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Variational Methods of Micro–Macro Transitions in Nonlinear Elasticity
Author(s) -
Bayreuther C.G.,
Miehe C.
Publication year - 2002
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/1617-7061(200203)1:1<217::aid-pamm217>3.0.co;2-f
Subject(s) - micromechanics , nonlinear system , boundary value problem , traction (geology) , mathematics , macro , constitutive equation , microstructure , displacement (psychology) , mathematical optimization , computer science , statistical physics , mechanics , mathematical analysis , materials science , physics , finite element method , algorithm , mechanical engineering , thermodynamics , engineering , quantum mechanics , programming language , psychology , composite number , metallurgy , psychotherapist
Continuum micromechanics deals with idealized materials where the macroscopic material response is modelled in an averaged or homogenized sense based on the information of the heterogeneous microstructure. In general, an efficient treatment of multiscale systems requires the application of equivalent structural problems where the constituents are governed by overall properties. The key contribution of this paper is the computational exploitation of variational methods for a numerical upper and lower bound estimation of the effective material response. We present aspects for the formulation of an appropriate minimizing principle yielding the displacement fluctuations on the microstructure and the local effective constitutive variables of the macrostructure depending on the choice whether we apply linear displacement, traction or periodic boundary conditions to the displacement fluctuations on the boundary of the microstructure. The proposed concept will be demonstrated in the scope of some representative model problems.