i c *' On the Thermodynamics and Approximation Theory of Viscoelastic Materials D. C. LEIGH University of Kentucky '7 Lexington, Kentucky A / 6 .a.
Summary A thermodynamic theory of materials u-:h memory fading memory was an approximation theorem for function- was developed in terms of difference histories by als by Coleman and No11 [3].
Coleman. The author has developed the theory in terms A natural next development was the application of of past histories. The second law of thermodynamics the approximation theory to the thermodynamics of mater- in the form of the Clausius-hhem inequality is taken ials with memory. In attempting to carry out this pro- as a resrriction on processes as well as on constitu- gram, the author found that Coleman's theory in terms tive equations. The Coleman-No11 approximation theory of difference histories [l, 23 was not entirely cokrect for materials with fading memory is developed for the (see the appendix Sect. 10 of this paper). It appeared thermodynamic theory. that the thermodynamic theory in terms of past histories
' A parallel development of the thetmodynamic theory could be a valid theory, however in [2, Sect. 91 Coleman and the approximation theory is carried out for fluids only implies a summary of that theory. The author has with fading memory. For the zero-order approximation, developed the thermodynamic theory in terms of past the pErfect fluid, the usual classical results are histories in Sect. 3 of this paper. Furthermore, in obtained. For the first-order approximation, corres- this theory the Clausius-hhem inequality is taken as a ponding to the Newtonian fluid, the constitutive equa- restriction on processes as well as on constitutive equa- tions differ from the traditional ones because of the tions. In Sect. 4 the Coleman-No11 approximation theory possibility of certain additional terms; restrictions for materials with memory is developed for the thermo- are obtained on the new coefficients as well as the dynamic theory of Sect. 3. The foregoing results are classical restrictions on the usual coefficients. For specialized to fluids. The restrictions of the Clausius- the second-order approximation some restrictions are Duhem inequality are determined for perfect fluids, linear obtained on the material coefficients. fluids and the second-order fluid approximation.*
1. Introduction 2. Review of preliminaries
Recently two fundamental and pioneering papers For the mst part this section will consist of a have appeared by Coleman [la23 on the development of review of appropriate definitions and notations from the thermodynamics of materials with memory. Prior to Sects. 2,3,4 and 6 of Colemn [l:. this, the mechanics of materials with memory, ignoring Following Coleman [laSect. 21, we codsider a thermodynamics , had been extensively developed; many references are given in references [I, 23. AII impor- *See also Eringen [4] which treats general non-polar materials and specific constitut)Veequations different tant development in the mechanics of materials with from those of this work. 4 body B with material points X. A thermodynamic process Couple stresses, body couples, and other echanical din B is described by eight functions of X and the tirme interactions not included in gor hare are assumed to
t. The values of these functions have the following be absent.
phys ica 1 interpretations: In order to specify a themdynamic process it
(1) spatial position vector = &(X,t); here the suffices to prescribe the six functions X&e,a,~ and 0.
function &, called the deformation function, describes The remaining functions kand r are then determined by
a mtion of the body. (2.1) and (2.2). (2) The synsnetric stress tensor 2- z(X,t). &re we depart somewhat from Coleman [l] and make (3) The body force vector k=k(X,t), per unit a new definition. We define an allowable thermodynamic
mass, exerted on B at X by the "external world". process for a body as a thermodynamic process which
(4) The specific internal enerE 8 = e(X,tj, per satisfier the C:zieius->&.zz3 i2eq.ality [SI:
unit nmss. (5) The specific entropy v = T(X,t), per unit mass. We take (2.4) as a staterent of the Second Law of Thenm-
(6) The local absolute temperature 8 = @(X,t), dynamtcs.
