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Hello everyone,
I'm studying the finite strain theory and have come across the maximum dissipation principle. It implies a dissipation function defined as
I quiet understood the physics but there is a mathematical relation that I don't understand. Given the above definition, we claim in a book that
?
I can't tell why, if you do please share it
Regards.
I'm studying the finite strain theory and have come across the maximum dissipation principle. It implies a dissipation function defined as
D=\tau:d-\dfrac{d\Psi}{dt}
\tau denotes the Kirchhoff stress tensor, d the eulerian deformation rate and \Psi=\Psi(b_e,\xi) the free energy, b_e the left Cauchy-Green tensor, and \xi an internal variable.I quiet understood the physics but there is a mathematical relation that I don't understand. Given the above definition, we claim in a book that
D=\Big (\tau-2\dfrac{\partial \Psi}{\partial b_e}b_e \Big) : d + 2\dfrac{\partial \Psi}{\partial b_e}b_e : \Big ( -\dfrac{1}{2} L_v(b_e) b_e^{-1} \Big )-\dfrac{\partial \Psi }{\partial \xi} \dfrac{d\xi}{dt}
where d is the symmetric part of the spatial velocity gradient L and L_v(b_e) denotes the Lie derivative of b_e. We can show that \dfrac{d}{dt}b_e=Lb_e+b_eL^t+L_v(b_e)
thus the both expressions given to D are equal if and only if \dfrac{\partial \Psi}{\partial b_e}:b_e L^t=\dfrac{\partial \Psi}{\partial b_e }:L^t b_e, but how can this last equality be true ? I can't tell why, if you do please share it
Regards.
