Energy Release Rate in Fracture Mechanics

This dissertation aims at developing theories and methods to simulate dynamic crack propagation in large deformation. The analytical solution of linear elastic fracture mechanics is utilized to find the shape of the crack surface after deformation. It is discovered that the deformed shape of a line crack is an ellipse and the deformed shape of an elliptical void is also an ellipse under mode I tensile loading. A numerical example shows that the small strain formulation in linear elastic fracture mechanics overestimates the crack tip stress compared to large strain formulation. Therefore stress intensity factor, as a conclusion from linear elastic fracture mechanics, is not accurate in large deformation or at the close neighborhood of crack tip. Energy release rate fracture criterion is revisited by taking molecular dynamics point of view. It is proved by theory and numerical example that the critical energy release rate is not related to surface energy. The energy release rate is the decrease of potential energy per unit crack advance. The surface energy is the increase of potential energy per unit crack advance. The two quantities always have an opposite signs therefore they cannot be equal. It is suggested that the energy release rate is also the kinetic and/or thermal energy generation rate according to conservation of energy. It is noticed that the plasticity exists even in perfect brittle fracture due to the lattice reorientation occurring at the edge of the material. Fracture is a dynamic and irreversible process in nature. It is emphasized that proper region should be chosen to calculate the energy release rate. Such region should enclose the crack tip plastic zone and the part of the elastic zone that involves lattice relaxation, but exclude the plastic zone not related to crack. The same principle also applies to J-integral. Therefore such integral is path dependent. A general method is proposed to calculate energy release rate in molecular dynamics for dynamic or static, brittle or ductile fracture. The equivalency between interatomic potential energy and strain energy is established with new definitions for stress and strain in molecular dynamics. Here, in molecular dynamics, strain energy is defined as the area under the stress-strain curve. The stress and the strain are defined on each atom by taking the summation and the average, respectively, of the influence from all other atoms. Such stress and strain satisfy conjugacy in multi-atom cases with arbitrary configuration under zero temperature. Due to the equivalency of strain energy and interatomic potential energy, the potential energy release rate calculated in molecular dynamics is the same as the strain energy release rate in continuum mechanics. The node release method in finite element analysis is rigorously formulated and verified. The formula to calculate energy release rate in node release method is proposed and validated. The methods to control the crack speed, energy release rate, crack initiation, crack propagation and crack direction are proposed. With such methods, the following conclusions are made: (1) Node release method does not influence material behavior prior to crack extension. (2) Node release method is independent of the steps chosen to release the force. (3) The force-displacement relationship of the releasing nodes, often referred as the cohesive law, depends on the crack speed. The linear relationship is only valid with slow crack speed. (4) Node release method is able to calculate static energy release rate with great accuracy. The static energy release rate is dynamic energy release rate with a slow crack speed. (5) Node release method can be applied to specimen with no initial crack. Therefore the fracture toughness can be obtained from simple tension test. (6) Node release method can simulate dynamics crack propagation. The linear relationship between static and dynamics energy release rates, G=G0 (1--v/vR) , is in good agreement with simulation result. In the simulation, the crack speed asymptotically reaches Rayleigh wave speed in elastic material with fixed grip boundary condition. (7) It is discovered that plasticity is one of the mechanisms that reduces the crack speed. The crack in elasto-plastic material will asymmetrically reach 60% of Rayleigh wave speed. (8) The plasticity influences the crack path in mixed mode fracture. For convenience, all units are normalized Unless otherwise specified.