  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
