Inl. J. NOr1·Lint'ar Mt'dwtiC'l. Vol. 1, pp. 439--50L PeTpmon Press 1972.. Printed in Great Brit_in

A VARIATIONAL PRINCIPLE FOR NON-LINEARTHERM OVISCOPLASTICITY

D. R. BHANDARIand J. T. aDENDepartment of Engineer;ng Mechanics. University of Alabama. Huntsville. Alabama. U.S.A.

Abstract-A general variational principle associated with the non-linear theory of thermoviscoplasticity ispresented. In particular. this paper shows that the use of Vainberg's theorem on non-linear potential operatorsand a special bilinear form using the convolution of two functions makes possible a variational formulationfor non-linear initial-and-boundary value problems. Several alternate principles and their relalionship With theexisting variational principles are also discussed.

I. INTROnl!(TION

IN THE following we formulate a general variational principle associated with the non-linear theory of thermoviscoplasticity. The principle embodies the general theory ofirreversible thermodynamics of dissipative continuous media in which a notion of finitepermanent deformation (plasticity) is accommodated. As such. the associated Eulerequations for the functional developed herein may be reduced as special cases to those des-cribing, for example. Coleman's thermodynamics of simple materials [1]. the generaltheory of thermoplasticity of Prezcnya and W onjo [21 the theory of elastoplasticity ofGreen and Naghdi [3], finite elasticity, heat conduction in dissipative media. etc. Toobtain this principle, we employ a generalization of Vainberg's theorem [4] of non-linearpotential operators and construct a functional which we show assumes a stationary valuewhenever the displacements, strains, stresses. dissipation. temperature gradients. heatflux, entropy and the temperature satisfy the complete collection offield equations (momen-tum, energy. constitutive etc.) governing the behavior ofthermoplastically simple materials.We have used a similar procedure successfully for constructing variational principles fornon-linear viscoelasticity [5]. and its application to the more difficult problem of visco-plasticity is discussed herein.

In Section 2, we briefly review the system of field equations of non-linear thermovisco-plastic materials. In describing the constitutive equations for this class of materials wehave chosen as independent variables ,. ". 0 and g, where' is called the difference straintensor. defined by , = y - " (y and" being the total and inelastic strain). However, wenote that the use of two tensors " " instead of y, " implies no loss of generality in a generaltheory (see [6] and [7]).

A review of the essential features of certain theorems on variational principles and poten-tial operators is given in Section 3 for completeness. We then demonstrate in Section 4that, under appropriate additional assumptions. the variational principles of Gurtin [8].Nickel [9]. Sandhu and Pister [10]. aden and Bhandari [5] can be obtained as specialcases of the general variational principle presented herein.

Finally, in Sections 5 and 6 we construct variational principles associated with thethermodynamic theory of elastoviscoplastic materials with internal state variables. We

489

490 D. R. BHANDARI and 1. T. ODEN

also demonstrate that Gurtin's method of transforming the initial valuc problems intoboundary value problems governed by integrodifferential equations is not necessary. Infact, as pointed out by Tonti [Illwe are able to show that with the direct use of the initialvalue problem in the differential form we can arrive at the variational principle. ThenGurtin's method becomes a special case of this formulation.

2. THE GOVERl'lING FIELD EQl!ATlOl'lS

For future reference, we rcfer here to a complete system of field cquations governingthe behaviour of thermoplasticaIIy simple materials. Let fli be a closed region of a three-dimensional Euclidean space occupied by a continuous body f!J and let (jt x (- oc,. 0:::)denote the domain of all functions of position x and time t, fli being the closure of an open,bounded and connected region fli of the three dimensional Euclidean space and ( - 00. (0)is the time interval. We assume for simplicity that the material coordinates Xi of the particlein the references configuration are rectangular cartesian. With this notation. the strain-displacement and the thermal gradient relations are

Yoo = -2t(1I . + II} . + II .11 .)

f) '.) • I m. I m,) (2.1a)

(2.1 b)

where y(x. t), u(x. t). g(x. t) and O(x, t) denote respectively, the Green strain tensor. thedisplacement vector, the thermal gradient and the absolute temperature. The commadenotes partial differentiation with respect to the material coordinates Xi'

The local forms of the laws of linear momentum, angular momentum and the energy arc

