Nonlinear Finite Element Modeling of Soft Tissue Cavitation and Dynamic Brittle Fracture

Abstract

This dissertation presents two nonlinear finite element methods: one for simulating cavitation in soft isotropic tissues and the other for dynamic fracture in brittle materials. Both finite element codes implement the open-source Deal.II C++ library and derive from the same tutorial program. Here we present our computational toolbox of compatible finite element models, developed with high-performance computing (HPC) in mind, featuring technical capabilities such as adaptive mesh refinement, adaptive mesh deletion, and adaptive time stepping. These models separately address distinct nonlinear behaviors involved in the multivariate biomechanical process of injection-driven cavitation and fracture in soft isotropic tissues. Although the combined phase-field fracture-cavitation model is outside the scope of this current dissertation, it concludes by clearly outlining the path toward its forthcoming completion.Soft isotropic tissues, e.g. brain and liver, deform in an exceedingly nonlinear manner due to their incompressible, biphasic, and viscous nature. Thus, we adapt a state-of-the-art material model which combines nonlinear poromechanics and finite viscoelasticity. This model also utilizes the vaunted Ogden constitutive relation which is the standard choice for simulating high-strain applications, e.g. deformation of soft tissues. Using this highly advanced model, we demonstrate several key experimentally observed mechanical behaviors in soft tissues: hysteresis, preconditioning, and tension-compression asymmetry. Our next numerical example simulates wetting cavitation in brain tissue, wherein fluid flows into the expanding cavity. Although this does not reflect our stated injection-driven application, wherein fluid flows out of the expanding cavity, this serves as an important proof of concept which further motivates development for the combined phase-field fracture cavitation model. We extensively detail this combined phase-field fracture cavitation model as the logical conclusion to this research in the future directions discussion.We simulate dynamic brittle fracture in polymethyl methacrylate (PMMA) using the phase-field method, integrating time using the implicit generalized-and explicit HHT-methods for their favorable high-frequency dissipation. Using an error convergence test, we validate our elastodynamic time integrator by observing the expected quadratic spatiotemporal error convergence rate. Our first physically motivated numerical example presents a mesh convergence study, featuring a classic dynamic brittle fracture benchmark problem. To showcase the favorable high frequency dissipation, we also simulate a highly complex fracture topology, featuring multiple branching and merging events. Our next numerical example investigates delamination fracture by considering discrete elastic heterogeneity. Lastly, we produce microbranching instabilities using pre-stretch loading and a random Gaussian distribution for the critical fracture energy release rate. We also detail our phase-field fracture modeling of dynamic brittle fracture in polyacrylamide in the appendix material.