Mechanical Behaviour of Engineering Materials - Metals, Ceramics, Polymers and Composites 2010 Part 10 pptx

Nevertheless, metals do have one advantage if loaded cyclically: Their fatigue strength is larger if they are designed against a small number of cycles since the Paris region can be expl

∆Kth(R) =

((1 − R)γ∆Kth|R=0 for R < Rt,

with Rt = 0.5 0.7 In low- to medium-strength ferritic steels, γ ≈ 1, inhigh-strength martensitic steels, γ → 0

Because the stress intensity factor Kopneeded to open the crack depends

on the deformation near the crack tip, it also depends on Young’s modulus,for the crack opening in a linear-elastic material is the smaller, the higherYoung’s modulus is (see equation (5.3)) Accordingly, Schwalbe [133] providesthe following approximation for the fatigue-crack-growth threshold in metals:

∆Kth(R) = (2.75 ± 0.75) × 10−5E(1 − R)0.31√

m for R < 1 (10.6)Equation (10.6) also shows the dependence of the fatigue-crack-growth thresh-old on the R ratio, which, however, is not in agreement with equation (10.5)above

Although equations like these exist, it should be kept in mind that Kopandthus ∆Kth depends on many other material parameters e g., the grain size.Accordingly, large differences in the exact values can be found even within

a certain material class Nevertheless, the equations are useful in estimatingthe order of magnitude of ∆Kth If we take steel as an example (with E =

210 000 MPa), we find ∆Kth= 5.8 MPa√

m for R = 0 This is more than oneorder of magnitude smaller than the static fracture toughness KIc of ductilesteels and thus illustrates how dangerous even small cracks can be under cyclicloads

Crack propagation

If the cyclic stress intensity factor ∆K exceeds the fatigue-crack-growth old ∆Kth (i e., if Kmax> Kop), the crack grows in every cycle The crack-growth rate da/dN is determined by those parts of ∆K that exceed Kop

thresh-i e., (for the case Kmin < Kop) by the effective cyclic stress intensity factor

∆Keff = Kmax− Kop Because Kop is usually unknown, da/dN cannot beplotted against ∆Keff Instead, its dependence on ∆K and R is used Asfigure 10.15 illustrates, ∆Keff increases with ∆K and with the mean stressintensity factor Km i e., the R ratio

During crack propagation, the cyclic stress intensity factor ∆K increasesdue to the increase of the crack length Therefore, the crack-growth rate da/dNalso increases even if the cyclic load of the component is constant If the max-imum stress intensity factor Kmax approaches the fracture toughness, thecrack accelerates rapidly and eventually becomes unstable after a few morecycles.13Final fracture of the component ensues Similar to the fatigue-crack-growth threshold, the transition to unstable crack propagation is determined

13

Because of the preceding cyclic crack propagation, the crack may not becomeunstable exactly when the stress intensity factor equals K (cf., for example,

t)(¢a/¢N)2>(¢a/¢N)1

a region marked ‘II’ where the dependence between log(da/dN ) and log(∆K)

is almost linear Accordingly, the crack-growth rate follows the so-called Parislaw

Fig 10.16 Crack-growth curve, plotting da/dN versus the cyclic stress intensityfactor ∆K in a double-logarithmic plot There are three characteristic regions asshown for the curve with the R ratio R1

in brittle materials it can be as large as 50 [120] For ferritic-pearlitic steels,Landgraf [87] states the following upper limit for the da/dN curve at R = 0:da

dN = 6.9 × 10

−9 mmcycle×

MPa√m

3

If we load a crack with a constant stress range ∆σ with ∆K > ∆Kth,the crack grows According to equation (10.3), ∆K increases, and the loadingpoint in the da/dN curve in figure 10.16 moves to the right The crack-growthrate increases in each cycle until ∆KIcis reached, and final fracture destroysthe component

