Expansion of a local dissipation function

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The discussion revolves around the maximum dissipation principle in finite strain theory, specifically the mathematical relationship involving the dissipation function D. The user seeks clarification on an equality related to the derivatives of the free energy with respect to the left Cauchy-Green tensor and the symmetric part of the spatial velocity gradient. They express confusion over why the equality holds, despite attempts to prove it. Another participant suggests considering the possibility of a transcription error regarding the use of L and its transpose. The conversation highlights the complexities of finite strain theory and the challenges in understanding its mathematical foundations.
trabo
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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
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 :confused: ?
I can't tell why, if you do please share it :wink:

Regards.
 
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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
Not yet I'm afraid. I have a book on line that states that, but I just can't figure it out yet !
 
I wasn't able to show that last equality either. But, are you sure that one of those L transposes is not an L? Just a thought.

Chet
 
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