Large deformation models of structures made of soft materials such as polymers or biological tissues are faced with mathematical and modelling issues. On the mathematical side, existence results are based on quasi convexity and coerciveness assumptions to be a priori imposed on the macroscopic free energy. On the modelling part, the challenge is to introduce a relevant choice of macroscopic internal variables and to postulate an adequate form of dissipation within the material. This global and phenomenological approach can successfully handle different levels of complexity to take into account electrochemical excitations, clusterisation, ageing, crystallization and so on. It is nevertheless faced with a high level of arbitration in the derivation of the model, and with the difficulty of handling evolving anisotropy or local heterogeneities as observed in damage or in phase transitions.
To overcome such limitations, one tries to relate the energy densities at the continuum level with the physically motivated free energy of polymer chains. The difficulty is to pass from one chain to a network of cross-linked chains, and to relate the evolution of this network to the macroscopic deformation. The use of a microscopic network problem minimizing the local free energy while imposing the macroscopic deformation through a far field microscopic boundary condition is a mathematically rigorous and attractive approach. It leads in theory to well behaved mathematical models but it is practically out of reach because of its complexity.
A simpler strategy which is yet to be mathematically justified is to reduce the microstructure to a distribution of one dimensional stress strain relations over the orientation space. Such microsphere approaches have been used successfully in the past to describe complex phenomena such as Mullins effect or strain induced crystallization. But in most cases, the different local orientations were related to the 3D deformation through a simplified affine network deformation assumption. The talk will explain why and how to go beyond the affine assumption through a local variational approach which minimizes the local free energy of the microstructure in the configuration space under a macroscopic deformation constraint to be expressed as a maximal path constraint.
About the Speaker
Patrick Le Tallec is a Professor with the Department of Mechanics at Ecole Polytechnique. His field of research is concerned with computational mechanics. A part of his career was devoted to Augmented Lagrangian and operator splitting methods in nonlinear mechanics, to the numerical analysis and simulation of nonlinear elastic problems, to domain decomposition techniques and to fluid structure interaction problems. His current interests concern the dynamics and control of nonlinear structures, the multiscale simulation of contact problems, and the development of multiscale and multimaterial modelling for nonlinear structures.
Host: Professor Yuhang Hu