Journal of Applied Mathematics and Physics (ZAMP) Vol. 23, 1972 Birkh~iuser Verlag Basel

Dynamically Possible Finite Deformations of Isotropic, Incompressible, Elastic-Inelastic Solids with Temperature Independent Response By Jan Kratochvil, Institute of Solid State Physics, Prague, Czechoslovakia

1. Introduction

Several families of exact solutions, involving large inhomogeneous deformations, have been obtained in finite elasticity theory for incompressible materials. The deformations involved include bending, stretching, shearing, torsion, and both cylindrical and spherical inflation. Detailed desription of those families, with refer- ences to the original papers, is contained in the books by Truesdell and Noll [1] and Green and Adkins [2] and also in the papers [3, 4]. Solutions which are similar to these elastic solutions have been obtained for incompressible materials with memory by Carroll [4-8] and Fosdick [9]. In the present paper it is shown that the same families of finite deformations are dynamically possible also in incompressible elastic-inelastic solids 1).

The finite theory of elastic-inelastic solids has been formulated in the frame- work of the thermodynamics with internal state variables in the recent paper [10] 2). The internal state variables are considered to be the inelastic strain tensor and variables (called here structural parameters e~i), i = 1, ..., n) which appear in phenome- nological theories of work-hardening (for details see [12]). Only a special case of elastic-inelastic materials, namely isotropic, incompressible materials are acceptable in problems discussed here. Moreover all thermal effects are excluded from the present consideration.

The mechanical theory of elastic-inelastic solids is summarized in Section 2. The five families of deformations with which the present paper is concerned are recorded in Section 3. In Section 4 the method of solution is described in detail for bending, stretching and shearing of a rectangular block. Results for other families of deformations are presented without detailed proofs.

The method of solution is an inverse one. The deformation is specified precisely at the outset, and the problem of finding the surface tractions which are required in order to support the deformation is reduced to the problem of solving a system of

1) The term 'inelastic' is used as a synonym of'visco-plastic'. 2) An internal state variable type theory cannot be regarded as a special case of a theory of materials

with memory (see e.g. Coleman, Gurtin [11]).

950 Jan Kratochvil ZAMP

ordinary differential equations. It is shown, by taking into consideration the special symmetry of the deformation gradient fields for each deformation and the incompres- sibility of the material (see I-8, 9]), that for all families the system consists at most of 3 + n scalar ordinary differential equations (n is the number of structural para- meters which characterize the material).

2. Elastic-Inelastic Materials

We now summarize the results of the theory of elastic-inelastic materials suggested in the previous paper [10]. Only mechanical response of elastic-inelastic materials will be considered.

A mechanical behaviour of an elastic-inelastic body, in which each particle will be identified with its position X in a fixed reference configuration, is assumed to be completely described by six functions of X and time t: the spatial position x = z(X, t) in the motion Z, the symmetric Cauchy stress tensor T= T(X, t), the specific body force per unit mass b = b(X, t), the elastic deformation gradient E=E(X , t), the inelastic deformation gradient P=P(X, t), and the structural parameter vector �9 =~(X, t), ~ = ( ~ ) , ..., ~"~).

These functions must satisfy the kinematical relation of the finite strain theory

F = E P , (2.1)

where the deformation gradient F = (c~/OX)x, and the dynamical equation (super- posed dots will indicate material time derivatives, p is the density at time t)

p J / - div T= p b. (2.2)

The relation (2.1) yields the decomposition rule for the velocity gradient L = grad 2 =/~F- 1

L = L~ + Le, (2.3)

where L~ =/~E-1 and Lp =EPP-~ E-~ are the elastic and inelastic velocity gradients respectively.

We assume that a knowledge of the present values of E and �9 suffices to determine the mechanical state of the body. That is, the functions T, Lp and f characterize the response of the material at the point X at time t as follows

T= T(E, ~), Le=Le(E, o0, d~=f(E, ~). (2.4)

We shall consider homogeneous materials, i.e. X does not enter (2.4) as an inde- pendent variable.

