An Approach Based on Integral Equations for Crack Problems in Standard Couple-Stress Elasticity

Abstract

The distributed dislocation technique proved to be in the past an effective approach in studying crack problems within classical elasticity. The present work aims at extending this technique in studying crack problems within standard couple-stress elasticity (or Cosserat elasticity with constrained rotations), i.e., within a theory accounting for effects of microstructure. This extension is not an obvious one since rotations and couple-stresses are involved in the theory employed to analyze the crack problems. Here, the technique is introduced to study the case of a Mode I crack. Due to the nature of the boundary conditions that arise in couple-stress elasticity, the crack is modeled by a continuous distribution of climb dislocations and wedge disclinations that create both standard stresses and couple stresses in the body. In particular, it is shown that the Mode I case is governed by a system of coupled singular integral equations with both Cauchy and logarithmic kernels. The numerical solution of this system shows that a cracked solid governed by couple-stress elasticity behaves in a more rigid way (having increased stiffness) as compared to a solid governed by classical elasticity.