Abstract
Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain. The method of wave splitting is employed to formulate a standard scattering problem, forming the mathematical basis for both direct and inverse problems. A quasi‐linear version of the Wendroff scheme (FDTD) is used to solve the direct problem. To solve the inverse problem, an asymptotic expansion is used for the wave field; this linearizes the order equations, allowing the use of standard numerical techniques. Analysis and numerical results are presented for three model inverse problems: (i) recovery of the nonlinear parameter in the stress‐strain relation for a homogeneous elastic rod, (ii) recovery of the cross‐sectional area for a homogeneous elastic rod, (iii) recovery of the elastic modulus for an inhomogeneous elastic rod.
Highlights
Wave propagation in nonlinear elastic and viscoelastic materials has been an area of interest for some time
These efforts fall into two classes: the modeling of the stress-strain relation, usually in conjunction with curve fitting to experimental data, and the modeling of wave propagation in nonlinear media, both in the time domain and in the frequency domain
Elastic wave propagation in weakly nonlinear elastic rods is considered in the time domain
Summary
Wave propagation in nonlinear elastic and viscoelastic materials has been an area of interest for some time. There has been an extensive amount of work done on the mathematical modeling of such materials (e.g., [2, 10, 17, 19, 20, 23]), as well as on the analysis of the behavior of specific materials (e.g., [11, 21, 26]) These efforts fall into two classes: the modeling of the stress-strain relation, usually in conjunction with curve fitting to experimental data, and the modeling of wave propagation in nonlinear media, both in the time domain and in the frequency domain. It is convenient to begin the study of wave propagation in nonlinear elastic media with the study of one-dimensional rods of finite length d, as is done here This makes the wave splitting analysis tractable, and still provides insight into meaningful applications
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