To date, there are essentially two different ways to postulate the foundations of continuum mechanics. The first method, called the non-variational approach, has been conceived mainly by Cauchy and assumes forces and moments as the elemental quantities. The second method, which traces back to Lagrange and Piola, is of variational nature and defines forces in a generalized sense as the quantities dual to the virtual displacement field and the gradients thereof. The variational approach has gained much attention during the last half century, especially in the field of generalized continua like gradient materials, polar and micromorphic media. Most of these generalized theories cannot be stated straightforwardly by using the non-variational approach. Nevertheless the 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.
Not widely accepted is in particular the internal virtual work contribution of a continuum, as being 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. Such a derivation can be achieved by 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 macro-model is identified with the virtual work of the discrete micro-model. In the course of this procedure, the stress tensor emerges directly as the quantity dual to the gradient of the virtual displacement field.
Eugster, S.R., Glocker, Ch.: “An alternative perspective on the concept of stress in classical continuum mechanics”, In Proceedings in Applied Mathematics and Mechanics. Vol. 16, pp. 331-332, 2016. PDF, PAMM