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Bathe (reference below) outlines the updated Lagrangian (UL) and total Lagrangian (TL) approaches using the second Piola Kirchhoff (PK2) stress. Others (i.e., Ji, et al. and Abaqus) define the UL and TL formulations in terms of the Kirchhoff or the Cauchy stress in rate form. This form requires consideration of an objective rate (since the rate of the Kirchhoff or Cauchy stress is not objective).
Why even mess with the Kirchhoff or Cauchy stress in the governing equation? Why not just cast the equation as Bathe did in terms of the PK2 stress, since the rate of the PK2 stress is objective? If your constitutive equation is in terms of the Kirchhoff or Cauchy stress could you not just transform it through the pullback operator to cast it in terms of the PK2 stress?
Bathe, http://web.mit.edu/kjb/www/Books/FEP_2nd_Edition_4th_Printing.pdf )
Ji, et al., https://pdfs.semanticscholar.org/d915/d7bb83fb2b5a9f3d41f699751eb8ac557e3d.pdf
Abaqus 6.11 Theory Manual, Section 1.5.1, http://130.149.89.49:2080/v6.11/books/stm/default.htm
Why even mess with the Kirchhoff or Cauchy stress in the governing equation? Why not just cast the equation as Bathe did in terms of the PK2 stress, since the rate of the PK2 stress is objective? If your constitutive equation is in terms of the Kirchhoff or Cauchy stress could you not just transform it through the pullback operator to cast it in terms of the PK2 stress?
Bathe, http://web.mit.edu/kjb/www/Books/FEP_2nd_Edition_4th_Printing.pdf )
Ji, et al., https://pdfs.semanticscholar.org/d915/d7bb83fb2b5a9f3d41f699751eb8ac557e3d.pdf
Abaqus 6.11 Theory Manual, Section 1.5.1, http://130.149.89.49:2080/v6.11/books/stm/default.htm