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When the fatigue occurs above 103 cycles (usually 104 or more), it is usually called High-cycle fatigue.
The S-N curve for a specific material is the curve of nominal stress S (y axis) against the number of cycles to failure N (x axis). The S-N curve of 1045 steel and 2014-T6 aluminum alloy is enclosed below to represent two tipical S-N curves of metal materials. In short, the S-N curve is used to predicts the number of cycles sustained under certain stress before failure.
While S-N curve is clear and straight forward on addressing the service life under fatigue, its accuracy leaves some room to be improved. The non-zero mean stress S-N relation requires huge amount of experiments to obtain the required data and form the mesh over a wide range of mean stresses. A more popular diagram for design purposes is called master diagram which accumulates fatigue data under different mean stresses and presents each line as the fatigue life under the net of maximum and minimum stresses in addition to mean stress and alternating stress as the reference axises.
Please note that the S-N curve and its elaborated master diagram require a lot of experiments to accumulate the necessary data. With the complexity of a master diagram, not to mention the time and effort to create one, the fatigue prediction is still less that perfect. Palmgren (1924) and Miner (1945) suggested an algorithm to combine individual contributions, known as Palmgren-Miner's linear damage hypothesis or Miner's rule.
Miner's rule assume the fatigue life is consumed by the linear combination of different portion of stress state, both cycles and magnitude. Based on the crack opening displacement, fatigue crack growth model is expressed as a function of mechanical properties. All structural component experiences cyclic stresses of sufficient magnitude during their life span.
Equation (5) and (6) are solved with the initial parameters as given by Virkler et al [3] and model predictions are compared in Fig 1. The stochastic crack growth model based on crack opening and random process parameter for the effect of material non-homogeneity is found to be suitable for crack growth prediction. Two components with exactly the same alloy can have vastly different mechanical propoerties.
ABSTRACT: In this paper the development of a torsional fatigue testing machine for metallic materials is presented. Most of the available fatigue properties on materials come from the rotating-beam tests and from axial loading tests; however, little data are obtained from torsional tests and most of them are for materials that behave as ductile under static loading [1]. To obtain more accurate information about torsional fatigue properties in metallic materials, a torsional fatigue machine was built at the Universidad Autónoma de Manizales.
According to Weibull [5], the machines used to perform torsional fatigue tests are classified depending on the type of load used to generate the failure.
Several kinds of machines have been built to characterize the mechanical behavior of materials that are subject to torsional fatigue. As will be shown in the kinematic analysis, the magnitude of the angular acceleration is a function of the angular position of the crank and is also a function of the AC motor angular speed.
Such independence is achieved by raising the torsional stiffness of the rotational elements of the machine. The alternating rotational movement of the testing specimen is generated by a plane four-bar crank and rocker linkage; this mechanism was adapted from the transmission of a commercial washing machine. Figure 3 shows the CAD mechanism as well as its symbolic description; this representation is needed for the kinematic analysis.
Figure 4 shows the mechanical drive system with the identification of its corresponding elements. In order to estimate the magnitude of the angular acceleration of the testing specimen, the equation for the analytical method of vector loop proposed by Norton [8] was used.
The input speed of the AC motor is transmitted to the crank of the four-bar linkage by the V-belt transmission and the two-stage speed reducer as shown in Fig. The variable represents the magnitude of the angular acceleration of the input link driving mechanism.
Since the test specimen is connected to the pinion gear meshed with rocker output 3 (see Figs. Because , c, J, and I values remain constant throughout the operation, it is concluded that the stress wave is a scaled function of the wave shown in Fig. Since and it can be seen that the specimen is subject to a completely alternating stress wave [13].