From the thermodynamic consideration and the concept of the symmetry of elastic-inelastic materials described in [10] it follows that P must be volume pre- serving deformation, i. e.

det P = 1, therefore tr Lp = 0. (2.5)

Vol. 23, 1972 Finite Deformations of Elastic-Inelastic Solids 951

The principle of material indifference and the concept of the isotropy group [13] (see also [10] ; we consider the isotropy group with respect to E and assume that the scalars e(0 are unaffected by the choice of the reference configuration) can be used to derive from (2.4) the reduced form of the constitutive equations. For an isotropic elastic-inelastic solid in a natural state we have

T= T(BE,~), Lp=Le(BE,~) , ~ = f ( B E , ~ ) , (2.6)

where ~ Le and f are isotropic functions of the left Cauchy-Green elastic tensor B E = E E r and the scalars ~(0.

Assuming that the material is elastically incompressible, i.e. d e t E = l , and using the representation theorems for tensor and scalar valued isotropic functions [14], from (2.6) we have

T= - p l + S , S=cI)IBE+qT)_IB~ 1,

D e = d o 1+ d 1 B E + d _ 1B~ 1,

d:(o = f(i), i= 1 , . . . , n.

(2.7) (2.8) (2.9)

In (2.7) T is determined by B E only to within an arbitrary hydrostatic pressure p, S is the extra stress tensor (see [1] w 49). ~1, ~-1, dt, d_ 1 and f(0 are scalar functions of ~(~), I and II , where I and I I are the principal invariants of B E (from the assumption of elastic incompressibility I I I = 1); the relation do = - (I d 1 + I I d_1)/3 respects the restriction (2.5). In (2.8) D p - ( L e + L T ) / 2 is the inelastic stretching; the skew sym- metric part of Lp, i.e. W e - ( L e - Lr)/2, is equal zero (this is the consequence of the representation theorem for skew-symmetric tensor-valued isotropic functions [14]).

The main result of this paper concerns five families of solutions, involving large inhomogeneous deformations. These solutions can be obtained by the inverse method, in which the motion is specified at the outset, and the corresponding tensors E, P, T and the vector e may be calculated from (2.1) (or (2.3)), (2.2), (2.7)-(2.9).

Full information will be obtained if corresponding E and 0~ are found as a solution of (2.9) and the Equation (2.10) ((2.10) follows from (2.3) and (2.8))

[~ = LE - d o E - d 1 EE T E - d _ 1 (E T)- - 1, (2.10)

where for each family L is a specified function of time. For known E and e the corre- sponding plastic deformation gradient P is obtained from (2.1) and the stress T follows from (2.7), where the pressure p is determined from (2.2).

If we want to calculate only the corresponding stress tensor T it is sufficient to solve a reduced problem: to find B~ and 0c from (2.9) and the Equation (2.11) ((2.11) follows from (2.10) and the relation/IE = LE BE + Be LrE).

BE = LBE + BE Lr + h2 B 2 + h~ B~ + ho 1, (2.11)

where h 2 = - 2 d l , h i= - 2 d o , ho= - 2 d _ 1. For known B E and �9 the stress tensor T is obtained from (2.2) and (2.7).

952 Jan Kratochvil ZAMP

3. Families of Deformations

The families of deformations discussed in this paper are those which are presently known to be dynamically possible in arbitrary isotropic, incompressible, simple materials. While these five families of isochoric deformations are contained in the works [8, 9], for future reference we shall list them here along with the com- ponents of their related deformation gradients F. The choice of coordinates in the reference state (X L) and coordinates in the present configuration (x k) is also recorded for each family.

For our consideration it is convenient to express tensors in terms of matrices of their physical components. These components are e.g. F<mL>=(gmmGLL)-~F~, B<~ ink> = (gram Skkl'V ~-~ B rake , m, k, L are not summed, gmr, are the covariant metric compo- nents in the system x k, and G LL are the contravariant metric components in the system X L.

