a question about the concepts related to the plasticity

I have a question on the topic plasticity. In the Chaboche mixed hardening plasticity model, the return mapping equations are being solved in the Cartesian coordinate space for a plastic criteria such as Von Mises plastic criteria. I have two main questions.How Von Mises plastic criteria can be written in the principal space ( for mixed hardening model including stress and back stress in the principal space)? and How Return mapping equations are solved in the principal space? 
Can the results from Ansys Von Mises plastic criteria of the Chaboche mixed hardening model (solved in the Cartesian space) compared with the results of the equations solved in the principal space?

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