To test the machine, 24 AISI 1045 steel specimens were tested whose scheme is shown in Fig. Figure 10 shows the relationship between the AC motor angular speed and the number of cycles to failure , and Fig. The correlation values show that there is adequate correspondence between the data tested and the regression line; although it is necessary to test a larger number of specimens to confirm this behavior. A machine for dynamic torsional fatigue testing with inertial loads was designed and built, with the adaptability to fasten metal samples with diameters between 5 and 15 mm, and a maximum length of 100 mm. Furthermore, regression of the data suggests a fatigue limit near the middle of the ultimate tensile strength (Fig. When the amplitude of repeat loading is below the fatigue limit, small stresses do not shorten the fatigue life of the material. The curve gives designers a quick reference of the allowable stress level for an intended service life. Partially because of the statistical nature of fatigue and Partially because of the difference between laboratory experiments and the real-life practice. However, it is more often the time-varying stresses are oscillating near a non-zero mean stress. On the other hand, Goodman and Gerber's approximations, although simple, they might not properly represent the specific material. In practice, a mechanical component is exposed to a complex, often random, sequence of loads, large and small and different mean values. This approximation, which is simple and straight forward, does not take the squences of loading history into account. The effect of material non-homogeneity is included in the model through a random process parameter of Gaussian type. Under fluctuating load condition the physical mechanisms leading to crack formation and failure is very complicated and difficult to model. The predicted a-N curve seems to be very close to the experimental results of Virkler data. 2 Prediction of mean a – N curve from the four generated sample curves at 90% probability and 95% confidence level within an error of 5%.
2 shows the predicted a - N curve from the four generated sample curves at 90% probability and 95% confidence level within an error of 5%. This machine is considered type I according to the Weibull classification because the fatigue failure is caused by a load of inertial origin.
This fact is peculiar since mechanical components in many industrial applications are operated under conditions of variable torsion load [2,3].
Therefore, the shear stress wave generated over the testing specimen surface in a single cycle can be estimated from the drive mechanism kinematic analysis of the machine and can be controlled by the manipulation of the AC motor frequency and by the selected flywheel.
The elevation of the torsional stiffness is also necessary in order to ensure that the elastic system specimen-flywheel operates on a level which is lower than its fundamental torsional frequency.
The cyclical movement of the testing specimen is generated by a pinion connected to the rocker. Figure 5 illustrates the vector loops and the variables used for the development of the kinematic analysis, and it also shows the two possible configurations of the four-bar linkage: the crossed-configuration and the opened-configuration.
This is attenuated during starting by the variable-frequency drive speed controller that generates a ramp function from 0 to the desired angular speed in 30 s. 6 correspond to the angular speeds of testing on the AC motor in which the first steel test specimens failed. 7, since in the analysis of fatigue without corrosion, the material's response is unaffected by the waveform but it is affected by peak values of stress [12]. 7 it is possible to obtain the maximum and minimum values of the magnitude of the angular acceleration in and which are independent of the AC motor angular velocity. 8 represents the torque's peak and the average applied values, depending on the flywheel used.
Standard Practice for statistical analysis of linear or linearized Stress-Life (S-N) and Strain-Life (?-N) fatigue data.
For example, a serial of high stress loading, which weaken the material, followed by a serial low stress loading may cause more damage than a serial of low stress loading followed by a serial of high stress loading. The model is validated through the experimental and predicted results from several data sets. The a – N curve thus obtained explains only the variability due to the material properties and specimen geometry.


The random factor X (a) now is added to the each normalized life data (normalized between -1 and 1) obtained for each crack increment from Eq. The machine operation is based on the alternating movement that the transmission of a washing machine produces.
Much torsion fatigue data are needed to determine the constants of materials used in multiaxial fatigue designs [4]. In this paper, a general description is presented about the machine operation as well as the mathematical formulation used to determine the fully reversed wave shape that rules the testing specimen movement. The system is fed by a variable function of the magnitude of the angular acceleration during the oscillation. This analysis assumes that the dynamic effects of rotation are independent of the specimen's torsional elasticity and independent of the rotational elements of the machine. The machine consists of two assemblies: the lower ranging from the AC motor to the lower sliding clamp where the testing specimen is subjected, and the superior ranging from the upper clamp to the flywheel which generates the inertial load. Two sensors are used: one to count the number of cycles and one to automatically turn off the machine when the testing specimen is broken. The clamp that supports the testing specimen is joined to a pinion (not shown) that is geared with output link 3 (the rocker) that produces the fully reversed movement.
From that time, the angular speed of the AC motor remains unchanged for the rest of the test. Figure 8 shows the dependence of the peak and average values of per cycle, depending on the angular speed of the AC motor. The estimation of alternating torque can be achieved through a kinematic analysis of the input mechanism according to the selected angular velocity of the AC motor. It is widely recognized that the fatigue crack growth is fundamentally a stochastic phenomenon.
Hence, the variability due to material non-homogeneity is to be incorporated to account this effect on the growth process. La carga inercial es producida por un volante ubicado en la parte superior de la máquina.
The paper also shows the general behavior of the failure specimens in the initial testing and its relationships to the data found in the literature. These speed values were estimated from the theoretical model developed from the stress-life approach and the failure theory of Hencky-von Mises [9,10,11], preventing the number of cycles of the first tests from being too high. The two main reasons for the randomness in fatigue crack growth behaviour are the random material resistance or inhomogeneous material properties and the random loading. 4-6 shows the predicted mean S-N curve along with the experimental data for different materials. Unlike many machines that generate fatigue-alternating stresses through relocation or the rotation of the specimen on loads that are consistent across time, the design of the machine clearly shows how inertial loads due to the acceleration of the mechanical elements generated loads of alternating character which eventually causes fatigue failures. Figure 12 shows the relationship between applied stress and life in cycles of the material or S-N curve, following the methodology proposed by Wöhler [15]. During the last three decades, the probabilistic aspect of fatigue crack growth has been addressed by many researchers [1-7]. The results shown in figures are very close to the experimental results which show the capability of the model presented in this study. These studies are based on Markov chain model, random process model or random variable model. Most of the stochastic crack growth models are based on the inclusion of a suitable stochastic random process either of stationary and ergodic process of Gaussian type or non Gaussian type in the deterministic crack growth model. Mostly the deterministic part of the model is based on Paris-Erdogan, Elber or polynomial models. Keeping in view these aspects, the present study is aimed at the development of a stochastic crack growth model and estimation of confidence and probability bounded crack growth relation (a-N) curve) coupled with stationary Gaussian random process. In the model the statistical scatter of the material properties between the specimens and the microstructural stochastic non-homogeneity of the material within a specimen are well addressed. The validity of the model is demonstrated through the comparison with an extensive amount of the published crack growth data.
The probability-confidence bounded prediction of a-N curves presented in this paper will be extremely helpful for the reliability assessment of structure.



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