Family 1. Bending, stretching and shearing of a rectangular block. In terms of rectangular Cartesian coordinates (XL)=(X, Y,,Z), and cylindrical coordinates (x k) = (r, O, z), the motion ;~ is defined through

r(t)=I/A(2X+D), O(t)=B(Y+E), z ( t )=Z/(AB)-BCY+F, (3.1)

where A = A(t), B = B (t), etc., are supposed to be scalar functions of time, and AB # 0 for any t. The corresponding matrix of the physical components of the deformation gradient F is obtained from (3.1) in the form

or 0 (Ai)- I " IIF<kL>II = Br - B C

(3.2)

Family 2. Straightening, stretching and shearing of a sector of a hollow cylinder. To characterize this family of deformations, we assume cylindrical coordinates (X L) = (R, O, Z) and rectangular Cartesian coordinates (x k) = (x, y, z) and write

x(t)=AB2RZ/2+D, y(t)= O/(AB)+E, z(t)=Z/B+ CO/(AB)+F, (3.3)

where A=A(t), B=B(t), etc., are scalar functions of time, and AB#O for any t. From (3.3) we obtain

AB2 R 0 OBO - IIF<kL>l[ = 0 (ABR) -1 .

0 C/(ABR) 1

(3.4)

Family 3. Inflation, bending, torsion, extension and shearing of an annular wedge. With respect to cylindrical coordinates (X a) = (R, O, Z) and (x k) = (r, 0, z) this family

Vol. 23, 1972 Finite Deformations of Elastic-Inelastic Solids 953

of deformation is characterized by the motion

r ( t ) = ~ , O(t)=CO+DZ+G, z ( t )=EO+FZ+H, (3.5)

where A=A(t), B=B(t), etc., and A(CF-DE)= 1 for all t. From (3.5) it follows

AR/r 0 0

[IF<kL>,]= 00 Cr/R D r . E/R

(3.6)

Family 4. Inflation or eversion of a sector of a spherical shell. This family of deformations is defined in terms of spherical coordinates (XL)=(R, O, 4~) and (x k) = (r, 0, ~o) by

r(t)=l/+_R3+A, O(t)=+_O, ~0(t)=~+ C, (3.7)

where A =A(t) and C = C(t). (3.7) yield

! R2/r2 0 0

fIF<~L>ll = 0 +_r/R 0 m

0 r/R (3.8)

Family5. Inflation, bending, extension and azimuthal shearing of an annular wedge. This family is defined by

r(t)=AR, O(t)=B lgR + CO+D, z(t)=EZ+F, (3.9)

where we employed cylindrical coordinates (xL)= (R, O, Z) and (x k) =(r, 0, z), and where again A =A(t), B=B(t), etc., and A 2 CE= 1 for all t. From (3.9) we obtain

[/F<kL>II = AC 0

(3.1o)

All five families of deformations possess a high degree of symmetry. This symmetry is reflected in the associated matrices of physical components of F. From (3.2), (3.4), (3.6), (3.8), and (3.10) we see that in all five families at least four off diagonal components of F vanish identically. In Family 4, all off diagonal components of F were shown to vanish, and F (22) = F <33> in the case of inflation (+ sign in (3.7)) or F <22>= - F <33> in the case of eversion ( - sign in (3.7)). In the next section we shall show that in all families of deformations the corresponding physical components of the elastic deformation gradient E, and the left Cauchy Green elastic strain tensor Be must possess the similar reduction properties (see (4.2), (4.3)).

954 Jan Kratochvil ZAMP

4. Method of Solution

The symmetry of each family of deformations will be reflected in the invariance of the associated deformation gradient F

F(X, t)= QF(X, t) R r, (4.1)

where Q and R are two special proper orthogonal transformations.

