Nonlinear Korn's Inequalities and Related Questions

The central objective of three-dimensional elasticity theory consists in finding the equilibrium position of an elastic body occupying a given reference configurationWÌR3in the absence of applied forces, when it is subjected to applied body and surface forces as well as specific boundary conditions. This objective is achieved by solving ad hoc equations (which take either the form of a boundary value problem or the form of a minimization problem), whose unknown is a vector fieldj:W®R3called a deformation of the reference configuration; this means that the vectorj(x) represents the new position occupied in the equilibrium position by each pointxÎW.It turns out that the unknown deformationjappears in the equations of nonlinear elasticity through the positive-definite, symmetric, tensor fieldÑjTjÑcalled the metric, or Cauchy-Green, tensor field, which “governs” the changes of volumes, surfaces, and lengths associated with the deformationj.It is thus natural to seek whether a deformation varies continuously as a function of its Cauchy-Green tensor field: This is the central theme of this proposal.One way to do so is by means of nonlinear Korn's inequalities, viewed as “nonlinear coun- terparts" of the linear Korn inequalities (which naturally arise in linearized three-dimensional elasticity, i.e., when only the “first-order parts" are kept in the equations), the genesis and analysis of which is on firm grounds.Another way to do so is by means of a detailed analysis of the way a deformation can be recovered from the knowledge of its Cauchy-Green tensor field.Such results (continuity results or new nonlinear Korn's inequalities) could shed a new light on the relatively new and promising theory of “intrinsic nonlinear elasticity", where the Cauchy-Green tensor is considered as the primary unknown.