A texture tensor to quantify deformations: the example of two-dimensional flowing foams

In a continuum description of materials, the stress tensor field $\bar{% \bar{\sigma}}$ quantifies the internal forces the neighbouring regions exert on a region of the material. The classical theory of elastic solids assumes that $\overline{\stackrel{ˉ}{\sigma }}$ determines the strain, while hydrodynamics assumes that $\overline{\stackrel{ˉ}{\sigma }}$ determines the strain rate. To extend both successful theories to more general materials, which display both elastic and fluid properties, we recently introduced a descriptor generalizing the classical strain to include plastic deformations: the statistical strain'', based on averages on microscopic details (A texture tensor to quantify deformations’’ M.Au., Y.J., J.A.G., F.G, companion paper, {\em Granular Matter}, same issue). Here, we apply such a statistical analysis to a two-dimensional foam steadily flowing through a constriction, a problem beyond reach of both theories, and prove that the foam has the elastic properties of a (linear and isotropic) continuous medium.