Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models

• Published in: Journal of inverse and ill-posed problems Vol. 30; no. 4; pp. 521 - 547
• Main Authors: Carrillo, Hugo, Waters, Alden
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.08.2022; Walter de Gruyter GmbH
• Subjects: 35B30; biological tissues; Displacement; Elasticity; Fluid flow; Hypotheses; Incompressible flow; Inverse problems; magnetic resonance elastography; Mathematical models; optimal control; Parameters; shear modulus reconstruction; Stability; Stability analysis
• ISSN: 0928-0219; 1569-3945
• DOI: 10.1515/jiip-2020-0142

We study some inverse problems involving elasticity models by assuming the knowledge of measurements of a function of the displaced field. In the first case, we have a linear model of elasticity with a semi-linear type forcing term in the solution. Under the hypothesis the fluid is incompressible, we recover the displaced field and the second Lamé parameter from power density measurements in two dimensions. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement, with the background pressure fixed. On the other hand, we prove in dimensions two and three a stability result for the second Lamé parameter when the displacement field follows the (nonlinear) Saint-Venant model when we add the knowledge of displaced field solution measurements. The Saint-Venant model is the most basic model of a hyperelastic material. The use of over-determined elliptic systems is new in the analysis of linearization of nonlinear inverse elasticity problems.