Explicit dispersion relation for strongly nonlinear flexural waves using the homotopy analysis method

Abstract

Using the homotopy analysis method, we present an explicit frequency-versus-wavenumber nonlinear dispersion relation for a flexural elastic beam. In our analysis, we employ the Euler–Bernoulli kinematic hypothesis and consider both a conventional transverse motion model and an inextensional planar motion model. As an example, we consider geometric nonlinearity in the form of Green–Lagrange strain. The underlying constitutive relation is formulated by linearly relating the second Piola–Kirchhoff stress to the Green–Lagrange strain, although the method is directly applicable to material nonlinearities as well. The derived analytical solution, for each model, is obtained to Mth-order accuracy and is verified by comparing with a numerical result brought about by laboriously finding the roots of the corresponding travelling-wave implicit relation governing the nonlinear elastodynamics. This is the first derivation of an explicit dispersion relation for an elastic beam undergoing strongly nonlinear finite flexural deformation. The derived relation characterizes the nature of a traveling cosine-like nonlinear wave throughout its stable pre-breaking state.

Appendices

Appendix A: Explicit first-order finite-strain dispersion relation for a flexural beam using the homotopy analysis method

From Sect. 4.1.2, we obtain the following explicit expressions for the first-order HAM nonlinear dispersion relation for a conventional beam:

$$\begin{aligned}&\omega _{{\text {HAM}}}^1(\kappa ;B)\nonumber \\&\quad =\sqrt{\frac{1-(B\kappa {})^2/4+ 3B^2/(8r_0^2)+3(1-(B\kappa )^2-15(B\kappa )^4/8)B^2\kappa ^4 r_f^2/(8r_0^2)}{1+r_0^2\kappa {}^2-r_0^2\kappa {}^4B^2(4+3 \kappa {}^2B^2)/4}} \kappa {}^2c_0r_0. \end{aligned}$$

(A.1)

Similarly, from Sect. 4.2.2 we obtain the following explicit expression for the first-order HAM nonlinear dispersion relation for an inextensional planar beam:

$$\begin{aligned}&\omega _{{\text {HAM}}}^1(\kappa ;B) \nonumber \\&\quad =\sqrt{\frac{1-B^2\kappa ^2 (-2+B^2\kappa ^2)/4+3(16+8B^2\kappa ^2-9B^4\kappa ^4)B^2\kappa ^4 r_f^2/(128r_0^2)}{1+\kappa ^2(8+B^2\kappa ^2-3B^4\kappa ^4)r_0^2/8}} \kappa {}^2c_0r_0, \end{aligned}$$

(A.2)

By taking the limit $\mathop {{\text {lim}}}\nolimits _{B \rightarrow 0}\omega _{{\text {HAM}}}^M(\kappa ;B)$, in Eq. (A.1) or Eq. (A.2), we obtain

$$\begin{aligned} \omega _{{\text {inf}}}(\kappa )=\frac{c_0r_0\kappa {}^2}{\sqrt{1+ r_0^2\kappa {}^2}}, \end{aligned}$$

(A.3)

which is the exact frequency-versus-wavenumber dispersion relation based on linear, infinitesimal deformation.

Appendix B: Explicit dispersion relation based on all $\xi =0$ conditions

An alternative approach for obtaining an explicit nonlinear dispersion relation is imposing a $\xi =0$ condition for the second derivative of ${\bar{v}}$ [39], similar to the choices made in Eq. (30) for the conventional beam model and Eq. (57) for the inextensional beam model for ${\bar{v}}(0)$ and ${\bar{v}}_{\xi }(0)$. In particular, we select ${\bar{v}}_{\xi \xi }(0)=-B\kappa$. Applying this condition—instead of the ${\bar{v}}(0)={\bar{v}}(2\pi )$ condition—to the inextensional beam implicit dispersion relation of Eq. (55) yields the following explicit nonlinear dispersion relation:

$$\begin{aligned} \omega = \sqrt{\frac{E J \kappa ^4}{\rho (A - A B^2 \kappa ^2 + J \kappa ^2)}} \end{aligned}$$

(B.1)

The nonlinear dispersion relation from the two approaches is compared in Fig. 9. It is seen that there is a slight deviation between the two approaches, particularly at high wavenumbers. The all $\xi =0$ approach has been numerically validated for a model of a thick rod with lateral inertia [39].

Fig. 9

Comparison of explicit nonlinear dispersion relation obtained using all $\xi =0$ conditions and the nonlinear dispersion relation obtained by the implicit approach of Ref. [13] where a spatial periodicity condition on the displacement gradient is assumed. These results are for the inextensional beam model