Numerical solution of crack problems in gradient elasticity

Abstract

Gradient elasticity is a constitutive framework that takes into account the microstructure of an elastic material. It considers that, in addition to the strains, second-order derivatives of the displacement also affect the energy stored in the medium. Three different yet equivalent forms of gradient elasticity can be found in the literature, reflecting the different ways in which the second-order derivatives can be grouped to form other physically meaningful quantities. This paper presents a general discretisation of gradient elasticity that can be applied to all three forms, based on the finite-element displacement formulation. The presence of higher order terms requires C1-continuous interpolation, and some appropriate two- and three-dimensional elements are presented. Numerical results for the displacement, stress and strain fields around cracks are shown and compared with available solutions, demonstrating the robustness and accuracy of the numerical scheme and investigating the effect of microstructure in the context of fracture mechanics.