Aspects of the nonlinear theory of type II thermoelastostatics

European Journal of Mechanics A/Solids 32 (2012) 109e117

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Aspects of the nonlinear theory of type II thermoelastostatics R. Quintanilla a, *, J. Sivaloganathan b a b

Matematica Aplicada 2, ETSEIAT-UPC, 08222 Terrassa, Barcelona, Spain Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 May 2011 Accepted 14 September 2011 Available online 22 September 2011

In this paper we study the equations of the nonlinear theory of type II Thermoelasticity proposed by Green and Naghdi. We propose a constitutive framework for the free energy function arising in this theory and study various consequences including the existence and uniqueness of quasistatic solutions and the possibility of thermally activated instability.
2011 Elsevier Masson SAS. All rights reserved.

Keywords: Type II thermoelastostatics Nonlinear thermoelasticity

1. Introduction James Clerk Maxwell was amongst the first to point out that the classical linear theory of heat conduction, which is based on Fourier’s law for the thermal flux, predicts that the effects of a thermal disturbance at some point in a material body will be felt instantly, but unequally, at all other points of the body, however distant. This behaviour is usually known as the “paradox of heat conduction” and is physically unrealistic since it implies that thermal signals propagate with infinite speed. It is therefore not surprising, given this non-causal aspect of the classical theory, that a number of alternative theories of heat conduction have been proposed. A survey of nonclassical thermoelastic theories was given by Hetnarski and Ignaczak (see (Hetnarski and Ignaczak, 1999), (Hetnarski and Ignaczak, 2000) and also (Hetnarski and Eslami, 2009), (Ignaczak and Ostoja-Starzewski, 2010)). In this paper we consider one of the thermoelastic theories proposed by Green and Naghdi in the latter years of the last century (Green and Naghdi, 1991; Green and Naghdi, 1992; Green and Naghdi, 1993; Green and Naghdi, 1995). In their theories, these authors employ a procedure which differs from the usual one by proposing a general entropy balance rather than the usual entropy inequality. The basic development is very general and the characterization of the material response for thermal phenomena is based on three different families of constitutive functions. They referred to them as type I, II and III respectively. Of these, type III is the most general and includes types I and II as special cases. The type I theory recovers the classical theory of Thermoelasticity. The type II theory is conservative and hence it is

* Corresponding author. E-mail address: [email protected] (R. Quintanilla). 0997-7538/$ e see front matter
2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2011.09.002

also called Thermoelasticity without energy dissipation. It is a fully consistent theory of Thermoelasticity which is capable of incorporating thermal pulse transmissions in a logical manner. Since the publication of the works of Green and Naghdi, several authors have been interested in studying the implications of these theories. However, it is apparent that further mathematical and physical studies are still needed to clarify the applicability of these alternative theories and our work is in this spirit. Many people have shown interest in the qualitative study of the solutions to these theories: for instance the studies (Chandrasekharaiah, 1996a,b; Iesan, 1998; Iesan, 2008; Lazzari and Nibbi, 2008; Quintanilla, 1999, 2002, 2007; Quintanilla and Straughan, 2000) and (Quintanilla and Straughan, 2005) are concerned with linear thermoelastic theories and contain results on existence, uniqueness and time or spatial decay of solutions, others concern conservation laws in this theory (Kalpakides and Maugin, 2004; Maugin and Kalpakides, 2002). Furthermore, nonlinear acceleration waves have been studied for type II and III nonlinear Thermoelasticity (Quintanilla and Straughan, 2004) and fluids without energy dissipation (Quintanilla and Straughan, 2008). In this paper we consider nonlinear type II Thermoelasticity. One of the main ingredients in this theory is to work with the thermal displacement (see (2.4)). This scalar on the macroscopic scale is regarded as representing, on the molecular scale, some “mean” thermal displacement magnitude. This nonlinear theory has received little attention to date and thus our contribution represents the first steps in developing a framework for this theory. Moreover, we will mainly work with what we call the “static problem” which has not been considered in the literature yet. More specifically, we consider a class of quasistatic solutions for the dynamical problem (see (2.5)). This assumption reduces the equations to that of a static problem (see (3.2)) the study of which will

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R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

be a main aim of this paper. The particular form of the constitutive equations for the type II theory allows us to apply several arguments which have previously been used in the context of the isothermal theories, however, this is the first time that such arguments have been applied to nonlinear Thermoelasticity. Though the type II theory was developed as a dynamic theory, a prerequisite for utilising it is a thorough understanding of the corresponding time independent problem. In the current paper we study some aspects of including the new thermal displacement variable by studying static solutions to the equations of type II Thermoelasticity. In this case, in contrast to the isothermal theory, we require an additional equilibrium equation for the thermal displacement (which relates to the “mean” thermal displacement magnitude on the molecular scale e see [(Green and Naghdi, 1991), Sections 7 and 8]). Though one might anticipate that in many cases the thermal displacement gradient is in fact zero (which should correspond to the classical isothermal problem) we have the added possibility within the type II theory that there may exist new solutions in which the thermal gradient is non-zero. Hence, the type II theory potentially allows for a richer class of solutions. We illustrate this, in particular, with a new example of thermally activated cavitation (see section nine) in which cavitation may occur when the thermal displacement gradient is non-zero but not when it is zero. Our intention in this paper is to highlight how existing analytical theory and concepts already used in isothermal nonlinear elasticity might be extended and adapted to provide a mathematical framework for the type II theory. These extensions are relatively straightforward from an analytic viewpoint, however, the type II theory we develop is potentially much richer than the isothermal case and we are led to further possible phenomena such as the possibility of thermally activated cavitation mentioned above. The plan of the paper is the following: In section two we recall several preliminaries concerning type II thermoelastic theory. In section three we present the class of static solutions which will be the main focus of this paper. In section four we consider some possible forms for the thermoelastic counterpart of the classical stored energy function known for the isothermal case. A general form for this function is obtained in section five that guarantees that the frameindifference principle is satisfied. In section six we recall different convexity assumptions which we will work with in the remainder of the paper and give a result on the existence of solutions to the equations of type II thermoelasticity which follows by applying the direct method of the Calculus of Variations under a suitably generalised polyconvexity assumption. Later in section seven, we give examples of functions satisfying the generalised polyconvexity assumption and in section eight we use an approach of Knops & Stuart (1984) to prove the uniqueness of solutions for the Dirichlet problem with affine boundary conditions, assuming polyconvexity of the counterpart of the stored energy function. In section nine we give an example where it is shown that the existence of a non-zero thermal gradient can produce qualitatively different behaviour in comparison to the case of zero thermal gradient: specifically, we give an example in which an underlying homogeneous deformation is globally minimising if the thermal displacement gradient is zero but which is unstable to cavitation in the case of non-zero thermal gradient. Finally, in section ten, we prove a stronger uniqueness result than that proved in section eight for the case of radial deformations of a ball of type II thermoelastic material. 2. Preliminaries In this paper we consider a nonlinear homogeneous type II thermoelastic material. That is we assume that the Helmholtz free energy J, the first Piola-Kirchhoff stress tensor T, the entropy density h and the entropy flux vector q all depend on the gradient of deformation

