Optimization of geometry and mechanical characteristics of elastic bodies in the vicinity of singular points

Abstract

For practical applications, singular solutions of classical problems in the theory of the elasticity associated with the infinite values of stresses in the vicinity of singular points can be viewed as an indication of specific zones with strong stress concentration at the points of disturbance of boundary smoothness, variation in the type of boundary conditions or contact points of dissimilar material. The existence or non-existence of singular solutions and their exponential dependence are defined by the geometry and mechanical characteristics of the material in the vicinity of singular points. The problem of optimization of geometry and material properties at singular points is considered. The objective of the optimization problem is minimization of the stress state in a solid body characterized by the stress intensity, a functional relation, which specifies the degree of nonhomogeneity of the stress state. A mathematical formulation of the examined optimization problem is considered and the key steps of the algorithm for its numerical implementation are discussed. For the sake of illustration, several optimization problems have been solved for different types of singular points. The analysis of solutions of optimization problems has shown that optimal solutions have the following common property: in the problems of geometry optimization in the vicinity of singular points, the parameters of optimal geometry and optimal characteristics of the material determine the boundary between the solutions with and without stress singularity, while in the problem of optimization of mechanical characteristics in the vicinity of singular points they determine absence of singular solutions.