constant, $\tilde { k } ,$ or speed of sound, . Data defined with the pair $\left( \tilde { Z } , \tilde { k } \right)$ or $( \tilde { Z } , \tilde { c } )$ can be converted into the complex density and bulk modulus form, beginning from the following standard formulae: $$ \tilde {Z} \equiv \sqrt {\tilde {\rho} _ {f} \tilde {K} _ {f}}, $$ $$ \tilde {k} \equiv \omega \sqrt {\frac {\tilde {\rho} _ {f}}{\tilde {K} _ {f}}}, $$ $$ \tilde {c} \equiv \sqrt {\frac {\tilde {\rho} _ {f}}{\tilde {K} _ {f}}}. $$ Consistent with the Abaqus sign conventions, the real parts of $\tilde { k }$ and must be positive; the imaginary part of must be negative, and the imaginary part of must be positive. In commonly observed materials, the ratio of the magnitude of the imaginary part to the real part for each of these constants is usually much less than one. # Input File Usage: Use the following option to use the general frequency-dependent model: \*ACOUSTIC MEDIUM, COMPLEX BULK MODULUS \*ACOUSTIC MEDIUM, COMPLEX DENSITY If desired, either complex material option can be used instead in conjunction with a real-valued, frequency-independent material option: \*ACOUSTIC MEDIUM, COMPLEX BULK MODULUS \*DENSITY or, alternatively, \*ACOUSTIC MEDIUM, BULK MODULUS \*ACOUSTIC MEDIUM, COMPLEX DENSITY # Abaqus/CAE Usage: General frequency-dependent acoustic material models are not supported in Abaqus/CAE. Conversion from complex material impedance and wavenumber Since $$ \tilde {\rho} _ {f} = \frac {1}{\omega} \tilde {k} \tilde {Z} $$ and $$ \tilde {K} _ {f} = \omega \frac {\tilde {Z}}{\tilde {k}}, $$ the real and imaginary parts of $\tilde { \rho } _ { f }$ are, respectively: $$ \Re (\tilde {\rho} _ {f}) = \frac {1}{\omega} \big (\Re (\tilde {k}) \Re (\tilde {Z}) - \Im (\tilde {k}) \Im (\tilde {Z}) \big), $$ $$ \Im (\tilde {\rho} _ {f}) = \frac {1}{\omega} \left(\Im (\tilde {k}) \Re (\tilde {Z}) + \Re (\tilde {k}) \Im (\tilde {Z})\right); $$ and the real and imaginary parts of $\tilde { K } _ { f }$ are, respectively: $$ \Re (\tilde {K} _ {f}) = \frac {\omega}{| \tilde {k} | ^ {2}} \big (\Re (\tilde {k}) \Re (\tilde {Z}) + \Im (\tilde {k}) \Im (\tilde {Z}) \big), $$ $$ \Im (\tilde {K} _ {f}) = \frac {\omega}{| \tilde {k} | ^ {2}} \big (\Re (\tilde {k}) \Im (\tilde {Z}) - \Im (\tilde {k}) \Re (\tilde {Z}) \big). $$ Conversion from complex impedance and speed of sound Since $$ \tilde {\rho} _ {f} = \frac {\tilde {Z}}{\tilde {c}} $$ and $$ \tilde {K} _ {f} = \tilde {Z} \tilde {c}, $$ the real and imaginary parts of $\tilde { \rho } _ { f }$ are, respectively: $$ \Re (\tilde {\rho} _ {f}) = \frac {1}{| \tilde {c} | ^ {2}} \big (\Re (\tilde {c}) \Re (\tilde {Z}) + \Im (\tilde {c}) \Im (\tilde {Z}) \big), $$ $$ \Im (\tilde {\rho} _ {f}) = \frac {1}{| \tilde {c} | ^ {2}} \big (\Re (\tilde {c}) \Im (\tilde {Z}) - \Im (\tilde {c}) \Re (\tilde {Z}) \big); $$ and the real and imaginary parts of $\tilde { K } _ { f }$ are, respectively: $$ \Re (\tilde {K} _ {f}) = \left(\Re (\tilde {c}) \Re (\tilde {Z}) - \Im (\tilde {c}) \Im (\tilde {Z})\right), $$ $$ \Im (\tilde {K} _ {f}) = \big (\Im (\tilde {c}) \Re (\tilde {Z}) + \Re (\tilde {c}) \Im (\tilde {Z}) \big). $$ In general, fluids cannot withstand any significant tensile stress and are likely to undergo large volume expansion when the absolute pressure is close to or less than zero. Abaqus/Explicit allows modeling of this phenomenon through a cavitation pressure limit for the acoustic medium. When the fluid absolute pressure (sum of the dynamic and initial static pressures) reduces to this limit, the fluid undergoes free volume expansion (i.e., cavitation), without a further drop in the pressure. If this limit is not defined, the fluid is assumed not to undergo cavitation even under a tensile, negative absolute pressure, condition. The constitutive behavior for an acoustic medium capable of undergoing cavitation can be stated as $$ p = \max \left\{p _ {v}, p _ {c} - p _ {0} \right\}, $$ where a pseudo-pressure $p _ { v }$ , a measure of the volumetric strain, is defined as $$ p _ {v} = - \mathrm{K} _ {f} \varepsilon_ {V}, $$ where $p _ { c }$ is the fluid cavitation limit and $p _ { 0 }$ is the initial acoustic static pressure. A total wave formulation is used for a nonlinear acoustic medium undergoing cavitation. This formulation is very similar to the scattered wave formulation except that the pseudo-pressure, defined as the product of the bulk modulus and the compressive volumetric strain, plays the role of the material state variable instead of the acoustic dynamic pressure and the