Fatigue crack growth test for 2A12-T4 aluminum alloy was conducted under constant amplitude loading, and the scatter of fatigue crack growth was analyzed by using experimental data based on mathematical statistics. A probabilistic modeling method was introduced to describe the crack growth behavior of 2A12-T4 aluminum alloy. The posterior distribution of model parameter is obtained based on diffuse prior distribution and fatigue crack test data, which is through Bayesian updating. Based on posterior samples of model parameter, the simulation steps and approach give us the crack length exceedance probability, the cumulative distribution function of loading cycle number, and scatter of crack length and loading cycle number, of which simulation results were used to verify the veracity and superiority of the proposed model versus the experimental results. In the present study, it can be used for the reliability assessment of aircraft cracked structures.
Based on aircraft structural life management viewpoint, the mechanical performance of metal materials is often considered homogeneous. However, a considerable amount of scatter has been observed in fatigue life (includes crack initiation life and crack propagation life) even under the same service (loading and environment) conditions.
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Based on (5) and (6), we can know that the reliability function of and the reliability function of can be determined by experimental crack propagation curves, which do not take the measure error and model parameter error into account. In other words, the probability distribution can be determined by the probability distribution of model matrix parameter . It is difficult to obtain the analytical solution form of (5) and (6), so the simulation approach will be given in the following work of this study.
(3) Based on the above 1000 simulation random (, ) samples, estimate the probability value of , which satisfies for different loading cycle number. For this case, is the given crack length, in this condition =3mm.
As is shown in Figure 11, for the observed crack length, the experimental CDF of loading cycle number fits the analytical CDF very well. It means that the modified probabilistic fatigue crack growth model and above simulation approach can describe the crack growth behavior pretty well.
As is shown in Figure 12, the experimental probability of crack exceedance fits the analytical probability exceedance very well. It means that the modified fatigue crack growth model and above simulation approach can predicate the actual crack growth behavior pretty well.
The validity of the introduced analytical solution was substantiated by convergence diagnose of the interested parameter. Accuracy of the probabilistic crack growth model was substantiated by designing simulation steps and approach, whose simulation results were compared with the experimental results. However, the scatter analysis is an important part for the validation of this study.
It is founded that the introduced probabilistic fatigue crack growth modeling method and simulation approach can predict the crack growth behavior pretty well. The proposed simulation framework provides a good approach to obtain the CDF of loading cycle number, the probability of crack exceedance, and the scatter of crack growth behavior.
The paper is devoted to theoretical study of the longitudinal shear (mode II) crack unstable growth dynamics in brittle materials. We considered two main regimes of the dynamic propagation of the crack (sub-Rayleigh and supershear) and their implementation conditions. The research was carried out by computer simulation with the Movable Cellular Automaton method, using the generalized kinetic fracture model, which takes into account the finite duration of local fracture (fracture incubation time). It is shown that the fracture incubation time is a key parameter, which determines the transition conditions of the shear crack growth process from the sub-Rayleigh regime to supershear.
Finite-element simulations comparing energy release rates for forward and sideways cracking. (Top) Finite-element simulations of the maximum principal stress distribution in a silicone elastomer with a precut at λ = 2.0 for a crack that has propagated by dc = 0 and dc = c0/10 (2.54 mm) in the forward direction (left images) and the sideways direction (right images). (Bottom) Energy release rate, G, for forward crack growth and sideways crack growth as a function of the length of newly created crack, dc.
Finite-element simulations showing the effects of crack-tip blunting. (Left) Finite-element simulations of the maximum principal stress distribution in a silicone elastomer with a precut at λ = 1.1 (Top) and λ = 2.0 (Bottom) for a crack that has propagated by dc = c0/10 (2.54 mm) in the forward direction (left images) and the sideways direction (right images). (Right) Energy release rate for sideways crack growth relative to that of forward crack growth as a function of the ratio of the sideways crack length cs to the forward crack length cf.
