The solid orthotropic materials give a perfectly good mode shape. Did you turn down the Deformation Scale Factor as shown in the image below? Displacements in a Modal analysis are arbitrary, so you might need to turn this down until you get normal looking displacements.
I suppressed the Fixed Support and requested 9 modes instead of 6 modes. With no support, the first six modes are rigid body modes with no deformation and a practically zero frequency. Modes 7 and up are bending modes.
Orthotropic materials are not required to satisfy the Isotropic relation you showed above.
Orthotropic materials are required to satisify this requirement:
The way I would get the 2t thickness is by doing a split body on the two planes that bound a column of 2t faces so that you have separate bodies for the t thickness walls from the 2t thickness walls. In SpaceClaim, on the Workbench tab, use the Share button to have the mesh connect the separate bodies together.
I don’t think it is necessary to build a honeycomb cell model to get good orthotropic material constants. What would be more useful is if you can do a 3 or 4 point bending test on a physical honeycomb sandwich sample, or do a shear modulus test on a physical sample of the core.
One way to make the model solve faster is to replace solid elements on the face sheets with shell elements, however, you will have to do some work to align the material properties to a coordinate system. The reason is solid elements automatically align the material direction to the global coordiantes by default. However, shell elements automatically align the material direction to the element coordinates by default, and that can result in the material direction randomly pointing in different directions depending on how the face was meshed. You are almost required to override the default direction and define the material direction you want.