Herbold E B, Nesterenko V F
Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, California 92093-0411, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Feb;75(2 Pt 1):021304. doi: 10.1103/PhysRevE.75.021304. Epub 2007 Feb 26.
The shock wave structure in a one-dimensional lattice (e.g., granular chain of elastic particles) with a power law dependence of force on displacement between particles (F proportional to delta(n)) 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 to investigate its influence on the shape of a steady shock. The critical viscosity coefficient p(c), defining the transition from an oscillatory to a monotonic shock profile in strongly nonlinear systems, is obtained from the long-wave approximation for arbitrary values of the exponent n. The expression for the critical viscosity is comparable to the value obtained in the numerical analysis of a discrete system with a Hertzian contact interaction (n=3/2) . The expression for p(c) in the weakly nonlinear case converges to the known equation for the critical viscosity. 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.
研究了一维晶格（例如弹性颗粒的颗粒链）中的冲击波结构，其中力与颗粒间位移呈幂律依赖关系（F 与δ(n)成正比）且存在粘性耗散，并将其与相应的长波近似进行了比较。包含一个取决于相邻颗粒相对速度的耗散项，以研究其对稳态激波形状的影响。从长波近似中得到了定义强非线性系统中从振荡激波剖面到单调激波剖面转变的临界粘度系数 p(c)，其中指数 n 为任意值。临界粘度的表达式与具有赫兹接触相互作用的离散系统（n = 3/2）数值分析中得到的值相当。弱非线性情况下 p(c)的表达式收敛到已知的临界粘度方程。离散系统中的初始扰动在传播一段与稳态激波前沿主导脉冲宽度相当的短距离后接近稳态激波剖面。当粘度等于其临界值时，激波前沿宽度最小。
