Some Issues in the Kinematics of Finite Deformations

This dissertation deals chiefly with various issues pertaining to the existence and uniqueness of a finite deformation that gives rise to a prescribed right or left Cauchy-Green strain-tensor field. Following a review and discussion of available existence and uniqueness theorems appropriate to a pre-assigned right strain field, the extent of uniqueness of a generating deformation is established under minimal smoothness and invertibility assumptions. Further, the compatibility equations of finite continuum kinematics are used to arrive at an analytical proof of Liouville's theorem on conformal deformations, which supplies an exhaustive classification of three-dimensional deformations that preserve all angles. The remainder of the dissertation is devoted to the more involved corresponding existence and uniqueness questions for a given left strain-tensor field. These questions are first discussed in a three-dimensional setting and are then resolved for the special class of plane deformations. The results thus obtained stand in marked contrast to their counterparts for a given right strain field.