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

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The Generalized Derivative and Its Use in the Analysis of the Microstructure of a Heterogeneous Medium. / Mishin, A. V.

In: Journal of Applied and Industrial Mathematics, Vol. 15, No. 4, 11.2021, p. 631-646.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{a5a166780f794746a93e5477c863a67a,

title = "The Generalized Derivative and Its Use in the Analysis of the Microstructure of a Heterogeneous Medium",

abstract = "We provide some analytical account of the influence of the internal boundaries ofa heterogeneous medium on the propagation of an elastic stress field through it. The generalizedderivative serves as the mathematical concept displaying the microstructure of a heterogeneoussystem. Using the generalized derivative we modify the operator in the initial model of linearelasticity. The Green{\textquoteright}s function (built on the operator) displays the microstructural features of thesystem. We use the method of conditional moments to obtain the effective coefficients of elasticitywhich are included in the averaged equations and describing the elastic properties ofa heterogeneous medium. This approach leads to the integrals containing the modified averagedGreen{\textquoteright}s function and the correlation function of the structure geometry. Using these terms, weintegrally take into account the microstructure of the system in the final effective elasticitycoefficients.",

keywords = "averaging, generalized derivative, Green{\textquoteright}s function, heterogeneous medium, microstructure, stochastic model",

author = "Mishin, {A. V.}",

note = "Funding Information: The author was supported by the Ministry of Education and Science of Russian Federation (project no. 075–15–2020–781). Publisher Copyright: {\textcopyright} 2021, Pleiades Publishing, Ltd.",

year = "2021",

month = nov,

doi = "10.1134/S1990478921040074",

language = "English",

volume = "15",

pages = "631--646",

journal = "Journal of Applied and Industrial Mathematics",

issn = "1990-4789",

publisher = "Maik Nauka-Interperiodica Publishing",

number = "4",

}

RIS

TY - JOUR

T1 - The Generalized Derivative and Its Use in the Analysis of the Microstructure of a Heterogeneous Medium

AU - Mishin, A. V.

PY - 2021/11

Y1 - 2021/11

N2 - We provide some analytical account of the influence of the internal boundaries ofa heterogeneous medium on the propagation of an elastic stress field through it. The generalizedderivative serves as the mathematical concept displaying the microstructure of a heterogeneoussystem. Using the generalized derivative we modify the operator in the initial model of linearelasticity. The Green’s function (built on the operator) displays the microstructural features of thesystem. We use the method of conditional moments to obtain the effective coefficients of elasticitywhich are included in the averaged equations and describing the elastic properties ofa heterogeneous medium. This approach leads to the integrals containing the modified averagedGreen’s function and the correlation function of the structure geometry. Using these terms, weintegrally take into account the microstructure of the system in the final effective elasticitycoefficients.

AB - We provide some analytical account of the influence of the internal boundaries ofa heterogeneous medium on the propagation of an elastic stress field through it. The generalizedderivative serves as the mathematical concept displaying the microstructure of a heterogeneoussystem. Using the generalized derivative we modify the operator in the initial model of linearelasticity. The Green’s function (built on the operator) displays the microstructural features of thesystem. We use the method of conditional moments to obtain the effective coefficients of elasticitywhich are included in the averaged equations and describing the elastic properties ofa heterogeneous medium. This approach leads to the integrals containing the modified averagedGreen’s function and the correlation function of the structure geometry. Using these terms, weintegrally take into account the microstructure of the system in the final effective elasticitycoefficients.

KW - averaging

KW - generalized derivative

KW - Green’s function

KW - heterogeneous medium

KW - microstructure

KW - stochastic model

UR - http://www.scopus.com/inward/record.url?scp=85133974269&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/b5258294-7496-3cea-a37c-76b2b0e7d970/

U2 - 10.1134/S1990478921040074

DO - 10.1134/S1990478921040074

M3 - Article

AN - SCOPUS:85133974269

VL - 15

SP - 631

EP - 646

JO - Journal of Applied and Industrial Mathematics

JF - Journal of Applied and Industrial Mathematics

SN - 1990-4789

IS - 4

ER -

ID: 36686877