I’m doing FEM analysis of stress and strain, but have no training.
I see subscripts of xy and yz, but for the combination of x and z I see subscripts xz and zx used (though not necessarily together). Googling the terms stress, xy, yz, and xz together produces almost twice the hits that substituting zx does.
Is the difference between xz and zx only a stylistic convention? Or does a distinction between the two ever mean anything, like maybe a sign convention?
Ok, so you have a differential element. The first subscript is the face that the stress acts on. This means’s it’s the vector that’s normal to that plane. The second subscript is the direction that the stress is pointing in.
A tensor an element for each combination of elements: There’s an xx component, and an xy, and a yx, and a yy, and so on. Stress tensors, though, are symmetric, so the xy element is the same as the yx component, and the xz is the same as the zx, and so on. So instead of specifying all nine elements, you really only need to specify six elements. The logical convention is to refer to two of the off-diagonal elements as xy and yz (because that’s the order the letters come in), but there are two different logical conventions for that last element. You could call it xz, since x comes before z, or you could call it zx, to go through the dimensions cyclically. Either one means the same, though (at least for symmetric tensors like the stress tensor).
Oh, good lord, that first sentence of mine is terrible. That’s what I get for trying to multitask. Try something more like “A tensor has an element for each combination of indices”.
Chronos, thanks. It is apparent that I’m manipulating stresses that are symmetric, and that zx and xz both get used (by different authors) in this context, so I think I am in the situation you describe, and your advice is for me. I’m going with xz because it’s more popular, just as a stylistic convention.
Snarky, thanks, I see where the two are different, but also conclude that’s not my situation, though from what I posted nobody can tell. It’s good to know about the Doper mechanical engineering (right?) crowd!
You are in that situation, and Chronos’s advice is for you.
And you are also in the situation Snarky_Kong describes. [symbol]s[/symbol][sub]zx[/sub] and [symbol]s[/symbol][sub]xz[/sub] really are seperate stresses acting on seperate faces of the element. Only thing is, as Chronos says, [symbol]s[/symbol][sub]zx[/sub] and [symbol]s[/symbol][sub]xz[/sub] are always equal, so it’s not necessary to specify both. You can specify just one, and the other will always be the same.
In other words, the “convention” decision isn’t that what to name the single xz/zx shear stress, it’s which of the two seperate stresses to explicitly specify. A little explanation is included on the Wikipedia pageSnarky’s pic is used on, and a clearer explanation on symmetry (and why the two stresses are equal) is here.
I believe Chronos is engaged in “non-applied engineering” (what they like to call “physics”).
Yep, I’m a theoretical physicist. Amusingly, though, I’m intimately familiar with the stress tensor, since (in slightly modified form) it’s one of the most important tensors in general relativity (basically, the stress tensor plays the role of “mass” in Einstein’s equations).
Damnit, a question that’s actually in my immediate field, and Chronos scoops me! It’s just not fair. But I can’t really add anything useful to his explanation.
Napier, what FE code and pre/post processor are you using for analysis? Although all of the training in the world for using a FEA code is no substitute for experience, mentoring, and correlation to physical test, there are some resources online (starting with the NAFEMS website) that can be helpful. This is especially important if you are venturing outside of basic linear sub-yield stress analysis in ductile, fully elastic materials. Once you start dealing with material nonlinear, fracture mechanics, composite or ceramic materials, non-linear dynamic modelling, et cetera, it’s ridiculously easy to get answers that look beautiful and are completely misleading.
Unfortunately there isn’t a textbook I’ve found that I can point to that wil really help you with the practicalities of doing FEM analysis. Bathe’s [url=http://www.amazon.com/Finite-Element-Procedures-Klaus-Jürgen-Bathe/dp/097900490X/]Finite Element Procedures (colloqually known as “The Red Book”) is the classic text that nearly everyone who has been academically educated in the finite element method has on their bookshelf, but its one of those things that seems rarely opened, and I have to admit to never having procured my own copy, and have only used it in the never-ending preparation of the theory section of my organization’s structural design and analysis handbook. The NAFEMS publications seem to be pretty good when it gets into the more advanced topic, but unless you can get your organization to spring for a membership ($1,400 for a site membership) then they’re pretty pricy.
That should be Finite Element Procedures by Bathe. Somehow I can find one syntax error in ten thousand lines of an input deck and call out card formats by memory, but I can’t do simple vBulletin code tags without pranging it up.
I’m using FlexPDE, which is a FEM solver for differential field equations. It frew out of PDEase software, which I think no longer exists, but which it resembles. This software package allows you to specify any field variables you want, and equations that correspond one-to-one with the field variables that are being adjusted to minimize errors in the equations. The equations need not contain the variables they correspond with. This software automatically handles meshing and produces a variety of graphic outputs as well as ASCII text tables. You can solve any equations in any variables with any coupling you like, as long as you can write the equations and boundary conditions and as long as you have some kind of luck and patience and skill in getting things to converge.
I have used this software for years, as I say, even taught some short courses about it, and one of my suggestions has become a feature in the last couple of versions. But I almost always use it with diffusion problems, especially heat conduction, or other things (including magnetism and radiation absorption and capacitance). I’m trying my first stress analysis problem ever, and coupling this software with another general purpose analytical package in an iterative automated search for optimal parameters.
I am, however, working with an engineer who is very good with stress and strain, and also pretty skilled with this software. So I am in good hands. Though, he’s 12 time zones away now, and I can sometimes get faster answers to simple isolated questions elsewhere. Here, for instance.
Well, I generally use cursing and Irish whiskey along with what passes for skill when dealing with model convergence problems, but to each his own. The scripting on FlexPDE looks really clean in comparison to the commercial finite element code I use (which is primarily a structural code with some multiphysics capability), and you’re clearly getting into coupled analysis areas that are outside my experience. I don’t mind a bit of thermal imbalance or things moving around a bit in a discrete mechanical fashion, but when it comes to continuua that ooze like an ichor or vaporize like a red-shirted away team member in the middle of your simulation I draw the line. That stuff just shouldn’t happen in a well-ordered, mechanistic universe. And when our CFD analyst gets up and starts talking about the accuracy of his results I usually leave the room.