�G%0( ~N � ~u)i+�= 0
�FS: uid-solidh�L + ~u � ~g0
i+�= 0h
�Ei+�= 0h
~N �Grad�L + ~N �Grad ~u � ~g0 + ~u �Grad�~g0 � ~Ni+�= 0h
~N �Grad�E + 4�G%0( ~N � ~u)i+�= 0
Table 8.3. Summary of linearized gravitational interface conditions.
8.9.6 Boundary-value problem for the Eulerian potential increment
The gravitational increments �E and �L can be obtained as a function of particle displacement~u in two ways. Either by solving the boundary-value problem for linearized Poisson's equationsor by the linearization of the exact Newton integrals. We consider both approaches here anddemonstrate their equivalence.
The Eulerian potential increment �E is the solution to the linearized Poisson's equation
r2�E = 4�G%E in V � � ; (8.125)
subject to the interface conditions derived above:
h�Ei+�
= 0 on � ; (8.126)h~N �Grad�E
i+�
= �4�G[%0]+�( ~N � ~u) on � : (8.127)
Outside the body B, where %E = 0, the incremental potential is harmonic, r2�E = 0. Thesolution to the boundary-value problem (8.127){(8.130) is
�E = �GZV
%E 0
LdV 0 +G
Z�
[%00]+�(~N 0 � ~u 0)L
d�0 ; (8.128)
136
where L is the distance between the computation point ~X and a dummy point of integration ~X 0,
L := k ~X � ~X 0k : (8.129)
The �rst term in (8.130) accounts for the volumetric density distribution %E in V and the secondterm accounts for the surface mass distribution due to the normal displacement of the boundary�. Substituting %E 0 = �Div 0(%00~u 0) in (8.130) and applying Green's theorem,
ZV
Div 0~v 0
LdV 0 = �
Z�
[ ~N 0 � ~v 0]+�L
d�0 �ZV~v 0 �Grad 0
� 1L
�dV 0 ; (8.130)
which is valid for a di�erentiable vector function ~v, we �nd that the surface integrals cancel,leaving simply
�E = �GZV%00~u
0 �Grad 0� 1L
�dV 0 : (8.131)
By direct di�erentiation we �nd that
Grad 0� 1L
�=
~X � ~X 0
k ~X � ~X 0k3 : (8.132)
Equation (8.133) is the most convenient explicit analytical representation of the Eulerian potentialincrement �E as a linear function of the particle displacement ~u. The corresponding representationof the incremental Eulerian gravitation ~gE = �Grad�E can be written in the form
~gE = G
ZV%00(~u
0 ��)dV 0 ; (8.133)
where
� :=I
k ~X � ~X 0k3 �3( ~X � ~X 0) ( ~X � ~X 0)
k ~X � ~X 0k5 : (8.134)
8.9.7 Boundary-value problem for the Lagrangian potential increment
The Lagrangian potential increment �L satis�es the linearized Poisson's equation
r2�L = �r2(~u � ~g0)� 4�GDiv (%0~u) in V � � ; (8.135)
subject to the interface conditions derived above:h�Li+�
= 0 on �SS ; (8.136)h~N �Grad�L
i+�
= � ~N � [H ]+� � ~g0 on �SS ; (8.137)h�Li+�
= �[~u]+� � ~g0 on �FS ; (8.138)h~N �Grad�L
i+�
= � ~N � [H ]+� � ~g0 � [~u]+� �Grad�~g0 � ~N on �FS : (8.139)
Outside the body B, where %0 = 0 and ~u = ~0, the incremental potential is harmonic, r2�L = 0.The solution to the boundary-value problem (8.137){(8.141) is
�L =1
4�
ZV
r2 0(~u 0 � ~g0 0) + 4�GDiv 0(%00~u0)
LdV 0 +
1
4�
Z�SS
~N 0 � [H 0]+� � ~g0 0L
d�0
137
(8.140)
+1
4�
Z�FS
~N 0 � [H 0]+� � ~g0 0 + [~u 0]+� �Grad0�~g0 0 � ~N 0
Ld�0� 1
4�
Z�FS
([~u 0]+� �~g0 0)�~N 0 �Grad 0
� 1L
��d�0 :
The �rst two surface integrals on the right-hand side of (8.142) can be regarded as the gravitationalpotentials of single layers placed on �SS and �FS with surface densities 1
4�G(~N � [H ]+� � ~g0) and
14�G(
~N � [H ]+� �~g0+[~u]+� �Grad�~g0 � ~N), respectively. The last surface integral in (8.142) represents
the potential of a double layer placed on �FS with the density � 14�G [~u]
+� � ~g0.
Applying Green's theorem (8.132) on the volume integral, we obtain
�L = � 1
4�
Z�SS
~N 0 � [Grad 0(~u 0 � ~g0 0) + 4�G%00~u0]+�
Ld�0
+1
4�
Z�SS
~N 0 � [H 0]+� � ~g0 0L
d�0
� 1
4�
Z�FS
~N 0 � [Grad 0(~u 0 � ~g0 0) + 4�G%00~u0]+�
Ld�0
+1
4�
Z�FS
~N 0 � [H 0]+� � ~g0 0 + [~u 0]+� �Grad0�~g0 0 � ~N 0
Ld�0
� 1
4�
Z�FS
([~u 0]+� � ~g0 0)�~N 0 �Grad 0
� 1L
��d�0
� 1
4�
ZV[Grad 0(~u 0 � ~g0 0) + 4�G%00~u
0] �Grad 0� 1L
�dV 0 :
It is necessary to apply Green's theorem to each sub-volumes of the volume V separately, and addthe results, since the expression r2(~u �~g0)+ 4�GDiv (%0~u) in the integrand may possess di�erentjumps at the discontinuities �SS and �FS. Making use of the identity
Grad 0(~u 0 � ~g0 0) =H 0 � ~g0 0 +Grad 0~g00 � ~u 0 ;
and the continuity of the initial gravitation, the surface integrals containing the displacementgradientH 0 cancel. Moreover, the last contribution in the volume integral is exactly the Eulerianpotential increment �E referred to the particle position ~X. Hence, we have
�L = �E � 1
4�
Z�SS
~N 0 � [Grad 0~g0 0 � ~u 0 + 4�G%00~u0]+�
Ld�0
� 1
4�
Z�FS
~N 0 � [Grad 0~g0 0 � ~u 0 + 4�G%00~u0]+�
Ld�0
+1
4�
Z�FS
[~u 0]+� �Grad0�~g0 0 � ~N 0
Ld�0
� 1
4�
Z�FS
([~u 0]+� � ~g0 0)�~N 0 �Grad 0
� 1L
��d�0
� 1
4�
ZVGrad 0(~u 0 � ~g0 0) �Grad 0
� 1L
�dV 0 :
138
The surface integral over �SS and the �rst two surface integrals over �FS vanish because of theinterface condition (8.120) and (8.125), respectively. As a result, we have
�L = �E � 1
4�
Z�FS
([~u 0]+� � ~g0 0)�~N 0 �Grad 0
� 1L
��d�0 � 1
4�
ZVGrad 0(~u 0 � ~g0 0) �Grad 0
� 1L
�dV 0 :
(8.141)We now intend to apply the following Green's theorem to the volume integral on the right-hand
side of (8.144). For a di�erentiable scalar function �, it holdsZVGrad 0�0 �Grad 0
� 1L
�dV 0 = �
Z�[�0]+�
�~N 0 �Grad 0
� 1L
��d�0 �
ZV�0r2 0
� 1L
�dV 0 : (8.142)
Since
r2 0� 1L
�= �4�Æ( ~X � ~X 0) ; (8.143)
where Æ( ~X � ~X 0) is the three-dimensional delta function with the propertyZV�( ~X 0)Æ( ~X � ~X 0)dV 0 = �( ~X) ; (8.144)
