Some Topics on a New Class of Elastic Bodies

Some Topics on a New Class of Elastic BodiesAuthor(s): Roger BustamanteSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 465, No. 2105 (May8, 2009), pp. 1377-1392Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/30243772 .

Accessed: 29/11/2014 18:01

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings:Mathematical, Physical and Engineering Sciences.

http://www.jstor.org

This content downloaded from 155.97.178.73 on Sat, 29 Nov 2014 18:01:47 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/30243772?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


PROCEEDINGS -OF- THE ROYAL SOCIETY A

Proc. R. Soc. A (2009) 465, 1377-1392 doi:10.1098/rspa.2008.0427

Published online 20 January 2009

Some topics on a new class of elastic bodies BY ROGER BUSTAMANTE*

Departamento de Ingenieri'a Mecadnica, Universidad de Chile, Beaucheff 850, Santiago 6511261, Chile

In this paper, we study the problem of prescribing deformation as a function of stresses. For the particular case of small deformations, we find a weak formulation, from which we define the constitutive equation of a Green-like material, where an energy function that depends on the Cauchy stress tensor is proposed. Constraints on the deformation are studied for this new class of elastic bodies.

Keywords: weak formulation; elasticity; Green elastic material; constraints

1. Introduction

In a recent paper, Rajagopal (2007) showed that the class of elastic bodies (bodies that do not dissipate energy) is larger than Cauchy or Green elastic materials (see also Rajagopal 2003; Rajagopal & Srinivasa 2007). A new class of a constitutive relationship between the Cauchy stress T and the left Cauchy stretch tensor B is1 (Rajagopal 2003)

g(T, B) = 0, (1.1)

from where we have, as special classes,

T = f(B) and B = t(T). (1.2)

In this paper, we investigate the class of materials described by (1.2)2; in particular, we consider the special case when the displacement gradient is small, but E) is, in general, a nonlinear function of T.

This problem has already been studied in a short note by Bustamante & Rajagopal (in press), where for a constitutive equation of the form2 E=-(T), they studied boundary-value problems for plane strain and stresses. They also found a weak formulation for such problems.

Here, we extend the results shown in that paper. In S2, we have a short review of some basic aspects of the theory of elasticity. We also give some comments on the theory of implicit constitutive equations developed by Rajagopal and co-workers (Rajagopal 2003, 2007; Rajagopal & Srinivasa 2007). In S3a, we study briefly a weak formulation for the boundary-value problem associated with *[email protected] 1 For an application, see Mollica et al. (2007). 2 This equation was derived from (1.2)2. The symbol e is the linearized strain.

Received 23 October 2008 Accepted 22 December 2008 1377 This journal is C 2009 The Royal Society



1378 R. Bustamante

the constitutive equation ey-c(T); this weak formulation is obtained for the plane strain problem using a general system of orthogonal coordinates. In S3b, we propose a weak formulation for the general three-dimensional problem. From these results, we define a new kind of Green-like (or hyperelastic-like) material in S4. In S5, we study briefly the problem of constraints on deformations and how they can be incorporated in materials described by B= j(T), and in particular for the subclass E = -(T). In S6, we comment about the results obtained, we speak in particular on the boundary conditions obtained from the weak formulation, and also on the complementary energy function in finite elasticity.

2. Fundamental relations

(a) Kinematics

Let B denote a body and X denote a particle in this body in the reference configuration Kr(B). In the current configuration Kt(B), the position of the same particle is denoted by x. We assume that there exists a mapping X, such that (Truesdell & Noll 2004)

x = X(X, t). (2.1) The displacement field u and the deformation gradient F are defined as

ox(2.2) u=x-X and F-X (2.2) dX

The left and right Cauchy-Green stretch tensors B and C are defined as

B=FFT and C=FTF. (2.3)

c = B-1 = F-TF-1. (2.4)

We define c as

The Green-St Venant strain E and the linearized strain E are defined, respectively, as

E and eI

au+ OuT.

(2.5) 2 2 ax (ax) (b) Balance and compatibility equations

The Cauchy stress tensor T must satisfy the balance equation (in the absence of body forces)

div T = 0. (2.6) If T" are the contravariant components of T, the above equation is equivalent to (Truesdell & Toupin 1960)

T = 0. (2.7) li

It is well known that to have deformations that satisfy (2.3)2, (2.4) or (2.5)1, some integration conditions must be satisfied (e.g. Truesdell & Toupin 1960; Fosdick 1966). These conditions are

(D) skrl = 0, (2.8)

Proc. R. Soc. A (2009)



A new class of elastic bodies 1379

where (D)R is Riemann's tensor, whose components are defined as (Fosdick 1966)

(D)Rskrl kls,r - krs,l - ( sp - sp) , (2.9)

kls (Dks,l + Dl,k- Dkl,) and r = Dpmklm. (2.10) 2 1 k where

The tensor D can be either C, c or E.

