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On page 587 of Finite Element Procedures by Bathe the author gives the following kinematic transformations

$$ {}^t\tau_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^t_oS_{rs} \; {}^t_ox_{j,s} $$

and

$$ {}^{t + \Delta t}_{\;\;\;\;t} S_{ij} = \frac{{}^t\rho}{{}^o\rho} \; {}^t_ox_{i,r} \; {}^{t + \Delta t}_{\;\;\;o}S_{rs} \; {}^t_ox_{j,s} $$

I can derive the first statement but not the second. Specifically, I do not understand how the LHS of the second equation is the 2nd Piola-Kirchhoff stress ${}^{t + \Delta t}_{\;\;\;\;t} S_{ij}$ and not of the Cauchy stress term ${}^{t + \Delta t}_{\;\;\;\;t} \tau_{ij}$.

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