Linear and nonlinear shear wave propagation in viscoelastic media

This dissertation covers material pertaining to direct and inverse problems for viscoealstic materials with applications to pulmonary fluids (e.g. mucus) and other biological materials. This research extends a classic viscoelastic characterization technique, originally developed by John D. Ferry, to small volume (micro-liter) samples of fluid through mathematical modeling and computation in the development of the micro-parallel plate rheometer (MPPR). Ferry’s inverse characterization protocol measures the attenuation length and wave length of shear waves in birefringent synthetic polymers, and uses the exact solution for a linear viscoelastic fluid undergoing periodic deformation in a semi-infinite domain to obtain formulas relating these values to the linear viscoelastic (material) parameters. In this dissertation, Ferry’s exact solution is extended to include both finite depth effects that resolve counter-propagating waves, and nonlinear effects that arise from Giesekus and upper-convected Maxwell constitutive equations. These generalizations of Ferry’s solution are used in the development of the MPPR, which generates a shear wave in a finite depth domain and uses micro-bead tracking and nonlinear regression on fluid displacement time series to extend Ferry’s protocol of viscoelastic characterization. The final topic of this dissertation is a predictive model for stress communication and filtering across viscoelastic layers. By focusing on stress signals arriving at either boundary in a finite depth domain, this work identifies a remarkable structure in boundary extreme stress signals which could play a key role in regulating certain aspects of lung function. This structure is derived from harmonic resonance in the elastic solid material limit, and demonstrated to persist and diverge from this limit as a function of a dimensionless parameter (the ratio of the attenuation length to the wave length). Analytical results demonstrate that this structure persists with respect to all material and driving parameters and in more general boundary value problems