# 26.2.3 SPECIFIC HEAT Products: Abaqus/Standard Abaqus/Explicit Abaqus/CFD Abaqus/CAE # References • “Material library: overview,” Section 21.1.1 • “Thermal properties: overview,” Section 26.2.1 • \*SPECIFIC HEAT • “Defining specific heat,” Section 12.10.6 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview A material’s specific heat: • is required for transient “Uncoupled heat transfer analysis,” Section 6.5.2; transient “Fully coupled thermal-stress analysis,” Section 6.5.3; transient “Coupled thermal-electrical analysis,” Section 6.7.3; and “Adiabatic analysis,” Section 6.5.4; • must be defined for an Abaqus/CFD analysis when the energy equation is active (“Energy equation” in “Incompressible fluid dynamic analysis,” Section 6.6.2); • must appear in conjunction with a density definition (see “Density,” Section 21.2.1); • can be linear or nonlinear (by defining it as a function of temperature); and • can be specified as a function of temperature and/or field variables. # Defining specific heat The specific heat of a substance is defined as the amount of heat required to increase the temperature of a unit mass by one degree. Mathematically, this physical statement can be expressed as: $$ c = \frac {\delta Q}{d \theta} = \theta \left(\frac {d s}{d \theta}\right), $$ where is the infinitessimal heat added per unit mass and is the entropy per unit mass. Since heat transfer depends on the conditions encountered during the whole process (a path function), it is necessary to specify the conditions used in the process to unambiguously characterize the specific heat. Thus, a process where the heat is supplied keeping the volume constant defines the specific heat as: $$ c _ {v} = \left(\frac {\delta Q}{d \theta}\right) \Big | _ {v} = \theta \left(\frac {\partial s}{\partial \theta}\right) \Big | _ {v} = \left(\frac {\partial u}{\partial \theta}\right) \Big | _ {v}, $$ where is the internal energy per unit mass. Whereas, a process where the heat is supplied keeping the pressure constant defines the specific heat as: $$ c _ {p} = \left(\frac {\delta Q}{d \theta}\right) \Big | _ {p} = \theta \left(\frac {\partial s}{\partial \theta}\right) \Big | _ {p} = \left(\frac {\partial h}{\partial \theta}\right) \Big | _ {p}, $$ where $h = u + p v$ is the enthalpy per unit mass. In general, the specific heats are functions of temperature. For solids and liquids, $c _ { v }$ and $c _ { p }$ are equivalent; thus, there is no need to distinguish between them. When possible, large changes in internal energy or enthalpy during a phase change should be modeled using “Latent heat,” Section 26.2.4, instead of specific heat. # Defining constant-volume specific heat The specific heat per unit mass is given as a function of temperature and field variables. By default, specific heat at constant volume is assumed. Input File Usage: \*SPECIFIC HEAT The following option can also be used in Abaqus/CFD: \*SPECIFIC HEAT, TYPE=CONSTANT VOLUME Abaqus/CAE Usage: Property module: material editor: Thermal→Specific Heat; Type: Constant Volume # Defining constant-pressure specific heat In Abaqus/CFD the constant-pressure specific heat is required when the energy equation is used for thermal-flow problems. Input File Usage: \*SPECIFIC HEAT, TYPE=CONSTANT PRESSURE Abaqus/CAE Usage: Property module: material editor: Thermal→Specific Heat; Type: Constant Pressure # Elements Specific heat effects can be defined for all heat transfer, coupled thermal-electrical-structural, coupled temperature-displacement, coupled thermal-electrical, and fluid elements in Abaqus. Specific heat can also be defined for stress/displacement elements for use in adiabatic stress analysis. Specific heat must be defined for all transient thermal analyses even if the only elements in the model are user-defined elements (“User-defined elements,” Section 32.17.1), in which case a dummy specific heat must be specified. # 26.2.4 LATENT HEAT Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE # References • “Material library: overview,” Section 21.1.1 • “Thermal properties: overview,” Section 26.2.1 • \*LATENT HEAT • “Specifying latent heat data,” Section 12.10.5 