Theorem: Suppose that we are given an initial time to, an initial elastic deforma- tion gradient Eo(X), and an initial structural parameter vector %(X) for each X, and a deformation gradient F(X, t) in the body. Suppose further that for all t in some interval <to, to + "c) the system (2.9), (2.10) has a unique solution E (X, t), ~t (X, t) with E(X, to)=Eo(X) and ~t(X, to)=eo(X ). If there exist proper orthogonal trans- formations Q and R such that (4.1) is satisfied for all re(to, to+Z) and Eo(X)= QE o (X) R r, then

E(X, t)=QE(X, t) R r, for all t~(to, to+Z). (4.2)

Proof Suppose that E(X, t), e(X, t) is the solution of the system (2.9), (2.10) for t~(to, to+Z) which corresponds to the deformation gradient F(X, t), and satisfies E (X, to) = Eo (X), ~t (X, to) = ~to (X). Then from L =/~'F- 1 and the Equations (2.9), (2.10) we see that QE(X, t)R r, e(X, t) is the solution of the system (2.9), (2.10) which corre- sponds to the deformation gradient QF(X,t)R r, and satisfies QE(X, to)R r= QEo (X) R r, ~t (X, to) = % (X). As F (X, t) satisfies (4.1) for all t ~ ( to, to + z ), QEo (X) R r = Eo (X), and the solution of the system (2.9), (2.10) is unique for all t~ (to, to + z), the statement (4.2) must be valid.

Under the conditions specified in theorem from BE = EE r and (4.2) we have

BE(X, t)=QBE(X, t) Qr for all rE(to, to+Z). (4.3)

We shall utilize the above invariance conditions (4.1)-(4.3) to find the explicit forms of the Equations (2.10) and (2.11) and the stress tensor T for the families of deformations. We shall use four orthogonal transformations Q and R. The matrices of their physical components IlQ<km>ll and IlR<L~t>ll will be identified with

1 0 01 - 1 0 ! I a) 0 - 1 0 , b) 0 - 1 ,

0 0 - 0 0 (4.4)

Z i0! c) 0 0 , d ) 0 - .

0 1 - 1

The method of solution is quit6 similar for each of the five families listed in Section 3. Family 1 is considered here in some detail and corresponding results for the other families are presented, without detailed proofs, at the end of this section.

Vol. 23, 1972 Finite Deformations of Elastic-Inelastic Solids 955

Family 1. From (3.2) it is clear that (4.1) is satisfied if both IIQ<k'~>ll and IIR<LM>I[ are identified with (4.4a). Then (4.2) implies that E<IE>=E<a3>=E<2I>=E<3'>=O. Since the component E <11> may be determined from the condition de tE- -1 , the only non-trivial components are E <22>, E <33>, E <23>, E <32>. When the material description is employed, i.e. E<kL> = E <kL> (X, Y, Z, t), the Equation (2.10) is reducible to the system of four ordinary scalar differential equations

dE<22> dE<a3> - - = a E <22> + M <22>,

dt dt

dE<23> dE<32> _ _ = a E<23> + M<23>,

dt dt

where

_ _ _ c E < 2 3 > _ b E < 3 3 > + M < 3 3 > '

_ _ _ c E<22>_b E<32> + M<32> '

(4.5)

a = [ A ( 2 X + D ) + A D ] [2A(2X+D)] -1 +[~/B, b=A/A+[~/B,

c = [(A/A + 2 [3/B) C + C] [A (2 X + D)] - ~

are the functions of time given by the motion (3.1), and

M <kL> = - (do E + d l t ~ U E + d _ l ( U ) - 1)<k%

If initially at t = to the components E <kL> and ~(i) depend on the coordinate X only, it is evident from (2.9) and (4.5) that the components E <kL> and a(0 may be regarded as functions of X and t only, i.e. in terms of the spatial description we have E <kL> ---- E<kL>(r, t), O~ (i) =a(i)(r, t).

Similarly, utilizing the ortbogonal transformation (4.4a) in (4.3) we arrive at the result: ~'E1:~<12>-- ]~<13>--0"--~'g - - Thus the reduced problem (the system (2.9), (2.11)) requires to solve (2.9) and three equations

dB<E22> - 2 a B<e 22> + h 2 [(B<~22 >) 2 + (B<~23 >) 23 + h 1 B<~ 22> + ho, dt

dS~<33> 2bB<E33>-2cS<E23>+h2[(B<E23>)2+(n<E33>)2]+hlS<E33>+ho, (4.6) dt

dB<E23 > = ( a - b) BE <23 > -- c n<~ 22 > + h2 [B<E 22 > + B<E 33 >1 S<e 23 > + hi U<e 23 >, dt

where h z, h,, h 0 are functions of ~(i), I=A -1 +B<E22>+B<~ 33>, and

I I = A + (S<~ 22> +B<E 33>) A- l ; A =B<~ 22> S<~ 33> - (B<~23>) 2.

