Boundary Element Methodfor Elasto-plastic Analysisof Plane Orthotropic Problems

Elasto-plastic analysis of orthotropic bodies is a theoretical and practical problem, which has attracted many interests on study due to the development of modern sciences and technologies. On the other hand, the development of computer technologies permits computational methods an effective approach to solve this kind of complicated problems.A general boundary element method (BEM) is proposed for elasto-plastic analysis of plane orthotropic bodies. Based on the Betti's reciprocal work theorem, the boundary integral equation of plane orthotropic elasto-plastic problem is established firstly, and then the displacement and stress formulae in internal points are derived according to the geometric relations and constitutive equations. For the integrals in domain resulted from the plastic deformation, the convected differentiation concept of singular integral by Mikhlin is adopted to analyze the kind of singularity, and the free term induced by the strong singularity in domain is introduced with analytical expressions. According to the fundamental solutions in the plane orthotropic elastic problem and expressions from the above integral equations, the stress, strain and traction fundamental solutions in the plane orthotropic elasto-plastic problems are derived, respectively, and the matrix denotation is introduced to formulate the unified relations among all the fundamental solutions.On the basis of above integral equations and fundamental solutions, the boundary and domain are discretized, respectively, and the discretized equations and iterative equations are established with the initial plastic stress to analyze the plane orthotropic elasto-plastic problems. All the kinds of singular integrals in the boundary and domain integral equations are numerically dealt with the schemes such as logarithmic quadratures, coordinate transformations, rigid-body displacement solutions, constant plastic strain solutions, etc, respectively. Hill-Tsai yielding criterion and the orthotropic elastic-perfectly plastic constitutive equations with incremental formulations are adopted in the numerical elasto-plastic analysis. The tangent predictor-radial return algorithm is used to determine the stress state in solving the nonlinear constitutive equations with the initial stress and incremental iteration method. Finally, numerical examples show that the BEM is effective and reliable in analyzing elasto-plastic problems of orthotropic bodies.As the BEM for plane orthotropic elasto-plastic problems is established on the basis of the corresponding elastic analysis, the elastic analysis of plane orthotropic problems with the BEM is also discussed before implementation of the elasto-plastic analysis. The existing displacement fundamental solutions are improved, and then the strain, stress and traction fundamental solutions in the plane orthotropic elastic problems are derived, respectively. All of the improved fundamental solutions can be exploited in the isotropic and orthotropic problems. Numerical examples show the improved fundamental solutions and BEM schemes are effective and reliable in the elastic analysis, which provides the foundations for the elasto-plastic analysis of orthotropic bodies.