the Green's theorem (8.144) is simpli�ed toZVGrad 0�0 �Grad 0
� 1L
�dV 0 = 4���
Z�[�0]+�
�~N 0 �Grad 0
� 1L
��d�0 : (8.145)
Applying (8.147) to the volume integral in (8.143), we �nd that the surface integrals over �FS
cancel, leaving
�L = �E � ~u � ~g0 + 1
4�
Z�SS
[~u 0 � ~g0 0]+��~N 0 �Grad 0
� 1L
��d�0 :
The surface integral over �SS vanishes because of the continuity of ~u and ~g0 on the discontinuity�SS. This equation becomes precisely �L = �E � ~u � ~g0 in agreement with the general relation(8.88). Substituting for �E from (8.133) and for ~g0 from (8.82), we �nally obtain
�L = �GZV%00(~u
0 � ~u) �Grad 0� 1L
�dV 0 : (8.146)
8.9.8 Linearized integral relations
Alternatively, we can obtain gravitational increments �L, ~gL, �E and ~gE by linearizing the exactNewton integrals (8.75) and (8.76). Considering (8.75), for example, we rewrite it in the form
�0( ~X) + �L = �GZV
%00
k ~X + ~u� ~X 0 � ~u 0k dV0 ; (8.147)
and expand the right-hand side in powers of ~u� ~u 0. The zeroth-order term is simply �0, whereasthe �rst-order term is
�L = �GZV
%00(~u0 � ~u) � ( ~X � ~X 0)
k ~X � ~X 0k3 dV 0 : (8.148)
139
which is identical to (8.148). The expansion of
~g0( ~X) + ~gL = �GZV
%00(~X + ~u� ~X 0 � ~u 0)
k ~X + ~u� ~X 0 � ~u 0k3 dV0 (8.149)
likewise leads to
~gL = G
ZV%00(~u� ~u 0) �� dV 0 ; (8.150)
which is identical to ~gL = ~gE + ~u �Grad~g0 = �Grad�E � ~u �GradGrad�0.
8.10 Equation of motion for a self-gravitating body
We now deduce the linearized equation of motion describing in�nitesimal deformation of a bodyinitially pre-stressed by its own gravitation. For this purpose, the general theory presented inSection 8.6 will be applied to a particular case that the gravitational force is the only force actingin the body. In particular, we substitute the initial gravitational attraction ~g0 for the initial bodyforce ~f0 in static linear momentum equation (8.1), that is
~f0 = �Grad�0 : (8.151)
The Lagrangian increments in body force �L, de�ned by (8.27)2, that occurs in the equation ofmotion will be replaced by the Lagrangian increments in gravitation ~gL which can be expressedeither in terms of the Eulerian increment in gravitational potential �E, see (8.92), or in terms ofthe Lagrangian increment in gravitational potential �L, see (8.93):
~fL = �Grad�E � ~u �GradGrad�0= �Grad�L +Grad ~u �Grad�0 : (8.152)
Likewise, the Eulerian increments in body force �E , de�ned by (8.27)1, is replaced by the Eulerianincrements in gravitation ~gE which can be expressed either in terms of the Eulerian increment ingravitational potential �E, see (8.90), or in terms of the Lagrangian increment in gravitationalpotential �L, see (8.88):
~fE = �Grad�E= �Grad�L + ~u �Grad�0 : (8.153)
All forms of the equation of motion derived in Section 8.6 can be expressed in terms of theLagrangian or Eulerian incremental gravitational potential using relations (8.153){(8.155). Sincethis substitution is straightforward, we shall not write explicitly the resulting relations.
140
Appendix A. Vector and tensor di�erential identities
Let � and be di�erentiable scalar �elds, ~u and ~v di�erentiable vector �elds, A and B dif-ferentiable second-order tensor �elds and I the second-order unit tensor. The following vectordi�erential identities can be veri�ed:
grad (� ) = � grad + grad� (A.1)
grad (�~u) = � grad ~u+ grad� ~u (A.2)
grad (~u � ~v) = grad ~u � ~v + grad~v � ~u (A.3)
grad (~u� ~v) = grad ~u� ~v � grad~v � ~u (A.4)
grad (~u ~v) = grad ~u ~v + (~u grad~v)213(A.5)
= grad ~u ~v + (grad~v ~u)132~w � grad (~u ~v) = (~w � grad ~u) ~v + ~u (~w � grad~v) (A.6)
grad (�A) = � gradA+ grad�A (A.7)
grad (A � ~u) = gradA � ~u+ grad ~u �AT (A.8)
grad (~u �A) = grad ~u �A+ ~u � (gradA)213 (A.9)
grad~v � ~u� ~u � grad~v = ~u� rot~v (A.10)
grad ~u� (grad ~u)T = �I � rot ~u (A.11)
div (�~u) = �div ~u+ grad� � ~u (A.12)
div (~u� ~v) = rot ~u � ~v � ~u � rot~v (A.13)
div (~u ~v) = (div ~u)~v + ~u � grad~v (A.14)
div (�A) = �divA+ grad� �A (A.15)
div (�I) = grad� (A.16)
div (A � ~u) = divA � ~u+AT :: grad ~u (A.17)
div (~u �A) = grad ~u :: A+ ~u � divAT (A.18)
div (A� ~u) = divA� ~u+AT :� grad ~u (A.19)
div (~u�A) = rot ~u �A� ~u � rotA (A.20)
div (I � ~u) = rot ~u (A.21)
div rot ~u = 0 (A.22)
div (A �B) = divA �B +AT :: gradB (A.23)
div (~u � gradA) = grad ~u :: gradA+ ~u � grad divA (A.24)
div [(grad ~u)T ] = grad div~u (A.25)
rot (�~u) = � rot ~u+ grad�� ~u (A.26)
rot (~u� ~v) = ~v � grad ~u� ~u � grad~v + ~udiv~v � ~v div~u (A.27)
rot (~u ~v) = rot ~u ~v � ~u� grad~v (A.28)
rot (�A) = � rotA+ grad��A (A.29)
rot (�I) = grad�� I (A.30)
rot (~u� I) = rot (I � ~u) = (grad ~u)T � (div~u)I (A.31)
rot rot ~u = grad div~u� div grad ~u (A.32)
141
rot grad� = ~0 (A.33)
rot (grad ~u)T = (grad rot ~u)T (A.34)
I :: grad ~u = div~u (A.35)
I:� grad ~u = rot ~u (A.36)
r2(� ) = r2�+ �r2 + 2(grad�) � (grad ) (A.37)
r2(~u � ~v) = (r2~u) � ~v + ~u � r2~v + 2(grad ~u)T :: grad~v (A.38)
r2(grad�) = grad (r2�) (A.39)
r2(rot ~u) = rot (r2~u) (A.40)
r2(rot rot ~u) = rot rot (r2~u) : (A.41)
The symbols :: and:� denote, respectively, the double-dot product of vectors and the dot-cross
product of vectors. The symbols ( )T , ( )213 and ( )132 denote, respectively, the transpose of adyadic, the left transpose of a triadic and the right transpose of a triadic, e.g. (~u ~v)T = ~v ~u,(~u ~v ~w)213 = (~v ~u ~w), (~u ~v ~w)132 = (~u ~w ~v).
