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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 -4, the velocity decreases with the increase of pressure in the 2D disc granular system, and when P > 10-4, the transversal wave velocity Vt and longitudinal wave velocity Vl show the scaling power laws, i.e., νt~P0.202 and vl~P0.338, 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 (αT, αL) of the horizontal excitation and vertical excitation also show the picecewise behaviors with the change of frequency f. When f f. When f > 0.05, α ∝ fTα, αL ∝ f. And when f > 0.35, αT ∝ f2 and αL ∝ f1.5. 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 -4, only the transversal nonlinear coefficient changes according to βT ∝ 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., βT ∝ P-0.703, βL ∝ P-0.684, αT ∝ P-0.099, αL ∝ 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 -4, l* ∝ P-0.595; when P > 10-4, l* ∝ P0.236.
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Keywords:
- granular matter /
- acoustic velocity /
- nonlinear /
- acoustic attenuation
[1] Liu A J, Nagel S R 1998 Nature 396 21
[2] Henkes S, Chakraborty B 2005 Phys. Rev. Lett. 95 198002
[3] O'Hern C S, Silbert L E, Nagel S R 2003 Phys. Rev. E 68 011306
[4] O'Hern C, Langer S, Liu A, Nagel S 2002 Phys. Rev. Lett. 88 075507
[5] Xu N 2011 Front. Phys. 6 109
[6] Ikeda A, Berthier L 2015 Phys. Rev. E 92 012309
[7] Wang X, Zheng W, Wang L, Xu N 2015 Phys. Rev. Lett. 114 035502
[8] Coulais C, Behringer R P, Dauchot O 2014 Soft Matter 10 1519
[9] Karimi K, Maloney C E 2015 Phys. Rev. E 92 022208
[10] Sussman D M, Goodrich C P, Liu A J, Nagel S R 2015 Soft Matter 11 2745
[11] Wyart M, Nagel S R, Witten T A 2005 Europhys. Lett. 72 486
[12] Wyart M, Silbert L, Nagel S 2005 Phys. Rev. E 72 051306
[13] Silbert L, Liu A, Nagel S 2005 Phys. Rev. Lett. 95 098301
[14] Vitelli V 2010 Soft Matter 6 3007
[15] Mizuno H, Silbert L E, Sperl M 2016 Phys. Rev. Lett. 116 068302
[16] Merkel A, Tournat V, Gusev V 2014 Phys. Rev. E 90 023206
[17] Zhang Q, Li Y, Hou M, Jiang Y, Liu M 2012 Phys. Rev. E 85 031306
[18] Zhang Q, Li Y C, Liu R, Jiang Y M, Hou M Y 2012 Acta Phys. Sin. 61 234501 (in Chinese)[张祺, 李寅阊, 刘锐, 蒋亦民, 厚美瑛 2012 物理学报 61 234501]
[19] Zheng H P 2014 Chin. Phys. B 23 054503
[20] Vitelli V, Xu N, Wyart M, Liu A J, Nagel S R 2010 Phys. Rev. E 81 021301
[21] Somfai E, Roux J N, Snoeijer J, van Hecke M, van Saarloos W 2005 Phys. Rev. E 72 021301
[22] Jia X, Caroli C, Velicky B 1999 Phys. Rev. Lett. 82 1863
[23] Zhang P, Zhao X D, Zhang G H, Zhang Q, Sun Q C, Hou Z J, Dong J J 2016 Acta Phys. Sin. 65 024501 (in Chinese)[张攀, 赵雪丹, 张国华, 张祺, 孙其诚, 侯志坚, 董军军 2016 物理学报 65 024501]
[24] Lherminier S, Planet R, Simon G, Vanel L, Ramos O 2014 Phys. Rev. Lett. 113 098001
[25] West B J, Shlesinger M F 1984 J. Stat. Phys. 36 779
[26] Langlois V, Jia X 2015 Phys. Rev. E 91 022205
[27] Hong J 2005 Phys. Rev. Lett. 94 108001
[28] Wang P J, Li Y D, Xia J H, Liu C S 2008 Phys. Rev. E 77 060301
[29] Wang P J, Xia J H, Li Y D, Liu C S 2007 Phys. Rev. E 76 041305
[30] Brunet T, Jia X, Johnson P A 2008 Geophys. Res. Lett. 35 L19308
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[1] Liu A J, Nagel S R 1998 Nature 396 21
[2] Henkes S, Chakraborty B 2005 Phys. Rev. Lett. 95 198002
[3] O'Hern C S, Silbert L E, Nagel S R 2003 Phys. Rev. E 68 011306
[4] O'Hern C, Langer S, Liu A, Nagel S 2002 Phys. Rev. Lett. 88 075507
[5] Xu N 2011 Front. Phys. 6 109
[6] Ikeda A, Berthier L 2015 Phys. Rev. E 92 012309
[7] Wang X, Zheng W, Wang L, Xu N 2015 Phys. Rev. Lett. 114 035502
[8] Coulais C, Behringer R P, Dauchot O 2014 Soft Matter 10 1519
[9] Karimi K, Maloney C E 2015 Phys. Rev. E 92 022208
[10] Sussman D M, Goodrich C P, Liu A J, Nagel S R 2015 Soft Matter 11 2745
[11] Wyart M, Nagel S R, Witten T A 2005 Europhys. Lett. 72 486
[12] Wyart M, Silbert L, Nagel S 2005 Phys. Rev. E 72 051306
[13] Silbert L, Liu A, Nagel S 2005 Phys. Rev. Lett. 95 098301
[14] Vitelli V 2010 Soft Matter 6 3007
[15] Mizuno H, Silbert L E, Sperl M 2016 Phys. Rev. Lett. 116 068302
[16] Merkel A, Tournat V, Gusev V 2014 Phys. Rev. E 90 023206
[17] Zhang Q, Li Y, Hou M, Jiang Y, Liu M 2012 Phys. Rev. E 85 031306
[18] Zhang Q, Li Y C, Liu R, Jiang Y M, Hou M Y 2012 Acta Phys. Sin. 61 234501 (in Chinese)[张祺, 李寅阊, 刘锐, 蒋亦民, 厚美瑛 2012 物理学报 61 234501]
[19] Zheng H P 2014 Chin. Phys. B 23 054503
[20] Vitelli V, Xu N, Wyart M, Liu A J, Nagel S R 2010 Phys. Rev. E 81 021301
[21] Somfai E, Roux J N, Snoeijer J, van Hecke M, van Saarloos W 2005 Phys. Rev. E 72 021301
[22] Jia X, Caroli C, Velicky B 1999 Phys. Rev. Lett. 82 1863
[23] Zhang P, Zhao X D, Zhang G H, Zhang Q, Sun Q C, Hou Z J, Dong J J 2016 Acta Phys. Sin. 65 024501 (in Chinese)[张攀, 赵雪丹, 张国华, 张祺, 孙其诚, 侯志坚, 董军军 2016 物理学报 65 024501]
[24] Lherminier S, Planet R, Simon G, Vanel L, Ramos O 2014 Phys. Rev. Lett. 113 098001
[25] West B J, Shlesinger M F 1984 J. Stat. Phys. 36 779
[26] Langlois V, Jia X 2015 Phys. Rev. E 91 022205
[27] Hong J 2005 Phys. Rev. Lett. 94 108001
[28] Wang P J, Li Y D, Xia J H, Liu C S 2008 Phys. Rev. E 77 060301
[29] Wang P J, Xia J H, Li Y D, Liu C S 2007 Phys. Rev. E 76 041305
[30] Brunet T, Jia X, Johnson P A 2008 Geophys. Res. Lett. 35 L19308
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