Equi-stress boundaries in two- and three-dimensional elastostatics: The single-layer potential approach

Abstract

The well-known developments in elastostatics concerning the equi-stressness criterion of optimality for two-dimensional multi-connected unbounded solids under the bulk-dominating load are generalized toward the transient three-dimensional case with rotational symmetry. This paper advances our previous work by focusing specifically on explicitly identifying the optimal equi-stress surfaces through a simple regular integral equation which involves the single-layer potential kernel associated with the axially symmetric Laplacian. Its two-dimensional analogue is also obtained as a competitive counterpart to the commonly used complex-variable formalism. In both cases, the equations are reformulated as a minimization problem, solved numerically with a standard genetic algorithm over a wide variety of governing parameters thus permitting comparison of the shape optimization results in spatial and plane elasticity for multi-connected domains.