I applied periodic boundary conditions (PBC) in Abaqus CAE using the *Equation constraint. After that, I performed an analysis on a single 8-node hexahedral element, applying pure shear strain in the xy direction. In Abaqus, I obtained a strain value of 1 for only the xy component, as expected. However, when I calculated the strain using the equation 
strain=𝐵×displacement
 I found values for components other than xy.

Could you help me understand how to resolve this issue? I’m wondering if it is due to the problem being an overdetermined system, since the periodic boundary conditions were applied without fixing any degrees of freedom, or if there might be another issue.

I have attached the node values, connectivity, and the displacement results from Abaqus. Any assistance would be greatly appreciated.

*NODE
     1,           1.,           1.,           1.
     2,           1.,           0.,           1.
     3,           1.,           1.,           0.
     4,           1.,           0.,           0.
     5,           0.,           1.,           1.
     6,           0.,           0.,           1.
     7,           0.,           1.,           0.
     8,           0.,           0.,           0.
*ELEMENT, TYPE=C3D8
1, 5, 6, 8, 7, 1, 2, 4, 3

---Result from rpt
      Node Label            U.U1            U.U2            U.U3
                         @Loc 1          @Loc 1          @Loc 1
-----------------------------------------------------------------
              1         3.04729        -2.91674        -1.52095
              2         2.54729        -2.91674        -1.52095
              3         3.04729        -2.91674        -1.52095
              4         2.54729        -2.91674        -1.52095
              5         3.04729        -3.41674        -1.52095
              6         2.54729        -3.41674        -1.52095
              7         3.04729        -3.41674        -1.52095
              8         2.54729        -3.41674        -1.52095