Elastic wave dispersion equation considering material and geometric nonlinearities

In this letter, we describe the propagation of longitudinal waves in one-dimensional nonlinear elastic thin rods with material nonlinearity and geometric nonlinearity. Mathematical analysis is used to derive the analytical dispersion relationship of longitudinal waves; subsequently, the numerical Fourier spectrum method is used to solve the nonlinear wave equation directly and the results are used to verify the correctness of the derived nonlinear dispersion relation.


Introduction
Wave motion is a ubiquitous phenomenon in nature as evidenced by sound waves, water waves, electromagnetic waves, waves in solids, and even chemical reactions, life process, population migration, and traffic flows. Understanding the governing principles and nature of wave propagation is of great significance for controlling wave propagation and benefiting human society 1 . Because of this, research on wave motion has attracted researchers from across a variety of fields.
The theory of linear dispersion waves is almost complete, but the nonlinear behavior has not yet been fully characterized.
In the study of waves, the dispersion relationship can provide valuable information about the dynamics of the propagation medium 2 . Most of the early studies on elastic wave dispersion were based on linear governing equations but now, nonlinear effects are deemed more important because they can describe the potential motion of waves more accurately. The first nonlinear wave theory was studied by Stokes (1847). However, because the study of nonlinear waves involves very complicated mathematical models, the progress of research is very slow. In this regard, G. B. Whitham systematically introduced different wave phenomena in physics and mechanics in his work, especially the problem of wave propagation in solids and fluids 2 . P. L. Bhatnager (1979) introduced the interaction between shock waves, solitary waves in a simple one-dimensional dispersion system in a monograph 3  rods. In recent years, scholars have mainly studied nonlinear elastic waves in phononic crystals or artificial metamaterial [10][11] [12][13] [14] .
However, research that considers both geometric nonlinearity and material nonlinearity in the nonlinear dispersion relationship is not available; in actual engineering, there are indeed two nonlinearities in a material, such as in the case of rubbers or in the case of explosions and high-speed impact, where two nonlinear factors must be considered simultaneously. In this study, we consider thin rods (ignoring lateral inertial effects) with two simultaneous nonlinearities: geometric nonlinearity and material nonlinearity and study the influence of these nonlinear factors on the dispersion relationship. Initially, the analytical method is used to derive the nonlinear dispersion relationship by considering the linear factors (lateral inertial effects) and nonlinear factors (geometric nonlinearities).
Subsequently, we ignore the linear factor and concentrate on the influence of the nonlinear factors on the wave displacement gradient solution. From three different instants, we can see that the nonlinear factor makes the wavefront steeper. Using sinusoidal excitation as the initial condition, the nonlinear dispersion relation with a specific excitation wavenumber is derived; subsequently, the nonlinear wave equation with initial excitation is solved directly, and finally, the correctness of the deduced dispersion relation is verified by wavenumber-frequency spectrum analysis.

Dispersion relationship including material nonlinearity and geometric nonlinearity
Consider an infinitely long one-dimensional rod with a radius r , where unidirectional axial stress ( , ) is the system kinetic energy density and 1 2 U  = is the elastic potential energy density of the system.
Here  and  are the mass density and Poisson's ratio, respectively, and the uniaxial force is a nonlinear relationship [9] and Hencky strain (HS): The Lagrangian density function is substituted into the following Euler-Lagrangian equation, namely: Take the derivative with respect to x on both sides of the above equation and set u u x  =  to obtain: To facilitate the description of the dispersion, the transformation kx t  =−is introduced, where k is the wave number and is the frequency of the traveling wave. Equation (4) can be written as: We integrate the two ends of equation (5) twice and set the integral constant term to 0 to ensure that the traveling wave solution of the initial bounded displacement field is bounded at infinity before solving the equation. After replacing the above variables, the initial displacement field [15] is a sinusoidal If the amplitude B of the wave and the initial excitation of the wave number e k are given, the initial conditions become: Then, we get the nonlinear dispersion with a single harmonic excitation: 4

Numerical verification
The Fourier spectrum method is used to solve equation (4) and the precise nonlinear dispersion relationship (10) and (11)  Three different snapshots of the simulated motion are shown in Fig. 1(a).
Performing Fourier analysis in space on the wave function at the time of excitation (t0) and two timesteps later (t1 and t2) shows the evolution of the energy spectrum, K. At time t0, only a single harmonic exists, which is the cosine excitation signal. As the wave evolves, the nonlinear effects increasingly cause distortion and generate higher harmonics, as shown in Fig.1(b). The WFS analysis results corresponding to this excitation is also presented in Fig.1(c). The frequency wavenumber spectrum analysis is performed, and the linear dispersion

Conclusion
In this letter, we describe how we considered material and geometric nonlinearities in a long, thin rod simultaneously. Based on the given material nonlinearity, we deduced the nonlinear longitudinal wave equations under two different geometric nonlinearities, GLS and HS, respectively, and used analytical mathematical methods to derive the nonlinear dispersion relationship. Finally, the obtained dispersion relationship was verified using a numerical simulation.