[uiJ(i> . + u .)]. + pF = pii' uiJ = (I}i (2.2a b)m) m. J 'I m m ~ •

uiJ). .. + ph + qi. = p€ (2.2c)J) ,I

where all quantities are referred to the reference configuration: u(x.1) being the secondPiola-KirchholT stress tensor. F(x, t) the body force vector. pIx) the mass density of thesolid, q(x, t) the heat flux vcctor, h(x, t) is thc internal heat generated per unit mass perunit time and f(X. t) the internal energy density. The superposed dot indicates differentiationwith respect to time t and i>i} denotes the Kronecker delta. Equation (2.2c) can be written inan alternate form by introducing the Helmholtz free energy density cp = f - SO(S(x, t) beingthe specific cntropy) and the internal dissipation fj = UiJ}'iJ - p(ip + slJ) into (2.2c): i.e.

(2.2d)

To complete the description of the behavior of a thermoplastically simple solid, weneed only add to the basic kinematical and physical laws the constitutive equations whichcharacterize the material of which thc solid is composed. As a definition. we say that amaterial is thermoplastically simple at a particle x if the response at x is detcrmined bythe histories of "(s) = y'(s) - ,,'(s) ('I being the total strain tensor). the inelastic strain,,'(s), the absolute temperature O'(s) and thc current value g of thc tempcrature gradient at x.We define such a material by a system of constitutive equations:

00

cp = cp [{'Is). ,,'(s). ()'(s); g(t)].=0

(2.3a)

,r

A I'llriarional principle for non-linear rlrermoviscoplllsricily

00

a = :F ["(s), ,,'(S), orIs); g(t)].=0

00

S = f/ [,Irs), ,,'(s). orIs); gU)].=0

'7)

;1 = .,V [,Irs). tl(s), O'(s); g(t)]s=o

00

q = :!J. [,Irs), ,,'(s). or(s); g(t)] .5=0

491

(2.3b)

(2.3c)

(2.3d)

(2.3e)

.. Here "~IS) = '(I - s). ,,'(s) = ,,(t - s) and or(s) = O(t - s) are, respectively. the total historiesof the "difference strain", the "inelastic strain" and the absolute temperature (0 < s < w);the dependence of the constitutive variables on x is understood. If the functional rp possessessufficient smoothness properties. then it can be shown that this class of materials is charac-terized by only three constitutive functionals one describing the free energy which isindependent of g and the other two being thc heat flux and the inelastic strain rate. Forexample.

00

q> = rp ['~. ,,~, O~:" ", 0]5=0

7)

;, = Ai' ['~. ,,~. O~;,. ". O.g].=0

00

q = 21 ['~. ,,~. O~;,. ". 0, g].s=o

(2.4a)

(l.4b)

(2.4c)

In (2.4), we have decomposed the total histories into the "past histories" and the ··currentvalues" of the respective argument functions; e.g. the total history of "(s) (0 ~ s < (0) canbe represented by a pair ['~(s), "(0)]. where '~(s) = 'Irs) [s E (0. (0)]. Then the stress. entropyand the internal dissipation are determined from (2.4) by the formulas

00 00

(1ii= §'ii [r~(s); r] = pc, rp [r~:r] (2.5a)s=O .'laO

00 00

S = g> [r~(s); r] = -00 rp [r~:r] (2.5b)$=0 ssO

00 00 00

u = 9 [r~(s): r] = tr{(a - a~ rp [ : ])in - p{(jrr rp [ : Ir~]}. (2.5c).5=0 .psO 5=0

00

Tn (2.5) we have used the notation r = (,.". 0) and (j r rp [ : I ] denotes the Frechetr 5=0

differentials which are linear in their arguments to the right of thc vertical stroke:00 '7.

0, tP [ :] and iio tP [ : ] denote usual partial differentials of tP with respect to , and 0s-o s-o

respectively.

