Finite element studies to understand reflection of nonlinear waves in a continuous functionally graded material

Abstract

The nonlinear pulse-echo ultrasonics technique is effective when only one side of pressure vessels mainly used in thermal and nuclear power plants is assessable. As another side is exposed to continuous thermal and nuclear damages, the early-stage damages accumulate as a function of position. Concluding experimental results from such a complex interaction of reflected nonlinear waves during pulse-echo testing is difficult. Obtaining analytical solutions to the reflected waves from a functionally graded nonlinear region with an extended linear part is challenging. This complexity in understanding is resolved through various computational studies presented in this article. The linear and exponential variation of nonlinear parameters, loss in elastic stiffness, and density of the damaged region are modeled by updating linear and nonlinear constitutive equations. The results demonstrate higher energy of reflected longitudinal harmonic waves (2f, 3f, 4f,….) from a fixed boundary than a free boundary. Amplitudes of higher harmonic responses as a function of the position are verified. Harmonically reflected higher harmonics are more sensitive to the spatial distribution of nonlinear parameters than the loss in linear elastic stiffness, whereas a significant change in the overall shapes of the pulses in time domains is observed. Self-wave mixing increases the number of higher harmonics.