relation between the Euler-Lagrange strain tensor and the 2nd Piola-Kirchhoff stress tensor

At finite deformations, the 2nd Piola-Kirchhoff stress tensor S_{IJ} is work-conjugate to the Euler-Lagrange strain tensor E_{IJ}, i.e. the work done by the stress S_{IJ} at developing a strain E_{IJ} per unit volume in the reference configuration is

W = 1/2 S_{IJ} E_{IJ}  (Einstein summation over repeated indices implied)

I would like to express this in the same form as used in linear elasticity, i.e. to write S_{IJ} = M_{IJKL} E_{KL}, which would give me

W = 1/2 M_{IJKL} E_{IJ} E_{KL}

It is clear to me that E_{IJ} is a measurable quantity, but S_{IJ} is not (a pull-back of the Cauchy stress to the reference configuration). I think M_{IJKL} basically reduces to the elastic stiffness tensor C_{IJKL} when the strains are infinitesimaly small. However, how I am to think about M_{IJKL} when the strains are finite? If I express S_{IJ} using the Cauchy stress tensor, the equation  S_{IJ} = M_{IJKL} E_{KL} leads to

sigma_{kl} = 1/J F_{Jk} F_{Ik} M_{IJKL} E_{KL} = N_{klKL} E_{KL}

which is a relation between two measurable quantities, E_{KL} on the right and the Cauchy stress sigma_{kl} on the left (small indices refer to the deformed configuration while capital ones to the reference configuration). The tensor N_{klKL} (referring simultaneously to the deformed and the reference configuration) will again reduce to the elastic stiffness tensor for small strains. I would expect N_{klKL} could me measured. Can you, please, point me in the right direction (a paper or a discussion on the web)? Many thanks,

Roman