The derivation of elasticity models for slender structures from 3d elasticity is a classical topic in elasticity theory. In 2003 Friesecke, James and Müller derived in a seminal paper the geometric rigidity estimate, which can be viewed as geometrically nonlinear version of Korn's inequality. Based on it, many nonlinear lower-dimensional theories for rods, plates and shells have been rigorously derived in the spirit of Gamma-convergence from nonlinear 3d-elasticity.
In the talk we consider this topic in a situation where the three-dimensional model features microstructure and prestrain. The presence of prestrain has a tremendous effect on the mechanical behavior of slender structures and may lead to complex equilibrium shapes that show spontaneous bending, symmetry breaking, and wrinkling. Our goal is to predicts equilibrium shapes for microheterogeneous, prestrained composite plates in the case of bending driven shape shifting. To this end we present an approach that blends analytic and numerical methods. We start with a 3d nonlinear elasticity model for a periodic material that occupies a plate-like domain with small thickness. We consider a spatially periodic prestrain modeled via a multiplicative decomposition of the deformation gradient. By simultaneous dimension reduction and homogenization, we rigorously derive (in the Γ-limit of vanishing thickness and period) a homogenized, nonlinear bending model for plates with an emergent spontaneous curvature term that may lead to non-flat equilibrium shapes. The effective properties of the plate model (bending stiffness and spontaneous curvature) are characterized via corrector problems.
These allow to connect the microstructural configuration of the 3d plate with its effective shapes with minimal energy. We study this microstructure-shape relation analytically on the level of a model problem of a parametrized sandwich plate with periodically distributed, prestrained fibres, and numerically for general composites. Our study reveals a rather complex dependence of the equilibrium shape on the considered parameters.
The talk is based on joint work with Klaus Böhnlein, Oliver Sander, and David Padilla-Garza.