Help: calculate elasticity tensor for hyperelastic material with plane stress configuration
I am confused with the calculation of elasticity tensor of hyperelastic material in plane stress configuration. Your helps are very much appreciated. Below is my question:
Incompressible strain energy function is expressed as, saying, U=***+p(J-1).
Therefore, the second piola-kirchhoff stress is: S=***+pC^-1 (the Lagrange muliplier p is treated as an arbitrary constant here, right?).
Using the plane stress condition, S<33>=0 so that p is updated as a function of deformation.
Now, to calculate the elasticity tensor, H=2*dS/dC. Should I continue to treat p as a constant, independent of deformation, and get:
(1): H=***+2*p*d(C^-1)/dC otherwise, should I treat it here as a function of deformation and get:
I am not sure which one is correct (or none of them?). The elasticity tensor H possesses both major and minor symmetry (is this always correct?), if H takes the form (1), major symmetry holds;
but if H takes the form (2), it lost its major symmetry. Any comment about this? Thank you all very much indeed!