The Progress of Fatigue - Essay Example

?Table of Contents Introduction 2 Characterization of Cyclical Loads 3 Fatigue-Life Methods 4 Stress-Life Method 4 Strain-Life Method 7 Linear-Elastic Fracture Mechanics (LEFM) 9 Endurance Limit Modifying Factors 10 Calculations 11 References 13 Introduction In contrast to failure through static loading, stress levels in fatigue failure are often well below the ultimate strength of the member in its original state. Fatigue failure occurs through the propagation of microcracks due to repeated cyclical stresses. This type of failure is not characterised by the same visible signs as static failure; there is neither buckling nor deflection, and the internal cracks that lead to fatigue failure are not always visible to the naked eye. Because of this subtlety, it is important to develop methods of predicting and controlling fatigue during the design process. The progress of fatigue is categorised into three stages (Budynas and Nisbett, 2006). Stage I is the presence of microcracks, which are the tiny regions of plastic deformation from which the failure process begins. During stage II, these cracks steadily grow and increase in length. It is during this stage that cracks can be detected and measured, and are apparent in post-failure analysis as visible ridges: How quickly a part reaches failure due to fatigue is an important question for mechanical design. It is a complicated process, influenced by many factors. A cornerstone of fatigue behaviour prediction is extensive materials testing. Whatever the mathematical framework used to predict fatigue behaviour, it invariably relies on empirical measurements of representative material samples. Because crack formation and propagation are stochastic processes, these tests must be repeated a large number of times and a statistical average used as a guideline. This is particularly true for low cycle fatigue. Characterization of Cyclical Loads In order to discuss fatigue and the types of loading that cause it, some standard terminology needs to first be established. Any cyclical stress can be characterized by the following parameters: ?a Stress amplitude ?r Stress range ?m Mid-range stress ?max Maximum stress ?min Minimum stress Additionally, there is the stress ratio R, and the amplitude ratio A defined by: In contrast to static failure, which occurs when a single threshold value is exceeded, fatigue failure can result from a near-endless combination of the above components. A number of different criteria have been developed to determine which combinations will lead to failure. The following diagram shows some of these criteria, with points on or above each line indicating failure. Some criteria are clearly more conservative than others, from the Soderberg line to the Gerber ellipse. These criteria are somewhat crude approaches that do little to describe the physical phenomenon; they are deterministic, whilst the phenomenon itself is stochastic. They were developed early in the study of fatigue, and are primarily useful for quick estimation. The following three sections define more rigorous approaches. Fatigue-Life Methods Stress-Life Method The stress-life method is very convenient and intuitive, and consists of applying repetitive stresses to a sample, then measuring the number of cycles to failure. A sample of material is loaded into a high-speed rotating-beam machine, which places the sample into a state of pure bending, then rotates it to create fully-reversed cyclical axial stresses within it. These samples are very carefully machined and polished to control surface defects that could initiate a crack. Testing begins at near-yield strength, and then gradually reduced in subsequent tests. The result is known as the “S-N curve” which shows the expected fatigue strength vs. the number of cycles to expected failure. This method of determining material fatigue properties is straightforward to implement, but lacks accuracy, particularly for determining low-cycle endurance. Fatigue failure is stochastic in nature, and even the most carefully-controlled experiments are not perfectly reproducible. This stems from the microscopic nature of fatigue; grain boundaries, dislocations, and slip planes in metal are randomly distributed, and no two samples can be the same. For this reason, stress-life tests have to be repeated a large number of times at each level of loading and the presented S-N curve often represents the 50% probability of failure. It has been recognized that the endurance limit of real parts can be very different from the polished laboratory samples used in rotating beam tests. In order to quantitatively correct for a wide-range of real-world conditions, Marin (1962) identified a set of multiplicative factors that change the endurance limit: Se = rotary-beam test specimen endurance limit S’e = endurance limit at the critical location of a machine part in the geometry and condition of use ka = surface condition modification factor In rotating beam tests the surface of the sample is highly polished in a manner that removes circumferential scratches. Real samples are generally not polished in this fashion, and this factor corrects for the effect of different finishes. Lipson and Noll (1946) gathered endurance limit vs. ultimate tensile strength to produce the following relation: Data for this typically contains a large amount of scatter, and so this factor should be considered as an approximation kb = size modification factor Mischke (1987) combined a large number of datasets and determined a set of empirical equations to account for the effect of size. kc = load modification factor The type of loading, whether bending, axial, or torsion can cause significant differences in endurance limit: kd = temperature