You should start with the strain energy density $\psi$, then define:
$$
C_{ijkl} = \frac{\partial^2 \psi}{\partial \epsilon_{ij}\partial \epsilon_{kl}}, $$
and then define $$
\sigma_{ij} = C_{ijkl} \epsilon_{kl}
$$
The remainder of my answer will be about explaining why you have to do it that way. Firstly it is physical, there really is energy associated with strain, and if there weren't there would not be any stress. Secondly, it is linear exactly because we are considering the changes in energy due to small strains.
But let's go back to the minor symmetries. We need $C_{ijkl}=C_{jikl}$ because otherwise $\sigma_{ij}\neq\sigma_{ji}$ (and then we get arbitrarily large, hence unphysical, angular velocities for smaller and smaller regions). But the other minor symmetry is not required. If someone handed you a random rank four tensor, let's call it $B_{ijkl}$, and called it an elastic tensor and it didn't have the second minor symmetry, you can define $C_{ijkl}=(B_{ijkl}+B_{ijlk})/2$ $D_{ijkl}=(B_{ijkl}-B_{ijlk})/2$ and then $B=C+D$ but when D is contracted with a symmetric rank two tensor (like the strain tensor) it gives zero. So the part of the rank four tensor without that second minor symmetry simply doesn't contribute, as an actor it does nothing (when you think all the elasticity does is give you stress from strain). So you might as well assume your tensor has both minor symmetries because it acts like it has the second ($B$ and $C$ act the same on symmetric tensors) and it has to have the first.
Did I bring that up to be pedantic? No, I brought it up because the same thing happens if you contract the elasticity tensor with a rank four symmetric combination of the strain tensor. The part of the elasticity tensor without the major symmetry doesn't contribute to the strain energy density. So a random tensor needs the first minor symmetry to be physical. But you might as well assume it has the second minor symmetry since it doesn't affect the stress-strain relationship. And you might as well assume it has the major symmetry because the part that doesn't will not contribute to the strain energy density.
But it is the strain energy density that is physical, and how it changes is what elasticity is. So you aren't really deriving these symmetries as much as saying that only the symmetric ones generate the physical things you want, energy when given strain. And a real derivation should start with strain energy density and strain, and then just define elasticity from that.
The answer you're looking for seems to be contained in
Rezzolla & Zanotti: Relativistic Hydrodynamics (Oxford U.P. 2013)
https://books.google.com/books/?id=KU2oAAAAQBAJ
but it is not a trivial generalization. Quoting Disconzi's On the well-posedness of relativistic viscous fluids (Nonlinearity 27 (2014) 1915, arXiv:1310.1954):
we still lack a satisfactory formulation of viscous phenomena within
Einstein's theory of general relativity. [... T]here have been different proposals for what the correct $T_{\alpha\beta}$ should be. [... A]ttempts to formulate a viscous relativistic theory based on a simple covariant generalization of the classical (i.e., non-relativistic) stress-energy tensor for the Navier-Stokes equations have also failed to produce a causal theory.
Similar problems exist for elastic materials.
You may also take a look at Chapter 15 ("Relativistic continuum mechanics") of Maugin's Continuum Mechanics Through the Twentieth Century (Springer 2013), and its references:
https://books.google.com/books?id=-QhAAAAAQBAJ
and at Bressan's Relativistic Theories of Materials (Springer 1978):
https://books.google.com/books?id=kMTuCAAAQBAJ
General-relativistic continuum mechanics unfortunately has not been given a clear mathematical and conceptual framework yet. Newtonian continuum mechanics is easy to summarize:
- We choose a reference frame (preferably but not necessarily inertial).
- We have a set of 11 spacetime-dependent fields with clear physical meaning: mass, momentum or deformation, stress, body force, internal energy, heating flux, body heating, temperature, entropy, entropy flux, body entropy supply. Of these, the "body" ones represent external interventions.
- We have 5 balance equations: mass, force-momentum, torque-rotational momentum, energy, entropy. They are clearly written in terms of the fields above and are valid for any material.
- We choose a set of independent fields (usually mass, momentum or deformation, temperature).
- We choose constitutive equations (compatibly with the balance ones) that relate the remaining fields to the independent ones. These equations express the peculiar properties (fluid, solid, elastic, plastic, with/without memory...) of the material under study.
And at this point we have a well-defined set of partial differential equations in a number of unknown fields, for which we can set up well-defined initial- & boundary-value problems to be solved analytically or numerically. (An expanded but analogous framework accommodates electromagnetism and continua with internal structure.)
This framework and steps are very neat – we clearly know what the fields are, which of them are dependent and which independent; what are the equations valid for all materials, and what are the equations constitutive to each material. I've never seen a clearly defined procedure like the one above for general relativity, although I believe it could be extracted from Rezzolla & Zanotti's or Bressan's books. Moreover, the core of general-relativistic community uses a different jargon and way of thinking.