If we consider a specific material and increase the mean stress intensityfactor Km (thus usually also increasing the R ratio, see section 10.1), Kmaxincreases as well The cyclic stress intensity factor ∆K that the componentcan bear decreases, shifting the curve to the left, as shown by the dashedline in figure 10.16 This is a direct consequence of what we discussed aboveconcerning the mean-stress dependence of ∆Kth, ∆KIc, and da/dN In equa-tion (10.8), this shift of the curve is accounted for by the R-dependence of thefactor C There are a large number of, sometimes contradictory, approaches

to describe the dependence of C on the R ratio and the da/dN curve in allthree regions (see, for example, Broek [23], Radaj [113], and Schott [130]).Some exemplary da/dN curves are shown in figure 10.17 Only in the case

of the steel was the range of the cyclic stress intensity factor sufficiently large

to capture all three regions of the curve The slope of the curve is much largerfor ceramics than in the Paris region of metals, resulting in a cyclic stressintensity factor that is almost the same for negligible and rapid crack growth.The reason for this is that the strength of ceramics is at most only slightly

Fig 10.17 Crack-growth curves of several materials: An aluminium alloy(AlZn 6 CuMgZr), a steel (20 MnMoNi 4-5), a nickel-base superalloy (Waspaloy), atitanium alloy (TiAl 6 V 4), silicon carbide, and a psz ceramic [6, 38, 53, 57, 87]

reduced by cyclic loading The fatigue-crack-growth threshold ∆Kthis almostthe same in ceramics and metals Nevertheless, metals do have one advantage

if loaded cyclically: Their fatigue strength is larger if they are designed against

a small number of cycles since the Paris region can be exploited in the design.Because of the extended Paris region, significant crack growth must take placebefore the component fails so that regular inspection cycles may detect thegrowing crack before failure

In polymers, the da/dN curves are similar to those of metals Below acertain threshold value ∆Kth, there is no crack growth, at larger values, threeregions can be distinguished, with region II being described by a Paris law.The exponent n takes a value of about 4 in many polymers [97]

In composites, the fatigue behaviour can frequently not be described quately by da/dN curves because the material usually fails by accumulatinglocal damage, not by propagation of a single crack Measuring da/dN curves

ade-is thus a rather involved procedure [29] If a single crack determines the failurebehaviour, the da/dN curves can be described with a Paris law Compared tothe matrix material alone, KIcis often reduced in polymer and metal matrixcomposites (see section 9.3.4), but ∆Kth is increased Despite the reducedfracture toughness, the fatigue life of a composite may thus be larger thanthat of the matrix material

Assessing life times

Using equation (10.3), we can calculate the critical crack length af at whichunstable or accelerated crack growth occurs (transition between regions II andIII in figure 10.16) If we require that this crack length must not be exceeded,

we can calculate the number of cycles to failure for a given initial crack length

a0< af

To do so, we exploit the equality ∆K = ∆KIc or ∆K = ∆Ktr (forthe transition between region II and III), respectively The number ofcycles until the critical crack length is reached can be estimated for theinitial crack length a0 by [8, 40]:

Nf(a0) =

Z Nf0

dN =

Zaf

a 0

1C

„1

Nf(a0) = 1

C

„1

∆σ√π

Growth of short cracks

As explained above, the crack-growth rate da/dN depends on the cyclic stressintensity factor ∆K = ∆σ√

πa Y and on the R ratio According to this, a shortcrack loaded with a large stress range will propagate with the same rate as along crack loaded with a small stress range provided the cyclic stress intensityfactor ∆K is the same In many cases, this simple picture is correct

However, the statement of the previous paragraph only holds for cracks Microcracks may grow faster than expected from the da/dN curve(figure 10.16), and they may even grow at a cyclic stress intensity factor below

macro-∆Kth[21, 113] On the one hand, this is due to the fact that the crack growthresistance of the material varies on the microscopic scale A microcrack that

is, for example, surrounded by favourably oriented grains may grow rather

14

For the case n = 2, we have to integrate 1/a, leading to ln a This case is dealtwith in exercise 30

low-cycle fatigue strength

high-cycle fatigue strength

endurance limit

Fig 10.18 Example of an S-N diagram with data points

quickly, whereas another crack is stopped at a grain boundary because bouring grains are less favourably oriented On the other hand, short cracksmay remain open even under compressive loads because they are embedded

neigh-in a plastic deformation field [113] This explaneigh-ins why crack propagation maytake place even below ∆Kth

If a component contains only microcracks or is not cracked at all, da/dNcurves cannot be used to assess the life time In this case, other methods arerequired that are the subject of the next section

