Abstract
In this study, we numerically investigated the mechanical responses and trajectories of frictional granular particles under oscillatory shear in the reversible phase where particle trajectories form closed loops below the yielding point. When the friction coefficient is small, the storage modulus exhibits softening, and the loss modulus remains finite in the quasi-static limit. As the friction coefficient increases, the softening and residual loss modulus are suppressed. The storage and loss moduli satisfy scaling laws if they are plotted as functions of the areas of the loop trajectories divided by the strain amplitude and diameter of grains, at least for small values of the areas.
Graphic abstract
Similar content being viewed by others
Notes
We have also checked that particles in the reversible phase exhibit almost the same trajectories for ten cycles in all samples.
References
M. van Hecke, J. Phys. Condens. Matter 22, 033101 (2009)
R.P. Behringer, B. Chakraborty, Rep. Prog. Phys. 82, 012601 (2019)
C.S. O’Hern, S.A. Langer, A.J. Liu, S.R. Nagel, Phys. Rev. Lett. 88, 075507 (2002)
B.P. Tighe, Phys. Rev. Lett. 107, 158303 (2011)
M. Otsuki, H. Hayakawa, Phys. Rev. E 95, 062902 (2017)
C. Coulais, A. Seguin, O. Dauchot, Phys. Rev. Lett. 113, 198001 (2014)
M. Otsuki, H. Hayakawa, Phys. Rev. E 90, 042202 (2014)
K.H. Nagamanasa, S. Gokhale, A.K. Sood, R. Ganapathy, Phys. Rev. E 89, 062308 (2014)
E.D. Knowlton, D.J. Pine, L. Cipelletti, Soft Matter 10, 6931 (2014)
T. Kawasaki, L. Berthier, Phys. Rev. E 94, 022615 (2016)
P. Leishangthem, A.D.S. Parmar, S. Sastry, Nat. Commun. 8, 14653 (2017)
A.H. Clark, J.D. Thompson, M.D. Shattuck, N.T. Ouellette, C.S. O’Hern, Phys. Rev. E 97, 062901 (2018)
J. Boschan, S. Luding, B.P. Tighe, Granul. Matter 21, 58 (2019)
J. Boschan, D. Vågberg, E. Somfai, B.P. Tighe, Soft Matter 12, 5450 (2016)
D. Nakayama, H. Yoshino, F. Zamponi, J. Stat. Mech. 2016, 104001 (2016)
T. Kawasaki, K. Miyazaki, arXiv:2003.10716 (2020)
S. Dagois-Bohy, E. Somfai, B.P. Tighe, M. van Hecke, Soft Matter 13, 9036 (2017)
M. Otsuki, H. Hayakawa, arXiv:2101.07473 (2021)
M. Lundberg, K. Krishan, N. Xu, C.S. O’Hern, M. Dennin, Phys. Rev. E 77, 041505 (2008)
C.F. Schreck, R.S. Hoy, M.D. Shattuck, C.S. O’Hern, Phys. Rev. E 88, 052205 (2013)
N.C. Keim, P.E. Arratia, Soft Matter 9, 6222 (2013)
N.C. Keim, P.E. Arratia, Phys. Rev. Lett. 112, 028302 (2014)
I. Regev, T. Lookman, C. Reichhardt, Phys. Rev. E 88, 062401 (2013)
I. Regev, J. Weber, C. Reichhardt, K.A. Dahmen, T. Lookman, Nat. Commun. 6, 8805 (2015)
N.V. Priezjev, Phys. Rev. E 93, 013001 (2016)
M.O. Lavrentovich, A.J. Liu, S.R. Nagel, Phys. Rev. E 96, 020101(R) (2017)
K. Nagasawa, K. Miyazaki, T. Kawasaki, Soft Matter 15, 7557 (2019)
P. Das, H.A. Vinutha, S. Sastry, Proc. Natl. Acad. Sci. USA 117, 10203 (2020)
M. Otsuki, H. Hayakawa, Phys. Rev. E 83, 051301 (2011)
S. Chialvo, J. Sun, S. Sundaresan, Phys. Rev. E 85, 021305 (2012)
E. Brown, H.M. Jaeger, Phys. Rev. Lett. 103, 086001 (2009)
