Second order formulation of boundary value problems in gradient elasticity

TL;DR

This paper introduces a second order formulation for boundary value problems in gradient elasticity, simplifying the typically fourth order equations and providing new methodologies for analysis.

Contribution

It develops two novel approaches—pseudo-differential calculus and variable augmentation—to reformulate gradient elasticity equations as second order systems.

Findings

Equivalent second order PDE systems are constructed.

The approaches handle non-local behavior effectively.

Existence, uniqueness, and stability are analyzed.

Abstract

A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth order equation of gradient elasticity. Two alternative methodologies are developed and presented in the present work. The first approach is based on the framework of the pseudo-differential calculus and exploits the special characteristics of this approach to deal with the non local behavior of gradient elasticity. The second implementation is purely differential and is based on the augmentation of the independent variables of the problem. Under this concept, the constitutive equations of gradient elasticity become part of the differential system itself and the whole framework is reminiscent of the transformation of the wave equation to a first order differential system. Crucial issues like existence, uniqueness and stability of the corresponding initial boundary value problems are also encountered via the new concepts.