# Numerical Analysis of Vibroacoustic Response of Timber Floor Panels Damped with Porous Materials
## 3 Finite Element Formulation of Sound Radiation of an Elastic Structure with a Poroelastic Layer  
### Local Equations  
We consider an elastic structure occupying the domain \(\Omega_{E}\), clamped on its boundary \(\Gamma_{u}\), and subject to external forces \(\overset{\rightarrow}{F}^{d}\) on its boundary \(\Gamma_{t}\) (Figure 4). A poroelastic layer of domain \(\Omega_{P}\) is fixed to the elastic structure in order to dampen its sound radiation. The structure-poroelastic interface is denoted \(\sum_{EP}\). \(\overset{\rightarrow}{n_{E}}\), and \(\overset{\rightarrow}{n_{P}}\) represent the exterior normals of the structure and poroelastic domains, respectively. The effects of gravity are neglected. Figure 3: Experimental and numerical force–displacement curve of GL24h beam. The porous material is modeled in this work using Biot theory [21]. This theory is applied when the porous material has an elastic matrix. In this case, the material is defined by its porosity \(\phi\), static flow resistance \(\sigma\), tortuosity \(\alpha_{\infty}\), viscous length \(\Lambda\), and thermal length \(\Lambda^{\prime}\). The elastic skeleton is characterized by its Young's modulus \(E_{5}\), Poisson's ratio \(\
u_{S}\), structural damping coefficient \(\eta_{S}\), and its density \(\rho_{S}\). In total, the poroelastic material is characterized by nine parameters. The local equations of the coupled elastic-poroelastic problem can be written in terms of the displacement of the elastic structure \(\overset{\rightarrow}{u}^{E}\), the displacement of the solid phase of the porous \(\overset{\rightarrow}{u}^{s}\), and the pressure for the fluid phase of the porous \(p^{F}\)[22,23]:  
* Elastic structure: \[\overset{\rightarrow}{div}\overset{=E}{\sigma}\left(\overset{-E}{u}\right)+ \rho_{E}\omega^{2}\overset{\rightarrow}{u}=0\ \ ;\ \forall\ \ M\ \in\Omega_{E}\] (2a) \[\overset{=E}{\sigma}\overset{\rightarrow}{n_{E}}=\overset{\rightarrow}{F^{ d}}\ \ ;\ \forall\ \ M\ \in\Gamma_{t}\] (2b) \[\overset{\rightarrow}{u}^{E}=\overset{\rightarrow}{0}\ \ ;\ \forall\ \ M\ \in\Gamma_{u}\] (2c)
* Poroelastic material (solid phase): \[\overset{\rightarrow}{div}\overset{=S}{\sigma}\left(\overset{\rightarrow}{u} ^{S}\right)+\omega^{2}\overset{\rightarrow}{\rho_{s}}\overset{\rightarrow}{u }^{S}+\overset{\rightarrow}{\gamma}\overset{\rightarrow}{\
abla}p^{F}=0;\ \forall\ \ M\in\Omega_{P}\] (3a)
* Poroelastic material (fluid phase): \[\overset{\phi^{2}}{\overset{\rightarrow}{\rho}}\Delta p^{F}+\omega^{2} \overset{\phi^{2}}{\overset{\rightarrow}{R}}p^{F}-\omega^{2}\overset{ \rightarrow}{\gamma}\overset{\rightarrow}{div}\overset{\rightarrow}{u}^{S}= 0;\ \forall\ \ M\in\Omega_{P}\] (3b) where \(\overset{=E}{\sigma}\) is the stress tensors of the elastic structure. \(\overset{=}{\sigma}^{S}\) is the in vacuo stress tensor of the porous material. It is related to the total stress tensor \(\overset{=}{\sigma}^{t}\) of the material through the following relation: \(\overset{=S}{\sigma}=\overset{=T}{\sigma}+\phi\left(1+\overset{\sim}{Q}/ \overset{\sim}{R}\right)p^{F}\overset{=}{1}\) where \(\overset{\sim}{Q}\) is an elastic coupling coefficient between the two phases, and \(\overset{\sim}{R}\) may be interpreted as the bulk modulus of the air occupying a fraction \(\phi\) of a unit volume aggregate. \(\omega\) in rad/s is the angular frequency. The continuity conditions between the elastic-solid domain and the porous medium are expressed as [24,25]:  
\[\overset{\rightarrow}{u}^{S}=\overset{\rightarrow}{u}^{E};\ \forall\ \ M\ \in\sum_{EP}\] (4a) \[\left(\overset{\rightarrow}{u}^{F}-\overset{\rightarrow}{u}^{S}\right) \overset{\rightarrow}{n_{P}}=0\ ;\forall\ \ M\ \in\sum_{EP}\] (4b) \[\overset{=T}{\sigma}\overset{\rightarrow}{n_{P}}=-\overset{=E}{\sigma} \overset{\rightarrow}{n_{E}};\ \forall\ \ M\ \in\sum_{EP}\] (4c) where \(\overset{\rightarrow}{u}^{F}\) is the displacement of the fluid phase of the porous material. Figure 4: Elastic structure coupled to poroelastic layer.