Enfoque variacional a la homogenización de una barra heterogénea en fase post-crítica

Abstract

In the present work we seek to explore concepts related to the one-dimensional fracture using variational homogenization as a conceptual framework for analysis. The variational approach opens the possibility of analyzing the post-critical development of deformation microstructures in inelastic materials based on the analysis of the energy density of the material itself. The proposed formulations offer two important perspectives on solid inelastic mechanics. First, the definition of the material stability of standard dissipative materials is based on convexity conditions of the incremental stress potential. Secondly, the developments of the microstructure in unstable inelastic solids are associated with non-convex incremental stress potentials similar to the problems of elastic phase transformation. The research focuses on the analysis of a homogeneous bar formed by components that exhibit softening that occurs only when a certain level of effort is reached. The phenomenon of deformation softening occurs when the plastic module considered in the constitutive law becomes negative generating the so-called loss of ellipticity of the constitutive equations. This problem is translated into a model which has a disturbance in the central element of the homogeneous bar, this disturbance is a reduction in the initial yield stress to activate the localization. The global solution is achieved through the implementation of finite elements with an algorithm that controls the displacements in each step of time.