Defining a Thermal Interaction for a Cohesive Element

You can define thermal interaction between the top and bottom surfaces of a coupled temperature-displacement cohesive element.

A thermal interaction between the top and bottom surfaces of a coupled temperature-displacement cohesive element:

can be included in a thermal-stress analysis (Fully Coupled Thermal-Stress Analysis);
can involve conductive heat transfer between the top and bottom surfaces;
can involve radiative heat transfer between the top and bottom surfaces when the surfaces are separated by a narrow gap; and
can be specified by a material model.

This page discusses:

Specifying Thermal Conductance of a Cohesive Element
Modeling Conductance between the Top and Bottom Surfaces of a Cohesive Element
Modeling Radiation between Surfaces When the Gap Is Small

Specifying Thermal Conductance of a Cohesive Element

The thermal conductance between the top and bottom surfaces of a coupled temperature-displacement cohesive element can be defined by specifying gap conductance as part of the material definition.

Input File Usage

Use the following options to define the thermal conductance with a material model:

COHESIVE SECTION, ELSET=name, MATERIAL=name
MATERIAL, NAME=name
GAP CONDUCTANCE

Modeling Conductance between the Top and Bottom Surfaces of a Cohesive Element

The conductive heat transfer between the top and bottom surfaces of a cohesive element is assumed to be defined by

q = k (θ_{A} - θ_{B}),

where q is the heat flux per unit area crossing the cohesive element from point A on the top surface to point B on the bottom surface, $θ_{A}$ and $θ_{B}$ are the temperatures of the points on the surfaces, and k is the gap conductance. Point A is a node on the top surface, and point B is the corresponding node on the bottom surface.

You can define k directly or in user subroutine GAPCON.

Defining Gap Conductance Directly

When defining k directly, define it as

k = k (d, \bar{θ}, \bar{| \dot{m} |}, {\bar{f}}_{γ}),

where

d: is the normal separation between A and B,
$\bar{θ} = \frac{1}{2} (θ_{A} + θ_{B})$: is the average of the surface temperatures at A and B,
$\bar{| \dot{m} |}$: is not considered here and should be set as zero, and
${\bar{f}}_{γ} = \frac{1}{2} (f_{γ}^{A} + f_{γ}^{B})$: is the average of any predefined field variables at A and B.

Defining Gap Conductance as a Function of Normal Separation

You can create a table of data defining the dependence of k on the variables listed above. The default in Abaqus is to make k a function of the separation d. When k is a function of separation, d, the tabular data must start at zero separation and define k as d increases. At least two pairs of k-d points must be given to define k as a function of the separation. The value of k drops to zero immediately after the last data point, so there is no heat conductance when the separation is greater than the value corresponding to the last data point.

Input File Usage

GAP CONDUCTANCE 
 $k$ , d,  $\bar{θ}$

Defining Gap Conductance as a Function of Predefined Field Variables

In addition to the dependencies mentioned previously, the gap conductance can be dependent on any number of predefined field variables, ${\bar{f}}_{γ}$ . To make the gap conductance depend on field variables, at least two data points are required for each field variable value.

Input File Usage

GAP CONDUCTANCE, DEPENDENCIES=n 
k, d,  $\bar{θ}$ ,  $\bar{| \dot{m} |}$ ,  ${\bar{f}}_{γ}$

Defining the Gap Conductance in User Subroutine GAPCON

You can define k in user subroutine GAPCON. In this case there is greater flexibility in specifying the dependencies of k. It is no longer necessary to define k as a function of the average of the two surface's temperatures, mass flow rates, or field variables:

k = k (d, p, θ_{A}, θ_{B}, {| \dot{m} |}_{A}, {| \dot{m} |}_{B}, f_{γ}^{A}, f_{γ}^{B}) .

The pressure and mass flow rates in user subroutine GAPCON are not used to model conductance in cohesive elements, and the variables should be set to zero.

Input File Usage

GAP CONDUCTANCE, USER

Modeling Radiation between Surfaces When the Gap Is Small

Abaqus assumes that radiative heat transfer between closely spaced surfaces occurs in the direction of the normal between the top and bottom surfaces.

The gap radiation functionality in Abaqus is intended for modeling radiation between surfaces across a narrow gap. A more general capability for modeling radiation is available in Abaqus/Standard (see Cavity Radiation in Abaqus/Standard).

Radiative heat transfer is defined as a function of normal separation between the top and bottom surfaces through the effective view factor. Abaqus maintains the radiative heat flux even when the surfaces are in contact. This causes only a minor inaccuracy since normally the heat flux from conduction is much larger than the radiative heat flux.

Abaqus defines the heat flow per unit surface area between corresponding points as

q = C [{(θ_{A} - θ^{Z})}^{4} - {(θ_{B} - θ^{Z})}^{4}],

where q is the heat flux per unit surface area crossing the gap at this point from surface A to surface B, $θ_{A}$ and $θ_{B}$ are the temperatures of the two surfaces, $θ^{Z}$ is the absolute zero on the temperature scale being used, and the coefficient C is given by

C = \frac{F σ}{1 / ϵ_{A} + 1 / ϵ_{B} - 1},

where $σ$ is the Stefan-Boltzmann constant, $ϵ_{A}$ and $ϵ_{B}$ are the surface emissivities, and F is the effective view factor, which corresponds to viewing the main surface from the secondary surface.

The view factor F must be defined as a function of the separation, d, and should have a value between 0.0 and 1.0. At least two pairs of F-d points are required to define the view factor, and the tabular data must start at zero separation (closed gap) and define the view factor as the separation increases. The value of F drops to zero immediately after the last data point, so there is no radiative heat transfer when the separation is greater than the value corresponding to the last data point.

Input File Usage

GAP RADIATION 
 $ϵ_{A}$ ,  $ϵ_{B}$  
 $F_{0}$ ,  $0.$  
 $F_{1}$ ,  $d_{1}$  
…

Specifying the Value of Absolute Zero

You must specify the value of $θ^{Z}$ .

Input File Usage

PHYSICAL CONSTANTS, ABSOLUTE ZERO= $θ^{Z}$

Specifying the Stefan-Boltzmann Constant

You must specify the Stefan-Boltzmann constant, $σ$ .

Input File Usage

PHYSICAL CONSTANTS, STEFAN BOLTZMANN= $σ$

Improving Convergence in Abaqus/Standard

Because the heat flux due to radiation is a strongly nonlinear function of the temperature, the radiation equations are strongly nonsymmetric and using the unsymmetric matrix storage and solution scheme for the step may improve the convergence rate in Abaqus/Standard (see Defining an Analysis).