The following equations describe the computation of orthotropic secant coefficients of thermal expansion. To perform this computation, one load case is needed in addition to the 6 load cases needed for computation of the orthotropic elasticity constants.
For a orthotropic linear elastic material with thermal strain, the strain is given by:
 | (3–34) |
where the thermal strain is given by:
 | (3–35) |
If we now fix the strain 
and enforce an increase of the temperature by 
from the zero-thermal-strain reference temperature, we
obtain:
 | (3–36) |
Inserting the definition of the thermal strain, this simplifies to:
 | (3–37) |
The values of the stresses 
are obtained by integrating and normalizing the force
reactions on the boundary of the RVE. The stiffness matrix [D] is known from the
other 6 load cases. From those values, we can compute the secant coefficients of
thermal expansion.
It only remains to specify how the boundary conditions are actually enforced, which is done in the following sections.
To enforce a vanishing macroscopic strain we enforce the following conditions:
On the faces normal to the X-axis, enforce:
 | (3–38) |
On the faces normal to the Y-axis, enforce:
 | (3–39) |
On the faces normal to the Z-axis, enforce:
 | (3–40) |
To avoid rigid body motions, enforce
To enforce a thermal strain, we fix the zero-strain reference temperature
and raise the temperature by a small fixed 
.
To enforce a vanishing macroscopic strain we enforce the following conditions:
On the faces normal to the X-axis, enforce:
 | (3–41) |
On the faces normal to the Y-axis, enforce:
 | (3–42) |
On the faces normal to the Z-axis, enforce:
 | (3–43) |
To enforce a thermal strain, we fix the zero-strain reference temperature
and raise the temperature by a small fixed 
.