Vx ¼ ðxi;A Þ, the temperature q and the gradient of the thermal displacement Va ¼ ða;A Þ. This assumption is fully consistent with the theories of Green and Naghdi (1993). In the absence of supply terms we recall that the equations of motion are given by

r€xi ¼ TKi;K ; i ¼ 1; ::; n5rx€ ¼ Div T

(2.1)

and that the balance of the entropy is given by

rh€ ¼ qA;A 5rh€ ¼ div q:

(2.2)

It follows from the arguments of (Green and Naghdi, 1993) that the following constitutive relations must hold

TKi ¼

vJ vJ vJ ; rh ¼  ; qK ¼  ; i; K ¼ 1; ::; n; vxi;K va;K vq

(2.3)

where J ¼ JðVx; q; VaÞ is the free energy. The thermal displacement and the temperature are related by the equation

Zt

aðX; tÞ ¼ aðX; 0Þ þ

qðX; sÞds:

(2.4)

0

If we substitute the constitutive equations into the evolution equation, we obtain the following system for the dynamical problem of nonlinear type II Thermoelasticity:

r€xi ¼

vJ vxi;K

! ; i ¼ 1; ::; n;  ;K

    d vJ vJ ¼ : dt vq va;K ;K

(2.5)

To complete the specification of the problem we need to impose initial and boundary conditions.

3. Static solutions Consider a nonlinear type II thermoelastic body occupying the bounded region U3Rn in its reference configuration. The aim of this section is to introduce a particular class of solutions for the system (2.5). Some of the functions which define these solutions are time independent and we will refer to these as static solutions. Let q ¼ q0 a positive constant and let v0 ¼ ðy01 ; .; y0n Þ be a constant vector. We seek solutions of the system (2.5) of the form

~ ðX; tÞ ¼ v0 t þ xðXÞ; a ~ ðX; tÞ ¼ q0 t þ aðXÞ: x

(3.1)

These satisfy system (2.5) provided that xðXÞ and aðXÞ satisfy

vWq0 vxi;K

!

 ¼ 0;

;K

vWq0 va;K

 ¼ 0;

(3.2)

;K

where Wq0 ðVx; VaÞ ¼ JðVx; q0 ; VaÞ; Vx ¼ ðxi;K Þ and Va ¼ ða;K Þ:1 Note that the unknowns in the system (3.2) are now the dependent variables x and a and that these do not depend on the time variable t. We note that system (3.2) always has non-trivial solutions. In fact, affine functions of the form

xh ðXÞ ¼ MX þ d; ah ðXÞ ¼ C:X þ D;

(3.3)

are always solution of the system (3.2) for arbitrary constants M ¼ ðMiA Þ; d ¼ ðdi Þ; C ¼ ðCA Þ and D.

1 Note that Vah0 in the case of constant thermal displacement and the fact that (3.1) generates a solution of (2.5) for any v0 whenever (3.2) is satisfied is then a consequence of frame-indifference.

R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

For the remainder of this paper, we present our results in three dimensions, with the exception of Section 8 in which we work in general dimension n. 4. The function W The aim of this section and of Section 7 is to analyse restrictions on the possible forms of the function Wq0 introduced in Section 3. To simplify the notation in this section we will denote by F the gradient of deformation and by b the gradient of the thermal displacement a. For ease of exposition we will work in dimension 3 and henceforth suppress the dependence of the free energy function (at constant temperature q0) Wq0 on the parameter q0 by writing WðF; bÞ in place of Wq0 ðF; bÞ. Thus, we are interested in the function

W ¼ WðF; bÞ:

(4.1)

whenever

vW ðVx; 0Þ vxi;K

! ¼ 0;

i ¼ 1; 2; 3:

(4.9)

;K

Hence (4.8) must correspond to a conservation law for solutions of (4.9). R 5. Conservation laws for the functional U WðVx; 0ÞdX It is a standard consequence of Nöether’s Theorem (see (Olver, 1986)) that, for an integrand WðVx; 0Þ, every variational R symmetry of U WðVx; 0ÞdX gives rise to a corresponding conservation law. We next list the conservation laws for solutions of (4.9) that follow from some of the commonly imposed symmetries. (i) If WðF; 0Þ is frame-indifferent, then WðQ F; 0Þ ¼ WðF; 0Þ for every Q ˛SOð3Þ and every F˛M33. The corresponding conservation laws are

A basic physical requirement is the postulate of frame-indifference which asserts that

WðQ F; bÞ ¼ WðF; bÞ;

111

(4.2)

2

3

for every Q ˛SOð3Þ and for every b˛R and We write

F˛M 33 .

vW WðF; bÞ ¼ WðF; 0Þ þ ðF; 0ÞbK þ EðF; bÞ; vbK

"

vW vW xj ðVx; 0Þ  xi ðVx; 0Þ vxi;K vxj;K

(4.3)

where EðF; bÞ ¼ Oðjbj2 Þ as jbj/0, for each F. On setting b ¼ 0 in (4.2) we see that