492 D. R. BHANDARI and 1. T. aDEN

To the system of field equations just citcd, we must adjoint the .initial and boundaryconditions. Let {fl. and !J4. denote thc disjoint sets whose union is afJP. Here {fl. is taken asthe portion of the boundary on which displacements are prescribed and ~., representsthe portion over which tractions are prescribed. Also let {flo and fllq denote disjoint setswhose union is 091.918 being the portion of the boundary on which temperatures arespecified and fJlq is the portion over which heat flux is prescribed. Further, let the outwardunit normal vector to the boundary a91be n. Then the displacemcnt boundary conditionsare

1/ = Um m on fJI. x (- 00.00) (2.6a)

the traction boundary conditions are

Tm = nuii(c5 . + II .) = tmI mJ m,)

the temperature boundary condition is

on {fl. x (- 00.00) (2.6b)

on (2.6c)

and the heat flux boundary condition is

Q = qinj = Q on ff1q x (- 00.00). (2.6d)

For simplicity, and without loss in generality, we take as associated initial conditions

andII = U = 0: u = , = '1 = () on .IJP x (- 00. 0) (2.7a. b)

on 91 x (- 00. 0). (2.7c)

Since the solution of initial-boundary value problems requires that we incorporate theinitial conditions explicitly into the field equations, it is convenient to consider the Laplacetransformation of (2.2a), (2.2d) and (2Ab). However, (2.2d) a<; it stands does not admitLaplace transfoqnation. Ip order that such a transformation be performed on (2.2d). wereplace OS by ~S, where S = O/ToS and write

q~j + ph + fJ = pToS. (2.8)

Now (2.2a). (2Ab) and (2.8) can be cast into a form incorporating the initial conditionsby applying the Laplace transform. re-arranging terms and finding the inverse. This yields

9 * [Uii(c5mi + lIm)l i + Plm = pllm (2.9a)

wherein g(e) = t, g'(t) = 1

1m = [g * F m] (x. t)

fj + t = g' * [h + fJ] Ix. t) + p(x) ToS(x, 0).

(2.9b)

(2.9c)

(2.9d)

(2.ge)

In summary. the behavior of thermoviscoplastic materials is described by the conditions(2.6), (2.7) and the system of equations

9 * [uii(c5mi + um)),; + plm - pllm = 0 (2.10a)

A variational principle for non-linear thermoviscop/asticitr

Y"-~2I/ .. +U .. +U .u .)=0I) I.J J, f m. I m. J

00

(j - £i) ['~, ,,~, if,: " ". 0] = 0.=0

00

S - g> ['~. ,,~, O~;"", 0] = 0.=0

00

qi _ !}I err n' 0"" n 0 g] = 0'»r~·'r~ ,"~, ~" ,.•=0

gj-O.i=O00

'1jj - g' * %ij ['~, ,,~.O~:'.", O. g] = 0.-IT

493

(2. lOb)

(2.1Oc)

(2.lOd)

(2.lOe)

(2.101)

(2. 109)

(2.10h)

(2.1Oi)

The symbol * appearing in (2.10) denotes the convolution operator: i.e. if 1/ and v arescalar-valued functions defined on 1/1 x (- 00, (0). then the convolution u. t' is a functionon 91 x [0, 00) defined by ,

1/ • V = J [I/(x. l - r) L'(X.T)] dTo

(x, t) E iJt x [0. 00). Then a bilinear map suggested by Gurtin [8] is defined as

(u. l') = J [1/ • l'] d91iJt

which satisfy the condition that if (u, v) = 0, either u = 0 or v = 0 for t, ~ O.

(2.11)

(2.12)

3. V ARIA TIONAL PRINCIPLE

To construct a variational principle associated with the non-linear theory of thermo-visco-plasticity. we consider in the abstract sense an equation of the form

&'(A) = () (3.1)

(3.2)

where &' is a non-linear operator from a dense set Q £; 1', into 1".,1' and 1" being realBanach spaces. A = A(x, t) is an element of Q. the domain of ~, and 0 E 1". Here thedomain of 9 consists of functions of position x and time l defined by the cartesian productgj x (- <Xl, 00) [x E 9f. t E (- 00, <Xl)]. Equation (3.1) is to be satisfied in the inlerior of 91;on the boundary we impose certain boundary conditions of the form .!l'(A) = O. where .!l'is also a non-linear operator.

