Deformation Modes of a Packing of Rigid Grains: Rotation,   Counter-rotation, dislocation field
  The properties of the processes of deformation of a packing of rigid grains are analysed when the exact distribution of the normal forces at contacts is known. Importance of grain rotation and of counter-rotation of adjacent grains is stressed. It is shown that all deformation modes form an incomplete vectorial space D. This allows the statistics of fluctuation to be determined. Parallel with anti-ferromagnetism is then drawn and frustration of rotation is shown to be the leading dissipating mechanism. However this does not allow making the parallel with spinglass due to the structure of the space of deformation modes which is vectorial, which ensures response linearity. In order to demonstrate experimentally the validity of the approach and the existence of the incomplete vectorial space of deformation, experimental examples are studied starting from regular arrays. They do demonstrate the mechanism of counter-rotation and the existence of the vectorial space of deformation. This theory is applied to compare different experiment and simulation results on the deformation of square and triangular lattices of rods, which can only be done through a statistical approach based on existence of different modes of deformation. Non linear behaviour of the modes is demonstrated experimentally; link with the continuous theory of dislocation is made. Pacs # : 5.40 ; 45.70 ; 62.20 ; 83.70.Fn 
P. Evesque