13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- The computational expense of Monte Carlo analysis can be high. The uncertainty in a predicted mean property, such as the expected strength in a distribution of strengths, will fall as the number of computed cases N rises. If σ denotes the uncertainty in a predicted mean property normalized by the width of the distribution of the property, σ ≈ N-1/2. For example, if strength is predicted to have a deviance of 10%, then determining the mean strength to 1% accuracy requires 100 simulations. 3(3´) 4(4´) 1 2(2´) (1´) 5 (5´) 6(6´) 7 (7´) 8(8´) 1 eΩ 2 eΩ cΓ 3 3´ 4 4´ 1 2 2 ´ 1´ 5 5´ 6 6´ 7 7´ 8 8´ (a) (b) Figure 6. (a) In the A-FEM formulation, a single element can split into several domains, accommodating a crack bifurcation event. (b) When a crack in a solid element impinges upon a material interface, the cohesive element governing fracture of the interface (grey domain, with width exaggerated for visibility) divides into two sub-elements, allowing the correct sign and magnitude of interfacial shear displacement to be computed on either side of the impinging crack. If the virtual test is used in materials design, a design optimization study addressing the statistics of failure might require computing the effects of 102 – 104 different material or architectural parameters. To establish reasonably accurate trends in mean strength, 104 – 106 virtual tests must be executed. For the computational time to remain within one week (106 seconds) for a relatively wide search, a single virtual test should execute in 1 sec, to order of magnitude. Current execution times for multiple crack evolution leading to ultimate failure in a single virtual test are orders of magnitude higher than this. Probabilistic models offer computational efficiency and the possibility of accurate predictions of rare events. In a probabilistic theory, instead of tracking damage evolution through a particular microstructure, one tracks the evolution of a probability distribution P(X) for a damage variable X through time or elapsed fatigue cycles. Because P(X) can be represented numerically with arbitrary precision over all values of X, accuracy in predicting the tails of the distribution is at least possible. However, is it not assured: accuracy also demands that the probabilistic theory incorporate a faithful representation of the details of the influence of the stochastic microstructure on the evolution of P(X). The first and simplest approach uses diffusion equations or discrete chain models to describe the evolution of a damage variable such as crack length, coupled to a simplified representation of the influence of some other factor, which could be a microstructural factor. However, such phenomenological models are limited in the degree of fidelity they can achieve in predicting the effects of microstructural variation within a single class of materials. They must use calibration data for the same materials as those for which predictions are to be made, with the same failure mechanisms and microstructural statistics being present. They cannot be used easily to make
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