Crack-like formation of failure-decided angle points on middle planes of the layers resistant to bending in multi-layer structure - a continuum model

Abstract

A theoretical model is considered that describes, in a continuum approximation, formation of a segment of angle points on the middle planes of thin layers forming a multi-layer structure. These points are associated with the jumps of the slope of the middle planes on the segment. A 2-D case is dealt with. The structure is assumed to be a half-plane with its boundary parallel to the layers and acted upon by a symmetric distribution of the displacements normal to the boundary. The layers forming the structure are assumed capable of mutually gliding with respect to each other and of revealing their flexure rigidity under the above loading. The continuum approximation to describe the above multi-layer structure has been applied. Physically the above mathematical angle points may (depending on the layer material properties) emerge either as a result of transverse fracture of the layers or as a result of intensive local plastic deformation (formation of the plastic `hinges'). As a result, the bending moment drops drastically, so that it is assumed dropping down to zero. This condition is employed to determine the distribution of the above slope jumps. The segment length is determined by equating the bending moment at the remote (from the boundary) end of the segment to a critical (specified) value of the bending moment. Thus, the problem of determining the slope jumps on the segment is reduced to a Fredholm integral equation of the first kind with the kernel having an integrable singularity. This equation has been solved numerically. The results of the calculations are presented.