Analysis of planar cracks in 3D elastic media with consideration of surface elasticity

Abstract

This paper presents an efficient numerical procedure for analysis of arbitrary shaped, planar cracks in a three-dimensional, linear elastic infinite medium by taking the surface elasticity into account. The concept of surface stresses via Gurtin–Murdoch surface theory is integrated with the classical theory of linear elasticity to form a mathematical model capable of simulating cracks under general loading conditions and nano-scale influences. The key governing equations for the bulk material are established in terms of weakly singular displacement and traction boundary integral equations involving only unknowns on the crack face, whereas that for a zero-thickness material layer on each crack face is established in a weak form following the weighted residual technique. The final governing equations for the bulk material and the two crack-face layers are fully coupled via the continuity condition along the interface. The solution of the resulting coupled system is determined numerically by the coupling of a standard finite element procedure and a symmetric Galerkin boundary element method. Owing to the weakly singular feature of all involved boundary integrals, C\(^{0}\) basis functions are adopted everywhere in the approximation of crack-face data and only the special quadrature for evaluating nearly singular and weakly singular double surface integrals is required. Once the implemented numerical scheme is fully tested with existing reference solutions, it is then applied to investigate the role and influence of surface parameters such as the residual surface tension and the in-plane modulus on the near-front field and the size dependency of predicted solutions. Results of several examples under various scenarios are reported not only to demonstrate the capability and robustness of the proposed method but also to indicate the significance of the surface stresses in enhancing the near-surface stiffness and reducing stresses in a local region ahead of the crack front.