Using the results of Exercise 13.20, continue the superposition process by combining three force doublets in each of the coordinate directions. This results in a center of dilatation at the origin as shown in the figure. Using spherical coordinate components, show that the stress field for this problem is given by:

Data from exercise 13.20

A force doublet is commonly defined as two equal but opposite forces acting in an infinite medium as shown in the following figure. Develop the stress field for this problem by superimposing the solution from Example 13.1 onto that of another single force of –P acting at the point z =–d. In particular, consider the case as d→0 such that the product Pd→D, where D is a constant. This summation and limiting process yield a solution that is simply the derivative of the original Kelvin state. For example, the superposition of the radial stress component gives:

Transcribed Image Text:

OR (1-2v)D 2T (1-V)R C R TRO = 0