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Discrepancy in radial displacement governing eq for a sphere from Dr. Allen Bower's text in section 4.1.3

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

AttachmentSize
PDF icon exam practice.pdf161.07 KB

Comments

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

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?

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

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

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