ABAQUS Benchmarks Manual  


1.6.3 Coupled temperature-displacement analysis: one-dimensional gap conductance and radiation

Products: ABAQUS/Standard  ABAQUS/Explicit  

This example illustrates two elementary nonlinear cases of one-dimensional, fully coupled, heat transfer and stress analysis. The problems are simple enough that exact solutions are obtained easily, thus providing verification of the numerical solutions obtained with ABAQUS.

Problem description

Solution

Mechanical equilibrium along the rod requires that

where S is the distance along the rod measured from the fixed end, Integrating along the rod, the stress is

where p is the pressure transmitted by contact between the end of the rod, B, and the adjacent fixed point,

Thermal equilibrium requires that the heat flux along the rod, q, has no gradient:

Integrating this along the rod and imposing the boundary condition that the flux at B is the same as the flux transmitted from B to C through the gap, , then gives the thermal equilibrium equation

Since we assume that the strains are small, the strain at any point in the rod is

and the displacement is

The rod is assumed to be made of a linear elastic material, so the stress constitutive equation is

where the modulus, E, and the thermal expansion coefficient, , are constants (they are not temperature dependent).

Heat conduction in the rod is assumed to be governed by Fourier's law, which states that the heat flux is determined by

where is the thermal conductivity of the rod and is also assumed to be constant. Combining thermal equilibrium with the Fourier law in the rod shows that is constant in the rod, so the temperature, , varies linearly along the rod:

where L is the length of the rod.

The heat flux in the gap, , between the end of the rod, B, and the fixed point C is assumed to be proportional to the difference in temperature between B and C:

Case 1

First we assume that the gap is open and that the gap thermal conductivity, , increases linearly as the gap reduces, so

where and are nonnegative constants and is the displacement of point Gap radiation is neglected in these calculations.

The thermal boundary conditions are that the temperatures at A and C, and , are held constant. The mechanical boundary conditions are that points A and C are fixed. Since in this case the end of the rod never touches C, force equilibrium requires that

The equations given above define the problem. Their solution is readily developed as follows. Combining integrated force equilibrium with the linear elastic constitutive equation and the displacement relationship, , gives

Thermal equilibrium combined with Fourier's law and the gap heat flux equation then gives

With the assumed form of the gap thermal conductivity, , and assuming 0, this is

Substituting for then gives a quadratic equation for :

The roots of this quadratic equation provide two solutions for The solutions for are available once is determined. Only one of the two solutions gives a value of for which 0; hence, this is the only physically acceptable solution.

As a numerical example the parameters are chosen in consistent units as  1.0; 10–5,  1.0;  100;  400°; and  200°.

These values give  285.4° or –4485°, so 3.427 × 10–3 or –2.043 × 10–2. The second solution must be rejected as it gives 0. The first solution is valid so long as

Next we assume that the gap is open and that the gap radiation viewfactor, , increases linearly as the gap reduces; so

where and are nonnegative constants and is the displacement of point In this case gap conduction is neglected.

The thermal and mechanical boundary conditions are the same as the gap conduction problem considered above. Force equilibrium requires that

Following a procedure similar to that used in the gap conduction problem and combining integrated force equilibrium with the linear elastic constitutive equation and the displacement relationship, , gives

Combining thermal equilibrium with Fourier's law and the gap heat flux equation then gives

With the assumed form of the gap radiation, , and assuming 0, this is

Substituting for then gives the following equation for :

is obtained by solving the above equation numerically. The solutions for are available once is determined.

As a numerical example the parameters are chosen in consistent units as  1.0; 10–5;  1.0;  50; 1.E–8; 1.0;  400°;  200°; and absolute zero –460.

These values give  222.4°, so 3.112 × 10–3. This solution is valid so long as All other solutions must be rejected since they give or

Case 2

In this case the rod is always in contact with C. Combining the integrated equilibrium equation with the mechanical constitutive model gives

Combining this with the temperature solution, , gives

Integrating this along the rod using the displacement relationship, , gives the pressure as a function of the temperature at points A and B,

since

In this case the conductivity of the closed gap is proportional to the contact pressure:

where and are nonnegative constants. Since , with this behavior for we have

Combining this with the equation for the pressure provides a quadratic equation for :

The roots of this equation provide two solutions for , and the corresponding values of p are then defined by Only one solution gives a positive value for p; the other must be rejected because it is inconsistent with the assumption that the gap is closed.

As a numerical example the parameters are chosen in consistent units as  10;  2;  0.2;  10–5;  105; 1.0;  200°; and  100°.

These values give  122.6° or –342.6°, so  161.3 or –71.3. The second solution must be rejected as it gives

Results and discussion

Input files

Figure

Figure 1.6.3–1 Coupled temperature-displacement analysis specifications.