The linearized Kirchhoff theory for plates with incompatible prestrain

TL;DR

This paper derives a linearized Kirchhoff model for thin plates with incompatible prestrain, based on three-dimensional nonlinear elasticity, using a variational approach and Gamma-convergence, applicable to plates with various Gaussian curvatures.

Contribution

It introduces a rigorous derivation of a linearized Kirchhoff theory for incompatible prestrain in plates, extending nonlinear elasticity models to the thin plate limit with different curvature conditions.

Findings

Derived the Gamma-limit of the nonlinear elastic energy for thin plates with prestrain.

Established the model for plates with positive, negative, or zero Gaussian curvature.

Provided a variational framework for analyzing incompatible prestrain in elasticity.

Abstract

In this paper, we derive a linearized Kirchhoff model from three dimensional nonlinear elastic energy of plates with incompatible prestrain as its thickness $h$ tends to zero and its elastic energy scales like $h^{\beta}$ with $2<\beta<4.$ The incompatible prestrain is given as a Riemannian metric $G(x')$ in the three dimensional thin plate which only depends on mid-plate of the thin plates. The problem is studied rigorously by using a variational approach and establishing the $\Gamma-$ limit of the non-Euclidean version of the nonlinear elasticity functional when the gauss curvature of the mid-plate $(\Omega, g=G_{2\times2})$ is always positive, negative or zero.