Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

TL;DR

This paper derives complete general solutions for the equilibrium equations in isotropic strain gradient elasticity, incorporating additional length-scale parameters, and provides methods to decompose and relate stress functions.

Contribution

It introduces extended Boussinesq-Galerkin and Papkovich-Neuber solutions for strain gradient elasticity with new decomposition and relation techniques.

Findings

Derived extended BG and PN solutions for strain gradient elasticity.

Established relations between stress functions and proved completeness.

Showed how to recover known fundamental solutions using the new framework.

Abstract

In this paper, we consider isotropic Mindlin-Toupin strain gradient elasticity theory in which the equilibrium equations contain two additional length-scale parameters and have the fourth order. For this theory we developed an extended form of Boussinesq-Galerkin (BG) and Papkovich-Neuber (PN) general solutions. Obtained form of BG solution allows to define the displacement field through the single vector function that obeys the eight-order bi-harmonic/bi-Helmholtz equation. The developed PN form of the solution provides an additive decomposition of the displacement field into the classical and gradient parts that are defined through the standard Papkovich stress functions and modified Helmholtz decomposition, respectively. Relations between different stress functions and completeness theorem for the derived general solutions are established. As an example, it is shown that a previously known fundamental solution within the strain gradient elasticity can be derived by using the developed PN general solution.