on a principle of virtual work for thermo-elastic bodies

Journal of Elasticity 21: 131-146, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands. 131

On a principle of virtual work for thermo-elastic bodies

G A U T A M BATRA Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, NE 68588-0347, U.S.A.

Received 24 April 1987; in revised form 29 August 1988

Abstract. A principle of virtual work is proposed for thermo-elastic bodies. From it are derived the equations of motion, the Cauchy stress principle and the Gibbs relations. The principle is also used to analyse the response of internally constrained bodies.

1. Introduction

The Principle of Virtual Work has often been used to establish the equations of motion of structural elements such as bars, plates and shells; it has also been used to determine the equations of motion of a general continuum. A

clear discussion of this and related principles is given by Truesdell & Toupin [16,§§231 et seq.], whose "presentation concerns solely the formal prob- l e m . . , of setting up expressions such that the vanishing of their first variation is equivalent to Cauchy's laws".

The validity of the Principle of Virtual Work or, more generally, that of any variational principle is usually accepted if it can be demonstrated that the local forms of the equations obtained from it coincide with those obtained from the integral form of some acceptable balance law. Thus to make this demonstra-

tion one assumes that the fields, which in combination constitute, say, the equations of motion, are sufficiently regular in order that these equations be well-defined in the classical sense.

Antman & Osborn [1] have recently re-examined the connection between the Principle of Virtual Work and the Principle of Balance of Linear Momen- tum for a general continuum. Using techniques from modern analysis they

established the equivalence of these two principles without having to introduce the local equations of motion in an intermediate step. A more restricted case of such a demonstration appears in a work by Carey & Dinh [5].

It is our intention to show here that the Principle of Virtual Work (henceforth simply referred to as The Principle) has application to thermo-

AMS (MOS) Subject Classification 73B05, 73B30, 73C20, 73UO5.

132 G. Batra

elastic bodies. We postulate a form of The Principle that holds, as do the balance principles, over an arbitrary part ~ t of a material body ~t , and not merely over the whole of it, viz.

~ V(~,) = 3-W(~,), (1.1)

where V(~,) denotes the potential energy of the part ~ , , 6 is the first variation as defined in the calculus of variations, and 3-W(~,) denotes the virtual work of body forces, surface tractions etc. over ~ , . By assuming now the existence of a specific internal energy function e = e(F, q) and a temperature function 0 = 0(F, q), in which F is the deformation gradient of the motion and ~/ the specific entropy field, we formally deduce from (1.1)

(i) the equations of motion in ~ t :

p~ = div T + pb; (1.2)

(ii) the stress and temperature relations

Tr(F, q) = pR(det F) 'F(t3Fe(F, r/)) r (1.3)

and

0(F, r/) = 8,e(F, q); (1.4)

(iii) the stress principle of Cauchy on any surface in ~',:

t(x, t, n) = T(x, 0n(x, t), (1.5)

t being the traction vector; and

(iv) the Principle of Determinism for Internally Constrained Bodies [15, §30], that is to say, a rule giving the stress and temperature relations when the body is internally constrained. To account for internal constraints, however, The Principle (1.1) is modified to (cf. ~4)

6 V(~t) - 6C(~,) = SW(~t), (1.6)

in which C(~,) denotes constraints; the minus sign in front of the second term in (1.6) is conventional.

The stress and entropy relations,

Tr(F, 0) = pR(det F) - 'F(gv~(F, 0)) r (1.7)

Virtual work for thermo-elastie bodies 133

and

. (F , 0) = -o0O(v , 0), (1.8)

are obtained instead of (1.2) and ( 1.3) when F and 0 are used as independent variables in constitutive equations, ~ being the specific free energy function,

0(F, 0) = e(F, 0) - 0r/(F, 0). (1.9)

In this case we assume the existence of the specific energy and entropy functions e(F, 0) and q(F, 0) respectively. We also show that the dependence of 0, e, and t t on the temperature gradient g = Vx0 is ruled out by our statement of The Principle.

It is worthwhile pointing out that although ~TW(~t) is equated to a first variation (cf. (1.1) and (1.6)), there is, in general, no function of which it is the first variation; that is to say, The Principle as proposed here does not give extremal values. For this reason, it may be called a quasi-variational principle. It is possible to give a variational principle (i. e., one that extremises a functional), but this would seem to require - (i) that only mechanical effects be considered, (ii) that the body be hyperelastic, (iii) that the body-force and prescribed surface-traction fields be conservative (el [3, pp. 233-234]), and (iv) that the principle be formulated so as to hold only over the whole body, but not over its arbitrary parts. Cf., e.g., Lee & Shield [11] for such a principle.

