invariants

Dear,
I am having problem to find "Cauchy stress(σ) " and " Spatial elastic Tensor (c)" from the "strain energy function (ψ)" using 2nd Piola Kirchhoff stress ( S) of transversely isotropic hyper-elastic material. as per the Mechanics the 

When you first start learning finite deformation plasticity, you will run into a plastic flow rate that can be derived from a flow potential such that