which is assumed to be positive: 8 >O. Following Coleman [l] again, it is often convenient (7) The heat flux vector --q = q(X,t). to identify the material point X with its position vector (8) The heat supply r = r(X,t); this is the radi- &in a fixed reference configuration R and to write ation energy, per unit mass and unit time, absorbed by z=;&cst> (2.5) B at X and furnished by the "external world". The gradient of x(X,t) with respect to X, i.e., 2 -u Such a set of eight functions is called a them- E= gst) - azq,t)iax_ (2.6) dynamic process if and only if it is conpatible with is the deformation gradient tensor at (i.e., at X)
the law of balance of linear momentum and the law of relative to the configuration R. It is well known that balance of energy: i=I,L i.e., k=t5-l (2.7) [&E da - f, pdv - - Jv pdv (2.1) It is assumed that --X(X,t) is always smDothly invertible in its first variable, Le., that the inverse E-' of 5 Jv tr{x gdv - japzda - fv; pdv = -jvr pdv (2.2) exists, or, equivalently, that det E# 0. where the conservation of mass is given by Since %/auoccurs often Coleman introduced the Jv pdv = constant (2.3) abbreviation -g = a/* (2.8) Ifre p denotes the mass density; E = @*is the ve- The mass density p is determined by EthrOugh the equa- locity gradient; a superinposed dot denotes the material t ion time derivative, i.e., the derivative with respect to P PrIIdet (2.9) t keeping X fixed; tr is the trace operator; V and dv where pr is a positive number, constant in time and indicate integration Over the volume occupied by the equal to the mass density in the reference configuration body B and aV and da indicate integration over the R. surface of the space occupied by B. me specific free energy is defined by The assumed synmetry of the stress tensor insures *=8-e~ (2.10) that the -nt of -ntum is automatically balanced. Using (2.2), (2.7) and (2.10), under appropriate moth- Let 3 = (3,01) and 5 - (%,%I. The sum & + 5 is nee& assump.tions the differential equation form of the defined to be
Clausius-Duhem inequality is & +&I = consider function which is determined A~(s>= &(t sj = ~t B(t) = f(cpt(S)) (2.13) Using this compact notation we can rewrite (2.14) as In wing (2.13) it will be understood that whenever a f = f(g(s);@, f = t,s,5a (2.21a,b,c,d) function of this case appears as the argu- s, in (pt(s), Introducing the generalized stress ment of a function, f, then f is to be considered a (2.22) Nc LE-T. -7) = 6QITlDT. s) functional of +(s). - (. - A material is defined by a set of constitutive we see that by (2.18) the Cleusiue-Duhem inequality assrrmptions, which are restrictions on thernudynandc (2.11) can be rewritten as -1 processes. Following Coleman we define a simple rpate- ey=- *+go Ip,-,a.po (2.23) rial as one for which where f = f(p,t(s),et(s); &, f 1 tsTs~~(2.14a,bSc,d) CIi=J &a) (2.24) Following Coleman a themdynamlc process is said It follows from (2.22) and (2.21b), (2.21~) that is to be admissible in B if it is compatible with (2.14) given by a functional of &'(s) which is also a function at each point X of B and at all times t. Thus an allow- of g, that is N able admissible thermodynamic process is an admissible N - gqt Duhem inequality. Coleman [l] shows that the following The past histolir g(s) is defined by Coleman [2, is an admissible thermodynamic process: E=&(X,t) and Sect. 91 as the history Ut(s) restricted to the open 0 = O(X,t) are assigned for all t and X in B. interval 0 < s C 01, that is, Coleman rl] introduces the following compact nota- d(s>= ~'(s), o c s < m (3.1) tion. We denote by &the ordered pair (FFO), where For the time being we will assme of $(s) only that it is any tensor and 0 any scalar: have a limit at s - 0 and we define & = (500) (2.15) A = lim At(s) (3.2) -c 6-0- Let LY be a scalar, the product Ah = & is defined to be The history At(s) is allowed to have a jump at s = 0 so *=(Qgg dJ) (2.16) that in general b !L+ 15 (3.3) there exist functionals ah+ and a,* such that N 5 pshere A is defined by (2.20). cc-, Pollawing Coleman 12, Sect. 91 we define a new functional for the free energy * by means of *(t)= *@(SI; &(t))- *(;?F(s>; &),&(t)) (3.4) (3.10) where the new functional form is indicated by $( ; , ) where is an arbitrary element