10.6.2 Stress-cycle diagrams (S-N diagrams)

At the beginning of the chapter, we already saw that the complex load-timecurves occurring in real life are usually replaced by simplified curves in thelaboratory e g., using sinusoidal loading Frequently, smooth specimens areused, similar to the tensile specimens discussed in section 3.2 They are loadedcyclically with a fixed period, prescribing the stress amplitude σaor the strainamplitude εa, and also the R ratio (R or Rε, respectively) The advantagesand disadvantages of these two experimental procedures will be discussed atthe end of this section; in the following, we will consider stress-controlledexperiments only

For each fatigue experiment, the number of cycles to failure15is measured

If several fatigue experiments are performed and the number of cycles tofailure Nf is plotted versus the stress amplitude σA or the stress range ∆σ,the resulting diagram is called a stress-cycle (or S-N ) diagram (sometimes alsostress-life or Wöhler diagram, see figure 10.18) We denote the stress values inthe S-N diagram with capitalised subscripts For example, we denote the stressamplitude that causes failure after Nfcycles as σAinstead of σa The number

of cycles can also be specified in the subscript, as in σAN f, stating, for example,

15

Failure can be defined as fracture of the specimen or occurrence of a crack

endurance limit

(b) Type IIFig 10.19 The characteristic types of S-N curves

σA(1.5×10 4 ) = 130 MPa The number of cycles to failure is always plottedlogarithmically in the S-N diagram; the stress can be plotted logarithmically

or linearly

Some materials exhibit a true fatigue limit (sometimes also called theendurance limit ) In this case, there exists an limiting number of cycles NE,with the S-N curve being almost horizontal at a larger number of cycles Inthis case, the S-N diagram is of type I (figure 10.19(a)) A specimen thathas survived NE cycles never fails The experiment can be stopped and thespecimen can be marked accordingly, usually with an arrow in the diagram(sometimes denoted as ‘run out’, see figure 10.18) Frequently, NEtakes valuesbetween 2 × 106 and 107, depending on the material The stress level thatcorresponds to NE in the S-N curve is called the fatigue strength, endurancelimit, or fatigue limit σE

In many materials, there is no horizontal part of the S-N curve (type II,figure 10.19(b)) Although the slope of the S-N curve becomes smaller beyond

a certain number of cycles, failure can still occur These materials thus have

no true fatigue limit To ensure safety of the component, a limiting number ofcycles of 108is often used, ten times larger than the usual value for materialswith a true fatigue limit To state explicitly that a fatigue strength correspondsonly to a certain number of cycles, not to a true fatigue limit, the number ofcycles can be added to the subscript, as in σE(108 )

So far, we have only looked at large numbers of cycles, the so-called cycle fatigue (hcf) regime As we already saw in the introduction of thechapter for the example of the car engine (section 10.1), it is sometimes nec-essary to design against a rather limited number of cycles If this number issmaller than about 104

high-, the term low-cycle fatigue (lcf) is used Howeverhigh-,the number of cycles that characterises the transition from low- to high-cycle

fatigue is not well-defined [130] A stress amplitude that causes failure in thelcf regime is called low-cycle fatigue strength, an amplitude causing failure

in the hcf regime is called high-cycle fatigue strength

As can be seen from figure 10.18, the slope of the S-N curve is usuallymuch smaller in the lcf than in the hcf regime so that a small change in thestress amplitude has a large effect on the number of cycles This phenomenon

is restricted to metals and polymers and will be discussed for the case ofmetals in the next section

If the maximum stress σmaxreaches the strength of the monotonous iment in the first cycle (the tensile strength Rm for the case of axial loading),the specimen fractures during this cycle Often, the number of cycles to failure

exper-is then taken to be Nf= 0.5 The left end of the S-N curve is thus determined

by σA(0.5)= 0.5(1 − R)Rm

Independent of the material tested, the scatter of the cycles to failure isusually rather large, for even small defects in the material or on the surfacecan have a strong effect on the life time Different specimens thus are neveridentical For this reason, several experiments have to be performed at eachstress level (usually 6 to 10) to allow ascertaining the width of the scatter band.Using statistical methods, limiting curves can be constructed that represent

a certain probability of failure (for example, 95%) This is elaborated on inForrest [50], Radaj [113] or Schott [130]

