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Nonlinear ultrasonic measurements enable the detection of the onset of plastic deformation and fatigue damage at an earlier stage than conventional linear nondestructive testing (NDT) techniques, which have insufficient sensitivity to the changes in the microstructure brought on by dislocation movements. Finite-deformation elastic theory introduces three independent constants, referred to as third order elastic constants (TOECs), which describe the nonlinear stress-strain behavior in an isotropic material.^1,2 Different sets of independent TOECs have been proposed by various authors, including A, B, and C used by Landau and Lifshitz,¹ which are a linear combination of the l, m, and n Murnaghan constants.²

Of practical interest is the dependence of TOECs on the level of plastic strain or fatigue damage induced dislocation accumulation in a material. Various ultrasonic methods of measuring material nonlinearity have been developed. The first makes use of the so-called acousto-elastic effect.^3,4 In this case the nonlinear behavior manifests itself through variations in ultrasonic propagation velocity with applied strain. Through the application of different wave types and the measurement of velocity in unstrained and strained states all three TOECs can be measured. One problem with this technique is the difficulty of measuring the small changes in propagation time and distance accurately enough to allow the velocity, and from that the TOECs, to be determined. A second problem is the necessity of loading a specimen to measure the changes in velocity.

The second and perhaps most widely reported method for interrogating material nonlinearity is the harmonic generation technique.^5,6,7,8 If ultrasonic energy at one frequency is injected into a material, harmonics of the input frequency are generated due to nonlinearity as the ultrasound propagates. By measuring the magnitude of the harmonics the degree of material nonlinearity can be quantified. There is a considerable body of experimental evidence that shows a strong correlation between the normalized harmonic amplitude and the amount of fatigue damage⁶ or plastic deformation⁷ in a material. The major measurement difficulty with the harmonic generation method as a NDT technique lies in isolating the causes of nonlinearity. Specifically, amplifiers, transducers, and coupling methods are all contributors to the measured harmonic, often on a scale greater than the material nonlinearity itself. Thus it is practically very difficult to determine if the measured nonlinearity is due to the material or the equipment.

A third technique, which is the main subject of this paper, for TOEC measurement was first proposed by Jones and Kobett,⁹ and experimentally observed by Rollins.¹⁰ This approach is based on the fact that material nonlinearities cause interaction between two intersecting ultrasonic waves.¹¹ Under certain circumstances, this can lead to the generation of a third wave with a frequency and wavevector equal to the sum of the incident wave frequencies and wavevectors, respectively. Theoretically, there are several incident wave combinations that can achieve this; however, practical material constraints to the theory lead to the interaction of two shear waves generating a longitudinal wave as the most useful case.

The non-collinear mixing technique has two important advantages over the conventional nonlinear ultrasonic harmonic generation technique. First, it is much less sensitive to system nonlinearities due to spatial selectivity (the nonlinear interaction is limited to the region where the incident beams intersect), modal selectivity (the nonlinear mixing signal is a different mode to the incident waves), frequency selectivity (the mixing signal frequency can be separated from harmonics of the incident waves if the driving frequencies are chosen to be unequal), and directional selectivity (the mixing signal propagates in a different direction form the mixed ones and their higher harmonics). Second, unlike the harmonic generation techniques, the level of the underlying system nonlinearity can be measured directly by summing the responses to each of the incident waves excited separately, that is, without the interaction present.

It is important to note that the evidence of correlation between material degradation (e.g., fatigue or plasticity) and nonlinear ultrasonic phenomena that has been reported is based mainly on evidence from the harmonic generation technique. In this configuration, only longitudinal waves can be used, and the harmonic amplitude is a function of all three TOECs (A, B, and C) or alternatively two of Murnaghan's three TOECs (l and m). However, the non-collinear technique based on the interaction of two shear waves to produce a longitudinal wave was shown by Jones and Kobett⁹ and Taylor and Rollins¹¹ to lead to a longitudinal wave amplitude that depends only on TOECs A and B (or the m and n Murnaghan TOECs).

What has not been studied to date is whether the particular combination of the two TOECs probed by the non-collinear technique is sensitive to fatigue and plasticity, and therefore whether the non-collinear technique can be used for NDT of fatigue damage. The purpose of this letter is to demonstrate that the non-collinear mixing technique can indeed detect changes due to plasticity and fatigue damage, and therefore has the potential to be used as a NDT technique.