To investigate crack initiation and propagation of rock mass under coupled thermo-mechanical (TM) condition, a two-dimensional coupled TM model based on the numerical manifold method (NMM) is proposed, considering the effect of thermal damage on the rock physical properties and stress on the heat conductivity. Then, the NMM, using empirical strength criteria as the crack propagation critical criterion and physical cover as the minimum failure element, was extended for cracking process simulation. Furthermore, a high-order cover function was used to improve the calculation accuracy of stress. Therefore, the proposed method consists of three parts and has a high accuracy for simulating the cracking process in the rock mass under the coupled TM condition. The ability of the proposed model for high accuracy stress, crack propagation, and thermally-induced cracking simulation was verified by three examples. Finally, the proposed method was applied to simulate the stability of a hypothetical nuclear waste repository. Based on the outcome of this study, the application of NMM can be extended to study rock failure induced by multi-field coupling effect in geo-materials.
The property change is significantly influenced by micro- and macrocracks induced by thermal loading. To obtain a deeper understanding of the thermal-induced damage of granite under elevated temperatures, numerical simulation has become an important method. Previous simulations have usually applied constant properties and low or fixed temperatures [30, 35, 40]. Yang et al. [37] used a particle-based method to investigate the failure behaviour of pre-holed granite specimens after elevated temperature treatment. Although they use heterogeneous models, the mineral properties (especially the thermal expansion coefficient) are temperature independent. Yan and Zheng [36] proposed a coupled thermo-mechanical model (FDEM-TM) for simulating thermal cracking of granite, but the correctness of this model is verified only in the temperature range between 0 and 100 C. By comparing temperature-dependent and temperature-independent rock properties applied in underground coal gasification reactor simulations, Otto and Kempka [19] found notable differences in rock failure behaviour and concluded that temperature-dependent parameters are important to obtain more reliable results. Wang and Konietzky [30] proposed a model with temperature-dependent thermo-mechanical parameters to simulate thermal cracking of granite. However, the quantitative analysis of thermal cracks is limited due to the fact that most model parameters are generalized values instead of specific laboratory-tested ones. Accurate temperature-dependent properties of granite exposed to high temperatures are necessary for building a reliable numerical model. Unfortunately, a reliable numerical model of thermal cracking of granite exposed to temperatures up to 1000 C is still missing so far.
Before mechanical loading, no obvious cracks caused by thermal stresses only are observed directly in the temperature range between 25 and 800 C. However, the samples experienced 1000 C heat treatment show obvious macrocracks which can be observed by the naked eye (see Fig. 12a, c). Although these cracks appear isolated, they interact with each other and are widespread across the whole sample. In a continuum numerical model, the macrocracks can be represented by a certain values of plastic strain (see Fig. 12b, d). It is visible that the crack patterns of the simulations are in good agreement with the laboratory test observations in terms of macrocracks, if a strain value of about 0.09 is interpreted as macroscopic crack.
Although the macrocracks are only visible above certain temperatures, change in p-wave velocity and open porosity (Fig. 3) indicate that microcracks induced by thermal stresses occur much earlier. Johnson et al. [15] has also found that thermal cracking occurs when a certain threshold temperature is exceeded, which is different for different rocks (e.g. 75 C for Westerly granite and 200 C for Sioux quartzite). They also found that thermal cracking increases progressively after threshold temperature, and the preponderance of cracking occurs below the quartz transition temperature of 573 C. This trend is also observed in our simulations. Figure 13 shows the variation of p-wave velocity (laboratory testing) and the number of failed elements (i.e. element with plasticity states) at different temperatures in the simulation. Failed elements can be interpreted as thermally induced microcracks. Their evolution in time shows a reverse tendency compared to the development of p-wave velocity. The quantity of induced cracks is increasing with increasing temperature, and more than about 80% of the cracks are induced before 600 C. Figure 14 shows the plasticity states (i.e. microcracks) and plasticity tension strain (cracks with certain widths) on the axial plane of the cylindrical sample. The crack initiation temperature is about 80 C with a few randomly induced microcracks across the sample. The element failures begin to occur progressively after the threshold temperature. Most elements fail in tension in the temperature range from 80 to 1000 C. However, at about 500 C shear failure begins to develop quickly with increasing temperature (see Fig. 14a). 2ff7e9595c
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