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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2014011234
pages 11-53

GENERAL INTEGRAL EQUATIONS OF MICROMECHANICS OF HETEROGENEOUS MATERIALS

Valeriy A. Buryachenko
Civil Engineering Department, University of Akron, Akron, Ohio 44325-3901, USA and Micromechanics and Composites LLC, 2520 Hingham Lane, Dayton, Ohio 45459, USA

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.


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