The premature fracture of materials under fluctuating load (stress/strain/ temperature) is known as fatigue. Fatigue is a sudden failure exhibiting no overall ductility in the component and is known to be the cause in 90% of the total failures of structures. During each fatigue cycle the material absorbs part of the applied energy and, when the accumulated strain energy reaches the value of the surface energy of the material (in that environ­ment), a surface forms (i. e. a crack appears). The accumulation of strain energy is facilitated by the presence of a notch or scratch and the surface energy is the minimum for the exposed crack than the embedded one. Often, the fracture surface is perpendicular to the direction of the applied stress and a compressive residual stress is beneficial in delaying the fatigue failure. Fatigue life is represented by a plot of applied stress (S) against the number of cycles to failure (Nf) known as the S-N curves. Figure 1.6 depicts the S-N curves for various metals8 and we note that ferrous metals exhibit a distinct ‘endurance’ limit below which fatigue failures do not occur whereas non­ferrous metals do not seem to exhibit such a limit, albeit the slope of the S-N curve decreases at very high cycles. The stress axis can also be either the stress amplitude (omax — omin)/2, the stress range (omax — omin) or mean stress (omax + omin)/2 and it is generally seen that the fatigue life depends weakly on the R ratio (R = omin/omax), where omax and omin represent the maximum and minimum stresses, respectively. Depending on the number of cycles to fail­ure the fatigue curve is classified as low cycle fatigue (LCF) and high cycle

Reversals to failure (log scale) 1.7 Де vs N curve showing plastic and elastic strain regimes.11

fatigue (HCF) regions, corresponding to the plastic and elastic deforma­tion ranges, respectively. LCF is characterized by macroscopic cyclic plastic strains and is generally limited to less than 104 cycles. LCF is controlled by the ductility and HCF by the strength of the material, and thus, cold-work and radiation hardening (both of which result in reduced ductility) result in decreased fatigue life in the LCF range while being beneficial in the HCF range, especially at low stresses/strains. Figure 1.7 shows a typical fatigue life plot as strain range (Де) against number of failure cycles (Nf) along with the corresponding stress-strain loops (broad in LCF and narrow in HCF). In the high cycle region corresponding to HCF, the Basquin equation relates the applied stress (Да) to the number of cycles:

Nf (Да)р = C or in terms of strains Nf (ЕДе)р = C, [1.10]

where C and p are material constants. LCF with inelastic strains is often described by the Coffin-Manson equation

Де = 2A(2Nf)c [1.11]

where A, a function of the ductility, and c (-0.5 to -0.7) are material con­stants and Nf is the number of stress/strain reversals. The Coffin-Manson equation is seen to be valid for many materials over a broad range of tem­perature, environment, stress history and microstructural conditions. The complete fatigue curve can be described by combining the LCF and the HCF

formulations by either the universal slopes equation (Equation [1.12a]) or the characteristic slopes equation (Equation [1.12b]):

where Su is ultimate tensile strength, ef is true fracture strain, af true frac­ture stress, and b and c are material constants. In terms of the characteristic slopes (Equation [1.12b]) the value of fatigue life at which the transition from low cycle (plastic) to high cycle (elastic) occurs is given by

[1.12c ]

Fatigue crack growth rate (FCGR, da/dN) is determined by measuring the extension of a pre-crack using visual, potential drop, unloading compliance or other techniques over the elapsed number of load cycles from stress con­trol tests conducted on either compact tension (CT) or three-point bend specimens and is related to the range of stress intensity factor (AK). Typical crack extension curves at two different starting stress ranges (Ac) versus number of cycles are shown in Fig. 1.8a and the slopes of the curve yields da/dN. The plot on logarithmic scale of (da/dN) versus AK (Fig. 1.8b) clearly reveals three stages. Stage I is associated with crack blunting with very little crack growth, while crack growth in stage II can be related using Paris’ law:

[1.13 ]

where p is the Paris parameter/constant with values ranging from 2 to 4; this covers the majority of the crack growth event before entering the final stage (stage III) where plastic fracture occurs as crack length reaches a crit­ical value (af) corresponding to the plane strain fracture toughness (KIC) value:

1.8 ( a) Crack extension with number of cycles and (b) log-log plot of da/dN vs AK.

In Equation [1.14], Y is the geometric factor which is a function of a/w (a is crack length and w is specimen width).

Stage I corresponds to formation of a fine crack from surface defects (such as scratches, key ways, stress concentrations) with slow initial propagation along specific crystallographic directions covering few grains before the growth enters stage II where the crack propagates at a relatively faster rate and on a plane perpendicular to the loading direction. In general, persistent slip bands (PSBs), beach marks and fatigue striations (Fig. 1.9a and 1.9b)

are characteristics of stage II crack propagation and the separation between striations depends on the stress range and frequency of loading. The total number of cycles to failure can be estimated as follows from Equations 1.15 and 1.16:

and

Another important aspect of considering the crack growth versus AK is to examine the effects of superimposed environment such as corrosion and radiation. The variation of da/dN with AK in these cases would shift the threshold stress intensity range to lower values and the critical crack length at fracture would be indicated by KISCC instead of KIC.

