Sources and References

• Dowling, N. E. Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue (4th ed., Prentice Hall, 2012).
• Bannantine, J. A., Comer, J. J., and Handrock, J. L. Fundamentals of Metal Fatigue Analysis (Prentice Hall).
• Stephens, R. I., Fatemi, A., Stephens, R. R., and Fuchs, H. O. Metal Fatigue in Engineering (2nd ed., Wiley).
• Anderson, T. L. Fracture Mechanics: Fundamentals and Applications (4th ed., CRC Press).
• Suresh, S. Fatigue of Materials (2nd ed., Cambridge University Press).
• Broek, D. Elementary Engineering Fracture Mechanics (Martinus Nijhoff).
• MIT OpenCourseWare 3.35 Fracture and Fatigue of Materials.
• Stanford ME 340 Elements of Fracture Mechanics.
• Cambridge Engineering Tripos Part IIA, Materials module notes on fracture and fatigue.

Chapter 1: Scope of Fatigue and Fracture Analysis

Fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading at stress levels that are, in most cases, far below the monotonic yield strength. The damage accumulates slowly as micro-plasticity rearranges dislocations, nucleates a crack at a stress riser, and then drives that crack through the remaining cross-section until sudden rupture. Fracture mechanics is the complementary discipline that begins once a crack exists: it treats the crack tip as a singular stress field and asks whether an applied load will drive stable growth, arrest, or catastrophic separation. Together the two disciplines furnish the quantitative tools used for life prediction, damage-tolerant design, inspection-interval planning, and forensic failure analysis of metallic components, including the welded joints that dominate ships, pressure vessels, bridges, and offshore structures.

The subject matter of ME 526 interlocks these two views. Review of test and design procedures frames the experimental basis for all subsequent models. Sources of cyclic loading, cycle-counting procedures, and cumulative damage rules provide the input stream of stress histories. Stress-life curves and the effects of mean stress, residual stress, and multiaxial stress establish the classical high-cycle tools. Stress concentrations and scatter in fatigue life distributions convert those curves into reliability-aware estimates. Transition temperature concepts and linear elastic fracture mechanics analysis of crack propagation and initiation introduce the defect-tolerant perspective, and crack arrest completes the picture by showing when a running crack can be stopped.

Chapter 2: Engineering Materials and Cyclic Behavior

Metals used in fatigue-critical applications span body-centred cubic ferritic steels, face-centred cubic aluminium and austenitic stainless steels, hexagonal close-packed titanium and magnesium alloys, and various cast irons and nickel superalloys. Each crystal structure produces a characteristic response to cyclic plasticity. FCC metals generally exhibit planar slip at low amplitudes transitioning to wavy slip, and they show a pronounced ratcheting sensitivity. BCC steels below the ductile-brittle transition temperature can fail by cleavage when a sharp crack is present, while above that temperature they tolerate extensive plastic flow. HCP metals display strong texture-induced anisotropy and twinning contributions that complicate life prediction.

Under fully reversed straining, most metals reach a stabilized hysteresis loop after a few percent of life. The stabilized loop defines the cyclic stress-strain curve, usually fitted in Ramberg-Osgood form as

\[ \varepsilon_a = \frac{\sigma_a}{E} + \left(\frac{\sigma_a}{K'}\right)^{1/n'} \]

where \( K' \) and \( n' \) are the cyclic strength coefficient and cyclic strain-hardening exponent. A material is cyclically hardening if the stabilized curve lies above the monotonic one, and cyclically softening if it lies below. Dowling emphasizes that softening is common in cold-worked metals and hardening in annealed ones, with the ratio of monotonic ultimate strength to yield providing a rough indicator. Masing behaviour, in which the ascending branch of every hysteresis loop traces a scaled copy of the cyclic curve, lets analysts reconstruct loops for arbitrary histories by simple memory rules.

Chapter 3: Stress, Strain, and Stress Concentrations

Before any life model is applied, the local stress state at the critical location must be resolved. Elastic stress analysis yields a nominal field; geometric discontinuities amplify that field by an elastic stress concentration factor \( K_t \), defined as the ratio of peak local stress to nominal stress. Tabulated values for holes, fillets, grooves, and keyways are collected in Peterson and reproduced in Dowling. The fatigue strength reduction factor \( K_f \) is usually smaller than \( K_t \) because the averaging length of the fatigue process zone smooths the peak. Neuber’s rule supplies the bridge between them,

\[ K_f = 1 + \frac{K_t - 1}{1 + a/r} \]

where \( a \) is a material constant with units of length and \( r \) is the notch root radius. For local strain calculations at yielded notches Neuber also proposed the equal-energy rule

\[ \sigma \, \varepsilon = \frac{(K_t \, S)^2}{E} \]

which, combined with the cyclic stress-strain curve, gives the actual local stress and strain at the notch root under elastic-plastic conditions. The Glinka energy-density method is an alternative that is preferred when plastic zones are extensive.

