Variational multiscale methods to embed the macromechanical continuum formulation with fine-scale strain gradient theories

Abstract

A variational basis is presented to link fine-scale theories of material behaviour with the classical, macromechanical continuum theory. The approach is based on the weak form of the linear momentum balance equations, and a separation of the weighting function and displacement fields into coarse and fine-scale components. Coarse and fine-scale weak forms are defined. The latter is used to introduce a strain gradient theory that operates at finer scales of deformation. Attention is focused upon applications requiring the enhanced physical accuracy of the fine-scale strain gradient theory, without the computational cost of discretization that spans the range from coarse to fine scales. A variationally consistent method is developed to embed the fine-scale strain gradient theory in the macromechanical formulation. The embedding is achieved by eliminating the fine-scale displacement field from the problem. Two examples demonstrate the numerical efficiency of the method, while retaining physical and mathematical properties of the fine-scale strain gradient theory. Copyright © 2003 John Wiley & Sons, Ltd.