Displacement general solutions in strain gradient elasticity: review and analysis

TL;DR

This paper reviews and extends displacement solutions in isotropic strain gradient elasticity, demonstrating how classical elasticity solutions can be generalized within the SGE framework and establishing their relationships and completeness.

Contribution

It provides a comprehensive review and new derivations of displacement solutions in strain gradient elasticity, connecting classical solutions with the SGE framework.

Findings

Classical elasticity solutions can be generalized to SGE.

Helmholtz decomposition facilitates solution generalization.

Established relationships and completeness of stress functions.

Abstract

In this work, we provide an overview of general solutions for displacement fields in static problems of isotropic strain gradient elasticity (SGE). We not only review existing solutions but also derive new representations, showing that all classical elasticity solutions - including those of Boussinesq-Galerkin, Papkovich-Neuber, Naghdi, Lame, Love and Boussinesq - can be simply generalized to SGE framework. In general, it is shown that SGE enables the use of any classical general solution representation combined with a Helmholtz decomposition for the gradient part of the displacement field. Consistency is also established between the presented Papkovich-Neuber representation and the general solutions of SGE proposed previously by Mindlin (1964), Lurie et al. (2006) and Charalambopoulos et al. (2020). Furthermore, we establish the relationships between the stress functions of different general solutions and show their completeness.