Singular solutions in gels: Cavitation and debodning

This award supports the ongoing research program of the Principal Investigator on modeling, analysis, and simulation of material failure in gels. Engineering devices generally consist of more than one material. Their interconnections, as well as the binding of the device to substrates, are susceptible to environmental agents that ultimately cause them to break down, leading to device failure. It is estimated that the cost of product recall in the medical device industry alone ranges between $2.5 and $5 billion annually, 50% of which is attributed to material failure. This project will investigate material fracture in polymeric materials, with a main focus on gels. The mechanism of failure in gels as well as the mathematical methods to address it are very different from those in crystalline materials. Upon placement of a polymeric material in a fluid-rich environment, such as an implanted medical device in the body, it tends to swell by absorbing water. Different materials have different swelling ratios, which cause stress to build up at the bonding interfaces and substrates. Upon reaching the stress threshold of the adhesive, the device may experience debonding. The research activities will consist of modeling, analysis, and numerical simulation of the phenomena that causes a gel to break apart from its substrate. There will be a strong and clear connection with laboratory experiments and industrial applications, especially those related to the medical device industry. Several graduate and undergraduate students will be involved in the research.

Understanding material fracture, its initiation and evolution, remains one of the main challenges in materials science. From the mathematical point of view, the study of debonding requires dealing with singular solutions, in cases when standard mechanical assumptions fail. This project will develop and apply combined tools from calculus of variations, geometric measure theory, asymptotic analysis, and free boundary problems for partial differential equations to provide a mathematical description of how microdefects present in the material may evolve into cavities that grow and cause the material to detach from a substrate. The main mathematical difficulty in the study of cavitation is the loss of injectivity of the deformation map that transforms a point in the reference configuration into a cavity surface. The application of tools from measure theory, such as the distributional determinant, allows validation of the governing equations in the singular frameworks. The research will provide a unified framework to study cavitation and debonding, and the evolution from the former to the latter. The project will focus on studying singularities that form in gels, assumed to be mixtures of polymer and fluid. Swelling of a gel sample attached to a substrate may produce the necessary force for singularities to form. The analysis of delamination will be based on the conjecture that there are two relevant parameters, geometric and material, able to describe the optimal energy of the body in terms of the length of detachment from the substrate. One goal is to show that stress concentrations at interfaces correspond to shearing.