A Homogenized Bending Theory for Prestrained Plates

Abstract The presence of prestrain can have a tremendous effect on the mechanical behavior slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for fabrication active and functional materials. In this paper we study microheterogeneous, prestrained feature non-flat equilibrium shapes. Our goal is to understand relation between properties microstructure global shape plate equilibrium. To end, consider three-dimensional, nonlinear elasticity model describes periodic material occupies domain with small thickness. We spatially described form multiplicative decomposition deformation gradient. By simultaneous homogenization dimension reduction, rigorously derive an effective as $$\Gamma $$ Γ -limit vanishing thickness period. That limit has energy emergent curvature term. homogenized (bending stiffness curvature) are characterized by corrector problems. For composite—a laminate composed isotropic materials—we investigate dependence parameters composite. Secondly, composite set shapes minimal energy. reveals rather complex these parameters. instance, principal directions depend discontinuous way; certain parameter regions observe uniqueness non-uniqueness also size effects: geometries aspect ratio As second application our theory, problem programming: prove any target (parametrized deformation) be obtained (up tolerance) minimizer plate, which simple sense consists only finitely many grains filled parametrized single degree freedom.