Most general-relativity books tell you that the Einstein equations determine everything, but they are not so clear about which fields in them are independent and which dependent; even Misner et al.'s Gravitation (ch. 21) has a long discussion and explanation about this point. It was only with 3+1 formulations and the work of Arnowitt, Deser, Misner, York, and others around the 1970s that this point got clarified. Then they tell you that we need "special" additional equations for the stress tensor – that is, constitutive equations. Sometimes other conservation equations, like baryonic number (basically rest-mass), are added with no real explanation. This is a sample of books where "constitutive equations" are mentioned explicitly (only once or twice in most of them):
- Rezzolla & Zanotti above (and they explain what a "constitutive equation" is as though it was an exotic concept)
- Choquet-Bruhat: General Relativity and Einstein's Equations
- Anile & Choquet-Bruhat: Relativistic Fluid Dynamics
- Bertotti et al.: General Relativity and Gravitation
- Puetzfeld et al.: Equations of Motion in Relativistic Gravity
- Tonti: The Mathematical Structure of Classical and Relativistic Physics
- Bini & Ferrarese: Introduction to Relativistic Continuum Mechanics
- Tolman (obviously): Relativity, Thermodynamics, and Cosmology
but they constitute a very small minority in the huge relativistic literature.
Yet, the general-relativistic community cannot be criticized for the confused conceptual state and somehow confused language of the subject. Newtonian continuum mechanics can be neatly formulated today because it has been refined over several centuries. General relativity is still very young instead, and its conceptual refinement still in progress. Some of the steps in the Newtonian framework become extremely complicated in general relativity. For example: step 1. (choose an inertial frame) cannot be done so simply. The Einstein equations, evolved from initial conditions, construct a reference frame "along the way", while they determine the dynamics. This gives rise to peculiar fields like "lapse" and "shift", which aren't really physical, and all sorts of redundancy (gauge freedom) in the equations.
Another example: the metric becomes a dynamical field variable, and you suddenly realize that it is hidden almost everywhere in the Newtonian framework – divergences, curls, vectors/covectors... So its evolution can't be easily divided among some new balance and constitutive equations (like we can do with electromagnetism instead). Are all of its appearances in the Newtonian framework dynamically significant? or can the metric be eliminated from some places? There's some research today on this "de-metrization" of Newton's equations; see for example Segev's Metric-independent analysis of the stress-energy tensor, J. Math. Phys. 43 (2002) 3220. This line of research has shown that some Newtonian physical objects actually don't need a metric: they be expressed via differential forms and other metric-free differential-geometrical objects (e.g., van Dantzig's On the geometrical representation of elementary physical objects and the relations between geometry and physics, Nieuw Archief voor Wiskunde II (1954) 73; there is a vast literature on this, let me know if you want more references). This is still work in progress – which means that it's obviously not completely clear yet how mass-energy-momentum-stress and metric are coupled.
To conclude, I think another good starting point to understand how things work in general-relativistic continuum mechanics is to look in books on numerical formulations of general relativity and matter dynamics. The conceptual framework in them is a bit confused, but you can see how they actually do it. If from the practice of these books you manage to reverse-engineer a framework like the Newtonian one above, please write a pedagogical paper about it!
Here are some books and reviews on numerical relativity with continua:
- Rezzolla & Zanotti above
- Gourgoulhon: 3+1 Formalism in General Relativity (Springer 2012, arXiv:gr-qc/0703035)
- Baumgarte & Shapiro: Numerical Relativity (Cambridge U.P. 2010)
- Alcubierre: Introduction to 3+1 Numerical Relativity (Oxford U.P. 2008)
- Palenzuela-Luque & Bona-Casas: Elements of Numerical Relativity and Relativistic Hydrodynamics (Springer 2009)
- Lehner: Numerical relativity: a review, Class. Quant. Grav. 18 (2001) R25, arXiv:gr-qc/0106072
- Guzmán: Introduction to numerical relativity through examples, Rev. Mex. Fis. S 53 (2007) 78
I'm happy to provide or look for additional references.
Best Answer
To my knowledge, I'm afraid it is not generally possible to compute $\frac{\partial\mathbf{F}}{\partial\mathbf{E}}$. Here's the reason:
Usually we compute the Green-Lagrange strain tensor from the deformation gradient with its definition $$ \mathbf{E}(\mathbf{F})=\frac{1}{2}(\mathbf{F}^T\mathbf{F}-\mathbf{I}) \tag{1} $$ It is easy to verify that $\mathbf{E}$ is symmetric, but $\mathbf{F}$ is not necessarily symmetric. Take the 3D space as example, one would have 9 independent components for $\mathbf{F}$, but only 6 independent components for $\mathbf{E}$. That is to say, one may not possible to obtain the inverse of Eq. (1), namely $\mathbf{F}(\mathbf{E})$, not even $\frac{\partial\mathbf{F}}{\partial\mathbf{E}}$.
Being symmetric is also the reason that people prefer to use the right Cauchy-Green tensor $\mathbf{C}$ as well as the 2nd Piola-Kirchhoff stress $\mathbf{S}$.
However, there are still formulations using the 1st Piola-Kirchhoff stress $\mathbf{P}$, which is computed as $$ \mathbf{P}=\frac{\partial\Psi}{\partial\mathbf{F}} \tag{2} $$ and the corresponding tangent modulus $$ \mathbb{A}=\frac{\partial\mathbf{P}}{\partial\mathbf{F}} \tag{3} $$ which is usually called the 1st elasticity tensor.
Maybe Eq.(2) and (3) are what you are looking for.