Consider for the moment that we want to compute stress-strain curves for a tension test in X direction for a periodic RVE with a periodic mesh. The basic principle is that we specify several values for the strain in the X direction and measure the force in the X direction. After normalization, this yields the corresponding stress values.
In contrast to the boundary conditions specified here, we allow the RVE to contract
macroscopically in Y and Z direction; in other words, we allow a non-zero lateral strain
and 
. To do so, we introduce 6 degrees of freedom 
, 
, 
, 
, 
, and 
(via an additional node in the finite element model) and let them correspond
to the macroscopic strains.
Assume that the RVE occupies the volume [0,Lx] x [0,Ly] x [0,Lz]. Then we enforce the following boundary conditions:
On the faces normal to the X-axis, enforce
 | (2–1) |
On faces normal to the Y-axis, enforce
 | (2–2) |
On faces normal to the Z-axis, enforce
 | (2–3) |
In addition to these periodicity conditions, rigid body motions must also be prevented. This is done by enforcing
The strain in X is enforced by setting
while uypivot, uzpivot are unconstrained, allowing lateral contraction of the RVE.
is successively set to the values specified by the user.
The remaining degrees of freedom , , and are not needed for this load case and are set to zero. They will be used in the
shear load cases to introduce shear strains.
Note that we aim to compute engineering strain vs. engineering stress (even in case large deformations are active). Thus, the force in the X direction is integrated and normalized by the original area Ly Lz to obtain σx.
The lateral strains can be extracted from the degrees of freedom of the pivot node:
In case true (logarithmic) strains are needed, you can compute them via
To compute true stresses, one needs to compensate the stress for area changes due to contraction, which leads to
The boundary conditions for the other test cases will be specified in the following.
Note that the stress-strain computation in Material Designer does not take into account (microscopic) damage, such as pull-out of fibers, micro-cracks, delamination, nonlinear interfaces, etc. If these effects are important, then Material Designer is not suitable to model and simulate this RVE.
Further note that it is not possible to setup a "clean" shear case if large deformations are active; in other words, there are always some other macroscopic strains that are non-zero (see Polar Decomposition of a Shearing Deformation and the corresponding description). For stress-strain curves, there will in general be a non-negligible effect given that the applied strains can be large.
Note that Material Designer is limited to rate-independent material behavior. You can use constituent materials that show a hyperelastic or a rate-independent plastic behavior. Rate-dependent material properties, like viscoplasticity, viscoelasticity, and creep are not supported.
For completeness, the boundary conditions used in the different load cases for the stress-strain curves are presented here as well.
In each load case, one of the quantities 
, 
, 
, 
, 
, and 
is successively set to the values specified by the user.
In tension and compression cases, the remaining two normal strains of 
, 
, 
are left free, while 
,
, and
are set to zero.
In the simple shear cases, all quantities , , , , , and except the one specified by the user are set to zero.
Then there is a one-to-one correspondence between the macroscopic strains and the degrees of freedom of the pivot node:
 | (2–4) |
 | (2–5) |
With these macroscopic strains in place, the boundary conditions are defined as follows:
On the faces normal to the X-axis, enforce
 | (2–6) |
On the faces normal to the Y-axis, enforce
 | (2–7) |
On the faces normal to the Z-axis, enforce
 | (2–8) |
To avoid rigid body motions, enforce
 | (2–9) |
Note that the compression and tension cases are the same, except that you are expected to specify different strain values.
For the tension and compression tests, one quantity of
, , and is nonzero, the other two normal strain are left free. All the shear
strains 
, 
, and 
are set to zero.
Once again there is a one-to-one correspondence between the macroscopic strains and the degrees of freedom of the pivot node:
 | (2–10) |
 | (2–11) |
On the faces normal to the X-axis, enforce
 | (2–12) |
On the faces normal to the Y-axis, enforce
 | (2–13) |
On the faces normal to the Z-axis, enforce
 | (2–14) |
For the shear XY case, the boundary conditions are set as follows
( a specified by the user):
On faces normal to the X-axis, enforce
 | (2–15) |
On faces normal to the Y-axis, enforce
 | (2–16) |
On faces normal to the Z-axis, enforce
 | (2–17) |
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).
Note that for the non-periodic shear test cases, we refrain from using additional degrees of freedom.