# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 3 Numerical Examples
### Wire-Reinforced Hard Rubber Pipe under Internal Pressure  
The cylinder analysed in this section is a complex, hard rubber pipe reinforced by a helical steel wire. The procedure presented in [32; 33] is here applied to take into account the helical pattern in the WFE formulation. The ratio between the thickness and the external radius of the pipe is \(h/R_{ext}=0.25\). The lay-up angle of the helix is \(\phi=20^{\circ}\). A schematic representation of the pipe structure and of the small segment taken for WFE analysis are shown in Figure 6. Both sides of the  
Figure 5: Air-filled sandwich cylinder, \(n=0\) and \(n=1\). (**a**) Fluid–filled and (**b**) empty:.... complex valued wavenumbers;.... pure real and pure imaginary wavenumbers; (**c**,**d**).... complex valued wavenumbers in the coupled system,.... pure real and pure imaginary wavenumbers in the coupled system,.... fluid in a hard–walled duct,.... fluid in a soft–walled duct. steel wire-reinforcement, denoted by 4 in Figure 6b, are filled with rubber. Three different hard rubber materials, numbered from 1 to 3, are used as depicted in Figure 6b. This type of cylindrical construction is used for applications such as pipes for oil transport or aircraft refuelling hoses. The specific construction might vary depending on the specific application, but typically strengthened wires are twisted around an inner synthetic rubber tube and then covered by another synthetic rubber layer. It is difficult to predict how these pipes behave mechanically, and typically several pipe specimens are experimentally tested. Although some mathematical models for these constructions have been studied, e.g., [34], developing numerical or analytical models is very difficult. Material characteristics are given in Table 1. The size of the segment modelled for this example is: \(L_{y}~{}=~{}8\)mm and \(\alpha~{}=~{}1^{\circ}\). The FE model of the segment in Figure 6b is realised using 30 solid elements of the type SOLID45 in ANSYS. In the following results, the wavenumber is nondimensional with respect to the internal radius of the pipe while the frequency is nondimensional with respect to \(\Omega_{ring}\) calculated for \(E\), \(\rho\) and \(\
u\) given by the values of material 3 in Table 1.  
Real-valued dispersion curves for circumferential orders \(n=0\), \(n=1\) are shown in Figure 7. Dispersion curves obtained for the same pipe with \(\phi=0^{\circ}\) and with no reinforcement are shown for comparison. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material 1** & **Material 2** & **Material 3** & **Material 4** \\ \hline \(E=10^{7}\) Pa & \(E=10^{8}\) Pa & \(E=10^{9}\) Pa & \(E=19.2^{10}\) Pa \\ \(\
u=0.49\) & \(\
u=0.3\) \\ \(\rho=1000\) kg/m\({}^{3}\) & \(\rho=1100\) kg/m\({}^{3}\) & \(\rho=1100\) kg/m\({}^{3}\) & \(\rho=7800\) kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of the wire-reinforced hard rubber pipe. Figure 6: **(a)** Schematic representation of the wire–reinforced pipe; **(b)** FE discretisation of a small skew segment with material properties numbered as in Table 1; **(c)** WFE super–element; **(d)** rotation of the local coordinates. Dispersion curves and their analysis can be used to improve the performance of these structures at a design stage. As an example, an increase in the slope of the dispersion curves generally implies increasing rigidity of the structure. The same consideration generally holds for increasing values of the cut-off frequency. Branches in the dispersion curves represent waves which are in general predominantly flexural, shear or extensional in nature. With reference to the global Cartesian coordinates in Figure 6, flexural waves involve motion primarily in the \(z\) direction while shear and extensional waves involve motion mostly in the \(x\) and \(y\) directions. However, due to the complicated construction of the pipe, the dispersive behaviour in Figure 7 cannot be described simply in terms of these torsional, extensional and flexural waves alone.