Fundamental solutions and dual boundary element methods for fracture in plane Cosserat elasticity

Abstract

Artículo de publicación ISIIn this paper, both singular and hypersingular fundamental solutions of plane Cosserat elasticity are derived and given in a ready-to-use form. The hypersingular fundamental solutions allow to formulate the analogue of Somigliana stress identity, which can be used to obtain the stress and couple-stress fields inside the domain from the boundary values of the displacements, microrotation and stress and couple-stress tractions. Using these newly derived fundamental solutions, the boundary integral equations of both types are formulated and solved by the boundary element method. Simultaneous use of both types of equations (approach known as the dual boundary element method (BEM)) allows problems where parts of the boundary are overlapping, such as crack problems, to be treated and to do this for general geometry and loading conditions. The high accuracy of the boundary element method for both types of equations is demonstrated for a number of benchmark problems, including a Griffith crack problem and a plate with an edge crack. The detailed comparison of the BEM results and the analytical solution for a Griffith crack and an edge crack is given, particularly in terms of stress and couple-stress intensity factors, as well as the crack opening displacements and microrotations on the crack faces and the angular distributions of stresses and couple-stresses around the crack tip.Fondecyt Chile 11130259 European Research Council 27957

Micro-structured materials: Inhomogeneities and imperfect interfaces in plane micropolar elasticity, a boundary element approach

“…Matrices of fundamental solutions D ij (x, y), P ij (x, y) are given in Atroshchenko [a]. According to their asymptotic behaviour in the vicinity of x = y, which explained in details in Atroshchenko and Bordas [2015] …”

“…Recently, the boundary element method Atroshchenko and Bordas [2015], Hadjesfandiari and Dargush [2012] has been emerging as a powerful alternative due to its advantage in treating problems with non-smooth boundaries and infinite domains. For example, in Atroshchenko and Bordas [2015] the dual boundary element method was applied to crack problems in plane strain micropolar continua.…”

“…For example, in Atroshchenko and Bordas [2015] the dual boundary element method was applied to crack problems in plane strain micropolar continua.…”

“…For the evaluation of all weakly-singular integrals Telles transform Telles and Oliveira [1994] is used, while the singular integrals are evaluated using the singularity subtraction technique (SST), based on the asymptotic expansions of the matrices of fundamental solutions, given in Atroshchenko and Bordas [2015]. After the values of displacements/microrotation and tractions/couple tractions are evaluated along the boundary, the values of u i , u e and t i , t e inside the inclusion domain S i and the matrix domain S e respectively can be calculated using the micropolar analogues of Somiglina's displacement and stress identities, described in Atroshchenko and Bordas [2015]. The expressions for two more matrices of fundamental solutions, used in Somiglina's representation of the stresses and couple stresses inside a domain, are provided in a ready-to-use form in Atroshchenko [a].…”

Singularity-free defect mechanics for polar media

“…Several numerical methods have been used to study cracks within micropolar elasticity. Discrete element method, boundary element method [3,51], extended finite element method [24,59,60], and pseudospectral technique [53] are among those. Comparing the dislocation-and disclination-based approach and other methods, it is understood that once the analytical solution to the line defect is available, the dislocationand disclination-based approach seems to be more efficient and computationally less expensive.…”