Visco-hyperelastic model with internal state variable coupled with discontinuous damage concept under total Lagrangian formulation L. Laiarinandrasana a, *, R. Piques a , A. Robisson a,b a Ecole des Mines de Paris, Centre des Mate ´riaux, UMR CNRS 7633, BP 87, 91003 Evry Cedex, France b Schlumberger Riboud Product Centre, 26, Rue de la Cave ´e, BP 202, 92142 Clamart Cedex, France Received in final revised form 20 May 2002 Abstract The silica-filled rubber material presented in this paper exhibits nonlinear elasticity, non- linear rate dependence and stress-softening effect under cyclic loading. In order to model the material behavior in a finite element code, the internal state variable concept is considered for finite deformation viscoelasticity. Moreover, the so-called Mullins’ effect is taken into account by using a discontinuous damage concept. A total Lagrangian formulation with incompres- sibility constraint is adopted in the finite element code. The constitutive equations with their optimized set of parameters are validated by comparing the simulated results with experi- mental data. This result is very useful for the fatigue lifetime analysis of the investigated silica- filled rubber material. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Constitutive behavior; B. Finite strain; C. Finite elements; Viscoelasticity; Hyperelasticity 1. Introduction The present work is part of a general study on the fatigue lifetime of a filled rubber material used in shoe-sole manufacturing. The scientific approach followed for this analysis is to derive a diagram enabling to predict the number of cycles to crack initiation. This diagram should display on its axes relevant parameters which have to be calculated at every location of the prescribed structure. This task requires a tool that is capable of computing, for every loaded sample, not only the global International Journal of Plasticity 19 (2003) 977–1000 www.elsevier.com/locate/ijplas 0749-6419/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(02)00089-X * Corresponding author. Tel.: +33-1-60-76-30-64; fax: +33-1-60-76-31-60. E-mail address: lucien.laiarinandrasana@mat.ensmp.fr (L. Laiarinandrasana).