in the ten-dimensional as compared with the old form *( ; ). vector space and :is an arbitrary vector. Now we can see that the time derivative of * is We also see from (3.5), (3.6) and (3.8) that $ required for use in (2.23). Let us then proceed to mst be differentiable in the function $s), that is formally take the time derivative of the last term in there exists a functional 6* such that (3.4) and then make the assumptions required to justify such a procedure. By definition 1 i(t) = lh~;[*(t + k) - *(t)] (3.5) where in order to be consistent with (3.2) L(s) must k-0 have a limit as s 4 0, where 6$ is a linear functional Then by (3.4) we have of IJs), and, I IJS) jjh is a suitable norm of L(s). $(t + k) = *(&+k(s); &(t+ k), g(t + k)) (3.6) As discussed by Coleman El], we may consider a class Now because of the possible jump in Lt(s) at s - 0 as of materials whose mem~ryfades gradually in time. In indicated by (3.3), AJt) and g(t) can only have deriv- that case a suitable norm I z(s) lh is one whose value atives for positive k, that is depends more on IJs) for small s than for large s. k >O Nh(t + k) = AJt) + g(t)+ o(k), (3.7a) Such a norm is = &(t) k >O (3.7b) ug(t + k) + kiL which goes to zero rapidly as s - 0). We summarize the above statements concerned with differe-tiation with where we have made use of (2.19) and where o(k,s) in- t respect to &(s) as an dicates that the error term depends on s as well being Assumption of fading memory. We consider a class small order k. Now since At(s) is allowed to suffer a of sixple materials for which the specific free energy jump discontinuity at s = 0, then at a later instant is differentiable in the past history in terms of the t+k t + k (k > 0) the function & (8) has a jump at s = k. norm (3.12) where the influence function decreases Hence the expansion (3.8) will only be meaningful for rapidly to zero as s -. m. We note especially that the s > k. This restriction has to be kept in mind in the past history must have a limit as s -L 0, but that it limit as s -. 0. To be consistent with the assumption may otherwise be badly behaved so long as the norm that the argurnent function ck(s)of (3.6) satisfy (3.12) exists. (3.2) we must ass- as s - 0 that the derivative simple material for which the above assumption t A d&(s)/ds exist and that o(k,s) exist. of fading memory holds will be called a viscoelastic We see from (3.5), (3.6) and (3.7) that we must material. ass- that * is differentiable in Land &. That is, b Now returning to the problem of evaluating (3.5), our discussion it is clear that &(t)in the positive be substitute (3.7a), (3.7b) and (3.8) into (3.6); we direction may be chosen arbitrarily for jump histories. then use (3.91, (3.10) and (3.11) and carry out formally Since can be assigned arbitrarily in (3.16) with N the limiting process of (3.5) and finally get everything else held fixed it is clear (see [laSect. 61 for a detailed argument) that $ = *(&s); ;?) (3.17) where the derivatives &(t),&t) and hence i(t) may only That is, the specific free energy cannot be a function exist in the positive direction. In carrying out the of the temperature gradient. Thus (3.16) reduces to above procedure it is necessary to make a weak addition- al assumption that* 1t lhi; 6Q(s); k(t>,&(t> I o(k,s)) = 0 (3.14) k-w Since & can be assigned arbitrarily holding every- We also write the entropy, stress, heat flux and thing else fixed we conclude that generalized-stress functionals in terms of the past .. c = g ($(SI; = a,&s); (3.19) history AL(s): N -r - We observe that the generalized stress cannot be a f = f@(s); 5, &I, f = h,%$Z (3.15a ,b ,c ,d) function of the temperature gradient 5 that the gener- Then substituting (3.13), (3.15~) and (3.15d) in alized stress is the derivative of the specific free the inequality (2.23) we get energy with respect to the present value of the total = g 6, B) aAi(g(s);N L ev { --..(gw; - d} 8+ + 6t(&s); & & I $s&s)) - g&p;&&)*&-1t history, b and that the assumption of fading memory t must apply to the generalized stress. Next we can - ag.c(&(s); & 8) jL* 0 (3.16) N assign &the value zero independently of the other Now Coleman [laSect. 