# ¼ 0;

(5.1)

;K

for i; j ¼ 1; 2; 3. (ii) Since

R U

WðVx; 0ÞdX is invariant under translations xi /xi þ ci ,

we obtain

WðQ F; 0Þ ¼ WðF; 0Þ;

(4.4) F˛M 33.

for every Q ˛SOð3Þ and for every Similarly, differentiating (4.3) with respect bK and setting b ¼ 0 yields that

"

# vW ðVx; 0Þ ¼ 0; vxi;K

i ¼ 1; 2; 3

(5.2)

;K

which is simply (4.9) itself.

vW vW ðQ F; 0Þ ¼ ðF; 0Þ: vb vb

(4.5)

(4.6)

Z WðVxðXÞ; VaðXÞÞdX;

(4.7)

U

defined on maps x : U/R3 ; a : U/R. Considering the system (3.2) in the case of constant thermal displacement (i.e., Va ¼ 0), we impose the requirement that



vW ðVx; 0Þ va;K

 ¼ 0;

WðVx; 0ÞdX is invariant under translations of reference

coordinates XI /XI þ CI , we obtain the energy-momentum equations

for every Q ˛SOð3Þ and for every b˛R3 and F˛M 33 . Hence each individual coefficient in the expansion of W in terms of b in (4.3) is frame-indifferent. We note that the solutions ðx; aÞ of the system (3.2) are the solutions of the EulereLagrange equations for the functional

Eðx; aÞ ¼

R U

It now follows from (4.2)e(4.5) that

EðQ F; bÞ ¼ EðF; bÞ;

(iii) Since

(4.8)

;K

2 The special orthogonal group of 3  3 orthogonal matrices with determinant equal to 1.

h i MKL ðVxÞ ¼ 0; ;K

L ¼ 1; 2; 3;

(5.3)

where L

MKL ðVxÞ ¼ WðVx; 0ÞdK  xi;L

vW ðVx; 0Þ; vxi;K

(5.4)

is the energy-momentum tensor. (iv) If WðF; 0Þ is isotropic then WðFQ ; 0Þ ¼ WðF; 0Þ, for every Q ˛SOð3Þ and for every F˛M33 and the corresponding conservation law is

h i XN MKL ðVxÞ  XL MKN ðVxÞ ¼ 0; ;K

(5.5)

for L; N ¼ 1; 2; 3. We will assume initially only that WðF; bÞ is frame-indifferent, vW so it follows from (4.8), (4.9) that in general ðF; 0Þ must be vbK

112

R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

a linear combination of the vector fields in the conservation laws (5.1)e(5.3). However, by the requirement of frame-indifference (4.5), it now follows that, for K ¼ 1; ::; 3,

  vW ðF;0Þ ¼ C L MKL ðFÞþdK ;for some C ¼ C L ;d ¼ ðdK Þ˛R3 ; vbK

(5.6)

since the bracketed terms in (5.1), (5.2) are not frame-indifferent. (Notice also that (5.1) is also inadmissible since the linear term in b in (4.3) does not explicitly depend on the variable x.) Hence, by expression (4.3), it now follows that

WðF; bÞ ¼ WðF; 0Þ þ C

L

MKL ðVxÞbK

þ dK bK þ EðF; bÞ;

(5.7)

This frame-indifferent structure now guarantees that (4.8) is satisfied by any solution of (4.9).3 Definition 5.1. We say that WðF; bÞ is isotropic if 33 and WðFQ ; Q T bÞ ¼ WðF; bÞ for every Q ˛SOð3Þ, every F˛Mþ 3 every b˛R . Hence, if we further require that the frame-indifferent W given by (5.7) is also isotropic, then the only possibility is

WðF; bÞ ¼ WðF; 0Þ þ EðF; bÞ;

(5.8)

with WðF; 0Þ and EðF; bÞ both being isotropic. For later use, we note that it follows from (5.8) that

v2 WðF;0Þ j vFKi vFL

li mK lj mL þ

v2 WðF;0Þ 2 I J g m m  0; cl; m˛R3 ; g˛R: vbI vbJ

Definition 6.2. A function WðF; bÞ is said to be quasiconvex at ðF; bÞ if the inequality

Z WðF þ VhðXÞ; b þ VgðXÞÞdX  WðF; bÞjVj;

(5.9)

Remark 5.2. We note that if WðF; bÞ is isotropic then, by the polar 33 we have F ¼ VR for some decomposition theorem, for each F˛Mþ R˛SOð3Þ with V a symmetric positive definite matrix. Hence pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi WðF; bÞ ¼ WðV; RbÞ ¼ Wð FFT ; RbÞ, where Wð FFT ; Q bÞ ¼ T ^ ; bÞ for every Q ˛SOð3Þ, F˛M 33 and b˛R3 . In particular, this WðFF þ

T T ~ ^ ; bÞ ¼ WðFF ; jbjÞ (see also condition is satisfied if WðF; bÞ ¼ WðFF the next example). Example 5.3. Other examples of frame-indifferent and isotropic energy functions are given by: (i) WðF; bÞ ¼ f ðjFbjÞ, (ii) WðF; bÞ ¼ f ðjðadjFÞT bjÞ and hence, in particular, it follows that not T ~ ; jbjÞ. every frame indifferent, isotropic W can be expressed as WðFF

6. Existence of solutions to the equations of type II thermoelasticity In this paper we will work with several notions of convexity and, for the convenience of the reader, we next recall some of these and refer to (Ciarlet, 1988) for further details. 1

Definition 6.1. A C function WðF; bÞ is said to be rank-one convex at ðF; bÞ if the inequality

WðFþ l5m;bþ gmÞ WðF;bÞþ

vW vW i ðF;bÞl mK þ ðF;bÞgmI ; vbI vFKi

(6.1)

holds for every l; m˛R3 ;and g˛R. We say that W is rank-one convex if (6.1) holds for all ðF;bÞ.4 If we further assume that W is C2, then it follows5 from (6.1), (5.9) that

3 We do not, at this stage, consider general null-lagrangian terms of the type considered in section 11. 4 Note that it follows on choosing g ¼ 0 that Wð,; bÞ is a rank-one convex function of F and, on choosing l ¼ 0, that WðF; ,Þ is a convex function of b. 5 This follows on rearranging (6.1), replacing l and g by εl, εg where ε > 0, then dividing by ε2 and letting ε/0.

