Poroelastic Material
Use the Poroelastic Material node to define the poroelastic material and fluid properties of an isotropic porous material, that is the properties of the porous matrix and the saturating fluid.
Poroelastic Model
Select the Model used to describe the losses to include in the saturating fluid of the porous material:
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Biot (viscous losses), this model is primarily used in earth sciences when the saturating fluid is a liquid like water or oil. The model is based on Biot’s original work and only includes the effects of viscous losses in the pores.
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Biot–Allard (thermal and viscous losses) (the default), this model is intended for simulating porous materials where the saturating fluid is air. This is for modeling sound absorbers, liners, foams used in headphones and loudspeakers, cloth, and much more. The model still relies on the mixed u-p formulation of Biot but includes additional losses.
When selecting a specific model, the required material input will change in order to align with the data normally available for the intended applications.
Three different waves can propagate in a poroelastic material; fast pressure waves, slow pressure waves, and shear waves. Their speed can be evaluated through the variables pelw.cp_fast, pelw.cp_slow, and pelw.cs_poro, respectively. These should not be confused with the material properties for the constituting solid (pelw.cp and pwel.cs) and fluid (pelw.c). All three poroelastic wave types should be resolved by the computational mesh.
Porous Matrix Properties
The default Porous elastic material uses the Domain material (the material defined for the domain). This setting should be different from the one for the Fluid material selected in the Fluid Properties section. Typically, select another material (a material defined in the Materials node) for the porous elastic material, and use the default Domain material option for the Fluid material.
First, enter the mechanical properties of the isotropic drained porous matrix. The drained parameters are also sometimes referred to as the in-vacuo elastic parameters; they are in principle measured without the presence of the saturating fluid. Then (below the dividing line) enter the porous (or poroacoustic) parameters of the material:
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Porous Parameters for Biot (viscous losses)
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Porous Parameters for Biot–Allard (thermal and viscous losses)
For the isotropic drained porous matrix, from the Specify list, select:
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Young’s modulus and Poisson’s ratio to specify drained Young’s modulus (elastic modulus) Ed (SI unit: Pa) and Poisson’s ratio νd (dimensionless). For an isotropic material, Young’s modulus is the spring stiffness in Hooke’s law, which in 1D form is σ = Edε, where σ is the stress and ε is the strain. Poisson’s ratio defines the normal strain in the perpendicular direction, generated from a normal strain in the other direction and follows the equation ε⊥ = −υε||
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Shear modulus and Poisson’s ratio (the default for the Biot–Allard model) to specify drained shear modulus Gd (SI unit: Pa) and Poisson’s ratio νd (dimensionless).
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Young’s modulus and Shear modulus to specify drained Young’s modulus (elastic modulus) Ed (SI unit: Pa) and drained shear modulus Gd (SI unit: Pa).
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Bulk modulus and shear modulus (the default for the Biot model) to specify the drained bulk modulus Kd (SI unit: Pa) and the drained shear modulus Gd (SI unit: Pa). The bulk drained modulus is a measure of the solid porous matrix’s resistance to volume changes. The shear modulus is a measure of the solid porous matrix’s resistance to shear deformations.
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Lamé parameters to specify the drained Lamé parameters λd (SI unit: Pa) and μd (SI unit: Pa).
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Pressure-wave and shear-wave speeds to specify the drained pressure-wave speed cp (SI unit: m/s) and the shear-wave speed cs (SI unit: m/s).
Then also enter:
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Drained density of porous material to specify the drained density of the porous material in vacuum ρd (SI unit: kg/m3). The drained density ρd is equal to (1 − εp) ρs where ρs is the density of the solid material from which the matrix is made and εp is the porosity.
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Isotropic structural loss factor (only for the Biot-Allard model) to specify the loss factor of the porous matrix ηs (dimensionless). This value introduces the damping due to losses in the porous structure by transform the elastic moduli into complex valued quantities. This quantity can be frequency dependent if necessary.
