Discrete rearranging disordered patterns: Prediction of elastic and plastic behaviour, and application to two-dimensional foams

We study the elasto-plastic behaviour of materials made of individual (discrete) objects, such as a liquid foam made of bubbles. The evolution of positions and mutual arrangements of individual objects is taken into account through statistical quantities, such as the elastic strain of the structure, the yield strain and the yield function. The past history of the sample plays no explicit role, except through its effect on these statistical quantities. They suffice to relate the discrete scale with the collective, global scale. At this global scale, the material behaves as a continuous medium; it is described with tensors such as elastic strain, stress and velocity gradient. We write the differential equations which predict their elastic and plastic behaviour in both the general case and the case of simple shear. An overshoot in the shear strain or shear stress is interpreted as a rotation of the deformed structure, which is a purely tensorial effect that exists only if the yield strain is at least of order 0.3. We suggest practical applications, including: when to choose a scalar formalism rather than a tensorial one; how to relax trapped stresses; and how to model materials with a low, or a high, yield strain.