570 IV International Conference on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2011 M. Papadrakakis, E. Oñate and B. Schrefler (Eds) APPLICATION OF THE PROPER GENERALIZED DECOMPOSITION METHOD TO A VISCOELASTIC MECHANICAL PROBLEM WITH A LARGE NUMBER OF INTERNAL VARIABLES AND A LARGE SPECTRUM OF RELAXATION TIMES M. HAMMOUD*, M. BERINGHIER* AND J.C. GRANDIDIER* * Institut P' -Département Physique et Mécanique des Matériaux UPR CNRS 3346 - ENSMA Téléport 2, 1 avenue Clément Ader, BP 40109, F-86961 Futuroscope Cedex, France. E-mail: {hammoudm, marianne.beringhier, J.C.grandider}@ensma.fr, Web page: http://www.ensma.fr Key words: PGD, Internal variables, Relaxation time. Abstract. We here extend the use of the PGD to the case of a viscoelastic mechanical problem with a large number of internal variables and with a large spectrum of relaxation times. Such a number of internal variables leads to solving a system of non linear differential equations which correspond to the return to the equilibrium state. The feasibility and the robustness of the method are discussed in a simple case; a future application is the simulation of a polymer reaction under cyclic loading. 1 INTRODUCTION To solve a problem with a large number of degrees of freedom (dofs), numerical techniques, as parallel computing and domain decomposition, can be used. In the case of a multiphysical problem or of a problem with a large number of internal variables, it leads to solving a large number of differential equations. The PGD method, based on the radial approximation [4], has proved to be efficient for solving problems with a large number of dofs [2, 3], and particularly in the case of a coupled thermo mechanical problem [1]. We here extend the use of the PGD [1] to the case of a viscoelastic mechanical problem with a large number of internal variables and with a large spectrum of relaxation times (50 to 100 times [5]). Such a number of internal variables leads to solving a system of non linear differential equations which correspond to the return to the equilibrium state. The feasibility and the robustness of the method are discussed in a simple case; a future application is the simulation of a polymer reaction under cyclic loading. Section 2 introduces the equations of the viscoelastic mechanical problem. In section 3, the PGD is used to solve the problem with internal variables. While in section 4, we present the numerical results of a problem with one internal variable but with different relaxation times.