General Integral Equations of Micromechanics of Composite Materials
In many problems the material may possess a periodic microstructure formed by the spatial repetition of small microstructures, or unit cells. Periodicity of the solution allows one to explore a type of modeling completely different from the average schemes of random media such as asymptotic expansion as well as the Fourier series expansion to field in a solid with periodic microstructure . Such a perfectly regular distribution, of course, does not exist in actual cases, although periodic modeling can be quite useful, since it provides rigorous estimations with a priori prescribed accuracy for various material properties. Another problem is that the development of the theory of homogenization of initially periodic structures with the introduced infinite number of imperfections of unit cells randomly distributed in space has an essentially practical meaning. For example, the accumulation of damage, such as the microcracks in the matrix and debonding of inclusions from the matrix, is essentially random in nature.
The final goal of micromechanical research of composites involved in a prediction of both the overall effective properties and statistical moments of stress-strain fields is based on the approximate solution of exact initial integral equations connecting the random stress fields at the point being considered and the surrounding points. This infinite system of coupled integral equations is well known for statistically homogeneous composite materials subjected to homogeneous boundary conditions , , , . The goal of this chapter is to obtain a generalization of these equations for the case of either the statistically inhomogeneous or doubly periodical structures of composite materials subjected to essentially inhomogeneous loading by fields of the stresses, temperature, and body forces. The case of triply periodic structures with random imperfections is also considered.
KeywordsBody Force Homogeneous Boundary Condition Curly Bracket Conditional Probability Density Locality Principle
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