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Some Contributions to the Homogenization of Macroscopically Isotropic Composites

Salit, Victor (2011):
Some Contributions to the Homogenization of Macroscopically Isotropic Composites.
Darmstadt, Techn. Univ., Studienbereich Mechanik, In: Forschungsbericht // Studienbereich Mechanik, Technische Universität Darmstadt, ISBN 978-3-935868-23-5,
[Online-Edition: urn:nbn:de:tuda-tuprints-26704],
[Book]

Abstract

The development of homogenization models and solutions, either analytic or numerical, is a difficult task. Due to the integral nature of equations, the result has a good chance to fall within the theoretical bounds and as long as it is there - there might be no indication for any mistake. The difficulty of the homogenization stems not from the involved mathematical concepts, but rather from the little, to at times, no difference between something correct and something that just happens to look plausible. In this work it is argued that the Hashin-Shtrikman expressions are not bounds, but rather boundaries of the solution (hyper)surface. It is also shown that the well known Hill condition is not a necessary ingredient for the homogenization. Using a few homogenization concepts, a complete system of equations describing the macroscopic behavior of the heterogeneous materials has been derived. This system possesses a simple solution in the isotropic case.

Item Type: Book
Erschienen: 2011
Creators: Salit, Victor
Title: Some Contributions to the Homogenization of Macroscopically Isotropic Composites
Language: English
Abstract:

The development of homogenization models and solutions, either analytic or numerical, is a difficult task. Due to the integral nature of equations, the result has a good chance to fall within the theoretical bounds and as long as it is there - there might be no indication for any mistake. The difficulty of the homogenization stems not from the involved mathematical concepts, but rather from the little, to at times, no difference between something correct and something that just happens to look plausible. In this work it is argued that the Hashin-Shtrikman expressions are not bounds, but rather boundaries of the solution (hyper)surface. It is also shown that the well known Hill condition is not a necessary ingredient for the homogenization. Using a few homogenization concepts, a complete system of equations describing the macroscopic behavior of the heterogeneous materials has been derived. This system possesses a simple solution in the isotropic case.

Series Name: Forschungsbericht // Studienbereich Mechanik, Technische Universität Darmstadt
Volume: 23
Place of Publication: Darmstadt
Publisher: Techn. Univ., Studienbereich Mechanik
ISBN: 978-3-935868-23-5
Divisions: 13 Department of Civil and Environmental Engineering Sciences
13 Department of Civil and Environmental Engineering Sciences > Mechanics > Solid Body Mechanics
13 Department of Civil and Environmental Engineering Sciences > Mechanics
Date Deposited: 03 Aug 2011 06:41
Official URL: urn:nbn:de:tuda-tuprints-26704
Additional Information:

[Darmstadt, TU, Diss., 2011]

License: only the rights of use according to UrhG
Referees: Gruttmann, Dr.-Ing Friedrich and Gross, Dr.-Ing Dietmar
Refereed / Verteidigung / mdl. Prüfung: 7 July 2011
Alternative keywords:
Alternative keywordsLanguage
homogenization, heterogeneous materials, bounds, Hill condition, embedded-cell model, iterative homogenizationEnglish
Alternative Abstract:
Alternative abstract Language
Die Entwicklung von analytischen und numerischen Homogenisierungsmethoden ist eine anspruchsvolle Aufgabe. Wenn die Ergebnisse sich trotz möglicher Fehler innerhalb der theoretisch zulässigen Schranken befinden, gibt es kaum Mittel um einen solchen Fehler zu ermitteln. Die Homogenisierung ist daher nicht nur wegen der mathematischen Komplexität eine anspruchsvolle Aufgabe, sondern auch aufgrund der Tatsache, dass keine zuverlässige Methode existiert, die es erlaubt die Modelle und ihre Ergebnisse zu validieren und zu verifizieren. In dieser Arbeit wird gezeigt, dass die Hashin-Shtrikman Ausdrücke keine Schranken sind, sondern eher die Grenzen der Lösungs(hyper)fläche. Es wird auch gezeigt, dass die bekannte Hillbedingung für die Homogenisierung nicht erfordlerlich ist. Mit Hilfe von einigen Homogenisierungskonzepten, wird ein vollständiges Gleichungssystem, das das makroskopische Verhalten im allgemeinen Fall beschreibt, zusammengestellt. Im Fall der Isotropie besitzt das Gleichungssystem eine einfache Lösung.German
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