Boundary element formulation for plane problems in couple stress elasticity

Couple-stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics. Linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level (e.g., as the radius a of a critical hole reduces to a size comparable to l). Consequently, this size-dependent elastic theory is essential to understand the behavior at micro- and nano-scales and to bridge the atomistic and classical continuum theories. Here we develop an integral representation for two-dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media. The resulting boundary-only formulation involves displacements, rotations, force-tractions and moment-tractions as primary variables. Details on the corresponding numerical implementation within a boundary element method are then provided, with emphasis on kernel singularities and numerical quadrature. Afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size-dependent couple stress elasticity.