Laurent Stainier

A Variational Approach to Modeling Coupled Thermo-Mechanical Nonlinear Dissipative Behaviors

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.

Effective transient behaviour of inclusions in diffusion problems

This paper is concerned with the effective transport properties of heterogeneous media in which there is a high contrast between the phase diffusivities. In this case the transient response of the slow phase induces a memory effect at the macroscopic scale, which needs to be included in a macroscopic continuum description. This paper focuses on the slow phase, which we take as a dispersion of inclusions of arbitrary shape. We revisit the linear diffusion problem in such inclusions in order to identify the structure of the effective (average) inclusion response to a chemical load applied on the inclusion boundary. We identify a chemical creep function (similar to the creep function of viscoelasticity), from which we construct estimates with a reduced number of relaxation modes. The proposed estimates admit an equivalent representation based on a finite number of internal variables. These estimates allow us to predict the average inclusion response under arbitrary time-varying boundary conditions at very low computational cost. A heuristic generalisation to concentration-dependent diffusion coefficient is also presented. The proposed estimates for the effective transient response of an inclusion can serve as a building block for the formulation of multi-inclusion homogenisation schemes.