Nanomechanical Burmister Solution of an Elastic Layer with Couple Stresses and Steigmann-Ogden Surface Elasticity

This study investigates the Burmister problem for a nanosized elastic layer under normal, tangential, and couple tractions, incorporating surface effects within the framework of classical couple stress theory. By integrating Gurtin-Murdoch and Steigmann-Ogden surface elasticity models, we derive a semi-analytical solution using stress function formulation and Fourier integral transforms, numerically evaluated via Gauss-Legendre quadrature. Comprehensive parametric analyses reveal significant surface effects on force stresses, displacements, and couple stresses near the layer's top surface. Both surface models mitigate singularities in stresses and displacements compared to classical theory, with the Steigmann-Ogden model achieving greater reduction under normal loading due to its bending rigidity. Under tangential loads, it smooths fields but yields slightly higher magnitudes than the Gurtin-Murdoch model. Couple traction induces nonzero surface stresses and alters displacement profiles, with Steigmann-Ogden reducing subsidence displacement more effectively. However, couple stress singularities persist, with surface effects playing a secondary role. As layer thickness increases, solutions converge to the half-plane case, with stresses converging faster than displacements. These findings highlight the critical role of surface mechanics in nanoscale structures, offering insights for designing robust nanomaterials in engineering applications. The results underscore the Steigmann-Ogden model's superior regularization of elastic fields, advancing the understanding of surface effects in couple stress layers.