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+<!-- source-page: 621 -->
+
+$$
+\eta = \eta_ {\infty} + \left(\eta_ {0} - \eta_ {\infty}\right) \left(1 + \left(\lambda \dot {\gamma}\right) ^ {a}\right) ^ {\frac {n - 1}{a}},
+$$
+
+where $\eta _ { 0 }$ is the low shear rate Newtonian viscosity, $\eta _ { \infty }$ is the infinite shear viscosity (at high shear strain rates), is the natural time constant of the fluid $( 1 / \lambda$ is the critical shear rate at which the fluid changes from Newtonian to power law behavior), and represents the flow behavior index in the power law regime. The coefficient is a material parameter. The original Carreau model is recovered when $a { = } 2$ .
+
+Input File Usage: \*VISCOSITY, DEFINITION=CARREAU-YASUDA
+
+Abaqus/CAE Usage: The Carreau-Yasuda model is not supported in Abaqus/CAE.
+
+# Cross
+
+The Cross model is commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity. The viscosity is expressed as
+
+$$
+\eta = \eta_ {\infty} + \frac {(\eta_ {0} - \eta_ {\infty})}{1 + (\lambda \dot {\gamma}) ^ {1 - n}},
+$$
+
+where $\eta _ { 0 }$ is the Newtonian viscosity, $\eta _ { \infty }$ is the infinite shear viscosity (usually assumed to be zero for the Cross model), is the natural time constant of the fluid $( 1 / \lambda$ is the critical shear rate at which the fluid changes from Newtonian to power-law behavior), and is the flow behavior index in the power law regime.
+
+Input File Usage: \*VISCOSITY, DEFINITION=CROSS
+
+Abaqus/CAE Usage: The Cross model is not supported in Abaqus/CAE.
+
+# Herschel-Bulkley
+
+The Herschel-Bulkley model can be used to describe the behavior of viscoplastic fluids, such as Bingham plastics, that exhibit a yield response. The viscosity is expressed as
+
+$$
+\eta = \left\{ \begin{array}{c c c} \eta_ {0} & \text {if} & \tau <   \tau_ {0}; \\ \frac {1}{\dot {\gamma}} \left(\tau_ {0} + k (\dot {\gamma} ^ {n} - (\tau_ {0} / \eta_ {0}) ^ {n})\right) & \text {if} & \tau \geq \tau_ {0}. \end{array} \right.
+$$
+
+Here $\tau _ { 0 }$ is the “yield” stress and $\eta _ { 0 }$ is a penalty viscosity to model the “rigid-like” behavior in the very low strain rate regime $( \dot { \gamma } \le \tau _ { 0 } / \eta _ { 0 } )$ , when the stress is below the yield stress, $\tau \leq \tau _ { 0 }$ . With increasing strain rates, the viscosity transitions into a power law model once the yield threshold is reached, $\tau \geq \tau _ { 0 }$ . The parameters and are the flow consistency and the flow behavior indexes in the power law regime, respectively. Bingham plastics correspond to the case $n { = } 1$ .
+
+Input File Usage: \*VISCOSITY, DEFINITION=HERSCHEL-BULKLEY
+
+Abaqus/CAE Usage: The Herschel-Bulkley model is not supported in Abaqus/CAE.
+
+<!-- source-page: 622 -->
+
+# Powell-Eyring
+
+This model, which is derived from the theory of rate processes, is relevant primarily to molecular fluids but can be used in some cases to describe the viscous behavior of polymer solutions and viscoelastic suspensions over a wide range of shear rates. The viscosity is expressed as
+
+$$
+\eta = \eta_ {\infty} + (\eta_ {0} - \eta_ {\infty}) \frac {\sinh^ {- 1} (\lambda \dot {\gamma})}{\lambda \dot {\gamma}},
+$$
+
+where $\eta _ { 0 }$ is the Newtonian viscosity, $\eta _ { \infty }$ is the infinite shear viscosity, and represents a characteristic time of the measured system.
+
+Input File Usage: \*VISCOSITY, DEFINITION=POWELL-EYRING
+
+Abaqus/CAE Usage: The Powell-Eyring model is not supported in Abaqus/CAE.
