Nonlinear elasticity in rocks: A comprehensive three-dimensional description

Abstract

We study theoretically and experimentally the mechanisms of nonlinear and nonequilibrium dynamics in geomaterials through dynamic acoustoelasticity testing. In the proposed theoretical formulation, the classical theory of nonlinear elasticity is extended to include the effects of conditioning. This formulation is adapted to the context of dynamic acoustoelasticity testing in which a low-frequency "pump" wave induces a strain field in the sample and modulates the propagation of a high-frequency "probe" wave. Experiments are conducted to validate the formulation in a long thin bar of Berea sandstone. Several configurations of the pump and probe are examined: the pump successively consists of the first longitudinal and first torsional mode of vibration of the sample while the probe is successively based on (pressure) P and (shear) S waves. The theoretical predictions reproduce many features of the elastic response observed experimentally, in particular, the coupling between nonlinear and nonequilibrium dynamics and the three-dimensional effects resulting from the tensorial nature of elasticity.

A Model for Nonlinear Elasticity in Rocks Based on Friction of Internal Interfaces and Contact Aging

“…Since M l has different sign for compression and dilation, it must result from compression and it should thus disappear for shear excitation. This was indeed observed by Lott et al (2017) as vanishing of the fundamental frequency component for shear excitation.…”

“…A series of DAET experiments investigated the detailed nonlinear behaviour of different rocks (Renaud et al 2012(Renaud et al , 2013Rivière et al 2013Rivière et al , 2015. Longitudinal, shear (Lott et al 2017) and complex flexural deformation from impacts (Eiras et al 2016) have been used as excitation. The response of a single fatigue crack was investigated (Rivière et al 2014) as well as the in situ nonlinearity of soil on the scale of metres using seismic equipment and a vibrator truck (Renaud et al 2014).…”

“…DAET experiments are performed with a pump and a probe wave of a certain wave type and polarization. This allows to investigate the complex anisotropic response of the material as a function of propagation directions and polarizations of the two participating waves (Lott et al 2017). A variant of the typical DAET set-up that measures the response during steady-state oscillation of a standing wave was used by Ten-Cate et al (2016).…”

Propagation of a Finite‐Amplitude Elastic Pulse in a Bar of Berea Sandstone: A Detailed Look at the Mechanisms of Classical Nonlinearity, Hysteresis, and Nonequilibrium Dynamics

“…This result was later confirmed by an independent set of experiments where it was demonstrated that, given appropriate corrections to account for the fact that the pump and the probe are in orthogonal directions, the material softening obtained from DAET in the conditioning phase is equivalent to that obtained from NRUS (Lott et al, ). Last, Lott et al () demonstrated experimentally that there are additional subtleties to consider that are related to the type of strain field induced by the LF pump (e.g., axial motion versus torsional motion) and the type of waves used as HF probes (e.g., P waves and S waves with various polarizations).…”

Separation of Damping and Velocity Strain Dependencies using an Ultrasonic Monochromatic Excitation

“…Memory effects are peculiar of hysteretic elastic media [7,8,34,35] and consist in the following. When a sample is excited at a fixed (high) amplitude of excitation, transition to a nonequilibrium state is observed (conditioning), which is manifested by a not instantaneous change of both velocity and damping.…”