Biot has also considered the Principle of Virtual Work for problems of thermo-mechanics. His formulations as presented in [4], are, at once, more general and more special than ours: more general because they account for dissipation (via dissipation potentials); and more special because when used to account for thermo-mechanical phenomena, the stress is assumed to be a linear function of infinitesimal strains and small char~ges of temperature. Moreover, the Gibbs relations ((1.3), (1.4), (1.7) and (1.8)) are not derived from his proposed principle; they are simply assumed to hold (cf [4, App. §3]).

Maugin [13] writes from a point of view somewhat different from our own. However, as our reviewer has pointed out, he does formulate his principle to hold over arbitrary parts, and he thereby recovers the "boundary conditions". For principles that account for singular surfaces, el, e. g., [3], [7] and [14]. In some of the works cited above, the Gibbs relations are not derived from the principle; in few is there any real attempt to derive the Cauchy stress principle. Internal constraints, the subject of §4, are also considered in [2], [8], [10], [15, §30] and [16, §233]. Relations similar to (1.3), (1.4), (1.7) and (1.8) can probably be traced to works of Maxwell and Gibbs. Cf., Carlson's article [6, Footnote 11] for a report on modern theorems which establish these results

134 G. Batra

using the Clausius-Duhem inequality. Results (1.3) and (1.7) have been obtained much earlier from a virtual work principle for hyperelastic bodies. Cf. e.g. [16, §232A].

Finally, we point out that the concept of the stress tensor in modern continuum mechanics is a derived one. Accordingly, the stress tensor does not appear explicitly in our formulation of The Principle. Rather, it is introduced in a manner similar to Love's third way of introducing the concept of stress; cf. [12, pp. 616 et seq.], also [16, §232A].

2. Preparatory results

We consider a body B which occupies a closed, bounded, regular region of Euclidean space in a fixed reference configuration ~R. The motion of B is given by a smooth, differentiable map X on ~R × (0, to), where to > 0 may be infinity:

x = x(X, t). (2.1)

The current configuration of B is the set ~ t consisting of all points x defined by (2.1). We assume the function X to be invertible with a smooth, differentiable inverse on ~t × (0, to). The fields of velocity v = ~, deformation gradient 17, and acceleration a = ~ are defined in the usual way, viz.

v(X, t) = 8,z(X, t), F(X, t) = VxZ(X, t), a(X, t) = ~3tv(X, t). (2.2)

We assume that det(17(X, t)) > 0. The Principle of Balance of Mass then gives the local form

p~ = p(det F), (2.3)

pn(X, t) and p(x, t) being, respectively, the density fields in the reference and current configurations, both of these being assumed to have strictly positive values.

In view of the assumed properties of the map ~ in (2.1), we may express any field either in terms of the co-ordinates (X, 0 or (x, t).

A comparison motion for B is defined by

x* = z(X, t) + ~ ( X , t) = z*(X, t; e), (2.4)

where Z* maps ~¢~ onto a configuration ~ * = ~t*(e), say, at each time t and for each value of the scalar parameter e. The deformation gradient of the

Virtual work for thermo-elastic bodies 135

comparison motion is given by

F*(X, t; ~) = VxZ*(X, t; ~) = F(X, t) + eVxi(X, t). (2.5)

To describe the thermal state of the body we introduce the smooth fields of specific entropy and absolute temperature on M, × (0, to), ~/(x, t) and 0(x, t), respectively, with 0(x, t ) > 0. We shall need to have comparison fields for either the specific entropy or the absolute temperature. These are defined on ~ , × ( O , to) × ~ b Y

r/*(x, t; e) = q(x, t) + eO(x, t) and 0*(x, t; e) = 0(x, t) + t0-(x, t), (2.6)

in which f /and ~ are arbitrary smooth fields over ~ t x (0, to). Let ~ be an arbitrary, regular part of ~ , and consider a smooth scalar

field ~b(x, t)on ~ t x (0, to). By the transport theorem, presuming conservation of mass, we have

d~ p~b dV = P -d~ dV, (2.7) t t

where d V denotes volume measure on ~t . Now consider a smooth scalar field ~b*(x, t; e). Let p* be the density in the configuration ~,*. Then it too satisfies an equation of the form (2.3); and so by the transport theorem (2.7) with e playing the role of t, and using the fact that ~,* is a material volume, we conclude that

df~, p*~b*dV*=f~ p, dtp* ~ee ,. ,. T dV*, (2.8)

where dV* denotes volume measure on ~,*, and ~ * is an arbitrary part of ~,*.

