Open Access

ARTICLE

A Variational Multiscale Method to Embed Micromechanical Surface Laws in the Macromechanical Continuum Formulation

K. Garikipati¹

1 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109

Computer Modeling in Engineering & Sciences 2002, 3(2), 175-184. https://doi.org/10.3970/cmes.2002.003.175

Abstract

The embedding of micromechanical models in the macromechanical formulation of continuum solid mechanics can be treated by a variational multiscale method. A scale separation is introduced on the displacement field into coarse and fine scale components. The fine scale displacement is governed by the desired micromechanical model. Working within the variational framework, the fine scale displacement field is eliminated by expressing it in terms of the coarse scale displacement and the remaining fields in the problem. The resulting macromechanical formulation is posed solely in terms of the coarse scale displacements, but is influenced by the fine scale; thereby it has a multiscale character. The procedure results in an embedding of the micromechanical model in the macromechanical formulation. In this paper, this general approach is presented for the special case of traction-displacement laws on internal surfaces. Numerical examples are presented that demonstrate the method for several benchmark problems.

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