modification factor The operating temperature affects the fatigue properties of a material. For metals, this means a decrease in ductility with temperature, and a tendency towards brittle fracture. Increased temperature creates a slightly more complicated situation. Fatigue strength is seen to increase at first, and for some materials the fatigue limit disappears entirely. As temperature continues to increase, the yield strength drops and thus the fatigue strength also drops. The following table was compiled from 145 tests of 21 carbon and alloy steels (Brandes, 1983). Temperature (°C) kd 20 1.000 50 1.010 100 1.020 150 1.025 200 1.020 250 1.000 300 0.975 350 0.943 400 0.900 450 0.843 500 0.768 550 0.672 600 0.549 Another important consideration at elevated temperatures is creep, and therefore time-dependence. At elevated temperatures, the effects of creep and fatigue compound each other and have to be jointly taken into consideration for failure criteria. ke = reliability factor This factor accounts for the fact that most endurance strength data are in fact an average of a large number of measurements. Haugen and Wirching (1975) have shown standard deviations of endurance strengths of less than 8%. Therefore the reliability factor is given by: In combination with the data from the following table, the endurance limit can be corrected to a particular level of reliability: Reliability % Transformation Variate Reliability factor ke 50 0 1 90 1.288 0.897 95 1.645 0.868 99 2.326 0.814 99.9 3.091 0.753 99.99 3.719 0.702 99.999 4.265 0.659 99.9999 4.753 0.620 kf = miscellaneous-effects modification factor Intended as a catch-all for other effects, really more of a reminder that other factors exist outside the previous 5, since values for kf may not be readily available. Examples of this include residual stresses, directional characteristics, case hardening, corrosion, electrolytic plating, metal spraying, cyclic frequency, and frettage corrosion The stress-life method gives the designer a straightforward means of estimating fatigue life, and the Marin factors introduce an extra degree of flexibility, so that data gathered in the laboratory can be extended into the real-world environment. A large body of data already exists for this method, and so it can be used to quickly get a basic estimate of fatigue life. Strain-Life Method Analysis of strain is another means of characterizing fatigue, and one which more accurately captures the physical process failure. It delivers better accuracy, particularly for low-cycle endurance, but is not as easy to implement. Low-cycle, high-stress loading causes plastic deformation that is revealed as hysteresis in a stress-strain plot: This hysteresis is not detected when the test is controlled only for applied stress. It shows a sample undergoing strain-hardening, which would skew the portion of the S-N curve for stresses above the elastic limit. It has been shown for most metals that a relationship exists between the width of this hysteresis loop (Benham, Crawford, and Armstrong 1996), corresponding to plastic deformation, and cycles to failure up to roughly 10­­5 cycles: In general, for metals at room temperature, the value of ? is between -0.5 and -0.6 and the value of the constant increases with ductility. A more detailed version of this relationship has been developed by Manson, Coffin, and Tavernelli (1962): -?’F is the fatigue strength coefficient which is the true stress corresponding to fracture in one reversal. -?’F is the fatigue ductility coefficient which is the true strain corresponding to fracture in one reversal -b is the fatigue strength exponent which is the slope of the elastic-strain line -c is the fatigue ductility exponent which is the slope of the plastic strain line -N in this case is the number of reversals, and so 2N is the number of cycles This relationship can be seen graphically as: This equation can be challenging to incorporate into the design process. It isn’t clear, for example, how to determine the total strain at a notch or discontinuity. However, finite element analysis can provide an approximation of this, and is gaining in popularity(Yang 2009). The application of the strain-life method is more complex than the stress-life method, but it more accurately models the fatigue process, and is much more suitable for low-cycle fatigue estimation. Linear-Elastic Fracture Mechanics (LEFM) The fracture mechanics approach provides an analytical tool for prediction of crack growth and cycles to failure. It pre-supposes the existence of a stage I crack, which at some threshold stress intensity will become a stage II crack and begin to propagate. The defining feature of a stage II crack is that it is large enough to be detected. During stage II, growth is actually rather orderly until stage III fatigue is reached and the final fracture occurs. This growth is characterised in the Paris equation, which relates the increase in crack length per stress cycle to the stress intensity at the crack tip. Crack nucleation and growth occurs when some part of the stress cycle includes tension, which pulls the two sides of the crack farther apart. Compressive stress does not contribute to propagation, and in some cases may even strengthen the material. The relation between the length of the crack and the stress intensity forms the basic relationship of LEFM: Here the ?MAX and ?MIN are the maximum and minimum stresses of the applied stress cycle. Compressive stresses in this case are taken to be zero. This change in stress intensity is the basis for the Paris equation: Again, extensive material testing is part of the process, as the constants c and m are empirically determined for a given material. LEFM is best used in combination with a regime of inspection, so