二维圆盘颗粒体系声学行为的数值研究

数值测量了卸载过程中二维单分散圆盘颗粒系统的横波、纵波声速、声衰减系数、非线性系数随压强的变化以及声衰减系数随频率的变化.结果表明,二维(2D)圆盘颗粒体系的横波、纵波声速均随压强呈分段幂律标度:当压强P10~(-4)时,横波、纵波声速随压强的增大而减小;当P10~(-4)时,有v_t~P~(0.202),v_l~P~(0.338).进一步得到其剪切模量和体积模量的比值G/B也随压强呈幂律标度,G/B~P~(-0.502),暗示在低压强下,与三维(3D)球形颗粒体系类似,2D圆盘颗粒体系也处于L玻璃态.水平激励和垂直激励下2D圆盘颗粒系统的衰减系数随频率变化也呈现分段行为:当频率f0.05时,衰减系数不随f变化;当f0.05时,横波纵波的衰减系数α~f;当f0.35时,横波衰减系数α_T~f~2,纵波衰减系数α_L~f~(1.5).此外,竖直水平激励下的2D圆盘颗粒系统的非线性系数和衰减系数随压强也呈现与声速类似的分段规律:当P10~(-4)时,横波非线性系数β_T~P~(-0.230),其余都不随压强变化.当P10~(-4)时,两者均随压强增大呈幂律减小:β_T~P~(-0.703),β_L~P~(-0.684),α_T~P~(-0.099),α_L~P~(-0.105).进而得到2D圆盘颗粒系统中散射相关的特征长度?~*随压强呈幂律标度,当P10~(-4)时,?~*~P~(-0.595);当P10~(-4)时,?~*~P~(0.236).

Numerical study on acoustic behavior of two-dimensional granular system

The transversal and longitudinal wave velocities, the acoustic attenuation coefficients, the nonlinear coefficients at different pressures and the acoustic attenuation coefficient as a function of frequency in a two-dimensional (2D) monodisperse disc system are numerically calculated. The results show that the transversal and longitudinal wave velocities both exhibit a piecewise power law with pressure P. When P < 10^-4, the velocity decreases with the increase of pressure in the 2D disc granular system, and when P > 10^-4, the transversal wave velocity V_t and longitudinal wave velocity V_l show the scaling power laws, i.e., ν_t~P^0.202 and v_l~P^0.338, respectively. The ratio of the shear modulus to the bulk modulus G/B shows a power law scaling with the pressure, G/B~P^-0.502, implying that the system lies in an L glass state at low pressure, similar to that of a three-dimensional (3D) spherical granular system. The attenuation coefficients (α_T, α_L) of the horizontal excitation and vertical excitation also show the picecewise behaviors with the change of frequency f. When f < 0.05, neither of the two attenuation coefficients changes with frequency f. When f > 0.05, α ∝ f_T^α, α_L ∝ f. And when f > 0.35, α_T ∝ f² and α_L ∝ f^1.5. In addition, the nonlinear coefficient and the attenuation coefficient of the 2D disc granular system under the vertical and horizontal excitation both also show a piecewise law behavior with pressure, similar to that of the acoustic velocity. When P < 10^-4, only the transversal nonlinear coefficient changes according to β_T ∝ P^-0.230, while the other coefficient has no change. When P > 10^-4, the attenuation coefficients and nonlinear coefficients decrease according to their power law with the increase of pressure, i.e., β_T ∝ P^-0.703, β_L ∝ P^-0.684, α_T ∝ P^-0.099, α_L ∝ P^-0.105. The characteristic length l^*, which characterizes the disordered structure responsible for the scattering, also decreases according to power law with the increase of pressure, when P < 10^-4, l^* ∝ P^-0.595; when P > 10^-4, l^* ∝ P^0.236.