## EN224: Linear Elasticity

##### Homework 2: Potential representations, Fourier Transforms, Singular
Solutions

##### Due Friday March 4, 2005

**Division of Engineering**

**Brown University**

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**1. The simplest possible
derivation of the Kelvin State.** Assume that the solution is to be generated from
Papkovich-Neuber potentials satisfying

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### Here, we have chosen the body force to be a Dirac delta sequence centered
at .
By taking Fourier transforms of the governing equations for above, find the Papkovich-Neuber potentials
that generate the required solution.

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**2.** **Papkovich-Neuber potentials for the Doublet states. **Let
denote the Papkovich-Neuber potentials for the normalized Kelvin
state, i.e. a point force of unit magnitude acting in the direction at the origin.
Let

denote
the doublet states, i..e.

Show
that may be generated from Papkovich-Neuber potentials

### Hence, verify that the Papkovich Neuber potentials for the doublet states
centered at the origin are

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**3. Center of Compression**. Using the results of the preceding section, find the
displacement, strain and stress fields associated with a center of compression
at the origin, i.e., find

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**4. Center of compression in
a sphere**. Using superposition and the results of
problem (3), find the displacement fields induced by a center of compression at
the center of a sphere of radius *a. *Assume that the surface of the sphere is
free of traction.

**5. Dilatation at the center
of a sphere due to arbitrary surface traction.** Using the result of problem
(4), show that the dilatation at the center of a sphere of radius *a* due to

a
self-equilibrating distribution of traction **t** acting on its surface is

where
**r** is the position vector of a point
on the sphere’s surface relative to the origin, and

*B* denotes the surface of the
sphere. Verify the predictions that
were made in our proof of Saint-Venants principle. What happens if the tractions act tangent to the surface of the
sphere?