that once small cracks have been detected, estimates can be made of how long until the component fails. Endurance Limit Modifying Factors The presence of irregularities or discontinuities in a part creates stress concentration in the vicinity of the discontinuity. Under static loading, this concentration is characterized by the factor (Budynas and Nisbett 2006): Where Kt is for normal stresses and Kts­ is for for shear stresses, and these two factors relate the actual maximum stress at the discontinuity to the nominal stress over the net cross-section. These are theoretical values only, and the results are summarized in tables and graphs for various geometries. Some materials are, however, not fully affected by this, and so a reduced value of the stress concentration factor, called the fatigue stress-concentration factor can be determined (Benham, Crawford, and Armstrong 1996): Alternatively, a notch sensitivity factor can be calculated: Analysis using these factors is carried out by first determining Kt from the geometry of the part, selecting a material, then using the associated q to solve for Kf. The analytical tools presented up to this point have generally been for uniaxial stresses cycled at a single frequency. In practice, many components are subjected to a variety of loads across a spectrum of frequencies and amplitudes. The following section expands on some techniques for treating fatigue caused by multi-axial or multi-frequency loads. The plotting of an S-N curve is straightforward when the applied stress is cycled at a constant amplitude and frequency. Clearly if the maximum stress is different from cycle to cycle, or if the stress does not fully reverse before returning to a maximum, the same straightforward analysis cannot be applied. Miner’s rule (Benham, Crawford, and Armstrong 1996) provides a means of algebraically tracking damage contributions when the amplitude and or frequency of loading vary over time: The numerator ni is the number of stress cycles carried out at stress level ?i, for which the expected cycles to failure is Ni. In this way, each component is seen to contribute the proportion of its fatigue life used up in the given number of cycles, which is termed cumulative fatigue damage. In practice the value of c ranges from 0.6 to as high as 2.0. A large caveat to Miner’s Rule is that not all stresses contribute in the same manner. It has been noted that cycling first at a high stress for a short number of cycles can induce strain-hardening, changing the behaviour for subsequent lower-stress cycling. Because of this, it should be used only as an approximation, and with a safety factor built in. Calculations Under consideration is a support bracket for a large air receiver that must withstand a tensile loading of 500 MN/m2 every day for 25 years. A decision must be made between a maraging steel and a medium strength steel for this application. In the parameters of the problem are the fracture toughnesses of the materials and the coefficients for the Paris Equation. The modelling approach taken is thus one of fracture mechanics. A part reaches failure when the stress intensity at the crack exceeds the critical stress intensity for the material: At this critical threshold, the crack achieves its final length. From the expression for stress intensity: The Paris Equation describes the rate of crack growth with each cycle: We now perform a substitution to remove the dependence on a. The ? is now suppressed for clarity, and because the minimum stress intensity in the present application is zero (since the minimum stress is also zero): This then substituted into the above expression for the Paris Equation: Then integrated to obtain an expression for the expected cycles to failure: Now, since both parts are subjected to the same conditions (stress, initial crack size, and stress intensity modification factor) the difference between the two can be taken to find the difference in lifetime. The initial crack size of 0.1mm corresponds to a stress intensity of 8.862 MN/m^-3/2: Substituting in the given values: This works out to a staggering 35200 cycle advantage for the medium strength steel. This is not altogether surprising, since the growth coefficient m in the Paris equation is larger for the maraging steel, meaning that the geometric contribution to crack growth is significantly larger. The individual lifetimes are approximately 8200 for the maraging steel, and 43400 for the medium strength steel. Given that the desired lifetime is just over 9000 cycles, the medium strength steel is clearly the correct choice. References Budynas, R. and Nisbett, K., 2006. Shigley’s Mechanical Engineering Design. 8th edition. New York: McGraw-Hill. Marin, J., 1962. Mechanical Behaviour of Engineering Materials. Englewood Cliffs: Prentice-Hall Noll, C.J. and Lipson, C. “Allowable Working Stresses,” Society for Experimental Stress Analysis, vol. 3, no. 2, 1946 Mischke, C.R. “Prediction of Stochastic Endurance Strength,” Trans. of ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 109, no. 1, January 1987 Brandes, E.A. (ed.), 1983. Smithells Metal Reference Book, 6th ed. London: Butterworth Haugen, E. B. and Wirsching, P. H. “Probabilistic Design,” Machine Design, vol. 47, no. 12, 1975 Benham, P.P., Crawford, R.J., Armstrong, C.G., 1996. Mechanics of Engineering Materials. 2nd edition. Essex: Pearson Education Limited Tavernelli, J. F. and Coffin, L. F., Jr., “Experimental Support for Generalized Equation Predicting Low Cycle Fatigue,’’ and S. S. Manson, discussion, Trans. ASME, J. Basic Eng., vol. 84, no. 4 1962 Yang, Z. “Stress and Strain Concentration Factors for Tension Bars of Circular Cross-Section with Semicircular Groove,” Engineering Fracture Mechanics, vol. 76, no. 11, 2009. Read More