142
Appendix B. Fundamental formulae for surfaces
In a three-dimensional space, let � be an oriented surface with the unit normal ~n. Let (#1; #2)be the orthogonal curvilinear coordinates on � with the coordinate lines tangent to � in thedirections of the principal curvatures of � with principal curvature radii %1 and %2. If #1 and #2are chosen as angles, the elements of length on the surface in the principal curvature directionsare given by
ds� = %�d#� ; � = 1; 2 : (B.1)
The vectors de�ned by
~e� :=@~n
@#�; � = 1; 2 : (B.2)
are tangent to the curvilinear coordinate lines on the surface. Due to the de�nition of (#1; #2),the coordinate lines on the surface coincide with two lines of principal directions. Hence, ~e1 and~e2 are mutually perpendicular. In addition, they are perpendicular to ~n,
~n � ~e� = ~e1 � ~e2 = 0 : (B.3)
We assume that the surface � is suÆciently smooth such that, for instance, the curvature of �is bounded at all points of �. Because of this regularity assumption, we can imagine an auxiliarysurface � + d� that is parallel with � at a di�erential distance dn reckoned along the normal~n. A local three-dimensional orthogonal curvilinear coordinates (n; #1; #2) with the coordinate nmeasured in the direction of the normal ~n can be employed to describe the position of a point inthe layer bounded by surfaces � and � + d�. Consequently,
@~n
@n= ~0 (B.4)
at all points in the layer.
B.1 Tangent vectors and tensors
An arbitrary vector de�ned on an oriented surface � with the unit normal ~n can be decomposedinto a normal and a tangential part:
~u = ~nun + ~u� ; (B.5)
where un = ~n � ~u and ~u� = ~u � (I � ~n ~n). Obviously, ~n � ~u� = 0. The quantity un is the normalcomponent of ~u. Any vector ~u� with the property ~n � ~u� = 0 is referred to as a tangent vector to�. The component form of a tangent vector is
~u� =X�
u�~e� ; (B.6)
where the summation index � takes values 1 and 2. An arbitrary second-order tensor T can bedecomposed in an analogous manner:
T = (~n ~n)Tnn + ~n ~Tn� + ~T�n ~n+ T�� ; (B.7)
where Tnn = ~n � T � ~n. The quantities ~Tn� and ~T�n are tangent vectors satisfying ~n � ~Tn� =~n � ~T�n = 0, whereas T�� is a so-called tangent tensor, de�ned by ~n � T�� = T�� � ~n = ~0. Thetranspose of T is given by
T T = (~n ~n)Tnn + ~Tn� ~n+ ~n ~T�n + (T��)T : (B.8)
143
If T is a symmetric tensor, then ~T�n = ~Tn� and (T��)T = T��. The component form of tangent
vectors ~Tn�, ~T�n, and tangent tensor T�� are
~Tn� =X�
Tn�~e� ; ~T�n =X�
T�n~e� ; T�� =X��
T��(~e� ~e�) : (B.9)
The scalar product of two vectors is
~u � ~v = unvn + ~u� � ~v� ; (B.10)
and the left or right scalar product of a tensor and a vector is
T � ~u = ~n(Tnnun + ~Tn� � ~u�) + ~T�nun + T�� � ~u� ;
~u � T = ~n(Tnnun + ~T�n � ~u�) + ~Tn�un + ~u� � T�� :(B.11)
Obviously, it holds
T � ~n = ~nTnn + ~T�n ;
~n � T = ~nTnn + ~Tn� ;(B.12)
and~n � T � ~u� (~n � T � ~n)(~n � ~u) = ~Tn� � ~u� ;
~u � T � ~n� (~n � T � ~n)(~n � ~u) = ~T�n � ~u� :(B.13)
B.2 Surface gradient
The three-dimensional gradient operator can also be decomposed into a normal and a tangentialpart:
grad = ~n@
@n+ grad
�; (B.14)
where @@n
:= ~n � grad and ~n � grad�= 0. The tangential part grad
�is called the surface gradient
operator; its component form is
grad�=X�
~e�%�
@
@#�: (B.15)
Since grad�involves only di�erentiation in directions tangent to the surface �, it can be applied
to any scalar, vector or tensor �eld de�ned on �, whether that �eld is de�ned elsewhere or not.Likewise, the three-dimensional divergence operator can be decomposed into a normal and atangential part:
div = ~n � @@n
+ div� : (B.16)
The component form of the surface divergence operator is
div� =X�
~e�%�� @
@#�: (B.17)
B.3 Identities
144
Let � and be di�erentiable scalar �elds, ~u and ~v di�erentiable vector �elds and T a di�erentiablesecond-order tensor �eld. The following vector di�erential identities can be veri�ed:
grad�(� ) = � grad
� + grad
�� ;
grad�(~u � ~v) = grad
�~u � ~v + grad
�~v � ~u ;
grad�(�~u) = � grad
�~u+ (grad
��) ~u ; (B.18)
div�(�~u) = �div�~u+ grad�� � ~u ;
div�(~u ~v) = (div�~u)~v + ~u � grad�~v ;
div�(�T ) = �div�T + grad�� � T :
B.4 Curvature tensor
The surface gradient of the unit normal grad�~n, the so-called surface curvature tensor, is often
used to classify the curvature of the surface �. Combining (B.2) and (B.15), we obtain
grad�~n =
X�
1
%�(~e� ~e�) : (B.19)
Obviously, grad�~n is a symmetric tensor,
(grad�~n)T = grad
�~n : (B.20)
The surface divergence of the unit normal can be obtained from (B.2) and (B.17):
div�~n =1
%1+
1
%2=: 2c ; (B.21)
where c is the mean curvature of �. Another relation of interest in this context is
div�(~n ~n) = (div�~n)~n ; (B.22)
where the identity (B.18)5 has been used.