The operator &'(A) defined by the formula

(&'(A),::1) = lim ~ [K(A + aA) - K(A)].-0 ex

494 D. R. BHANDARI and 1. T. ODEN

wherc K(A) is a functional oni/' (i.e. a mapping from t' into the real numbcrs R), is calledthe grodient of K(A) and we write 91'(.1) = grad K(A). The operator 91' from 1/ into 1/*(1'* being the conjugate of 1/) is referred to as a potential opcrator. whenever exists afunctional K(A) whose gradient is 91'(.1). Indeed. Vainbcrg [4] has shown that if &'(.1) ispotential. there exists a unique functional whose gradient is gil(A) given by

I

K(A) = f (gil(Ao + A(A - .10)). (A - A» dA + Koo

(3.3)

where Ko = K(Ao) is a constant and), a real parameter.In the following. wc shall identify non-linear operators appcaring in the governing

cquations (2.6), (2.7) and (2.10) of the theory of thermoviscoplasticity and apply the abovetheory for special definitions of thc bilinear form appearing in (3.2). We note that if thegiven operator gil is not potential, then gil can be transformed into a potential operator byusing any of the schemes described by aden [12l

In order to construct the variational statcment of the problcm described by (2.6), (2.7)and (2.10), we set .

&>1.1)= &'(.1) - r = II (3.4)where

_ { ij i }TA - IIi' 'Iii' (1 ,8. q . gj' 0, S, 'IiiI. 00 00r = {pi;. _g*?,ii[: ].0. -g* ~ [: ],0, -g* .!!ll[ : 19*[ph + Xl

.=0 .=0 .=0

IX) (7)

-g* '-:0 [:]. -g*~~[:]f (3.5)

Then. using the definition (2.12) for the bilinear mapping, it follows that in this case (allowingfor non-homogeneous boundary conditions) we havc

<~()A) . .1) = (.'P(},A) - r.A)

=J{),II *PII -14 *g*[Mii(bj+A.u j)],.-II *p!. _),g*Y .. *(J'ij3t m m m ~ . m nI, I m m • IJ

'CIJ

+ 9 * Iii· ~ij[ r~:1'1 - ),g. (1ii* "Ii) + fg • [).lIi.j + luj. I..=0

(7)

+ ;'?u .lIn .] * (1ii - ),g. 8.8 + 9 * 8 * fZ [r~.r] - ),g. S * Sm,l I,).=0

'T.

+ 9 * S * 09' [ r:.: r] - ).g. (g. - 0 .) * qi _ ).g * g .• qiI ,I I

.•=0IX)

+ 9 * gi * .~~[r~: r,g] - ),g * g' * .q~i* 0 - 9 * (pji + t) • 0 + Tog·S.e(7)

- ).g. 'Iii. rJjj + 9 * g'. 'Iij * ,~.;iJr~:r, g]} dA d9f

+ 1,{um• g. [A.uijn/lbmj + ).14",.)] - 9 * 11m * f"'} d;,dtB

- J {Tm * g. (AUm - lim}} d)' dt:4l'+ J {g' * O. g().Q - Q)) dAd""~" ~q

- J {g. Q. ().O - O)} d)' d~~. (3.6)

A variational principle for non-/i~ear thermoviscoplasticity 495

Now introducing (3.6) into (3.3). using the divergence theorem and the commutativity andassociativity of the convolution operator. we obtain the functional

K(A) = 12J{u * plI - 2g * II * ct'?i - 211 * p'.f - 9 * II * (aiju .). - 2g * y .. * aijm m m ,I m '.1m m m,j .1 fJ.If

00 ~,

+ 2g*}'ij*§'ji[r~:r] - g*lhft + 2g*ft* <!) [r::r] - g*S*Ss~O 5:0

aJ aJ

+ 2g * S * [/ [r~: r] + 2g * fi [r~.rJ* gj - 2g * qi * Oi - 2g * (y' * q~js=o s=o

<I)

+ ph + t - pToS) * 0 - 9*11ij*11ij + 2g*9'*17/i* ~iI[r~:r,g]}d9l'

+ t J {g * [lIjaji(2bmi + 11m) - 21"'] * 11m)d.~ + J g * Urn * Tm d 9d~s ~u

+ J}g*g'*(Q - Q)*O}d~+ Jog*g'*O*Qd£j (3.7)

In arriving at (3.7) from (3.3), we have set Ko = O.Ao = O.Thus, we have dcrived a variational principle associated with the non-linear operator

equations (2.6) and (2.10) for the case in which it corresponds to thc equations governingthe non-linear theory of thermoplasticity: that is. K(A) of (3.7) assumes a stationary valuewhen A satisfies (2.6) and (2.10) provided 9 is potential. That & is indeed potential. isverified by the following theorem.

