Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Abstract

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach.

Sensing linear viscoelastic constitutive parameters with a Timoshenko beam on a multi-layer foundation: Modeling and simulation

“…Since no ambiguity arises, in the following we will drop the hat for nondimensional boundary forces and torques. By assuming that only the portion of the substrate in contact with the sensor experiences deformation, we adopt the following boundary conditions for the field g gðx; tÞ ¼ 0 at x ¼ 0 and x ¼ 1: We obtain a reduced order model for the coupled system (10) through Galerkin projections of the fields w; w, and g on suitable bases. By separation of variables, the kinematic fields are expressed as…”

“…The model for the coupled system is given by the set of Eqs. (10), which describe the coupled evolution of the beam and of the interface between the substrate and the coupling elements. We formulate an inverse problem based on the model (10) to estimate the material parameters of the substrate given displacement and/or force measurements of the beam.…”

“…(10) and premultiplication by the sets of test functions w; w, and g respectively. Integration of the domain of the projected governing equations and integration by parts give…”

“…An algorithm to solve inverse problems in linear elasticity is presented in [7], and an in inverse problem to estimate constitutive parameters in elastoplastic solids in presented in [8]. Full-field measurements techniques are presented in [9,10] to estimate elastic parameters in solids, and a model-based approach is used in [11] to estimate viscoelastic response constitutive parameters. A numerical technique based on finite elements method is presented in [12] to estimate material parameters from displacement and force measurements.…”

A Joint Sparse Recovery Framework for Accurate Reconstruction of Inclusions in Elastic Media

“…Many dedicated mathematical and computational algorithms for the reconstruction of location and parameters of anomalies of different geometrical nature (cavities, cracks, and inclusions) have been proposed over the past few decades (see, for instance, [1,4,6,12,13,14,15,21,26,27,29,31,33,34,42,43,55], the survey articles [16,11], and the monograph [5]). Most of the classical techniques are suited to continuous measurements, in other words, to experimental setups allowing to measure continuum deformations inside the elastic body or on a substantial part of its boundary.…”