+
+# Ellis-Meter
+
+The Ellis-Meter model expresses the viscosity in terms of the effective shear stress, ${ \boldsymbol { \tau } } = { \sqrt { { \frac { 1 } { 2 } } \mathbf { S } : \mathbf { S } } } .$ , as:
+
+$$
+\eta = \eta_ {\infty} + \frac {(\eta_ {0} - \eta_ {\infty})}{1 + (\tau / \tau_ {1 / 2}) ^ {(1 - n) / n}},
+$$
+
+where $\tau _ { 1 / 2 }$ is the effective shear stress at which the viscosity is 50% between the Newtonian limit, $\eta _ { 0 }$ and the infinite shear viscosity, $\eta _ { \infty }$ , and represents the flow index in the power law regime.
+
+Input File Usage: \*VISCOSITY, DEFINITION=ELLIS-METER
+
+Abaqus/CAE Usage: The Ellis-Meter model is not supported in Abaqus/CAE.
+
+# Tabular
+
+In Abaqus/Explicit the viscosity can be specified directly as a tabular function of shear strain rate and temperature. In Abaqus/CFD only shear strain rate dependence is supported.
+
+Input File Usage: \*VISCOSITY, DEFINITION=TABULAR
+
+Abaqus/CAE Usage: Specifying the viscosity directly as a tabular function is not supported in Abaqus/CAE.
+
+# User-defined (Abaqus/Explicit only)
+
+In Abaqus/Explicit you can specify a user-defined viscosity in user subroutine VUVISCOSITY (see “VUVISCOSITY,” Section 1.2.26 of the Abaqus User Subroutines Reference Guide).
+
+Input File Usage: \*VISCOSITY, DEFINITION=USER
+
+Abaqus/CAE Usage: User-defined viscosity is not supported in Abaqus/CAE.
+
+<!-- source-page: 623 -->
+
+The temperature-dependence of the viscosity of many polymer materials of industrial interest obeys a time-temperature shift relationship in the form:
+
+$$
+\eta (\dot {\gamma}, \theta) = a _ {T} (\theta) \eta (a _ {T} (\theta) \dot {\gamma}, \theta_ {0}),
+$$
+
+where $a _ { T } ( \theta )$ is the shift function and $\theta _ { 0 }$ is the reference temperature at which the viscosity versus shear strain rate relationship is known. This concept for temperature dependence is usually referred to as thermo-rheologically simple (TRS) temperature dependence. In the Newtonian limit for low shear rates, when $\dot { \gamma } \to 0$ , we have
+
+$$
+\eta_ {0} (\theta) = \lim _ {\dot {\gamma} \rightarrow 0} \eta (\dot {\gamma} \theta) = a _ {T} (\theta) \eta_ {0} (\theta_ {0}).
+$$
+
+Thus, the shift function is defined as the ratio of the Newtonian viscosity at the temperature of interest to that of the chosen reference state: $a _ { T } ( \theta ) = { \eta _ { 0 } ( \theta ) } / { \eta _ { 0 } ( \theta _ { 0 } ) }$ .
+
+See “Thermo-rheologically simple temperature effects” in “Time domain viscoelasticity,” Section 22.7.1, for a description of the different forms of the shift function available in Abaqus.
+
+Input File Usage: Use the following options to define a thermo-rheologically simple (TRS) temperature-dependent viscosity:
+
+\*VISCOSITY
+
+\*TRS
+
+Abaqus/CAE Usage: Defining a thermo-rheologically simple temperature-dependent viscosity is not supported in Abaqus/CAE.
+
+# Use with other material models
+
+Material shear viscosity in Abaqus/Explicit must be used in combination with an equation of state to define the material’s volumetric mechanical behavior (see “Equation of state,” Section 25.2.1).
+
+# Elements
+
+Material shear viscosity can be used with any solid (continuum) elements in Abaqus/Explicit except plane stress elements and with any fluid (continuum) elements in Abaqus/CFD.