41 has shown that at a given quantities and get material point X and time t we can arbitrarily choose t dt t 80 I 6S(&(s); &(s) 2 0 (3.20) the past history &(s) and the quantities AJt),&(t),~(t) That is, the internal dissipation o [l, Sect. 61 cannot and be certain that there exists at least one admiss- be negative. Then finally we must have ible thermodynamic process corresponding to this. From Nq(&(s); L, Ng 5 pea, (3.21) our discussion in this section, i(t)may only be defined N The results (3.19). (3.20) and (3.21) are similar in the positive direction for jump histories. Also from to those obtained by Coleman [la Sect. 61. The equi- *Dr. C.4. Wang has kindly pointed out to me that Prof- valent of our equation (3.19) is also given by Coleman essor V. J. Mizel and he proved in a private discussion and Gurtin [SI. A jump in 0 is, according to Coleman the following much mre complete theorem regarding (3.13); and Gurtin, a therm-mechanical acceleration wave. Thus If + satisfies (3.11) for all L(s) which obeys equations (3.19)to (3.21) mey be said to hold for mater- k(s>lh < = and, if one writes + L(s)=(&(s),b(s), ials in which acceleration waves are allowed. However det k(s) # 0, b(s) > 0 then (3.13) holds for all &(s) for a given class of constitutive equations the allow- provided that dsll, < = in the sense that M;(s)/ ance of acceleration waves may impose intolerable t A (s) is absolutely continuous, and, the derivative Y restrictions on the constitutive equations. Let us now d$(s)/ds, which exists almst everywhere, has a finite examine the inequality (3.18) with the assumption that h-norm. 4 we shall only insist that it hold for emooth admissible *(&(E); &,$ = )(&(S) - 3 & + $ 9 (4.3) thedynamic processes. For solooth processes &(s),&(t) for arbitrary E. Further, we split (4.2) in the follow- and i(t)may be assigned arbitrarily, but then ing way: dt &= b,Q= - & !pls=o (3.22a ,b) * *O(&, 9 + *'(&d(s); &* g (4.4) For smth processes the result (3.17) still applhs. where Nothing further can be deduced from (3.18) in a general *o&, @ *Cs &, @, &, @ = 0(4*5a,b) way, unless we ass- that the generalized-stress func- With (4.5b) and the fact that tional gdoes not depend on the tenperature gradient, lim At (8) = (4.6) s- .ud 2 that is we are in position to apply the Coleman-No11 appraxi- &= z (&); &I (3.23) mation theorem for functionals 131 to (4.4). We first In such a case we make the following deductions from the have to extend the previousiy postulated assumption of inequality (3.18) : fading memory by assuming that the functional + is t n-times differentiable in terms of Then assum- 00' = {z(&s); - a,t@cs);N a>} i+ 4d(~). + a*(&); & 2 0 (3.24) ing that &(s) is differentiable n-times at 8 = 0, the -q&(s); h, E) E* (3.25) Coleman-No11 approximation theorem for slow mtione where we shall call 0' the total dissipation. Note that yields (3.23), (3.24) and (3.25) also apply to adiabatic mater- ials Q(t) = OJ with & a paremeter in and to the where i special case of homthermal deformations (g(t) = 0). &:(s)lg=O = (-l) dti (4.8) CI - 2 $ 5 We have observed that the Clausius-l)uhem inequal- and each term in the sumnation is linear in each of the ity may be taken to impose restrictions on allwable -9. For the n-th order approximation the swtion in processes as well as on constitutive equations. We find (4.7) is over all sets of integers (jl,...jk) satisfy- that there mst be a give and take between constitutive ing equations and allowable processes : the mre allowable 1 * jl %..s fk s 11, jl +..a+ jk s n (4.9) processes one wishes to consider the greater are the re- The asymptotic approximation sign -has the folloving strictione on the constitutive equations whereas for specific meaning in this article: The retardation t mre general constitutive equations one has fewer allow- QJs) of the past history &(e) is defined by able processes. &e) = &m), 0 CY < 1 (4.10) 4. doproximation Theoq We see that as the retardation factor CY- 0 the history t We introduce the difference past history kd(s) varies mre and mre slowly. Then the Coleman-No11 def ined by theorem applied to (4.4) is At(S> (4.1) -T &a(') + & We may rewrite (3.17) in the form * *(d(e);@ &, b) (4.2) where the sumPation is governed by (4.9). 