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Effective medium theory (EMT) predicts a scaling relation between sound velocity c and pressure P as c (Z)1/3 (P/E0)1/6, where and Z are respectively the packing fraction and the mean coordination number of granular material. In this relation, the granular contact network is represented via two simple parameters and Z stemming theoretically from a strong approximation that microscopic and macroscopic granular displacements remain affine. This hypothesis simplifies tremendous computations for sound wave in a granular system, however some experimental results show that the scaling relation is recovered only for the case of very high pressure confinement (larger than 106 Pa for a glass bead system), but for the lower pressure case (less than 106 Pa) the relation does not hold. Owing to the fact that the change of microscopic granular displacement relates to the contact network variation of granular sample, and for better understanding the effect of the variation of contact network on the sound propagation in granular system, we conduct uniaxial shear experiments, in which the granular solid sample, composed of 0.28-0.44 mm glass beads, is cyclically compressed under a series of axial loadings (denoted as Pcomp). After these axial loadings, different contact networks of the sample are formed. Ultrasonic waves are then measured in the granular sample with these different contact networks under a constant axial pressure (denoted as Pobse). It is found that the axial deformation of the granular sample apparently affects the incoherent part of ultrasonic wave, but not the coherent part. A resemblant parameter is introduced to quantitatively discuss the variations of incoherent parts of sound waves in different axial deformations. In this paper, we also compare the frequency and the energy spectra of the sound waves, and find that the tendencies of their varying with the increase of axial deformation are nearly the same. This indicates that during the sound wave propagation in the granular solid sample, the processes of wave scattering and dissipation on particle contacted occur at the same time and the energy dissipation of sound wave in the air among particles can be neglected. In our experiments, compressional wave velocities based on time-of-flight method are also explored. The experimental results show that the velocity increases rapidly at the beginning of the axial deformation, and then tends to a steady value which is predicted by EMT. These illuminate that the variation of contact networks of granular sample may contribute to the deviation of velocity-pressure exponent from the prediction of EMT in low confining pressure.
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Keywords:
- granular material /
- ultrasonic wave /
- effective medium theory
[1] Liu C H, Nagel S R, Schecter D A, Coppersmith S N, Majumdar S, Narayan O, Witten T A 1995 Science 269 513
[2] Jacco H S, Thijs J H V, van Martin H, van Wim S 2004 Phys. Rev. Lett. 92 054302
[3] Bi D P, Zhang J, Chakraborty B, Behringer R P 2011 Nature 480 355
[4] Makse H A, Gland N, Johnson D L, Schwartz L M 1999 Phys. Rev. Lett. 83 5070
[5] Tournat V, Gusev V E 2009 Phys. Rev. E 80 011306
[6] Jia X, Brunet Th, Laurent J 2011 Phys. Rev. E 84 020301
[7] Caroli C, Velick B 2003 Phys. Rev. E 67 061301
[8] Khidas Y, Jia X P 2012 Phys. Rev. E 85 051302
[9] Zhang Q, Li Y C, Hou M Y, Jiang Y M, Liu M 2012 Phys. Rev. E 85 031306
[10] Domentico S N 1977 Geophysics 42 1339
[11] Yin H 1993 Ph. D. Dissertation (Stanford: Stanford University)
[12] Majmudar T S, Sperl M, Luding S, Behringer R P 2007 Phys. Rev. Lett. 98 058001
[13] Jia X, Caroli C, Velick B 1999 Phys. Rev. Lett. 82 1863
[14] Owens E T, Daniels K E 2011 Eur. Phys. Lett. 94 54005
[15] Liu C H, Nagel S R 1992 Phys. Rev. Lett. 68 2301
[16] Yacine K, Jia X P 2010 Phys. Rev. E 81 021303
[17] Wambaugh J F, Hartley R R, Behringer R P 2010 Eur. Phys. J. E 32 135
[18] Corwin E I, Jaeger H M, Nagel S R 2005 Nature 435 1075
[19] Nicolas V, Giammarinaro B, Derode A, Barrire C 2013 Phys. Rev. E 88 023201
[20] Makse H A, Gland N, Johnson D L, Schwartz L M, Schwartz L 2004 Phys. Rev. E. 70 061302
[21] Vitelli V 2010 Soft Matter 6 3007
[22] Walton K 1987 J. Mech. Phys. Solids 35 213
[23] Lherminier S, Planet R, Simon G, Vanel L, Ramos O 2014 Phys. Rev. Lett. 113 098001
[24] Gilles B, Coste C 2003 Phys. Rev. Lett. 90 174302
[25] Goddard J D 1990 Proc. R. Soc. Lond. Ser. A 430 105
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[1] Liu C H, Nagel S R, Schecter D A, Coppersmith S N, Majumdar S, Narayan O, Witten T A 1995 Science 269 513
[2] Jacco H S, Thijs J H V, van Martin H, van Wim S 2004 Phys. Rev. Lett. 92 054302
[3] Bi D P, Zhang J, Chakraborty B, Behringer R P 2011 Nature 480 355
[4] Makse H A, Gland N, Johnson D L, Schwartz L M 1999 Phys. Rev. Lett. 83 5070
[5] Tournat V, Gusev V E 2009 Phys. Rev. E 80 011306
[6] Jia X, Brunet Th, Laurent J 2011 Phys. Rev. E 84 020301
[7] Caroli C, Velick B 2003 Phys. Rev. E 67 061301
[8] Khidas Y, Jia X P 2012 Phys. Rev. E 85 051302
[9] Zhang Q, Li Y C, Hou M Y, Jiang Y M, Liu M 2012 Phys. Rev. E 85 031306
[10] Domentico S N 1977 Geophysics 42 1339
[11] Yin H 1993 Ph. D. Dissertation (Stanford: Stanford University)
[12] Majmudar T S, Sperl M, Luding S, Behringer R P 2007 Phys. Rev. Lett. 98 058001
[13] Jia X, Caroli C, Velick B 1999 Phys. Rev. Lett. 82 1863
[14] Owens E T, Daniels K E 2011 Eur. Phys. Lett. 94 54005
[15] Liu C H, Nagel S R 1992 Phys. Rev. Lett. 68 2301
[16] Yacine K, Jia X P 2010 Phys. Rev. E 81 021303
[17] Wambaugh J F, Hartley R R, Behringer R P 2010 Eur. Phys. J. E 32 135
[18] Corwin E I, Jaeger H M, Nagel S R 2005 Nature 435 1075
[19] Nicolas V, Giammarinaro B, Derode A, Barrire C 2013 Phys. Rev. E 88 023201
[20] Makse H A, Gland N, Johnson D L, Schwartz L M, Schwartz L 2004 Phys. Rev. E. 70 061302
[21] Vitelli V 2010 Soft Matter 6 3007
[22] Walton K 1987 J. Mech. Phys. Solids 35 213
[23] Lherminier S, Planet R, Simon G, Vanel L, Ramos O 2014 Phys. Rev. Lett. 113 098001
[24] Gilles B, Coste C 2003 Phys. Rev. Lett. 90 174302
[25] Goddard J D 1990 Proc. R. Soc. Lond. Ser. A 430 105
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