Abstract
A physical model of stress-strain dynamics and long-time relaxation (slow time) in structured media is proposed. The model is based on the analysis of inter-grain contacts and the resulting surface force potential with a barrier. The result is a unified description of the classical acoustic nonlinearity, stress-strain hysteresis, and logarithmic relaxation law for sound velocity (and, hence, for the frequency of nonlinear resonance in samples of structured materials). Estimates of a characteristic volume of interacting contacts give close values for the variety of consolidated materials. For weak (linear) testing waves, the logarithmic relaxation occurs if a classical quadratic nonlinearity is added to the stress-strain relation.
Similar content being viewed by others
References
L. A. Ostrovsky and P. A. Johnson, Riv. Nuovo Cimento, 24, 1 (2001).
R.A. Guyer and P. A. Johnson, Nonlinear Mesoscopic Elasticity: The Complex Behavior of Rocks, Soil, Concrete (Wiley-VCH, 2009).
L. A. Ostrovsky, J. Acoust. Soc. Am. 116, 3348 (2004).
A. V. Nazarov, A. V. Radostin, L. A. Ostrovsky, and I. A. Soustova, Acoust. Phys. 49, 444 (2003).
V.E. Nazarov, Acoust. Phys. 57, 192 (2011).
E. H. Field, P. A. Johnson, I. A. Beresnev, and Y. Zeng, Nature 390, 599 (1994).
P. A. Johnson, P. Bodin, J. Gomberg, F. Pearce, Z. Lawrence, and F.-Y. Menq, J. Geophys. Res. 114, B05304 (2009).
O. V. Rudenko, Giant nonlinearities in structurally inhomogeneous media and the fundamentals of nonlinear acoustic diagnostic techniques, Phys.-Usp. 49, 69 (2006).
A. V. Lebedev, L. A. Ostrovsky, and A. M. Sutin, Akust. Zh. 51, S88 (2005).
V. S. Averbakh, V. V. Artel’ny, B. N. Bogolyubov, V. V. Bredikhin, A. V. Lebedev, A. P. Maryshev, and V. I. Talanov, Acoust. Phys. 54, 71 (2008).
V. Gusev and V. Tournat, Phys. Rev. B: Condens. Matter Mater. Phys. 72, 05104 (2005).
J. A. TenCate and T. J. Shankland, Slow dynamics in the nonlinear elastic response of Berea sandstone, Geophys. Res. Lett. 23, 3019 (1996).
K. W. Winkler and L. Xingzhou, J. Acoust. Soc. Am. 100, 1392 (1996).
J. A. TenCate, E. Smith, and R. Guyer, Phys. Rev. Lett. 85, 1020 (2000).
P. A. Johnson and A. M. Sutin, J. Acoust. Soc. Am. 117, 124 (2005).
D. Pasqualini, K. Heitmann, J. A. TenCate, S. Habib, D. Higdon, and P. A. Johnson, J. Geophys. Res. 112, B01204 (2007).
V. S. Averbakh, A. V. Lebedev, A. P. Maryshev, and V. I. Talanov, Acoust. Phys. 55, 211 (2009).
V. S. Averbakh, A. V. Lebedev, S. A. Manakov, and V. V. Bredikhin, Radiophys. Quantum Electron. 56, 135 (2013).
K. L. Johnson, K. Kendall, and A. D. Roberts, Proc. R. Soc. Lond. A 324, 301 (1971).
J. N. Israelachvili, Intermolecular and Surface Forces (Academic, New York, 1992).
D. Maugis, J. Colloid Interface Sci. 150, 243 (1992).
Y. L. Chen, C. A. Helm, and J. N. Israelachvili, J. Phys. Chem. 95, 10736 (1991).
M. M. Sharma and A. N. Tutuncu, Geophys. Res. Lett. 21, 2323 (1994).
V. A. Aleshin and K. Van Den Abeele, J. Mech. Phys. Solids 53, 795 (2005).
K. Kendall, Energy analysis of adhesion, in The Mechanics of Adhesion, Ed. by D. A. Dillard and A. V. Pocius (Elsevier, New York, 2002), vol. 1, p. 77.
N. A. Burnham and A. J. Kulik, Surface forces and adhesion, in Handbook of Micro/Nano Tribology, Ed. by B. Bhushan (CRC, Boca Raton, 1999), 2nd ed., p. 263.
K. L. Page, Th. Proffen, S. E. McLain, T. W. Darling, and J. A. TenCate, Geophys. Res. Lett. 31, L24606 (2004).
W. Stiller, Arrhenius Equation and Non-Equilibrium Kinetics: 100 Years of Arrhenius Equation (Teubner, Leipzig, 1989).
Y. Estrin, P. G. McCormic, and R. Street, J. Phys.: Condens. Matt. 1, 4845 (1989).
J. A. TenCate, in Proc. Int. Conf. on Nonlinear Elasticity in Materials (Ticino, Switzerland, 2013), p. 84.
V. Aleshin and K. Van Den Abeele, J. Mech. Phys. Solids. 55, 765 (2007).
V. Aleshin and K. Van Den Abeele, J. Mech. Phys. Solids. 55, 366 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article is published in the original.
Rights and permissions
About this article
Cite this article
Lebedev, A.V., Ostrovsky, L.A. A unified model of hysteresis and long-time relaxation in heterogeneous materials. Acoust. Phys. 60, 555–561 (2014). https://doi.org/10.1134/S1063771014050066
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063771014050066