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Computational Microstructures in Phase Transition Solids and Finite‐Strain Elastoplasticity
Author(s) -
Carstensen C.,
Hackl K.,
Hoppe U.,
Orlando A.
Publication year - 2006
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201490031
Subject(s) - quasiconvex function , microstructure , convexity , relaxation (psychology) , length scale , phase transition , rank (graph theory) , statistical physics , scale (ratio) , strain energy , phase (matter) , mathematics , mathematical optimization , mathematical analysis , physics , materials science , mechanics , condensed matter physics , thermodynamics , geometry , finite element method , regular polygon , combinatorics , convex optimization , quantum mechanics , psychology , social psychology , convex set , metallurgy , economics , financial economics
Aim of this paper is to show that the life‐golden rule ‘If you can relax, do it’ applies also to the numerical analysis of microstructures. In broad terms relaxation aims to characterize the weak limits of infimizing sequences with fine scale oscillations of gradients. The latter are identified with the fine‐scale microstructure observed in phase transition solids and in inelastic materials in the form of shear bands. An effective density energy is introduced that allows characterization of the admissible microstructure. This equals the least value that the energy of the system can reach among all the microstructures and defines the difficult notion of quasiconvexity. For vector nonconvex energy densities W , the quasiconvex hull W qc is indeed known only in very few cases, therefore, a numerical relaxation has to be involved. This poses however a computational challenge, because the solution of a nonconvex optimization problem is required. Other notions, such as finite laminates, rank‐one convexity, and polyconvexity are introduced and a new class of algorithms for their approximations is introduced. This and other issues will be discussed and illustrated on model examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)