Elastic Deformation
In this section we want to examine the deformation of a linear elastic body caused by expansion through a heat distribution. We want
a displacement field
which solves the momentum equation
:
 |
|
|
(45) |
where the stress
is given by
 |
|
|
(46) |
In this formula
and
are the Lame coefficients,
is the
temperature expansion coefficient,
is the temperature distribution and
a reference temperature. Note that
Equation (1.47) is similar to Equation (1.31) introduced in section Section 1.4 but the
inertia term
has been dropped as we assume a static scenario here. Moreover, in
comparison to the Equation (1.32)
definition of stress
in Equation (1.48) an extra term is introduced
to bring in stress due to volume changes trough temperature dependent expansion.
Our domain is the unit cube
 |
|
|
(47) |
On the boundary the normal stress component is set to zero
 |
|
|
(48) |
and on the face with
we set the
-th component of the displacement to 0
 |
where |
 |
(49) |
For the temperature distribution we use
 |
|
|
(50) |
with a given positive constant
and location
in the domain.
When we insert Equation (1.48) we get a second order system of linear PDEs for the displacements
which is called
the Lame equation. We want to solve
this using the LinearPDE class to this. For a system of PDEs and a solution with several components the LinearPDE class
takes PDEs of the form
 |
(51) |
is of rank-4 Data object and
is of rank-2 Data object. We show here the coefficients relevant
for the we trying to solve. The full form is given in Equation (4.4).
The natural boundary conditions take the form:
 |
(52) |
Constraints take the form
where  |
(53) |
and
are each rank-1 Data object.
We can easily identify the coefficients in Equation (1.53):
 |
|
|
(54) |
 |
|
|
(55) |
The characteristic function
defining the locations and components where constraints are set is given by:
 |
(56) |
Under the assumption that
,
,
and
are constant we may use
. However,
this choice would lead to a different natural boundary condition which does not set the normal stress component as defined
in Equation (1.48) to zero.
Analogously to concept of symmetry for a single PDE, we call the PDE defined by Equation (1.53) symmetric if
 |
|
|
(57) |
This Lame equation is in fact symmetric, given the difference in
and
as compared to the scalar case.
The LinearPDE class is notified of this fact by calling its setSymmetryOn method.
After we have solved the Lame equation we want to analyse the actual stress distribution. Typically the von-Mises stress defined by
 |
(58) |
is used to detect material damage. Here we want to calculate the von-Mises and write the stress to a file for visualization.
The following script, which is available in heatedbox.py in the example directory, solves the Lame equation
and writes the displacements and the von-Mises stress into a file deform.xml in the VTK file format:
Figure 1.11:
von-Mises Stress and Displacement Vectors.
|
Finally the the results can be visualize by calling
Note that the filter CellToPointData is applied to create smooth representation of the
von-Mises stress. Figure 1.11 shows the results where the vertical planes showing the
von-Mises stress and the horizontal plane shows the vector representing displacements.
esys@esscc.uq.edu.au