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview A material’s latent heat: • models large changes in internal energy during phase change of a material; • is active only during transient heat transfer, coupled thermal-stress, coupled thermal-electricalstructural and coupled thermal-electrical analysis in Abaqus (see “Heat transfer analysis procedures: overview,” Section 6.5.1); • must appear in conjunction with a density definition (see “Density,” Section 21.2.1); and • always makes an analysis nonlinear. # Defining latent heat Latent heat effects can be significant and must be included in many heat transfer problems involving phase change. When latent heat is given, it is assumed to be in addition to the specific heat effect (see “Uncoupled heat transfer analysis,” Section 2.11.1 of the Abaqus Theory Guide, for details). The latent heat is assumed to be released over a range of temperatures from a lower (solidus) temperature to an upper (liquidus) temperature. To model a pure material with a single phase change temperature, these limits can be made very close. As many latent heats as are necessary can be defined to model several phase changes in the material. Latent heat can be combined with any other material behavior in Abaqus, but it should not be included in the material definition unless necessary; it always makes the analysis nonlinear. # Direct data specification If the phase change occurs within a known temperature range, the solidus and liquidus temperatures can be given directly. The latent heat should be given per unit mass. Input File Usage: \*LATENT HEAT Abaqus/CAE Usage: Property module: material editor: Thermal→Latent Heat # User subroutine In some cases it may be necessary to include a kinetic theory for the phase change to model the effect accurately in Abaqus/Standard; for example, the prediction of crystallization in a polymer casting process. In such cases you can model the process in considerable detail using solution-dependent state variables (“User subroutines: overview,” Section 18.1.1) and user subroutine HETVAL. Input File Usage: Use the following options: \*HEAT GENERATION \*DEPVAR Abaqus/CAE Usage: Property module: material editor: Thermal→Heat Generation General→Depvar # Elements Latent heat effects can be used in all diffusive heat transfer, coupled temperature-displacement, coupled thermal-electrical-structural and coupled thermal-electrical elements in Abaqus but cannot be used with convective heat transfer elements. Strong latent heat effects are best modeled with first-order or modified second-order elements, which use integration methods designed to provide accurate results for such cases. See “Freezing of a square solid: the two-dimensional Stefan problem,” Section 1.6.2 of the Abaqus Benchmarks Guide, for an example of a heat conduction problem involving latent heat. # 26.3 Acoustic properties • “Acoustic medium,” Section 26.3.1 # 26.3.1 ACOUSTIC MEDIUM Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE # References • “Acoustic, shock, and coupled acoustic-structural analysis,” Section 6.10.1 • “Acoustic and shock loads,” Section 34.4.6 • “Material library: overview,” Section 21.1.1 • “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1 • \*ACOUSTIC MEDIUM • \*DENSITY • \*INITIAL CONDITIONS • “Defining an acoustic medium,” Section 12.12.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide # Overview An acoustic medium: • is used to model sound propagation problems; • can be used in a purely acoustic analysis or in a coupled acoustic-structural analysis such as the calculation of shock waves in a fluid or noise levels in a vibration problem; • is an elastic medium (usually a fluid) in which stress is purely hydrostatic (no shear stress) and pressure is proportional to volumetric strain; • is specified as part of a material definition; • must appear in conjunction with a density definition (see “Density,” Section 21.2.1); • can include fluid cavitation in Abaqus/Explicit when the absolute pressure drops to a limit value; • can be defined as a function of temperature and/or field variables; • can include dissipative effects; • can model small pressure changes (small amplitude