Vol. 5, No. 3-4, 2017

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On the notion of stress in classical continuum mechanics

Simon R. Eugster and Christoph Glocker

Vol. 5 (2017), No. 3-4, 299–338
DOI: 10.2140/memocs.2017.5.299

A variational formulation of continuum mechanics, in which the principle of virtual work and the variational law of interaction are postulated as the basic axioms, is still controversially discussed. In particular, not widely accepted is the internal virtual work contribution of a continuum, as postulated as a smooth density integrated over the deformed configuration of the body, in which the stress field is defined as the quantity dual to the gradient of the virtual displacement field. The question arises whether this internal virtual work can be deduced, rather than just postulated, from already known mechanical concepts completely within the variational framework. To achieve such a derivation, we give in this paper an interpretation of Piola’s micro-macro identification procedure in view of the Riemann integral, which naturally provides in its mathematical definition a micro-macro relation between the discrete system of infinitesimal volume elements and the continuum. Accordingly, we propose a definition of stress on the micro level of the infinitesimal volume elements. In particular, the stress is defined as the internal force effects of the body that model the mutual force interaction between neighboring infinitesimal volume elements. The internal virtual work of the continuum is then obtained by Piola’s micro-macro identification procedure, where in the limit of vanishing volume elements the virtual work of the continuous macromodel is identified with the virtual work of the discrete micromodel. In the course of this procedure, the stress tensor emerges directly as the quantity dual to the gradient of the virtual displacement field. Furthermore, we try to gather important results of variational continuum mechanics, which have appeared here and there in very diverse forms, in order to underline once more the strength of a variational formulation of continuum mechanics.

continuum mechanics, virtual work, law of interaction, stress, micro-macro identification
Mathematical Subject Classification 2010
Primary: 70A05, 74A10
Received: 18 November 2016
Revised: 22 May 2017
Accepted: 19 August 2017

Communicated by Francesco dell'Isola
Simon R. Eugster
Institute for Nonlinear Mechanics
University of Stuttgart
Christoph Glocker
ETH Zürich