The Generalized Derivative and Its Use in the Analysis of the Microstructure of a Heterogeneous Medium

We provide some analytical account of the influence of the internal boundaries of a heterogeneous medium on the propagation of an elastic stress field through it. The generalized derivative serves as the mathematical concept displaying the microstructure of a heterogeneous system. Using the generali...

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Bibliographic Details
Published in:Journal of applied and industrial mathematics 2021, Vol.15 (4), p.631-646
Main Author: Mishin, A. V.
Format: Article
Language:English
Subjects:
Derivatives
Elastic properties
Elasticity
Green's functions
Mathematics
Mathematics and Statistics
Microstructural analysis
Microstructure
Operators (mathematics)
Stress distribution
Stress propagation
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Summary:We provide some analytical account of the influence of the internal boundaries of a heterogeneous medium on the propagation of an elastic stress field through it. The generalized derivative serves as the mathematical concept displaying the microstructure of a heterogeneous system. Using the generalized derivative we modify the operator in the initial model of linear elasticity. The Green’s function (built on the operator) displays the microstructural features of the system. We use the method of conditional moments to obtain the effective coefficients of elasticity which are included in the averaged equations and describing the elastic properties of a heterogeneous medium. This approach leads to the integrals containing the modified averaged Green’s function and the correlation function of the structure geometry. Using these terms, we integrally take into account the microstructure of the system in the final effective elasticity coefficients.
ISSN:1990-4789
1990-4797
DOI:10.1134/S1990478921040074