Finite elements for materials with strain gradient effects

Abstract

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. In conventional displacement-based approaches, the interpolation of displacement requires C1-continuity in order to ensure convergence of the finite element procedure for higher-order theories. Mixed-type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C0-continuous shape functions and can achieve the same convergence as C1 elements. These C0 elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.