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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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Fast Calculation of Elastic Fields in a Homogeneous Medium with Isolated Heterogeneous Inclusions

Volume 7, Issue 4, 2009, pp. 263-276
DOI: 10.1615/IntJMultCompEng.v7.i4.30
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ABSTRACT

This work is devoted to the calculation of static elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. First, the problem is reduced to the solution of integral equations for strain fields inside the inclusions. Then, Gaussian approximating functions are used for discretization of these equations. For such functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms. The method is mesh-free, and the coordinates of the approximating nodes is the only geometrical information required in the method. If such nodes compose a regular lattice, the matrix of the discretized problem will have Toeplitz structure. By the calculation of matrix-vector products with such matrices, the fast Fourier transform technique may be used. The latter essentially accelerates the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in a 2-D medium with an isolated heterogeneous inclusion and with several inclusions are presented.

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CITED BY
  1. Kanaun S., Pervago E., Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures, International Journal of Engineering Science, 49, 5, 2011. Crossref

  2. Zeman Jan, Vondřejc Jaroslav, Novák Jan, Marek Ivo, Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients, Journal of Computational Physics, 229, 21, 2010. Crossref

  3. Kanaun S., Martinez R., Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions, Computational Materials Science, 55, 2012. Crossref

  4. Kanaun S., On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities, International Journal of Engineering Science, 48, 2, 2010. Crossref

  5. Kanaun S., An efficient homogenization method for composite materials with elasto-plastic components, International Journal of Engineering Science, 57, 2012. Crossref

  6. Kanaun S., Scattering of monochromatic elastic waves on a planar crack of arbitrary shape, Wave Motion, 51, 2, 2014. Crossref

  7. Kanaun S., Levin V., Scattering of elastic waves on a heterogeneous inclusion of arbitrary shape: An efficient numerical method for 3D-problems, Wave Motion, 50, 4, 2013. Crossref

  8. Kanaun Sergey, An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions, World Journal of Mechanics, 01, 02, 2011. Crossref

  9. Kanaun S., Scattering of acoustic waves on a planar screen of arbitrary shape: Direct and inverse problems, International Journal of Engineering Science, 92, 2015. Crossref

  10. Kanaun S., Markov A., An efficient numerical method for the solution of the problem of elasticity for 3D-homogeneous elastic medium with cracks and inclusions, 1836, 2017. Crossref

  11. Markov A., Kanaun S., An Efficient Numerical Method for the Solution of the Problem of Elasticity and Poroelasticity for 3D-Homogeneous Elastic Medium with Cracks and Inclusions, Poromechanics VI, 2017. Crossref

  12. Markov A., Kanaun S., Interactions of cracks and inclusions in homogeneous elastic media, International Journal of Fracture, 206, 1, 2017. Crossref

  13. Trofimov A., Markov A., Abaimov S.G., Akhatov I., Sevostianov I., Overall elastic properties of a material containing inhomogeneities of concave shape, International Journal of Engineering Science, 132, 2018. Crossref

  14. Markov A., Trofimov A., Abaimov S., Akhatov I., On the applicability of replacement relations to tetrahedron-like inhomogeneities, International Journal of Solids and Structures, 167, 2019. Crossref

  15. Kanaun S., Levin V., Effective Field Method in the Theory of Heterogeneous Media, in Effective Properties of Heterogeneous Materials, 193, 2013. Crossref

  16. Trofimov A., Markov A., Sevostianov I., Replacement relations for elastic composite materials having different matrices and related problems, Acta Mechanica, 231, 6, 2020. Crossref

  17. Kanaun Sergey, Numerical solution of volume integral equations for static fields in heterogeneous media, in Heterogeneous Media, 2021. Crossref

  18. Markov A., Trofimov A., Sevostianov I., A unified methodology for calculation of compliance and stiffness contribution tensors of inhomogeneities of arbitrary 2D and 3D shapes embedded in isotropic matrix – open access software., International Journal of Engineering Science, 157, 2020. Crossref

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