Second Order Nonlinearities and their Application in NDE

Abstract

Although it has been recognized for quite some time that solid materials do not strictly obey Hooke’s law, linear elastic wave studies provided valuable tools to obtain both microscopic and macroscopic material parameters. Ultrasonic wave velocities can be used to obtain second order elastic constants while attenuation measurements can be related to many parameters, e.g. grain size in metals. It is quite recent that the NDE community turned its attention to investigate the possibility of using nonlinear acoustic techniques to measure properties of solids and interfaces [1–8]. The so-called nonlinearity parameter which is the combination of second and third order elastic constants has been extensively studied for crystalline solids [1] and has been related to microscopic behavior of the solid, e.g. a harmonic behavior of the interatomic potential as well as to macroscopic quantities e.g., thermal expansion, residual stress and material hardness. Basically, there are two approaches to obtain third order elastic constants and subsequently the so-called nonlinearity parameter from ultrasonic measurements. Either from the second harmonic generation [1] of a finite amplitude elastic wave or from the so-called acousto-elastic [9] effect, i.e. from stress dependent velocity data. An extensive list is available for the third order elastic constants in the literature [10]. There are several indications that higher than third order elastic constants could play a role in material characterization, e.g. fourth and fifth order elastic constants enter in the temperature dependence of velocity as functions of stress [11,12]. It has been suggested also that higher order nonlinearities may be more strength related in an adhesive joint than linear parameters [5]. There are practically no reported measurements on fourth order elastic constants available although theoretical considerations show that there are 4 fourth order elastic constants which describe an isotropic solid. It is not evident how harmonic generation could be developed for fourth order elastic constant measurement. On the other hand, the stress dependence of the velocity when carried out up to a second order offers several possibilities for measuring second order nonlinearities, and hence perhaps fourth order elastic constants. The sound velocity in a stressed solid may be expressed

$$C\left( \sigma \right) = C_0 + C_1 \sigma + C_2 \sigma ^2 + \cdots$$

((1))

where C₀ is the velocity in the unstressed material. C₁ is actually the first order a cousto-elastic constant and is a combination of the second and third order elastic constants from which first order nonlinearity parameters can be obtained. C₂ may be called a second order acoustoelastic constant and it is given as the combination of second, third, and fourth order elastic constants and related to second order nonlinearity parameters. Depending on the direction of σ with respect to wave propagation and polarization direction, both C₁ and C₂ have different forms. Methods to determine C₁ from acousto-elastic measurement are well documented in the literature. In order to obtain C₂ from Eq. 1 one can use extremely high shock compression stresses as was done for fused quartz and sapphire [13]. This approach however is clearly not applicable to many NDE applications. The possibility of measuring simultaneously both C₁ and C₂ at relatively low stress levels (below 15% of the yield) was reported last year [5] using a dynamic a cousto-elastic measurement technique. In this technique a high amplitude, low frequency external excitation provides stresses in an adhesive bond. A high frequency broadband shear wave transducer in a pulse-echo mode excites waves in the bond. The recorded velocities are measured as functions of first and second harmonic of the external load thus providing both stress and stress square dependence of the velocities. In this paper we are proposing two new measurement techniques to provide second order nonlinearity parameters. Both methods are based on the elimination of the first order term in the stress dependence of the velocity; therefore, any nonlinear effect measured will be due to a second (or higher) order. The first method eliminates the linear stress dependent term by utilizing the effect of “acoustic birefringence” at a proper polarization angle of the shear wave and we call it polarization technique. The second method is a modified a cousto-elastic measurement by applying a “pure” shear stress to the sample. It was pointed out before [5] that due to symmetry considerations the lowest order nonlinear terra appears only as the stress square term with C₂ as coefficient. This second method is referred to as “pure” shear stress technique.