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Prediction of microstructure in a Cosserat continuum using relaxed energies
Author(s) -
Khan Muhammad Sabeel,
Hackl Klaus
Publication year - 2012
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201210123
Subject(s) - microstructure , kinematics , nonlinear system , phase field models , classical mechanics , compressibility , energy functional , regular polygon , phase (matter) , physics , mechanics , statistical physics , mathematics , materials science , mathematical analysis , geometry , composite material , quantum mechanics
We develop a nonlinear incompressible multiphase material model in a Cosserat continuum with microstructure. The free energy of the material is enriched with an interaction potential taking into account the intergranular kinematics at the continuum scale. As a result the total energy becomes non‐convex, thus giving rise to the development of microstructural phases. To guarantee the existence of minimizers an exact quasi‐convex envelope of the corresponding energy functional is derived. As a result a three phase material energy appears, among them two of the phases are with microstructure in the translational motion (displacment field) and micromotion (microrotation field), whereas the third phase is without internal structure. The corresponding relaxed energy is then used for finding the minimizers of the two field minimization problem corresponding to a Cosserat continuum. Results from a numerical example predicting the development of microstructure in the material are presented. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)