Memory effect in homogenization of a visoelastic Kelvin-Voigt model with time-dependent coefficients

Z Abdessamad, I Kostin, G Panasenko, V P Smyshlyaev

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.
Original languageEnglish
Pages (from-to)1603-1630
Number of pages28
JournalMathematical Models & Methods in Applied Sciences
Volume19
Issue number9
DOIs
Publication statusPublished - 2009

Fingerprint

Memory Effect
Kelvin
Homogenization
Solidification
Data storage equipment
Mechanical Properties
Coefficient
Oscillating Coefficients
Mechanical properties
Memory Term
Periodic Coefficients
Regularity Properties
Asymptotic Solution
Composite Materials
A Priori Estimates
Unique Solution
Initial-boundary-value Problem
Forcing
Error Bounds
Boundary value problems

Cite this

Memory effect in homogenization of a visoelastic Kelvin-Voigt model with time-dependent coefficients. / Abdessamad, Z; Kostin, I; Panasenko, G; Smyshlyaev, V P.

In: Mathematical Models & Methods in Applied Sciences, Vol. 19, No. 9, 2009, p. 1603-1630.

Research output: Contribution to journalArticle

Abdessamad, Z ; Kostin, I ; Panasenko, G ; Smyshlyaev, V P. / Memory effect in homogenization of a visoelastic Kelvin-Voigt model with time-dependent coefficients. In: Mathematical Models & Methods in Applied Sciences. 2009 ; Vol. 19, No. 9. pp. 1603-1630.
@article{f705fccbf18c432c869cb0c92b4170ee,
title = "Memory effect in homogenization of a visoelastic Kelvin-Voigt model with time-dependent coefficients",
abstract = "This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.",
author = "Z Abdessamad and I Kostin and G Panasenko and Smyshlyaev, {V P}",
year = "2009",
doi = "10.1142/S0218202509003905",
language = "English",
volume = "19",
pages = "1603--1630",
journal = "Mathematical Models & Methods in Applied Sciences",
issn = "0218-2025",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "9",

}

TY - JOUR

T1 - Memory effect in homogenization of a visoelastic Kelvin-Voigt model with time-dependent coefficients

AU - Abdessamad, Z

AU - Kostin, I

AU - Panasenko, G

AU - Smyshlyaev, V P

PY - 2009

Y1 - 2009

N2 - This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.

AB - This paper is motivated by modeling the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which thereby influence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin-Voigt type with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution.

UR - http://www.scopus.com/inward/record.url?scp=72549118624&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1142/S0218202509003905

U2 - 10.1142/S0218202509003905

DO - 10.1142/S0218202509003905

M3 - Article

VL - 19

SP - 1603

EP - 1630

JO - Mathematical Models & Methods in Applied Sciences

JF - Mathematical Models & Methods in Applied Sciences

SN - 0218-2025

IS - 9

ER -