COMMUNICATIONS IN APPLIED NUMERICAL METHODS, zyxwvuts Vol. 4, 731-740 (1988) AN INTEGRATION ALGORITHM AND THE CORRESPONDING CONSISTENT TANGENT OPERATOR FOR FULLY COUPLED ELASTOPLASTIC AND DAMAGE EQUATIONS AHMED BENALLAL, RENE BILLARDON AND ISSAM DOGHRI zyxwv Laboratoire de Mtkanique et Technologie, E. N.S. de Cachan, C. N. R.S., Llniversiti Paris 6, 61, avenue du Prlsident Wilson, 94230 Cachan, France SUMMARY This paper deals with the implementation in finite element calculations of complex elastoplastic constitutive equations exhibiting non-linear kinematic and isotropic hardening, and fully coupled to a continuous ductile damage evolution model. The Newton method is used to solve the non-linear global equilibrium equations as well as the non-linear local equations obtained by fully implicit integration of the constitutive equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. The procedure described has been implemented in the general purpose code ABAQUS. 1. INTRODUCTION In the past twenty years, ever more sophisticated constitutive equations have been developed to model the complex non-linear behaviour of engineering materials. During this time a considerable amount of knowledge has been acquired on the accuracy and stability of the various efficient algorithms implemented in most industrial finite element packages. However, the class of constitut- ive equations generally implemented in these codes is restricted to simple models such as isotropic hardening, or linear kinematic hardening for elastoplasticity zyxw . These models fail to represent phenomena such as the microscopic effects of damage, for instance the effects of growing micro- voids in ductile damage. Within the framework of continuum damage theory, damage is described by a continuous (in space and time) variable. The evolution of ductile damage associated with elastoplasticity can be modelled by a simple continuous differential equation. The coupling of this damage evolution model with the elastoplastic strain constitutive equations makes it possible to describe experimen- tally observed phenomena such as the decrease of the elasticity modulus with increasing plastic strain, and also the softening effect of damage which competes and eventually supersedes the classical strain hardening. Such constitutive equations have been identified for many industrial materials.* The algorithms generally implemented in finite element programs for integrating elastoplastic constithtive equations are the so-called return mapping algorithms commonly associated with incremental methods based on the use of the classical elastoplastic modulus. In this paper, a generalization of the return mapping algorithm recently proposed by Simo and Taylor3 is applied to complex elastoplastic constitutive equations exhibiting non-linear kinematic and isotropic hard- ening and fully coupled to a continuous ductile damage evolution model. 2. CONSTITUTIVE EQUATIONS 2,l. Thermodynamics zyxwvut of irreversible processes4 It is assumed that the mechanical state of the material at a given time is completely described by a finite number of state variables defined at each point. Here, the set of state variables is defined as 074&8025/88/060731-10$05 zyxwvut .OO Received August 1987 @ 1988 by John Wiley & Sons, Ltd. Revised February 1988