On Two Theories for Brittle Fracture: Modeling and Direct Numerical Simulations

The work presented in this dissertation focuses on extending a recent effort of developing brittle fracture theory with an aim of achieving bounded crack-tip stress and strain without the use of extra near-tip cohesive surface. The first model studied in this dissertation is attributed to modeling the bulk material response in the context of nonlinear strain-limiting theory elasticity. The second model is a theory of fracture developed by Sendova and Walton based on incorporation of surface mechanics. In the first part of this dissertation, we analyze the nonlinear fracture model using a combination of asymptotic and numerical arguments. We find that the use of nonlinear response relations, for a special case of plane-strain fracture, leads to a highly nonlinear partial differential equation. We obtain an asymptotic solution to this nonlinear boundary value problem and subsequently develop a numerical model using an adaptive finite element method. In the second part of the dissertation, a main focus is to implement the surface-mechanics class of fracture theory developed by Sendova and Walton using a stable numerical method such as finite elements. If the surface-tension is assumed to be dependent linearly on the in-plane curvature, the resulting jump momentum balance boundary condition will contain higher-order tangential derivatives. We present a reformulation of the crack-surface boundary condition using the boundary Green's function and Hilbert' transform (as Dirichlet-to-Neumann map) and subsequently implemented the model using an adaptive finite element method. Both the models, studied in this work, predict a physically reasonable crack-tip strain compared to the singular prediction from the linearized elasticity model. Moreover, the crack-tip stress predicted by both the models remain smaller in magnitude compared to the corresponding prediction from the classical linearized model. Finally, since the two models studied in this dissertation do not indicate the singular stress growth in the vicinity of the crack-tip, the crack-tip is not a singular energy sink. Therefore, the classical fracture criterion based upon the singular solution such as Stress Intensity Factor (SIF) or local Energy Release Rate (ERR) is not available. For the nonlinear plane-strain fracture model, we study the behavior of the Critical Crack-Tip Stress as a possible fracture criterion. The numerical results indicate that the cleavage stress is maximum along the line directly ahead of the crack-tip and this result is in agreement with the classical linearized elastic fracture mechanics solution for pure mode-I loading