An Example - The Potential Energy. Consider the potential energy V(~t) of the part ~t:

V(~,) = f~ pe(F, 7) dV, (2.9) t

where e = e(F, 7) is the specific internal energy function, assumed to be differentiable with respect to each of its arguments, ~ being a scalar field denoting either the entropy or temperature. The potential energy of the part ~,* is similarly given by

V(~',*) = f~ p*e(F*, 7*) dV*. (2.10) ~*

136 G. Batra

By (2.5) and (2.6) we obtain

e(F*, ~*) = e(F + ~Vx~, 7 + e~7)

= e(F, 7) + g{(0Fe(F, 7)) " Vx:~ + (0re(F, ?))~} + 0(5) as ~ -~0. (2.11)

Thus by (2.8), (2.10), (2.11), the identity Vx~ = (Vx~)F, and the divergence theorem, it is now readily shown that

6V(~t)= f {-:~ • div(p(0ve)FT") + p(0~e)~} d V + ~ ~" (p(OFe)FT)n dA, d~ t J 0 ~ t

(2.12)

where ~ , denotes the boundary of ~ , , and where we have introduced the notation

. d 6V(~,) = ~+rn ° ~ V(~*). (2.13)

In §4 we shall assign a similar meaning to 6C(~,). If e = e(F, 7, h), where h = V~7, a calculation similar to the one above gives

6V(~,) = ~ { - ~ " div(p(OFe)F r) + p(0~e)~} dV + f Y~ " (p(OFe)FT)n dA .)~ t J O l t

( p(div 0,e)~,7 dV + ~ p(0,e) • w7 dA. (2.14) d~ l ~JO'~t

3. Thermo-e las t i c sol ids

We consider a thermo-elastic solid for which the fields of internal energy and temperature are given by the constitutive equations

e = e(F, q), 0 = 0(F, q). (3.1)

On the boundary 0~t, partitioned into the complementary subsets (0~,)1 and (0~,)2, we have the boundary conditions

z(X, t) = ~(X, t) on (0~R) 1 × (0, to), (3.2)

t(x, t) = i(x, t) on (0~t)2 x (0, to), (3.3)


where (O~R)l is the part of the boundary in the reference configuration of the body which is mapped by (2.1) into (O~,)l, ~ is a prescribed motion, and i is a prescribed traction field. A motion is said to be kinematically admissible if the comparison field (2.4) satisfies (3.2); that is to say, if ~(X, t) vanishes on (O~s)~ x (0, to).

Let

V(~,) = f~ pe dV (3.4) t

and

gW(~,) = fe p((b - i ) ' ~ + Ot~) dV + fo t" ~ dA. t ~ t

(3.5)

Here V(~t) is recognised as the potential energy of the part ~, , and 3-W(~,) as the virtual work of the 'total body-force' [9, p. 180], surface-traction and temperature fields through changes of position and entropy. Naturally, when ~, is taken to be ~, we replace ~t by ~, and OPt by (O~t)2 in (3.5), so as to only admit kinematically admissible motions that also satisfy (3.3).

We consider now the implications of The Principle- The actual motion of the thermo-elastic body is such as to satisfy the condition

6 V(~,) = SW(~,) (3.6)

for all adm&sible variations of the motion and entropy field. With ~t contained within the interior of .~,, (3.6) implies (cf also (2.12))

f~ {~, " ( -OK + pb + div(p(OFe)Fr)) - fl(pO - p c~,e) } dV t

+ fo (t - p ( ~ r e ) F r n ) • ~ d A = 0 .~ t

(3.7)

for ~ arbitrary in ~, and i arbitrary in ~t and on ~ , . Thus

pf¢ = div(p(Ove)F T) + pb (3.8)

and

0 = 0 , e (3.9)

138 G. Batra

on # , x (0, to), and

t = Tn on d ~ , x (0, to), (3 .10)

where we have used the notation

Tr=pF(t~Fe) r on t?N, x (0, to). (3.11)

Let us now observe that since ~ t c ~ , is arbitrary (~t denotes the interior of ~,), any point in ~ , can be made to either lie within N, or on its boundary O~, by a suitable choice of ~, . Hence relations (3.8)-(3.11) must also hoM on

~ , x (0, to). Repeating the above argument with Nt = ~ t and i = 0 on (dNt)1 merely confirms (3.3), and extends (3.10) and (3.11) up to (O~t)2.

