The effect of cubic material nonlinearity on the propagation of torsional wave modes in a pipe

Abstract

The effect of cubic material nonlinearity on the propagation in a pipe of the lowest axially symmetric torsional wave mode has been investigated in this paper. Two cases, one that the material of the whole pipe is nonlinear, and the second that a small segment of the pipe is nonlinear, have been considered. For the first case, a first and a third harmonic have been obtained by the perturbation method. Analytical expressions for the two cumulative harmonics have been derived. The second case leads to a scattering problem. The segment produces nonlinear terms in the equation of motion, which can be regarded as a distribution of body forces. The problem is then reduced to a linear scattering problem. An analytical expression for the backscattered wave can be easily obtained by using the elastodynamic reciprocity theorem. Due to the low amplitude of the backscattered wave, the authors propose to add another higher frequency wave to the primary wave, to increase the total magnitude of the scattered wave. An example that the originally scattered wave is amplified 50 times by selecting proper frequencies is presented. Both cases considered here have a potential application to determine the material properties in a region of nonlinear material behavior.

On the harmonic scattering amplitudes from a nonlinear elastic spherical inclusion

“…The secondary field generated from the region of nonlinearity contained first-, second-, and third-harmonic components. The second-harmonic component was found to be a result of the quadratic nonlinearity while the first-and third-harmonic components were generated by the effects of cubic nonlinearity [3], which parallels the finding of Wang and Achenbach [2]. This paper considers the interaction of an incident longitudinal wave with a spherical region of constant quadratic and cubic elastic nonlinearity.…”

“…Recently, a number of investigators have considered the interaction of an elastic wave with a localized region of elastic nonlinearity [1,2,3]. Tang et al modeled the interaction of an incident longitudinal wave with a region with spatially-dependent quadratic nonlinearity, which was contained in a linear elastic host medium [1].…”

“…Wang and Achenbach modeled the interaction of an incident torsional guided wave on a local annular region of cubic nonlinearity [2]. Solutions based on reciprocity found backward and forward propagating waves from the region of nonlinearity, which contained first-and third-harmonic components [2]. Kube then considered the interaction of an incident longitudinal or shear wave with a region of spatially-dependent and generally anisotropic quadratic and cubic nonlinearity encompassed in an otherwise linear host medium [3].…”

SH wave generated Lamb wave in a plate with a localized region of material nonlinearity

“…The problem stated in Fig. 1 can be solved by using the reciprocity theorem in an elegant manner [2,3]. The virtual wave can be selected as the free Lamb wave propagating in the plate without material nonlinearity, which is denoted by the state B.…”

Ultrasonic harmonic generation from materials with up to cubic nonlinearity

“…Recently, a number of investigators have turned to exploiting the effects of cubic material nonlinearity in an effort to measure material degradation of solids using nonlinear ultrasonics. [1][2][3][4] While attention has been primarily given to third-harmonic generation, Wang and Achenbach 3,4 unveiled a significant increase to the fundamental harmonic amplitude because of the combinations of quadratic and cubic nonlinearity. The possibility of extracting material nonlinearity information from the fundamental wave for applications in nondestructive evaluation (NDE) is especially promising.…”