A homogenized bending theory for prestrained plates

TL;DR

This paper develops a rigorous homogenized bending theory for prestrained elastic plates, linking microstructure properties to global shape and spontaneous curvature, with applications to composite materials.

Contribution

It derives an effective nonlinear bending model for prestrained plates via homogenization and dimension reduction, revealing how microstructure influences shape and curvature.

Findings

Homogenized properties depend on microstructure parameters.

Spontaneous curvature emerges from microstructural prestrain.

Shape minimizers are complex functions of composite parameters.

Abstract

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium -- a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature nonflat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a {\Gamma}-limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite -- a prestrained laminate composed of isotropic materials -- we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters.