Midwest Mechanics Seminar Series
Tuesday, March 20, 2017
10:15 a.m., 3540 Engineering Building
Refreshments Served at 10:00 a.m.
Variational Multiscale MODELLING of soft materials BASED on ELASTIC NETWORKS MODELS
Patrick Le Tallec
Ecole Polytechnique, F91 128 Palaiseau Cedex, France
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 is 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 .
- GLORIA, A.; LE TALLEC, P.; VIDRASCU, M.: Foundation, Analysis, and Numerical Investigation of a Variational Network Based Model for Rubber. Continuum Mech. Thermodyn., (2013), 1-31.
- DIANI, J.; BRIEU, M.; VACHERAND, J.M.: A damage directional constitutive model for Mullins effect with permanent set and induced anisotropy. European Journal of Mechanics A/Solids, 25, (2006), 483–496.
- GUILIE, J., LÊ, T.N, LE TALLEC, P.: Microsphere model for strain-induced crystallization in rubber, Journal of the Mechanics and Physics of Solids, Volume 81, August 2015, Pages 58-74.
- RASTAK R., LINDER, C.: A non-affine micro-macro approach to strain-crystallizing rubber-like materials, Journal of the Mechanics and Physics of Solids, Volume 111, 2018, Pages 67-99.
Patrick LE TALLEC is a Professor in the Department of Mechanics at the Ecole Polytechnique where he is the holder of the Chair André Citroën. Since 2010 he has been the Director of the Laboratory of Solid Mechanics at the 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. He is a past Editor of the International Journal for Numerical Methods in Engineering, and serves on the editorial board of Computer Methods in Applied Mechanics and Engineering and Computer and Structures. He has been awarded the Prize Blaise Pascal of the French Academy of Sciences (1985), the Chevalier de la légion d'honneur (2004), the Chevalier des Palmes Academiques (2010), and is an Officier de l’Ordre National du Mérite (2015). He has supervised 35 Ph.D. students; his own Ph.D. is from the University of Texas at Austin. His current interests concern the dynamics and control of nonlinear structures, the multiscale simulation of contact problems and of nonlinear structures, and the development of multiscale modelling of soft materials.