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Time‐dependent fibre reorientation of transversely isotropic continua—Finite element formulation and consistent linearization
Author(s) -
Himpel G.,
Menzel A.,
Kuhl E.,
Steinmann P.
Publication year - 2007
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.2124
Subject(s) - transverse isotropy , finite element method , linearization , isotropy , tangent , mathematical analysis , exponential function , nonlinear system , mathematics , constitutive equation , field (mathematics) , classical mechanics , physics , geometry , quantum mechanics , pure mathematics , thermodynamics
Transverse isotropy is realized by one characteristic direction—for instance, the fibre direction in fibre‐reinforced materials.Commonly, the characteristic direction is assumed to be constant, but in some cases—for instance, in the constitutive description of biological tissues, liquid crystals, grain orientations within polycrystalline materials or piezoelectric materials, as well as inoptimization processes—it proves reasonable to consider reorienting fibre directions. Various fields can be assumed to be the driving forces for the reorientation process, for instance, mechanical, electric or magnetic fields. In this work, we restrict ourselves to reorientation processes in hyper‐elastic materials driven by principal stretches. The main contribution of this paper is the algorithmic implementation of the reorientation process into a finite element framework. Therefore, an implicit exponential update of the characteristic direction is applied by using the Rodriguez formula to express the exponential term. The non‐linear equations on the local and on the global level are solved by means of the Newton–Raphson scheme. Accordingly, the local update of the characteristic direction and the global update of the deformation field are consistently linearized, yielding the corresponding tangent moduli. Through implementation into a finite element code and some representative numerical simulations, the fundamental characteristics of the model are illustrated. Copyright © 2007 John Wiley & Sons, Ltd.