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