As the introductory example of a car engine (see section 10.1) shows, life fatigue loads can be stress- or strain-controlled Stress-controlled loadsoccur if the loads are determined by external forces, strain-controlled loads,for example, if there are temperature changes causing thermal strains Inmany cases, lcf loads are strain-controlled and hcf loads stress-controlled.This, however, cannot be used as a rule For example, loads in a rotating discare determined by centrifugal forces Since these are constant during rota-tion, switching the device on and off corresponds to a single cycle The load isthus stress-controlled, but the number of cycles is low (lcf) Usually, stress- orforce-controlled experiments are easier to perform than strain-controlled exper-iments and are thus often preferred This is especially true in the hcf regime

real-If we look at an S-N curve (figure 10.18), we can see that the number

of cycles to failure strongly depends on the stress in the lcf regime Smallscatter in the stress-strain properties of different specimens (due to scatter inthe material properties, for example) would cause large changes in the number

of cycles to failure measured in the experiment The scatter band would thus

be rather wide In this regime, strain-controlled experiments are more usefulsince, with a prescribed strain amplitude, the scatter of the stress amplitude

is small Furthermore, stress-controlled experiments would also cause morerapid failure due to the reduction in the cross section of the specimen caused

by crack propagation [113]

To assert the influence of notches and inhomogeneous stress distributions

on fatigue life, experiments can also be performed with notched components

or specimens, resulting in specific S-N curves The influence of notches onfatigue life is discussed in more detail in section 10.7.

S-N curves of metals

In a double-logarithmic plot, the S-N curve of many metals is a straight linefor a wide range of the number of cycles (see figure 10.18) This line can bedescribed by the Basquin equation [14]

The fatigue strength coefficient σf0 is related to the tensile strength In plaincarbon and low-alloy steels, a rule of thumb states σ0f= 1.5Rm; in aluminiumand titanium alloys, σf0 = 1.67Rm holds approximately [113] The fatiguestrength exponent a depends on the material and the specimen geometry; inmany materials, it takes values between 0.05 and 0.12 if smooth specimensare used [8, 113]

In plain carbon steels and titanium alloys with body-centred cubic lattice,there is a true fatigue limit with a horizontal S-N curve at a number of cyclesbeyond 2 × 106 to 107 [130] (type I, figure 10.19(a)) This, however, is nottrue for notched specimens (and thus also for components) or if corrosion oroxidation occur during the experiment

Face-centred cubic metals and hardened steels do not have a true fatiguelimit (S-N curve of type II, figure 10.19(b)) At a number of cycles beyond

107, the slope of the S-N curve is rather small and a limiting number of cycles

of NE= 107 to 108 can be used to design safely against fatigue [130]

Recently, it has been found even in body-centred cubic metals that aspecimen can fail in fatigue even beyond the limiting number of cycles(107) At a very large number of cycles (more than 1010), the S-N curvemay drop again [93, 135] This is called ultra-high-cycle fatigue (uhcf)

or very-high-cycle fatigue

In contrast to failure at smaller numbers of cycles, which ally start from the surface, failure in the uhcf regime is caused bymicrocracks being initiated at microscopic inclusions slightly belowthe surface of the specimen, visible as so-called fish eyes at the sur-face [135, 139]

usu-S-N curves of metals have a small slope at low numbers (Nf 103) of cles as well as in the regime Nf > NE In this region, the yield strength ofthe material is exceeded, and the strain amplitude increases rapidly with thestress amplitude A slight increase of the stress causes much larger plasticdeformations and thus strongly reduces the life time

cy-If we plot the strain amplitude εA(Nf) versus Nf in a double-logarithmicplot, we get a strain-cycle diagram as shown in figure 10.20 Two linear regimes,with a smooth transition between them, can be discerned As we will see soon,

is given by the Coffin-Manson equation [32, 94, 95].