(6.3)

V

holds, for all non-empty open bounded subsets V3R3 and all Lipshitz continuous functions h; g which vanish on the boundary of V. In addition, we say that WðF; bÞ is strictly quasiconvex at ðF; bÞ if and only if WðF; bÞ is quasiconvex at ðF; bÞ and equality in (6.3) holds only when h ¼ 0; g ¼ 0. We recall that quasiconvexity implies rank-one convexity (see, e.g., (Ball, 1977)). In general, this quasiconvexity condition will always be satisfied by the polyconvex functions that we next introduce (see the arguments used in (Ball, 1977)). Definition 6.3. We assume that the function WðF; bÞ is polyconvex in the sense that 33 WðF; bÞ ¼ GðF; adjF; detF; bÞcF˛Mþ ;

v2 WðF; 0Þ ¼ 0: vFvb

(6.2)

þ

cb˛R3 ;

(6.4)

3

where G: M 33  M 33  R  R /R (see (Ball, 1977)) is convex. Assume further that GðFn ; An ; dn ; bn Þ/N as ðFn ; An ; dn ; bn Þ 33  M 33  R3 and that G /ðF; A; 0; bÞ for any ðF; A; bÞ˛Mþ þ satisfies the growth hypotheses

  GðF; adjF; detF; bÞ  K0 þ K1 jFjp þ jadjFjq þ jbjr ;

(6.5)

for some constants K1 > 0; K0 , with p  2; q  3=2 and r > 1. Then the functional Eðx; aÞ defined by (4.7) attains a minimum on the set

A ¼

n x˛W 1;p ðUÞ; a˛W 1;r ðUÞ; xðXÞ ¼ x0 ðXÞ; o aðXÞ ¼ a0 ðXÞ for X˛vU; detVx>0 a:e: ;

(6.6)

where x0 and a0 are specified (such that A is non-empty). The proof of this result follows, for example, from a straightforward modification of the proof of Theorem 4.2 in (Muller et al., 1994). Remark 6.4. If we make a separability assumption that

GðF; adjF; detF; bÞ ¼ Gð1Þ ðjFj; jadjFj; detFÞ þ Gð2Þ ðjbjÞ

(6.7)

where Gð1Þ ; Gð2Þ are convex, with Gð2Þ ð,Þ a non decreasing function and Gð1Þ a non decreasing function in its first two arguments, with Gð1Þ ; Gð2Þ satisfying the appropriate growth hypotheses, then the resulting G will be frame-indifferent, isotropic and satisfy (6.5). Remark 6.5. Sufficiently regular minimisers of (4.7) will give rise to weak solutions of the Eq. (3.2). For general results on the different forms of the EulereLagrange equations satisfied by minimisers of (3.2) we refer to (Ball, 2002). Remark 6.6. We note that the combination of dependent variables given by ðadjVxÞT Va (which was used in Example 5.3 for constructing a frame-indifferent isotropic free energy) has good sequential weak continuity properties with respect to weak convergence in W 1;p (this follows by the div-curl lemma of (Murat, 1978)) and hence this combination could be incorporated as one of the arguments of the free energy in the definition of polyconvexity and in the associated existence theorems for minimisers (i.e., we could replace the structural assumption (6.4) by

R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

  ~ F;adjF;detF;b;ðadjFÞT b cF˛M 33 ; cb˛R3 ; WðF;bÞ ¼ G þ

(6.8)

33 M 33 Rþ R3 R3 /R is a convex function.) ~ where G:M In the next section we consider in more detail the possible interaction terms that can appear between the deformation gradient variable F and the thermal gradient variable b in the stored energy function WðF; bÞ and which are still consistent with the existence theory given above.

7. Examples of frame-indifferent polyconvex energy functions WðF; bÞ We first consider the case in which the free energy WðF; bÞ is frame-indifferent and expressible as a polynomial in the components of b with coefficients depending on F so that, by (5.7), N P

WðF;bÞ ¼

K ¼0;jIj¼K

FðIÞ ðFÞbi11 bi22 bi33

¼ WðF;0ÞþC L MKL ðVxÞbK þdK bK N X

þ

K ¼2;jIj¼K

FðIÞ ðFÞbi11 bi22 bi33 ;

(7.1)

where we have used the multi-index notation I ¼ ði1 ;i2 ;i3 Þ, jIj ¼ i1 þi2 þi3 , ik ˛NWf0g, k ¼ 1;2;3. By earlier arguments of Section 4, it follows that W is frameindifferent if and only if each coefficient function FðIÞ ðFÞ is frameindifferent. As previously noted in Remark 6.4, separable energy functions of the form (6.7) can give rise to (jointly) polyconvex energy functions of F and b. In this section we consider the possible interaction (i.e., mixed) terms in F and b that can occur in the last term of a polyconvex function of the form (7.1). The following Lemma, based on standard concavity arguments, will be used to illustrate some of the restrictions that must be placed on such interaction terms. Lemma 7.1. Let f : ½0; NÞ/ð0; NÞ, f ˛C 2 ð½0; NÞÞ satisfy 2

f ðxÞf 00 ðxÞ  ð1 þ gÞðf 0 ðxÞÞ  0 on ½0; NÞ

(7.2)

for some g> 0. Then there can be no point x0 ˛½0; NÞ for which f 0 ðx0 Þ> 0. Proof. We show that if f 0 ðx0 Þ > 0 at some point, then f cannot be smooth on the whole of ½0; NÞ and must have a singularity at some point x˛½x0 ; xmax , where

xmax ¼ x0 þ

f ðx0 Þ

gf 0 ðx0 Þ

:

This result follows from a standard estimate

f g ðxÞ 

f gþ1 ðx0 Þ ; f ðx0 Þ  ðx  x0 Þgf 0 ðx0 Þ

which is derived using concavity arguments (see Section 1.2.1 in (Straughan, 1998)). As an application of this result we obtain the following lemma. Lemma 7.2. Let Hðx; yÞ ¼ f ðxÞyn for ðx; yÞ˛½0; NÞ  R be convex on its domain, where the positive function f ˛C 2 ð½0; NÞÞ is convex and n ¼ 2m> 1 , n; m˛N. Then there can be no point x0 ˛½0; NÞ for which f 0 ðx0 Þ> 0. Proof. This follows from requiring that the determinant of the Hessian matrix of second order partial derivatives of H be non negative, which yields

f 00 ðxÞf ðxÞ 

113

n 2 ðf 0 ðxÞÞ  0: n1

(7.3)

1 then yields the result. n1 Hence, convexity of H requires that the convex function f be non-increasing. Hence, for example, considering terms of the form WðF; bÞ ¼ f ðjFjÞjbjn or f ðjadjFjÞjbjn, we see that such individual terms will not generally yield polyconvex functions,6 though we countenance the possibility that such terms could appear as part of an overall polyconvex function. However, there can be non-trivial dependence on the variable det F and examples of polyconvex functions to which the existence theory discussed in the previous section applies include Applying the previous lemma with g ¼

 WðF; bÞ ¼ GðF; adjF; detF; bÞ þ

aij bi bj

p

ðdetFÞq

;

where q  0; p  2ðq þ 1Þ, A ¼ ðaij Þ is positive definite and the function G satisfies the hypotheses of Section 6. Remark 7.3. In the one-dimensional case, the above discussions (see (7.3)) illustrate that the only jointly convex functions of the form Wðux ; ax Þ ¼ f ðux Þðax Þn , n ¼ 2m> 1, arise from convex decreasing functions f. Example 7.4. An example of a jointly polyconvex function which is not a finite polynomial in the components b, and hence not of the form (7.1), is the following: let W ð1Þ ðFÞ; W ð3Þ ðFÞ be polyconvex functions and W ð2Þ ðbÞ; W ð4Þ ðbÞ be convex functions. Then it is easily verified that

h i WðF;bÞ ¼ W ð1Þ ðFÞ þ C L MKL ðFÞbK  h   i þ C1 exp W ð3Þ ðFÞ exp W ð4Þ ðbÞ  1 þ W ð2Þ ðbÞ ð7:4Þ where ðMKL Þ is the energy momentum tensor given by (5.4), C ¼ ðC L Þ is a constant vector and C1 > 0 is a constant, is an example of a class of frame-indifferent polyconvex functions in the sense of (6.4). It is interesting that, e.g., in the case W ð4Þ ðbÞ ¼ jbj2 , the fourth term on the righthand side of (7.4) vanishes when Va ¼ 0. Hence this term is “switched on” by non-constant thermal displacements aðXÞ (see Section 9 for an interesting example relating to this observation). If we impose the further restriction that W in (7.4) be isotropic for all values of b, it then follows that each of W ð1Þ ðFÞ; W ð3Þ ðFÞ is also isotropic and that the constant vector C must be zero. 8. Uniqueness for the displacement boundary-value problem In this section we adapt an approach of (Knops and Stuart, 1984), (Knops and Trimarco, 2006) to prove that the affine solutions (3.3) are the unique solutions to the Dirichlet boundary value problem for the system (3.2) with the boundary conditions

xðXÞ ¼ xh ðXÞ; aðXÞ ¼ ah ðXÞ for x˛vU:

(8.1)

We assume that U3R is a bounded domain with C boundary, which is star-shaped with respect to the point X0 ˛U. Hence, in particular, it follows that n

NðXÞ$ðX  X0 Þ>0; for every X˛vU;

1

(8.2)

6 Typical conditions imposed on a stored energy function WðFÞ ¼ GðjFj; jadjFj; detFÞ to ensure polyconvexity are that G be convex and monotone increasing in its first two arguments. (See also (Mielke, 2005) on necessary and sufficient conditions for polyconvexity.)

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R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

where NðXÞ denotes the outward pointing unit normal to U at the point X˛vU. The main tool to obtain the result is the following divergence identity which is satisfied for every solution ðx; aÞ˛C 2 ðUÞ of (3.2):

nWðVx; VaÞ ¼ Div QA ;

    TKi ðVx; VaÞ xi  XA xi;A NK  TKi ðVy; VbÞ yi  XA yi;A NK    ¼ TKi ðVx; VaÞXA yi;A  xi;A NK þ TKi ðVx; VaÞ   NK  TKi ðVy; VbÞ yi  XA yi;A

(8.3)

vðy  xi Þ ¼ ðXL NL Þ TKi ðVx; VaÞ i NK vN    þ TKi ðVx; VaÞ  TKi ðVy; VbÞ yi  XA yi;A NK :

where

  vW ðVx; VaÞ xi  XK xi;K vxi;A   vW þ ðVx; VaÞ a  XK a;K : va;A

QA ¼ XA WðVx; VaÞ þ

Similarly,

(8.4)

nþ1

nðEðx; aÞEðy; bÞÞ  Z    ðTKi ðVx; VaÞ  TKi ðVy; VbÞÞ yi  XA yi;A NK da vU

Z

þ

    qK ðVx; VaÞ a  XA a;A NK  qK ðVy; VbÞ b  XA b;A NK ¼ qK ðVx; VaÞXA ðb;A a;A Þ NK þ ðqK ðVx; VaÞ

Lemma 8.1. Let ðx; aÞ: U/R ; ðy; bÞ: U/R be smooth static solutions (i.e., solutions of (3.2)) such that x ¼ y and a ¼ b on the boundary of U. If W is C1 on its domain and rank-one-convex, we have that, nþ1

 qK ðVy; VbÞÞ ðb  XA b;A Þ NK ¼ ðXL NL Þ qK ðVx; VaÞ   þ ðqK ðVx; VaÞ  qK ðVy; VbÞÞ b  XA b;A NK

vðb  aÞ NK vN ð8:10Þ

We then obtain

Z nðEðx; aÞ  Eðy; bÞÞ ¼ Z þ

   ðqK ðVx; VaÞ  qK ðVy; VbÞÞ b  XA b;A NK da;

vU

Z

vU

þ

(8.5)