Theorem. Let !/j(x,t), Yij(x,t). aii(x,t). ft(x,t). l/(X.t), gj(x,t). O(x,t), S(X.I), and 11jix,t). aU definedfor (x,t) E 9j X (- ce. ce), satisfy the field equations (2.10) and the boundary conditions(2.6) and (2.7). Then the functional K(A) of (3.7~ assumes a stationary value at A = {u. I', (1, ft.q. g, O. S. ,,}.

Proqf. Let If = fii.Y.i1.~,q,g.O.S.~} denote an arbitrary elemcnt in the domain of operator. Then, using (3.2). we have. lim l/1X[K(A + aIf) - K(A)] = o/ih. K(A + x1f11

2=0

2-0

/)AK(A) = S {(plI - Y * [alj(b j + II .)]. - PI. 1* U + 9 * (t[lI.. + II ..;11 m m m.l ,I m m I, J J.I

O?

+ II .u .] - 'I..)* ali + 9 * (ffji[T': r] - 0'1) * y.tn, I m. J I) 5 = 0 r IJ

<I) o?

- q * g' * Wi - g) * il + 9 * ( .Cf' [r~:r]- S) * S + 0 * ( q; [r~: r] - a) * ~s=o s=o

aJ

- 9 * (qi - iii [r:.: r.g]) * 9j - g * (g' * q~i + pJi + t - pToS) * ().=0<Xl

+ {J*(g'* 15d[r~:r.g] -11/j)*ijjj}d9P + J. {g*(Tm - fm)*Um}d..;i

+ r {Y*(Um - Urn) * T"}d.~+ J {O*g'*(Q - I*O}d.Sit~" 91.

+ J {g * g' * (0 - 0) * Q} d.s1l= (9(A);1) = 0JI.

Since If is an arbitrary elcment of Q. it follows that grad K(A) = 9(A) = O.This proves thatthe functional K(A) of (3.7) assumes a stationary value. when A satisfied (2.10) and (2.6).

496 D. R. BHANDARI and 1. T. OOEN

that is (2.10) and (2.6) are indeed the Euler's equations of (3.7). All none fields u, 'I, (t. U,q, g, 0 and" are varied independently in the interior of fJI whereas u. T, 0 and Q are vari'edon!B , ~ ,1M and 940 respectively.s u q

4. ALTERNATE PRINCIPLES

A number of alternate principles are discussed in this section. We show that underappropriate additional assumptions variational principles for non-linear thermovisco-elasticity. non-linear viscoelasticity and numerous other existing principles for linearizedtheories can bc derived from the general principles developed herein.

First, we delete the inelastic strain rate equation (2.lOh) from the list of field equationsand also assume that ,,'(s) = 0, then the functional (3.7) corresponds to the theory governingthe non-linear behavior of thermoviscoelastic solids with non-homogeneous boundaryconditions. Furthermore. when thermal effects are also neglected in (3.7) i.e. if O'(s) isignored in (2.10). a theory of non-linear isothermal viscoelasticity arises as a spccial casediscussed in [5].

Now consider the case when the displacement field u(x.t) and temperature field O(x,t)arc kinematically and thermally admissable [i.e. the displacements u(x,t) and temperatureO(x,t) satisfy the strain-displacement rclation (2.1Oc) and the temperature-thermal gradientrelation (2.lOg)J. Then the resulting functional yields

00

H{ u. O} = tHum• pUm + 2y * Yij * §ij[,~,~:" 0] - 2um• pim + 2g. g' * qi * 0, i~ s=o

- 2y * Ii. {}- 2g * t * 0+ 2y * pros. 0] dfJI - f 9 * tm * um d3191,

(

•

- J g' * 9 * Q • 0 daJ91•

(4.1)

'"For the special case in which the internal dissipation t is zero and§ij[,~. O~; (. OJ.""Ii 00 5=0

