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.


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