# Visco-inertial effects

In 1987, Johnson Koplik and Dashen [JKD87] proposed a semi-phenomenological model to describe the complex density of an acoustical porous material with a motionless skeleton having arbitrary pore shapes.
This model is further refined by Pride, Morgan & Gangi [PMG93] (and corrected by D. Lafarge) in 1993 to account for pores with possible constrictions between them. The final expression for $\widetilde{\rho}$ obtained is:

\begin{align} &\widetilde{\rho} =\displaystyle\frac{\rho_0\widetilde{\alpha}\left(\omega\right)} { \phi} \\ &~~ \\ &~~ \\ &\widetilde{\alpha}\left(\omega\right)=\displaystyle\alpha_\infty\left[1+\displaystyle\frac{1}{j \bar{\omega}}\widetilde{F}\left(\omega\right)\right]\\ &~~ \\ &~~ \\\ &\widetilde{F}\left(\omega\right)=1-P+P\sqrt{1+\displaystyle\frac{M}{2{P~}^2}j\bar{\omega}}\\ &~~ \\ &~~ \\ &\bar{\omega}=\displaystyle\frac{\omega \rho_0 k_0 \alpha_\infty}{\eta \phi}\\ &~~ \\ &M=\displaystyle\frac{8 k_0 \alpha_\infty}{\phi\Lambda^2} \\ &~~ \\ & P=\displaystyle\frac{M}{4\left(\displaystyle\frac{\alpha_0}{\alpha_\infty}-1\right)}\\ \end{align}

5 parameters are involved in the calculation of this dynamic density: the static air flow resistivity $\sigma$ (or the static viscous permeability $k_0=\eta/\sigma$), the open porosity $\phi$, the high frequency limit of the tortuosity $\alpha_{\infty}$, the viscous characteristic length $\Lambda$ and the static viscous tortuosity $\alpha_0$.

# Thermal effects

In 1991, Champoux and Allard [CA91] introduced an expression for the dynamic bulk modulus based on the previous work by Johnson et al. This model is further refined by Pride, Morgan & Gangi [PMG93] (and corrected by D. Lafarge) in 1993 to account for pores with possible constrictions between them. Finally, after the work by [LLAT97] (introducing a new parameter to describe the thermal behavior at low frequencies: the static thermal permeability $k'_0$), the final expression obtained for $\widetilde{K}$ is:

\begin{align} & \widetilde{K} =\displaystyle\frac{\gamma P_0}{\phi}\frac{1}{\widetilde{\beta}\left(\omega\right)}\\ &~~ \\ &~~ \\ &\widetilde{\beta}\left(\omega\right)=\displaystyle\gamma - \left(\gamma-1\right)\left[1+\displaystyle\frac{1}{j \bar{\omega}'}\widetilde{F'}\left(\omega\right)\right]^{-1}\\ &~~ \\ &~~ \\ &\widetilde{F'}\left(\omega\right)=1-P~'+P~'\sqrt{1+\displaystyle\frac{M'}{2{P~}'^{2}}j\bar{\omega}'}\\ &~~ \\ &~~ \\ &\bar{\omega}'=\displaystyle\frac{\omega \rho_0 C_P k'_0}{\kappa\phi}\\ &~~ \\ &M'=\displaystyle\frac{8 k'_0}{\phi \Lambda'^2} \\ &~~ \\ &P~'=\displaystyle\frac{M'}{4\left(\alpha'_0-1\right)}\\ \end{align}

4 parameters are invloved in the calculation of this dynamic bulk modulus: the open porosity $\phi$, the thermal characteristic length $\Lambda'$, the static thermal permeability $k'_0$ and the static thermal tortuosity $\alpha'_0$.

# Links with JCA and JCAL models

The Johnson-Champoux-Allard (JCA) and Johnson-Champoux-Allard-Lafarge (JCAL) models are recovered from the JCAPL one by setting, respectively, $M'=P=P~'=1$ and $P=P~'=1$ in the expressions of $\widetilde{\rho}$ and $\widetilde{K}$ reported above.