(c) Constitutive equations As mentioned in S1, Rajagopal (2003, 2007) considered a new class of elastic

bodies, whose constitutive relation is of the form g(T,B)= 0. Assuming g to be isotropic, Rajagopal (2007) obtained the implicit

constitutive relation

0oI + ~IT + f0B + f3T2 + 4B2 + 05(TB + BT) + 6(T2B + BT2)

+ P7(B2T + TB2) + 0s(T2B2 + B2T2)= 0, (2.11)

where pi, for i= 1, ..., 8, are scalar functions that depend on the invariants obtained from the pair T, B.

From the above relation, it is possible to see that what we know as Cauchy isotropic elastic material, T = f(B) =- I + qjB + q2B2, is in fact a special case of (2.11). Moreover, a constitutive equation of the form (1.2)2

B = ao0I + aoT + a2T2 (2.12)

is also a subclass of (2.11), and can be seen as a new class of elastic bodies. Let us assume that (see Rajagopal 2007) maxxGLBIVuI = 0(6) with 6 <<1, this

tEIW implies that maxxellell = O(b). Note that, in general, the converse is not true.

From (2.2) and (2.3)1, we have

B = I + 2E + Vu(VU)T, (2.13)

which, as a result of the previous assumption, gives the approximation

B - I + 2e, (2.14)

and, from (2.12), we obtain the following constitutive equation valid for an isotropic material in the case maxxslIVulI = 0(6) with 6 <<1:

e = olI + y7T + 72T2. (2.15)

In (2.12) and (2.15), we can assume that ao, a1, a2, yo, 1 and 72 are only scalar functions of the invariants of T.

Invariance under superposed rigid motions should be studied directly, for example, from (2.12). We need QBQT=QI)(T)Q =I=(QTQT) for all orthogonal Q with det Q= 1. Using (2.13), for example, in (2.12), after some manipulations




1380 R. Bustamante

we have

QEQT + Q(VU)(Vu)TQT = QIQT + 1QTQT + 2QT2T. (2.16) 2 If maxx~ llVull= 0(6), we see from the above equation that (2.15) is

teli approximately invariant to rigid motions. Note that, in general, this is not the case if we would just assume I|ell to be small.

On assuming maxxes||Vull= 0(6), Rajagopal (2007) showed that the

appropriate approximation, when we consider small deformations but arbitrarily large stresses, is of the form

E = 1(T), (2.17) and not of the form T - f(e), which is the classical form used to model nonlinear behaviour of solids for infinitesimal deformations.

A final remark about t and the scalar functions 71, 72 and 73 that appear in (2.15). The constitutive equations (2.15) and (2.17) are valid for the case where we have small deformations and arbitrarily 'large' stresses; therefore, ., 71, 72 and 73 Should be such that E remains small for arbitrary T. This could be seen as a restriction on the constitutive equation.

There is another more subtle and important restriction to consider. As we saw previously, I1ell small does not imply necessarily I Vull small. The condition

maxxey|| Vul = 0(6) is important in order to have invariance to rigid motions; as

a result, this condition should be seen as an additional restriction on the constitutive equations.

(d) Boundary-value problem In general, from (1.2)2 or (2.17), we will not be able to obtain explicit

expressions for T as a function of B or E. We could still try to obtain T numerically from (1.2)2 or (2.17), and then to solve (2.6) to find x= X(X, t) using the usual procedures. However, we can try a different method, as shown in the paper by Bustamante & Rajagopal (in press, S1), where we can express T as a function of Airy's stress function, and then from (1.2)2 or (2.17), we could obtain either B or E as, in general, nonlinear functions on this stress potential. But if B or E (or eventually C or c) are known and given from (2.3), (2.4) or (2.5), then to have continuous displacement fields, the compatibility equation (2.8) must be satisfied.

For the plane strain and stress cases, using Cartesian coordinates and working with the constitutive equation (2.15), Bustamante & Rajagopal (in press, S1) obtained a highly nonlinear partial differential equation for the stress potential (eqn (36) of that paper). That equation is a generalization of the well-known biharmonic equation, which is found following a similar procedure for the linearized theory (Muskhelishvili 1953).

Owing to the present difficulties to solve (2.8) for e using, for example, (2.17), Bustamante and Rajagopal developed a weak formulation for the plane problem, only considering Cartesian coordinates. This weak formulation will be used to solve some boundary-value problems working with the finite-element method. In this paper, we study more general cases.