S = 9' ['~.~ :(.0] and qi = ~i ['~. 0:.; ,. 0, g] are linear in the strain history" and thes=o .=0

temperature history ()', (4.1) is identical with (IV. J) of [9].For linearized coupled thermoeJasticity. the system of field equation in terms of displace-

ment vectors u and temperature 0 as given by Nickell and Sackman [13] arc

9 * (C}"IU" I - /3..0) . - PUI + J: = 0 (4.2a)I A A. IJ.J 1

if P hT"* (K} }) . - - CiJ - /3..U .. + - = O. (4.2b)10 I •• J To J) I.} To

Associated with this system of field equations are the initial conclitions (2.7) and the linear-ized boundary condition (2.6). Then following the procedure adopted in section 3. we obtain

J{ u, O} = -21 f {".* pu. + 9 * II.. * C}"IUL I - 2u. * J: - 2g • 0 * B.II. }J' t.) I" A, I I 'J I,

!II

A I'arialional principle for non-linear Ihermoviscoplaslicil,l' 497

The above problem can also be written in a form admitting the displacement vector, thestrain tensor, the stress tensor. the heat flux vector, the temperature gradient and thetemperature as a sct of dependent variables. Again, several other variational principlescould be derived. We also observe that the variational principles of Gurtin [8], Sandhu andPister [10] for the quasi-static motion of linear ,'iscoelastic and linear elastic solids becomespecial cases of (4.1) and (4.3) discussed herein.

We note that much literature has been devoted to the question of complementary cnergytheorems for finite deformations of elastic bodies and, indeed, whether or not such variationalprinciples exist. That they do indeed exist follows immediately from (3.7) by ignoring allthermal effects, assuming cach of the remaining mechanical constitutive equations aresatisfied. ignoring rate terms in (2.10) (and, hence, avoiding the necessity of the operationg *). and setting

2'\!..(ffii - uij) = -24J(ulj) + C('\! .. )Iy Iy'

Here 4J(uii) is the complementary energy density; i.e, if W("Iij) is the strain energy function.

4J = uli,\! .. - W.I,)

The quantity C(i'ij) depends onl~ on Yij and can be omitted since we plan to admit onlyvariations in the stress tensor (1

). Incorporating all of these restrictions. and applying theGreen-Gauss theorem. the functional (3.7) reduces to

K(u) = J [ -4J(u) + tUijllmll .J d9t + J II Tm d8d.!It . • I m. ) .:iI1~ m

(4.4)

It is easily verified that if variations in K(u) are considered due to variations in stress whichsatisfy the equilibrium equations (2.1a) and the traction boundary conditions (2.6b). then15"K(a) vanishes if. and only if. (2.1Oc) holds.

5. \1ATERIALS OF EVOLUTION TYPE

An altcrnative approach to the continuum theory of thcrmoviscoplastic materials isbased on the work of Coleman and Gurtin [14] and involves the concept of hidden variables.Recently, several investigators, including Kratochvil and Dillon [15]. Tseng [16], andHahn [17] have used Coleman-Gurtin type thermodynamics for the study of thermoplasticmaterials. The resulting collection of constitutive equations describe what is generallyreferred to as materials of evolution type. Here separate constitutive equations are givenfor the rate of change of hidden variables which are known as equations of evolution.

In this theory a t~crmopIastic material at point x is characteriz.ed by .s~ responsefunctions cpo fl. S, q. Nand l")which determine the values of tp, a. S. q. ". and (X'II at X when(, ". O. g and (X(i) are known at x and t: •

cp = cp(,. 0,,,. g, rxlil)

a = oK 8.". g. (X(i,)

S = S(,. O. fl. g, xli)~

q = q(', 0.". g. (Xli))

. - .i-(r 0 (iI)" - .;l ... ,,,.g.rx

(Xli) =Jli)(" 0, '1. g. (X(/)

(5.1a)

(5.1b)

(5.1c)

(5.1d)

(S.le)

(5.1 f)

498 D. R. BHANDARI and 1. T. ODI'N

Here 1X(i). i = 1. 2, ... ,/I denotes a finite set of internal (or hidden) state variables which maybe thought of as representing such phenomena as latticc distortions. internal slipping ofgrains, twinning, etc. Then making use of the positive entropy production inequality, it canbe shown that this material can be characterized by only four response functions one des-cribing the free energy {p which is independent of g and the other three being 4. ,AI' andjli! For example.

lfJ = 4>('.0. ", (tlil)

q = qK 0.". g. (t(i)

~ = ,.(l'.0, ". g. lX(i))

ali) = jil(', 0,,,. g, IX(/).

