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An incremental theory of large strain and large displacement problems and its finite element formulation

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Summary

This paper is concerned with the development of an incremental finite'element theory for the large strain and the large displacement problems, referred to the current configuration of the body. Using the convected coordinate system which is embedded in the body, the incremental representations of strain and stress tensors and the energy relations are presented, and then the general procedure to construct the so-called element stiffness matrix in incremental form is considered. The finite element formulation is developed for a typical constitutive relation and it is shown that some correction matrices, some of which have been omitted in the previous works, are to be added to the element stiffness matrix. Finally the method to assemble the equations of the element to the global system is discussed and a simple finite element model satisfying the compatibility condition is presented.

The finite element theory developed in this paper is able to be extended to the problems for the general thermodynamical process of a broad class of nonlinear materials.

Übersicht

Mit Hilfe der Methode der finiten Elemente wird eineZuwachstheorie zurBehandlung von Problemen mit endlicher Verformung abgeleitet. Dabei wird ein im Körper eingebettetes, der momentanen Form angepaßtes Bezugssystem verwendet. Es werden Ausdrücke für die Energie sowie für die Änderungen der Spannungs- und Verformungs-Tensoren abgeleitet und es wird ein Verfahren zur Konstruktion der Steifigkeitsmatrix für ein Element angegeben. Ein typisches Stoffgesetz wird dabei zugrundegelegt. Dabei zeigt es sich, daß einige in früheren Arbeiten vernachlässigte Korrektur-Matrizen zu der Steifigkeits-Matrix des Elementes hinzugefügt werden müssen. Die Möglichkeiten der Zusammenfassung der für die Elemente geltenden Gleichungen zu einem globalen Gleichungssystem werden diskutiert und es wird ein den Verträglichkeitsbedingungen genügendes Elemente-Modell angegeben.

Das angegebene Verfahren kann für allgemeine thermodynamische Prozesse in einer breiten Klasse nichtlinearer Materialien erweitert werden.

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Authors and Affiliations

  1. Faculty of Engineering, Osaka University, 565, Yamada-Kami, Osaka, Japan

    H. Kitagawa (Associate Professor)

  2. Faculty of Engineering, Kobe University, 657, Rokko-Dai, Nada-Ku, Kobe, Japan

    Y. Seguchi (Associate Professor)

  3. Faculty of Engineering, Osaka University, 565, Yamada-Kami, Osaka, Japan

    Y. Tomita

Authors
  1. H. Kitagawa
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  2. Y. Seguchi
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  3. Y. Tomita
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Kitagawa, H., Seguchi, Y. & Tomita, Y. An incremental theory of large strain and large displacement problems and its finite element formulation. Ing. arch 41, 213–224 (1972). https://doi.org/10.1007/BF00532634

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  • DOI: https://doi.org/10.1007/BF00532634

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