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A second-order two-scale homogenization procedure using \(C^{1}\) macrolevel discretization

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Abstract

The present study deals with a second-order two-scale computational homogenization procedure for modeling deformation responses of heterogeneous materials at small strains. The macro to micro transition and the application of generalized periodic boundary conditions on the representative volume element (RVE) at the microlevel are investigated. The structure at macroscale level is discretized by the \(C^{1}\) two dimensional triangular finite elements, while the \(C^{0}\) quadrilateral finite element is used for the discretization of the RVE. The finite element formulations and the new proposed multiscale scheme have been implemented into the finite element software ABAQUS using user subroutines derived. Due to the \(C^{1}-C^{0}\) continuity transition, an additional integral condition on microlevel fluctuation field has to be imposed, as expected. The integration has been performed using various numerical integration techniques and the results obtained are compared in a few examples. It is concluded that only trapezoidal rule gives a physically based deformed shape of the RVE. Finally, the efficiency and accuracy of the proposed multiscale homogenization approach are demonstrated by the modeling of a shear layer problem, usually used as a benchmark in multiscale analyses.

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Acknowledgments

The authors express their gratitude to the Ministry of Science, Education and Sports of the Republic of Croatia for their financial support. The investigations described in this paper are also part of the project “Centre of Excellence for Structural Health” (CEEStructHealth) supported by the European Union under contract IPA2007/HR/16IPO/001-040513.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 , Zagreb, Croatia

    Tomislav Lesičar, Zdenko Tonković & Jurica Sorić

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  1. Tomislav Lesičar
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  2. Zdenko Tonković
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  3. Jurica Sorić
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Correspondence to Jurica Sorić.

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Lesičar, T., Tonković, Z. & Sorić, J. A second-order two-scale homogenization procedure using \(C^{1}\) macrolevel discretization. Comput Mech 54, 425–441 (2014). https://doi.org/10.1007/s00466-014-0995-3

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  • DOI: https://doi.org/10.1007/s00466-014-0995-3

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