Hyperelasticity: relating two PK2 stress tensors in terms of two nonlinear displacements

I am working on numerical analysis for a nonlinear hyperelasticity problem. Given that the second Piola Kirchhoff stress tensor $$S$$ depends on the Green strain tensor $$E$$, which in turn depends on the nonlinear displacement $$u$$, is there a way to relate two values of $$S$$ that are obtained for different values of $$u$$ e.g. given $$E(u_1)$$ and $$E(u_2)$$, is there an approach to relate $$S(E(u_2))$$ to $$S(E(u_1))$$ in general terms?

Edit: I am also informed that a "Hill inequality" may be used for this issue, and I checked some research papers but I can't seem to connect the dots.

Context: We have it that the elastic work done is an integral of the double-dot contraction of an energetically conjugate pair i.e. $$W = \int{S:dE}$$ where $$dE$$ is a Gateaux derivative of $$E$$ at $$u$$ and in the direction of $$u$$.

My analysis has got me to a point where for displacements at two consecutive time steps, $$u_{n+1}$$ and $$u_n$$, I have an energy term $$S:\frac12 \Bigl(dE(u_{n+1}) - dE(u_n) + dE(u_{n+1}-u_n)\Bigr)$$ but I still need to obtain energetically conjugate terms for each tensor double dot product.

E.g. choosing $$S$$ at $$E(u_{n+1})$$ will only produce an energetically conjugate term for $$dE(U_{n+1})$$. As such I am thinking of a way to split up the terms, hence my original question.