In Biot's poroelasticity theory, the frequency-dependent effective densities are defined as:

(8–261)

where:

= porosity

= fluid mass density (air)

= tortuosity

= the bulk density of the solid phase of the poroelastic material

= solid mass density

= fluid resistivity

= fluid dynamic (shear) viscosity

= viscous characteristic length

ω = angular frequency

In the poroelastic acoustic material, the mixed displacement and pressure (u-P) governing equations are as follows:

(8–262)

(8–263)

where:

= solid stress tensor (eliminating the dependency of fluid displacement) defined as:

= displacement of the solid (skeleton)

= strain tensor of the solid

N = complex shear modulus in vacuo (second Lame coefficient) of the solid

= effective density defined as:

p = pressure in the pores

Ks = bulk modulus of the elastic solid (frame)

Kb = bulk modulus of the porous material in vacuo

P0 = static reference pressure

= specific heat ratio of air

Prt = Prandtl number

= thermal characteristic length

For the majority of poroelastic materials used in acoustics, . The Biot's constant is defined by:

(8–264)

In most cases, the Biot's constant equals 1.

Introducing variable transformation,

(8–265)

the finite element formulations are obtained by testing Equation 8–262 and Equation 8–263 using the Galerkin procedure. Equation 8–262 and Equation 8–263 are multiplied by testing function and q, respectively, and integrated over the volume of the domain with some manipulation to yield the following:

(8–266)

(8–267)

where:

= total stress tensor, defined by:

= fluid phase displacement

The fluid phase displacement is cast by:

(8–268)

In poroelastic material, the total displacement is written as:

(8–269)

Boundary conditions are applied to the finite element model to solve poroelastic acoustic equations according to the application.

Natural Boundary Condition

(8–270)

where subscript n = normal direction of the interface.

In a finite element model, the boundary integral terms are not considered with the natural boundary condition.

Fixed Edge

(8–271)

A poroelastic medium is bounded onto a rigid impervious wall. The solid phase displacement and normal relative displacement vanish at the rigid wall.

Free Porous Surface

(8–272)

On the free porous surface, the pressure is set to zero.

Sliding Edge

The sliding edge condition indicates that the normal displacements on an impervious rigid wall are constrained, and the tangential displacements are free.

(8–273)

The boundary integrations vanish because the normal solid displacement is constrained.

Pervious Porous Surface with Permeability K

The boundary condition imposed with the difference in displacements between solid phase, fluid phase, and pressure on the pervious porous surface is written as:

(8–274)

Imposed Pressure

(8–275)

For the case of imposed pressure, the pressure is set to p⁰.

Imposed Displacement Field

(8–276)

The continuity between the imposed displacement and the solid phase displacement is maintained. The normal displacements between solid phase and fluid phase continues.

The couplings occur at the interfaces between the poroelastic material and other materials.

Poroelastic-Elastic Coupling Conditions

The coupling conditions at the interface between poroelastic and elastic media are given by:

where:

= elastic stress tensor

The coupling between poroelastic and elastic media is natural.

Poroelastic-Acoustic Coupling Conditions

The coupling conditions at the interface between poroelastic and acoustic media are given by:

where: