Recovery of nonlinear material parameters in a quasilinear Lam\'e system

TL;DR

This paper addresses the inverse problem of uniquely and stably recovering nonlinear elastic parameters in quasilinear Lamé systems from boundary measurements, advancing material characterization techniques.

Contribution

It proves the unique and stable recovery of a broad class of nonlinear elastic tensors, including isotropic and certain anisotropic cases, from limited boundary data.

Findings

Successful recovery of nonlinear elastic parameters from boundary stress data.

Applicable to both isotropic and anisotropic nonlinear elastic tensors.

Boundary measurements can be obtained at finite points, even a single point in isotropic cases.

Abstract

We investigate the inverse problem of determining nonlinear elastic material parameters from boundary stress measurements corresponding to prescribed boundary displacements. The material law is described by a nonlinear, space-independent elastic tensor depending on both the displacement and the strain, and gives rise to a general class of quasilinear Lam\'e systems. We prove the unique and stable recovery of a wide class of space-independent nonlinear elastic tensors, including the identification of two nonlinear isotropic Lam\'e moduli as well as certain anisotropic tensors. The boundary measurements are assumed to be available at a finite number of boundary points and, in the isotropic case, at a single point. Moreover, the measurements are generated by boundary displacements belonging to an explicit class of affine functions. The analysis is based on structural properties of nonlinear Lam\'e systems, including asymptotic expansions of the boundary stress and tensorial calculus.