B.5 Divergence of vector, Laplacian of scalar
In applying the operators (B.15) and (B.17) to decomposed vectors and second-order tensorsof the form (B.5) and (B.7), it must be remembered that the surface gradient acts on the unitnormal ~n. For instance, the surface divergence of a vector �eld is given by
div�~u = div�(~nun + ~u�) = (div�~n)un + grad�un � ~n+ div�~u� ;
where the identity (B.18)4 has been used. In view of the orthogonality property (B.3)1, the secondterm on the right-hand side vanishes, and we obtain
div�~u = (div�~n)un + div�~u� : (B.23)
This enables us to express the three-dimensional divergence of a vector �eld as
div ~u = ~n � @~u@n
+ div�~u = ~n ��@~n@n
un + ~n@un@n
�+ div�~u :
145
In view of (B.4), this can be written in the form
div ~u =@un@n
+ div�~u : (B.24)
The equations derived above will now be employed to decompose the three-dimensional Lapla-cian operator of a three-dimensional scalar �eld �. Since the normal and tangential part of
gradient of � is @�@n
and grad��, respectively, (B.23 and (B.24) yield
div grad� =@2�
@n2+ (div�~n)
@�
@n+ div�grad�� ; (B.25)
where div�grad� is usually called the Beltrami surface operator.
B.6 Gradient of vector
The surface gradient of a decomposed vector �eld of the form (B.5) is given by
grad�~u = (grad
�un) ~n+ ungrad�~n+ grad
�~u� ; (B.26)
where the identity (B.18)3 has been used. This enables us to express the three-dimensionalgradient of a vector �eld in the form:
grad ~u = ~n @~u
@n+ grad
�~u = ~n
�@~n@n
un + ~n@un@n
+@~u�@n
�+ grad
�~u ;
or, in view of (B.4),
grad ~u = (~n ~n)@un@n
+ ~n @~u�@n
+ grad�~u : (B.27)
Combining (B.26) and (B.27), grad ~u can be put to a general form (B.7):
H := grad ~u
= (~n ~n)Hnn + ~n ~Hn� + ~H�n ~n+H�� ;(B.28)
whereHnn := @un
@n ; ~H�n := grad�un ;
~Hn� :=@~u�@n
; H�� := ungrad�~n+ grad�~u� :
(B.29)
Note that ~n � H�� = ~0, but, in general, H�� � ~n 6= ~0 since the vector @~e�=@#� may havea component in the direction normal to �. Equations (B.24) and (B.27) can alternatively bewritten in the form
div�~u = div ~u� (~n � grad ~u � ~n) ; (B.30)
grad�~u = grad ~u� ~n (~n � grad ~u) : (B.31)
The right scalar product of the last equation with the unit normal ~n results in an importantidentity
grad ~u � ~n� grad�~u � ~n = ~n(~n � grad ~u � ~n) : (B.32)
146
Obviously, ~n � (grad�~u � ~n) = 0, and vector grad
�~u � ~n is tangent to �. This implies that
grad�~u � ~n = (grad
�~u � ~n) � (I � ~n ~n) : (B.33)
B.7 Divergence of tensor
The surface divergence of a decomposed second-order tensor �eld of the form (B.7) is given by
div�T = div� [(~n ~n)Tnn + ~n ~Tn� + ~T�n ~n+ T��]
= Tnndiv�(~n ~n) + (grad�Tnn) � (~n ~n) + (div�~n)
~Tn� + ~n � grad�~Tn�
+(div�~T�n)~n+ ~T�n � grad�~n+ div�T�� ;
where the identity (B.19)5;6 have been used. The second and the fourth terms vanish because ofthe orthogonality property (B.3)1. Simplifying the �rst term according to (B.22), we obtain
div�T = Tnn(div�~n)~n+ (div�~T�n)~n+ (div�~n)
~Tn� + ~T�n � grad�~n+ div�T�� : (B.34)
B.8 Vector ~n � grad grad�
We are now interested in changing the order of di�erentiation of a scalar �eld � with respect ton and with respect to surface curvilinear coordinates #�. Taking the derivative with respect to nof (B.15) applied to �, we obtain
@
@n
�grad
���=X�
@
@n
�~e�%�
@�
@#�
�=X�
h @@n
�~e�%�
� @�@#�
+~e�%�
@2�
@n@#�
i:
Introducing tensor C�,
C� :=X�
%�@
@n
�~e�%�
� ~e� ; (B.35)
that classi�es the curvature of the surface �, we obtain
@
@n
�grad
���= grad
�
�@�@n
�+C� � grad�� ; (B.36)
which shows that the operators @@n and grad
�do not, in general, commute.
We are now ready to decompose vector (~n � grad grad�) into a normal and a tangential part.We have
~n � grad grad� =@
@n(grad�) =
@
@n
�~n@�
@n+ grad
���=@~n
@n
@�
@n+ ~n
@2�
@n2+
@
@n
�grad
���:
Making use (B.4) and (B.35), the last equation can be put to the form
~n � grad grad� = ~n@2�
@n2+ grad
�
�@�@n
�+C� � grad�� : (B.37)
The �rst term on the right-hand side of (B.37) is a normal part and the sum of the second andthe third terms is a tangential part of vector (~n � grad grad�), respectively. The �nal relation ofinterest in this context arises when the second derivative of � with respect to n is eliminated from(B.37) by means of (B.25):
~n �grad grad� = ~ndiv grad��(div�~n)@�
@n~n�~ndiv�grad��+grad
�
�@�@n
�+C� �grad�� : (B.38)
147
Appendix C. Orthogonal curvilinear coordinates
The concepts of strain, deformation, stress, general principles applicable to all continuous mediaand the constitutive equations were developed in Chapters 1 through 6 using almost exclusivelyrectangular Cartesian coordinates. This coordinate system is the simplest system in which thethree Cartesian unit base vectors have a �xed orientation in space. These coordinates, however,are suitable only for the solution of boundary-value problems involving bodies of rectangularshape. We often need to treat �nite con�gurations with spherical, spheroidal, ellipsoidal, oranother type of symmetric boundaries. We must therefore learn how to write the componentsof vectors and tensors in a wider class of coordinates. Here, we make a section on orthogonalcurvilinear coordinates.