We re-write (3.8) and (3.11) as

p i = div T + pb (3.12)

and

Tr(F, rt) = p~(det F ) - ~F(cqFe(F, r/))r; (3.13)

and thus recognise T as the Cauchy stress, and (3.10) as the Cauchy Stress Principle. Although we have heretofore referred to 0 as the tempe, ature and to r/as the entropy, the preceding analysis does not require this interpretation. Indeed, we should not make this identification unless these variables appear in an appropriate way in a statement of the Second Law of Thermodynamics, since, for instance, r/ could simply be an 'internal state variable', and 0 a 'generalized force' conjugate to it such that they together contribute to the virtual work according to (3.5).

If (F, 0) are the independent variables in the constitutive equations, one can show (cf. [15, §82]) that (3.9) and (3.13) imply the stress and entropy relations (1.7) and (1.8) so long as the relation 0 = 0(F, r/) can be inverted to give ~/= r/(F, 0). However, we prefer to give below an alternative formulation of The Principle which solves this problem directly.

Assume that

e = e(F, 0), t /= q(F, 0), (3.14)

and that (3.2) and (3.3) hold as before. Introduce the specific free-energy ~,

~t(F, 0) = e(F, 0) - 0~/(F, 0), (3.15)


and re-write (3.4) in terms of this function. Since we are now considering the variations

x * = x + ~ and 0 " = ~ + ~ 0 - , (3.16)

evaluate the quantity O in (3.5) according to

0 = (0Fq) • ~ + (C3or/)0-, (3.17)

in which ~ = Vx~. By repeating the argument leading to (2.12), verify that

~v(~,) = - ~ ~. d i v ( ~ ( ~ + o ~ ) r ~ ) ~ v + ~ ~(~o~ + 0 ~o~ + ~)o dV d~ t d ~ t

+ ~ ~" p(dv¢ + 0 Ovn)Frn dA, (3.18) do ~ t

SW(~,) = - ~ ~'div(pO(Ovq)F T) dV + ~ ~" pO(Ov,)FTndA J~ t ~ t

+f~ t '~dA+f~p(b-~ , '~dV+;~ pO(Oo,,~dV; (3.19) ~ t t t

and thus from the statement of The Principle given above, replacing the word entropy by temperature, obtain the stress and entropy relations

T~(F, 0) = o~(det F ) - IF(Ov@(F, 0))r, (3.20)

and

q(F, 0) = - Oo@(F, 0). (3.21)

Using either relations (3.12) and (3.13) or (3.20) and (3.21), along with the equation for balance of energy,

pk = T . D - div q + pr, (3.22)

in which q is the heat-flux vector and r is the specific heat supply, it can be shown that

pO~ = - d i v q + pr. (3.23)

140 G. Batra

That is to say, a thermodynamic process for a thermo-elastic body is adiabatic if and only if it is isentropic. The entropy production inequality

Pfl -(r/O) + div(tl/0 ) ~> 0 (3.24)

places a restriction on the heat-flux vector; for by (3.23) and (3.24) we have

-~l" grad 0 ~> 0. (3.25)

This inequality has the interpretation that heat flows from a hot spot to a cold o n e .

We re-iterate that the representations (3.12), (3.13), (3.20) and (3.21) have been here derived by use of The Principle, without recourse to (3.22) and (3.24).

We have assumed from the outset that the functions e, ¢, and r/ do not depend upon the temperature gradient. In modern works on thermo-elasticity (cf., e.g., [6, ~4-6], [9; ~j7, 8]) it is customary to assume that they do; that is to say,

e = e(F, 0, g), ~k = ~(F, 0, g), r /= q(F, 0, g), (3.26)

where g = Vx0. The Second Law of Thermodynamics is then used to rule out such a dependence. We show below that this dependence is also ruled out by our statement of The Principle.