'he form (4.2) can be written even if (4.6) below did It follows from (4.7) that not hold, so that by (4.1) the form (4.2) bas the pro- perty blogous appropimations can be written for the general- Each of the term If::::::: (pJ,q,~O)( 1 is a k-linear fzed-stress functional and for the heat-flov fuactioual. fuxtfon. Appropriate "component1*fortm of (4.12), Using the property (4.3) it can be Shawn that the differ- (4.13) and the other approximation forms can be written. ential in (3.20) has the approximation form 5. Thennodynamics of Viscoelastic Fluids Ue wish to apply the foregoing theory to visco- elastic fluids. Purely pechanical sirpple fluids were first defined and discussed by No11 [7]. The general- ization of the constitutive equatiom to include the- We note that on substituting the approximations into dynamics is, following Coleuun [l, Sect. 133. (5. la, b ,c ,d) (3.18) sd tskiq e11 e-mmmtims to the n-th order, all f = fGt(S), ot(s); P(t), dt)), f *,s,sa terms in (3.18) have been approximated to order (n + 1). where For the jump history results (3.19), (3.20) and (3.211, we see, first of all, by substituting the approximations in (3.19) that The functionals in (5.1) are isotropic in zt(s) and (5.3) (3.20). The approximation for the heat flux ouet then and A to be ordered pair (p,O) with a scalar product satisfy (3.21). defined by For the case of histories smooth at s = 0 the approx- A1 Ib = PlPa + 01% (5.4) t imations mist all substituted in (3.18) which then The past history A (e) and the quantity & are again be Y must be satisfied. Note that even though we consider defined according to (3.1) and (3.2). Corresponding smooth histories, the jlmp-history operator a, may yield to (3.3) we note in particular that N non-eerc values. We now put the above results in "c- =sp # E=g(0) (5.5) ponent" form, that is in terms of the deformation grad- Now ient tensor and the termperature. Equation (4.7) becomes _a f (s)lSd 5 (5.6) the 1-et Bivlin-Ericksen tensor. Thus Ni= (3, 4) (5.7) The Clausius-lkrhrp irrquality retains the form (2.23) if the generalized stress is nmi deftned to be where the surrmation is over all sets of indices E= (#p b 7) (5.8) (jl,...Djk) satisfying (4.9) and over all sets of - Then in term of past histories the constitutive indices (Il,. .. , ik) satisfying equations (5.1)D can be written in the form f f (&(E); A, E), f $, 53 (5.9aDb, c) We consider now mdifications of the general theory of Sect. 3 which are needed for fluids. Ue note in (5.2) that &(I))depends on the tine t through the subscript t of &(s) indicating the changing reference configur- b ation ae well as through the history superscript t. It can be easily shown that 6. Approximation Theory for Viscoelastic Fliuds Basically the approximation theory of Sect. 4 applies to viscoelastic fluids with &replaced €or the most part corresponding to (4.13) takes on a somewhat different replaced by 1.t is clear that the equivalent for fluids of the where on and as well as on &s). Since &and b also depend on Qand Ewe can deduce no relation on the generalized stress corresponding to (3.19) of the general (: (: Iklt i1 .ik theory. On the other hand, since &(s) does not depend (s,Qr, p,Q) 48P & restrictions plaFed on the approximate consititutive that it must be isotropic and thus from (7.6) we have eq*%ations for viscoele-tic fluids by the ClaUsiUs-hh,~~~~ T-- p2+$O(A)L (7.12) inequality (5.17). We will assume that the generalized From (7.7) we have stress is not a function of the temperature gradient T - - a,tO(A) (7.13) and we shall not study here the restrictions placed by We note that the results (7.11), (7.12) and (7.13) (3.25) on the constitutive approximation for the heat define the classical perfect fluid. Also these results flux. do not depend on the jump (which may be a shock wave as In terms of the retardation factor (Y of Sect. 4, well as an acceleration wave) and therefore it can easily be verified that these results are necessary as well as sufficient conditions that (5.17) be satisfied for smooth histories as well as jump histories. 