Abstract

This chapter provides a general and self-contained overview of the variational approach to nonlinear dissipative thermo-mechanical problems initially proposed in Ortiz and Stainier (1999) and Yang, Stainier, and Ortiz (2006). This approach allows to reformulate boundary-value problems of coupled thermo-mechanics as an optimization problem of an energy-like functional. The formulation includes heat transfer and general dissipative behaviors described in the thermodynamic framework of Generalized Standard Materials. A particular focus is taken on thermo-visco-elasticity and thermo-visco-plasticity. Various families of models are considered (Kelvin–Voigt, Maxwell, crystal plasticity, von Mises plasticity), both in small and large strains. Time-continuous and time-discrete (incremental) formulations are derived. A particular attention is dedicated to numerical algorithms which can be constructed from the variational formulation: for a broad class of isotropic material models, efficient predictor–corrector schemes can be derived, in the spirit of the radial return algorithm of computational plasticity. Variational approximation methods based on Ritz–Galerkin approach (including standard finite elements) are also described for the solution of the coupled boundary-value problem. Some pointers toward typical applications for which the variational formulation proved advantageous and useful are finally given.

Keywords

Thermo-mechanics; Variational principles; Nonlinear dissipative behavior; Thermo-visco-elasticity; Thermo-visco-plasticity; Constitutive update algorithms; Ritz–Galerkin approaches

1 Introduction

Variational principles have played an important role in mechanics for several decades, if not more than a century (see for example Lanczos, 1986 or Lippmann, 1978). They have been mostly developed, and widely used, for conservative systems: the most eminent examples are Hamilton’s principle in dynamics and the principle of minimum potential energy in statics. Some variational principles with application to dissipative systems have been around for a long time as well, such as principles of maximum dissipation for limit analysis (notably in plasticity).

Variational approaches present many attractive features, especially regarding the possibilities that they offer for mathematical analysis, but also for numerical approaches. They open an easier way to unicity, convergence, and stability analysis of mathematical formulations and associated numerical methods. This has motivated a very large quantity of published work and an exhaustive review is thus out of the scope of this chapter. To directly focus on the category of variational approaches envisioned here, let us then simply say that, following the pioneering work of Biot (1956, 1958), the variational form of the coupled thermo-elastic and thermo-visco-elastic problems has been extensively investigated (see for example Batra, 1989; Ben-Amoz, 1965; Herrmann, 1963; Molinari & Ortiz, 1987; Oden & Reddy, 1976). On the other hand, several authors have proposed variational principles for the equilibrium problem of general dissipative solids in the isothermal setting: see for example Carini (1996), Comi, Corigliano, & Maier (1991), Hackl (1997), Han, Jensen, & Reddy (1997), Martin, Kaunda, & Isted (1996), Mialon (1986), Ortiz & Stainier (1999), in elasto-visco-plasticity, and also Balzani & Ortiz (2012), Bourdin, Francfort, & Marigo (2008), Francfort & Marigo (1998), Kintzel & Mosler (2010, 2011), in brittle and ductile damage. By contrast, the case of thermo-mechanical coupling (i.e. with conduction) in these latter classes of dissipative materials has received comparatively less attention (cf. Armero & Simo, 1992, 1993; Simo & Miehe, 1992, for notable exceptions).

This chapter is intended to provide an overview of recent and less recent work by the author and colleagues on a specific variational approach (initially described in Yang et al., 2006) to coupled thermo-mechanical problems involving nonlinear dissipative behaviors, such as thermo-visco-elasticity and thermo-elasto-visco-plasticity. It will also be the occasion to fill a few gaps between previously published material, in particular by providing a more detailed account of thermal coupling aspects for a variety of constitutive models written under variational form. Links toward closely related work by other researchers are also provided.

We start in Section 2 by setting the general thermodynamic modeling framework serving as a foundation for the proposed variational formulation of coupled thermo-mechanical boundary-value problems. This framework follows closely that of Generalized Standard Materials (GSM) (Halphen & Nguyen, 1975), with a local state description based on internal variables. We will also recall some elements of finite transformations kinematics, as well as balance equations in Lagrangian and Eulerian formulations (although we will mostly work in a Lagrangian setting). This part is quite standard, with a few departures from the mainstream approach (e.g. the use of a Biot conduction potential). We then proceed (Section 3) to reset the constitutive and balance equations defining thermo-mechanical boundary-value problems under a variational form. By variational, we here understand that the problem is formulated as an optimization (or at least a stationarity) problem, with respect to fields of state variables. Since we work with a local state description based on internal variables, we will split the presentation in two parts: first, the local constitutive problem (determination of internal variables or their rate), and, second, the boundary-value problem (determination of fields of external variables). Each part is itself structured in two subparts: we first present the time-continuous evolution problem and its variational formulation, followed by the time-discrete (or incremental) variational formulation. This structure of presentation, which somewhat differs from that adopted in Yang et al. (2006), allows to show that the variational boundary-value problem is formally identical to a thermo-elastic problem, internal variables being handled locally through a nested constitutive variational problem. This is probably the most interesting result presented in that specific part of the paper. After presenting the variational formulation for the incremental boundary-value problem, we show how to add more complex thermo-mechanical boundary conditions, such as mixed thermal conditions (e.g. convective exchange). The variational formulation of coupled thermo-mechanical boundary-value problems is initially presented within a quasi-stationary context (yet including combined heat capacity and conduction effects), but we show in subsection 3.3 how it can be extended to account for inertia effects in the time-discrete framework. We conclude this part by describing linearization procedure in the case of infinitesimal (small) displacements and temperature variations. Note that nonlinearities can remain within this “linearized” context, due to the presence of thermo-mechanical coupling terms.

In Sections 4 and 5, we look in more details at the variational formulation of (continuous and incremental) constitutive equations for some specific models in thermo-visco-elasticity and thermo-elasto-visco-plasticity. We start with the simplest thermo-visco-elastic model possible: linearized kinematics Kelvin–Voigt model with linear elasticity and viscosity. Given its relative simplicity of formulation, the variational update for this constitutive model is treated in details, including all possible temperature dependence effects (thermo-elasticity, thermal softening of elastic and viscous moduli). Note that this is the only model for which a complete treatment is provided here, some simplifying hypotheses being taken for later, more complex, models, for the sake of clarity in the presentation. We then move on to generalized Maxwell models, which introduce internal variables (viscous strains). The analysis of Kelvin and Maxwell models is then repeated for finite strains kinematics. Maybe the most interesting point in that part is the fact that, for isotropic materials at least, constitutive updates can be reduced to solving a reduced number of scalar equations by adopting a spectral approach [as given in Fancello, Ponthot, and Stainier (2006)], independently of the complexity of elastic and viscous potentials adopted in the models. Section 5 deals with thermo-elasto-visco-plasticity. We start with crystal plasticity, which describes fine scale behavior of crystalline materials (mostly metals for our purpose, but also some organic materials and rocks). For this class of models, the variational formulation offers the power of optimization algorithms to solve the complex problem of determining (incremental) plastic slip activity, a problem which can become quite acute when considering complex latent hardening models. The section continues by considering plasticity models at the macroscopic scale, chiefly von Mises (J2) plasticity. We look at both the time-continuous and time-discrete variational formulations of linearized...