The two stiffnesses denote the different configurations in which the stress is calculated. In the first case, when you use the Cauchy stress, it is in the current configuration (similar to true stress-strain) while in the second case when you use 2-PK, the equation is in employed in reference configuration. So you have to use the stiffnesses in the respective configurations as well.

Stiffness matrix

Thank you Aritra Chakraborty

You mean that I use tangent modulus of true stress-strain curve when using the first equation, but in the second one which curve could I use ?

Thanks.

At each step strain increments has to be used to compute C (for instance using mechanical power density ), this depends on how the code is organised and if you are in large displacement, hyperelasticity or have internal parameters in the strain definition etc.., refer to following (paragraph 2.2.2 and 2.2.3 and 3.1) :

http://tfel.sourceforge.net/documents/mfront/finitestrain.pdf

(in french)

The primary difference between the two equations is

the first one is defined for small deformation framework (reference configuration remains same as deformed configuration and the decomposition of strain is additive)

whereas the 2nd equation indicates large deformation framework (decomposition of strain or deformation gradient is multiplicative, there exists an intermediate configuration in between reference and deformed configuration) where we define Hooke's law in intermediate configuration or plastic configuration, that is why stress definition is 2nd Piola-Kirchhoff stress and strain definition is finite green Lagrangian strain.

hope it helps

Thank you Satyapriya Gupta

But do both C have the same value?

Khaled Megahed , C, stiffness tensor is material property (number of independent components depends on isotropic or anisotropic assumption), so I believe it should not change with formulation.

thank you Satyapriya Gupta