Shock Wave Structure in a Strongly Nonlinear Granular Lattice with   Viscous Dissipation
  The shock wave structure in a one-dimensional lattice (e.g. granular chain) with a power law dependence of force on displacement between particles with viscous dissipation is considered and compared to the corresponding long wave approximation. A dissipative term depending on the relative velocity between neighboring particles is included in the discrete model to investigate its influence on the shape of steady shock profiles. The critical viscosity coefficient is obtained from the long-wave approximation for arbitrary values of the exponent n and denotes the transition from an oscillatory to a monotonic shock profile in stronly nonlinear systems. The expression for the critical viscosity coefficient converges to the known equation for the critical viscosity in the weakly nonlinear case. Values of viscosity based on this expression are comparable to the values obtained in the numerical analysis of a discrete particle lattice with a Herzian contact interaction corresponding to n = 3/2. An initial disturbance in a discrete system approaches a stationary shock profile after traveling a short distance that is comparable to the width of the leading pulse of a stationary shock front. The shock front width is minimized when the viscosity is equal to its critical value. 
E. B. Herbold and V. F. Nesterenko