The following equations describe the computation of orthotropic elasticity constants for a periodic unit cell. To perform this computation, 6 load cases are considered: 3 tensile tests (X, Y, Z) and 3 shear tests (XY, YZ, XZ). A corresponding macroscopic strain is applied in each case, and reaction forces in the boundary faces of the RVE are used to assemble the stiffness matrix. Engineering constants are then extracted.
Consider the tensile test in the X-direction. For an orthotropic material, the following relation exists:
 | (3–16) |
If the strain in the X-direction is fixed to 
= 0.001 and all other strains are set to zero, the first
column of the stiffness matrix is obtained:
 | (3–17) |
Making use of the periodic structure, this is reached in the following way
(compare also [Li,(2008)] or [Li, et. al (2015)] for a more detailed discussion of
boundary conditions for unit cells). Assume the RVE occupies the volume
. On the faces normal to the X-axis, enforce
 | (3–18) |
On faces normal to the Y-axis, enforce
 | (3–19) |
On faces normal to the Z-axis, enforce
 | (3–20) |
In addition to these periodicity conditions, rigid body motions must also be prevented. This is done by enforcing
 | (3–21) |
There are alternatives to these periodic boundary conditions. Unless there exist enough symmetries, these alternatives lead to boundary effects. On periodic structures, periodic boundary conditions should be used.
To compute macroscopic stresses, the forces on the top faces are integrated.
Consider 
. The force in the X-direction at the face 
is integrated. 
is obtained by normalizing with the face area. 
and 
are obtained similarly. The entries for 
, 
, and 
in the stiffness matrix are easily obtained.
By repeating the steps for all the other load cases (see Periodic Boundary Conditions), all the entries for the stiffness matrix are obtained. The stiffness matrix is inverted to obtain the compliance matrix:
 | (3–22) |
Finally, the engineering constants 
, 
, 
, 
, 
, 
, 
, 
, and 
are computed from the relationship
 | (3–23) |
For completeness, the boundary conditions used in the different load cases
are presented here as well. In each load case, one quantity of
, 
, 
, 
, 
, and 
is set to a predefined value (0.001) and all the other
quantities are set to 0.
On the faces normal to the X-axis, enforce
 | (3–24) |
On the faces normal to the Y-axis, enforce
 | (3–25) |
On the faces normal to the Z-axis, enforce
 | (3–26) |
To avoid rigid body motions, enforce
 | (3–27) |
For the tensile tests, one quantity of 
, 
, and 
is nonzero, all the others are set to zero.
On the faces normal to the X-axis, enforce
 | (3–28) |
On the faces normal to the Y-axis, enforce
 | (3–29) |
On the faces normal to the Z-axis, enforce
 | (3–30) |
For the shear XY case, the boundary conditions are set as follows (with
).
On faces normal to the X-axis, enforce
 | (3–31) |
On faces normal to the Y-axis, enforce
 | (3–32) |
On faces normal to the Z-axis, enforce
 | (3–33) |
The boundary conditions for shear XZ can be obtained by switching the roles of y and z.
The boundary conditions for shear YZ can be obtained by switching the roles of x and y (starting from the shear XZ case).