For each pair of properties, select from the applicable list to use the value From material or enter a User defined value or expression. Each of these pairs define the drained elastic properties and it is possible to convert from one set of properties to another.
Porous Parameters for Biot (viscous losses)
Enter the following (remaining) parameters necessary to define the properties of a Biot (viscous losses) porous material model. The defaults use values From material. For User defined, enter other values or expressions as needed.
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Porosity to specify the porosity of the material εp (dimensionless). It defines the amount of void volume inside the porous matrix and takes values between 0 (fully solid material no fluid) and 1 (no porous material only fluid).
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Permeability to specify the permeability of the porous material κp (SI unit: m2). The permeability is a measure of the ability of the porous material to let fluid pass through it. It hence gives some measure of the pore size and thus correlates to the viscous damping experienced by pressure waves propagating in the saturating fluid.
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Biot–Willis coefficient to specify the Biot–Willis coefficient αB (dimensionless). This coefficient relates the bulk modulus (compressibility) of the drained porous matrix to a block of solid material. It is defined as
where Kd is the drained bulk modulus and Ks is the bulk module of a block of solid material (made of the matrix material). The drained bulk modulus is related to the stiffness of the porous matrix, while the solid bulk modulus is related to the compressibility of the material or grains from which the porous matrix is made. The Biot–Willis coefficient is bound by εp ≤ αB ≤ 1. A rigid porous matrix (Voigt upper bound) has αB = εp and a soft or limp porous matrix (Reuss lower bound) has αB = 1.
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Tortuosity factor (high frequency limit) or the structural form factor τ∞ (dimensionless). This is a purely geometrical factor that depends on the microscopic geometry and distribution of the pores inside the porous material. It is independent of the fluid and solid properties and is normally >1. The default is 2. The more complex the propagation path through the material, the higher is the absorption. The tortuosity partly represents this complexity.
Porous Parameters for Biot–Allard (thermal and viscous losses)
Enter the following (remaining) parameters necessary to define the properties of a Biot–Allard (thermal and viscous losses) porous material model. The defaults use values From material. For User defined enter other values or expressions as needed.
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Porosity to specify the porosity of the material εp (dimensionless). It defines the amount of void volume inside the porous matrix and takes values between 0 (fully solid material no fluid) and 1 (no porous material only fluid).
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Flow resistivity to specify the (static) flow resistivity of the porous material Rf (SI unit: Pa·s/m2). The flow resistivity is a measure of the ability of the porous material to let fluid pass through it. It hence gives some measure of the pore size and thus correlates to the viscous damping experienced by pressure waves propagating in the saturating fluid. The flow resistivity is also sometimes denoted σ (using the unit N·s/m4) and it is related to the permeability through κp = μ/Rf.
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Tortuosity factor (high frequency limit) or the structural form factor τ∞ (dimensionless). This is a purely geometrical factor that depends on the microscopic geometry and distribution of the pores inside the porous material. It is independent of the fluid and solid properties and is normally >1. The default is 2. The more complex the propagation path through the material, the higher the absorption. The tortuosity partly represents this complexity.
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Viscous characteristic length to specify the viscous length scale Lv (SI unit: m). This value is sometimes denoted Λ and replaces the hydraulic radius used in simpler models to account for the viscous losses that appear in the acoustic boundary layer at pore walls.
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Thermal characteristic length to specify the thermal length scale Lth (SI unit: m). This value is sometimes denoted Λ’ and replaces the hydraulic radius used in simpler models to account for the thermal losses that appear in the acoustic boundary layer at pore walls.
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Biot–Willis coefficient how to calculate the Biot–Willis coefficient αB by selecting From material, Rigid assumption (the default), General model, or User defined. This coefficient relates the bulk modulus (compressibility) of the drained porous matrix to a block of solid material.
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From material to pick up the value from the domain material.
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Rigid assumption (the default) and the model defines a rigid porous matrix (Voigt upper bound) where αB = εp.
