Relaxation element method in mechanics of deformable solid

Abstract

In this chapter a new method - the relaxation element method is justified. The definition of the changing of stress fields in solids under loading as a result of the change of elastic energy in a local volume, undergoing plastic deformation, is laid down at the basis of the method. The Relaxation Element Method (REM) solves effectively two problems of a deforming solid (DS): 1. The construction of the different distributions of plastic deformation in local regions of various geometrical shape. 2. Modelling of the consequent involvement of separate structural elements into plastic deformation, operating on the principle of an inverse task of mechanics of deforming solids. With this method a stress-strain state of the elastic plane with the sites of plastic deformation in the form of a circle, rectangle, and a localized shear band is analytically described. Examples of the construction of the sites of plastic deformation with gradients are given. The stress-strain state of a plane with a round inclusion is considered. Examples of the simulations by the REM of the effects of Lüders band formation and interrupted flow in polycrystals are given. The analysis of the influence of rigidity of a testing device on qualitative and quantitative characteristics of the loading diagram is presented. The effect of the gradients of plastic deformation on the stress of Lüders band initiation is analyzed. It is shown that the dependence of the stress of Lüders band initiation on grain sizes is the consequence of the independency of the gradient of plastic deformation under the changing of grain sizes. A modified model of Griffith crack surrounded by a layer of plastically deformed material is proposed. Plastic deformation is shown to eliminate the singularity at the crack tip. The maximum stresses are observed at the boundary of the plastic zone in an elastically deformed matrix. The stress concentration increases as the thickness of the plastic layer decreases. The obtained results testify to high predictable possibilities of the developed method. They are in a good agreement with known experimental data.