Nonlinear elasticity in rocks: A comprehensive three-dimensional description

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.

Figure 1

Schematic representation of the experimental setup.

Figure 2

Time histories of the LF pump and HF probe DAE data in the compressed stroboscopic time frame. (a) Time history of the strain ${\epsilon }_{11}$ when the sample is driven at the resonance frequency of its first longitudinal mode of vibration (LF pump). The LF pump is activated at time $t=0.5\mu \mathrm{s}$. (b) Time history of the RVM of the $P$ wave in the “$e_2$” direction (HF probe).

Figure 3

DAE data for the configuration where the LF pump is the first longitudinal mode of vibration of the sample and the HF probe is a $P$ wave. (a) RVM of the $P$ wave in the “$e_2$” direction as a function of strain for the largest drive amplitude of the pump. (b) Time history of the RVM of the $P$ wave in the “$e_2$” direction. (c) Time history of the strain induced by the LF pump at the axial position of the HF probe. Time histories are displayed in the compressed stroboscopic time frame: to obtain the three cycles depicted in (b) and (c), 300 cycles of real data were acquired. The blue and red colors indicate increasing and decreasing strain, respectively.

Figure 4

Normalized strain field for a torsional motion in polar (left) and Cartesian (right) coordinates.

Figure 5

DAE data for the configuration where the LF pump is the first torsional mode of vibration of the sample and the HF probe is a $P$ wave. (a) RVM of the $P$ wave in the “$e_2$” direction as a function of strain for the largest drive amplitude of the pump. (b) Time history of the RVM of the $P$ wave in the “$e_2$” direction. (c) Time history of the strain induced by the LF pump at the axial position of the HF probe. Time histories are displayed in the compressed stroboscopic time frame: to obtain the three cycles depicted in (b) and (c), 300 cycles of real data were acquired. The blue and red colors indicate increasing and decreasing strain, respectively.

Figure 6

Two-dimensional simulation of a $P$ wave propagating from an immersed transducer through the cylindrical sample. Two instants are depicted: soon after the wave penetrates the sample ($t=5\mu \mathrm{s})$ and soon before the wave reaches the receiving transducer ($t=15\mu \mathrm{s})$. The slowness surface of the $P$ wave in the strain field of a torsional mode and a schematic representation of the probed region are also depicted to understand the experimental observations.

Figure 7

Magnitude of the FFT of the RVM of the $P$ wave in the “$e_2$” direction as a function of frequency. (a) The LF pump is the first longitudinal mode of vibration so that $f_0\approx 3\mathrm{kHz}$. (b) The pump is the first torsional mode of vibration so that $f_0\approx 1.8\mathrm{kHz}$.

Figure 8

Amplitudes of the fundamental frequency ($f_0$), second ($2f_0$), fourth ($4f_0$), sixth ($6f_0$) harmonics as a function of the strain amplitude. The LF pump consists of the (a) longitudinal mode of vibration and (b) torsional mode of vibration. Amplitudes are shown in log scale, relative to their respective amplitude at the lowest strain value.

Figure 9

Schematic representation of the polarization angle of the HF $S$-wave probe.

Figure 10

DAE data for the configuration where the LF pump is the first longitudinal mode of vibration of the sample and the HF probe is an $S$ wave. (a) Time history of the RVM of the $S$ wave in the “$e_2$” direction. (b)–(d) RVM of the $S$ wave in the “$e_2$” direction as a function of strain for the largest drive amplitude of the pump at polarization angles of the probe of $\theta =0^{\circ }, {40}^{\circ }$, and ${90}^{\circ }$. (e) Polar plot (function of the polarization angle of the probe) of the measured average value of the RVM (conditioning) around times $t_1$ and $t_2$. (f) Polar plot of the predicted instantaneous value of the RVM (conditioning) at the largest strain amplitude $A_L$.

Figure 11

DAE data for the configuration where the LF pump is the first longitudinal mode of vibration of the sample and the HF probe is an $S$ wave. (a) Time history of the RVM of the $S$ wave in the “$e_2$” direction. (b)–(d) RVM of the $S$ wave in the “$e_2$” direction as a function of strain for the largest drive amplitude of the pump at polarization angles of the probe of $\theta =0^{\circ }, {40}^{\circ }$, and ${90}^{\circ }$. (e) Polar plot (function of the polarization angle of the probe) of the measured average value of the RVM (conditioning) around times $t_1$ and $t_2$. (f) Polar plot of the predicted instantaneous value of the RVM (conditioning) at the largest strain amplitudes $A_T$, at the radial positions $r=-R$ and $R$.