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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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GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

Volume 13, Issue 1, 2015, pp. 11-53
DOI: 10.1615/IntJMultCompEng.2014011234
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ABSTRACT

One considers a linear composite medium, which consists of a homogeneous matrix containing either the periodic or random set of heterogeneities. An operator form of the general integral equation (GIE) is obtained for the general cases of local and nonlocal problems, static and wave motion phenomena for composite materials with periodic and random (statistically homogeneous and inhomogeneous, so-called graded) structures containing coated or uncoated inclusions of any shape and orientation with perfect and imperfect interfaces and subjected to any number of coupled or uncoupled, homogeneous or inhomogeneous external fields of different physical nature. The GIE, connecting the driving fields and fluxes in a point being considered and the fields in the surrounding points, are obtained for both the random and periodic fields of heterogeneities in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and taking into account a possible imperfection of interface conditions. The mentioned perturbations can be found by any available numerical method which has advantages and disadvantages and it is crucial for the analyst to be aware of their range of applications. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs which are presented in two equivalent forms for both the driving fields and fluxes. Some particular cases, asymptotic representations, and simplifications of proposed GIE are presented for the particular constitutive equations such as linear thermoelastic cases with the perfect and imperfect interfaces, conductivity problem, problems for piezoelectric and other coupled phenomena, composites with nonlocal elastic properties of constituents, and the wave propagation in composites with electromagnetic, optic, and mechanical responses.

CITED BY
  1. Buryachenko Valeriy A., Solution of general integral equations of micromechanics of heterogeneous materials, International Journal of Solids and Structures, 51, 23-24, 2014. Crossref

  2. Buryachenko Valeriy A., Statistical average of residual stresses in elastically homogeneous medium with random field of noncanonical inclusions, Computers & Structures, 187, 2017. Crossref

  3. Buryachenko Valeriy A, Effective properties of thermoperistatic random structure composites: Some background principles, Mathematics and Mechanics of Solids, 22, 6, 2017. Crossref

  4. Buryachenko Valeriy A., Method of fundamental solutions in micromechanics of elastic random structure composites, International Journal of Solids and Structures, 124, 2017. Crossref

  5. Buryachenko Valeriy, General Interface Integral Equations in Elasticity of Random Structure Composites, in Micromechanics and Nanomechanics of Composite Solids, 2018. Crossref

  6. Buryachenko Valeriy A., Effective field hypothesis in Hashin–Shtrikman bounds estimations for effective moduli of composites with noncanonical shape of inclusions, Mechanics of Materials, 119, 2018. Crossref

  7. Buryachenko Valeriy A, Interface integral technique for the thermoelasticity of random structure matrix composites, Mathematics and Mechanics of Solids, 24, 9, 2019. Crossref

  8. Buryachenko Valeriy A., Computational homogenization in linear elasticity of peristatic periodic structure composites, Mathematics and Mechanics of Solids, 24, 8, 2019. Crossref

  9. Buryachenko Valeriy A., Generalized Mori–Tanaka Approach in Micromechanics of Peristatic Random Structure Composites, Journal of Peridynamics and Nonlocal Modeling, 2, 1, 2020. Crossref

  10. Buryachenko Valeriy A., Peridynamic Micromechanics of Random Structure Composites, in Local and Nonlocal Micromechanics of Heterogeneous Materials, 2022. Crossref

  11. Buryachenko Valeriy A., Background of Peridynamic Micromechanics, in Local and Nonlocal Micromechanics of Heterogeneous Materials, 2022. Crossref

  12. Buryachenko Valeriy A., Critical analysis of generalized Maxwell homogenization schemes and related prospective problems, Mechanics of Materials, 165, 2022. Crossref

  13. Buryachenko Valeriy A, Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems, Mathematics and Mechanics of Solids, 2022. Crossref

  14. Buryachenko Valeriy A., Generalized effective fields method in peridynamic micromechanics of random structure composites, International Journal of Solids and Structures, 202, 2020. Crossref

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