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Fatigue Analysis for Extreme Environments Stress-Life Approach Stress-Life Approach Presented by Calvin M Fatigue Analysis For Extreme Environments Stress-Life Approach Stress-Life Approach Presented by Calvin M. Stewart, PhD MECH 5390-6390 Spring 2021 Outline • Fatigue Loading • Stress-Life Approach • Unmodified Endurance Limit • Endurance Limit Modifiers • Mean Stress • Solving Problems Fatigue Loading Fatigue Loading • Components, structures, and vehicles are subjected to quite diverse load histories. • Some load histories may be simple and repetitive. • Others may be completely random. • The randomness may contain substantial portions of more deterministic loading. • For example, the ground-air-ground cycle of an aircraft has substantial similarity from flight to flight. Fatigue Loading Repetitive Randomness Ground-Air-Ground Cycle of AirCraft Fatigue Loading Environmental Effects Offshore Structures Aircraft Fatigue Loading Typical Load Histories From Actual Ground Vehicle Components Fatigue Loading A Typical Load History of Short-Span Bridge Fatigue Loading • These load histories are typical of those found in real-life engineering situations. • Fatigue from variable amplitude is discussed in Chapter 9. • Constant amplitude loading is introduced in this chapter. • Constant amplitude loading is used: • To obtain material fatigue behavior/properties for use in fatigue design, • Some real-life load histories can occasionally be modeled as essentially constant amplitude. Real World Laboratory Fluctuating stress with high Sinusoidal frequency Fluctuating ripples Stress Nonsinusoidal fluctuating stress Pulsating Tension Nonsinusoidal Completely fluctuating Reversed stress (S-N Curve) Shigley’s Mechanical Design Fig. 6–23 Fatigue Loading Shigley’s Mechanical Design Period 1 = f Fatigue Loading Frequency • Def’s: 1 f = – Stress range = r = max − min – Stress amplitude − = max min a 2 –Mean stress + = max min m 2 – Stress ratio, R R = min max – Amplitude ratio, A A = a m Shigley’s Mechanical Design Fatigue Loading R = 0 • Fully Reversed: R = -1; A = ∞ • Zero to Tension: R = 0; A = 1 • Zero to Compression: R = ∞; A = -1 • R w.r.t A 1− A R = -1 R = 1+ A • A w.r.t. R 1− R A = 1+ R Shigley’s Mechanical Design Fatigue Loading • R= -1 and R= 0 are two common reference test conditions used for obtaining fatigue properties. • R= -1 is called the fully reversed condition since Smin= -Smax • R = 0, where Smin = 0, is called pulsating tension. • One cycle is the smallest segment of the stress versus time history which is repeated periodically. • Under variable amplitude loading, the definition of one cycle is not clear and hence reversals of stress are often considered. • In constant amplitude loading, one cycle equals two reversals. Stress-Life Approach Stress-Life Approach: (S-N) • The Stress-Life, S-N, method was the first approach used to understand and quantify metal fatigue. • It was the standard fatigue design method for almost 100 years. • The S-N approach is still widely used in design applications where the applied stress is primarily within the elastic range of the material and the resultant lives (cycles to failure) are long, such as power transmission shaft. • The Stress-Life method does not work well in low-cycle fatigue, where the applied strain have a significant plastic component. • In low-cycle fatigue the Strain-Life method is more appropriate. Stress-Life Approach: Typical Curve Stress-Life Approach • Typical schematic S-N curve obtained under axial load or stress control test conditions with smooth specimens. • Constant amplitude S-N curves of this type are plotted on semi-log or log- log coordinates. • S-N curves obtained under torsion or bending load-control test conditions often do not have data at the shorter fatigue lives (say 10^3 or 10^4 cycles and less) due to significant plastic deformation. • Torsion and bending stress equations τ=T r / J and σ= M y /I can only be used for nominal elastic behavior. Stress-Life Approach: Variability • Typical variability with less variability at shorter lives and greater variability at longer lives. • Variability in life for a given stress level can range from less than a factor of two to more than an order of magnitude. • Variability and statistical aspects of fatigue data are discussed in Ch. 13. Stress-Life Approach: Endurance Limit • Fig. (a) shows a continuous sloping curve, while Fig. (b) shows a discontinuity or “knee” in the S-N curve which indicates an endurance limit. • This endurance limit has been found in only a few materials (i.e. low and medium strength steels) between 10^6 and 10^7 cycles in non-corrosive conditions. • Most materials do not contain the endurance limit even under controlled environments. • Under corrosive environments all S-N data have a continuous sloping curve. • When sufficient data are available, S-N curves are usually drawn through median points and thus represent 50 percent expected failures. S-N Diagram for Steel in Laboratory Air 0 (10 , Sut ) Endurance Limit Shigley’s Mechanical Design S-N Diagram for Steel in Laboratory Air 0 (10 , Sut ) Finite Life Low Cycle Fatigue High Cycle Fatigue (LCF) Endurance Limit Infinite Life Shigley’s Mechanical Design Stress-Life Approach: Definitions • Definitions • Cycles to Failure, Nf – the number of cycles a material (or a component) can resistance fatigue failure connoted by fatigue life • Fatigue Strength, Sf – the value of fully reversed alternating stress, σa that corresponds to a particular number of Nf • Endurance/Fatigue Limit, Se – a stress level below which a material can be said to have infinite life 0 3 • Low Cycle Fatigue (LCF) - when Nf is in the range of 10 to 10 . • High Cycle Fatigue (HCF) - when the number of Cycles to failure, Nf , is in the range of 103 to 106. 