Extended Finite Element Method Based On Couple Stress Theory

Finite element method (FEM) is a computer-based numerical method, but elemental boundary needs conformation to the discontinuity of displacement and material boundary in classical FEM, which ensures the continuous displacement-field function and means re-meshing is frequent during every crack propagation. Sufficiently refine meshes are essential in regions of interest to obtain high-precision results, as well. Extended finite element method (XFEM) was proposed to effectively present discontinuous fields without re-meshing or conforming to the elemental edges by taking both nodal and elemental enrichments into account. Based on this superiority, XFEM has been implemented in analysis of shear band, procedure dislocation and crack propagation. The distinctive feature of this new method is to introduce enrichment functions to capture the discontinuity, thus, unlike the classical FEM, the discontinuous boundary could pass through the elements arbitrarily.Micro-curvature and material length scale parameter are taken into account in couple stress theory, so it can depict the size effect in micro-structure, which classical continuous theories fail to predict. Many researchers have applied this micro-dipolar theory to Micro-Electric-System, crack propagation and nanomaterial.In this paper, extended finite element method based on couple stress theory is derived utilising features of both theories. The finite element formulation can be derived by applying energy variation principle to geometric equations and constitutive equations, and then stiffness matrix can be obtained by using the Hermite interpolation. In the discontinuous fields, level set method is presented for judging and describing the type of nodal and elemental enrichment, and then Heaviside function describing disjunction of crack and asymptotic nearfield for crack tip are added to the interpolation functions.Numerical results are achieved by coding programme in FORTRAN compiler, in which visualisation of the results is developed at the same time after the implementing stress recovery.Comparison of.J-integral and stress intensity factor for mode I crack between the results and those in references shows simplicity, high efficiency and high precision of XFEM with limited regularly rectangular meshes.XFEM numerical results based on couple stress theory indicate that crack initiation angle remains the same for mode I crack compared to classical continuous theory, while the stress field near the crack tip reduces, contributing to higher cracking load. Data based on couple stress also predicts the same tendency as the increase in micro material length scale parameter.For built-in oblique crack with an angle of 45°, couple stress theory has the size dependent effect on both crack initiation angle and stress field, which become smaller simultaneously then those predicted in classical theory.