Transversely Isotropic Problems

A transversely isotropic medium is a medium, which has a favored direction and is isotropic in the plane perpendicular to this direction. Among crystalline media, all materials with a hexagonal crystal system belong to this class: they are elastically isotropic in the plane perpendicular to the hexagonal axis. Fiber composites with the fibers arranged in parallel in one direction also represent a transversely isotropic medium, which is isotropic in the plane perpendicular to the fiber direction. Many functional materials exhibiting a preferred direction can also be classified as such, e.g. some piezo-electric materials. We can find many more examples in biological media. A linear transversely isotropic medium is fully defined by five elastic constants. The fundamental solution for transversely isotropic media was found by Michell. He demonstrated that the normal displacement of the surface of a transversely isotropic elastic half-space under the effect of a force acting on the origin is given by the same equation as in the case of isotropic media; only a modified definition of the effective elastic modulus has to be used. Consequently, there is no need for a special consideration of all normal contact problems for transversely isotropic media. We will simply refer to the results from Chap. 2 to 4, which are equally valid for transversely isotropic media.