The last non zero component B<E 11> may be determined from the condition det Be = 1. It is evident from (2.9), (4.6) and (3.1)1 that B<~ k"> and ~(0 may be regarded as functions of r and t.

As long as the right-hand sides of the equations of the system (4.6) satisfy the requirement of the CarathOodory existence theorem [151 the solution of the system (4.6) exists. For a known solution of (4.6) the extra-stress tensor S can be determined

956 Jan Kratochvil Z A M P

from (2.7)2. The pressure p in (2.7)1 can be obtained from (2.2) if two additional assumptions are introduced.

First we shall admit only those body forces b which possess a single valued potential v, i.e. b = - g r a d v. Further we shall assume 3) that time dependent scalar functions A(t), B(t), etc. which occur in (3.1), (3.3), (3.5), (3.7), and (3.9) are such that the related accelerations possess a potential f2, i.e. ~ = g r a d f L The condition under which 2 = grad ~2 is satisfied as well as a derivation of the potential function ~2 appropriate to each family have been given previously and can be found in [ l l for Families 1-4 and in [4 3 for Family 5. Therefore we shall assume that the poten- tials v and ~2 are given.

Introducing the extra-stress S in (2.2) and using both new assumptions the dynamical Equation (2.2) takes the form (p is a constant)

grad [p + p (v + f2)] = div S. (4.7)

For all five families the Equation (4.7) is readily solvable for p. For Family 1 we have from the invariance condition (4.3) and the relation (2.7)2

that S<~2>=S<13>=0. Moreover from (2.7)2 and the consequence of (2.9), (4.5) (or (2.9), (4.6)) may be deduced that S (kn> are functions of r and t only. Thus the Equation (4.7) in terms of the physical components of extra stress in cylindrical coordinates (r, 0, z) takes the form

D~p+p(v+O)] OS <11> 5<11>--S (22)

~r Or r

a [p + p (v + o)] a[p + p (v + o)] =0, --0.

00 ~z

(4.8)

This system is satisfied if p is a function of r and t, of the form

p = - - D (v q- ~~) -1-- S <11 > q:- S S < l l > _ S (227

dr + K(t), (4.9)

where K(t) is an arbitrary scalar function of time. The physical components of stress T at time t may be then obtained from (2.7) and (4.9).

Other Families. The main results for other families are presented below. To shorten the text the explicit forms of the Equation (2.10) will be not introduced. The forms of the dynamical equation (4.7) and the solutions for p appropriate to each family are not listed here, they can be found in Fosdick [9].

Family 2. From (3.4) and (4.1) we see that ll(2<mk>ll, IfR<ML>[I may be identified with (4.4a). Hence, it follows from (4.3) that B~12>=Bz<13>=0. The Equation (2.11)

a) If we abandon this assumption and suppose that A(t), B(t), etc., are arbitrary functions of time, there result states of deformation in the quasi-static theory of elastic-inelastic materials, i.e. the theory in which the effects of inertia are not considered.

Vol. 23, 1972 Finite Deformations of Elastic-Inelastic Solids 957

is reducible to the system of three ordinary differential equations

dB<E22 ) = (a + b) BE <22 > + h2 [(B<~22 )) 2 + (B<223)) 2] + hi B<~ 22 > + h o, dt

dB (33) dt -- b B~ 33 ) -+- 2 c B(E 23 ) -}- h 2 [(B<E 23 ))2 At_ (B<E33))2] .j_ hi B133 ) q._ ho '

dB(23 > dt = c BE <22> + (a/2 + b) B<E 23 > + h2 [Bi 22) + B<E 33>3 B<E 23 > + h, B<~ 23>,

where a = -2A/A, b= -2[~/B, c= d-AC/A; I= A -~ +B(E22>+B(E 33>,

II=A+(B~22) + B<~33>) A-1 ' A=B<E22) B~33>_(B~23>) 2.