Fig 10.21 Dependence of the fatigue limit for fully reversed loading (R = −1) onthe tensile strength in different material classes [113]

the strain-cycle diagram shown in figure 10.20 This equation frequently iscalled Coffin-Manson-Basquin equation

As we saw in section 10.2.1, the crack initiation – and thus the fatiguestrength – of smooth specimens of ductile materials is determined by accu-mulated plastic deformation which usually occurs at the surface For thisreason, the fatigue limit σE for fully reversed loading, R = −1, is related

to the strength under static loads The most suitable parameter to quantifythis relation is not the yield strength Rp, as might be expected, but thetensile strength Rm or a combination of both, as already used in the failure-assessment diagram in section 5.2.3 [113]

In low-strength materials, the fatigue limit is usually proportional tothe static strength In high-strength metals, the fatigue limit increases onlyslightly within a material class (figure 10.21) The reason for this is thathigh-strength materials are very notch-sensitive, and the fatigue limit is thusdetermined by surface or inner defects A large number of approximation for-mulae for the fatigue strength can be found in the literature [90, 113, 130, 147];some of them are listed in table 10.2, taken from Radaj [113]

S-N diagrams of ceramics

As already explained in section 10.3, many ceramics do not exhibit any cycliceffects and can thus bear infinitely many cycles of any load that is smallerthan the static strength (e g., under tension or bending) In these ceramics,the S-N curve is simply a horizontal line at σMax= Rmor σA= 0.5(1 − R)Rm(for the example of a uniaxial load) For a fully reversed stress, this results in

Table 10.2 Approximate fatigue limit of some metals

material class fatigue limit σE for R = −1

steels = (0.35 0.65) × Rm for Rm< 1 400 MPa

≈ 700 MPa for Rm≥ 1 400 MPacast irons = (0.3 0.4) × Rm for Rm< 500 MPa

aluminium alloys = (0.3 0.5) × Rm for Rm< 325 MPa

≈ 130 MPa for Rm≥ 325 MPatitanium alloys = (0.45 0.65) × Rm for Rm< 1 100 MPa

≈ 620 MPa for Rm≥ 1 100 MPa

Fig 10.22 S-N diagram of Si3N4 at different temperatures (measured in bending

at R = −1) [113] The dashed line is a fit according to the Basquin equation, common

to temperatures of 20℃ and 1000℃, whereas the dotted line is valid at 1200℃

This can be exploited to test components with the proof test (section 7.4) Ifthe component does not fail during the test, it can be assumed that it willnot fail by fatigue in service

If mechanical fatigue occurs in a ceramic, equation (10.17) does not holdanymore, and an S-N curve is useful Figure 10.22 shows such a curve forsilicon nitride at three different temperatures As it is usual for ceramics,the slope of the S-N curve is small Slightly reducing the stress thus cansignificantly increase the life time The fatigue limit is only slightly below thestatic strength Fatigue occurs in this ceramic because the crack propagates

on the glassy phase of the grain boundaries (see section 7.5.2), resulting incrack bridging effects as explained in section 10.3 However, the effect is rathersmall

The number of cycles to failure is almost identical for 20℃ and 1000℃

so that the same fit curve can be used to describe both Raising the

ture to 1200℃ has a marked influence on the fatigue strength of the ceramicbecause creep (see chapter 11) occurs in this case This small temperaturedependence over a wide temperature range is also typical of ceramics.

S-N curves of polymers

We already saw in section 10.4 that the fatigue behaviour of polymers stronglydepends on the load frequency because of their viscoelastic properties If thefrequency is sufficiently large, the polymer can fail by thermal fatigue due tothe heat generated during deformation This is shown for the example of athermoplastic polymer in figure 10.23 At low frequencies, the thermoplasticfails by crack formation and propagation, similar to a metal, at higher frequen-cies, thermal fatigue occurs (section 10.4.1), and the fatigue strength stronglydecreases The load frequency is for this reason usually limited to 10 Hz.S-N curves of different polymers are depicted in figure 10.24 In manypolymers (e g., pvc, pp, pa), the S-N curve is horizontal at a large number

of cycles, corresponding to a curve of type I (see figure 10.19(a)) However, asfigure 10.23 shows, this may be due to thermal fatigue, and in this case thehorizontal part of the curve meets the curve for true mechanical fatigue athigher numbers of cycles

S-N curves of polymers have to be used with caution in designing nents The fatigue strength depends much more strongly on the load frequencythan in metals because the equilibrium between heat production and dissipa-tion plays a crucial role To design components, experiments should be asclose to real service conditions as possible

