Posted on 07/18/2007 5:22:46 PM PDT by charming_harmonica
Yup. Trying to look ahead.
Then join us on the Undead Thread!
88 here now. :P
free dixie HUGS/SMOOCHES,sw
Good morning, General!
Morning, sw!
85 here with 55%. Ugh.
How are things in your neck of the woods?
where i am,the AIR is so still/hot/humid & generally "icky", that it seems to me that you should be able to SEE it.
otoh, BREATHING it is NOT a really good plan.
YETCH!
free dixie SMOOCH,sw
96 & 90+ % humidity. it's a "code orange" day.
YETCH!
free dixie SMOOCH,sw
When we lived in Bealeton, I could hang my clothes outside at 0700 and by 1900 they were still wet. Air you could wear.
Good Morning!
Alrighty then, sign me up!
\epsilon_{ij} = \frac{1}{2} (u_{i,j} + u_{j,i}), (5.19)
the linear elastic isotropic relation between stress and strain
\sigma_{ij} = \lambda \delta_{ij} \epsilon{kk} + 2\mu\epsilon_{ij}, \end{displaymath} (5.20)
So again the answer is "49" and the equilibrium condition
\sigma_{ji,j} + f_i = 0. (5.21)
Substituting 5.19 into 5.20 into 5.21 gives the simplified Navier equations in terms of displacement $u_i$:
\mu u_{i,jj} + (\mu+\lambda) u_{j,ji} + f_i = 0. (5.22)
This can also be written in terms of the modulus of elasticity $E$ and Poisson's ratio $\nu$, where we have the relations
E = \frac{\mu(3\lambda + 2\mu)}{\lambda+\mu}; \nu = \frac{... ... \frac{\nu E}{(1-2\nu)(1+\nu)}; \mu = G = \frac{E}{2(1+\nu)}. (5.23)
But I like the simplicity of the Navier equations better, though in terms of $\mu$ and $\nu$, things aren't so bad:
\mu u_{i,jj} + \frac{\mu}{1-2\nu} u_{j,ji} + f_i = 0 (5.24)
So, this is trouble because the three equations are fully coupled. To separate them, we define the Galerkin vector $g_i$ such that
2\mu u_i = c g_{i,jj} - g_{j,ji}, (5.25)
where $c$ is a scalar constant whose value will be determined later. Substituting this into equation % latex2html id marker 1126 $\ref{eq:munu}$ gives
\left[\frac{c}{2}g_{i,kk} - \frac{1}{2} g_{k,ki}\right]_{,j... ...}{2} g_{j,kk} - \frac{1}{2} g_{k,kj} \right]_{,ji} + f_i = 0, (5.26)
and since $g_{k,kijj} = g_{j,kkji} = g_{k,kjji}$, this rearranges to
\frac{c}{2}g_{i,kkjj} + g_{k,kjji} \left[-\frac{1}{2} + \frac{c}{2(1-2\nu)} - \frac{1}{2(1-2\nu)}\right] + f_i = 0. (5.27)
Notice that if we choose $c=2(1-\nu)$, then an entire term disappears! So equation 5.25 becomes
2\mu u_i = 2(1-\nu) g_{i,jj} - g_{j,ji}, (5.28)
and the elastic equilibrium equation becomes
(1-\nu) g_{i,kkjj} + f_i = 0. (5.29)
*HUG*
How are you today?
My one cup of coffee before my long commute to work this morning hasn’t been enough....need more....
Plenty to do today though. Has a mission yesterday morning early, so I didn’t get in until 1130am. Hence I couldn’t leave until 11pm.
What a disagreeable place! It’s hot here, but not very humid. Hasn’t rained for a few days.
Welcome, KoRn! This is not just the Undead Thread, but also the Flying Castle. We have most of what we need, but we’re still trying to figure out how to build a viable dragon.
You can take a tour if you want, but if you go into the lower levels, be sure to take a guide, as the walls and doors have a tendency to shift.
(I don’t think we built that in.)
Hey sionsar!
Haven’t seen you guys lately.
I thought about coffee this morning, but it just wasn’t the day for it. I had chocolate raspberry cocoa, instead. It’s my “cure-all.”
(Thanks for yesterday. You do good work!)
Hi, Pax! I should probably ping when change threads, as sion’s ping list and mine are different.
Good to see you!
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