acoustic dynamic pressure is readily available from this pseudo-pressure subject to the cavitation condition. Input File Usage: \*ACOUSTIC MEDIUM, CAVITATION LIMIT Abaqus/CAE Usage: Fluid cavitation is not supported in Abaqus/CAE. # Defining the wave formulation In the presence of cavitation in Abaqus/Explicit the fluid mechanical behavior is nonlinear. Hence, for an acoustic problem with incident wave loading and possible cavitation in the fluid, the scattered wave formulation, which provides a solution for only a scattered wave dynamic acoustic pressure, may not be appropriate. For these cases the total wave formulation, which solves for the total dynamic acoustic pressure, should be selected. See “Acoustic and shock loads,” Section 34.4.6, for details. Input File Usage: \*ACOUSTIC WAVE FORMULATION, TYPE=TOTAL WAVE Abaqus/CAE Usage: Any module: Model→Edit Attributes→model\_name. Toggle on Specify acoustic wave formulation: Total wave # Defining the initial acoustic static pressure Cavitation occurs when the absolute pressure reaches the cavitation limit value. Abaqus/Explicit allows for an initial linearly varying hydrostatic pressure in the fluid medium (see “Defining initial acoustic static pressure” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). You can specify pressure values at two locations and a node set of the acoustic medium nodes. Abaqus/Explicit interpolates from these data to initialize the static pressure at all the nodes in the specified node set. If the pressure at only one location is specified, the hydrostatic pressure in the fluid is assumed to be uniform. The acoustic static pressure is used only for determining the cavitation status of the acoustic element nodes and does not apply any static loads to the acoustic or structural mesh at their common wetted interface. Input File Usage: \*INITIAL CONDITIONS, TYPE=ACOUSTIC STATIC PRESSURE Abaqus/CAE Usage: Initial acoustic pressures are not supported in Abaqus/CAE. # Defining a steady flow field Acoustic finite elements can be used to simulate time-harmonic wave propagation and natural frequency analysis in the presence of a steady mean flow of the medium. For example, air may move at a speed large enough to affect the propagation speed of waves in the direction of flow and against it. These effects are modeled in Abaqus/Standard by specifying an acoustic flow velocity during the linear perturbation analysis step definition; you do not need to alter the acoustic material properties. See “Acoustic, shock, and coupled acoustic-structural analysis,” Section 6.10.1, for details. # Elements An acoustic material definition can be used only with the acoustic elements in Abaqus (see “Choosing the appropriate element for an analysis type,” Section 27.1.3). In Abaqus/Standard second-order acoustic elements are more accurate than first-order elements. Use at least six nodes per wavelength in the acoustic medium to obtain accurate results. # Output Nodal output variable POR (pressure magnitude) is available for an acoustic medium in Abaqus (in Abaqus/CAE this output variable is called PAC). When the scattered wave formulation is used with incident wave loading in Abaqus/Explicit, output variable POR represents only the scattered pressure response of the model and does not include the incident wave loading itself. When the total wave formulation is used, output variable POR represents the total dynamic acoustic pressure, which includes contributions from both incident and scattered waves as well as the dynamic effects of fluid cavitation. For either formulation output variable POR does not include the acoustic static pressure, which is used only to evaluate the cavitation status in the acoustic medium. In addition, in Abaqus/Standard nodal output variable PPOR (the pressure phase) is available for an acoustic medium. In Abaqus/Explicit nodal output variable PABS (the absolute pressure, equal to the sum of POR and the acoustic static pressure) is available for an acoustic medium. # Additional references • Allard, J. F., M. Henry, J. Tizianel, L. Kelders, and W. Lauriks, “Sound Propagation in Air-Saturated Random Packings of Beads,” Journal of the Acoustical Society of America, vol. 104, no. 4, p. 2004, 1998. • Attenborough, K. F., “Acoustical Characterisitics of Rigid Fibrous Absorbents and Granular Materials,” Journal of the Acoustical Society of America, vol. 73, no. 3, p. 785, 1982. • Craggs, A., “A Finite Element Model for Rigid Porous Absorbing Materials,” Journal of Sound and Vibration, vol. 61, no. 1, p. 101, 