Complete general solutions for equilibrium equations of isotropic strain gradient elasticity

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.

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

“…Therefore, we can immediately transfer all the known solutions for the infinite domains from the general theory to the presented one. These are the fundamental solution and Green function [42], solutions for dislocations problems [1], representation of general solution by using extended Papkovitch-Neuber form [5,43], and so forth.…”

“…Substituting (43) into constitutive equations (7) and then into boundary conditions (39), (40), one can find that three of these conditions will be satisfied identically (𝜎 𝑟𝜙 ≡ 0, 𝑝 𝜙 ≡ 0, 𝜇 𝑟 ≡ 0), and last condition for the absence of normal stresses on the crack faces results in the relation:…”

The method of developing the complex stress tensor by basic states for the construction solutions of spatial boundary problems of the linear elasticity theory

“…The founders of this method of constructing solutions were Airy, Boussinesq, Papkovich, Neuber and Dougall. Proving the completeness of general The method of developing the complex stress tensor by basic states for the construction solutions of … 67 solutions, the existence of relationships between them, and the construction of solutions of boundary value problems were discussed by Eubanks and Sternberg, Timoshenko and Goodier, and others [1][2][3][4][5][6]. In particular, [7,8] created a universal design scheme for the development of general solutions and assessment of their completeness and non-unity within the framework of the classical theory of elasticity.…”