An important question about equation (2.8) is: what boundary conditions do we have to use to solve this problem? We will partially answer this question in the following sections.

3. Weak formulations

(a) Plane case, general coordinates

We work with an arbitrary system of orthogonal coordinates in a general plane strain problem. In such a case, the compatibility equations (2.8) for E, using E E with 1IeF1 small, reduces to (Eringen 1980; Saada 1993)

811,22 + 822,11 - 2E12,12 = 0. (3.1)

Consider an Airy stress function F, the equilibrium equation (2.7) (plane strain) is satisfied if T11 = (,22, T22 _ (I11 and T12 - -(,12 (Love 1944). Taking the product of (3.1) with F (a virtual Airy stress function), integrating in the volume Kt(B) and using the divergence theorem, after some manipulations, we obtain

el T + 22T + 2e12T12 dv +12da

+ J ))1 e)(l )da= 0, (3.2) where T are the components of the virtual stress tensor associated with the virtual

S11 ^ 22 12 Airy stress function through = 1,22, T = F,11 and T = -(,12. On the surfaces OKt(l)b and

OKt())a, we have assumed Eij, k and e8j as given data, respectively. We

have OKt(B) U OKt()b =

OKt(B) and aKt(B)0 n OKt(b = 0.

(b) Three-dimensional case

The equilibrium equations (2.7) in the three-dimensional problem (no body forces) are satisfied if (see S227 of Truesdell & Toupin (1960), and the paper by Finzi (1934))

Tk = elp emsq ,pq, (3.3) where e"" is the permutation symbol and a,8 are the components of a tensor potential a. This tensor is symmetric. From (3.3), we have

T11 = a22,33 -2a23,23 + a33,22, (3.4)

T22 = a11,33-2a13,13 + a33,11, (3.5) T33 = a11,22 -2a12,12 + a22,11, (3.6) T23 = a12,13 + a13,12 - a11,23 - a23,11, (3.7) T13 = a12,23 - a23,12 - a22,13 - a13,22 (3.8)

T12 = a23,13 + a13,23 - a12,33 - a33,12. (3.9)

and




1382 R. Bustamante

If we consider maxxeBllVull = 0(6) with 6<<1, the general form of the

compatibility equation (2.8) (E)Rskl = 0, with E= , is (Eringen 1980; Saada 1993)

ekl,lm + Elmy,kn - Ekm,ln - Eln,km = 0. (3.10)

Only six of the above equations are independent; these are

811,22 + 822,11 - 2 12,12 = 0, (3.11)

811,33 833,11 - 213,13 = 0, (3.12) 822,33 + 833,22 - 2823,23 = 0, (3.13)

813,12 + 813,12 - 811,23 - 823,11 = 0, (3.14)

823,12 + 812,23 - 822,13 - 813,22 = 0 (3.15)

823,13 + 813,23 - 833,12 - 812,33 = 0. (3.16) and

Consider a virtual tensor potential a. Consider the following integral, where we have taken the multiplication of the above equations by different components of a, adding them in one expression and integrating:

(E1122 + 822,11 - 212,12) 33 + (11,33 833,11 - 213,13)22

+ (822,33 + 833,22 -2 2E2323)l 11 2(E13,12

2-+ 813,12 811,23- 23,11)a23 + 2(823,12 + 812,23 - 822,13 - 13,22)a 13 + 2(823,13 + 813,23 - 833,12 - 812,33) a12} dv = 0.

(3.17) We can prove, for example, that after some manipulations, the multiplication of equation (3.11) by a33 becomes

e811 a33,22 + 822a33,11 -2812 33122 + [(11,2 - 812,1) a33 +

812a33,1- -

11633,21,2 + [(822,1 - 812,2) 33 + 812 33,2- 33,1],1= 0. (3.18)

Similar expressions can be found for the multiplication of equation (3.12) by a22, equation (3.13) by a11, equation (3.14) by 2 a23, equation (3.15) by 2 a13 and equation (3.16) by 2a12. These are the expressions that appear in (3.17). As a result, (3.17) is equivalent to

IK t( 11(33,22 22,33 -2 23,23)

- 822(33,11 +

a1133 -2&13,13)