(5.2a)

(5.2b)

(5.2c)

(5.2d)

The stress a. the entropy S and the internal dissipation fI are then determined from (5.2) bythe formulaes

AI... r () 'I)' o' (r (J (il)a = 3" ('t. .", IX' ) = P ,lfJ 't,", ,IX

• (i) 0 • (i)S = 9'(,. O. ", IX ) = - olfJ(',", O. IX )

fj = ~(,. 0.". (t(i) = (a - po,{P(')) . ~ - pc.'" (P(.) . ~(iJ

(5.3a)

(5.3b)

(5.3c)

wherein (5.3) the quantities c:{p('), 0o{P(·). o,{P(') and a,,,, (P(.) denote differentiation of (prespect to '.0,,, and 1X(i) respectively.

Then the behavior of thermoplastic materials is described by the conditions (2.6H2.7)and the system of equations

Y'" [aij(<5mj+ urn)}; + prill - pUm = 0 (5.4a)

aij - j;ij(" 0,,,, 2(i) = 0 (5.4b)

'1.- -M21U .. + tt.. + II .U .) = 0 (5.4c)

I) I.) J. I m. I m. )

S - S(" O. ". !.l(i) = 0 (5.4d)

fj - §t(,. 0. ". (t(iI) = 0 (5.4e)

qi _ tt("0,,,, g, 2(i) = 0 (5.4l)

Yfij - g' ......f;i'. 0,,,, g, 1X(i1) = 0 (5.4g)

(i) 'iil}r 0 iii) 0 (54h)!Xkj- g *Jkj\'t. ,,,,g.1X = .gi - 0.1 = 0 (5.4i)

g' * q~1 + piJ + t - pToS = 0 (5.4j)

where the quantities fm, Ii + t, g. g' and the convolution symbol * are defined in (2.9) and(2.11).

Now following the procedure of section 3, we construct the variational statement of theproblem described by (2.6H2.7) and (5.4) in the form below.

cfJ1(A)=tJ{u *pu -2y*u *cr"'.i-2u *pl. -g*u *(aijll .).-2g*y.*aij_ !it m m m. t In m • m m, J ., lJ

+ 2g '" ')Ii; * ffij(T) - {/ * iT * iJ + 2g '" iJ ... ~(n- g * S ...S + 2g * S ",g(n - 2g * qi * gl

2 Ai(T ) + 2' A> (T) (il (il+ g*q .g *fJI-g*'lij"'Yfij g*g*Yfij .... /'ij ,g -g*~; ...!Xkj

A \'arialional principle for "on-linear Ihl'rmo\'iscoplasliClly

+ 2(/* g' *'XrJ *~jJ(r, g) - 2g * (y' * qii + ph + t - pToS) * O} d9l'

+! J {g * [/Jiaii(2bmj + um•j) - 2fm] * um} d~ + J {j:~um * Tm

} daJ!II. !itl.

+ J {g * g' * (Q - (2) * O} ~ + J {g * g' * e * Q} d9.J.~. ~.

499

(5.5)

Here for the sake of conciseness we have used in the notation r = (,. O. ". :xi) and A is givcnby

,{ = {u.y.a.h.q.g.O.S.".cx(iI}T. (5,6)

It is now a simple matter to verify that thc functional cP I(A) assumes a stationary valuewhenever (5.4) are satisficd. In other words. equations (2.6) and (5.4) are the Euler's equationsof the functional cfJ 1 (A) of (5.5).

6. ALTERNAn: FORMULATION

The fact that the bilinear form (2.12) using the convolution of two functions give avariational formulation to the initial value problems makes it possible to use directly theinitial value problem in the differcntial form. In this formulation the transformation ofinitial value problem into an equivalent boundary value problem (obtained by taking theLaplace transform and then finding the inverse transform) that is characteristic of Gurtinmethod, is not really necessary, In fact, it can be shown that Gurtin's method becomes aa particular case of this formulation.

