Tensors in GR and in mechanics

[Chestermiller]

Is that the same as index-free index notation? E.g. $\nabla_{\mu}\xi^{\mu}$ would correspond to $\vec{\nabla} \cdot \vec{\xi}$ and $\nabla_{[\mu}\xi_{\nu]}$ would correspond to $d\xi$. However I should note that when it comes to relativistic fluid dynamics, most GR texts will use either index notation or index-free notation via differential forms and in my opinion the former is much easier to work with when doing calculations. MTW uses both.

Nevertheless, take a look at this: http://arxiv.org/pdf/gr-qc/0605010v2.pdf

Yes WBN. That's what I meant. I will check out this reference. Thanks again.

Chet

 

[TrickyDicky]

Yes. If you are working with the deformational behavior of curved composite membrane type structures (tires) and large deformation behavior of non-linear rheological materials like rubber membranes (e.g., balloons), you would model them as curved deformable 2D structures. In such cases, you usually treat the stress tensor as approximately exhibiting plane strain (i.e., within the 2D tangent space). I used this extensively when I worked on the mechanics of Kevlar reinforced radial tires in support of Kevlar tire cord sales at DuPont.

Miller, C., Popper, P., Gilmour, P.W., and Schaffers, W.J., Textile Mechanics Model of a Pneumatic Tire, Tire Science and Technology, 13, 4, 187-226 (1985)

Popper, P., Miller, C., Filkin, D.L., and Schaeffers, W.J., A Simple Model for Cornering and Belt-Edge Separation in Radial Tires, Tire Science and Technology, 14, 1 (1986)

Thanks for the info and references.

 

[Chestermiller]

Thanks for the info and references.

Incidentally, I meant plane stress, not plane strain. Sorry for the error.

Chet

 

[TrickyDicky]

... continuum mechanics is in and of itself a powerful and ubiquitously used calculational and conceptual tool in GR.

It really is, and you've given a few examples of it in this thread.
I was wondering if this is a two-way fruitful relation. Are there useful ways to apply concepts and insights from relativity in the continuum mechanics- mechanical engineering context?

 

[WannabeNewton]

I was wondering if this is a two-way fruitful relation. Are there useful ways to apply concepts and insights from relativity in the continuum mechanics- mechanical engineering context?

I'm afraid I can't help you there as I know nothing about mechE or even engineering as a whole

 

[TrickyDicky]

I'm afraid I can't help you there as I know nothing about mechE or even engineering as a whole

No problem, maybe Chestermiller? (being a experienced engineer and versed in relativity)

 

[Chestermiller]

No problem, maybe Chestermiller? (being a experienced engineer and versed in relativity)

Yes I can. On a practical basis, applying what you learn in relativity to attack engineering problems wouldn't be much help, but, from the standpoint of unifying concepts, relativity fills some important gaps.

Before studying relativity, I always thought that, because of the similarity in the differential form of the continuity equation and the equation of motion, there must be a way of uniting these into a single equation. However, it wasn't until I studied relativity that I was able to learn how this could be done. Einstein figured it out because he must have had some background in continuum mechanics and recognized the value in modeling at the continuum level. In relativity, the equation of continuity automatically combines with the equation of motion when one takes the divergence of the stress-energy tensor. This is usually set equal to zero. However, in engineering, we deal with mechanical stresses in elastic solids and viscous fluids that must be added to the right hand side of the equation. I don't know how to do this yet, especially for the case of solids where deformations are referenced to some initial state (i.e., at a specific constant time for some frame of reference). However, my not knowing how to do it doesn't mean that it can't be done.

Another unifying area is in electrostatics/electromagnetism. In engineering, we deal with the force of an electrical field on a stationary charged particle, and we deal separately with the force of a magnetic field on a moving charged particle. Relativity automatically unifies both these relationships into a single equation involving the dot product of the Faraday tensor with the 4 velocity vector. It's really quite dazzling mathematically.

Hope this helps.

Chet

 

[vanhees71]

Viscosity occurs in relativistic and non-relativistic fluid mechanics in principally similar ways, but there are also some impotant differences.

Fluid mechanics can be derived from kinetic theory as an aproximation that the mean free path is small compared to the typical macroscopic scales over which the collective observables like the flow velocity of the fluid changes. Then it is a good approximation to assume that each fluid cell is in thermal equilibrium. Then the continuity equations for conserved quantities like energy, momentum, and conserved charges (in the nonrelativistic case also mass) leads to ideal fluid dynamics (Euler's equations).

In the next step one considers small deviations from local thermal equilibrium, leading to viscous corrections (Navier-Stokes equations), heat flow, etc.

In relativistic fluid dynamics, however, the NS-approximation leads to instabilities and acausalities. Thus one has to go at least to 2nd-order corrections in the non-equilibrium expansion. In relaxation-time approximation this leads to the Israel-Stewart equations of relativistic viscous hydro.

Recently this expansion has been extended to higher orders. Look for papers by Gabriel Denicol, Andre El et al.

Relativistic hydro doesn't play a role in mechanical engineering, I guess, because there the fluids are well in the non-relativistic regime. It finds however important applications in ultra-relativistic heavy-ion collisions or supernova explosions and neutron-star formation. The latter includes even general relativistic hydro.

 

[robphy]

Some thoughts on the original question...

The dimensionality mentioned in the original question is a big difference. Some lower-dimensional analogues of tensors used in GR simplify greatly (sometimes to zero). (Sometimes exploiting symmetries in a three-dimensional space can hide the distinction between different kind of "vectors"... vectors, pseudovectors, bivectors, two-forms.)

The signature of GR is also a big difference. Null vectors (vectors perpendicular to itself) are foreign to Euclidean space.

Torsion is zero in GR. But folks are using torsion to study dislocations.

(A while back I was wondering how one would formulate continuum mechanics in n-dimensions [no longer relying on the special properties of three-dimensional space... no more "axis" of rotation].)