+ E33 (22,11 + a11,22- 2& 12,12) + 2812(23,13 + a13,23 - 633,12 - h12,33) +

2E13((23,12 + 12,23- 22,13 - 13,22) + 223(12,13 + 13,12- a23,11-

-11,23)dv

Kt + d2 + d33 }dv = 0, (3.19) + f;~:t (B) 11 12 13119

where the contravariant components d' of the vector d that appear in the second integral are





d = (E22,1 - e12,2)a33 + (F33,1- 813,3) a22 + (23,2 - 822,3)a13 +

(823,3- 33,2)12

+ (813,2 + 812,3 - 223,1)23 - 33 22,1 + 13 22,3 - 22 33,1

+ '12 a33,2 + 2E23 623,1 - 13 23,2 - 12 23,3 - 823 a13,2

+ 822 a13,3 - 823 a12,3 + 33 12,2, (3.20)

d2 (33,2 - e23,3)a 11 + (811,2- 812,1) 33 + (813,1- 11,3)a23

+ (813,3- 833,1)a 12

+ (E23,1 + 812,3 -

2813,2)a 13 - 33 11,2

+ 823 11,3 - 11 33,2 + 12 33,1 1323,1

+ 811 a23,3 - 23 a13,1 - 812 a13,3 + 2813 13,2 - 813a12,3 +

833 a12,1 (3.21) and

d3 = (E22,3-

823,2)11 + (11,3 - 13,1)a22 -+ (812,1 - 11,2) 23

+ (12,2 - E22,I)13 (23,1 + 813,2 - 212,3 12 22 22,3 S8_23

a11,2 - 11 e 22,3 + 813 J22,1 - 12 a23,1 + 11 8a23,2 - 812 a13,2

+ 822 a13,1 - 23 a12,1 - 13 a12,2 + 2E 12,3. (3.22) From (3.19), using (3.4)-(3.9) and the divergence theorem, we obtain

SKt()ij dv dini da = 0, (3.23)

where T is the virtual stress tensor, whose components are calculated using af from (3.4)-(3.9).

The components dX of the vector d can be expressed as

di (i) Ajk + (i) "jkljk, (3.24) where (i) Ajk and (i) jkl are the components of the second- and third-order tensors (i)A and (i)Y, respectively. The matrix forms of the tensors (A are

0 - (823,3 - 833,2) - (823,2 - 822,3) 2 2

1 1 (1) - -

(23,3 - 833,2) 833,1 - 813,3 - (813,2 + 133 - 223,1) 2 2

1 1 - (823,2 - 22,3) - (813,2 + 13,3 - 223,1) 822,1- 812,2

(3.25) 1 1

833,2 - 823,3 2 (8133 - 33,1) 2 (8231 + 812,3

- 2813,2)

(2) - - (13,3 - 33,1) 0 - (F13,1 - 811,3) 2 2

1 1 (23,1 + 12,3

- 2813,2) (13,1 - 811,3) 811,2 - 812,1 2 2(3.26)

(3.26)




1384 R. Bustamante

and

S1 1 F22,3 - 823,2 - (E23,1 + '13,2 - 2F12,3) - (c12,2

- 822,1)

1 1 (3)A

- (23,1 + 13,2 - 2E12,3) '11.3 - 813,1 (12,1 - 811,2) 2 2

1 1

S(c12,2 - c22,1) -(12,1- 0

(3.27) The non-zero components of the third-order tensors (i)Y are

(1)-22 - (1)212 = 33 (1) 123 (1)213 123 (3.28) 2 2

(1)y132 - (1) 312 = 23 (1)

y33 _ (1)-313 22 (3.29) 2 2 (1)y221 033 (1)=223 13 (1)231 - (1)y321 = 23, (3.30) (1)y232 = (1)-322

- 13) (1)y233 _ (1)'323 __ 1(3.31) 2 2

(1)-331= - 22 (1) 332 12, (3.32)

(2)-112

_- 33 (2)-113 = 23 (2) 121 __ (2),211 33, (3.33) 2

(2)-123 _

(2)y213 (2)y131 _ (2)y311 -- 23, (3.34) 2 2

(2)-132 (2)y312 = 13 (2)y133 = (2)Y313 _ 12, (3.35) 2

(2)-f231 (2)y321 = 13 (2)y233 _ (2)y323 11, (3.36) 2 2

(2)331 12, (2)y332 = _ 11, (3.37)

(3)yr112 = 23, (3)-113 = -E22 (3)-121 -

(3)8211 823, (3.38)

(3)-f 122

(3)y212 _ I13

(3)-y123

- (3)y213 12, (3.39) 2

1 ( 1 - 311 221 (3)-132 _ (3) 312 12, (3.40) 2 2

(3) 221 813 (3)Y.223 = 811 (3),f231 = (3)y321 _ 12 (3.41) 2 and

(3)y232 = (3)y322 _ 1 (3.42) 2





In coordinate-free notation, (3.23) can be written as

: T dv + n da = 0. (3.43) JKt(!) J

.4 From (3.24), we see that

d-n = ((')An1 + (2)An2 + (3)An3) : + (1)n1 + (2)2 (n3) Va. (3.44) In (3.43) and (3.44), the products e: T, (i)A : a and (Y : Va are interpreted as

eijJ.3 (i) Ajk kj and (i) jkljk, respectively. In (3.44), the term (1)An1 + (2)An2 + (3)An3 could be viewed as the dot product

A -n of a third-order tensor A with n. The components of this tensor are defined as

Aijk = (i) Ajk . (3.45) The dot product A - n is equivalent to Aijkni.

