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Hi Davide,
Some information is missing from your description. What are the boundary conditions? It sounds like there are 3 bodies, two 1 mm plates and the 6 mm thick cylinder. Is the bottom face of the bottom plate a Fixed Support? How did you apply the displacement to the top plate and how much displacement did you apply?
Did you use Contact between the cylinder and the plates? What type of Contact?
What material properties did you define for the cylinder besides the Young's Modulus (E)?
Please show a screen shot of the Analysis Settings.
Please show a screen shot of the Mesh on the Cylinder.