To this end, assume that the relations (3.26) hold. Then ~ in (3.17) is now given by

f /= (t~r/)- ~ + (Oor/)O - + (c~gr/) • vxO-; (3.27)

and to correctly evaluate 6V(~t) in (3.18) and ~-W(~,) in (3.19) we must now add the integrals

-f t p(div og . + o div O.,)edV + P( .O + 00 . , ) e ' .

and

- f~, pO( div ?,tl)~ d V + fo~, pO(~,q)g " n dA,

respectively, to their right-hand sides. We verify that the arbitrariness of 0- now implies

~0~ - div(OgO) + r /= 0 on ~ , x (0, to), (3.28)

and

(d~O) " n = 0


on 8~t x (0, to). (3.29)

Using our earlier argument we obtain from this last equation our desired result,

~g~ = 0 on ~ , x (0, to), (3.30)

which, along with (3.15) and (3.28), implies that e and r/ must also be independent of g.

4. Internal constraints in thermo-e last ic bodies

Incompressibility, inextensibility of fibres in a fibre-reinforced body etc. are examples of internal constraints. We are interested here in the internal constraints (4.1) and (4.10) to the extent that they modify the stress and temperature/entropy relations (1.3), (1.4), (1.7) and (1.8). However, a Prin- ciple of Determinism for Internally Constrained Bodies is also proposed, which determines the effects of the constraint on the thermo-mechanical response of a body of a more general type.

Let the response of a thermo-elastic body be constrained so as to satisfy the condition

qS(F, 0) = O. (4.1)

We incorporate this condition into our statement of The Principle by the method of Lagrange multipliers. To this end, consider the integral

C(~i~t) = ;~ p~) dV, (4.2) ¢

where m is a smooth Lagrange multiplier on ~ , x (0, to). We vary to according to

to*(x, t; e) = re(x, t) + ee3(x, t), (4.3)

t~ being assumed smooth and arbitrary on ~ t x (0, to). Thus C(~'t*) is given by

C(~*) = ;~ p*to*q~* dV*, (4.4) ,*

142 G. Batra

whence

6C(~t )=- f~ , i ' d i v ( p m ( a F $ ) F T ) d V + f o i'pto(OF~b)F~ndA t ~ t

+ f~ pt~(8o(~)8 dV + f p(off~ dV. (4.5) ~ d ~

Let F and 0 be the independent variables in the constitutive equations; let

V(N,) = f~ P(O + Oq) d V, (4.6) t

and 3-W(~,) be as given in (3.5). Extending the applicability of the previous statement of The Principle, we

say that - The actual motion of the thermo-elastic body is such as to satisfy the condition

3 v ( ~ , ) - , ~ c ( # , ) = 3 - w ( ~ , ) (4.7)

for all admissible variations of the motion, temperature field, and Lagrange multiplier m. As in the previous section, by the arbitrariness of 0- and ~ in ~ , , of i on ~ , and on ~ , , and of ~ t in ~ , , we obtain the equations of motion, The Stress Principle of Cauchy, (4.1), and the modified stress and entropy relations

Tr(F, 0) = pR(det F) - 1 F ( O F O ( F , 0 ) ) T _ pR~v(det r ) - 'r(dr~b(r, 0)) r (4.8)

and

n(F, 0) = -ao0 (F , 0) + mao$(F, 0), (4.9)

the terms on the right-hand sides of (4.8) and (4.9) being, respectively, designated T~, Tb, r/o and qc, and the subscripts D and C being chosen to recall 'determinate' and 'constraint', respectively (e.g., qo = qo(F, 0) = -d0ff(F, 0)).

Similarly, when F and ~/ are the independent variables in constitutive equations and the equation of constraint is written as

O(F, rt) = 0, (4.10)

we obtain the modified stress and temperature relations

Tr(F, r/) = p~(det F) - IF(0Fe(F, r/)) r _ pn~(det F) - 'F(C3F(I)(F , r/)) r (4.1 I)


and

0(F . r/) = 0 , e (F . r/) - ~? ,q ) (F . q). (4.12)

7t being a Lagrange multiplier as before. We similarly designate the terms on the right-hand sides of (4.11) and (4.12) T~, T~, 0o and Oc.

A thermo-elastic body is said to be perfect if its response functions satisfy either (4.8) and (4.9), or (4.11) and (4.12), as appropriate.

To illustrate the use of these results, consider the following example. Suppose that the body is incompressible, so that the equation of constraint is, say,

~b(F, 0) = det F - 1 = 0. (4.13)

Then by the identity 8F(det F) = (det F)F -T (cf. [15, §9]), (4.8) and (4.9), we have

Tb(F, 0) = - p R m I and t/c(F, 0) = 0 (4.14)

in which I denotes the identity tensor; that is to say, for an incompressible body the stress is determined by the deformation gradient and temperature only to within a hydrostatic pressure (cf. [15, §30]).