S-N curves of fibre composites

According to section 10.5, the fatigue strength of fibre composites is usuallyhigher than that of the matrix material alone This is shown in figure 10.25,using the S-N curves of unreinforced and short-fibre reinforced polysulfone.The increased fatigue strength is apparent from the figure Long carbon fibresare especially efficient in increasing the fatigue strength of polymer matrixcomposites, not only because of their high stiffness, but also because of theirthermal conductivity The picture is similar in metal matrix composites Forexample, adding 20% silicon carbide fibres to an aluminium matrix doublesthe fatigue strength [140]

Because fibre composites usually do not fail by formation and growth of

a single crack, but by accumulating damage, their stiffness decreases withincreasing number of cycles This is similar in unreinforced materials since thegrowing crack reduces their stiffness as well However, a significant reduction

in stiffness is usually observed only shortly before ultimate failure, whereas adamaged composite may have a long life time despite its reduced stiffness

10.6.3 The role of mean stress

In the S-N diagram (see the previous section), we plot all values at constant

R ratio To quantify the influence of the mean stress or the R ratio for thewhole range of numbers of cycles, from the lcf regime to the fatigue limit,

an extensive number of experiments are required If this is done, the result

is as should be expected: The curves shift to smaller stress amplitudes withincreasing mean stress In many cases, only the dependence of the fatiguelimit is of interest In this case, the fatigue strength diagrams after Smith andHaigh can be used

Smith’s fatigue strength diagram

To draw a Smith’s fatigue strength diagram, the stress amplitude at the tigue limit σE is measured for different values of the mean stress σm.16 Themaximum and minimum stress, σMax and σMin, are plotted in a diagram asshown in figure 10.26 As can be seen from the figure, the stress amplitude σEdecreases with increasing mean stress as expected Because plastic flow ofthe material is not allowed, the diagram is limited horizontally by the yieldstrength Rp or Re in the tensile, and by the compressive strength Rc (inductile metals, this is usually equal to Rp) in the compressive region

fa-Haigh’s fatigue strength diagram

A Haigh’s fatigue strength diagram, or Haigh’s diagram for short, corresponds

to a Smith’s fatigue strength diagram in which the stress amplitude σE isplotted versus the mean stress, instead of the maximum and minimum stress.The distance of a data point from the abscissa in Haigh’s diagram thus cor-responds to the vertical distance of the point from the diagonal in Smith’sdiagram Because the stress amplitude is the same above and below the meanstress, only the upper part of the curve is drawn (figure 10.27(a)) To plot the

R ratio R = σmin/σmax= (σm− σa)/(σm+ σa) in the figure, this equation isrewritten as

16

Lowercase subscripts are used for the prescribed quantities, e g., the mean stress

σm, while uppercase subscripts are used for the resulting quantities for endurance,

e g., the stress amplitude σE Or, in other words, for a given mean stress σm, wesearch the fatigue limit σ

The limits at +Rp and −Rc are not as easy to draw in Haigh’s diagram

as in Smith’s The relation σMax= σm+ σE= Rpyields σE= Rp− σm; from

σMin= σm− Rc we get σE= Rc+ σm Both limits thus correspond to straightlines with a slope of ±45°, intersecting the axis at Rp and Rc, respectively.These are shown in figure 10.27(a)

Different approximations can be used to describe the fatigue strength gram between these limits [130] Frequently, a linear Goodman equation isused, corresponding to a straight line that connects the fatigue strength σE

dia-at R = −1 and the tensile strength Rm at R = 1 (figure 10.27(b)) This proximation is valid for positive mean stresses (−1 ≤ R ≤ 1) Mathematically,

ap-it can be described by the equation [8, 76]

in grey

Fig 10.27 Haigh’s fatigue strength diagram

∗ 10.6.4 Fatigue assessment with variable amplitude loading

The S-N diagram plots the life time of a material at constant stress amplitudeand R ratio It is, however, not possible to assert the life time, using thediagram, if the load amplitude changes The most obvious way to determinethe life time in this case is to simulate the service load history in the laboratory.Unfortunately, this is a rather involved procedure that is not feasible in mostcases It would be helpful if it were possible to estimate the life time directlyfrom the S-N curves One way to do this is to use Miner’s rule (also known

as Palmgren-Miner rule) [99] that will be explained now The rule is rathersimple and thus easy to employ, but it has some disadvantages, discussed atthe end of this section