excitation); • can model waves in the presence of steady underlying flow of the medium; and • is active only during dynamic analysis procedures (“Dynamic analysis procedures: overview,” Section 6.3.1). # Defining an acoustic medium The equilibrium equation for small motions of a compressible, inviscid fluid flowing through a resisting matrix material is taken to be $$ \frac {\partial p}{\partial \mathbf {x}} + \gamma \dot {\mathbf {u}} ^ {f} + \rho_ {f} \ddot {\mathbf {u}} ^ {f} = 0, $$ where p is the dynamic pressure in the fluid (the pressure in excess of any initial static pressure), is the spatial position of the fluid particle, $\dot { \mathrm { \mathbf { u } } } ^ { f }$ is the fluid particle velocity, $\ddot { { \mathbf { u } } } ^ { f }$ is the fluid particle acceleration, $\rho _ { f }$ is the density of the fluid, and is the “volumetric drag” (force per unit volume per velocity) caused by the fluid flowing through the matrix material. The d’Alembert term has been written without convection on the assumption that there is no steady flow of the fluid, which is usually considered to be sufficiently accurate for steady fluid velocities up to Mach 0.1. The constitutive behavior of the fluid is assumed to be inviscid and compressible, so that the bulk modulus of an acoustic medium relates the dynamic pressure in the medium to the volumetric strain by $$ p = - \mathrm{K} _ {f} \varepsilon_ {V}, $$ where $\varepsilon _ { V } = \varepsilon _ { 1 1 } + \varepsilon _ { 2 2 } + \varepsilon _ { 3 3 }$ is the volumetric strain. Both the bulk modulus $\mathrm { K } _ { f }$ and the density $\rho _ { f }$ of an acoustic medium must be defined. The bulk modulus $\mathrm { K } _ { f }$ can be defined as a function of temperature and field variables but does not vary in value during an implicit dynamic analysis using the subspace projection method (“Implicit dynamic analysis using direct integration,” Section 6.3.2) or a direct-solution steady-state dynamic analysis (“Direct-solution steady-state dynamic analysis,” Section 6.3.4); for these procedures the value of the bulk modulus at the beginning of the step is used. Input File Usage: Use both of the following options to define an acoustic medium: \*ACOUSTIC MEDIUM, BULK MODULUS \*DENSITY Abaqus/CAE Usage: Property module: material editor: Other→Acoustic Medium: Bulk Modulus General→Density # Volumetric drag Dissipation of energy (and attenuation of acoustic waves) may occur in an acoustic medium due to a variety of factors. Such dissipation effects are phenomenologically characterized in the frequency domain by the imaginary part of the propagation constant, which gives an exponential decay in amplitude as a function of distance. In Abaqus the simplest way to model this effect is through a “volumetric drag coefficient,” (force per unit volume per velocity). In frequency-domain procedures, may be frequency dependent. can be entered as a function of frequency— , where f is the frequency in cycles per time (usually Hz)—in addition to temperature and/or field variables only when the acoustic medium is used in a steady-state dynamics procedure. If the acoustic medium is used in a direct-integration dynamic procedure (including Abaqus/Explicit), the volumetric drag coefficient is assumed to be independent of frequency and the first value entered for the current temperature and/or field variable is used. In all procedures except direct steady-state dynamics the gradient of $\gamma / \rho _ { f }$ is assumed to be small. Input File Usage: \*ACOUSTIC MEDIUM, VOLUMETRIC DRAG Abaqus/CAE Usage: Property module: material editor: Other→Acoustic Medium: Volumetric Drag: Include volumetric drag # Porous acoustic material models Porous materials are commonly used to suppress acoustic waves; this attenuating effect arises from a number of effects as the acoustic fluid interacts with the solid matrix. For many categories of materials, the solid matrix can be approximated as either fully rigid compared to the acoustic fluid or fully limp. In these cases a mechanical model that resolves only acoustic waves will suffice. The acoustic behavior of porous materials can be modeled in a variety