We use the results of (4.8), (4.9), (4.11) and (4.12) to propose a rule which determines the 'constraint response' (i.e., T'c, qc, T~ and Oc) when the body is not necessarily thermo-elastic, that is to say, when T~, r/o, T~ and 0o are not necessarily determined by the expressions given above. For a purely mechanical theory such a rule is given by Truesdeil & Noll's Principle of Determinism for Simple Materials Subject to Internal Constraints [15, §30]:- "The stress T at time t is determined by the history F(t)(s) of the deformation gradient only to within a stress N that does no work in any motion satisfying the constraints."

If a scalar-valued field 2 =2(17, y), 7 ~ , is indifferent to changes of frame

I*(X) = c + Q~(X), (4.15)

for orthogonal tensors Q and points c, then (0v2(F, 7))F r is symmetric and frame indifferent (el [15, §84]). We assume that q~(F, 0), r/(F, 0) and ~b(F, 0), or e(F, r/), 0(F, r/) and ~(F, r/), as appropriate, are frame indifferent so as to conclude that the stress tensors T(F, 0) and T(F, r/) in (4.8) and (4.11) are symmetric and frame indifferent.

By differentiating ~b(F, 0) = 0 and ~(F, 0), and using the above-mentioned symmetry, (4.8) and (4.9), we get

Tb" D + pR(det F) 'tlcO = 0, (4.16)

144 G. Batra

and an equation for the growth for the free energy,

p,~(det F)-~6 = T~" D + p,~(det F) lqoO (4.17)

in which D = sym(~'F- 1). Similarly from *(F, r/) = 0 and e = e(F, q) we get

T~" D + pR(det F) -~Ocil = 0, (4.18)

and an equation for the growth for the internal energy,

pR(det F) -1~ = T~ • D + pR(det F) - ~0Dr}. (4.19)

Equations (4.16)-(4.19) have the following interpretation: For a perfect thermo-elastic body, e i ther-

(i) (independent variables F, 0) - The fields of stress and entropy are determined by the deformation gradient and temperature only to within parts thereof which cause no change in the free energy for motions and temperature fields satisfying the constraints; or

(ii) (independent variables F, q ) - The fields of stress and temperature are determined by the deformation gradient and entropy only to within parts thereof which cause no change in the internal energy for motions and entropy fields satisfying the constraints.

We now say a few words on a matter to which we have already adverted. Namely, we wish to determine T~ and ~/c for a body whose response is constrained by (4.1) and for which the determinate parts of the stress and entropy are given, within the framework of the theory of simple materials, in terms of the histories of the deformation gradient and temperature, viz.

T~ = T~(F(°(s), O(°(s)) and rlz) = qD(F~°(s), O")(s)). (4.20)

Towards this end, if we admit our observation (i) above as a rule which also holds for this case, then we can show by a deliberate calculation (cf. [2]) that it implies

(T~.)r= --pRto(det F)-IF(c~F~b(F, 0)) r and rtc = t~0o~b(F, 0), (4.21)

the same result as in (4.8) and (4.9). Moreover, by the considerations of balance of energy and the Second Law of Thermodynamics, it can be shown, in respect of the heat-flux vector, that qc = 0 (cf. [2]). Similarly, when F and ~/are the independent variables and (4.10) is the constraint, we may use (ii) to determine Tb and Oc, the results thus obtained being the same as in (4.11) and (4.12).

Virtual work for thermo-elastie bodies 145

Conditions (i) and (ii) above, obtained by restricting attention to perfect thermo-elastic bodies, can thus be shown to determine the constraint response of bodies of a more general type (cf. [2]). It is recommended that these be used in lieu of Truesdell & Noll's "Principle of Determinism for Simple Materials Subject to Internal Constraints" [15, §30], cited above, when thermo-mechanical effects are to be accounted for.

We generalise the constraint equation (4.1) and express it in terms of the same variables as we do the constitutive equations in the last part of §3 (el (3.26)), viz.

