Fundamental solutions for isotropic size-dependent couple stress elasticity

TL;DR

This paper derives fundamental solutions for isotropic size-dependent couple stress elasticity, providing unique displacement, rotation, stress, and traction solutions crucial for boundary integral methods in elastic materials with size effects.

Contribution

It introduces the first fully determinate couple stress theory solutions, including displacements, rotations, and stresses, for infinite elastic domains under concentrated forces and couples.

Findings

Provides solutions for displacements, rotations, and stresses in size-dependent couple stress elasticity.

Establishes unique solutions for force-stresses and couple-stresses.

Enables boundary integral analysis for size-dependent elastic materials.

Abstract

Fundamental solutions for two- and three-dimensional linear isotropic size-dependent couple stress elasticity are derived, based upon the decomposition of displacement fields into dilatational and solenoidal components. While several fundamental solutions have appeared previously in the literature, the present version is for the newly developed fully determinate couple stress theory. Within this theory, the couple stress tensor is skewsymmetrical and thus possesses vectorial character. The present derivation provides solutions for infinite domains of elastic materials under the influence of unit concentrated forces and couples. Unlike all previous work, unique solutions for displacements, rotations, force-stresses and couple-stresses are established, along with the corresponding force-tractions and couple-tractions. These fundamental solutions are central in analysis methods based on Green's functions for infinite domains and are required as kernels in the corresponding boundary integral formulations for size-dependent couple stress elastic materials.