Chapter 4: Stress-Life (S-N) Approach

The stress-based or S-N approach is the oldest quantitative fatigue method and remains the workhorse for high-cycle designs such as rotating shafts, springs, and welded structures. Rotating-bending and axial fatigue tests on smooth polished specimens yield the basic curve of stress amplitude versus cycles to failure on log-log axes. For ferrous metals the curve shows a knee near \( 10^6 \) to \( 10^7 \) cycles beyond which the stress amplitude is called the endurance limit; aluminium alloys and many non-ferrous metals lack a sharp knee and are characterized by a fatigue strength at a specified life. The linear portion of the curve is captured by Basquin’s relation

\[ \sigma_a = \sigma_f' \, (2 N_f)^b \]

where \( \sigma_f' \) is the fatigue strength coefficient and \( b \) is Basquin’s exponent, typically between \( -0.05 \) and \( -0.12 \) for structural metals.

Laboratory endurance limits must be corrected before use in design. Marin’s modifying factors account for surface finish, size, loading type, temperature, reliability, and miscellaneous effects such as corrosion and residual stress, giving

\[ S_e = k_a \, k_b \, k_c \, k_d \, k_e \, k_f \, S_e' \]

Mean stress raises or lowers the allowable alternating stress. The Goodman line, Gerber parabola, Soderberg line, and Smith-Watson-Topper parameter all attempt to fold a nonzero mean into an equivalent fully reversed amplitude. Goodman’s relation,

\[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = 1 \]

is conservative for ductile metals and is the baseline in many design codes. Compressive means are generally beneficial and can be introduced deliberately through shot peening or surface rolling.

Chapter 5: Cycle Counting and Cumulative Damage

Service loads are rarely sinusoidal. A variable-amplitude history must first be reduced to a spectrum of closed stress-strain loops, and the rainflow counting algorithm of Matsuishi and Endo is the accepted procedure. Rainflow counting extracts cycles consistent with the material’s memory behaviour, pairing each reversal with a later reversal of opposite sign that forms a closed hysteresis loop. The result is a histogram of amplitude-mean pairs that can be fed into any life model.

Cumulative damage is most often assessed through the Palmgren-Miner linear rule

\[ D = \sum_i \frac{n_i}{N_i} = 1 \]

where \( n_i \) is the number of applied cycles at amplitude level \( i \) and \( N_i \) is the allowable life at that amplitude from the S-N curve. The rule is sequence-insensitive and ignores load-interaction effects such as the beneficial retardation following a tensile overload or the damaging effect of a compressive overload on a subsequent tensile cycle. Nonlinear damage accumulation laws (Marco-Starkey, double linear) refine the picture but are seldom used outside of research because the linear rule is simple and, with sensible safety factors, adequate for most design.

Chapter 6: Fatigue of Welded Structures

Welded joints are the dominant fatigue concern in ships, bridges, offshore platforms, pressure vessels, and rail vehicles. The fusion zone and heat-affected zone contain metallurgical notches, residual tensile stresses approaching yield, slag inclusions, undercut, and geometric stress risers at the weld toe. Smooth-specimen S-N data cannot be applied directly; instead, design codes such as IIW, Eurocode 3, BS 7608, and AWS D1.1 classify joints into detail categories, each with its own nominal-stress S-N curve of the form

\[ \log N = \log C - m \, \log \Delta\sigma \]

with \( m \) typically equal to three for steel in the high-cycle regime. The fatigue life of a welded detail is governed by crack propagation from the toe rather than by initiation, which justifies a purely stress-range formulation independent of mean stress in as-welded condition because residual stresses are already near yield. Post-weld treatments such as grinding, peening, hammer peening, and TIG dressing upgrade the detail category by reducing the notch severity or introducing compressive residual stress. Hot-spot stress and effective notch stress methods provide finer resolution when the geometry departs from the standard catalogue.