Regarding the term (1)~ nI

+ (2)Vn2 + (3) n3, this could be viewed as the dot product of a fourth-order tensor Y with n, where the components of this tensor are defined as

Sijkl -= (i)jkl. (3.46) The dot product Y -n is interpreted as ijklni.

Therefore, (3.43) can be written as

: Tdv + (A-n + (Y-n): Va da = 0. (3.47) Kt(B) RK8 Finally, if a= a on OKt(B)a and Va= A on

OKt(B)b, with

OKt(3) n OKt(B)b- 0 and OKt(B)a U OKt(B)b = OKt(B), where a and A are prescribed values of a and Va on OKt(B), then a= 0 on OKt(B)a and Va- 0 on OKt(l3)b, (3.47) becomes

e: Tdv + (A-n): a:Ada+ (Y-n): Vaida = 0. (3.48) K,() JoKt( 3)b J, t ()a

4. A Green-like material

As in the classical theory of nonlinear elasticity (Ogden 1997; Truesdell & Noll 2004), from (3.48) we could assume that there might be some materials for which there exists an 'energy' function3 W= W(T), such that

aW = e. (4.1) OT

" In the classical linear theory of elasticity, from the volume integral of the dot product of a virtual displacement ui field with the balance equation div T+b= 0, we obtain

T: T

dv . (Tn) -fi

da + b-fi dv, o,(u8)(f 1 (8) aK,(B)

where F would be the virtual deformation. The above equation can be seen as a balance of internal and external virtual work. Equation (3.48) could be seen as an equivalent expression for the balance of virtual work, working now with a virtual tensor potential a. In (3.48), the surface integrals aK,(B), (A-n) : a da and faK,(B),, (Tn) : Va da would contribute to the external virtual work; however, their exact physical meanings are not clear yet. This is why we put the word 'energy' in quotation marks.




1386 R. Bustamante

From the above definition, we have, in the first integral on the left-hand side of (3.48),

. T tr(eT )-=r T 1-T =aW, (4.2)

where now ba W could be interpreted as the variation in a of W; T as the variation in a of T; and af as the variation of a.

For an isotropic material, we have that W should be a function of three invariants. We choose two different sets of invariants

1 2 1 I, = tr T, I2 = -trT, 3 = - tr T3 (4.3) 2 3

1 I1 =tr T, 12 - [(tr T)2 -trT2], 13 = det T. (4.4) 2

and

These two sets are not independent. For an isotropic material, we have W = W(T) = W(11, I2, 13). Using the chain

rule in (4.1), considering (4.3) and (4.4) we obtain, respectively,

= W11 + W2T + W3T2 (4.5)

E (WI + 1 W2)I- W2T + 13 W3T-1. (4.6)

and

In the above expressions, W1, W2 and W3 are the partial derivatives of Win 1, I2 and 13, respectively.

In the classical theory of elasticity, for Green materials, the independent variable is the deformation gradient F. The class of deformations that, in general, are admitted are such that det F > 0; as a result, F -1 exists. But in our case, in which W= W(T), it is perfectly possible from the physical point of view to have a stress tensor such that det T=O0; in fact, T=O0 for all x E Kt,(B) would correspond to the case that there is no external load and no residual stress. Therefore, for some problems, T- might not exist, and this could create problems in the evaluation of the constitutive equation (4.6); however, we have that 3T-1= (cof T)T, where cof T is the cofactor matrix associated with T, and this matrix can always be defined.

For later reference, we also illustrate the case of a transversely isotropic material, which is symmetric with respect to a field e, with ell| =-1. In such a case, we have W= W(T,e) = W(11, I2, 13, 14, 15), where the invariants I, i= 1, ..., 5, are defined as (e.g. Spencer 1971; Zheng 1994)

T2 =-trT3 =e(Te) and I= e-(T2e). ItrT,2 I2=-tr 2 133-trT,4

(4.7) From (4.1) and the chain rule, we obtain

E = WII + W2T + W3T2 + W4e~e + W5[e0(Te) + (Te)0e], (4.8)

where Wi are the partial derivatives of W in Ij, i= 1, ..., 5.





