Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz-Mindlin balls: Mean-field description

We study the dependence on the external pressure P of the velocities $v_{L,T}$ of long wavelength sound waves in a confined two-dimensional hexagonal close-packed lattice of 3D elastic frictional balls interacting via one-sided Hertz-Mindlin contact forces, whose diameters exhibit mild dispersion. The presence of an underlying long range order enables us to build an effective medium description, which incorporates the radial fluctuations of the contact forces acting on a single site. Due to the nonlinearity of Hertz elasticity, self-consistency results in a highly nonlinear differential equation for the “equation of state” linking the effective stiffness of the array with the applied pressure, from which sound velocities are then obtained. The results are in excellent agreement with existing experimental results and simulations in the high- and intermediate-pressure regimes. It emerges from the analysis that the departure of $v_L\mathrm{}\left(P\right)\mathrm{}$ from the ideal $P^{1/6}$ Hertz behavior must be attributed primarily to the fluctuations of the stress field, rather than to the pressure dependence of the number of contacts.