NL XL Lda vU

  ðTKi ðVx; VaÞ  TKi ðVy; VbÞÞ yi  XA yi;A NK da 

    qK Vx; Va;B  qK ðVy; VbÞ b  XA b;A NK da

vU

where

(8.11)

Z Eðx; aÞ ¼

ð8:9Þ

WðVx; VaÞdy;

(8.6)

U

and

TKi ðVx; VaÞ ¼

vW vW ðVx; VaÞ; qK ðVx; VaÞ ¼ ðVx; VaÞ: vxi;K va;K

Proof. To simplify the calculation we assume, without loss of generality, that X0 ¼ 0. From the relation (8.3) we have

Z nEðx; aÞ ¼

   NK XK WðVx; VaÞ þ TKi xi  XA xi;A NK (8.7)

vU

   þ qK a  XA a;A NK da

and then

þ

ð8:12Þ

By our assumption of rank-one convexity (6.1) it follows that L  0 on vU. The lemma then follows from the assumption that U is starshaped with respect to 0, which implies that X:N  0 for X˛vU. , We note that the lemma also holds if we only assume that W satisfies the rank-one convexity condition on the set of the surface values ðVxðXÞ; VaðXÞÞ for X˛vU. In the particular case that we consider that ðy; bÞ is of the form (3.3) and ðx; aÞ is an arbitrary solution satisfying the boundary conditions (8.1), then we have

nðEðx; aÞEðy; bÞÞ 

NK XK ðWðVx;VaÞWðVy;VbÞÞda vU

  vðy  xÞ vðb  aÞ L ¼ WðVx; VaÞ  W Vx þ 5N; Va þ N vN vN vðyi  xi Þ vðb  aÞ þ TKi ðVx; VaÞ NK þ qK ðVx; VaÞ NK : vN vN

Z

Z nðEðx; aÞEðy; bÞÞ ¼

where

Z 

ðTKi ðVx;VaÞTKi ðVy;VbÞÞdi NK da vB

Z

  TKi ðVx;VaÞ xi XA xi;A NK

ðqK ðVx;VaÞpK ðVy;VbÞÞDNK da

þ

(8.13)

vB

vU

   TKi ðVy;VbÞ yi XA yi;A NK da Z    þ qK ðVx;VaÞ a XA a;A NK

As the following equalities

Z

Z TKi ðVx; VaÞNK da ¼

vU

vU

   qK ðVy;VbÞ b XA b;A NK da;

(8.8)

As ðx; aÞ agrees with ðy; bÞ at the boundary, for X˛vU, we can write7

ðTKi ðVy; VbÞÞNK da ¼ 0

(8.14)

vU

Z

Z ðqK ðVy; VbÞÞNK da ¼

vU

ðqK ðVy; VbÞÞNK da ¼ 0;

(8.15)

vU

hold for all solutions of (3.2), we obtain that 7

In

obtaining the last line of (8.9) we have vðx  yÞ 5n on vU since x  y ¼ 0 on vU. vN

Vðx  yÞ ¼

used

the

fact

that

Eðx; aÞ  Eðy; bÞ; whenever ðx; aÞ satisfies the boundary condition (8.1).

(8.16)

R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

Now, we obtain the main result of this section. 2

Theorem 8.2. Let us assume that W is a C rank-one convex function and suppose further that W is strictly quasiconvex at ðM; CÞ ¼ ððMiA Þ; ðCA ÞÞ. Let ðx; aÞ in C 2 ðUÞ be a solution to the system (3.2) that satisfies the boundary condition (8.1). Then ðx; aÞ is the solution given by (3.3). Remark 8.3. The strict quasiconvexity condition is necessary to obtain the uniqueness result. We next give an example where the strictness condition does not hold and the uniqueness of solutions fails. Let us consider the 1-D example given by the function

Wq ðux ; ax Þ ¼ u2x þ

a2x 4

 f ðqÞux ax  gqux  gðqÞ;

satisfy condition (INV) of Muller and Spector (Muller and Spector, 1995).8 First suppose that C ¼ 0 (the case of zero thermal 1 1 1 displacement gradient) then, using the inequality   2 ðd  aÞ d a a for d; a > 0, we obtain

axx 2

¼ 0:

Whenever f 2 ðqÞs1 the only possible solution for this system is uxx ¼ axx ¼ 0. Thus, if we impose Dirichlet boundary conditions the solution is completely determined. However in case that f ðqÞ ¼ 1, we can only conclude that the function f ¼ 2u  a must be linear. Hence we can obtain an infinite family of solutions. We note that in this case Wq ð,; ,Þ is convex but not strictly convex. 9. An example of thermally activated cavitation In this section we illustrate the theoretical possibility that a nonzero thermal displacement gradient can produce qualitatively different behaviour in comparison with the case of a zero thermal displacement gradient. In this particular example we construct a family of energy functions which have homogeneous solutions (3.3) as the only globally minimising equilibria when the thermal displacement gradient is identically zero but for which the homogeneous solutions are unstable to cavitation when the thermal displacement gradient is non-zero. We assume that W is of the form

WðF; bÞ ¼ GðF; det F; bÞ ¼ c1 jFjp þ c3 jbj2 i h 2 c þ 2 þ c4 edet F ejbj  1 ; det F

(9.1)

where the ci ; i ¼ 1; 2; 3 are nonnegative constants and c4 > 0. Then W is polyconvex of the form (7.4) since Gð,; ,; ,Þ is convex. 33 ; C˛R3 ; p˛ð2; 3Þ be given and consider Now let M˛Mþ minimising

Z Eðx; aÞ ¼

WðVx; VaÞdX

(9.2)