C.1 Coordinate transformation
In a three-dimensional space of elementary geometry, we can de�ne a system of curvilinear coor-dinates by specifying three coordinate transformation functions xk of the reference system of therectangular Cartesian coordinates yl:
14
xk = xk(y1; y2; y3); k = 1; 2; 3: (C.1)
We assume that the three functions xk of yl have continuous �rst partial derivatives and that thejacobian determinant does not vanish, that is,
j := det
�@xk@yl
�6= 0; (C.2)
exceptions may occur at singular points or curves but never throughout any volume. Then thecorrespondence between xk and yl is one-to-one and there exists a unique inverse of (C.1) in theform
yk = yk(x1; x2; x3); k = 1; 2; 3: (C.3)
If x1 is held constant, the three equations (C.3) de�ne parametrically a surface, giving itsrectangular Cartesian coordinates as a function of the two parameters x2 and x3. The �rstequation in (C.1), x1 = x1(y1; y2; y3), de�nes the same surface implicitly. Similarly, for �xedvalues of x2 and x3 we obtain other two surfaces. The three surfaces so obtained are the curvilinearcoordinate surfaces. Each pair of coordinate surfaces intersect at a curvilinear coordinate line,along which only one of the three parameters xk varies. In contrast to the rectangular Cartesiancoordinates, the curvilinear coordinate lines are space curves. The three coordinate surfaces (andalso all three coordinate lines) intersect each other at a single point P marked with speci�c valuesof x1, x2 and x3. We may take the values of x1, x2 and x3 as the curvilinear coordinates of pointP (Figure C.1). If at each point P the coordinate lines through point P are mutually orthogonal,we have a system of orthogonal curvilinear coordinates.
Example. The spherical coordinates r; #; �, (r � x1; # � x2; � � x3), are de�ned by theirrelations to the rectangular Cartesian coordinates yk by equations
y1 = r sin# cos �; y2 = r sin# sin�; y3 = r cos#; (C.4)
14More generally, the coordinate transformation functions can be introduced between any two curvilinear systems.
148
�Oy1
y2
y3
~i1
~i2
~i3
P
~p
x1
x2
x3
~e1
~e2
~e3
Figure C.1. Curvilinear coordinates.
or, inversely,
r =qy21 + y22 + y23; # = arctan
qy21 + y22
y3
!; � = arctan
�y2y1
�: (C.5)
The ranges of values are 0 � r < 1, 0 � # � �, and 0 � � � 2�. The inverse of the jacobian jis given by
j�1 =
�������sin# cos � sin# sin� cos#
r cos# cos � r cos# sin� �r sin#�r sin# sin� r sin# cos � 0
������� = r2 sin#: (C.6)
Hence, the unique inverse of (C.4) exists everywhere except at r = 0, # = 0 and # = �. Thecoordinate surfaces are the concentric sphere r=const. centered at the origin, the circular cones#=const. centered on the y3-axis, and the half planes �=const. passing through the y3-axis. Ther-coordinate lines are half straights, the #-coordinate lines are circles, the meridians, and the�-coordinate lines are again circles, the parallels.
149
C.2 Base vectors
We now intend to introduce a set of base vectors ~ek in the curvilinear system. We observe thatif we move along a curvilinear coordinate line, only one of the three curvilinear coordinates varies.
For instance, along the x1-coordinate line only x1varies while x2 and x3 are kept �xed. In analogyto the de�nition of the base vectors in the rect-angular Cartesian coordinates, the unit base vec-tor ~e1 in curvilinear coordinates will be de�nedas the tangent vector to the x1-coordinate line.Similarly, ~e2 and ~e3 are the unit tangent vectorsto the curvilinear coordinate lines of varying x2and x3, respectively. The direction cosines of thebase vector ~e1 are given by (see Figure C.2)
cos� =1
h1
@y1@x1
; cos � =1
h1
@y2@x1
; cos =1
h1
@y3@x1
;�y1
y2
y3
�
�
~e1
x1-coordinateline
Figure C.2. The unit base vector ~e1.
where, for instance, � is the angle between the positive x1-direction and the positive y1-direction.A scale factor h1 is introduced to satisfy the orthonormality condition for the direction cosines,
cos2 �+ cos2 � + cos2 = 1:
In view of this, the scale factor is
h1 =
vuut 3Xl=1
�@yl@x1
�2:
Here and throughout this appendix, we will suspend the Einstein summation convention andexplicitly indicate all summations over the range (1,2,3) when it is necessary. The unit basevector ~e1 can now be written in the form
~e1 =1
h1
�@y1@x1
~i1 +@y2@x1
~i2 +@y3@x1
~i3
�;
where ~i1, ~i2 and ~i3 are the rectangular Cartesian unit base vectors. If we express the positionvector ~p of point P in terms of the rectangular Cartesian coordinates by
~p =3X
k=1
yk~ik; (C.7)
the curvilinear coordinate unit base vector ~e1 can be expressed in the form
~e1 =1
h1
@~p
@xk:
Similar consideration can be carried out for the unit base vectors ~e2 and ~e3. Hence, for k = 1; 2; 3,we have
~ek =1
hk
@~p
@xk: (no summation over k !) (C.8)
150
The function hk are called the scale factors, or the Lam�e coeÆcients. They are de�ned by relation
hk :=
s@~p
@xk� @~p@xk
: (C.9)
They are non-negative functions of positions for a given curvilinear coordinate system.In what follows, we shall consider only orthogonal curvilinear coordinates, and we will order
the three curvilinear coordinates so that when each base vector ~ek points in the direction ofincreasing xk, the three base vectors form a right-hand system:
~ek � ~el = Ækl; ~ek � ~el = "klm~em; (C.10)
where Ækl and "klm is the Kronecker delta and the Levi-Civita permutation symbol, respectively.An in�nitesimal vector at any point P can be expressed as
d~p =3X
k=1
@~p
@xkdxk =
3Xk=1
hk~ekdxk; (C.11)
where we have substituted for @~p=@xk from (C.8). The scalar product of this with itself gives thesquare of the distance between two neighboring points, that is, the in�nitesimal element of arclength on an arbitrary curve through point P :
(ds)2 = (h1dx1)2 + (h2dx2)
2 + (h3dx3)2: (C.12)
In particular, along the xk-coordinate line the only increment dxk di�ers from zero, while theothers are equal to zero. The elementary distance along the xk-coordinate line is then
dsk = hkdxk: (no summation over k): (C.13)
Note that three curvilinear coordinates xk need not be lengths and the scale factors hk may havedimensions. The products hkdxk must, however, have dimension of length.
