The Principle of Virtual Work and a Symplectic Reduction of Non-Local Continuum Mechanics

Binz, Ernst ; Schwarz, Günter

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URN: urn:nbn:de:bsz:180-madoc-16753
Document Type: Working paper
Year of publication: 1991
Publication language: English
Institution: School of Business Informatics and Mathematics > Sonstige - Fakultät für Mathematik und Informatik
MADOC publication series: Veröffentlichungen der Fakultät für Mathematik und Informatik > Institut für Mathematik > Mannheimer Manuskripte
Subject: 510 Mathematics
Subject headings (SWD): Kontinuumsmechanik , Riemannscher Raum
Abstract: In this paper we derive a symplectic description for systems in continuum mechanics and a representation of the coresponding dynamics by exact one forms on a body manifold. In order to formulate the weak balance laws for the motion we start with the principle of virtual work and obtain thereby a generalization of the Hamiltonian approach. This makes it possible to include also non-hyperelastic media into the Hamiltonian description. For our considerations the virtual work functional is given in a very general form, i.e. we need not impose any restrictions on the constitutive behavior of the material in view. In symplectic framework we study the effect of the group of rigid translations by means of the Marsden-Weinstein reduction with respect to that group. Thereby the deformation gradient appears as a natural geometric object on the reduced phase space. On the other hand the existence of a stress tensor can shown to be a consequence of a (rigid) translational invariance for a general virtual work. In contrast to the classical approaches to continuum dynamics, we need not to impose any locality assumptions to find the stress tensor of such a system.[...]
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