# Wave Propagation in Laminated Cylinders with Internal Fluid and Residual Stress
## 2 2D WFE Formulation for Fluid-Filled Cylinders  
Consider a fluid-filled cylinder as depicted in Figure 1a, where cylindrical coordinates (\(y\), \(r\), \(\theta\)) and geometrical parameters are defined. The cylinder is of thickness \(h\) and mean radius \(R\). At frequency \(\omega\), time-harmonic waves propagate in the structure as helical waves so that \(w(r,\theta,y,t)=W(r)e^{(i\omega t-k_{\theta}\theta-k_{y}y)}\), where \(W(r)\) is the complex wave amplitude, \(t\) is time, while \(k_{\theta}\) and \(k_{y}\) are the wavenumber components in the \(y\) and \(\theta\) directions. The cylinder is homogeneous in the axial and circumferential directions, with an arbitrary period. Exploiting this periodicity, a very small segment of angle \(\alpha\) and length \(L_{y}\) is taken, Figure 1a. The segment is then meshed by a number of elements as shown in Figure 1b, using standard FE method. The elements used depend on the nature of the structure: typically solid, 8-noded solid elements might be used to model the structure, with acoustic elements for the fluid. The FE equations of motion of the discretised model in Figure 1b can be obtained using standard FE analysis, and vary according to the assumed field variables. The FE equation of motion in their standard, unsymmetrical formulation for acoustic FSI, when the structural and acoustic variables are the nodal displacements, \(\mathbf{q}_{s}\), and acoustic pressures, \(\mathbf{q}_{f}\), is  
\[\left[ \begin{array}{cc}\mathbf{M}_{s}&\mathbf{0}\\ \rho_{f}\mathbf{G}^{T}&\mathbf{M}_{f}\end{array} \right]\left[ \begin{array}{ cc}\ddot{\mathbf{q}}_{s}\\ \ddot{\mathbf{q}}_{f}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{C}_{ s}&\mathbf{0}\\ \mathbf{0}&\mathbf{C}_{f}\end{array} \right]\left[ \begin{array}{cc}\dot{ \mathbf{q}}_{s}\\ \dot{\mathbf{q}}_{f}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{K}_{s} &-\mathbf{G}\\ \mathbf{0}&\mathbf{K}_{f}\end{array} \right]\left[ \begin{array}{cc}\mathbf{q} _{s}\\ \mathbf{q}_{f}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{f}_{s}\\ \mathbf{f}_{f}\end{array} \right]. \tag{1}\]  
In Equation (1), the subscripts \(s\) and \(f\) refer to structure and fluid respectively, \(\rho_{f}\) is the fluid density, \(\mathbf{M}_{s}\), \(\mathbf{M}_{f}\), \(\mathbf{K}_{s}\), \(\mathbf{K}_{f}\) are mass and stiffness matrices, \(\mathbf{G}\) and \(\rho_{f}\mathbf{G}^{T}\) are coupling submatrices, \(\mathbf{f}_{s}\) and \(\mathbf{f}_{f}\) are the nodal forces, \(\mathbf{C}_{s}\) is the viscous damping matrix for the  
Figure 1: (**a**) Schematic representation of a laminated fluid-filled cylinder; (**b**) FE discretisation of the arbitrary small segment of length \(L_{y}\) and subtending an angle \(\alpha\); (**c**) WFE super–element; (**d**) rotation of the local coordinates. structure, while \(\mathbf{C}_{f}\) is a dissipation term added to account for absorption of the acoustical pressure waves. Note that Equation (1) can alternatively be rewritten using acoustic potential variables for the fluid, e.g., [15]. Residual radial and hoop stresses in the structure can be found by evaluating the FE pre-stress stiffness matrix, which is then added to the original stiffness matrix. In particular, the effect of internal pressure can be modelled by the pre-stress matrix \(\mathbf{K}_{llr}\), which is obtained simply by solving a FE static analysis for a slice of the cylinder to which the internal pressure is applied. The effect of axial stress can be also considered by evaluating the pre-stress stiffness matrix, \(\mathbf{K}_{ax}\), obtained through a FE static analysis of the segment of the cylinder loaded, for example, by the axial stress  
\[\sigma_{ax}=pR_{int}^{2}/(R_{ext}^{2}-R_{int}^{2}), \tag{2}\]  
which is applied on the faces parallel to the \(y-z\) plane. The resulting total stiffness matrix is  
\[\mathbf{K}=\mathbf{K}_{u}+\mathbf{K}_{hr}+\mathbf{K}_{ax}, \tag{3}\]  
where \(\mathbf{K}_{u}\) is the stiffness matrix of the unloaded segment.