From (C.13) we may immediately develop the area element of coordinate surface and thevolume element,
dakl = dskdsl = hkhldxkdxl; (C.14)
anddV = ds1ds2ds3 = h1h2h3dx1dx2dx3: (C.15)
Example. The scale factors in the spherical coordinates r; #; � are
hr = 1; h# = r; h� = r sin#: (C.16)
The square of the arc length is given by
(ds)2 = (dr)2 + r2(d#)2 + r2 sin2 #(d�)2: (C.17)
The unit vectors ~er, ~e# and ~e� in the direction of increasing r; #; � form a local right-handedorthonormal basis related to the rectangular Cartesian unit base vectors ~i1, ~i2 and ~i3 by
~er = sin# cos �~i1 + sin# sin�~i2 + cos#~i3;
~e# = cos# cos�~i1 + cos# sin�~i2 � sin#~i3; (C.18)
~e� = � sin�~i1 + cos �~i2:
151
Conversely,~i1 = sin# cos �~er + cos# cos �~e# � sin�~e�;
~i2 = sin# sin�~er + cos# sin�~e# + cos �~e�; (C.19)
~i3 = cos#~er � sin#~e#:
C.3 Derivatives of unit base vectors
Since the unit base vectors ~ek are functions of position, they vary in direction as the curvilinearcoordinates vary. Hence they cannot be treated as constants in di�erentiation. We shall nextevaluate the derivatives @~ek=@xl. Because this is a vector quantity, it can be represented as alinear combination of the base vectors ~em:
@~ek@xl
=3X
m=1
�mk l
�~em; (C.20)
where the expansion coeÆcients�mk l
�are known as the Christo�el symbols. Their meaning is
simply the mth components of the derivative of the kth unit base vector along the lth coordinate.Another common notation for the Christo�el symbols is
�mkl ��mk l
�;
but we will not use it in this text because the �mkl look like mixed components of a third-ordertensor, which they are not. 15
Each Christo�el symbol is a function of the scale factors only. To establish the explicit formof this functional dependence, we take the scalar product of (C.20) with the unit base vector~en, use the orthonormality relation (C.10)1, and denote the index n again by m. We obtain theexplicit expression for the Christo�el symbol:
�mk l
�=@~ek@xl
� ~em: (C.21)
By substituting for the unit base vectors from (C.8), we can successively write
�mk l
�=
@
@xl
�1
hk
@~p
@xk
�� 1
hm
@~p
@xm
= � 1
h2khm
@hk@xl
�@~p
@xk� @~p
@xm
�+
1
hkhm
@2~p
@xl@xk� @~p
@xm
!
= � 1
hk
@hk@xl
Ækm +1
hkhm
@2~p
@xl@xk� @~p
@xm
!;
15In general curvilinear coordinates, the Christo�el symbols of the �rst kind, [kl;m], and the Christo�el symbolsof the second kind, fkl;mg, are usually introduced. In orthogonal curvilinear coordinates, we can avoid the need
of the two kinds of Christo�el's symbols, and can operate only with the Christo�el symbol�mk l
�. It should be,
however, emphasized that neither the symbol [kl;m] nor the symbol fkl;mg does not reduce to the symbol�mk l
�when general curvilinear coordinates are orthogonal.
152
since the base vectors are orthogonal. Let us �nd an explicit form for the second term on theright-hand side. Di�erentiating the orthonormality relation
@~p
@xk� @~p
@xm= hkhmÆkm
with respect to xl, we have
@2~p
@xl@xk� @~p
@xm+
@~p
@xk� @2~p
@xl@xm=@(hkhm)
@xlÆkm:
By permuting indices in the last equation, we obtain other two equations,
@2~p
@xm@xl� @~p@xk
+@~p
@xl� @2~p
@xm@xk=@(hlhk)
@xmÆlk:
@2~p
@xk@xm� @~p@xl
+@~p
@xm� @2~p
@xk@xl=@(hmhl)
@xkÆml:
The last three equations can be combined to yield
@2~p
@xk@xl� @~p
@xm=
1
2
�@(hkhm)
@xlÆkm +
@(hmhl)
@xkÆml � @(hlhk)
@xmÆlk
�:
The Christo�el symbols can now be written in the form
�mk l
�= � 1
hk
@hk@xl
Ækm +1
2hkhm
�@(hkhm)
@xlÆkm +
@(hmhl)
@xkÆml � @(hlhk)
@xmÆlk
�:
Simple manipulation with this expression �nally results in
�mk l
�=
1
hk
@hl@xk
Ælm � 1
hm
@hk@xm
Ækl: (C.22)
If k, l and m are all di�erent, k 6= l 6= m, then
�mk l
�=� kk k
�=� kk l
�= 0: (C.23)
The last equality is the consequence of the fact that the vector @~ek=@xl is orthogonal to thexk-coordinate line and, thus, has no component in the direction of ~ek (but may have componentsin both directions orthogonal to ~el). Because of (C.23), at most 12 of the 27 Christo�el symbolsare non-zero: � l
k l
�=
1
hk
@hl@xk
;� lk k
�= � 1
hl
@hk@xl
if k 6= l: (C.24)
Of these, only six can be independent since it holds:
� lk l
�= �
� kl l
�: (C.25)
153
We have shown that the Christo�el symbols are fully de�ned in terms of the scale factors. Sincethe scale factors are constant in a Cartesian coordinate system (rectangular or skew), (C.24)shows that all the Christo�el symbols vanish in a Cartesian system.
Example. Only six Christo�el symbols are non-zero in the spherical coordinates r; #; �, namely:
� #r #
�= 1;
� �r �
�= sin#;
� �#�
�= cos#;
� r##
�= �1;
� r� �
�= � sin#;
� #��
�= � cos#:
(C.26)
The partial derivatives of the unit base vectors ~er, ~e# and ~e� are given by
@~er@r
= 0;@~e#@r
= 0;@~e�@r
= 0;
@~er@#
= ~e#;@~e#@#
= �~er; @~e�@#
= 0;
@~er@�
= ~e� sin#;@~e#@�
= ~e� cos#;@~e�@�
= �~er sin#� ~e# cos#:
(C.27)
C.4 Derivatives of vectors and tensors
With the formulae available for di�erentiating the unit base vectors we can write the partialderivatives of any vectors or tensors as follows.
Let ~v be an arbitrary vector represented in the form ~v =P
k vk~ek. Then the partial derivativeof ~v can be calculated by
@~v
@xl=
@
@xl
�Xk
vk~ek
�=Xk
�@vk@xl
~ek + vk@~ek@xl
�=Xk
"@vk@xl
~ek + vkXm
�mk l
�~em
#:
By interchanging the summation indices k andm in the last term we may write the vector @~v=@xlin a compact form,
@~v
@xl=Xk
vk;l~ek; (C.28)
where
vk;l :=@vk@xl
+Xm
� km l
�vm (C.29)
is the balanced or (neutral) derivative of vk with respect to xl.The partial derivatives of higher-order tensors are de�ned in a similar fashion. For example,
if we represent a second-order tensor T as a dyadic, that is, as the linear combination of the ninedyads formed from three curvilinear coordinate unit base vectors 16 ,
T =Xkl
Tkl(~ek ~el); (C.30)
16This is the Gibbs dyadic notation applied to orthogonal curvilinear coordinates.
