Thermal interface definition

Abaqus/Standard allows heat flow across an interface via conduction or radiation. Generally, both modes of heat transfer are present to some degree. Regardless of the mode, heat transfer across the interface is assumed to occur only in the normal direction.

This page discusses:

Conduction
Radiation
Jacobian matrix

Conduction

Heat conduction across the interface is assumed to be defined by

q = k (θ_{A} - θ_{B}),

where $q$ is the heat flux per unit area crossing the interface from point $A$ on one surface to point $B$ on the other, $θ_{A}$ and $θ_{B}$ are the temperatures of the points on the surfaces, and $k$ is the gap conductance.

The derivatives of $q$ are

\frac{\partial q}{\partial θ_{A}} = k + \frac{1}{2} \frac{\partial k}{\partial \bar{θ}} (θ_{A} - θ_{B})

and

\frac{\partial q}{\partial θ_{B}} = - k + \frac{1}{2} \frac{\partial k}{\partial \bar{θ}} (θ_{A} - θ_{B}),

where

\bar{θ} = \frac{1}{2} (θ_{A} + θ_{B}) .

Radiation

The heat flow per unit area between corresponding points is assumed to be given by

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 expression above accurately represents the radiation heat exchange between two infinite plates that are close to each other; in this case the effective view factor is F = 1.0. In all other cases the effective view factor serves as a scaling factor that approximates the radiation heat exchange between the two finite surfaces.

The derivatives of $q$ are

\frac{\partial q}{\partial θ_{A}} = 4 C {(θ_{A} - θ^{Z})}^{3}

and

\frac{\partial q}{\partial θ_{B}} = - 4 C {(θ_{B} - θ^{Z})}^{3} .

Jacobian matrix

The contribution to the variational statement of thermal equilibrium is

δ Π = q A δ θ_{A} - q A δ θ_{B},

where $A$ is the area. The contribution to the Jacobian matrix for the Newton solution is

d δ Π = d q A δ θ_{A} - d q A δ θ_{B},

where

d q = \frac{\partial q}{\partial θ_{A}} d θ_{A} + \frac{\partial q}{\partial θ_{B}} d θ_{B} .

For “tied” thermal contact the temperature at point $A$ is constrained to have the same temperature as point $B$ . The Lagrange multiplier method is used to impose the constraint by augmenting the thermal equilibrium statement as follows:

δ Π = δ λ (θ_{A} - θ_{B}) + λ (δ θ_{A} - δ θ_{B}),

where $λ$ is the Lagrange multiplier. The contribution to the Jacobian matrix for the Newton solution is

d δ Π = δ λ (d θ_{A} - d θ_{B}) + d λ (δ θ_{A} - δ θ_{B}) .