8. Linear Approximation Using the results (7.11), (7.12) and (7.13), the approximations for $, ;and 11 to order n = 1 are given below. It will be assumed that all coefficients are functions of & and A unless indicated otherwise. t - to(A) + A(trg) + Be: (8.1) -T - - p2+fo(A)I- + h (trG)&+ hr + &:I- (8.2) ll --a8fo(A) + a(tr&) + be: (8.3) From (7.1) it follows that We have used the porperty that the above equations nust iimr(eO')1 = a, 2 o (7.4) r Q-0 be isotropic in b. For histories with a jump at s = 0, each of the The conditions (7.5) to (7.10) made a, in (7.1) variables A, (3-ik-T %(&,A) - ae$o(&,A) (7.7) trsto(&,A)g- 0 (7.8) tr$to(&sA) 0 (7.9) a,r$o(&,A) = 0 (7.10) From (7.10) we conclude that to is not a function of 8,. From (7.8) and (7.9) we conclude that to cannot be a function of 5. Therefore we have r t - lo(A) (7.11) e,, while holding everything else fixed, we must have Since &(A,A) cannot vanish we conclude from (7.5) A=B=O (8.7) 489 Then the tneq~li~(8.6) reducce to with caution to probleps such an boundary-layer flow r paet a send-infhite plate or "shock structure". Questions concerning varioua types of approximation to - [a(tr&) + bo:] 6 2 0 viscoelastic flows and their respective ranges have been discussed recently r8.93. For the funp history case, that Is, for acceler- 9. Second-order Approximation ation waves, and 6 can be varied independently of Since the linear approximation does not sustain everything else, so that it can easily be seen that all acceleration waves, we consider only snmoth uotions the first-order coefficients vanish and the n = 1 ca8e here. reduces to the perfect fluid. Thus we conclude that From the results (8.10) and (8.11) we see by (8.9) t'ne iinear iiuid cannot satisfy t'ne Ciausius--em that al is not identically zero. We now rewrite (8.4) inequality for arbitrary acceleration waves, which is in the form the well-known result of Duhem for the classical linear 00' = ald + hd' + o(2) (9.1) viscous fluid [SI. For those cases where al vanishes, that is & - %and For the suooth history caee we drop the superscript 0, - 0, it follows that r and we have, using 0 O,, lid (00') = + 2 0 09a9 (9.3) al = [ AA(trh)a + btrg] + 2P 2 We write the secoui-order approximation to $ in the form rt - to + f(%,@d + c tr&a + Do, (9.4) where now the coefficients are functions of A only. A men by (6.4) it can be seen that has the form simple analysie wing (7.3) yields the classical results 80 g(k,o1abaoa) - ctr(& - tkmTh- r~~'lr, + 2 (9.5) pro, A+-pZO (8.10) 3 Since every term in g is of degree three, that is the as well as the results subscripts of the factors in each term mst total three, (8.11) we can see that g vanishes when h and 0, vanish. We can Thus, in sumary, we have for n = 1 conclude that the condition (9.3) requires that rt - rto(M (8. U) C-D=O (9.6) paa,rt0u);L + UA)(tr&),€ + P(A& + uT - - dl& (8.13) and that there are no other restrictions placed by (9.3). 9 - - berto(A) + a(h)tr& + b(h) 0, (8.14) 10. Appendix on Coleman's Difference-History Theory with the restrictlone (8.10) and (8.11). We observe The difference history &(e) of the history kt(s) that these constitutive equations differ from the tradi- is defined by t tional constitutive equations for linear fluids because -At(s) - + & &- $(O) = &(t> (10.1) of the possibility of the presence of the last term in Coleman [1,2] writes for the free energy functional and the last two terns (8.14). It rmst be (8.13) in rt - rt(g(S)) - t(&(s); g (10.2) kept inmind that (8.12),(8.13) and (8.14) is a first- Coleman [1,(5.4)](see (10.6) below) does not use (10.2) order slaw motion approximation to viscoelastic fluids. with the restriction that the functional argument vanish Whether or not these equations could represent a fluid at s - 0. Clearly then the right-hand side of (10.2) undergoing rapid motions ig a question that remains to should have the property be investigated. Thus these equations should be applied *(&(e); ;?) = rt(&(s) - 9 k+9 (10.3) 490 .. b for arbitrary Q. (10.14) Nou let us &ne equation [l, (5.14)] which defines the differential operator V (omitting the dependence on Lt*(s) is called the jump continuation of kt(s) with : jump J,. Naw applying the property (10.3) we see that v*(&);@ Q = 6*(p); &Jr?'