1. Reference problem#

1.1. Geometry#

_images/Cadre2.gif

Figure 1.1-1: Geometry (not scaled) We consider a flat metal ring immersed in a fluid medium with an external border for the latter (figure).

In the metal ring, a visco-elastic layer is modeled over a quarter of the circumference.

1.2. Material properties#

For the steel part, an isotropic elastic material is considered:

  • Modulus of elasticity: \({E}_{a}=177\mathit{GPa}\)

  • Poisson’s ratio: \({\mathrm{\nu }}_{a}=0.3\)

  • Density: \({\mathrm{\rho }}_{a}=7450\mathit{kg}\mathrm{.}{m}^{-3}\)

    For the fluid, it is water with the following characteristics:

  • Speed of sound: \(c=1500m\mathrm{.}{s}^{-1}\)

  • Density: \({\mathrm{\rho }}_{e}=1000\mathit{kg}\mathrm{.}{m}^{-3}\)

    For the visco-elastic part, we consider a material of the fractional derivative model type (Zener model) whose shear modulus depends on the pulsation:

(1.1)#\[ {G} ^ {\ text {*}}}\ left (\ omega\ right) =\ frac {{G} _ {0} + {G} _ {\ infty} {(\ mathit {i\ omega\ tau})}}}}} ^ {omega\ tau})}} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})} ^ {\ omega\ tau})}\]

The characteristics are as follows:

  • Poisson’s ratio: \({\mathrm{\nu }}_{v}=0.49\) (almost incompressible material)

  • Density: \({\mathrm{\rho }}_{v}=1460\mathit{kg}\mathrm{.}{m}^{-3}\)

  • Coefficient: \({G}_{0}=\mathrm{2,11}\mathit{MPa}\), \({G}_{\mathrm{\infty }}=\mathrm{0,59}\mathit{GPa}\), \(\mathrm{\tau }=\mathrm{0,44}\times {10}^{-6}\), and \(\mathrm{\alpha }=\mathrm{0,53}\)

Note: to improve the calculation of the modes, all these quantities are a-dimensioned in relation to water.

1.3. Boundary conditions and loads#

For the structure part, an embedment condition is imposed on the segment \(\mathit{AB}\) and a horizontal nodal force \(\mathit{Fx}=A\mathrm{,0}\times {10}^{-5}N\) is imposed on the point \(A\).

A fluid-boundary interaction condition is achieved by the use of 2D_ FLUI_STRU elements at the interface between the structure and the fluid. Moreover, an impedance condition is defined on the external border of the fluid, by a loading of type IMPE and the calculation of the impedance matrices that are linked to it.