# Multi-field asymptotic homogenization of thermo-piezoelectric materials with periodic microstructure

## TL;DR

This paper develops a multi-field asymptotic homogenization method for thermo-piezoelectric materials with periodic microstructures, capturing microstructural effects and coupling to derive effective macroscopic properties.

## Contribution

It introduces a novel multi-field asymptotic homogenization approach that accounts for microstructural heterogeneity and strong coupling in thermo-piezoelectric materials with periodic microstructures.

## Key findings

- The homogenized model accurately predicts the behavior of heterogeneous microstructures.
- The derived constitutive tensors effectively relate micro and macro fields.
- Validation shows good agreement with finite element solutions.

## Abstract

This study proposes a multi-field asymptotic homogenization for the analysis of thermo-piezoelectric materials with periodic microstructures. The effect of the microstructural heterogeneity is taken into account by means of periodic perturbation functions, which derive from the solution of non homogeneous recursive cell problems defined over the unit periodic cell. A strong coupling is present between the micro displacement field and the micro electric potential field, since the mechanical and the electric problems are fully coupled in the asymptotically expanded microscale field equations. The micro displacement, the electric potential, and the relative temperature fields have been related to the macroscopic quantities and to their gradients in the derived down-scaling relations. Average field equations of infinite order have been obtained and the closed form of the overall constitutive tensors has been determined for the equivalent first-order homogenized continuum. A formal solution of such equations has been derived by means of an asymptotic expansion of the macro fields. The accuracy of the proposed formulation is assessed in relation to illustrative examples of a bi-material periodic microstructure subjected to harmonic body forces, free charge densities, and heat sources, whose periodicity is much greater than the characteristic microstructural size. The good agreement obtained between the solution of the homogenized model and the finite element solution of the original heterogeneous material problem confirms the validity of the proposed formulation.