U

on

A ¼

n

ðx; aÞ x˛W 1;p ðUÞ; a˛W 1;2 ðUÞ; xðXÞ ¼ MX o aðXÞ ¼ C:X on vU; detVx>0 a:e:; ;

 xh ðXÞ; ah ðXÞ hðMX; C:XÞ

Z

U

 c2 þ c3 jVaj2 dy detðVxÞ

c c2 c1 jVxj þ 2   ðdetðVxÞ  detMÞ detM ðdetMÞ2 U ! Z   c c1 jVxjp þ 2 þ c3 jVaj2 dy þ c3 jVaj2 dy  detM U  Z    1 dy ¼ E xh ; 0 ; c1 jMjp þ c2  ð9:5Þ detM p

U

R

where in obtaining the last line we have used the fact that

U detðVxÞ  det MjUj (see (Muller and Spector, 1995)) and the

convexity of the integrands jFjp ; jbj2 . Hence ðxðXÞ; aðXÞÞhðMX; 0Þ is the unique global minimiser in this case.9 (Since the inequality in (9.5) is strict unless ðVxðXÞ; VaðXÞÞ ¼ ðM; 0Þ almost everywhere). It can be shown in contrast that if Cs0 (non-zero thermal displacement gradient) then, for M ¼ tB, with B fixed and t˛R sufficiently large, the solutions (9.4) are no longer energy minimisers of E. The idea behind the proof is to construct a test defor~ such that for t large Eðx; ~ ah Þ< Eðxh ; ah Þ and is exactly mation x analogous to the argument given in step 2 of the proof of Theorem 2 in (Sivaloganathan and Spector, 2004). Roughly speaking, the idea is that for Cs0 the energy can be further lowered from that of the ~ to produce a cavity or hole in the solution (9.4) by choosing x deformed body (see (Sivaloganathan and Spector, 2004) for further details). We refer to (Ball, 1982) for a study of radial cavitation in nonlinear elasticity without thermal effects. Remark 9.1. We note that the free energy functions (9.1), though polyconvex, do not satisfy the growth hypotheses of the existence theory given in Section 6. An existence theory for minimisers in type II thermoelasticity which allows cavitation to occur should follow on adapting the methods of (Muller and Spector, 1995; Sivaloganathan and Spector, 2000). It is clear that the above example of cavitation induced by a non-zero thermal gradient can be readily extended to more general free energy functions (for example, to suitable energy functions of the form (7.4)). 10. Radial solutions In this section we prove a uniqueness result for radial deformations of a ball, analogous to that given in Theorem 8.2, under the assumption that WðF; bÞ is strictly rank-one convex (see Definition 6.1).10 We assume further that W is frame-indifferent and isotropic so that we can write WðF; bÞ ¼ Fðy1 ; y2 ; y3 ; bÞ where F is a symmetric function of the yi which are the principal stretches (i.e., pffiffiffiffiffiffiffiffi the eigenvalues of FT F). We restrict our attention to the case where U ¼ B is the unit ball in R3 and we consider radial deformations of the form

(9.3)

We note that the pair of affine maps



c1 jVxjp þ

WðVx; VaÞdy  U

(8.17)

(8.18)

Z 

Z

Eðx; aÞ ¼

where f and g are given functions and q; g are constants. The system of Eq. (3.2) becomes

2uxx  f ðqÞaxx ¼ 0; f ðqÞuxx þ

115

(9.4)

is always a solution of the corresponding EulereLagrange equations. We assume henceforth that the deformations x also

8 Technically, we need to first extend the deformations u by the homogeneous deformation Ax to an open set containing U and require that the extended function satisfies (INV). 9 See also the related general results in (Spector, 1994) for the isothermal problem. 10 Notice, in particular, that in this section we do not assume that W is strictly quasiconvex as in the hypotheses of Theorem 8.2 (see also the corresponding result in (Knops and Stuart, 1984)).

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R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

UðrÞ X; r

xrad ðXÞ ¼

(10.1)

where r ¼ jXj and U : ½0; 1/½0; NÞ. We correspondingly assume that the thermal displacement is given by rad

a

F;4 ðl; l; l; 0Þh0:

It follows from the assumption of rank-one convexity of W11 that F satisfies

Fðw; y2 ; y2 ; cÞ  Fðy2 ; y2 ; y2 ; 0Þ þ F;1 ðy2 ; y2 ; y2 ; 0Þðw  y2 Þ þ F;4 ðy2 ; y2 ; y2 ; 0Þðc  0Þ;

ðXÞ ¼ aðrÞ

(10.9)

(10.10)

X where a : ½0; 1/½0; NÞ. In this case Varad ðXÞ ¼ a0 ðrÞ and, under jXj the assumption of isotropy (see Remark 5.1), it follows that W rad depends on Va only through jVaj. Thus we write

cw; yi > 0; cb; c˛R (i.e., Fð,; y2 ; y2 ; ,Þ is a convex function). We strengthen the assumptions on F by requiring that (10.10) holds with strict inequality whenever wsy2 or cs0. This holds, for example, provided that we further assume

    W Vxrad ðXÞ; Varad ðXÞ ¼ W rad Vxrad ðXÞ; Varad ðXÞ

F;11 ðy1 ; y2 ; y3 ; cÞ>0 and F;44 ðy1 ; y2 ; y3 ; cÞ>0 for all yi >0; c  0:

¼F

rad

0

ðy1 ; y2 ; y3 ; a ðrÞÞ;

(10.11) (10.2)

where F must satisfy the extra symmetry F ðy1 ; y2 ; y3 ; bÞ ¼ Frad ðy1 ; y2 ; y2 ; bÞ. The energy functional (4.7) then takes the form (up to a multiplicative constant 4p) rad

rad

Z1 IðU; aÞ ¼

  UðrÞ UðrÞ 0 r 2 Frad U 0 ðrÞ; ; ; a ðrÞ dr r r

(10.3)

0

and is defined in general on the admissible set

n o A ¼ U; a˛W 1;1 ðð0;1ÞÞ : Uð1Þ ¼ l;U 0 >0 a:e:;Uð0Þ  0; að1Þ ¼ a0 ; (10.4)

Thus, it follows from (10.10) that for any admissible functions in (10.4),

 

     U U U U U U U U U r 2 F U 0 ; ; ; a0 r 2 F ; ; ; 0 þ F;1 ; ; ;0 U0  r r r r r r r r r    U U U 0 ð10:12Þ ; ; ; 0 ða  0Þ : þ F;4 r r r Next, using (10.9) and the symmetry of F, note that the righthand side of the above expression is equal to

  d r3 U U U F ; ; ;0 : dr 3 r r r

It therefore follows by (10.3), (10.12), (10.13) that for C1 radial deformations

where l > 0; a0  0 are given constants.