Family 3. Under the orthogonal transformation (4.4a) the invariance condition (4.3) yields u<12)_u<13>_a ~E - ' - ' e - , , . Then we have from (2.11)

dB<~22> = 2 ( a + e) B<E 22) + 2b B~ <23) + h2 [(B<~22>) 2 + (B<223)) 2] + h I B<E 22> + ho, dt

dB(E 33 ) dt = 2 c B<E 33 > + 2 d B~ 23 ) + h 2 [(B<E a3 >)2 + (B~33 >)2] + hi B133 ) + ho '

dB(E 23)=dB(E22)+bB(E33>+(a+c+e)B~z3>+h rB(22>+B(33>qB(E23>+h B <23) dt 2k E e J 1 E ,

where

a = ( C F = D E ) ( C F - D E ) -1, b = ( C D , C D ) ( C F - D E ) -~ (AR 2 +B) ~,

c = ( P C , - [ ~ D ) ( C F - D E ) -~, d = ( E F - P E ) ( C F - D E ) - I ( A R 2 + B ) - ~ ,

e =(AR 2 +/3) [2(AR2+B)]-~; I=-A -~ +B<~22>+B<~33>,

lI=A+(B(EZ2>+BI33))A -1, A-B(22> B (33) /B(23)12 - - E E ~ k E ! "

Family 4. For this family all off-diagonal components of F (kL) are zero. While under the orthogonal transformation (4.4a) the invariance condition (4.3) implies B<12>-n<13)=0, it follows from a similar application of the orthogonal trans- E -- UE

formation (4.4b) to the invariance condition (4.3) that B<~23>=0. Further, since F<22)=F (33) in the case of inflation (or F<a2>= - F <33> in the case of eversion), the invariance (4.1) is satisfied if [IQ<mk>l] and IIR<UL)[] are identified as (4.4c) (or for eversion IrQ<mk>lJ and IIR<ML)II are identified as (4.4c) and (4.4d) respectively). In both cases we have from the invariance condition (4.3) that n<22>_u<33) Then UE -- L'E �9 (2.11) is reduced to one ordinary differential equation

dB<E 22) aB<Ea2> +h2(B<E22))Z +hl B<222) + ho , dt

where a = 2 A [ 3 ( + R 3 + A)]-a ; I = (B<~Z2)) -2 +2B~ 22), II=(B<e22>) 2 + 2(B~22>) -~. ZAMP 23/61

958 Jan Kratochvil ZAMP

Family 5. An application of the orthogonal transformation (4.4 b) to the in- variance condition (4.3) shows that u < ~ 3 > _ o < 2 3 > _ . .~ -,-,~ - 0 . The Equation (2.11) takes the form

dB~l 1 > dt

dB(E 22 )

dt

dB<l 2 >

dt where

a=2A/A , b = B - riB~C,

I I=A +(B<E n> + B(E 22>) A - 1

Fs <II>~2~_/1~<12>~2qA_/~ IQ<II>A_ - -=aB~l l>+h2L~BE J ~ ' e I a - , u ~ - h o ,

- - = (a + 2 c) B~ 22 > + 2 b B~ 12 > + h2 [(B(r 12 >)2 + (B~Z2 >)2] + hi B(E 22 > + ho,

= (a + c) B<E 12 > + b B<~ H > + h 2 [B~ 11 > + BE <22>] B<~ 12 > + hi B<I 2 >,

c--C/C; I=A-I+B<E~>+B<E 22>,

A =Bi 11> B i 22> - (B~12>) z.

A c k n o w l e d g e m e n t s

The author wish to express his sincere thanks to Dr. I. Hlavfi~ek and Dr. V. Kafka for their valuable comments.

References

[1] C. TRUESDELL and W. NOLL, Nonlinear Field Theories of Mechanics, in: Handbuch der Physik II1./3, S. Fliigge, ed. (Springer, Berlin/Heidelberg/New York 1965).

