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Stupid Question Regarding Deformation Gradient


       I have a stupid question, may I know how to determine the increment of deformation gradient tensor, dF from equation 

            dF/dt = L * F_old               (1)

where dF/dt is rate of deformation gradient tensor, L is velocity gradient tensor, and F is deformation gradient tensor at beginning of a time increment. It is not simply 

            dF = dt * L * F_old            (2)

right? Because I think if a specimen do not undergo any deformation in a time increment dt, the velocity gradient, L should be a zero matrix. Thus according to equation (2), dF will become a zero matrix too. But this should not be right because from equation (3), if a specimen do not undergo deformation in a time increment, the deformation gradient at start of increment, F_old should be the same with deformation gradient at end of increment, F_new, this mean dF must be an identity matrix, which is difference with equation (2), may I know where is the mistake?


            F_new = dF  * F_old         (3)   


            Thank you.



In (3) don't you mean F_new=dF + F_old? Then dF=dt*L*F_old=0 is consistent with F_new = F_old.

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