 Multiplicative decomposition of the deformation gradient

The factor $A$ can encode an internal relaxed configuration, while $F_{\ast }=F{A}^{-1}$ determines the instantaneous elastic energy. Differentiating $F_{\ast }$ at fixed $F$ or fixed $A$ produces the corresponding nominal stress tensor and thermodynamic force conjugate to an internal variable. This is a kinematic decomposition, not an assertion that each factor separately derives from a global deformation map.