The tikhonov regularization method in elastoplasticity

Abstract

The numeric simulation of the mechanical behaviour of industrial materials is widely used in the companies for viability verification, improvement and optimization of designs. The eslastoplastic models have been used for forecast of the mechanical behaviour of materials of the most several natures (see [1]). The numerical analysis from this models come across ill-conditioning matrix problems, as for the case to finite or infinitesimal deformations. A complete investigation of the non linear behaviour of structures it follows from the equilibrium path of the body, in which come the singular (limit) points and/or bifurcation points. Several techniques to solve the numerical problems associated to these points have been disposed in the specialized literature, as for instance the call Load controlled Newton-Raphson method and displacement controlled techniques. Although most of these methods fail (due to problems convergence for ill-conditioning) in the neighbour of the limit points, mainly in the structures analysis that possess a snapthrough or snap-back equilibrium path shape (see [2]). This work presents the main ideas formalities of Tikhonov Regularization Method (for example see [12]) applied to dynamic elastoplasticity problems (J2 model with damage and isotropic-kinetic hardening) for the treatment of these limit points, besides some mathematical rigour associated to the formulation (well-posed/existence and uniqueness) of the dynamic elastoplasticity problem. The numeric problems of this approach are discussed and some strategies are suggested to solve these misfortunes satisfactorily. The numerical technique for the physical problem is by classical Gelerkin method.