The elastic constants of a natural-quartz sphere using resonance-ultrasound spectroscopy (RUS) are measured. The measurements of the near-traction-free vibrational frequencies of the sphere are matched with the predicted frequencies from the dynamic theory of elasticity, with optimized estimates for the elastic constants driving the differences between these sets of frequencies to a minimal value. The present computational model, although based on earlier approaches, is the first application of RUS to trigonal-symmetry spheres. Quartz shows six independent elastic constants, and our estimates of these constants are close to those computed by other means. Except for C14, after a 1% mass-density correction, natural quartz and cultured quartz show the same elastic constants. Natural quartz shows higher internal frictions.
A noncontacting resonant-ultrasound-spectroscopy (RUS) method for measuring elastic constants and internal friction of conducting materials is described, and applied to monocrystalline copper. This method is called electromagnetic acoustic resonance (EMAR). Contactless acoustic coupling is achieved by energy transduction between the electromagnetic field and the ultrasonic vibrations. A solenoidal coil and static magnetic field induce Lorentz forces on specimen surfaces without using a coupling agent. By changing the field direction, a particular set of vibration modes can be selectively excited and detected, an advantage in identifying the vibration modes of the observed resonance peaks. Contactless coupling allows the measure of intrinsic internal friction free from energy loss associated with contact coupling. The elastic constants and internal friction measured by EMAR are compared with those by the usual RUS method for a rectangular-parallelepiped copper monocrystal. Both methods yielded the same elastic constants despite fewer resonant peaks in the EMAR case. The two methods gave essentially the same shear-mode internal friction, but the RUS method gave higher volume-mode internal friction.
Nine elastic stiffness coefficients, c ij , of a mullite single crystal (2Al 2 O 3 ⅐SiO 2 ) are measured using acoustic resonance spectroscopy. The obtained values are similar to those of the structurally related aluminosilicate phase sillimanite (Al 2 O 3 ⅐SiO 2 ). Characteristic elastic properties of the two minerals are interpreted with the help of their crystal structures and atomic force constants for sillimanite. The high longitudinal stiffness coefficients, c 33, of mullite (ϳ352 GPa) and sillimanite (ϳ388 GPa) are caused by continuous "stiff" load-bearing tetrahedral chains parallel to c-axis, while the "soft" octahedral chains have minor direct influence. They stabilize the tetrahedal chains against tilting. The lower c 33 value of mullite in comparison to the sillimanite value may be caused by a weakening of the load-bearing tetrahedral chains which are parallel to c-axis because of partial replacement of silicon by the weaker-bonded aluminum. The longitudinal stiffness coefficients perpendicular to c-axis are significantly lower, because of sequences of alternating "soft" octahedral and "stiff" tetrahedral units. Within the plane (001), the compliant octahedra exhibit stiffness-controlling influence with coefficients parallel to b-axis (c 22 Ϸ 233 GPa) being somewhat lower than parallel to a-axis (c 11 Ϸ 291 GPa). This is explained with the occurrence of compliant octahedral Al(1)-O(D) bonds, which are more effective parallel to b-axis rather than to a-axis. Because octahedra are unaffected by the aluminum to silicon substitution, c 11 and c 22 coefficients of mullite and sillimanite are very similar. Shear stiffness coefficients of mullite increase from c 55 (ϳ77 GPa) to c 66 (ϳ80 GPa) to c 44 (ϳ110 GPa), indicating increasing resistance against shear deformation within the planes (010), (001), and (100). The lattice plane of the highest shear stiffness (100) is built up of an oxygen-oxygen network, diagonally braced along ͗011͘ ("Jägerzaun"). This network with short oxygen-oxygen distances can be sheared by compression and elongation along oxygen-oxygen interaction lines only which is rather unlikely. Because of the lack of such networks in the planes (010) and (001), bending and deformation of structural units become easier, and consequently c 55 and c 66 are
Using ultrasonic methods, we determined the ambienttemperature elastic constants of dense (99.7%) hot-pressed polycrystalline 3:2 mullite (3Al 2 O 3 ؒ2SiO 2 ). We report the usual polycrystal elastic constants: Young's, shear, and bulk moduli, and the Poisson's ratio (). The Poisson's ratio ( = 0.280) suggests interatomic bonding that is very different from that of either alumina or silica. Temperature dependences of the elastic constants show a strong irregularity; for example, decreases as the temperature increases. This irregularity suggests an internal-state change that is perhaps related to the incommensurate structural modulation.
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