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International Journal of Solids and Structures 50 (2013) 40434054

Contents lists available at ScienceDirect

International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

An incremental variational approach to coupled thermo-mechanicalproblems in anelastic solids. Application to shape-memory alloys

0020-7683/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijsolstr.2013.08.013

Corresponding author. Tel.: +33 1 6415 3746; fax: +33 1 6415 3741.E-mail address: michael.peigney@polytechnique.org (M. Peigney).

M. Peigney a,, J.P. Seguin ba Universit Paris-Est, Laboratoire Navier (Ecole des Ponts ParisTech, IFSTTAR, CNRS), F-77447 Marne la Valle Cedex 2, Franceb Laboratoire dIngnierie des Systmes de Versailles, Universit de Versailles, 45 Avenue des Etats-Unis, 78035 Versailles Cedex, France

a r t i c l e i n f o

Article history:Received 27 March 2013Received in revised form 10 July 2013Available online 21 August 2013

Keywords:Incremental variational principlesThermo-mechanical couplingNon-smooth mechanicsShape memory alloys

a b s t r a c t

We study the coupled thermo-mechanical problem that is obtained by combining generalized standardmaterials with Fouriers law for heat conduction. The analysis is conducted in the framework of non-smooth mechanics in order to account for possible constraints on the state variables. This allows modelsof damage and phase-transformation to be included in the analysis. In view of performing numerical sim-ulations, an incremental thermo-mechanical problem and corresponding variational principles are intro-duced. Conditions for existence of solutions to the incremental problem are discussed and compared withthe isothermal case. The numerical implementation of the proposed approach is studied in detail. In par-ticular, it is shown that the incremental thermo-mechanical problem can be recast as a concave maximi-zation problem and ultimately amounts to solve a sequence of linear thermal problems and purelymechanical (i.e. at a prescribed temperature field) problems. Therefore, using the proposed approach,thermo-mechanical coupling can be implemented with low additional complexity compared to the iso-thermal case, while still relying on a sound mathematical framework. As an application, thermo-mechan-ical coupling in shape memory alloys is studied. The influence of the loading strain-rate on the phasetransformation and on the overall stressstrain response is investigated, as well as the influence of thethermal boundary conditions. The numerical results obtained by the proposed approach are comparedwith numerical and experimental results from the literature.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

This paper focuses on coupled thermo-mechanical evolutions ofdissipative solids, in the geometrically linear (small strains) setting.The framework of generalized standard materials in non-smoothmechanics is considered (Halphen and Nguyen, 1975; Moreauand Panagiotopoulos, 1988; Frmond, 2002). In that framework,the local state of the material is described by the strain e, the tem-perature h, and an internal variable a. The constitutive laws aredetermined from the Helmholtz free energy w and a convex dissipa-tion potential U. In its original form (Halphen and Nguyen, 1975),that framework covers a wide range of elastoplastic models,including limited and nonlinear hardening. Its extension to non-smooth mechanics has been extensively studied by Frmond(2002) and allows constraints on the internal variable a to be takeninto account in a rigorous fashion. That feature is crucial for themodelling of such phenomena as damage or phase-transformation,as the internal variable in such cases is typically bounded. Thethermodynamic analysis of the media considered is presented inSection 2, leading to a boundary value problem for the mechanical

and thermal fields. As pointed out by Yang et al. (2006), the time-discretization of the thermo-mechanical evolution problem is asensitive issue because of the coupling between mechanical andthermal equations. For instance, the Euler implicit scheme leadsto an incremental thermo-mechanical problem for which existenceof solutions cannot generally be ensured. This is in contrast with theisothermal case, for which the Euler implicit scheme provides awell-posed incremental problem under standard assumptions ofconvexity on the functions w and U.

