Variational upscaling for modeling state of strain-dependent behavior and stress-induced crystallization in rubber-like
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O R I G I NA L A RT I C L E
Patrick Le Tallec
· Julie Diani
Variational upscaling for modeling state of strain-dependent behavior and stress-induced crystallization in rubber-like materials
Received: 20 December 2019 / Accepted: 9 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this paper is to present a general upscaling strategy for deriving macroscopic constitutive laws for rubber-like materials from the knowledge of the network distribution and a mechanical description of the individual chains and of their free energy. The microscopic configuration is described by the position of the cross-links and is not obtained by an affine assumption but by minimizing the corresponding free energy on stochastic large representative volume elements with adequate boundary conditions. This general framework is then approximated by using a microsphere (directional) description of the network. It is presented in a global setting and is extended in order to handle situations with tube-like constraints and stress-induced crystallization. Keywords Microscopic free energy · Variational upscaling · Microsphere · Stress-induced crystallization
1 Introduction The purpose of this paper is to present and illustrate a general upscaling strategy for deriving macroscopic constitutive laws for rubber-like materials from the knowledge of the network distribution and a mechanical description of the individual chains and of their free energy. It is based on the variational approach which has been formalized in [7] in which the microscopic configuration is described by the position of the cross-links and is obtained not by an affine assumption but by minimizing the corresponding free energy on stochastic large representative volume elements (RVE) with adequate boundary conditions. This general framework is then approximated by using a microsphere (directional) description of the network and by performing a local minimization of the network free energy on this simplified configuration space while respecting the macroscopic boundary conditions. Microsphere models have been introduced for a long time for rubber-like materials, together with free energy minimization [8–15]. Herein, boundary conditions are expressed as a maximal advance path constraint [18]. This approximation framework handles the network free fluctuations and the configurational constraints applied to the different chains. It has already been introduced within a directional framework in [5] for taking into account anisotropic damage. We propose herein to give a general introduction of this upscaling strategy for network mechanics while trying to stay simple and constructive. For illustration, we extend this approach in two directions: Communicated by Andreas Öchsner. P. Le Tallec (B) · J. Diani Laboratoire de Mécanique des Solides, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau, France E-mail: [email protected] J. Diani E-mail: [email protected]
P. Le Tallec, J. Dia
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