[-2] A.E. GREEN and .I.E. ADKINS, Large Elastic Deformations and Nonlinear Continuum Mechanics (Clarendon Press, Oxford 1960).

[3] M. SrNGH and A.C. PIPKIN, Note on Ericksen's Problem, Z. angew. Math. Phys. 16, 706 (1965). [4] M.M. CARROLL, Controllable Deformations of Incompressible Simple Materials, Int. J. Engng. Sci. 5,

515 (1967). [5] M.M. CARROLL, Finite Deformations of Incompressible Simple Solids I. Isotropic Solids, Quart. J.

Mech. Appl. Math. 2I, 147 (1968). [6] M.M. CARROLL, Finite Deformations of Incompressible Simple Solids II. Transversely Isotropic Solids,

Quart. J. Mech. Appl. Math. 21,269 (1968). [-7] M.M. CARROLL, Finite Bending, Stretching and Shearing of a Block of Orthotropic, Incompressible

Simple Solid, J. Appl. Mech. 35, 495 (1968). [8] M.M. CARROLL, Controllable Motions of Incompressible Non-simple Materials, Arch. Rat'l Mech.

Anal. 34, 128 (1969). [9] R.L. FOSDICK, Dynamically Possible Motions of Incompressible, Isotropic, Simple Materials, Arch.

Rat'l Mech. Anal. 29, 272 (1968). [10] J. KRATOCHVIL, On a Finite Strain Theory of Elastic-Inelastic Materials, Acta Mech., in press. [11] B.D. COLEMAN and M.E. GURTIN, Thermodynamics with Internal State Variables, J. Chem. Phys. 47,

597(1967). [-12] J. KRATOCHViL and R.J. DE ANGELIS, Torsion of a Titanium Elasto-Visco-Plastic Shaft, J. Appl. Phys.

42, 1091 (1971). [13] W. NOLL, A Mathematical Theory of the Mechanical Behavior of Continuous Media, Arch. Rat'l

Mech. Anal. 2, 197 (1958). [14] C.-C. WANG, A New Representation Theorem for Isotropic Functions: An Answer to Professor G.F.

Smith's Criticism of my Papers on Representations for Isotropic Functions. Part 1 and Part 2. Arch. Rat'l Mech. Anal. 36, 166, 198, (1970).

[15] E.A. CODDINGTON and N. LEVINSON, Theory of Ordinary Differential Equations. (Mc Graw-Hill Book Co., New York/Toronto/London 1955.)

Vol. 23, 1972 Finite Deformations of Elastic-Inelastic Solids 959

Summary In isotropic, incompressible simple materials there are five known families of dynamically possible

inhomogeneous finite deformations. It is shown that these deformations are also possible in isotropic, elastic-inelastic materials with temperature independent response. The method of solution is an inverse one. The deformation is specified precisely at the outset, and the problem of finding the surface tractions which are required in order to maintain the deformation is reduced to the problem of solving a system of ordinary differential equations.

Zusammenfassung In isotropen inkompressiblen einfachen Substanzen mit Ged~ichtnis gibt es ftinf Gruppen yon

dynamisch m~glichen nicht-homogenen Deformationen. In der Arbeit wird gezeigt, dass diese Gruppen yon Deformationen auch in isotropen inkompressiblen elastisch-inelastisehen Substanzen dynamisch m6glich sind. Die vorgeschlagene Methode stellt einen inversen L6sungstyp dar. Die Deformationen werden als gegebene Funktionen der Zeit vorausgesetzt, und das Problem der Berechnung der Ober- fliichenspannungen, welche diese Deformationen erzeugen, wird auf die L6sung eines Systems von gew6hn- lichen Differentialgleichungen reduziert. Es wird auch gezeigt, dass die Anzahl der Gleichungen in diesem System ftir keine der erwShnten Gruppen von Deformationen 3 + n iibersteigt, wobei n die Zahl der Struk- turparameter ist, die die elastisch-inelastische Substanz charakterisieren.

(Received: February 24, 1972; Revised:August 28, 1972)