The uniqueness of an inverse problem of isotropic incompressible three dimensional elasticity aimed at reconstructing material modulus distributions is considered. We show that given a single strain field and no boundary conditions, arbitrary functions may have to be prescribed to make the solution unique. On the other hand, having two linearly independent strain fields leads to a favorable solution space where a maximum of five arbitrary constants must be prescribed to guarantee a unique solution. We solve inverse problems with two strain fields given using the adjoint weighted equation method and impose five discrete constraints. The method exhibits good numerical performance with optimal rates of convergence.
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