6 • Infinite Life - when Nf is equal to or greater than 10 . Stress-Life Approach: FCG • Physically, Fatigue crack growth (FCG) typically consists of crack nucleation, growth, and final fracture. • The Stress-Life Approach measures Total Life and cannot distinguish the individual stages of FCG. • When tracking crack length is important, the LEFM approach should be applied. Stress-Life Approach: Basquin’s Equation • Basquin in 1910 suggest a log-log straight line S-N relationship Basquin Equation B SANff= ( ) Tri-Slope Model A How to Find A & B Constants B AS= u Basquin Model BSS= (1 6) log( eu) Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes. Shigley’s Mechanical Design Stress-Life Approach • The S-N diagram is applicable for completely reversed stresses (R=-1) only! • We must use Fatigue Failure Criterion to correct for mean stresses! • Assumptions: – High cycle fatigue regime – Strain range is dominated by elastic rather than plastic strain – Peak and valley stresses are constant – Only one component of loading (e.g., axial, torsional, bending) acting at a time – Stress amplitude is plotted against cycles to failure (e.g. S-N diagram) Unmodified Endurance Limit The Endurance Limit, Se • Def: Endurance limit, Se the stress below which a material can be consider to have infinite life. • The material can be subject to an infinite number of cycles without fatigue failure. INFINITE LIFE The Endurance Limit, Se • For a given material the endurance/fatigue limit has an enormous range depending on: • surface finish, • size, • type of loading, • Cyclic rate (frequency), • temperature, • corrosive, and other aggressive environments, • mean stresses, • residual stresses, and • stress concentrations. Unmodified Endurance Limit, S’e • The unmodified Endurance limit, S’e is measured in stress amplitude, σa under completely reverse stress (R=-1) with small highly polished unnotched specimens based on 106 to 5x108 cycles to failure in laboratory air environment. • Can range from essentially 1 to 70 percent of the ultimate tensile strength. • Example of 1 percent of Su is a high strength steel with a sharp notch subjected to a high mean tensile stress in a very corrosive atmosphere. • Example of 70 percent of Su is a medium strength steel in an inert atmosphere containing appreciable compressive residual stresses. Test Specimen • Test specimens are subjected to repeated stress while counting cycles to failure • Most common test machine is R. R. Moore high-speed rotating-beam machine • Subjects specimen to pure bending with no transverse shear • As specimen rotates, stress fluctuates between equal magnitudes of tension and compression, known as completely reversed stress cycling • Specimen is carefully machined and Fig. 6–9 polished Test Machine Demonstration Unmodified Endurance Limit, S’e • In metals, the unmodified Endurance limit, S’e is due to interstitial elements, such as carbon or nitrogen in iron which pin dislocations. • This prevents the slip mechanism that leads to the formation of microcracks. • Care must be taken when using the endurance limit since it can disappear due to 1. Periodic overloads (which unpin dislocations) 2. Corrosive environments (due to fatigue corrosion interaction) 3. High temperature (which mobilize dislocations) The Endurance Limit, Se: Steel Unmodified Endurance Limit, S’e: Nonferrous Alloys • Nonferrous metals often do not have an endurance limit. • Fatigue strength Sf is reported at a specific number of cycles Plot shows typical S-N diagram for aluminums Unmodified Endurance Limit, S’e: Polymers Relationship of Se and Su Steel Wrought & Cast Iron Aluminum Alloys Copper Alloys Estimates for Steel • Steels data clustered near the • For unnotched, highly polished, fatigue ratio Se/Su≈0.5 for the small bending specimen fatigue low- and medium-strength limits for steels are. steels. The data however, SSS0.5 for 1400 MPa (200ksi) actually falls between 0.35 and e u u SSeu700 MPa for 1400 MPa (200ksi) 0.6 for Su <1400MPa. For Su > 1400 MPa, Se does not increase significantly. • The UTS can be approximated using Brinell Hardness Su 3.45 HB for MPa units Su 0.5 HB for ksi units Unmodified Endurance Limit, S’e: Limits Table of Properties • Representative monotonic tensile properties and bending fatigue limits of selected engineering alloys obtained under the above conditions are given in Table A.1.
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