A von Karman plate analogue for solving anti-plane problems in couple stress and dipolar gradient elasticity

Abstract

The purpose of the present work is to present a direct analogy regarding the formulation and solution of anti-plane problems in the context of couple stress and dipolar gradient elasticity and the von Karman plate framework. We show aspects of boundary conditions in both theories of elasticity and propose a robust Finite Element methodology based on the von Karman plate theory in order to solve complex anti-plane (mode III) crack problems. Furthermore, we establish the equivalence between the anti-plane gradient elasticity J-Integral and the plate Ic-Integral (Sanders’ plate integral). In passing, we prove the path independency of the Ic-Integral. Finally we examine the near tip fields for both anti-plane and plate problems and establish possible strengthening effects due to the underlying microstructure.The proposed analogy is achieved through the von Karman plate theory where the plate is pre-stressed by a constant biaxial tension. The plate theory involves properties such as the plate thickness h, the Poisson's ratio ν and the bending stiffness D. This information, together with the pre-stress N transforms into properties required by the anti-plane couple stress and dipolar gradient elasticity problems: the shear modulus G, the internal length ℓ and the coefficient η. In both problems the two dimensional space remains the same, including the presence of cracks and other defects. The analogy permits numerical and analytical solutions of demanding anti-plane problems of gradient elasticity (couple stress and dipolar) utilizing the von Karman plate corresponding, and vice versa.