In the updated Lagrange formulation, the principle of virtual power is $$ \int_{v}^{}\,\left[\overset{\triangle}{T}\cdot\breve{D}+(LT)\cdot\breve{L}\right]dv = \int_{a}\dot{t}\cdot\breve{v}da+\int_{v}\rho\dot{f}\cdot\breve{v}dv, $$ where $\overset{\triangle}{T}$ is Truesdell rate of Cauchy stress, $D$ is the deformation rate tensor and $L$ is the velocity gradient tensor. The variable with $\breve{}$ means the virtual value.
From some textbook, I learned that if the neo-Hookean type of free energy like $$ \mathit{\Psi}=\frac{\mu}{2}\left(\mathrm{tr}(C)-3\right)-\mu\ln(J)+\frac{\lambda}{2}\left(\ln(J)\right)^2, $$ Cauchy stress can be obtained by $$ T=\frac{\mu}{J}(B-I)+\frac{\lambda}{J}\ln(J)I $$ where $B$ is the right Cauchy-Green tensor and $\lambda$ and $\mu$ are material parameters. Truesdell rate of Cauchy stress can be calculated by the rate form of elastic constitutive equation as follows. $$ \overset{\triangle}{T}=C^{e}:D $$ $C^e$ is the elastic coefficient tensor and can be obtained by the following formula. $$ C^{e}_{ijkl}=\frac{\lambda}{J}\delta_{ij}\delta_{kl}+\frac{\mu-\lambda\ln(J)}{J}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}) $$
My questions are as follows:
- Which equations should I use for obtaining the current Cauchy stress, the third equation or fourth equation (with time-integration)?
- If I use the third equation, I have to use as $B$ the right Cauchy-Green tensor of the deformation occurs between the current step and initial step($B^{n}_{0}$), not the right Cauchy-Green tensor of the deformation occurs between current step and last step($B^{n}_{n-1}$)?
- When I obtain $C^{e}$ with the fifth equation, should I use the $J=\mathrm{det}F^{n}_{0}$, not $J=\mathrm{det}F^{n}_{n-1}$?
