Guo and nairn solved the j integral equation using

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methods. (Guo and Nairn) solved the J-integral equation using the material point method. In their method, virtual work principle was implemented to calculate the nodal acceleration. Fatigue Crack Behavior Repeated fluctuating loads have always caused drastic failures and the investigation of fatigue failure would be a key task in designing structures. Most of fatigue loading tests apply sinusoidal
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P a g e | 7 cyclic loads to obtain a description of stress as a function of number of cycles. A stress was defined that below which no failure occurred. Fatigue crack prediction has the same three steps as static loading steps had i.e. initiation, propagation, and failure. However, the difference is that the number of cycles in fatigue loadings plays an important role to define those three steps. The crack length and stress intensity factor are the parameters defined as a function of number of cycles. The results in the fatigue fracture mechanics are mainly present by log (da/dN) and log ( K) parameters. As mentioned, the cyclic loads have usually the sinusoidal behavior, and K is the difference between the maximum and minimum values of stress intensity factors in each cycle: K = Kmax-Kmin Some crack propagation laws have been developed by researchers to relate the stress intensity factor to the change in the crack length value. (Seitl and Kne´sl) considered a tensile specimen test with holes under the fatigue loadings and showed the effect of T-stress on the crack path. T- stress is a constant tensile stress parallel to the crack face described in the Williams equation of stress field. The stability of crack propagation under the mode I loading can be derived by T- stress. They compared the numerical simulation of fatigue crack propagation using the maximum tensile stress criterion with the experimental results. It was concluded that existence of holes in the specimen reduces the crack propagation rate. In addition, KI and T-stress had to be defined
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P a g e | 8 for curved cracks in fatigue loadings. (Ghidini and Donne) predicted the fatigue lives of pre- corroded and friction stir-welded materials with the residual stress consideration using the aerospace fracture-based packages. The cracks were formed at particle inclusion places and simulated as semi elliptical surfaces. The strength of friction stir welded materials and the base materials were compared and an experimental procedure was applied to their framework. They noted that the friction stir welded materials have higher fatigue life comparing to that of the base materials. In addition, 77% of pre-corroded friction stir welded joints were damaged at the fine- grained space on the weld nugget with some pits in the material. They observed the cracks under an optical microscope. (Dong and Atluri) compared XFEM with symmetric Galerkin boundary element method (SGBEM) for the fatigue crack propagation and solved many problems with their numerical simulation. They asserted that SGBEM has more advantages comparing to XFEM in predicting the fatigue crack growth. SGBEM was more accurate for computing the
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