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International Journal for Multiscale Computational Engineering

Published 6 issues per year

ISSN Print: 1543-1649

ISSN Online: 1940-4352

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ESSENTIAL FEATURES OF FINE SCALE BOUNDARY CONDITIONS FOR SECOND GRADIENT MULTISCALE HOMOGENIZATION OF STATISTICAL VOLUME ELEMENTS

Volume 10, Issue 5, 2012, pp. 461-486
DOI: 10.1615/IntJMultCompEng.2012002929
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ABSTRACT

A second gradient continuum description developed, for example, by Germain, Toupin and Mindlin, and Eringen, gives rise to strain gradient plasticity, and is becoming a common coarse scale basis for multiscale homogenization of material response that respects the non-local nature of heterogeneous fine scale material response. Such homogenization approaches are developed to build either concurrent or hierarchical multiscale computational models for the second gradient response at the coarse scale that represent salient aspects of material response at the fine scale. Typically, the homogenization procedure consists of solving an initial boundary value problem for a statistical volume element of heterogeneous material at the fine scale and computing coarse scale stresses and strains using various volume averaging procedures. By enforcing a kinematically consistent description of the deformation field at each scale and asserting invariance of linear momentum with respect to scale of observation of a fixed set of mass particles, critical features of the boundary conditions and computation of homogenized stresses are revealed. In particular, an internal constraint on the higher-order fluctuation field is required to ensure orthogonality between that part of the fine scale deformation attributed to the second gradient and the part associated with higher-order fluctuations. Additionally, the body forces resulting from such internal constraints must be included in the computation of coarse scale stresses to respect scale invariance of linear momentum at each scale. Numerical implementation of fine scale fluctuation constraints employs linear constraint equations; the computation of coarse scale stresses is facilitated through a multiscale statement of principle of virtual velocities. Example fine scale simulations and associated coarse scale homogenization are presented to illustrate aspects of the boundary conditions.

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CITED BY
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  3. McDowell David L., Multiscale Modeling of Interfaces, Dislocations, and Dislocation Field Plasticity, in Mesoscale Models, 587, 2019. Crossref

  4. Yeager John D., Manner Virginia W., Stull Jamie A., Walters David J., Schmalzer Andrew M., Luscher Darby J., Patterson Brian M., Importance of microstructural features in mechanical response of cast-cured HMX formulations, 1979, 2018. Crossref

  5. Khoei A.R., Sameti A. Rezaei, Kazerooni Y. Nikravesh, A continuum-atomistic multi-scale technique for nonlinear behavior of nano-materials, International Journal of Mechanical Sciences, 148, 2018. Crossref

  6. Rivarola F.L., Benedetto M.F., Labanda N., Etse G., A multiscale approach with the Virtual Element Method: Towards a VE2 setting, Finite Elements in Analysis and Design, 158, 2019. Crossref

  7. Rivarola F. L., Labanda N., Etse G., Thermodynamically consistent multiscale homogenization for thermo-poroplastic materials, Zeitschrift für angewandte Mathematik und Physik, 70, 3, 2019. Crossref

  8. Walters David J., Luscher Darby J., Yeager John D., Considering computational speed vs. accuracy: Choosing appropriate mesoscale RVE boundary conditions, Computer Methods in Applied Mechanics and Engineering, 374, 2021. Crossref

  9. Zuanetti Bryan, Luscher Darby J., Ramos Kyle, Bolme Cynthia, Unraveling the implications of finite specimen size on the interpretation of dynamic experiments for polycrystalline aluminum through direct numerical simulations, International Journal of Plasticity, 145, 2021. Crossref

  10. Lesičar Tomislav, Tonković Zdenko, Sorić Jurica, Two-scale computational approach using strain gradient theory at microlevel, International Journal of Mechanical Sciences, 126, 2017. Crossref

  11. Selimov Alex, Chu Kevin, McDowell David L., Coarse-grained atomistic modeling of dislocations and generalized crystal plasticity, Journal of Micromechanics and Molecular Physics, 07, 02, 2022. Crossref

  12. Hii Aewis K.W., El Said Bassam, A kinematically consistent second-order computational homogenisation framework for thick shell models, Computer Methods in Applied Mechanics and Engineering, 398, 2022. Crossref

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