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General model to define the Biot–Willis coefficient αB according to its general definition
where Kd is the drained bulk modulus and Ks is the bulk module of the skeleton material (bulk modulus of a block of solid material made of the matrix material). When this option is selected, also enter the Bulk modulus of skeleton material Ks (taken from material as default). The drained bulk modulus Kd is related to the stiffness of the porous matrix, while the skeleton bulk modulus Ks is related to the compressibility of the material or grains from which the porous matrix is made. The Biot–Willis coefficient is bound by εp ≤ αB ≤ 1. A rigid porous matrix (Voigt upper bound) has αB = εp and a soft or limp porous matrix (Reuss lower bound) has αB = 1.
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User defined enter a value for the Biot–Willis coefficient αB (dimensionless). In the limp limit when αB = 1 it is recommended to use the Poroacoustics feature of The Pressure Acoustics, Frequency Domain Interface instead of the Poroelastic Waves interface. In poroacoustics, the limp limit, when the structure is so “fluffy” that it moves with the fluid, has the assumption Kd = 0 and αB = 1 included explicitly.
Fluid Properties
The default Fluid material uses the Domain material. This is the material defined for the domain in the Materials node.
Define the properties of the saturating fluid in terms of its density, viscosity, and compressibility but also the viscosity model. The defaults use values for the material parameters are From material. For User defined enter other values or expressions as needed.
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Fluid Parameters for Biot (Viscous Losses) Model
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Fluid Parameters for Biot–Allard (Thermal and Viscous Losses) Model
Fluid Parameters for Biot (Viscous Losses) Model
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Density defines the density of the saturating fluid ρf (SI unit: kg/m3).
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Dynamic viscosity to define the dynamic viscosity of the saturating fluid μf (SI unit: Pa·s). The parameter is important for the amount of viscous damping experienced by the acoustic waves.
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Compressibility of the saturating fluid χf (SI unit: 1/Pa). Remember that the fluid compressibility χf is related to the fluid bulk modulus Kf (SI unit: Pa) and the speed of sound c, through the relation
The compressibility of the fluid also enters the expression for Biot’s module M, given by
It should be noted that Biot–Willis coefficient αB only depends on the properties of the porous matrix while Biot’s module M depends on both fluid and porous matrix properties.
Select a Viscosity Model, either Biot’s low frequency range or Biot’s high frequency range.
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Biot’s low frequency range models damping at low frequencies where the acoustic boundary layer (the viscous penetration depth) is assumed to span the full width of the pores. This is also the so-called Poiseuille limit.
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For Biot’s high frequency range also select Specify as Reference frequency or Characteristic pore size. Either enter a Reference frequency fr (SI unit: Hz) or a Characteristic pore size a (SI unit: m). This model implements a correction factor to the viscosity that accounts for the relative scale difference between a typical pore diameter and the acoustic boundary layer thickness. The modified viscosity is of the form
where fr is the reference frequency and a is a characteristic size of the pores. The expression for fr is one typically used in the literature but it is often measured or empirically determined. The expression for fr corresponds to finding the frequency at which the viscous boundary layer thickness is of the scale a.
See High Frequency Correction (Biot Model) for more details.
Fluid Parameters for Biot–Allard (Thermal and Viscous Losses) Model
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Density to define the density of the saturating fluid ρf (SI unit: kg/m3).
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Dynamic viscosity to define the dynamic viscosity of the saturating fluid μf (SI unit: Pa·s). The parameter is important for the amount of viscous damping experienced by the acoustic waves.
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Ratio of specific heats to define the ratio of specific heats (adiabatic index) of the saturating fluid γ (dimensionless).
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Heat capacity at constant pressure to define the (specific) heat capacity at constant pressure of the saturating fluid Cp (SI unit: J/(kg·K)).
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Thermal conductivity to define the thermal conductivity of the saturating fluid k (SI unit: W/(m·K)). The parameter is important for the amount of thermal damping experienced by the acoustic waves.