13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- predictions for materials with different microstructural statistics or different loading conditions. Improved fidelity requires incorporation of the details of how microstructure controls the evolution of local damage events. A formulation that can improve fidelity in a probabilistic framework was presented by Pardee and colleagues [38, 39]. The Pardee formulation recognizes the discontinuous nature of crack growth by associating each of the three phases of damage development in the example of Fig. 7 with a distinct domain ΩI, ΩII, or ΩIII in the space in which probability density is defined. Thus, at a point in ΩI one defines the probability density PI(XI = a/D; t)da that a crack of length XID ≤ a ≤ XID + da with XI < 1 is propagating through the matrix at time t; at a point in ΩII one defines the probability density PII(XII = β; t)d β that an arrested crack has damage parameter XII ≤ β ≤ XII + d β at time t; and at a point in ΩIII one defines the probability density PIII(XIII = a; t)da that a crack of length XIII ≤ a ≤ XIII + da with XI > 1 is propagating into the fiber tow at time t. With growth laws predicted by a small number of virtual tests and statistical distributions calibrated by experiments, the Pardee problem can be solved to generate rapid predictions of the full distribution function for engineering properties, such as the time to failure of the reinforcing tow in Fig. 7. Serving within such an efficient formulation, virtual tests can relate the statistics of rare failure events to microstructural variance with the least possible computational effort. a D matrix tow coating a =D D matrix tow coating a D matrix tow coating 0 1 a/D 0 1 β 1 ΩI ΩII ΩIII a/D (a) (b) (c) Figure 7. A microcrack (a) originating in a superficial matrix layer might (b) arrest temporarily at a fracture resistant coating that protects a tow before (c) propagating into the tow and exposing the fibers in the tow to the environment. In correspondence with the three stages of microcrack growth, there exist three domains ΩI, ΩII, and ΩIII, through which probability density flows in conservation equations describing their evolution; density flux is indicated by arrows. In domains I and III, the state variable is crack length a, while in domain II it is a damage parameter β that increases with time from 0 to unity. References 1. Cox, B.N. and Q.D. Yang, In quest of virtual tests for structural composites. Science, 2006. 314: p. 1102-1107. 2. Ashby, M.F., Physical modelling of materials problems. Materials Science and Technology, 1992. 8: p. 102-111. 3. Llorca, J., et al., Multiscale modeling of composite materials: a roadmap towards virtual testing.
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