(Solved): A modified St. Venant-Kirchhoff constitutive behavior is defined by its corresponding strain energ ...

A modified St. Venant-Kirchhoff constitutive behavior is defined by its corresponding strain energy functional \( \Psi \) as \[ \Psi(J, \boldsymbol{E})=\frac{\kappa}{2}(\ln J)^{2}+\mu I_{\boldsymbol{E}}, \] where \( I I_{\boldsymbol{E}}=\operatorname{tr}\left(\boldsymbol{E}^{2}\right) \) denotes the second invariant of the Green's strain tensor \( \boldsymbol{E}, J \) is the Jacobian of the deformation gradient, and \( \kappa \) and \( \mu \) are positive material constants. (a) Obtain an expression for the second Piola-Kirchhoff stress tensor \( S \) as a function of the right Cauchy-Green strain tensor \( C \). (b) Obtain an expression for the Kirchhoff stress tensor \( \tau \) as a function of the left Cauchy-Green strain tensor \( \boldsymbol{b} \). (c) Calculate the material elasticity tensor.