3D elasto-plastic stress analysis by the method of arbitrary lines

The method of arbitrary lines (MAL) constitutes a general dimensional reduction methodology for elliptic boundary value problems (BVP) in arbitrary two- and three-dimensional domains by solving systems of one-dimensional boundary value ordinary differential equations (ODEs). It has been already applied to two-dimensional problem, and the good results have been reported. In this work, we consider the extension of the MAL to three-dimensional elasto-plastic stress analysis. We first give the MAL formulation of three-dimensional elasto-plastic problems. Although the MAL formulation is derived from the principle of three-dimensional increment virtual work as well as the finite element method (FEM), the MAL is different from FEM in that displacement increment and virtual displacement increment are expressed continuous functions along one direction and shape-functions along other two directions. Substituting displacement increment and virtual displacement increment into the principle of three-dimensional increment virtual work, we have a system of ODEs. The three-dimensional elasto-plastic analysis of BGA model, which was a method of the solder joints of electronic component, was carried out. As results, it was confirmed that to solve 3D elasto-plastic problem at the good accuracy was possible by the MAL.