Open Access

ARTICLE

A Hybrid Variational Formulation for Strain Gradient Elasticity Part I: Finite Element Implementation

N.A. Dumont ¹, D. Huamán¹

1 PUC-Rio – Pontifical Catholic University of Rio de Janeiro, Brazil.

Computer Modeling in Engineering & Sciences 2014, 101(6), 387-419. https://doi.org/10.3970/cmes.2014.101.387

Abstract

The present paper starts with Mindlin’s theory of the strain gradient elasticity, based on three additional constants for homogeneous materials (besides the Lamé’s constants), to arrive at a proposition made by Aifantis with just one additional parameter. Aifantis’characteristic material length g², as it multiplies the Laplacian of the Cauchy stresses, may be seen as a penalty parameter to enforce interelement displacement gradient compatibility also in the case of a material in which the microstructure peculiarities are in principle not too relevant, but where high stress gradients occur. It is shown that the hybrid finite element formulation – as proposed by Pian and generalized by Dumont for finite and boundary elements – provides a natural conceptual framework to properly deal with the interelement compatibility of the normal displacement gradients, in which “corner nodes” are not an issue. Nonsingular fundamental solutions – domain interpolation functions – are presented for two-dimensional (2D) and three-dimensional (3D) problems, with the generation of families of finite elements that may be implemented in a straightforward way. Since the experimental data available in the technical literature are still scarce and the numerical results are in part questionable, consistency is assessed by means of patch tests and by investigating the spectral properties of the matrices derived for some 2D plane strain elements. The present developments, although of academical relevance, involve too many degrees of freedom to be considered for practical applications and are actually intended as a step toward a boundary-only implementation in terms of singular fundamental solutions.

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