of ways in Abaqus/Standard. # Craggs model The model discussed in Craggs (1978) is readily accommodated in Abaqus. Applying this model results in the dynamic equilibrium equation for the fluid expressed in this form: $$ \nabla^ {2} \tilde {p} + (k a) ^ {2} K _ {s} \Omega \tilde {p} - i k a \frac {R \Omega}{\rho_ {f} c _ {f}} = 0, $$ where is the real-valued resistivity, is the real-valued dimensionless porosity, $K _ { s }$ is the dimensionless “structure factor,” and $\textstyle k a = { \frac { \omega } { c _ { f } } }$ is the dimensionless wave number. This equation can be rewritten as $$ \nabla^ {2} \tilde {p} + \frac {(\omega) ^ {2}}{\rho_ {f} c _ {f} ^ {2}} \left[ \rho_ {f} K _ {s} \Omega + \frac {R \Omega}{i \omega} \right] = 0. $$ This model, therefore, can be applied straightforwardly in Abaqus by setting the material density equal to $\rho _ { f } K _ { s } \Omega$ , the volumetric drag equal to , and the bulk modulus equal to $\rho _ { f } c _ { f } ^ { 2 }$ . The Craggs model is supported for all acoustic procedures in Abaqus. # Delany-Bazley and Delany-Bazley-Miki models Abaqus/Standard supports the well-known empirical model proposed in Delany & Bazley (1970), which determines the material properties as a function of frequency and user-defined flow resistivity, ; density, $\rho _ { f } ,$ and bulk modulus, $K _ { f }$ . A variation on this model, proposed by Miki (1990) is also available. These models are supported only for steady-state dynamic procedures. Both models compute frequency-dependent material characteristic impedance, $\tilde { Z } _ { : }$ , and wavenumber or propagation constant, $\tilde { k } ,$ according to the following formula: $$ \tilde {k} \equiv \frac {\omega}{c _ {0}} \big (1 + C _ {1} x ^ {C _ {2}} - i C _ {3} x ^ {C _ {4}} \big), $$ $$ \tilde {Z} \equiv \rho_ {f} c _ {0} \big (1 + C _ {5} x ^ {C _ {6}} - i C _ {7} x ^ {C _ {8}} \big), $$ where $$ c _ {0} \equiv \sqrt {\frac {K _ {f}}{\rho_ {f}}}, $$ and $$ x \equiv \frac {\rho_ {f} \omega}{2 \pi R}. $$ The constants are as given in the table below:

	$C_1$	$C_2$	$C_3$	$C_4$	$C_5$	$C_6$	$C_7$	$C_8$
Delany-Bazley	0.0978	-0.7	0.189	-0.595	0.0571	-0.754	0.087	-0.732
Miki	0.1227	-0.618	0.1792	-0.618	0.0786	-0.632	0.1205	-0.632

The material characteristic impedance and the wavenumber are converted internally to complex density and complex bulk modulus for use in Abaqus. The signs of the imaginary parts in these formulae are consistent with the Abaqus sign convention for time-harmonic dynamics. Input File Usage: Use the following options to use the Delany-Bazley model: \*DENSITY \*ACOUSTIC MEDIUM, BULK MODULUS \*ACOUSTIC MEDIUM, POROUS MODEL=DELANY BAZLEY Use the following options to use the Miki model: \*DENSITY \*ACOUSTIC MEDIUM, BULK MODULUS \*ACOUSTIC MEDIUM, POROUS MODEL=MIKI Abaqus/CAE Usage: Porous acoustic material models are not supported in Abaqus/CAE. # General frequency-dependent models For steady-state dynamic procedures, Abaqus/Standard supports general frequency-dependent complex bulk modulus and complex density. Using these parameters, data from a wide range of models can be accommodated in an analysis; for example, see Allard, et. al (1998), Attenborough (1982), Song & Bolton (1999), and Wilson (1993). These models are used in a variety of applications, such as ocean acoustics, aerospace, automotive, and architectural acoustic engineering. The signs of these parameters must be consistent with the sign conventions used in Abaqus, and with conservation of energy. Abaqus uses a Fourier transform pair formally equivalent to assuming time dependence. Consequently, the real parts of the density and bulk modulus are positive for all values of frequency, the imaginary part of the bulk modulus must be positive, and the imaginary part of the density must be negative. A linear isotropic acoustic material can be fully described with the two frequency-dependent parameters: bulk modulus, ${ \tilde { K } } _ { f } ,$ and density, $\tilde { \rho } _ { f }$ . It is common, however, to encounter materials defined in terms of other parameter pairs, such as characteristic impedance, ${ \tilde { Z } } ,$ wave number or propagation