Nonlinear interaction of plane elastic waves

The paper presents basic first order results of nonlinear elastic theory by Murnaghan for elastic wave propagation in isotropic solids. The authors especially address the problem of resonant scattering of two collimated beams and present analytical solutions for amplitudes of all possible types of resonant interactions for elastic plane waves. For estimation of nonlinear scattered waves they use measured elastic parameters for sandstone. The most profound nonlinear effect is expected for interactions of two SH waves generating compressional P wave at sum frequency. Estimations show that nonlinear phenomena is likely to be observed in seismic data. Basic equations of nonlinear five-constant theory by Murnaghan are also presented

Time domain Solutions for Nonlinear Elastic 1-D Plane Wave Propagation

Time-domain solutions are obtained for 1-D nonlinear elastic wave propagation problems using a five-constant nonlinear theory. The assumption of weak attenuation was used throughout the development. The strongest nonlinear effects are obtained for the case of single compressional wave propagation, for single compressional or shear wave propagation through a longitudinally pre-stressed elastic material, and for shear wave propagation in a shear pre-stressed elastic material. Estimates of the size of these effects indicate that nonlinear phenomena are likely to be observable in real seismic data. The results may be useful for the measurement of nonlinear constants in elastic materials, for explaining the frequency content of seismograms, and for monitoring strain fields in the earth`s crust

Possible second-order nonlinear interactions of plane waves in an elastic solid

There exist ten possible nonlinear elastic wave interactions for an isotropic solid described by three constants of the third order. All other possible interactions out of 54 combinations (triplets) of interacting and resulting waves are prohibited, because of restrictions of various kinds. The considered waves include longitudinal and two shear waves polarized in the interacting plane and orthogonal to it. The amplitudes of scattered waves have simple analytical forms, which can be used for experimental setup and design. The analytic results are verified by comparison with numerical solutions of initial equations. Amplitude coefficients for all ten interactions are computed as functions of frequency for polyvinyl chloride, together with interaction and scattering angles. The nonlinear equation of motion is put into a general vector form and can be used for any coordinate system

Noncollinear wave mixing for measurement of dynamic processes in polymers: Physical ageing in thermoplastics and epoxy cure

Elastic wave mixing using an immersion method has shown effective monitoring and scanning capabilities when applied to thermoplastic ageing, epoxy curing, and non-destructive testing. In water, excitation and reception of waves do not require physical contact between the tools and the specimen, making the acquisition of high-resolution C-scans possible. The nonlinear material parameters exhibit a much higher sensitivity to the specimen state compared to linear ones. Thus, the nonlinear data for polymethyl methacrylate (PMMA) have a 40% difference between zones of “young” and “aged” material, while the linear data show no difference at all. Methodology and logistics of the immersion wave-mixing method are discussed in detail. Monitoring of epoxy curing has also revealed a good sensitivity of the method to this complex process including several characteristic stages, such as the time of maximal viscosity, the gel time, and the vitrification time. These stages are independently verified in separate rheometry measurements. The presented method allows for a number of possibilities: wave-mode and frequency separations, elimination of surrounding medium influence, “steering” (scanning) a scattered wave, controlling the location of the intersection volume, single-sided or double-sided measurements, and operation in detector mod

Isothermal epoxy-cure monitoring using nonlinear ultrasonics

Isothermal curing of LY 1564SP resin in an aluminium-adhesive-aluminium laminate is investigated, using a nonlinear ultrasonic immersion technique, to prove its applicability for this type of dynamic material transformation. For verification and comparison, epoxy-cure kinetics and rheological behavior are measured using differential scanning calorimetery (DSC) and dynamic mechanical analysis (DMA). Results reveal that the nonlinear ultrasonics, based on noncollinear wave mixing, can successfully be applied to in situ epoxy-cure monitoring—for example, to adhesive bonds—with reliable detection of gelation and vitrification time instant