Uniqueness of the elastography inverse problem for incompressible nonlinear planar hyperelasticity

The uniqueness of several 2D inverse problems for incompressible nonlinear hyperelasticity is studied. These problems are motivated by elastography, in which one is given a measured deformation field in a 2D domain Omega and seeks to reconstruct the pointwise distribution of material parameters within Omega. Two classes of models are considered. The simpler class is material models characterized by a single material parameter exemplified by the Neo-Hookean model. The second class of material models considered is characterized by two material parameters, and includes a simplified Veronda-Westmann model, a Blatz model and a modified Blatz model. Consistent with the results in linear elasticity, we find that significantly fewer data are required to determine the material properties under plane stress conditions than under plane strain conditions. The results show that, roughly speaking, one needs one measured deformation for each material parameter sought under plane stress conditions, and twice as much data for plane strain conditions.