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Archive of Applied Mechanics

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12.08.2017 | Special

Computational first-order homogenization in chemo-mechanics

verfasst von: Stefan Kaessmair, Paul Steinmann

Erschienen in: Archive of Applied Mechanics | Ausgabe 1-2/2018

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Abstract

In the present contribution, we consider species diffusion coupled to finite deformations in strongly heterogeneous microstructures. A semi-dual energy formulation parameterized in terms of the chemical potential is obtained by Legendre transformation of the free energy. Doing so avoids the presence of higher gradients of the deformation field. The constitutive response at the macroscopic level is obtained using variationally consistent homogenization (Larson et al. in Int J Numer Method Eng 81(13):1659–1686, 2010. doi:10.1002/nme.2747). This approach allows to treat transient microscale problems on a representative volume element which has a finite size, i.e., the scales are not clearly separated. Full details of the implementation are provided. A series of numerical examples compares the homogenization results to single-scale formulations which fully resolve all microstructural features.

Vorheriger Artikel Hysteretic beam element with degrading smooth models

Nächster Artikel Material point method for crack propagation in anisotropic media: a phase field approach

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It is assumed that no solid mass is produced, that is the identity \(\rho _0 \, \hbox {d}V= \rho _t \, \hbox {d}v\) relates the solid mass densities at the reference and current configuration via \(\rho _0 = J \rho _t\).

Quantities labeled by \({}^{\text {M}}{\{\bullet \}}\) belong to the macroscale.

Note that a physically motivated choice of the free energy function takes the form \(\psi ^{\text {chem}}(c) = A c\ln c + B c\). Nonetheless, the polynomial approach is computationally less expensive and is used in the following. This can be compared to the phenomenological approach of using a fourth-order polynomial to approximate the configurational free energy of a Cahn–Hilliard system rather than a logarithmic function, see the discussion in [28].

The concentration is evaluated in a nested loop at every quadrature point using a nonlinear Newton–Raphson solver. The update procedure is given by \(c \leftarrow c - [\psi ,{}_{cc}]^{-1}[\psi ,{}_c-\mu ]\).

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Titel: Computational first-order homogenization in chemo-mechanics
verfasst von: Stefan Kaessmair
Paul Steinmann
Publikationsdatum: 12.08.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Archive of Applied Mechanics / Ausgabe 1-2/2018
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-017-1287-0

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