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the TO amplitude dependence is hysteretic at low tem- peratures. If a sample is cooled at high oscillation ampli- tude, the apparent NCRI is small. When the amplitude is reduced at low temperature, the TO frequency (NCRI) rises, becoming constant below some critical amplitude. When the drive is then increased at low temperature, the NCRI does not begin to decrease at the critical am- plitude - it remains essentially constant at substantially larger drives. This hysteresis between data taken while decreasing and increasing the drive amplitude disappears at temperatures above about 70 mK. 

Figure 3 shows the corresponding hysteresis in the shear modulus. At 120 mK the maximum stiffening is about half as large as at 36 mK and already depends on amplitude at the lowest strains shown. The modu- lus measured when the amplitude is reduced (open cir- cles) and when it is subsequently increased (solid circles) agree. Hysteresis appears when the sample is cooled be- low 60 mK and is nearly temperature independent below 45 mK. Figure 3 shows this hysteresis at 36 mK. The sample was cooled from high temperature while driving at high amplitude (3 V$_{pp}$). The amplitude was then low- ered at 36 mK (open circles) which resulted in a shear modulus increase Δ µ / µ$_{o}$ of about 15%. When the am- plitude was then raised (solid circles), the modulus re- mained constant to much higher amplitude, the same behavior seen for the NCRI in TO experiments. At very high amplitudes (above about 1 V$_{pp}$, correspond- ing to ϵ ≈ 10 $^{$_{−}$6}$, σ ≈ 20 Pa) the modulus decreased, nearly closing the hysteresis loop. After each change we waited 2 minutes for the modulus to stabilize at the new amplitude. The only region where we observed fur- ther time dependence was while increasing the amplitude at drive levels above 1 V$_{pp}$. The modulus decrease was sharper when we waited longer at each point. In acoustic resonance measurements[17], we found that even larger stresses ( σ ≈ 700 Pa) produced irreversible changes which only disappeared after annealing above 0.5 K. 

The only obvious mechanism which can produce shear modulus changes as large as those shown in Figs. 1 to 3 involves the motion of dislocations[24]. At low tempera- tures, dislocations are pinned by $^{3}$He impurities and the intrinsic modulus is measured[17]. As the temperature is raised, $^{3}$He impurities thermally unbind from the disloca- tions, allowing them to move and reducing the modulus. At high amplitudes, elastic stresses can also tear the dis- locations away from the impurities. The hysteresis seen in Fig. 3 can be understood if, when a crystal is cooled at high strain amplitudes, the rapid motion of dislocations prevents $^{3}$He atoms from attaching to them. When the drive is reduced at low temperatures, impurities can bind, thus pinning the dislocations and increasing the modu- lus. Once the $^{3}$He impurities bind, the pinning length of dislocations is smaller and larger stresses are required to unpin them so the modulus retains its intrinsic value to much higher amplitudes. The critical amplitude for 

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this stress-induced breakaway can be estimated[25, 26] if the dislocation length and impurity binding energy are known. In single crystals of helium, a typical dislocation network pinning length[27] is (L ∼ 5 µ m), and disloca- tions would break away from a $^{3}$He impurity at strain ϵ ≈ 3x10 $^{$_{−}$7}$. Our crystals are expected to have higher dis- location densities and smaller network lengths, so break- away would occur at higher strains as the amplitude is increased. Measurements with different $^{3}$He concentra- tions would be useful to confirm that the amplitude de- pendence is due to this mechanism. 

This interpretation of the shear modulus behavior in- volves elastic stress (which is proportional to strain) rather than velocity, and so is at odds with the inter- pretation of the TO amplitude dependence in terms of a superfluid-like critical velocity. Since we can make mod- ulus measurements over a wide frequency range, we can unambiguously distinguish between an amplitude depen- dence which scales with stress (or strain) and one which depends on velocity. Figure 4 shows the modulus at 18 mK, measured at three different frequencies (2000, 200 and 20 Hz) as the drive amplitude was reduced from its maximum value. In Fig. 4a the modulus is plot- ted vs. shear strain ϵ (calibrated using the low tempera- ture piezoelectric coefficient of the shear stack, d 15 =1.25 nm/V) and in Fig. 4b the same data is plotted versus the corresponding velocities (v= ωϵ D). The amplitude depen- dence scales much better with strain than with velocity (and the scaling with acceleration is even less satisfac- tory). The critical strain appears be slightly larger at lower frequency. 

It is clear that the amplitude dependence of the shear modulus is most closely associated with stress (or strain) amplitude, rather than with a superfluid-like critical ve- 

