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Discrepancy in radial displacement governing eq for a sphere from Dr. Allen Bower's text in section 4.1.3
Sun, 2010-04-25 20:07 - tsunamiBTP
Attached is a pdf file indicating a discrepancy in the governing eq for radial displacements in a sphere. I am + Dr. Bower's text is correct but I cannot reconcile my derivation with his so I am assuming there must be some underlying assumption or approximation I am not aware of.
Can anyone provide me with greater insight?
my email: jberg3@unl.edu
Attachment | Size |
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exam practice.pdf | 161.07 KB |
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Comments
Re:Discrepancy in Bower
Check your equilibrium equations in spherical coordinates. You should get an equation of the form
\sigma_{rr,r} + 1/r(2\sigma_{rr} - \sigma_{tt} - \sigma_{pp}) = 0
-- Biswajit
I am not exactly familiar with your notation-->
What does the forward slash \ denote?
I did apply the divergence of the stress field to = 0. I presume you saw my attachment. Did I not execute the divergence in spherical coord's correctly?
Re: Bower and unfamiliarity with notation
What does the forward slash \ denote?
The \ is to make it easy for you to process it in Latex (with Lyx for example) since iMechanica does not allow Latex embedding.
I did apply the divergence of the stress field to = 0. Did I not execute the divergence in spherical coord's correctly?
That is correct. See e.g. Sadd, p. 70; Slaughter p. 189; Stulazec, p. 28; Richards, p. 58; These turn up for a Google scholar search with keywords "equilibrium spherical coordinates".
-- Biswajit
Re: Divergence operator in spherical coordinates
Just thought that I should remind students of mechanics that the divergence operator that appears in the equilibrium equations cannot be written in curvilinear coordinates as
sigma_{ij,i} = 0
Spherical coordinates are a special form of orthogonal curvilinear coordinate. The appropriate form of the divergence operator for general curvilinear coordinates can be found at
http://en.wikipedia.org/wiki/Curvilinear_coordinates#Divergence_of_a_second-order_tensor_field
It's a good exercise to work out the details for the special case of spherical coordinates.
-- Biswajit