Nonlinear elasticity and slow dynamics: physical and numerical modeling

International audienceVibrational experiments in rocks or concrete reveal that two different dynamics coexist [2,3]. First, a "fast" elastic dynamics occurs with a time scale ruled by the frequency of the excitation. Second, a "slow" dynamics governs the relaxation of the elastic modulus. Here, a "soft-ratchet" model [4] is preferred to the phenomenological Preisach-Mayergoyz model commonly used. The softening/recovering is related to the concentration of defects that evolves dynamically with the stress. This relaxation mechanisms is coupled to nonlinear elasticity. Lastly, viscoelasticity is introduced. Our contribution is two-fold [1]: • improvement of the physical model. A non-physical feature of the soft-ratchet model [4] is fixed. Moreover, the generalized Zener model introduced; • construction of a numerical scheme. Analytical tools used in [4] were unable to solve the full coupled system. On the contrary, we develop here a numerical strategy that enables to solve the whole equations. First, we introduce the physical model and its basic features: evolution of defects, nonlinear elasticity, and attenuation. Second, the evolution equations are written as a first-order system of partial differential equations. Third, the numerical method is introduced, based on a splitting strategy. The hyperbolic step is solved by a Godunov scheme, whereas the relaxation step is solved exactly. Fourth and last, numerical experiments show that the experimental observations performed by DAET are qualitatively recovered. [1] N. Favrie, B. Lombard, C. Payan, Fast and slow dynamics in a nonlinear elastic bar excited by longitudinal vibrations, submitted to Wave Motion (2015). [2] R.A. Guyer, P.A. Johnson, Nonlinear mesoscopic elasticity: Evidence for a new class of materials, Physics Today 52 (1999), 30-35. [3] J.A. Ten Cate, T.J. Shankland, Slow dynamics in the nonlinear elastic response of Berea sandstone, Geophys. Res. Lett., 23-21 (1996), 3019-3022. [4] O.O. Vakhnenko, V.O. Vakhnenko, T.J Shankland, Soft-ratchet modeling of end-point memory in the nonlinear resonant response of sedimentary rocks, Physical Review B71 (2005), 174103