Elastic Boundary Conditions in the Theory with Second Gradients in the Thermodynamic Potential

TL;DR

This paper investigates elastic boundary conditions within second gradient elasticity theory, highlighting the importance of strain gradients for phenomena like flexoelectricity and addressing the complex derivation of boundary conditions.

Contribution

It provides a detailed analysis of boundary conditions in second gradient elasticity, emphasizing the role of strain gradients and their impact on physical phenomena.

Findings

Derived boundary conditions for second gradient elasticity

Showed significance of strain gradients in flexoelectricity

Clarified the variational formulation involving second gradients

Abstract

When describing elastic deformations of a body sometimes it is worth to take in account elastic spatial dispersion. If spatial dispersion is weak, as usually happens, then it can be reduced to dependence of thermodynamic potential on strain gradients. Such a dependence may be worth in association with small body size which imply large gradients. Besides, the inclusion of this dispersion leads to physical phenomena absent without it. An example of the latter is flexoelectricity. Remarkable fact is that while the derivation of differential equations of elastic equilibrium can be made by ordinary means in this case, the derivation of boundary conditions for them is less trivial task. This is due to the fact that strain gradients should be represented in terms of second gradients of independently varied elastic displacements. Detailed consideration of this problem is the subject of this paper.