Poroelastic Acoustic Medium

Frequency Response of a Poroelastic Acoustic Medium

The frequency domain theory of the poroelastic acoustic medium is presented in Poroelastic acoustic medium in frequency domain. The primary variables in the formulation are structural displacements $u_{i}^{s}$ and fluid pore pressure $p$ . The dynamic equilibrium equations are

{\hat{σ}}_{i j, j}^{s} + ω^{2} \tilde{ρ} u_{i}^{s} + \tilde{γ} p_{, i} = 0,

Δ p + \frac{{\tilde{ρ}}_{22}}{\tilde{R}} ω^{2} p - \frac{{\tilde{ρ}}_{22}}{ϕ^{2}} \tilde{γ} ω^{2} Θ^{s} = 0.

Here, $ω$ is the steady-state dynamics excitation frequency, $ϕ$ is the porosity defined as the ratio of the connected volume fully saturated by fluid to the whole volume, ${\hat{σ}}_{i j}^{s}$ is the structural stress, $Θ^{s}$ is the structural dilatation, and parameters with an overtilde (^~) are derived from the material properties of the solid structural skeleton and fluid. The $\tilde{γ}$ terms in the equations above represent the fluid-structure volumetric coupling terms. Without these terms the equations are the regular dynamic equilibrium equations for the separate solid and fluid phases. For the solid skeleton, there is no $(1 - ϕ)$ factor in the stiffness or density structural terms.

Generally, complex solid and fluid properties and coupling terms are determined from classical isotropic linear elastic solid properties, including the skeleton density plus the fluid real and complex material densities, complex bulk modulus, and the medium tortuosity. In the case of no damping, all properties are real valued with zero imaginary parts. For the solid, the damping can come from structural damping; for the fluid, from nonzero imaginary parts of the complex density and bulk modulus.

There are two ways to obtain the fluid complex density and bulk modulus: either directly from input data as implemented in the Biot-Atalla acoustic medium porous model or using special theory under the Biot-Johnson model. The Biot-Atalla model is a convenient way to compare poroelastic acoustic results versus pure acoustic results.

Poroelastic Acoustic Material Behavior

The response of the solid skeleton is assumed to be isotropic linear elastic with (optionally) structural damping. In addition, the porosity of the medium must be specified as an initial condition.

For the pore fluid, two poroelastic acoustic material models are available: the Biot-Atalla model and the Biot-Johnson model. You must specify the following parameters for both models: fluid density, tortuosity at infinite frequency, skeleton material bulk modulus, and the fluid-structure coupling factor (the default is one). Specifying a coupling factor value of zero leads to an uncoupled fluid and structural response. Additional input is necessary for both models.

Biot-Atalla Model

The Biot-Atalla model relies on direct specification of the complex bulk modulus and complex density of the pore fluid.

Biot-Johnson Model

To compute the complex bulk modulus and complex density of the pore fluid, the model requires you to define the following material parameters:

static flow resistivity,
viscous characteristic length,
thermal characteristic length,
fluid dynamic shear viscosity,
ambient fluid standard pressure,
ambient fluid heat capacity ratio (the ratio of the specific heat per unit mass at constant pressure over the specific heat per unit mass at constant volume), and
Prandtl number (the ratio of the viscous diffusion rate over the thermal diffusion rate).

Elements

An acoustic material definition can be used only with three-dimensional first-order poroelastic acoustic elements in Abaqus (see Choosing the Appropriate Element for an Analysis Type).

Output

Nodal output variable POR (real and imaginary pore pressure components) is available for poroelastic acoustic elements in Abaqus. Similar to regular acoustic finite elements, the output variable POR does not include the acoustic static pressure. In addition, the nodal output variable PPOR (pressure magnitude and phase) is available for a poroelastic acoustic medium.

Element integration point output variable GRADP (pore pressure spatial gradient) is available for the fluid phase of poroelastic acoustic elements. Other element integration point output variables are ACDISP (pore fluid displacement), ACV (pore fluid velocity), and ACADMIT (poroelastic acoustic admittance, a reciprocal of impedance).

The solid phase output is the same as for regular solid elements. It includes nodal displacements, velocities, accelerations, element integration point strains and stresses, etc., all as complex variables.

The stresses in the solid phase include only structural terms, without fluid phase pore pressure contribution. Similarly, output variable POR is the pure fluid pore pressure.