(s>> (10.4) (10.13) reduces to where $(e) is the constant history v = t(&(s);g (10.15) t z(s)=& OSs The right-hand side of (10.4) is defined by [l, (5.4)]: '&is is a physically unacceptable result. Equation C(L(s) + Q5d;9 (10.3) was brought about by the equivalence [2,(5.5)]: t t = *Cb>;W + 6tC(s);hJQt(s)) + 41QtCs>l 1 &*(SI = - 2 (81, = L (10.16) (f0.6) which in turn was due to the use of the norm (3.12). where the norm 1 $(e) 1, is defined by (3.12). From In the author's opinion, the use of the norm (3.12) (10.5) and (3.12) we have for histories with a jump at s = 0 is not physically reasonable. Furthermore, it is the use of the norm and therefore (3.12) along with our result (10.12) that leads to the o(fl llh) = O(1 11 (10.8) previous deduction that the generalized stress is iden- Also by (10.5) and the property (10.3). the left-hand tically zero. Coleman's reasoning on pp. 17 and 18 of side of (10.6) can be written as [l], which leads to the generalized stress relation *(gs) + ,.(SI;@ - #(gs);A+ 9 (10.9) [l, (6.24)], hinges on considering a jump in qt(s)/de Substituting (10.8) and (10.9) in (10.6) we have at s - 0 and the use of the norm (3.12) with such junp *(@I + $(s);@ histories. a *(x(s);$ + ~+(L(S);~Q,(S)) + Gt(s) 1,) (10.10) Acknasledgnents But by comparison with [1, (5.5)] we see that This research was supported by the National Aeron- ~(p);&~w- aL*(mwg p. (10.11) autics and Space Administration Grant NsG-664 and National Now comparing (10.4) and (10.11) we see that Science Foundation Grant GK-621. Professor C. Truesdell vt(~(~);a= aA*(p);u- (10.12) and Dr. C.4. Wang made helpful criticisme and conments That is, the differential operators V and a0 are identi- on several points in Sects. 3 and 5 of an earlier version cal. The main consequence of this result in Coleman's of this paper. theory is that the generalized stress by [I, (6.24)], is identically zero. Of course, such a result is un- 1. Coleman, B.D., Thermodynamics of materials with memory. Arch. Rational Hech. Anal. l.7, 1-46(1964). acceptable. Let us leave this dilemma for the moment 2. Coleman B.D., On thenmdynamics, strain impulses, and turn to another point. and viscoelasticity. Arch. Rational Mech. Anal. l.7, 230-254 (1964). Let us examine equations [2, (5.6111: 3. Coleman, B.D., and W. Noll, An approximation theorem f - *(.&(e) - Lt(s);A+ 8 - *'*(a (10.13) for functionale, with applications in continuum mechanics. Arch. Rational Mech. Anal. 5. 355-370 This equation is supposed to give the value ** of the (1960). free energy corresponding to the history Aye) defined 4. Eringen, A. C., A unified theory of thermDmechanica1 materials, Technical Report No. 30, School of Aero- by nautics, Astronautics and Engineering Sciences, Purdue University, September 1965. e. c 5. Truesdell, C., and R. A. Toupin, The Classical Field Theories.Encyclopedia of Physics, Vol. III/l, edited by S,. Flugge, Berlin-Gottingen-bidelberg: Springer 1966. 6. Coleman, B. D., and M. E. Gurtin, Waves in materials with emry IV. Thermodynamics and the velocity of general acceleration waves. Arch. Rational Mech. Anal. l9, 317-338 (1965). 7. Noll, W. , A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2, 197-226 (1958). 8. Pipkin, A. C., Approximate constitutive equations for viscoelastic flows. Report M-G484/9, Division of Applied Mathematics, Brown University, September 1965. 9. Metzner, A. B., J. L. White and M. M. Denn, Consti- tutive equations for viscoeiastic fiuids r'or short deformation periods and for rapidly-changing flows- significance of the Deborah number, manuscript. . c Corrigenda and Addenda to ON THE THERMODYNAMIC AND APPROXIMATION THEORY OF VISCOELASTIC MATERIALS D. C. Leigh Equation (3.8j shouid be repiaced by Now the linear functional 64 of (3.11) can be represented by an integral: m t t 6g(Lr (3.18) is replaced by (5 I and (3.20) is replaced by We observe that the second terrr, iil tile internal dissipation ine_gality (6) is non-zero only if shock walres (jumps in A) occur in the processes H under cons ideration. The remainder of 4 3 is unchanged; several consequent changes should be made iri #$ 4, 5, 5 and 7; $4 8 and 9 are unchanged. Actually from44 oil is valid if we limit our considerations to processes in which shocls do not occur, but acceleration waves may or may not occur as the case may be in the paper. ,ct;s) (2.25) thedynamic process that is allowed by the Clausiun- 3. ThermDdynamlc Theory in terms of Past Histories ;L,E/IJS)) 5 $'(AT(z)y s ; N-A a g> * Nr