Are the same elastic properties used for large deformation theory as for small deformations?

I'm modeling some composites where I may need to formulate with large deflections and I can't seem to find if I should use the same elastic properties for when I'm using the cauchy strain definition and when I'm using the Green-Lagrange strain definition.
I have previously done tensile tests on steel specimen where the calculated young's modulus were different if they were calculated from cauchy strains or from Hencky strain (also called logarithmic- or true strain) but then, the young's modulus varied more from specimen to specimen than the influence of which strain definition was used.

Does anyone know if it is correct to use the same numeric values of elastic properties for different strain definitions and if not, how one can estimate the error or how one can calculate the numeric value of elastic properties of another strain definition from the values found in a data sheet?

submitted by /u/Partykongen
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