To illustrate this we consider the problem described in Section 5 and. for the sake ofsimplicity. we consider homogeneous boundary conditions. Making use of definition(2.12) for thc bilinear mapping together with (3.3) we obtain thc functional

,- 1 f f { du (x, t - I) dll (x. I) .cfJ,(A) = - p m --'''-- - 2Un (x, ( - r) am.'(x, r) - 2u (x. I - r) pF (x, r)

- d(1 - r) dr I ./ m m

~ 0

+ 2Yiix, t - r) #ij[r(X. I)] - h(x. t - r) 8(X. T)+ 2a(x. t - I) ~[r(X. r)]

- SIx, t - I) SIx, r) + 2S(x, ( - r) Y'[r(X. r)] - 2yj(x, I - r) qi(X. r)

i[ . ] d17j ~X, r)+ 2Yi(X. t - I) q r(X. r), g(X, r) - 17ij(X, I - r).::...!.1J.

d (i)

2 I ).r [ ] (i) ~(X. r)+ tljj\X. t - r ./J ij r(X. r). g(X, r) - ry;Aj(X. I - r)

2 (i) r(iJ[ ]+ :xAj(X, t - r) J Aj r(X. r), g(X. r) - 20(x, t - r)

r' dS(x. r)J}x _q~j(X, I) + ph(x. r) + h(x, r) - pO(X. r) dr dBP dr. (6.1)

500 D. R. BHANDARIand 1. T. ODEN

We observe that the variation of the functional 4>2Vi) isr

c54>zVi) = J J {c5um(x,t-1:)(piim(x,1:)- [aij(x, r)(um)x, r) + 15m)).;- (pFm(x, r»)~o

+ t5ajj(x, t - r)[lIj)x, r) + uix, r) + Uj. j(x, r) 11m,ix, 1:) - 2·/iiX. 1:)]

- t5Yi/..X.t - r) [aij(x.1:) - #jj[f(x, r»] - t5h(X, t - r) [a(X. 1:) - ~(f(x. r»)]

- t5S(X, t - r) [S(X, r) - Y[f(x, rl] - Jgj(X. t - r) [qi(X, r) - ll(f(x. r). g(X. r»)]

- t5'TifX. t - r) [iTiiX, 1:) - A~j(f(x, r). g(X, r»)]

(II [ (I) J(I)(f )]- t5'XliX, t - r) 'Xl/X, r) - lj (X, r), g(X, r)

- t5lJ(X.t - r) [q~ j(X, r) + ph(x. r) + a-(X. r) - p{}(X. r) S(X. r)]

- t5c/(X, t - r) [gi(X. r) - 8)x, r)} d9l dr (6.2)

wherein (6.2) the superposed dot denotes differentiation with respect to r (i.e. ii (x, r) =_ m

d2um(x. r)/dr2 etc). If we require that this variation vanishes for every A( r) that satisfy theinitial condition 1/ = U = 0, a = 'I = 'T = 2

i = O. we obtain equations (5.4) as the Euler'sequations of the functional 4>iA} We note that the functional 4>2(;1) of (6.1) is identicalto that of (5.5) with homogeneous boundary conditions.

Acknowledgement-The support of this research by the U.S. Air Force Office of Scientific Research throughContract F44620-69-C-OI24 is gratefully acknowledged.

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A Iwiational principle for non-linear thermoviscoplasticit)' 501

...

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(Received 6 Dect'l1tber 1971)

Resume On donne un principe variationnel general associe ala theorie non lincairc de la thermoviscoplasticite.En particulier. cel article montre que J'ulilisation du theoremc de Vainberg sur les opCrateurs dc potcntiel nonlineaires ainsi que d'une forme bilineaire speciale cmpJoyant la convolution de deux fonctions rend possibleune formulation variationellc des problemes non lineaires avec des conditions initiales et aux Iimites. On discuteegalcment plusieurs varianles de ce principe ct leurs relations avec les principes variationnels cxislants.

Zosammenfassung-Ein a\lgemcines Variationsprinzip im Zusammenhang mit der nichtlinearen Theoric derThermoviskoplastizitiit wird dargesteill. 1m besonderen zeigt diese Arbeit, dass die Anwendungdes VainbergschenTheorems iiber nichtlineare Potentialoperatoren und ciner spezie\len bilinearen Form, die die Faltung zweierFunktionen verwendel, eine Variationsdarste\lung fUr nichtlineare' Anfangs- und RandbedingungsaufgabenermogIicht. Einige Alternativprinzipien und deren Beziehung zu den bestehcnden Variationsprinzipien werdeneben falls diskutiert.

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