Chapter 7: Strain-Life Approach

Low-cycle fatigue, where significant plastic strain occurs each cycle, is the regime of notches, thermal transients, and high-performance components such as turbine discs and crankshafts. The strain-life or local strain approach treats the critical location as a small specimen undergoing the local cyclic stress and strain computed from Neuber’s rule. Coffin and Manson showed that the plastic strain amplitude is linearly related to cycles to failure on log-log axes, and Basquin’s elastic relation still applies, giving the combined Coffin-Manson equation

\[ \frac{\Delta \varepsilon}{2} = \frac{\sigma_f'}{E} (2 N_f)^b + \varepsilon_f' (2 N_f)^c \]

where \( \varepsilon_f' \) is the fatigue ductility coefficient and \( c \) is the fatigue ductility exponent, usually near \( -0.5 \) to \( -0.7 \). The transition life, at which elastic and plastic terms are equal, separates stress-dominated long lives from strain-dominated short lives.

Mean stress effects are incorporated through the Morrow correction, which replaces \( \sigma_f' \) with \( \sigma_f' - \sigma_m \), or through the Smith-Watson-Topper parameter

\[ \sigma_{\max} \, \frac{\Delta \varepsilon}{2} = \frac{(\sigma_f')^2}{E} (2 N_f)^{2b} + \sigma_f' \, \varepsilon_f' (2 N_f)^{b+c} \]

The SWT parameter is particularly useful for tension-dominated histories because it sets the damage to zero whenever the maximum stress is compressive. Multiaxial generalizations use critical-plane quantities such as the Fatemi-Socie parameter or the Brown-Miller shear-plus-normal strain formulation, each seeking a scalar damage measure that correlates with experiments across a range of loading modes.

Chapter 8: Linear Elastic Fracture Mechanics

A crack in a linearly elastic body produces a stress field whose leading term is singular as the inverse square root of distance from the tip. The amplitude of that singularity is the stress intensity factor

\[ K = Y \, \sigma \, \sqrt{\pi a} \]

where \( \sigma \) is the far-field stress, \( a \) is the crack length, and \( Y \) is a dimensionless geometry factor tabulated for centre-cracked plates, edge cracks, semi-elliptical surface flaws, corner cracks, and many configurations in the handbooks of Murakami and Tada-Paris-Irwin. Three loading modes are distinguished: opening (Mode I), in-plane shear (Mode II), and out-of-plane shear (Mode III). For isotropic metals Mode I dominates because tensile stresses drive most service failures.

Fracture occurs when the stress intensity factor reaches a material property called the fracture toughness \( K_{Ic} \), measured under plane-strain conditions on specimens thick enough to suppress plastic constraint relaxation. The ASTM E399 standard prescribes a compact-tension or single-edge-notch-bend specimen with a through-thickness fatigue precrack. A valid \( K_{Ic} \) requires that the specimen thickness satisfy

\[ B \ge 2.5 \, \left( \frac{K_{Ic}}{\sigma_y} \right)^2 \]

so that the plastic zone is small compared with all in-plane dimensions. Irwin’s plastic zone estimate \( r_p = (1/2\pi)(K/\sigma_y)^2 \) under plane stress and half that under plane strain delineates the region where LEFM is valid: the crack tip plastic zone must be much smaller than the crack length and the remaining ligament. When this condition fails, elastic-plastic parameters such as the J-integral and the crack-tip opening displacement (CTOD) replace \( K \).

The J-integral, introduced by Rice, is a path-independent contour integral around the crack tip whose value equals the energy release rate per unit of crack extension in an elastic or deformation-plasticity material, and reduces to \( J = K^2 / E' \) under LEFM, with \( E' \) equal to \( E \) in plane stress and \( E / (1-\nu^2) \) in plane strain. The HRR singularity of Hutchinson, Rice, and Rosengren characterizes the fully plastic crack-tip stress field by a power-law exponent set by the strain-hardening exponent, and \( J \) is its amplitude. CTOD is related to \( J \) by

\[ \delta_t = d_n \, \frac{J}{\sigma_y} \]

where \( d_n \) is a dimensionless factor depending on strain hardening. Standardized toughness tests (ASTM E1820) measure \( J_{Ic} \) or \( \delta_c \) and use resistance curves to track stable crack growth before instability.

Chapter 9: Transition Temperature and Mixed-Mode Considerations

Body-centred cubic steels show a sharp drop in toughness as temperature falls through the ductile-to-brittle transition. The Charpy V-notch impact test is the classical screen, producing an energy-versus-temperature curve from which a transition temperature is extracted at a specified energy level or percent shear fracture. The Pellini drop-weight test defines a nil-ductility temperature below which small flaws propagate catastrophically under nominal yield stress, and the Master Curve approach of Wallin provides a statistical \( K_{Jc}(T) \) for ferritic steels with a single reference temperature \( T_0 \). Service conditions that push a vessel or pipeline below its transition temperature, such as rapid depressurization or Arctic deployment, are a principal driver of brittle failure.