~b(F, 0, g) = 0, (4.22)

in which g = V,,O. We also assume that

e = e(F, 0, g) and r /= q(F, 0, g), (4.23)

so that

~ = ¢(F, 0, g). (4.24)

Then following the calculations leading to (2.14) we have

~c(~,) = ~ 2 . .~(~)~ . d~ + ~ p~(Og~)~ . n dA do ~t Js~t

- ~, ~. ~ i v ( . ~ ( o ~ , ) ~ ) ~ v - ~., .~(~iv o , ~ ) ~ ~ ~

+ f~,p~(Oo~)ffdV + f ~ , p ~ dV. (4.25)

When we apply The Principle (4.7) with this generalization and use results similar to (2.14), we find that

~o~ - div(0~) + to div(~b) + r /= 0

(~g~k - todg~,b) • n = 0 on d~,.

o n ~t (4.26)

(4.27)

But since ~t is an arbitrary part of ~t , we may adjust O~t so that any point of ~ , may be made to lie on it; moreover, n at that point may be directed arbitrarily. Thus, by (4.27),

~ = mO~b in ~ , ; (4.28)

146 G. Batra

tha t is to say, the strain energy funct ion ~k depends upon the temperature

gradient g i f and only i f the constraint equation does. The e n t r o p y r e l a t ion can

n o w be wr i t t en in the fo l lowing forms:

r / = -~o~k + div(~gg~) - m div(~g~b)

= - t~0~ + g rad m • ~3gtp

= - ~ 0 ~ ' + (g r ad In ttr) • d~k. (4.29)

O u r t r e a t m e n t here o f i n t e rna l c o n s t r a i n t s has been brief. F o r a ful ler

a c c o u n t see G r e e n , N a g h d i & T r a p p [8] a n d G u r t i n & P o d i o - G u i d u g l i [10].

5. References

1. Antman, S. & J.E. Osborn: The principle of virtual work and integral laws of motion. Arch. Rational Mech. Anal 69 (1979) 231-262.

2. Batra, G.: On Hamilton's principle for thermo-elastic fluids and solids, and internal constraints in thermo-elasticity. Arch. Rational Mech. Anal. 99 (1987) 37-59.

3. Batra, G., A. Bedford & D.S. Drumheller: Applications of Hamilton's principle to continua with singular surfaces. Arch. Rational Mech. Anal 93 (1986) 223-251.

4. Biot, M.A.: Variational Principles in Heat Transfer. Oxford: Clarendon Press (1970). 5. Carey, G.F. & H. Dinh: Conservation principles and variational problems. Acta Mech. 58

(1986) 93-97. 6. Carlson, D.E.: Linear thermoelasticity. In: C. Truesdell, S. FliJgge (eds), Handbuch der Physik

Via/2; Berlin: Springer-Verlag (1972). 7. Daher, N. & G.A. Maugin: The method of virtual power in continuum mechanics. Applica-

tion to media presenting singular surfaces and interfaces..4cta Mech. 60 (1986) 217 240. 8. Green, A.E., P.M. Naghdi & J.A. Trapp: Thermodynamics of a continuum with internal

constraints. Int. J. Engng. Sci. 8 (1970) 891-908. 9. Gurtin, M.E.: Modern continuum thermodynamics. In: S. Nemat-Nasser (ed.), Mechanics

Today 1; New York: Pergamon Press (1972). I0. Gurtin, M.E. & P. Podio-Guidugli: The thermodynamics of constrained materials. Arch.

Rational Mech. Anal. 51 (1973) 192-208. i 1. Lee, S.J. & R.T. Shield: Applications of variational principles in elasticity, Zeit. angew. Math.

Phys. 31 (1980) 454-472. 12. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity; Fourth edn. London:

Cambridge University Press (1927). Reprinted, New York: Dover (1944). 13. Maugin, G.A.: The method of virtual power in continuum mechanics: Application to coupled

fields. Acta Mech. 35 (1980) 1-70. 14. Taub, A.H.: On Hamilton's principle for perfect compressible fluids. Proc. Symp. Appl. Math.

i, Non-Linear Problems in Mechanics. New York: Am. Math. Soc. (1949). 15. Truesdell, C. & W. Noll: The non-linear field theories of mechanics. In: S. Fliigge (ed.),

Handbuch der Physik Ill/3; Bedim Springer-Verlag (1965). 16. Truesdell, C. & R.A. Toupin: The classical field theories. In: S. Fliigge (ed.), Handbuch der

Physik III/1; Berlin: Springer-Verlag (1960).

on a principle of virtual work for thermo-elastic bodies

Documents