Collaborative Research: Fracture in Soft Organic Solids --- The Variational View

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

This award supports a research collaboration on mathematical and numerical modeling and analysis of failure in soft materials. This research project concerns the derivation and numerical implementation of a mathematical theory capable of describing, explaining, and predicting the initiation and propagation of fracture in soft organic solids---namely, solids made up of networks of long carbon-based macromolecules such as elastomers, gels, and biological tissues---when subjected to arbitrarily large mechanical forces. Soft organic solids are known to fracture in a very different manner than standard hard solids (such as metals and ceramics). The defining difference is that internal fracture in soft organic solids initiates through the sudden growth of inherent defects into large enclosed cavities/cracks (a phenomenon popularly referred to as cavitation). With the ever-increasing use of soft materials in new technologies, a fundamental and quantitative understanding of when and how organic solids fracture is of utmost importance for their advancement. Likewise, such a fundamental and quantitative understanding is critical in advancing medical treatments involving soft biological tissues, such as shock-wave lithotripsy, or treatments dealing with aneurysms. This project centers on a novel variational theory of fracture for finitely deformable solids that is consistent with the principle of conservation of mass (a highly non-trivial feature that has been overlooked in the literature by related formulations) and wherein the newly created surfaces (by fracture) are not restricted to be hypersurfaces (as in classical brittle fracture) but can also be the boundaries of N-dimensional cavities, N being the spatial dimension. The main objectives of the project are: (1) to develop a formulation in terms of variational evolutions for the initiation and propagation of fracture in soft organic solids under arbitrarily large quasi-static deformations, and (2) to implement this formulation numerically and confront its predictions with emerging experimental evidence of high spatio-temporal resolution. Objective (1) entails rigorous existence results, while objective (2) entails the construction of appropriate approximate functionals (of the phase-field type) and their stable and convergent numerical implementation in the non-convex context of finite deformations with constraints (in particular, incompressibility).