What is the goal or purpose of this analysis? What is it that you want the results to tell you? Are you looking to calculate the shear modulus of the core in a specific direction?
In the graphics, I can see a row of ux, uy constraints on one edge of the top facesheet and I understand the corner node is constrained in x,y, and z. Having one straight line of nodes like that creates a hinge.
In the second image, it's not clear which nodes are given a displacement. I also don't know how thick is the core compared with the 10 mm displacement you mentioned.
I recommend switching to applied forces and don't use any applied displacements.
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