What is the Theory of Elasticity?

[Garth]

Off-topic minirant: Wikipedia has let me down even as a purveyor of absurd minutae! Yes, WP offers an article on Rendevous with Rama, a scifi novel by Arthur C. Clarke which features a kilometer scale cylinder spinning at a rate sufficient to induce a Earth gravity on the inner surface, but I can't recall the mass or the exact dimensions, which I believe were specified in the novel. Does anyone know the dimensions of this fictional space artifact? Needless to say, the application is to a spinning cylinder deformed by its own weight and by centrifugal forces. I guess the previous exercise would suggest that for Rama we can neglect weight

Rama was a cylinder fifty kilometres long and twenty kilometres in diameter rotating once every four minutes. The interior was sixteen kilometres wide.

An interesting historical quote from the beginning of the story (set in 2130) in the book (published in 1973), when the intruder had first been detected:

Although Professor Davidson took a very jaundiced view of the Neptune probe, it had already been approved and he saw no point in sending more good money after bad. He spoke eloquently on the follies of asteroid-chasing and the urgent need for a new high-resolution interferometer on the Moon to prove the newly-revived Big Bang theory of creation, once and for all. That was a grave tactical error, because the three most ardent supporters of the modified Steady State Theory were also members of the Council...

BTW Chris, thank you for a very informative and impressive series of posts in this thread.

Garth

 

[Chris Hillman]

Rama was a cylinder fifty kilometres long and twenty kilometres in diameter rotating once every four minutes. The interior was sixteen kilometres wide.

Perfect! Thanks, Garth!

BTW Chris, thank you for a very informative and impressive series of posts in this thread.

And thanks for that too!

 

[Chris Hillman]

12. EXAMPLES OF STATIC EQUILIBRIA: ADDENDUM

EXAMPLE TWO.FIVE (was exercise after EXAMPLE TWO): Consider a hollow cylinder, made of a homogenous isotropci material, with radii R_2 > R_1 > 0, rotating with constant angular velocity \Omega about the axis of symmetry. To determine the deformation, strain and stresses due to centrifugal force, we proceed as before, except that we determine the two constants of integration using the fact that the normal stress S^{22} must vanish on the inner and outer surfaces of the cylinder. I'll leave the details as an exercise.

In Arthur C. Clarke's scifi novel Rendevous with Rama, Rama is a hollow cylinder 50 km long with R_1 = 8 \, {\rm km}, \; R_2 = 10 \, {\rm km} which rotates once every four minutes, i.e. \Omega=\frac{2 \pi}{4 \cdot 60} \, \frac{{\rm rad}}{{\rm sec}}. This gives a centrifugal acceleration at the inner surface of about 2/3 Earth gravity. The cylindrical shell is not solid, but ignore that and plug in the numbers assuming Rama is made of something with material properties similar to structural steel. We find that the inner surface is displaced outward by nearly 20 m, and the maximal strain is about 0.0025, which is too large for a steel object. The maximal stress would be about 5000 atmospheres, also too large for steel. Repeating the computation assuming that Rama is a "double hulled" steel cylinder with much thinner skins doesn't greatly affect this conclusion. So, Rama is not made of steel!

Now consider a steel cylinder 5 \, {\rm km} long, with inner radius R_1 = 1 \, {\rm km}, with a thickness of 5 \, {\rm m}, rotating once every 90 seconds. This gives a centrifugal force at the inner surface of about 1/2 Earth gravity. The inner surface is displaced outward by about 18 cm, the strains are reasonable, and the maximal stress in the cylindrical walls again is the stress along latitudes at the inner surface and is about 385 atmospheres. Applying our result for a spinning shaft to the end caps, we find reasonable strains and a maximal stress of about 165 atmospheres along the axis of symmetry, within each cap. So, such a cylinder could in principle be made of steel. The assumed thickness of the skin matters little, since the stresses and strains are nearly constant. The cylinder would however require stiffening hoops or girders to prevent initially small vibrations (say due to thermal expansion due to differential heating) from rupturing the skin.

 

[Chris Hillman]

12. EXAMPLES OF STATIC EQUILIBRIA: ADDENDUM

Add after EXAMPLE FIVE.FIVE: exercise for steel ball with concentric spherical cavity at center, with radii R_1= 1 \, {\rm km}, \; R_2 = 10 \, {\rm km}:

Partial solution: radial displacement, strain tensor components, and stress tensor components are plotted for this example. The displacement is purely radial and inward, with maximal displacement occurring near the outer surface, with magnitude less than one cm. The maximal strain is on the order of 3 \times 10^{-6}. The red curve in the middle figure is the radial strain component, and the plot shows radial expansion near the inner and outer surface of the body. The maximal stress is orthogonal (green curve in the rightmost figure) and occurs at the inner surface, and has a magnitude of about 10 atmospheres. Notice that the radial stress component (red curve in the rightmost figure) vanishes on the inner and outer surfaces, as required by the boundary conditions of elastostatics.

(Security note: Never download files you don't trust! These are 12-13 K jpeg files produced using Maple and vetted using clamscan.)

 

[Vr6Fidelity]

Interesting. From this thread I was able to deduce I am supposed to be using E the Modulus of Elasticity for the stress, and not the ultimate or yield stress. Although I see Poisson's ratio as a term that did not immediately trigger me to go to "E". should it?

Fyi, this 500Mpa stress is easily attained with high quality steels. For example I am currently analyzing a disk material with an ultimate strength of 2030 Mpa. You could go faster than fiction!

Steel strengths are always in Mpa, I see here atmospheres is the unit you are using.

1Mpa = 10 Atmospheres.

I would be VERY interested to discuss a few things with Mr. Chris Hillman if he still exists on here.

 

[adnan jahan]

i need to know the basic of elasto dynamics the symmetri of the materials and the variation in the matrix of stiffness tensor of isotropic and other materials related.

thanks.