In strain controlled fatigue tests for life evaluation, it may be noted that the cyclic stress-strain curve leads to a hysteresis loop as depicted in Fig. 1.10a where O-A-B is the initial loading curve11 and, on unloading, the yielding occurs at lower stress (point C as compared to A) which is known as the Bauschinger effect. The material may undergo cyclic hardening or softening; in rare cases it remains stable (Fig. 1.10b). This behaviour depends on the initial metallurgical condition of the material. According to Fig. 1.10b, as the number of cycles increases cyclic hardening leads to decreasing peak strains while the peak strains increase in the case of cyclic softening. In general, the hysteresis loop stabilizes after about 100 cycles and the stress-strain curve obtained from cyclic loading will be different from that of monotonic load­ing (Fig. 1.10c), but the stress-strain follows a power law relationship similar to that in monotonic loading (Equation [1.3]):

Ac = K'(Ae)n

where the cyclic hardening coefficient n’ ranges from 0.1 to 0.2 for many metals and is given by the ratio of the parameters (b/c) (Equation [1.12b]). In some cases fatigue ratchetting occurs resulting in an increase in strain as a function of time when tested under a constant strain range (Fig. 1.11); this is often referred to as cyclic creep. i2 In a stress controlled test with non-zero mean stress, the shift in the hysteresis loop along the strain axis, as depicted in Fig. i. ii, is attributed to thermally activated dislocation movement at stresses well below the yield stress and/or due to dislocation pile up result­ing in stress enhancement. Fatigue ratchetting may also occur in the pres­ence of residual stress and in cases where microstructural inhomogeneities exist such as in welded joints.

In real situations stresses change at random frequencies and, in gen­eral, the percentage of life consumed in one cyclic loading depends on

(c) a

1.10 ( a) Cyclic stress-strain curve illustrating hysteresis loop.11

(b) Hysteresis loops during cyclic hardening and cyclic softening.12

(c)

Comparison of cyclic stress-strain curve for cyclic hardening and stress-strain curve under monotonic loading.11

1.11 An example of ratcheting fatigue.12

the magnitude of stress in subsequent cycles. However, the linear cumu­lative damage rule, known as Miner’s rule, assumes that the total life of a component can be estimated by adding up the life fraction consumed by each of the loading cycles. If Nfi is the number of cycles to failure at the ith cyclic loading and Nt is the number of cycles experienced by the structure then

although Miner’s rule is too simplistic and fails to predict the life when notches are present. Further, it fails to predict the life when mean stress and temperature are high or cyclic frequency is low where creep deformation dominates over fatigue loading. In such situations a better approximation is given by combining Robinson’s rule for creep fracture with Minor’s rule;

[1.19]

where (fi) and fracture time (t) corresponding to the ith creep conditions.

It turns out that many materials exhibit deviations from this linear addi­tion depending on whether it is cyclically hardening or softening.13 In par­ticular the predictions tend to be highly non-conservative for cyclically softening materials.

Fatigue strength or life of structures can be improved by reducing the mean positive stress, through appropriate design with no stress raisers and by surface finish and modifications. In particular, case hardening by carbu­rizing and nitriding as well as shot-peening, which increase surface residual compressive stresses, result in distinct improvements in fatigue life. In com­parison to pure metals, solid solution has been found to improve fatigue strength. Other factors such as interstitials inducing strain ageing could also improve fatigue life.

Environmental effects on creep-fatigue are quite complex and each case needs to be considered separately. While Equation [1.19] gives an approxi­mate assessment, the mechanistic explanations of high-temperature fatigue effects are corrosion — or creep-related. Coffin considered the time dependent fatigue to be essentially SCC and formulated frequency-modified fatigue life-time correlations for crack initiation and propagation.12 Manson pro­posed a plasticity oriented fatigue model using a strain-range partitioning method.13 Fatigue crack growth assisted by creep cavitation at grain bound­aries was considered by Majumdar and Maiya14 to model high-temperature fatigue crack growth.

As described in Section 1.2.1, exposure of materials and structures to high energy neutrons leads to the creation of microscopic defects such as vacan­cies, interstitials, Frenkel defects, dislocations and faulted loops, as well as voids and cavities. Figure 1.12a depicts voids and precipitates in irradiated stainless steel1 5 while large Frank loops are shown in Fig. 1.12b. 1 6 Similar faulted Frank loops are noted in irradiated aluminium and copper as well as iron (Fig. 1.12c).17

Materials undergo many changes on exposure to neutron radiation: defect concentration increases, neutron transmutation occurs, chemical reactivity changes (generally gets enhanced), diffusion of the elements increases and new phases (both equilibrium and non-equilibrium) form. The extent of change in properties is, in general, proportional to radiation flux, particle energy and irradiation time, while it decreases with an increase in irradia­tion temperature. The creation of voids, cavities and depleted zones leads to decreased density of the material with a corresponding increase in vol­ume known as radiation swelling. Increased defect concentration leads to increased electrical resistivity and decreased thermal conductivity while magnetic susceptibility decreases. The threshold neutron fluence or dpa that leads to extensive degradation in a material depends on the crystal structure and nature of atomic bonding — semiconductors and polymers degrade at much lower neutron fluences compared to ceramics and metals. The reader is referred to various monographs on nuclear materials and radiation effects for more details.18 These defects result in hardening and embrittlement of the material with an increase in strength and accompanying decrease in ductility commonly referred to as radiation hardening and radiation embrittlement; strain hardening in the material decreases accompanied by a decreased uni­form elongation and an increase in DBTT (or RTndt), which decreases the fracture toughness. The increased defect density enhances the diffusivity in the material which in turn increases the creep rates and reduces the rupture time. These various phenomena will be discussed in detail in the following sections.