154
the partial derivatives of T can then be calculated by
@T
@xm=
Xkl
�@Tkl@xm
(~ek ~el) + Tkl@~ek@xm
~el + Tkl~ek @~el@xm
�
=Xkl
"@Tkl@xm
(~ek ~el) + TklXn
� nkm
�(~en ~el) + Tkl
Xn
� nlm
�(~ek ~en)
#;
where we have substituted for the derivatives of the unit base vectors from (C.20). We interchangethe summation indices n and k in the second term and n and l in the last term to obtain
@T
@xm=Xkl
Tkl;m(~ek ~el); (C.31)
where
Tkl;m =@Tkl@xm
+Xn
� knm
�Tnl +
Xn
� lnm
�Tkn (C.32)
is the balanced or (neutral) derivative of Tkl with respect to xm. Higher-order tensors can betreated in the same way by writing them as polyadics.
C.5 Invariant di�erential operators
The results of previous section on derivatives of unit base vectors, arbitrary vectors and tensorswill now be used to formulate expressions for the gradient, divergence and curl in orthogonalcurvilinear coordinates.
C.5.1 Gradient of a scalar
We start to derive the basic di�erential expression for the gradient operator. Let �(x1; x2; x3) bea scalar function. Then the gradient of a scalar �, denoted by grad�, is de�ned as a product ofthe nabla operator with �,
grad� := r�: (C.33)
Note that this de�nition makes no reference to any coordinate system. The gradient of scalaris thus a vector invariantly independent of coordinate system. To �nd its components in anyorthogonal curvilinear system, we �rst express the nabla operator in the rectangular Cartesiancoordinates,
r =Xk
~ik@
@yk: (C.34)
The gradient of scalar function can then be calculated by the chain rule of calculus,
grad� =Xl
~il@�
@yl=Xkl
~il@�
@xk
@xk@yl
;
where � = �(xk(yl)) in the second term and � = �(xk) in the third term.Let us now �nd the explicit expression for @xk=@yl. In view of (C.7) and (C.8), the orthogo-
nality relation (C.10)1 has the form
Xm
@ym@xk
@ym@xl
= hkhlÆkl:
155
Multiplying this by @xl=@yn and summing the result over l, we obtain
Xm
@ym@xk
Xl
@ym@xl
@xl@yn
=Xl
hkhlÆkl@xl@yn
:
Because of the one-to-one mappings (C.1) and (C.3), it holds
Xl
@ym@xl
@xl@yn
= Æmn;
so that, we have@yn@xk
= h2k@xk@yn
: (C.35)
The gradient of a scalar � can now be arranged as follows
grad� =Xkl
1
h2k
@�
@xk
@yl@xk
~il =Xk
1
h2k
@�
@xk
@~p
@xk:
Substituting for the orthogonal curvilinear unit base vector ~ek, we �nally obtain
grad� =Xk
1
hk
@�
@xk~ek: (C.36)
Thus, the nabla operator can be expressed in an orthogonal curvilinear coordinate system in theform
r =Xk
~ekhk
@
@xk: (C.37)
Example. The gradient of scalar � in the spherical coordinates r; #; � is
grad� =@�
@r~er +
1
r
@�
@#~e# +
1
r sin#
@�
@�~e�: (C.38)
C.5.2 Divergence of a vector
The divergence of vector ~v is a scalar function, de�ned in any orthogonal curvilinear coordinatesystem as the scalar product of the nabla operator and vector ~v:
div~v := r � ~v: (C.39)
This de�nition can successively be arranged as follows.
div~v =Xk
~ekhk� @~v@xk
=Xk
~ekhk�Xl
vl;k~el =Xkl
vl;khk
Ækl =Xk
vk;khk
=Xk
1
hk
"@vk@xk
+Xm
� kmk
�vm
#
=Xk
1
hk
@vk@xk
+Xm
m 6= k
1
hm
@hk@xm
vm
!=
1
h1
@v1@x1
+1
h1h2
@h1@x2
v2 +1
h1h3
@h1@x3
v3
+1
h2
@v2@x2
+1
h2h3
@h2@x3
v3 +1
h2h1
@h2@x1
v1 +1
h3
@v3@x3
+1
h3h1
@h3@x1
v1 +1
h3h2
@h3@x2
v2;
156
or, in a more compact form,
div~v =1
h1h2h3
�@
@x1(h2h3v1) +
@
@x2(h3h1v2) +
@
@x3(h1h2v3)
�: (C.40)
Example. The divergence of vector ~v in the spherical coordinates r; #; � is
div~v =1
r2@
@r(r2vr) +
1
r sin#
@
@#(sin# v#) +
1
r sin#
@v�@�
: (C.41)
The divergence of the spherical unit base vectors is presented in the form
div~er =2
r; div~e# =
1
rcot#; div~e� = 0: (C.42)
C.5.3 Curl of a vector
The curl of vector ~v is a vector, de�ned by the cross-product of the r and ~v:
rot~v := r� ~v: (C.43)
We can successively write
rot~v =Xk
~ekhk� @~v
@xk=Xk
~ekhk�Xl
vl;k~el =Xklm
vl;khk
"klm~em =Xklm
"klmhk
"@vl@xk
+Xn
� ln k
�vn
#~em
=Xklm
"klmhk
"@vl@xk
+� lk k
�vk
#~em +
Xklm
"klmhk
Xn
n 6= k
� ln k
�vn~em =
Xklm
"klmhk
�@vl@xk
� 1
hl
@hk@xl
vk
�~em
=Xklm
�"klmhk
@vl@xk
+"lkmhkhl
@hk@xl
vk
�~em =
Xklm
�"klmhk
@vl@xk
+"klmhkhl
@hl@xk
vl
�~em;
or, in a more compact form,
rot~v =Xm
"Xkl
"klmhkhl
@(hlvl)
@xk
#~em: (C.44)
Thus, for example, the coeÆcient of ~e1 is
(rot~v)1 =1
h2h3
�@(h3v3)
@x2� @(h2v2)
@x3
�:
The other components can be written down by the cyclic permutation of indices. It is oftenconvenient to write the curl of a vector in determinant form:
rot~v =1
h1h2h3
���������
h1~e1 h2~e2 h3~e3@
@x1
@
@x2
@
