The behavior of a 3D numerical model made of spherical elements is investigated in terms of the sign of the second order work. The numerical experiments consist of stress probes in the Rendulic plane of stress increments. Initial stress states, from which stress probes are performed, are obtained by axi-symmetric triaxial tests. From stresses and strains determined at boundaries of the numerical specimen, the second order work (macroscopic form) is computed for each stress state and each loading direction. Thus, for stress states approaching failure, some loading directions are unstable and a limit of potential failure/bifurcation domain is exhibited. Then comparisons between values of second order work obtained from a macroscopic description and a description by contacts are carried out. Equality of the two expressions is observed for elastic strains.

Luc Sibille's Ph.D thesis. Collaboration with Félix Darve and François Nicot