One objective of this paper is to propose a sound time-discret-ization scheme for coupled thermo-mechanical problems, retain-ing some essential features displayed by the Euler scheme in theisothermal case (most notably the consistency with the rate prob-lem and the existence of solutions). A central idea is the use of avariational formulation for the incremental problem. Incrementalvariational principles for dissipative solids have been the focus ofa lot of attention in recent years, offering new perspectives in var-ious topics such as finite-strains elasto-viscoplasticity (Ortiz andStainier, 1999), homogenization (Miehe, 2002; Lahellec andSuquet, 2007), formation and stability of microstructures (Ortizand Repetto, 1999; Miehe et al., 2004). Incremental variationalprinciples for coupled thermo-mechanical problems have beenproposed by Yang et al. (2006) in the case where the heat flux q

http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijsolstr.2013.08.013&domain=pdfhttp://dx.doi.org/10.1016/j.ijsolstr.2013.08.013mailto:michael.peigney@polytechnique.orghttp://dx.doi.org/10.1016/j.ijsolstr.2013.08.013http://www.sciencedirect.com/science/journal/00207683http://www.elsevier.com/locate/ijsolstr

4044 M. Peigney, J.P. Seguin / International Journal of Solids and Structures 50 (2013) 40434054

derives from a potential v in rh=h, i.e. when the heat conductionlaw takes the form q v0rh=h. In this paper, we stick with thestandard Fouriers law of heat conduction q Krh, which doesnot fall in the format considered by Yang et al. (2006).

In Section 3 we introduce an incremental problem for the classof coupled thermo-mechanical problems considered, along with acorresponding variational formulation. The variational formulationof the incremental thermo-mechanical problem serves two pur-poses. First, it allows the existence of solutions to be studied, asdiscussed in Section 3. Second, the variational formulation leadsto a convenient and efficient way of solving the incremental ther-mo-mechanical problem. The latter can be indeed be recast as aconcave maximization problem, for which well-known algorithmsare available. As detailed in Section 4, an advantage of that ap-proach is that the solution of the thermo-mechanical problemcan be obtained by solving a sequence of linear thermal problemsand purely mechanical (i.e. at prescribed temperature) problems.This calls for an easy implementation in an existing finite-elementcode. A crucial point in the analysis lies in the introduction of anauxiliary linear problem, akin to the adjoint state used in optimalcontrol problems (Lions, 1968).

As an application, the proposed method is used in Section 5 tostudy thermo-mechanical coupling in shape-memory alloys. Thesignificant role of thermal effects in shape-memory alloys hasnotably been put forward by Peyroux et al. (1998) and Chrysochooset al. (2003). The solid/solid phase transformation that occurs inthose materials is known to produce significant amounts of heat,associated both with recoverable latent heat effects and irrevers-ible frictional contributions. Depending on the rate of loadingand on the thermal exchange conditions, the heat produced bythe phase transformation may not have time to diffuse in the bodyand the temperature field may become inhomogeneous. In suchconditions, the overall stressstrain response becomes signifi-cantly different from its isothermal counterpart, and it is manda-tory to take the thermo-mechanical coupling into account.Therefore, shape-memory alloys offer a particularly relevant appli-cation of the general methods presented in this paper. In Section 5,the influence of thermal effects on the phase-transformation andon the overall stressstrain curve is investigated in detail.

2. Thermo-mechanical evolutions of continuous media

2.1. Thermodynamic principles

Consider the evolution (on a time interval 0; T) of a continuousmedium occupying a domain X in the reference configuration. Werestrict our attention to the geometrically linear setting, definingthe strain e as e 1=2rurT u where u is the displacement.The first principle of thermodynamics givesZ t0

t

_EdsZ t0

t

_Kds Z t0

tPds

Z t0t

Qdt for all 0 6 t 6 t0 6 T: 1

In Eq. (1), K and E are respectively the kinetic and the internal en-ergy of the system. The internal energy E can be written in the formE

RX edx where e is the internal energy density. In the right-hand

side of (1), P denotes the power of external loads, and Qis the rateof heat received by the system. The upper dot in (1) denotes left-time derivative.1 The principle of virtual power gives the relation

_K P Z

Xr : _edx 2

1 In non-smooth mechanics, left- and right-time derivative of physical quantitiesmay not be equal. In order to respect the principle of causality, the constitutiverelations need to be written in terms of left-time derivatives (see e.g. Frmond, 2002).

where r is the stress. Expressing Qas

Q Z@X

q:ndxZ

Xrdx 3

where q is the heat flux and r a heat source, the relation (1) can berewritten asZ t0

t

ZX _e r : _e div q rdxds 0 for all 0 6 t 6 t0 6 T: 4

The relation (4) also holds when replacing X with an arbitrary sub-domain X0 X. Therefore, we obtain the local equation_e r : _e div q r 0 a:e: in X 0; T 5

where the ab