R. Seto, R. Mari, J.F. Morris, M.M. Denn, Phys. Rev. Lett. 111, 218301 (2013)
N. Fernandez, R. Mani, D. Rinaldi, D. Kadau, M. Mosquet, H. Lombois-Burger, J. Cayer-Barrioz, H.J. Herrmann, N.D. Spencer, L. Isa, Phys. Rev. Lett. 111, 108301 (2013)
C. Heussinger, Phys. Rev. E 88, 050201 (2013)
M.M. Bandi, M.K. Rivera, F. Krzakala, R.E. Ecke, Phys. Rev. E 87, 042205 (2013)
M.P. Ciamarra, R. Pastore, M. Nicodemi, A. Coniglio, Phys. Rev. E 84, 041308 (2011)
R. Mari, R. Seto, J.F. Morris, M.M. Denn, J. Rheol. 58, 1693 (2014)
M. Grob, C. Heussinger, A. Zippelius, Phys. Rev. E 89, 050201(R) (2014)
T. Kawasaki, A. Ikeda, L. Berthier, EPL 107, 28009 (2014)
M. Wyart, M.E. Cates, Phys. Rev. Lett. 112, 098302 (2014)
M. Grob, A. Zippelius, C. Heussinger, Phys. Rev. E 93, 030901(R) (2016)
I.R. Peters, S. Majumdar, H.M. Jaeger, Nature 532, 214 (2016)
A. Fall, F. Bertrand, D. Hautemayou, C. Mezière, P. Moucheront, A. Lemaître, G. Ovarlez, Phys. Rev. Lett. 114, 098301 (2015)
S. Sarkar, E. Shatoff, K. Ramola, R. Mari, J. Morris, B. Chakraborty, EPJ Web Conf. 140, 09045 (2017)
A. Singh, R. Mari, M.M. Denn, J.F. Morris, J. Rheol. 62, 457 (2018)
T. Kawasaki, L. Berthier, Phys. Rev. E 98, 012609 (2018)
J.E. Thomas, K. Ramola, A. Singh, R. Mari, J.F. Morris, B. Chakraborty, Phys. Rev. Lett. 121, 128002 (2018)
D. Bi, J. Zhang, B. Chakraborty, R. Behringer, Nature 480, 355 (2011)
J. Zhang, T. Majmudar, R. Behringer, Chaos 18, 041107 (2008)
J. Zhang, T.S. Majmudar, A. Tordesillas, R.P. Behringer, Granul. Matter 12, 159 (2010)
D. Wang, J. Ren, J.A. Dijksman, H. Zheng, R.P. Behringer, Phys. Rev. Lett. 120, 208004 (2018)
Y. Zhao, J. Barés, H. Zheng, J.E.S. Socolar, R.P. Behringer, Phys. Rev. Lett. 123, 158001 (2019)
S. Sarkar, D. Bi, J. Zhang, R.P. Behringer, B. Chakraborty, Phys. Rev. Lett. 111, 068301 (2013)
S. Sarkar, D. Bi, J. Zhang, J. Ren, R.P. Behringer, B. Chakraborty, Phys. Rev. E 93, 042901 (2016)
R. Seto, A. Singh, B. Chakraborty, M.M. Denn, J.F. Morris, Granul. Matter 21, 82 (2019)
H. Pradipto Hayakawa, Soft Matter 16, 945 (2020)
M. Otsuki, H. Hayakawa, Phys. Rev. E 101, 032905 (2020)
D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, 2nd edn. (Cambridge University Press, Cambridge, 2008)
M. Doi, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986)
Acknowledgements
The authors thank K. Saitoh and D. Ishima for fruitful discussion. This work was supported by JSPS KAKENHI Grant Nos. JP16H04025, JP19K03670, and JP21H01006, and ISHIZUE 2020 of the Kyoto University Research Development Program.
Author information
Authors and Affiliations
Contributions
M.O. carried out the simulations. Both authors interpreted the results and wrote the manuscript.
Corresponding author
Rights and permissions
About this article
Cite this article
Otsuki, M., Hayakawa, H. Shear modulus and reversible particle trajectories of frictional granular materials under oscillatory shear. Eur. Phys. J. E 44, 70 (2021). https://doi.org/10.1140/epje/s10189-021-00075-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/s10189-021-00075-0