Variational Formulations and Galerkin Methods for Strain Gradient Elasticity

Synonyms

Energy methods; Gradient elasticity; Ritz–Galerkin methods; Weak form

Definitions

Variational formulations express pointwise differential boundary value problems (requiring n times continuously differentiable solutions) as weighted integral equations (with essentially less regular solutions), typically indicating a balance of internal and external energies. Galerkin methods search for approximate solutions for variational problems, as linear combinations of simple basis functions. Strain gradient elasticity theories generalize classical elasticity toward multi-scale modeling by incorporating higher-order displacement gradients into the strain and kinetic energies.

Introduction

Thin or thin-walled structures such as bars, beams, membranes, plates, and shells are the key structural components in engineering designs – practically at any length scale: load-bearing and facade structures in civil and mechanical engineering, as one extreme, and sensors and actuators of micro- and...