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We consider the nonlinear,inverse problem of computing the stored energyfunction of a hyperelastic material from the full knowledge of the displacementfield. The displacement field is described as solution of the nonlinear,dynamic, elastic wave equation, where the first Piola-Kirchhoff stress tensoris given as the gradient of the stored energy function. We assume that we havea dictionary at hand such that the energy function is given as a coniccombination of the dictionary's elements. In that sense the mathematical modelof the direct problem is the nonlinear operator that maps the vector ofexpansion coefficients to the solution of the hyperelastic wave equation. Inthis article we summarize some continuity results for this operator and deduceits Fr\'{e}chet derivative as well as the adjoint of this derivative. Since thestored energy function encodes mechanical properties of the underlying,hyperelastic material, the considered inverse problem is of highest interestfor structural health monitoring systems where defects are detected fromboundary measurements of the displacement field. For solving the inverseproblem iteratively by the Landweber method or Newton type methods, theknowledge of the Fr\'{e}chet derivative and its adjoint is of utmostimportance.

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field