Summary
Led by an interest into the concept of "Material Objectivity", I am asking myself:
Are there objective time rates that are not Lie derivatives?
The long read
I am trying to understand the notion of "Material Objectivity" and "Objective Time Derivatives" (invariance of rheological equations under arbitrary time dependent rotations and translations) in Continuum Mechanics.
What I have researched so far:
The two works I mainly used and consider relevant to my question are:
Marsden und Hughes, Mathematical Foundations of Elasticity, Mineola: Dover Publications, 1994
Bampi and Morro, “Objectivity and Objective Time Derivatives in Continuum Physics,” Foundations of Physics, vol. 10, no. 11/12, pp. 905-920, 1980
In Marsden and Hughes work, it is shown that Lie derivatives along the flow field of the material body have the objectivity property (p. 101). They ask us to consider the fact that Lie derivatives do not commute with the metric $g$ by computing the Lie derivative for four different associated tensors $T_1 = T^{ij} e_i \otimes e_j,\phantom{a} T_2 = T_i^{\phantom{a} j} e^i \otimes e_j,\phantom{a} T_3 = T^i_{\phantom{a} j} e_i \otimes e^j, \phantom{a} T_4 = T_{ij} e^i \otimes e^j$. The Lie derivatives are $$(L_V T_1)^{ab}=\dot{T}^{ab}-T^{cb} \partial_c V^a- T^{ac} \partial_c V^b$$ $$(L_V T_2)_c^{\phantom{a}b} g^{ac}=\dot{T}^{ab}+T^{cb} \partial^a V_c- T^{ac} \partial_c V^b$$ $$(L_V T_3)^{a}_{\phantom{a}c} g^{cb}=\dot{T}^{ab}-T^{cb} \partial_c V^a + T^{ac} \partial^b V_c$$ $$(L_V T_4)_{cd} g^{ac}g^{db}=\dot{T}^{ab}+T^{cb} \partial^a V_c-+ T^{ac} \partial^b V_c$$ Then we can observe that e.g. $(L_V T_1)^{ab}$ and $(L_V T_4)_{cd} g^{ac}g^{db}$ are usually known as (upper and lower) convected or Oldroyd derivatives, while $1/2 ((L_V T_1)^{ab}+(L_V T_4)_{cd} g^{ac}g^{db}) = 1/2 ((L_V T_2)_c^{\phantom{a}b} g^{ac}+(L_V T_3)^{a}_{\phantom{a}c} g^{cb})$ is often called corotational derivative and in general any combination of the derivatives above with factors adding up to 1 constitutes an objective time derivative.
Bampi and Morro ask "what is the most general time derivative we can define on a tensor algebra". To cut things short they find that it is: $$Â^i = \dot{A}^i+K^i_{\phantom{i}j}A^j$$ with $K^i_{\phantom{i}j}$ being decomposable into an objective part $S^i_{\phantom{i}j}$ and a nonobjective part $H^i_{\phantom{i}j}$ which transforms according to: $$H'^i_{\phantom{i}j} = Q^i_{\phantom{i}p} Q_j^{\phantom{i}q} H^p_{\phantom{i}q}- \dot{Q}^i_{\phantom{i}p} Q_j^{\phantom{i}p} $$ Where $Q$ is the rotation matrix of the change of reference. Further they show that $H^i_{\phantom{i}j}$ needs to be skew-symmetric.
I have in general not been too lucky in finding good literature on the subject with an adequate mathematical depth (my dream would be a rigorous classification ...). Any help in that direction, i.e. any literature sources which deal with the topic on a reasonable level, would already be greatly appreciated.
My question is:
Are there objective time derivatives that are not Lie derivatives? To me it seems that there should be more possibilities to choose $S^i_{\phantom{i}j}$ and $H^i_{\phantom{i}j}$ than there are combinations of the four above Lie derivatives, but the objective time rates I usually encounter are all Lie derivatives ...
In addition, as has been pointed out by heather in the comments, Marsden and Hughes write that "All so-called objective rates of second-order tensors are in fact Lie derivatives." However, they do not provide any proof of this statement. So maybe, there are in fact no such derivatives but then my question is - is it possible to prove this?
