Modelling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits. Models for inclusions larger or comparable to the thickness of the plate

TL;DR

This paper develops effective models for thin elastic plates with periodically distributed piezoelectric inclusions, considering different size relationships between the plate thickness and inclusion dimensions, to better understand their coupled mechanical-electrical behavior.

Contribution

It introduces new homogenized models for thin plates with piezoelectric inclusions whose sizes are comparable or smaller than the plate thickness.

Findings

Derived effective models for different size regimes of inclusions.

Analyzed the coupling of elasticity and electrostatics in thin plates.

Provided mathematical frameworks for future numerical simulations.

Abstract

This paper is the second part of a work devoted to the modelling of thin elastic plates with small, periodically distributed piezoelectric inclusions. We consider the equations of linear elasticity coupled with the electrostatic equation, with various kinds of electric boundary conditions. We derive the corresponding effective models when the thickness $a$ of the plate and the characteristic dimension $\varepsilon $ of the inclusions tend\ together to zero, in the two following situations: first when $a \simeq \varepsilon$, second when $a/\varepsilon$ tends to zero.