Problems in Spherical Coördinates
The spherical harmonics of Chapter 24 can be used in combination with Tables 21.2, 22.2 to treat problems involving bodies whose boundaries are surfaces of the spherical coördinate system, for example the spherical surface R = a or the conical surface β=β0, where a, β0 are constants. Examples of interest include the perturbation of an otherwise uniform stress field by a spherical hole or inclusion, the stresses in a solid sphere due to rotation about an axis and problems for a conical beam or shaft.
The general solution for a solid sphere with prescribed surface tractions can be obtained using the spherical harmonics of equation (24.24) in solutions A,B and E. The addition of the singular harmonics (24.25) permits a general solution to the axisymmetric problem of the hollow sphere, but the corresponding non-axisymmetric solution cannot be obtained from equations (24.24, 24.25). To understand this, we recall from §20.4 that the elimination of one of the components of the Papkovich-Neuber vector function is only generally possible when all straight lines drawn in a given direction cut the boundary of the body at only two points. This is clearly not the case for a body containing a spherical hole, since lines in any direction can always be chosen that cut the surface of the hole in two points and the external boundaries of the body at two additional points.
KeywordsSpherical Harmonic Hollow Sphere Stress Concentration Factor Maximum Tensile Stress Solid Sphere
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