Page 34 - Summer2019
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Peculiar Acoustics of Rocks
10 relevant to most acoustic measurements. To clarify discus-
Fontainebleau sandstone sions about rocks, the rate-dependent effects were dubbed
5 ,“.\ “slow dynamics” i.n the geophysics literature quite a while ago
C ‘ ‘ ’-‘.~\\ and the name stuck (TenCate and Shankland, 1996).
"E 5 i  . . . . .
3 f  For the acoustician then, questions remai.n. Are nonlinear,
9 i T  hysteretic, and slow-dynamic effects large i.n rocks? Can they
.2 , ‘ t , \ _ _ .
2 4 y , V I \ be seen in wave propagation down a long thin bar or maybe
i i  i.n the acoustic waves sent across a block of sandstone? What
2 :»/’,/.  i ,/ ‘ 1,’ iii y,/J /’\ §e about nonlinear wave interactions in rocks; do they occur
, .‘-/ ////  I 7/ ’  /"/,“J’/‘y too? These kinds of questions have been the focus of roughly
25 years of research of a group of researchers and colleagues
0 '7 4 V 7 7 V V V at the Los Alamos National Laboratory and many others
1020 1040 1060, ‘PO80 1 100 1 120 114° 1 160 elsewhere. The acoustical experiments to date fall i.nto two
Dnvlng Frequency (Hz) dmind tyPes_
Figure 4. A series afresonance curves at increasing amplitudes for the
lowest resonance frequency ofa long thin bar afFontainelzleau sandstone. Two Kinds of Experiments
The driving frequency is swept upward (red) and downward (blue), Resonance
and the response of the sample is measured with a laser vibrameter The first type of experiment, resonance measurements on
and converted to strain. Note the softening nonlinearity (resonance long thin bars, is occasionally used to characterize the
frequency drops) with increasing excitation amplitude and the difierent mechanical properties of rock cores. Long thin cores have
up—ani1—down curves that result fram slaw dynamics. geometries that make it easy to find and precisely locate
  resonance frequencies. These are useful to characterize the
elastic moduli and intrinsic attenuation of the core.
done complicates the behavior of a rock much more. A rock is
not only nonlinear; there is hysteresis and there is some kind However, with increasing excitation amplitudes, some-
ofti_me-dependent behavior present as well. thing surprising happens. A series of resonance curves
taken around the lowest resonance frequency, 1-D wave
The measurements described above are quasi-static and at motion that in a solid resembles a snake swallowing, takes
fairly large stresses and strains. What about at lower strains on peculiar shapes with increasing drive amplitude. Figure
and higher frequencies? Does hysteresis still persist, can it be 4 shows a family of up-a.nd-down sweeps of frequency
measured, and does it really matter for acoustics? Lest you versus amplitude for yet another sedimentary rock, this one
think this hysteretic behavior isn’t present in wave propaga- from Fontainebleau, south of Paris, France. The resonance
tion, it is. At much lower strai.ns and at higher frequencies, frequency drops, and the upward and downward curves
those typical for seismic waves, you see “cuspy" triangu- are not the same. These curves resemble those of soften-
lar waveforms, also indicating the presence of hysteresis ing nonlinear spring-mass systems, such as those whose
(Mcl(avanagh and Stacey, 1974). Some of these waveforms motion is described by a Duffing equation. The springs
are seen at even higher frequencies and are shown in Wave in a Dufling equation do not obey (linear) Hooke’s law
Propagation. Hysteresis is present at many frequencies and and have a cubic nonlinear spring constant. Upward a.nd
amplitudes of i.nterest to the acoustician. downward sweeps in frequency (versus amplitude) produce
different resonance curves. Even the jump in the upward
Perhaps it should not be surprising that the hysteresis sweeps resembles a Dufling oscillator spring-mass system.
described above and seen i.n Figure 3 was also found to be In fact, a great number of early studies went into trying to
time dependent. There is a term for this behavior, elastic fit the resonance curves of rocks with the Dufling models.
afte1eflecI(e.g., see Becker, 1925), and it has an analogy with However, although a nonlinear spring-mass description
magnetic materials where the effect takes place over years works quite well down at very low strains, where nonlin-
instead of days. Elastic afteretfect is present in rocks, and it earity dominates, above a certain drive level, slow dynamics
fortunately (or unfortunately) manifests itself on timescales matters more.
:2 | AA:auIIiI:l‘I'b:Iay| Summer zoi9

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