Nonlinear Ultrasonic and Vibro-Acoustical Techniques for Nondestructive Evaluation by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9783319944760
Publisher: Springer International Publishing

Specifically, Fig. 9.31a plots the relative change in wave speed , as a function of interatomic distance r*. The temperature T0 is the stress-free value (corresponding to equilibrium distance r0 * = 4.489 Angstroms—minimum of the assumed Lennard-Jones potential). The independent variable r* represents the (prevented) thermal expansion rABD(T) for the fully constrained case, or the (partially prevented) thermal expansion rABD-PC(T) for the partially constrained case. The trend in Fig. 9.31a clearly indicates that the wave speed decreases with increasing (prevented) thermal expansion. This indicates a “softening” effect, consistently with the decrease in curvature of the interatomic potential, C(T), when moving slightly to the right of the equilibrium position r0 * (see, for example, Fig. 9.27).

Figure 9.31b plots the same velocity change as a direct function of the temperature change, ΔT = T–T0. This plot was obtained from the previous values in Fig. 9.31a by simply using the linear thermal expansion relation in Eq. (9.149), and assuming a thermal expansion coefficient for steel of α = 11 × 10−6/°C and a fully constrained case. The range of temperature excursion considered in Fig. 9.31b was arbitrarily chosen as 100 °C. For the case considered, for example, the longitudinal wave velocity is expected to decrease by about 1% for a temperature increase of 100 °C in the fully constrained solid.

The parameter of Eq. (9.162) contains the nonlinear portion of the potential through the cubic term O(r3), D(T). As mentioned above, the difference from classical nonlinear wave theory is that the cubic O(r3) energy term arises from the prevented thermal expansion due to the asymmetry of the interatomic potential, rather than from applied finite deformations. The nonlinear parameter is discussed more in depth in the next section.

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