1978. • Craggs, A., “Coupling of Finite Element Acoustic Absorption Models,” Journal of Sound and Vibration, vol. 66, no. 4, p. 605, 1979. • Delany, M. E., and E. N. Bazley, “Acoustic Properties of Fibrous Absorbent Materials,” Applied Acoustics, vol. 3, p. 105, 1970. • Miki, Y., “Acoustical Properties of Porous Materials - Modifications of Delany-Bazley Models,” Journal of the Acoustical Society of Japan (E), vol. 11, no. 1, p. 19, 1990. • Song, B. H., and J. S. Bolton, “A Transfer-Matrix Approach for Estimating the Characteristic Impedance and Wavenumbers of Limp and Rigid Porous Materials,” Journal of the Acoustical Society of America, vol. 107, no. 3, p. 1131, 1999. • Wilson, D. K., “Relaxation-Matched Modeling of Propagation through Porous Media, Including Fractal Pore Structure,” Journal of the Acoustical Society of America, vol. 94, no. 2, p. 1136, 1993. # 26.4 Mass diffusion properties • “Diffusivity,” Section 26.4.1 • “Solubility,” Section 26.4.2 # 26.4.1 DIFFUSIVITY Products: Abaqus/Standard Abaqus/CAE # References • “Mass diffusion analysis,” Section 6.9.1 • “Material library: overview,” Section 21.1.1 • \*DIFFUSIVITY • \*KAPPA • “Defining mass diffusion,” Section 12.12.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview # Diffusivity: • defines the diffusion or movement of one material through another, such as the diffusion of hydrogen through a metal; • must always be defined for mass diffusion analysis; • must be defined in conjunction with “Solubility,” Section 26.4.2; • can be defined as a function of concentration, temperature, and/or predefined field variables; • can be used in conjunction with a “Soret effect” factor to introduce mass diffusion caused by temperature gradients; • can be used in conjunction with a pressure stress factor to introduce mass diffusion caused by gradients of equivalent pressure stress (hydrostatic pressure); and • can produce a nonlinear mass diffusion analysis when dependence on concentration is included (the same can be said for the Soret effect factor and the pressure stress factor). # Defining diffusivity Diffusivity is the relationship between the concentration flux, , of the diffusing material and the gradient of the chemical potential that is assumed to drive the mass diffusion process. Either general mass diffusion behavior or Fick’s diffusion law can be used to define diffusivity, as discussed below. # General chemical potential Diffusive behavior provides the following general chemical potential: $$ \mathbf {J} = - s \mathbf {D} \cdot \left[ \frac {\partial \phi}{\partial \mathbf {x}} + \kappa_ {s} \frac {\partial}{\partial \mathbf {x}} \bigg (\ln (\theta - \theta^ {Z}) \bigg) + \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}} \right], $$ where

$\mathbf{D}(c,\theta,f_i)$	is the diffusivity;
$s(\theta,f_i)$	is the solubility (see “Solubility,” Section 26.4.2);
$\kappa_s(c,\theta,f_i)$	is the Soret effect factor, providing diffusion because of temperature gradient (see below);
$\kappa_p(c,\theta,f_i)$	is the pressure stress factor, providing diffusion because of the gradient of the equivalent pressure stress (see below);
$\phi \stackrel{\text{def}}{=} c/s$	is the normalized concentration;
$c$	is the concentration of the diffusing material;
$\theta$	is the temperature;
$\theta^Z$	is the temperature at absolute zero (see below);
$p \stackrel{\text{def}}{=} -\text{trace}(\sigma)/3$	is the equivalent pressure stress; and
$f_i$	are any predefined field variables.

Input File Usage: \*DIFFUSIVITY, LAW=GENERAL (default) Abaqus/CAE Usage: Property module: material editor: Other→Mass Diffusion→Diffusivity: Law: General # Fick’s law An extended form of Fick’s law can be used as an alternative to the general chemical potential: $$ \mathbf {J} = - \mathbf {D} \cdot \left(\frac {\partial c}{\partial \mathbf {x}} + s \kappa_ {p} \frac {\partial p}{\partial \mathbf {x}}\right). $$ Input File Usage: \*DIFFUSIVITY, LAW=FICK Abaqus/CAE Usage: Property module: material editor: Other→Mass Diffusion→Diffusivity: Law: Fick # Directional dependence of diffusivity Isotropic, orthotropic, or fully anisotropic diffusivity can be defined. For non-isotropic diffusivity a local orientation of the material directions must be specified (see “Orientations,” Section 2.2.5). # Isotropic diffusivity For isotropic diffusivity only one value of diffusivity is needed at each concentration, temperature, and field variable value. Input File Usage: \*DIFFUSIVITY, TYPE=ISO Abaqus/CAE Usage: Property module: material editor: Other→Mass Diffusion→Diffusivity: Type: Isotropic