A Riemannian approach to reduced plate, shell, and rod theories

We derive a dimensionally-reduced limit theory for an n-dimensional nonlinear elastic body that is slender along k dimensions. The starting point is to view an elastic body as an n-dimensional Riemannian manifold together with a not necessarily isometric W^1,2-immersion in n-dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally-reduced limit theory views the elastic body as a k-dimensional Riemannian manifold along with an isometric W^2,2-immersion in n-dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionally-reduced limit is obtained using a Γ-convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to "standard" elasticity and to "incompatible" elasticity, thus including as particular cases so-called non-Euclidean plate, shell, and rod theories.