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Recovery of nonlinear material parameters in a quasilinear Lamé system

David Johansson, Yavar Kian

TL;DR

This work tackles the inverse problem of identifying nonlinear elastic parameters in a stationary quasilinear Lamé system from boundary stress measurements produced by prescribed boundary displacements. The authors develop an infinite‑order linearization framework around a constant boundary state to derive explicit boundary‑stress asymptotics and relate them to derivatives of the nonlinear elastic tensor. They prove local uniqueness and Lipschitz stability for recovering the displacement–dependent part $C(\\lambda,0)$ from finite boundary data, and then extend the analysis to recover higher‑order derivatives with respect to the strain, $D_{\\eta}^{k}C(\\lambda,0)$, up to order $N-1$, with the full tensor $C(\\lambda,\\eta)$ determined under real‑analyticity for the tensor classes considered. These results provide a rigorous pathway to reconstruct nonlinear material properties from restricted boundary measurements, with potential impact on applications in soft tissues, polymers, and aerospace materials, and lay the groundwork for reconstruction algorithms based on linearized maps.

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 Lamé 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 Lamé 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 Lamé systems, including asymptotic expansions of the boundary stress and tensorial calculus.

Recovery of nonlinear material parameters in a quasilinear Lamé system

TL;DR

This work tackles the inverse problem of identifying nonlinear elastic parameters in a stationary quasilinear Lamé system from boundary stress measurements produced by prescribed boundary displacements. The authors develop an infinite‑order linearization framework around a constant boundary state to derive explicit boundary‑stress asymptotics and relate them to derivatives of the nonlinear elastic tensor. They prove local uniqueness and Lipschitz stability for recovering the displacement–dependent part from finite boundary data, and then extend the analysis to recover higher‑order derivatives with respect to the strain, , up to order , with the full tensor determined under real‑analyticity for the tensor classes considered. These results provide a rigorous pathway to reconstruct nonlinear material properties from restricted boundary measurements, with potential impact on applications in soft tissues, polymers, and aerospace materials, and lay the groundwork for reconstruction algorithms based on linearized maps.

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 Lamé 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 Lamé 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 Lamé systems, including asymptotic expansions of the boundary stress and tensorial calculus.
Paper Structure (10 sections, 18 theorems, 173 equations)

This paper contains 10 sections, 18 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Dependence with respect to the displacement
  4. Dependence with respect to the strain tensor
  5. Outline
  6. Forward problem
  7. Recovery of dependence on the displacement
  8. Recovery of dependence on strain tensor
  9. Complementary results
  10. The linear equation

Key Result

Theorem 2.2

Let $\lambda\in\mathbb{R}^n$, $\Omega$ satisfy assumption assumption-domain and, for $m=1,2$, let $C^m$ satisfy assumptions assumption-nonlinearity, with $k=1$, assumption-ellipticity and assumption-symmetry. If for some $\delta>0$ it holds that then $C^1(\lambda,0) = C^2(\lambda,0)$.

Theorems & Definitions (33)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 23 more