5. Constraints

Constraints are defined as restrictions on deformations, which can be described as (Truesdell & Noll 2004)

y(F) = 0. (5.1) The function y must be frame indifferent; therefore, (5.1) is written as

A(C) = 0. (5.2) In the case where we work with the constitutive equation (1.2)2, from (5.1) an

isotropic constraint could be written as

(B) = 0, (5.3)

where we must impose the restriction y(QBQT) = 0 for all Q orthogonal with det Q = 1.

Now, for a function B= )(T), an isotropic constraint of the form (5.3) would just introduce a restriction in the form of 1) in a direct way, which means that we would not need to assume some sort of decomposition of a stress in two parts, with one of them that does not do work with any deformation compatible with the constraint, which is the classical, somehow controversial, argument used in the classical theory of elasticity (Rajagopal & Saccomandi 2005).

Consider, for example, an isotropic material (2.12), where B = ao0I+ alT+ a2T2. Let us study, as illustration, the incompressibility constraint det F -1= 0, which can be described as

det B -1 = 0. (5.4)

Using (2.12) in (5.4), we obtain

det(aoI + a1T + a2T2) - 1 = 0, (5.5)

which could be seen as a cubic equation we could use to find, for example, ao as a function of al, a2 and T.

Let us study now how constraints can be incorporated in our constitutive equation (2.17), which is a special case of (1.2)2. If maxxeB|| Vull = O(6), 6<<1, we

have B I + 2E and C = I + 2e, then (5.2) is equivalent to

A(I + 2E) = 0. (5.6)

Expanding the function 2 as a Taylor series, we get

A(I) + : 2 + 0(62) = 0, (5.7) ac

which, if we neglect the terms 0(62), becomes

2(I) + (I) 2e = 0. (5.8)

Let us study two examples where we apply (5.8).




1388 R. Bustamante

(a) Incompressibility Consider an incompressible material, from A(C) = det C - 1, we have 2(I) = 0,

02/aC = det(C)C-1, and so 02/aC(I) = I; therefore, (5.8) is equivalent to I:e=0, which is equal to

tr(E) = 0. (5.9)

This is a classical result, equivalent to div u= 0, which could have been obtained directly from det F -= 1, using F = I+ Vu, with maxxEsBlVul -= O() and 6 < 1.

If we apply (5.9) to the constitutive equation of our new elastic isotropic material (2.15) = y0I + yTiT + 72T2, we obtain

Yo - (yltr T + 72tr ), (5.10) 3

where, for example yo has been expressed explicitly as a function of y1, y2 and T. Interesting results are found for a Green-like material. Consider the set of

invariants (4.3) and its corresponding constitutive equation (4.5). The restriction (5.9) implies that

3 W, + W2h, + 2 W312 = 0. (5.11) The solution of this equation is w 1 23-h2 W = W 12 1 3 + 13- _ 1112 , (5.12) 6 27 3

where W is a function in two variables 2 - (1/6)12 and 13 + (2/27)13 - (2/3)1112. For this case, we use the set of invariants (4.4), from (4.6) and (5.9) we obtain

the partial differential equation

3 W, + 2 W211 + W312 = 0, (5.13) whose solution is W = W(I2 -(1/3)172, 3 + (2/27)13 -(1/3)112).

(b) Inextensibility Let us consider an additional case, where we have a material that is

inextensible in the direction e with lel- =1. This is an example of a non-isotropic constraint. In this case, (5.2) becomes (e.g. Truesdell & Noll 2004)

2(C) = e.(Ce) -1 = 0. (5.14)

In this case, we have 2(I) =0 and a02/C(I)= e0e; therefore, (5.8) is equivalent to (e.g. S17.2 of Podio-Guidugli (2000))

e. (Ee) = 0. (5.15) Let us consider only a Green-like material. If the direction in which this

material is inextensible is the same field e we used in the definition of the transversely isotropic material (4.8), then the restriction (5.15) implies

W1++ WA + W315 + W4 + W514 = 0. (5.16)





The solution of this equation is of the form

W = W(I11,2,3,14), (5.17)

where

I1 14)11, 12 -2 1 2 IQ1 , (5.18)

I I 5 I1 ,1 4

and I4 - I 1 2 l4 I5 . (5.19)

6. Some further remarks

In this paper, we explored some issues concerning a constitutive equation of the form e = -(T). We discussed the boundary-value problem that we should solve for such a constitutive law. We obtained a weak formulation for the general three-dimensional case, extending some results presented in a recent paper by Bustamante & Rajagopal (in press, S1). From this weak formulation, for the case in which I1j11 is small, we were able to define a Green-like material, through the use of an energy function W= W(T). Additional issues concerning constraints were studied, in particular how constraints on deformations impose restrictions on either bI(T) or W(T).

