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.
Highlights
This paper addresses material bodies complying with a response function for free energy, which depends on both the strain tensor and its gradient
Usual gradient elasticity theories are of Toupin–Mindlin’s type, i.e., they are characterized by response functions for Cauchy stress tensors, which obey the structure of a space-like
In Broese et al [5], a gradient elasticity model, referred to as Version 3, has been proposed for the first time, which is characterized by a classical constitutive law for the Cauchy stress at every material point in the interior of the body, i.e., the Cauchy stress is given by the usual derivative of the free energy with respect to the strain
Summary
This paper addresses material bodies complying with a response function for free energy, which depends on both the strain tensor and its gradient. One might ask, if it is possible to develop gradient elasticity theories of non-Toupin–Mindlin’s type, where the response function of the Cauchy stress tensor is not necessarily expressed in terms of space-like Euler–Lagrange derivative of the free energy with respect to the strain tensor. In order to address non-localities over space and time, a non-conventional thermodynamics has been proposed in Alber et al [9] It has been shown in Broese et al [5,10], that this thermodynamics provides a proper framework for discussing gradient elasticity models. The main issue is that the hypothesis of the local equilibrium state of the usual irreversible thermodynamics is extended in order to capture non-local effects This is achieved by using an energy transfer law in addition to the conventional energy balance law.
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