# Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

## TL;DR

This paper develops a homogenization approach to interpret the micro-displacement tensor in stress-gradient theories, linking micro-scale behavior to macro-scale properties and validating with elastic porous materials and experimental data.

## Contribution

It introduces a homogenization framework that interprets micro-displacements in stress-gradient theories, enabling the derivation of constitutive relations from micro-scale analysis.

## Key findings

- Homogenization links micro-displacements to microscopic strain moments.
- The approach predicts negative size effects in elastic porous materials.
- Results agree with experimental data and numerical simulations.

## Abstract

The stress-gradient theory has a third order tensor as kinematic degree of freedom, which is work-conjugate to the stress gradient. This tensor was called micro-displacements just for dimensional reasons. Consequently, this theory requires a constitutive relation between stress gradient and micro-displacements, in addition to the conventional stress-strain relation. The formulation of such a constitutive relation and identification of the parameters therein is difficult without an interpretation of the micro-displacement tensor. The present contribution presents an homogenization concept from a Cauchy continuum at the micro-scale towards a stress-gradient continuum at the macro-scale. Conventional static boundary conditions at the volume element are interpreted as a Taylor series whose next term involves the stress gradient. A generalized Hill-Mandel lemma shows that the micro-displacements can be identified with the deviatoric part of the first moment of the microscopic strain field. Kinematic and periodic boundary conditions are provided as alternative to the static ones. The homogenization approach is used to compute the stress-gradient properties of an elastic porous material. The predicted negative size effect under uni-axial loading is compared with respective experimental results for foams and direct numerical simulations from literature.