To use Miner’s rule, a partial damage of the component is calculated foreach loading step Assume that the component is loaded with k different stressamplitudes σa,j, j = 1, , k and that the number of cycles for amplitude j is

nj We can then use the S-N curve to determine the number of cycles to failurefor each of the stress amplitudes, N (σ ) Miner’s rule now assumes that

each step ‘uses up’ part of the components life time, with a partial damage

Dj= nj/Nf,j The component fails when the total damage D equals one:

The sequence of the load steps is not taken into account in equation (10.20)

It is easy to see that this is not valid in some cases Assume that we use twostress amplitudes, σa,1, being smaller than the fatigue limit σE, and σa,2, beinglarger If we start with a sufficient number of cycles at the higher load σa,2,microcracks will form according to section 10.2.1 Afterwards, the smallerstress amplitude σa,1 is sufficient to further propagate the crack Miner’s rule,however, associates no damage with the smaller load because the number ofcycles the component would live at this load alone is Nf,1= ∞, corresponding

to D1= 0 Thus, a calculational damage of D = 1 does never occur, and thespecimen fails although D stays smaller than 1

If we load another specimen with the reversed sequence, no damage will

be caused by the first stress amplitude σa,1, and we find D1 = 0 for any n1.All damage in the material is accumulated during the second loading step, atstress amplitude σa,2, causing failure at n2 = Nf,2 when D = D2 = 1 holds

As the example shows, the sequence of loading may influence the fatigue lifestrongly, an effect neglected in Miner’s rule

Sometimes, cyclic hardening occurs during the first loading step (seesection 10.6.5 below) The specimen will then yield less in the nextloading steps This so-called coaxing is especially important if the stressamplitude is increased in many loading steps If a training effect occurs,the specimen may fail at damage values D > 1

To summarise, it can be stated that Miner’s rule is useful to provide a firstapproximation of the life time, but it has to be used with great care, for thecalculated life time may not be a conservative estimate

∗ 10.6.5 Cyclic stress-strain behaviour

We already mentioned in section 10.2.1 that the stress-strain behaviour of als may change during cyclic loading Depending on the initial state, differenteffects may occur

met-∗ Cyclic hardening and softening

If we perform, for example, a fatigue experiment at constant strain tude εa, the stress amplitude σa changes during the experiment Figure 10.28shows typical cases If the stress increases during loading, cyclic hardening oc-curs (figure 10.28(a)); if the stress decreases, the phenomenon is called cyclic

t

σ

εσ

(a) Cyclic hardening

tε

t

σ

εσ

(b) Cyclic softening

Fig 10.28 Cyclic stress-strain behaviour at the beginning of strain-controlled tigue experiments (after [130]) The controlled variable is shown on the left, thematerial answer in the centre

fa-"

cyclicmonotonous

¾

Fig 10.29 Cyclic stress-strain diagram

A static stress-strain curve is shown forcomparison

softening Frequently, the stress amplitude changes only initially and thenstabilises to a constant value

If we perform cyclic experiments at different strain amplitudes and plotthe stabilised values of the stress amplitude, we arrive at the cyclic stress-strain diagram sketched in figure 10.29 Usually, it does not coincide with theresult of a monotonous tensile or compressive test If cyclic hardening occurs,

The microstructure of the material may also change under cyclic loads Inprecipitation-hardened alloys with underaged precipitates (see sections 6.3.1and 6.4.4), the precipitates may be destroyed by repeated cutting, reducingthe strength of the material In ferritic steels, the dislocations may detach fromtheir surrounding carbon atoms so that no apparent yield strength exists inthe cyclic stress-strain diagram, and the cyclic curve lies below the static one

in this strain range (see section 6.4.3) Cyclic softening results

To reduce the experimental efforts in measuring cyclic stress-straincurves, the incremental-step test can be used In this test, the strainamplitude is varied block-wise between zero and a maximal value assketched in figure 10.30 After the block has been repeated several times,the material behaviour does not change anymore and a stationary state

is arrived at If the stress is measured at each of the strain maxima, the