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Variational multiscale methods to embed the macromechanical continuum formulation with fine‐scale strain gradient theories
Author(s) -
Garikipati K.
Publication year - 2003
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.727
Subject(s) - discretization , length scale , scale (ratio) , continuum mechanics , weighting , displacement field , finite strain theory , continuum hypothesis , embedding , range (aeronautics) , displacement (psychology) , field (mathematics) , classical mechanics , mathematics , finite element method , statistical physics , mathematical analysis , mechanics , physics , materials science , computer science , psychology , quantum mechanics , artificial intelligence , acoustics , pure mathematics , composite material , psychotherapist , thermodynamics
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.