Mixed-mode loading requires a criterion combining \( K_I \), \( K_{II} \), and \( K_{III} \). Maximum tangential stress, minimum strain energy density, and maximum energy release rate criteria each predict both the direction of crack kinking and the critical load. For fatigue crack growth under mixed modes, an equivalent stress intensity range is commonly formed and substituted into the Paris law.

Chapter 10: Fatigue Crack Propagation

Once a crack exists, cyclic loading advances it by a small increment each cycle. Paris and Erdogan observed that, in a mid-range of growth rates, the advance per cycle correlates with the stress intensity range \( \Delta K = K_{\max} - K_{\min} \) through the power law

\[ \frac{da}{dN} = C \, (\Delta K)^m \]

where \( C \) and \( m \) are material constants; \( m \) is usually between two and four for metals. Below the threshold \( \Delta K_{th} \) no measurable propagation occurs; above the Paris regime the growth rate accelerates as \( K_{\max} \) approaches the fracture toughness, producing the characteristic sigmoidal \( da/dN \) versus \( \Delta K \) curve. The NASGRO equation and Walker, Forman, and Elber modifications incorporate load ratio \( R = \sigma_{\min}/\sigma_{\max} \) and crack closure effects that alter the effective driving force.

Crack closure, discovered by Elber, recognizes that the crack tip remains closed for part of the cycle because of plastic wake, roughness, or oxide debris. Only the portion of the cycle during which the crack is fully open contributes to propagation, leading to an effective range

\[ \Delta K_{eff} = K_{\max} - K_{op} \]

Load interaction effects such as retardation after a tensile overload are explained by the enlarged plastic zone left in the wake, which increases closure and temporarily lowers the effective driving force. Models by Wheeler and Willenborg quantify retardation for design purposes. Integration of the Paris law from an initial flaw size \( a_0 \) to a critical size \( a_c \) defined by \( K_{Ic} \) gives the propagation life

\[ N_p = \int_{a_0}^{a_c} \frac{da}{C \, [Y(a)\,\Delta\sigma\,\sqrt{\pi a}]^m} \]

This integral is the core of damage-tolerant design and sets inspection intervals in aerospace and pressure-vessel industries.

Chapter 11: Fatigue Fracture and Crack Arrest

The final separation that ends fatigue life is often described as fatigue fracture to distinguish it from purely monotonic overload fracture. A fatigue surface typically shows three zones: the initiation site, often at a surface flaw or inclusion; the propagation zone marked by striations and beach marks; and the final overload region whose texture reflects whether the remaining ligament failed by dimpled microvoid coalescence, cleavage, or mixed modes. Forensic interpretation of these features supports failure analysis and feeds back into design.

Crack arrest exploits the fact that toughness can rise steeply with temperature or with distance into a tougher microstructure. A running cleavage crack in a pressure vessel, initiated by a thermal shock, may arrest when it encounters warmer material whose arrest toughness \( K_{Ia} \) exceeds the dynamic driving force. Arrest strakes in ship hulls and arrest rings in pipelines are engineered discontinuities of tougher material placed at intervals along a structure so that an unstable crack decelerates and halts within a specified distance. The design problem is to bound the dynamic driving force during arrest and to ensure that the available arrest toughness exceeds it with a margin.

Chapter 12: Scatter, Reliability, and Design Philosophies

Fatigue lives at a fixed stress amplitude scatter over one to two orders of magnitude, and this scatter must be modelled to produce defensible designs. Log-normal and two-parameter Weibull distributions are both common; the Weibull form with shape parameter between two and four captures the early-failure tail well. Reliability-based design selects an allowable stress corresponding to a specified survival probability, typically 95 or 99 percent, and combines it with a confidence level on the parameter estimates. The \( P-S-N \) curve is a family of S-N curves indexed by survival probability.

Three design philosophies organize the application of all the preceding tools. Safe-life design retires a component at a fraction of its predicted life and assumes no inspection; fail-safe design provides redundant load paths so that failure of one member does not compromise the structure; damage-tolerant design assumes the presence of initial flaws and uses fracture mechanics to guarantee that any undetected crack will not reach critical size between inspections. Modern aircraft certification under FAR 25.571 and equivalent standards combines damage tolerance with scheduled nondestructive inspection, an approach that owes its existence to the integrated framework covered in this course.