A polygonal element for couple stress/strain gradient elasticity based on SBFEM and spline interpolation
Abstract
The couple stress and strain gradient elasticities have been developed to extend the continuumelasticity to the micro-scale enabling describing the scale effects. The basic equations are fourth-order differential equations by adding the scaled parameter tothe conventional second-order elasticity equations. To pass the $C^{0-1}$ and $C^1$ enhanced patch test of the convergence condition, the elements of the couple stress or the strain gradient elasticity must satisfy $C^{0}$ continuity and possess second-order completeness for displacements simultaneously with the nodal parameters,including the displacements and their first-order derivatives.Meanwhile, polygonal elements are suitably flexible for meshing of complex regions.However, no polygonal element is available for couplestress or strain gradient elasticity, because polygonal shape functions with high-order completeness are difficult to construct.This paper presents a polygonal element for couple stress/strain gradient elasticity, which combines the scaled boundary finite element method (SBFEM) with a polygonal spline thin-plate element based on discrete Kirchhoff theory (DKPS).The stiffness matrix of the element includes two parts, one for standard strains and the other for strain gradients,which are computed by the SBFEM and DKPS elements, respectively. This approach avoids the computation of shape functions.The new combined element (SBFEMd3-DKPSd2) satisfies the patch-test conditions of couple stress/strain gradient elasticity.In numerical examples, the proposed element can obtain good accuracy for both couple stress and strain gradient elasticities on convex, non-convex, and irregular polygonal meshes.