@x3h1v1 h2v2 h3v3
���������: (C.45)
157
Example. The curl of vector ~v in the spherical coordinates r; #; � is
rot~v =
�1
r sin#
@(sin# v�)
@#� 1
r sin#
@v#@�
�~er+
�1
r sin#
@vr@�
� 1
r
@(r v�)
@r
�~e#+
�1
r
@(r v#)
@r� 1
r
@vr@#
�~e�;
(C.46)or, in determinant form,
rot~v =1
r2 sin#
���������
~er r ~e# r sin#~e�@
@r
@
@#
@
@�
vr r v# r sin# v�
���������: (C.47)
The curl of the spherical unit base vectors is presented in the form
curl~er = 0; curl~e# =1
r~e�; curl~e� =
1
rcot#~er � 1
r~e#: (C.48)
C.5.4 Gradient of a vector
The gradient of vector ~v is a non-symmetric, second-order tensor, de�ned by the left-dyadicproduct of the nabla operator with a vector,
grad~v := r ~v: (C.49)
In any orthogonal curvilinear coordinates, we can successively write,
grad~v =Xk
~ekhk @~v
@xk=Xk
~ekhkXl
vl;k~el =Xkl
1
hk
"@vl@xk
+Xm
� lmk
�vm
#(~ek ~el)
=Xk
1
hk
"@vk@xk
+Xm
� kmk
�vm
#(~ek ~ek) +
Xk
Xl
l 6= k
1
hk
"@vl@xk
+Xm
� lmk
�vm
#(~ek ~el)
=Xk
1
hk
@vk@xk
+Xm
m 6= k
1
hm
@hk@xm
vm
!(~ek ~ek) +
Xk
Xl
l 6= k
1
hk
@vl@xk
� 1
hl
@hk@xl
vk
!(~ek ~el):
Hence, the orthogonal curvilinear components of the second-order tensor grad~v are
(grad~v)kl =
8>>>>>><>>>>>>:
1
hk
@vk@xk
+Xm
m 6= k
1
hm
@hk@xm
vm
!if l = k;
1
hk
@vl@xk
� 1
hl
@hk@xl
vk
!if l 6= k:
(C.50)
The symmetric part of grad~v is the tensor 12 [grad~v + (grad~v)T ]; its components are given by
1
2[grad~v + (grad~v)T ]kl =
8>>>>><>>>>>:
1
hk
@vk@xk
+Xm
m 6= k
1
hm
@hk@xm
vm
!if l = k;
1
2
�1
hk
@vl@xk
+1
hl
@vk@xl
� 1
hkhl
@hk@xl
vk � 1
hkhl
@hl@xk
vl
�if l 6= k:
(C.51)
158
Example. The symmetric part of the gradient of vector ~v in the spherical coordinates r; #; �is
1
2[grad~v + (grad~v)T ] =
@vr@r
(~er ~er) + 1
r
�@v#@#
+ vr
�(~e# ~e#)
+1
r
�1
sin#
@v�@�
+ vr + cot # v#
�(~e� ~e�)
+1
2
�@v#@r
+1
r
@vr@#
� v#r
�(~er ~e# + ~e# ~er) (C.52)
+1
2
�@v�@r
+1
r sin#
@vr@�
� v�r
�(~er ~e� + ~e� ~er)
+1
2r
�@v�@#
+1
sin#
@v#@�
� cot# v�
�(~e# ~e� + ~e� ~e#):
C.5.5 Divergence of a tensor
The divergence of second-order tensor T is a vector, given in orthogonal curvilinear coordinatesas follows.
divT = r � T =Xm
~emhm
� @T@xm
=Xm
~emhm
�Xkl
Tkl;m(~ek ~el) =Xklm
Tkl;mhm
(~em � ~ek)~el =Xkl
Tkl;khk
~el
=Xkl
1
hk
"@Tkl@xk
+Xm
� kmk
�Tml +
Xm
� lmk
�Tkm
#~el
=Xkl
1
hk
"@Tkl@xk
+Xm
m 6= k
� kmk
�Tml +
� lk k
�Tkk +
Xm
m 6= k
� lmk
�Tkm
#~el
=Xkl
1
hk
"@Tkl@xk
+Xm
m 6= k
� kmk
�Tml
#~el +
Xl
Xk
k 6= l
1
hk
� lk k
�Tkk~el +
Xk
Xm
m 6= k
1
hk
� kmk
�Tkm~ek
=Xkl
1
hk
"@Tkl@xk
+Xm
m 6= k
� kmk
�Tml
#~el +
Xl
Xk
k 6= l
"1
hk
� lk k
�Tkk +
1
hl
� lk l
�Tlk
#~el
=Xkl
1
hk
@Tkl@xk
+Xm
m 6= k
1
hm
@hk@xm
!~el +
Xl
Xk
k 6= l
1
hkhl
�@hl@xk
Tlk � @hk@xl
Tkk
�~el:
The sum of the �rst two terms can be arranged in the same way as the divergence of a vector,see Section C.5.2. The lth component of the result is given by
(divT )l =1
h1h2h3
�@
@x1(h2h3T1l) +
@
@x2(h3h1T2l) +
@
@x3(h1h2T3l)
�+Xk
1
hkhl
�@hl@xk
Tlk � @hk@xl
Tkk
�:
(C.53)We should emphasize that tensor T has not been assumed symmetric.
159
Example. The divergence of a second-order tensor T in the spherical coordinates r; #; � is
divT =
�1
r2@
@r(r2Trr) +
1
r sin#
@
@#(sin#T#r) +
1
r sin#
@T�r@�
� 1
r(T## + T��)
�~er
+
�1
r2@
@r(r2Tr#) +
1
r sin#
@
@#(sin#T##) +
1
r sin#
@T�#@�
+1
r(T#r � cot#T��)
�~e# (C.54)
+
�1
r2@
@r(r2Tr�) +
1
r sin#
@
@#(sin#T#�) +
1
r sin#
@T��@�
+1
r(T�r + cot #T�#)
�~e�:
C.5.6 Laplacian of a scalar and a vector
We may obtain the Laplacian of scalar � by combining (C.36) and (C.40), using ~v = grad�. Thisleads to
r2� � div grad � =1
h1h2h3
�@
@x1
�h2h3h1
@�
@x1
�+
@
@x2
�h3h1h2
@�
@x2
�+
@
@x3
�h1h2h3
@�
@x3
��:
(C.55)Example. The Laplacian of scalar � in the spherical coordinates r; #; � is
r2� =1
r2@
@r
�r2@�
@r
�+
1
r2 sin#
@
@#
�sin#
@�
@#
�+
1
r2 sin2 #
@2�
@�2: (C.56)
In curvilinear coordinates, the Laplacian of a vector is more complicated than the Laplacianof a scalar due to the spatial dependence of the unit base vectors. Occasionally, the Laplacianof a vector is needed in spherical coordinates. It is best obtained by using the vector di�erentialidentity (A.31). Without detailed derivation, we introduce
(r2~v)r = r2vr � 2
r2vr � 2
r2@v#@#
� 2
r2cos#
sin#v# � 2
r2 sin#
@v�@�
;
(r2~v)# = r2v# � 1
r2 sin2 #v# +
2
r2@vr@#
� 2
r2cos#
sin2 #
@v�@�
; (C.57)
(r2~v)� = r2v� � 1
r2 sin2 #v� +
2
r2 sin#
@vr@�
+2
r2cos#
sin2 #
@v#@�
:
160
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