Some Formation Problems for Linear Elastic Materials

Abstract

Some equations of linear elasticity are developed, including those specific to certain actuator structures considered in formation theory. The invariance of the strain-energy under the transformation from rectangular to spherical coordinates is then established for use in two specific formation problems. The first problem, involving an elastic structure with a cylindrical equilibrium configuration, is formulated in two dimensions using polar coordinates. It is shown that L² controls suffice to obtain boundary displacements in H1/2. The second problem has a spherical equilibrium configuration and utilizes the elastic equations in spherical coordinates. Results similar to those obtained in the two dimensional case are indicated for the three dimensional problem.