Augmented Lagrangian Variational FormulationFor A Plastic-hardening-damage Coupled Model

Abstract

A model for elastic-plastic hardening materials with coupled damage based on internal variables is described. Each kinematic variable is partitioned in the sum of a reversible and an irreversible part. The first enters as argument of the free energy, the other enters as argument of a dissipation functional D. that generates a generalised elastic domain K for the thermodynamic variables as the level set at 1 of the dissipation function. Following a general procedure for multivocal potentials, a family of variational principles can be obtained, whose stationary points represent the solution of the structural problem. 1. Aims and Limitations of the work It is presented a new phenomenological model for a class of elasticplastic-damaging materials, based on an internal variable description of the modifications of the microstructure. No attempt is made to give any physical interpretation for the internal variables. A consistent thermodynamic development of the basic equations is presented, based on the introduction of two functional ruling the reversible and the irreversible phenomena. Mild regularity hypotheses are made on these functional, that guarantee the mathematical treatment of the equations with the tools of convex analysis. The most significant point of the model is the consistent derivation of a (convex) elastic region from a dissipation functional of the kinematic variables (internal and external). In this way it is obtained a single elastic domain in the extended space of stresses and thermodynamic forces dual of the hardening and damage internal variables -, for which Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533 98 Damage and Fracture Mechanics normality rule holds naturally. According to the choice of the dissipation functional, and therefore of the generalised elastic domain, coupling between plasticity and damage can be easily modelled. Various kinds of behaviour can be recovered, as the decrease of the yield strength or the degradation of the hardening modulus. The different damages are clearly distinguished in the model according to whether they influence the reversible or the irreversible part of the constitutive equations. Principal aim of the model is to use it in developing variational principles amenable for simple numerical treatment. It is shown that it is possible to obtain a series of energy functional as straightforward generalisation of those valid in the field of plasticity with internal variables. Particular attention is devoted to mixed principles, that the authors have shown in previous works to be particularly useful in the numerical analysis (Cuomo ). The paper will present the main features of the model, while the consequences of particular choices of the form of the governing potentials will not be discussed. The identification of the model parameters is not faced in the present work 2. State variables and ambient spaces Kinematic internal variables are introduced for describing both material hardening and damage, defined in suitable vector spaces. Conjugate thermodynamic variables are defined in adjoint spaces. Each kinematic variables is partitioned as the sum of a reversible and an irreversible part. The first enter as argument in the Helmoltz free energy F, that rules the generalised elastic (reversible) behaviour. The irreversible parts enter as arguments in a dissipation functional D, that rules the dissipating phenomena. Schematically it is: £ = £^,+£ E%> deformation space a = «^+cx^ e9 internal strain space co = (0 v (4) Linearity assumptions have been introduced, so small deformation only are considered. The operator C accounts for the possibility that internal microdeformations result into overall deformations, Owen; v is an external damage potential. In most of the classic works on damage and hardening it is assumed that d, C? reduce to the null operator. From the virtual power equality the equilibrium conditions follow in the usual way, Cc+C|'x=/ Ci'C = f (5) The second equation is non classical; in it t are external source of damage work. This contribution has been examined by Fremond 4. Constitutive equations 4.1 Free energy Many different forms of the free energy have been assumed in the literature. Following the pioneering works of Kachanov it is often introduced a damaging effect on the stored elastic energy of the type Transactions on Engineering Sciences vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533