Abstract:

The mathematical modeling of microdevices, in which structure and microstructure have approximately the same scale of magnitude, as well as of macrostructures of markedly granular or crystal nature (microcomposites), demands a nonlocal approach for strains and stresses. The present paper starts from a strain gradient theory firstly proposed by Mindlin, based on three additional constants for homogeneous materials (besides the Lamé’s constants), which was subsequently simplified by Aifantis and ended up with just one additional parameter. This simplified proposition has also been applied mainly by Beskos and collaborators in the context of the boundary element method. The extended version of the present contribution proposes and outlines in detail some novel concepts both in terms of a variational approach and in the treatment of singular and hypersingular integrals that arise in a general implementation for curved boundaries. The basic formulation is presented for 3D and 2D problems, but the numerical issues and the displayed examples only concern the 2D case. A concise hybrid boundary element formulation of gradient elasticity problems is proposed on the basis of two virtual work principles that stem from the Hellinger-Reissner potential for classical elasticity. The most important contribution seems to be the evidence that the proposed hybrid formulation naturally approximates normal displacement gradients along the boundary independently from displacements, which would improve Mindlin''s pioneer proposition (Mindlin and Esher, 1968), and for which “corner nodes” would require no additional treatment other than the demand of the classical elasticity. The singular fundamental solutions needed in a boundary element formulation are rederived from the developments made by Polyzos and co-workers and conceptually assessed. Concepts related to the evaluation of hypersingular integrals (as by Guiggiani and also by Mukherjee, for instance) had to be newly assessed in order to cope with the strict requirements of the variational formulation (there seems to be a conceptual contribution also with respect to the conventional boundary element method). Linear, quadratic and cubic elements are implemented, for which several results are assessed both conceptually (fulfilment of spectral properties) and in terms of numerical convergence for some academic applications.