*A statistical approach in the theory of fluctuations for a random array of identical spheres.*

**Abstract:** We study the mechanical response of a random arrays of identical spherical grains, which interact through a non central force. We focus on the first increment in strain after having compressed isotropically the aggregate. The simplest approach to the problem is to assume that the contact displacement is derived by an average strain. The consequent result is that when we compare the model with numerical simulation an important difference in the prediction of the elastic moduli, overall for the shear modulus, arises. This difference is related to the value of the isotropic compression.
Numerical results show that, in case of frictionless particles, there is almost no resistance to shear despite a far non zero value obtained by the theoretical model.
Our aim is to build up a theoretical incremental stress– strain relation that might capture the essential features of this aggregate. We firstly develop a model, which may be assumed as basis for the mechanical behavior of the aggregate, for the first increment of shearing.
We give up the average strain assumption considering that the contact displacement of a typical pair depends upon the average strain and fluctuations. The introduction of fluctuations relaxes the system giving more degree of freedom. We employ force and momentum equilibrium for the particles of the pair, assuming that the average strain provides a good approximation for their interactions with their neighbors. The above equilibrium equations allow us to determine the fluctuations as functions of the contact stiffness, determined by the average strain, and tensor depending on the packing geometry.
As example of our activity we restrict our attention on a frictionless aggregate. This is a direct consequence of our curiosity in finding plausible physical reasons that might justify a so low value of the shear modulus as predicted by numerical simulation and the reasons that justify why the bulk modulus is well described by the average strain assumption and only a few percent of difference results from the comparison of theory with simulation.
In this context we show how fluctuations are sensitive to the packing geometry. Therefore we employ a suitable statistical model in order to derive an explicit solution for the unknowns fluctuations.
The result is that the incorporation of these additional degree of freedom lead to a value more consistent to what numerical simulation predicts and more we furnish mechanical interpretation to justify this prediction.