(a) Boundary conditions

An important issue that appeared in this paper corresponds to the boundary conditions found from the weak formulation (3.48). In that expression, we have the terms A-n and .-n product a and Va, respectively; these terms are integrated over the surface OKt(B) of the body. From there, we could assume that boundary conditions associated with the partial differential equations (3.11)- (3.16) could be

A-n= A-n and T-n= T-n, (6.1)

where A and Y would be the prescribed values of A and Y, which from (3.25)-(3.27) to (3.28)-(3.42) would be calculated from prescribed values of E and its first derivatives in x on aKt(B).

In the classical theory of elasticity, the two important boundary conditions considered in most problems are a prescribed external traction Tn= t and a prescribed displacement x = x on the boundary of the body (Truesdell & Noll 2004). These are not the only boundary conditions we can find, other types can be found, for example, in SS5.1 and 5.2 of Ciarlet (1988) and in the paper by Noll (1978).

The condition (6.1) would correspond to a new kind of boundary conditions for the strains. In fact, even for the linearized theory, where, for (2.15), we would have yo = -(v/E)tr T, yj = (1 + v)/E and 72=0, working, for example, in the plane case, for the third integral on the left-hand side of (3.2) we would get




1390 R. Bustamante

8- 22 12 1 1 da JOKb) ( 812 - 81 2 ) 2

ST22-T12 n 1 v T1 T12 1 n + da.

aK T12 11 Tll n2 E T12 T22 n22

(6.2) We see that only for the second term inside the square brackets on the right-hand side of (6.2) could we identify the condition Tn = t.

From the physical point of view: could we actually apply, for example, a prescribed strain on the boundary of a body?

Bustamante & Rajagopal (in press, S1) developed explicitly the partial differential equation that must be solved to find the Airy stress function Q( (see S3a) for the plane case, when we use Cartesian coordinates (see eqn (36) of that paper). A boundary condition for that equation could be the classical traction on the boundary Tn= t, where T is expressed in terms of the second derivatives of (F. We then see that if we want to solve the strong form of the boundary-value problem, we do not have difficulties imposing directly Tn= t on OKt(3).

(b) On the Green-like material

We developed a weak formulation for the case E = I(T); from this formulation in S4, we were able to define a new class of Green-like elastic material. From those results, we could be tempted to do the same for the more general nonlinear problem B =- (T); however, this has been proved to be much more difficult.

In the case maxxe I|Vu| = 0(6) with 6<<1, we have E=E and from (2.9),

()Rskrl " Fkls,r- Fkrs,l, with Fkls calculated from (2.10) with D= . The equation Fkls,r~-- krs,1=0

was multiplied by appropriate components of the virtual stress potential ai, from where we obtained (3.48), which was used to propose the existence of an energy function W(T), such that e = (0 W/&T).

Now, for the nonlinear case B= b(T), we have tried to do the same; however, in the expression for the Riemann tensor, we have to deal with nonlinear terms of the form FlFr p.

We have not been able, so far, either to incorporate these terms in e : T, or to incorporate them in boundary conditions of the form (6.1). Of course, we could leave these terms aside, considering for : T only the contribution of the linear terms that come from Fkls~,r and leaving terms that come from 1Fl as some sort of extra contribution for the virtual work. But evidently if we do so, any function W= W(T) will not actually consider the whole energy stored by the body.

In the classical theory of nonlinear elasticity, there has been a long discussion on the existence of complementary energy functions (e.g. S5.4.3 of Ogden (1997); see also Ogden (1975), Dill (1977) and Zhong-Heng (1980)). This complementary energy function, which can be denoted as = (S), should be such that F = M,0/0S, where S would be the nominal or first Piola-Kirchhoff stress tensor. This complementary energy function has been proposed directly from the classical energy function 0-= (F), through the use of a Legendre transformation. The important assumption made is the invertibility of the function Q(F), which in general is not possible (Dill 1977).





Although the strong forms (2.7) and (2.8) are different, and so we cannot expect that weak forms derived from them could be directly comparable; it is remarkable to note that similarly we have not been able to find an equation for a Green-like elastic material, for the case in which, for example, B is given as a function of T (or for any other pair of strains and stresses); in the general case, we work with large deformations and displacements.

