Non-Conventional Thermodynamics and Models of Gradient Elasticity

Entropy (Basel). 2018 Mar 8;20(3):179. doi: 10.3390/e20030179.

Abstract

We consider material bodies exhibiting a response function for free energy, which depends on both the strain and its gradient. Toupin-Mindlin's gradient elasticity is characterized by Cauchy stress tensors, which are given by space-like Euler-Lagrange derivative of the free energy with respect to the strain. The present paper aims at developing a first version of gradient elasticity of non-Toupin-Mindlin's type, i.e., a theory employing Cauchy stress tensors, which are not necessarily expressed as Euler-Lagrange derivatives. This is accomplished in the framework of non-conventional thermodynamics. A one-dimensional boundary value problem is solved in detail in order to illustrate the differences of the present theory with Toupin-Mindlin's gradient elasticity theory.

Keywords: boundary conditions; energy transfer law; gradient elasticity; interstitial working; non-equilibrium thermodynamics.