4 not reflected in the corresponding TO behavior[18]. The existence of a critical velocity for superflow in solid he- lium can only be definitively shown if TO measurements can be made over a wide range of frequency and/or TO geometries. This work was supported by the Natural Sciences and Engineering Research Council of Canada. FIG. 4: Scaling of the shear modulus amplitude dependence with (a) strain and (b) velocity for three different frequencies: 20 Hz (triangles), 200 Hz (squares) and 2000 Hz (circles). locity. The many similarities to the TO behavior (e.g. the dependence on temperature, $^{3}$He concentration and frequency, the amplitude dependence and its hysteresis) suggest that the apparent velocity dependence of the NCRI may have a similar origin, e.g. in inertial stresses which exceed the critical value for the shear modulus. However, estimates of the inertial stress corresponding to TO critical velocities give values significantly lower than the critical stress for the shear modulus. For an an- nular TO geometry, the maximum inertial stress can be estimated as σ = ρ t ω v/2, where ρ is the helium density, t is the width of the annular channel, ω is the angular frequency of the TO and v is the oscillation velocity. For the oscillator of ref. 1, we estimate σ ≈ 0.002 to 0.015 Pa at the apparent critical velocities of 5 to 38 µ m/s. Measurements in an open cylindrical TO[23] show am- plitude dependence at velocities corresponding to iner- tial stresses below 0.01 Pa, also much smaller than the stresses at which we observe amplitude dependence in the shear modulus. There is also other evidence that the TO frequency changes and dissipation are not just me- chanical consequences of the modulus changes. First, the apparent NCRI is too large to be explained simply by mechanical stiffening of the torsional oscillator[18, 23]. Secondly, comparable modulus changes in hcp $^{3}$He are [1] E. Kim and M. H. W. Chan, Science 305 , 1941 (2004). [2] E. Kim and M. H. W. Chan, Phys. Rev. Lett. 97 , 115302 (2006). [3] A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 98 , 175302 (2007). [4] M. Kondo, S. Takada, Y. Shibayama, and K. Shirahama, J. Low Temp. Phys. 148 , 695 (2007). [5] A. Penzev, Y. Yasuta, and M. Kubota, J. Low Temp. Phys. 148 , 677 (2007). [6] Y. Aoki, J. Graves, and H. Kojima, Phys. Rev. Lett. 99 , 015301 (2007). [7] M. Keiderling, Y. Aoki, and H. Kojima, J. Phys. Conf. Ser. 150 , 032040 (2009). [8] B. Hunt, E. Pratt, V. Gadagkar, M. Yamashita, A. V. Balatsky, and J. C. Davis, Science 324 , 632 (2009). [9] X. Lin, A. Clark, and M. H. W. Chan, Nature 449 , 1025 (2007). [10] X. Lin, A. C. Clark, Z. G. Cheng, and M. H. W. Chan, Phys. Rev. Lett. 102 , 125302 (2009). [11] J. Day and J. Beamish, Phys. Rev. Lett. 96 , 105304 (2006). [12] S. Sasaki, R. Ishiguru, F. Caupin, H. J. Maris, and S. Bal- ibar, Science 313 , 1098 (2006). [13] S. Sasaki, F. Caupin, and S. Balibar, Phys. Rev. Lett. 99 , 205302 (2007). [14] M. W. Ray and R. W. Hallock, Phys. Rev. Lett. 100 , 235301 (2008). [15] M. W. Ray and R. W. Hallock, Phys. Rev. B 79 , 224302 (2009). [16] J. Day and J. Beamish, Nature 450 , 853 (2007). [17] J. Day, O. Syshchenko, and J. Beamish, Phys. Rev. B 79 , 214524 (2009). [18] J. T. West, O. Syshchenko, J. Beamish, and M. H. W. Chan, Nature Physics 5 , 598 (2009). [19] A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 101 , 155301 (2008). [20] Kojima, H. (private communication). [21] Http://www.piceramic.com/ Model 141-03. [22] N. Shimizu, Y. Yasuta, and M. Kubota, arXiv:cond- mat/09031326 (2009). [23] A. C. Clark and M. H. W. Chan, Phys. Rev. B 77 , 184513 (2008). [24] A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids (Academic Press, New York, 1972). [25] I. Iwasa and H. Suzuki, J. Phys. Soc. Japan 49 , 1722 (1980). [26] M. A. Paalanen, D. J. Bishop, and H. W. Dail, Phys. Rev. Lett. 46 , 664 (1981). [27] I. Iwasa and H. Suzuki, J. Phys. Soc. Japan 46 , 1119 (1979).