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A LBIE Method for Solving Gradient Elastostatic Problems

Sellountos, Euripides; Tsinopoulos, S. V.; Polyzos, D.

CMES: Computer Modeling in Engineering & Sciences, 86(2) (2012), 145-169

A Local Boundary Integral Equation (LBIE) method for solving two dimensional problems in gradient elastic materials is presented. The analysis is performed in the context of simple gradient elasticity, the simplest possible case of Mindlin's Form II gradient elastic theory. For simplicity, only smooth boundaries are considered. The gradient elastic fundamental solution and the corresponding boundary integral equation for displacements are used for the derivation of the LBIE representation of the problem. Nodal points are spread over the analyzed domain and the moving least squares (MLS) scheme for the approximation of the interior and boundary variables is employed. Since in gradient elasticity the equilibrium equation is a partial differential equation of forth order, the MLS is ideal for solving these problems since it holds the C(1) continuity property. The companion solution of displacements is explicitly derived and introduced in the LBIEs for zeroing the tractions and double tractions on the local circular boundaries. Two representative numerical examples are presented to illustrate the method, demonstrate its accuracy and assess the gradient effect in the response.