(c) Final comments

When we work with e = )(T), assuming maxxeB |IVul = 0(6) with 6 << 1, the tER

displacement field u can be calculated easily from (2.5)2 and is unique if (2.8) holds. But it is interesting to note that when we work with the more general case B = tj(T), once the compatibility equation (2.8) (with D= c=B- ) would be satisfied, we could say that X(X,t) is unique, but in order to calculate explicitly this field, we would need to solve (2.3)1 (ax/OX)(Ox/OX)T = B. This problem is not trivial (e.g. Blume 1989).

In a future communication, we expect to show solutions of some simple boundary-value problems, solving directly (2.8) with D= e, for the plane stress case, and for an isotropic material (2.15) e=70oI+y1T+72T2, considering some simple forms for yi, i= 0, 1, 2.

References

Blume, J. A. 1989 Compatibility conditions for left Cauchy-Green strain field. J. Elast. 21, 271-308. (doi:10.1007/BF00045780)

Bustamante, R. & Rajagopal, K. R. In press. A note on plain strain and stress problems for a new class of elastic bodies. Math. Mech. Solids.

Ciarlet, P. G. 1988 Mathematical elasticity Volume I: three dimensional elasticity. Amsterdam, The Netherlands: Elsevier/North Holland.

Dill, E. H. 1977 The complementary energy principle in nonlinear elasticity. Lett. Appl. Eng. Sci. 5, 95-106.

Eringen, A. C. 1980 Mechanics of continua, 2nd edn. New York, NY: Krieger Publishing Company. Finzi, B. 1934 Integrazione delle equazioni indefinite della mecanica dei sistemi continui. Rend.

Lincei. 19, 578-584. Fosdick, R. L. 1966 Remarks on compatibility. In Modern developments in the mechanics of

continua (ed. S. Eskinazi), pp. 109-127. New York, NY: Academic Press. Love, A. E. H. 1944 Treatise on the mathematical theory of elasticity, 4th edn. New York, NY:

Dover. Mollica, F., Ventre, M., Sarracino, F., Ambrosio, L. & Nicolais, L. 2007 Implicit constitutive

equations in the modeling of bimodulars materials: an application to biomaterials. Comput. Math. Appl. 53, 209-218. (doi:10.1016/j.camwa.2006.02.020)

Muskhelishvili, N. I. 1953 Some basic problems of the mathematical theory of elasticity. Groningen- Holland, The Netherlands: Noordhoff.

Noll, W. 1978 A general framework for problems in the statics of finite elasticity. In Contemporary developments in continuum mechanics and partial differential equations (eds G. M. de La Penha & L. A. Medeiros), pp. 363-387. Amsterdam, The Netherlands: North Holland.

Ogden, R. W. 1975 A note on variational theorems in non-linear elastostatics. Math. Proc. Camb. Philos. Soc. 77, 609-615. (doi:10.1017/S0305004100051422)

Ogden, R. W. 1997 Non-linear elasticity. Mineola, NY: Dover. Podio-Guidugli, P. 2000 A primer in elasticity. J. Elast. 58, 1-104. (doi:10.1023/A:1007672721487)




1392 R. Bustamante

Rajagopal, K. R. 2003 On implicit constitutive theories. Appl. Math. 48, 279-319. (doi:10.1023/ A:1026062615145)

Rajagopal, K. R. 2007 The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309-317. (doi:10. 1007/s00033-006-6084-5)

Rajagopal, K. R. & Saccomandi, G. 2005 On internal constraints in continuum mechanics. Diff. Equat. Nonlin. Mech. 2006, 18572. (doi:10.1155/DENM/2006/18572)

Rajagopal, K. R. & Srinivasa, A. R. 2007 On the response of non-dissipative solids. Proc. R. Soc. A 463, 357-367. (doi:10.1098/rspa.2006.1760)

Saada, A. S. 1993 Elasticity: theory and application. Malabar, FL: Krieger Publishing Company. Spencer, A. J. M. 1971 Theory of invariants. In Continuum physics I (ed. A. C. Eringen),

pp. 239-535. New York, NY: Academic Press. Truesdell, C. & Noll, W. 2004 The non-linear field theories of mechanics (ed. S. S. Antman),

3rd edn. Berlin, Germany: Springer Truesdell, C. & Toupin, R. 1960 The classical field theories. In Handbuch der Physik, vol. III/1.

Berlin, Germany: Springer. Zheng, Q. S. 1994 Theory of representations for tensor functions-a unified invariant approach to

constitutive equations. Appl. Mech. Rev. 47, 545-587. Zhong-Heng, G. 1980 The unified theory of varational principles in nonlinear elasticity. Arch.

Mech. 32, 577-596.




Documents

Some Topics on a New Class of Elastic Bodies