+
+<!-- source-page: 624 -->
+
+<!-- source-page: 625 -->
+
+# 26.2 Heat transfer properties
+
+• “Thermal properties: overview,” Section 26.2.1   
+• “Conductivity,” Section 26.2.2   
+• “Specific heat,” Section 26.2.3   
+• “Latent heat,” Section 26.2.4
+
+<!-- source-page: 626 -->
+
+<!-- source-page: 627 -->
+
+# 26.2.1 THERMAL PROPERTIES: OVERVIEW
+
+The following properties describe the thermal behavior of a material and can be used in heat transfer and thermal stress analyses (see “Heat transfer analysis procedures: overview,” Section 6.5.1):
+
+• Thermal conductivity: When heat flows by conduction, the thermal conductivity must be defined (“Conductivity,” Section 26.2.2).   
+• Specific heat: In transient heat transfer analyses as well as adiabatic stress analyses the specific heat of a material must be defined (“Specific heat,” Section 26.2.3).   
+• Latent heat: When a material changes phase, the change in internal energy can be significant. The amount of energy liberated or absorbed can be defined by specifying a latent heat for each phase change a material undergoes (“Latent heat,” Section 26.2.4).
+
+<!-- source-page: 628 -->
+
+<!-- source-page: 629 -->
+
+# 26.2.2 CONDUCTIVITY
+
+Products: Abaqus/Standard Abaqus/Explicit Abaqus/CFD Abaqus/CAE
+
+# References
+
+• “Material library: overview,” Section 21.1.1   
+• “Thermal properties: overview,” Section 26.2.1   
+• \*CONDUCTIVITY   
+• “Specifying thermal conductivity,” Section 12.10.1 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
+
+# Overview
+
+A material’s thermal conductivity:
+
+• must be defined for “Uncoupled heat transfer analysis,” Section 6.5.2; “Fully coupled thermal-stress analysis,” Section 6.5.3; and “Coupled thermal-electrical analysis,” Section 6.7.3;   
+• 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);   
+• can be linear or nonlinear (by defining it as a function of temperature);   
+• can be isotropic, orthotropic, or fully anisotropic; and   
+• can be specified as a function of temperature and/or field variables.
+
+# Directional dependence of thermal conductivity
+
+Isotropic, orthotropic, or fully anisotropic thermal conductivity can be defined. Only isotropic thermal conductivity can be defined for an incompressible fluid dynamic analysis that includes an energy equation. For orthotropic or anisotropic thermal conductivity, a local orientation (“Orientations,” Section 2.2.5) must be used to specify the material directions used to define the conductivity.
+
+# Isotropic conductivity
+
+For isotropic conductivity only one value of conductivity is needed at each temperature and field variable value. Isotropic conductivity is the default.
+
+Input File Usage: \*CONDUCTIVITY, TYPE=ISO
+
+Abaqus/CAE Usage: Property module: material editor: Thermal→Conductivity: Type: Isotropic
+
+# Orthotropic conductivity
+
+For orthotropic conductivity three values of conductivity $( k _ { 1 1 } , k _ { 2 2 } , k _ { 3 3 } )$ are needed at each temperature and field variable value.
+
+Input File Usage: \*CONDUCTIVITY, TYPE=ORTHO
+
+<!-- source-page: 630 -->
+
+Abaqus/CAE Usage: Property module: material editor: Thermal→Conductivity: Type: Orthotropic
+
+# Anisotropic conductivity
+
+For fully anisotropic conductivity six values of conductivity $( k _ { 1 1 } , k _ { 1 2 } , k _ { 2 2 } , k _ { 1 3 } , k _ { 2 3 } , k _ { 3 3 } )$ are needed at each temperature and field variable value.
+
+Input File Usage: \*CONDUCTIVITY, TYPE=ANISO
+
+Abaqus/CAE Usage: Property module: material editor: Thermal→Conductivity: Type: Anisotropic
+
+# Elements
+
+Thermal conductivity is active in all heat transfer, coupled temperature-displacement, coupled thermalelectrical-structural, and coupled thermal-electrical elements in Abaqus. Isotropic thermal conductivity is active in fluid (continuum) elements in Abaqus/CFD for incompressible fluid dynamic analyses that include an energy equation.