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In classical continuum mechanics, equations of motion (balance equations) are Galilean invariant, i.e. they have the same form in all inertial reference frames. The fact that these reference frames are inertial means that transformations between them can have a time dependence only up to constant velocity boosts and constant rotations, but they are not necessarily Galilean invariant if the time dependence of these transformations is more general than that.

On the other hand, the principle of material frame indifference (also known as the principle of material objectivity) is saying something about the constitutive equations (these are needed to close the balance equations and fully describe the system one studies). It requires that the constitutive equations are invariant or unaffected by an arbitrary time-dependent translation, rotation, and reflection of the coordinate axes and by an arbitrary translation in time. This is now a much more general set of motions than those involved between the Galilean inertial frames, involving basically any motions of the rigid body.

My questions:

  • First of all, is it really true that the principle is saying something only about the constitutive equations, as opposed to also affecting the balance equations?
  • Is this principle, strictly speaking, really needed? Could one not do without it? What would go wrong? Why is, for example, spinning an object so hard that it is torn apart by inertial forces not a counterexample of this principle?
  • Why does it hold? It seems like it is trying to generalize the Galilean relativity principle to noninertial frames, but just for a subset of equations. But how does it know which equations to pick? Can one show it holds explicitly in some examples, e.g. theory of gases as derived from the kinetic theory? What about electromagnetism (for the permittivity, permeability, conductivity etc.)?
  • Why is it a principle and not a hypothesis or assumption? Surely it can not be a fundamental thing... in the sense that for each material, one could, in principle, imagine deriving the constitutive equations from the underlying, small-scale physics (atomic physics, solid state physics, plasma physics, nuclear etc.)
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2 Answers 2

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Constitutive equations

  • physically speaking, are needed to describe the behavior of a medium;
  • mathematically speaking, are needed to write a well-defined mathematical problem that can be solved, where the number of unknowns is equal to the number of equations.

Equations of Physics, like mass, momentum, energy balance equations, are independent on the coordinates you use to describe it, i.e. they have absolute/invariant nature, and tensors are the proper mathematical tools to write them.

Answer to your questions

  • the principle is saying something about the constitutive equations, but it directly affects balance equations, since you introduce the constitutive equations in them (if you replace some variable appearing in them, or you need to add the constitutive equations to an extended system of equations composed by both the balance equations and the constitutive equations);

  • this principle is required, since physical processes don't depend on the coordinates you're using to describe it. Whenever you write a constitutive law, this law representing the behavior of a medium can't depend on the coordinates as well;

  • it must hold just because Physics has an absolute/invariant nature and can't depend on the coordinates you use to describe it; in electromagnetism, with linear constitutive equations, the material objectivity is granted if you write tensor equations, like

    • $\mathbf{d} = \mathbb{\epsilon} \cdot \mathbf{e}$ where $\mathbf{d}$ is the displacement field, $\mathbf{e}$ is the electric field, $\mathbb{\epsilon}$ is the permittivity tensor, in general a second order tensor, and $\cdot$ is the tensor dot-product (for general linear non-isotropic media), that is a tensor operation, i.e. it's absolute itself; if the medium is isotropic, the second order tensor is proportional to the identity vector, $\mathbf{d}$ and $\mathbf{e}$ are everywhere aligned, and you can write the constitutive equation as $\mathbf{d} = \epsilon \,\mathbf{e}$ ($\epsilon$ here is a scalar, the permittivity of the medium);
    • similarly for liner relations involving permeability and conductivity/resistivty, you can introduce permeability and conductivity/resistivity second order tensors, that reduces to a multiple of the identity tensor for isotropic media, $\mathbf{b} = \mathbb{\mu} \cdot \mathbf{h} \qquad , \qquad \mathbf{j} = \mathbb{\sigma} \cdot \mathbf{e} \qquad , \qquad \mathbf{e} = \mathbb{\rho} \cdot \mathbf{j}$;

    For nonlinear constitutive equations: nonlinear constitutive equations must be expressed as a function of invariants of the tensors involved in these equations;

  • it's a fundamental thing, indeed. As already state above, all the physical processes are independent on the coordinates you use to describe them, they're absolute, and thus equations that govern them must be expressed using tensors and the constitutive equations involved must be independent on the choice of the coordinates as well.

A constitutive law that doesn't satisfy material objectivity, introduced in an equation governing a physical process, would make them dependent on the choice of the coordinates, and thus is not compatible with a correct description of Physics

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  • $\begingroup$ Tnx! This helps a lot! Unfortunately, I'm still a bit confused. I understand one wants to write the equations in a covariant way. However, if we are dealing with classical continuum mechanics, I'd think this limits us to the invariance of inertial frames. Requiring general covariance, i.e. fact that all frames, including noninertial ones, are allowed, would lead us to the general relativistic equations of motion. But if I want to stick to the non-relativistic stuff, I need to worry about the inertial forces, no? And why would those not affect constitutive equations then? What am I missing? $\endgroup$
    – z.v.
    Aug 29, 2022 at 19:42
  • $\begingroup$ To say it another way, if I put a solid rod in a strong gravitational field (next to the black hole for example), and the rod gets spaghettified, won't its Young modulus change? $\endgroup$
    – z.v.
    Aug 29, 2022 at 19:50
  • $\begingroup$ When I wrote my answer I was thinking at Classical Mechanics. Anyway, I guess that the very same principle holds in relativity, whenever you write equations and constitutive equations using 4D-tensors. Remember that in Relativity, spacetime is absolute, not space and time independently. It's likely that 3D-space deformations, and time/gravity are coupled in a "classically" unexpected manner, but I can't be more precise now about the spaghettification and the Young's module. $\endgroup$
    – basics
    Aug 29, 2022 at 20:00
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My conclusion, when I thought about this during my PhD work in rheology, was that the principle of material objectivity is not a principle but a (very good) assumption. It is more general than frame invariance. For rotations, it also states that the rotation rate does not end up in, e.g., a constitutive equation relating a stress tensor to a deformation history. My conclusion was that this is not necessarily true. In a rotating frame of reference, fictitious forces (i.e. centrifugal and Coriolis forces) might contribute to stresses. However, in all practical applications, on the scale of a representative volume where the transition from molecular interactions to constitutive relations is made, the fictitious forces are completely negligible compared to contributions from intra- and intermolecular forces. Therefore, for all purposes in not very extreme environments, the assumption can be treated as a principle and be used to pose constraints on permissible forms of constitutive relations, e.g., giving rise to convected derivatives.

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  • $\begingroup$ Thank you! That is very helpful. It is quite good to hear clear and straightforward statements like that. It completely makes sense to me for it to be an approximation (albeit an extremely useful one). Do you perhaps know some literature that discusses the conditions of the validity of the "principle"? I.e. quantifies the "extreme" conditions where one might start to notice the deviations. Thanks again. $\endgroup$
    – z.v.
    Jan 20, 2023 at 22:41
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    $\begingroup$ One case where it is very much wrong is turbulence modeling. Although the goal of rheology and turbulence modeling is quite similar (give constitutive equations for the Reynolds stress tensor in TM), the application of frame invariance to the latter has led to quite a few misunderstandings and problems. $\endgroup$
    – Johannes C
    Jul 5, 2023 at 19:20
  • $\begingroup$ Very interesting! Do you perhaps have some references to the articles? It would be quite instructive for me to see how exactly this happens. $\endgroup$
    – z.v.
    Jul 11, 2023 at 5:56

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