Pseudo-planar deformations of a linearized elastic solid

The equations of motion for the pseudo-planar motions of a classical linearized elastic solid and an incompressible linearized elastic solid undergoing non-uniform rotation about a vertical axis are derived. The pseudo-planar motions for both a classical linearized and an incompressible linearized elastic solid are determined numerically. For a classical linearized elastic solid, the non-uniform rotation is time-dependent and is specified. We derive a wave equation that models the non-uniform rotation for an incompressible linearized elastic solid. A pressure Poisson equation is derived and depends on the time derivative of the non-uniform rotation. The locus of the equations of motion coupled with the pseudo-planar motions of a cylindrical solid are plotted and the results are discussed. We show that the pseudo-planar motions of a classical linearized elastic solid with zero rotation are translations of the pseudo-planes about the locus. The pseudo-plane motions for classical and incompressible linearized elastic solids undergo translations and rotations about the locus. The motions are bound and stable when the pressure is symmetric. Unsymmetric pressure, which is just the mechanical pressure, results in a distortion of the pseudo-planar curves.