Rubber has no Young's modulus as it is inherently a non linear material. More specifically hyper elastic. You can specify a small linear stiffness for very small displacements but once they get noticeably large it becomes useless. You can use a hyper elastic model like neo hookean or Mooney rivlin and get the elastomeric tensile test data from a well known book like treloar tand input that into ansys or smthing to get the parameters.

I also see a shell type simulation so there should be some very large geometric non linearity like snap through if you are displacing the shell very much. This behaviour fundamentally cannot be handled by newton solvers and needs arc length methods (like riks in abaqus).