Beltrami's solutions of general equilibrium equations in continuum mechanics

M. Gurtin has proved that the Beltrami representation, $\mathbf{S}=\mathrm{rot}\mathrm{rot}\mathbf{A}$, of a smooth, divergence-free stress tensor in a smooth domain, is verified if and only if S is self-equilibrated. Here, Gurtin's conditions are extended to the case of a bounded domain with a Lipschitz-continuous boundary, for a tensor field $\mathbf{S}\in L^2(\Omega ;M_{\mathrm{sym}}^3)$. We apply this result to obtain an extension of the Saint Venant's equations of compatibility to non necessarily simply-connected domains. To cite this article: G. Geymonat, F. Krasucki, C. R. Acad. Sci. Paris, Ser. I 342 (2006).