Displacement general solutions in strain gradient elasticity: review and analysis

TL;DR

The paper surveys displacement general solutions in isotropic strain gradient elasticity (SGE) and shows that all classical elasticity representations can be generalized by coupling a classical general-solution form with a gradient-part Helmholtz decomposition. It introduces and interrelates ten representations (Mindlin, LBVB, CGP, BG, PN, TNH) and specialized boundary-value forms (Lamé, Love, Boussinesq), establishing their completeness and equivalence through mappings between their stress functions. A key result is the equivalence between Mindlin-type solutions and the generalized Papkovich–Neuber form, along with a nontrivial BG–TNH relationship, enabling automatic translation across representations. The framework supports systematic analytical solutions and serves as a benchmark for validating gradient-elasticity models in micromechanics, metamaterials, and nanoscale problems, where gradient effects are non-negligible.

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.