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

TL;DR

This paper investigates the inverse problem of determining a hyperelastic material's stored energy function from displacement data, providing mathematical insights into the operator's properties and derivatives essential for iterative reconstruction methods.

Contribution

It derives the Fréchet derivative and its adjoint for the nonlinear operator mapping energy function coefficients to displacement solutions, aiding inverse problem solutions.

Findings

Continuity results for the nonlinear operator

Explicit formulas for the Fréchet derivative and its adjoint

Foundation for iterative reconstruction algorithms

Abstract

We consider the nonlinear,inverse problem of computing the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as solution of the nonlinear, dynamic, elastic wave equation, where the first Piola-Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article we summarize some continuity results for this operator and deduce its Fr\'{e}chet derivative as well as the adjoint of this derivative. Since the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton type methods, the knowledge of the Fr\'{e}chet derivative and its adjoint is of utmost importance.