Reconstruction of the shear modulus of viscoelastic systems in a thin cylinder: an inversion scheme and experiments

Abstract

We consider a problem of reconstructing the shear modulus of an viscoelastic system in a thin cylinder from the measurements of displacements induced by torques applied at the bottom of the cylinder. The viscoelastic system is a mathematical model of a pendulum-type viscoelastic spectrometer (PVS). We first compute in an explicit form the solution of the viscoelastic system, and then derive with an error estimate the leading order term of the average of the solution. This leading order term yields a nonlinear inversion scheme to determine the shear modulus from the measurements of displacements. We apply the inversion scheme to determine the shear modulus using experimental data acquired from a PVS system.

Topological sensitivity based far-field detection of elastic inclusions

“…The following remarks are in order. The covariance of weighted location indicator I n W has the same form as expression (39) in Lemma 3.6 except for a modulation by Cσ 2 ξ /n, where C is a positive constant independent of n and σ ξ . This elucidates that the shapes of the hot spots in the speckle field are identical to the shape of z S → I n W (z S , α) in the noise-free case and their amplitudes are √ Cσ ξ / √ n times that of the main peak of I n W .…”

“…It is worthwhile highlighting that the inverse elastic scattering problem caters to various applications including non-destructive evaluation of an elastic structure for integrity and material impurities [39], prospecting of mineral reservoirs [40], and medical diagnostics for detecting and classifying small tumors and locating tissue abnormalities of vanishing sizes [41,42].…”

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

“…Elasticity imaging frameworks cater to a broad range of applications, for example, nondestructive testing of elastic objects for material impurities and structural integrity [26], exploration geophysics for mineral reservoir prospecting [50,53], and medical diagnosis, in particular, for detection and characterization of potential tumors of diminishing sizes [47,48,44]. In the perspectives of medical diagnosis elasticity imaging aims to fathom spatial variations in the material parameters of human tissues by harnessing the interdependence between elastic field and tissue elasticity.…”

“…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.…”

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

“…The inverse problem of quantitative evaluation of constitutive parameters is notorious for its complexity and ill-posed character. 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.…”