Remark 10.1. For later use we note that if ðUðrÞ; aðrÞÞ in A corresponds to a radial deformation ðxðXÞ; aðXÞÞ which is in C 1 ðBÞ and satisfies detðVxðXÞÞ> 0 for X˛B, then by (10.1),

2

jVxðXÞj2 ¼ ½U 0 ðrÞ þ2



UðrÞ 2 /jVxð0Þj2 s0 as jXj/0: r

 Liminf r/0

   UðrÞ UðrÞ >0; Limsupr/0 < N: r r

, The EulereLagrange equations for the functional (10.3) are

  d 2 UðrÞ UðrÞ 0 r F;1 U 0 ðrÞ; ; ; a ðrÞ dr r r   UðrÞ UðrÞ 0 ¼ 2r F;2 U 0 ðrÞ; ; ; a ðrÞ ; r r

which implies in particular that, for all l > 0,

3   1 r U U U Fðl; l; l; 0Þ  Limr/0 F ; ; ; 0 ¼ Iðlr; 0Þ; 3 r r r 3 (10.14)

where we have used Remark 10.1, and so the radial homogeneous solution (10.8) is the global minimiser in the class of C1 radial maps (10.1). Notice also that the inequality in (10.14) is strict unless U is a radial homogeneous deformation and a is constant. The argument that leads to (8.16) shows that the reverse inequality also holds which leads to the following result. Theorem 10.2. . Let Fðy1 ; y2 ; y3 ; bÞ correspond to a rank-one convex energy function W, let ðUðrÞ; aðrÞÞ˛C 2 ðð0; 1ÞXC 1 ð½0; 1Þ satisfy the radial Eqs. (10.6), (10.7) and the boundary conditions Uð1Þ ¼ l; að1Þ ¼ a0 . Suppose further that F satisfies (10.11), then ðUðrÞ; aðrÞÞhðlr; a0 Þ. 11. Concluding remarks

(10.7)

for r˛ð0; 1Þ, where F;i denotes differentiation of F with respect to its i-th argument. We next note that the system (10.6), (10.7) should always have the homogeneous solutions

UðrÞhlr; aðrÞha0 ;

¼

3  r¼1 r U U U F ; ; ;0 3 r r r r¼0

(10.6)

and

  d 2 UðrÞ UðrÞ 0 r F;4 U 0 ðrÞ; ¼ 0 ; ; a ðrÞ dr r r

IðU; aÞ 

(10.5)

Hence, in particular,

(10.13)

(10.8)

In the development of the existence theory in Section 6 it may be of interest to incorporate more dependent variables, to represent the interaction between the strain and thermal displacement gradient variables, into our definition of polyconvexity (6.4). As noted in Remark 6.6, one such possibility is given by the combination ðadjVxÞT Va. A more general class of free energy functions can be obtained by adjoining the thermal displacement variable a to the deformation variable x, then defining x~ : U/R4 by x 11 , l ¼ m; b ¼ 0; g ¼ c in (6.1) and Set F ¼ y2 I þ ðw  y2 Þer 5er , where er ¼ jxj use (10.2).

R. Quintanilla, J. Sivaloganathan / European Journal of Mechanics A/Solids 32 (2012) 109e117

0

1

x1 ðXÞ B x2 ðXÞ C ~ B C xðXÞ ¼ @ x3 ðXÞ A aðXÞ

(11.1)

and then considering a polyconvex function consisting of a convex ~ of function of all k  k sub-determinants of the 4  3 matrix Vx order k ¼ 1; 2; 3. However, the frame-indifference properties of this class are at present unclear. (Though it appears that the use of ~ such as the 2  2 sub“mixed” sub-determinants of Vx determinants

va vXj vxi vX j

va vXk vxi i; j; k ¼ 1; 2; 3; jsk vX k

will not in general give rise to a frame-indifferent energy function.)12 It would also be of interest to incorporate boundary conditions other than the Dirichlet boundary conditions considered in (6.6): for example, the natural boundary condition arising when the thermal displacement a is left unprescribed on vU. Finally, we mention the interesting possibility that the structure underlying the example of thermally activated cavitation considered in Section 9 might be extended to give examples of free energy functions in the type II theory for which other instabilities (already known to occur in the nonlinear theory without thermal effects) will only occur in the case of non-zero thermal gradients (or thermal displacement gradients of sufficiently large magnitude). Acknowledgements The work of R.Q. is supported by the project “Partial Differential Equations in Thermomechanics. Theory and Applications (MTM2009-08150)” of the Spanish Ministry of Science and Innovation. References Ball, J.M., 1977. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337e403. Ball, J.M., 1982. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. R. Soc. Lond. A306, 557e611. Ball, J.M., 2002. Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics. Springer, New York, pp. 3e59. Chandrasekharaiah, D.S., 1996a. A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. J. Elasticity 43, 279e283. Chandrasekharaiah, D.S., 1996b. A uniqueness theorem in the theory of thermoelasticity without energy dissipation. J. Thermal Stresses 19, 267e272. Ciarlet, P.G., 1988. Mathematical Elasticity, vol. 1. Elsevier. Green, A.E., Naghdi, P.M., 1991. A re-examination of the basic postulates of thermomechanics. Proc. Royal Soc. London A 432, 171e194.

12 ~ correIt is well known that the kk sub-determinants of the 43 matrix Vx spond to null lagrangians (i.e, to integrands whose Euler-Lagrange equations are ~ identically satisfied by all smooth maps x).

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