NUMERICAL AND ANALYTICAL INVESTIGATION OF THE J-INTEGRAL VALUE IN A DOUBLE CANTILEVER BEAM SAMPLE WITH AN EXTREMELY THIN ELASTIC-PLASTIC ADHESION LAYER

Abstract

The loading of a double cantilever beam sample, which is a composition of cantilevers and an adhesive layer, is considered, which leads to deformation of the layer by normal rupture. The basis of the mathematical model describing the equilibrium of the composite is the variational equilibrium equation connecting the stresses averaged over the thickness of the layer and the stress-strain state of the cantilever by a linear parameter. The behavior of the cantilever is considered within the framework of a linearly elastic body, the adhesive layer is considered elastic-plastic without hardening. The composite is considered in a state of plane deformation. Assuming the fulfillment of the condition of complete plasticity of the equality of two principal stresses acting orthogonally to separation and elastic volumetric deformation, the constitutive relations of the behavior of the layer at the stage of elastoplastic deformation are obtained. Using the distribution of the displacement field according to the theory of Mindlin-Reisner plates, a system of ordinary differential equations, conjugation conditions for solutions, and boundary conditions are obtained from the variational formulation of the problem. It is shown that, depending on the ratio of the linear parameter and the height of the console, the characteristic equation of the corresponding homogeneous system of differential equations has both real and complex roots. The corresponding general solutions of the system are found. Based on the consideration of one region of elastoplastic deformations, an iterative procedure for finding the length of the plastic zone is proposed, taking into account the satisfaction of boundary conditions. From the solution of the problem for a given value of the linear parameter and a critical external load, the stress-strain state of the layer is found, on the basis of which the J-integral is calculated. It is shown that as the linear parameter tends to zero, the J-integral asymptotically tends to the value found in the framework of the linearly elastic behavior of the layer material. For a certain range of values of the linear parameter, the obtained solution was compared with the finite element solution of the 2D problem.