Kjell Simonsson 1

A brief introduction to the fatigue phenomenon

(last updated 2011-09-21)

Kjell Simonsson 2

Aim

The aim of this presentation is to give a short introduction to the fatigue phenomenon.

In addition we will take a quick glance at stress concentrations, cracks and cyclic crack growth.

Kjell Simonsson 3

What is fatigue?

A body or structure subjected to repeated loadings can fail, even though themagnitude of the applied loading is far below any yield- or static failure limit-we refer to this phenomenon as fatigue.

Generally, the higher the load amplitude is, the shorter the life will be(in terms of loading cycles).

By obvious reasons fatigue failures may be both dangerous and expensive,and must therefore be avoided. An additional complicating factor is that fatiguefailures often occur without any previous "warnings", and are thereforeespecially awkward. As a result, fatigue design has become an increasinglyimportant task in the modern product design process.

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An example of a real fatigue failure case

As an example of a real fatigue case, we can note the front wheelfailures of a motorbike from the 1980's. Without any previous warnings, asimultaneous failure of all spikes around the center of the wheel could happen(e.g. when using the brakes), see below.

It was later concluded that the failures were due to fatigue. It may be noted,that in order to avoid further accidents on existing motorbikes, it wasrecommended to continuously check the spikes for cracks (this is for instancedone in the Swedish national vehicle test programme for motorbikes in use).

Part of circumferentialbreakage illustrated.

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The physical nature of fatigue

The fatigue process (in metallic materials) is typically divided into an

• initiation phase, where micro-cracks are nucleated, cyclically growing, and finally merging into one dominant macro-crack

and a

• propagation phase, where the created macro-crack cyclically grows until it reaches a critical size, when the final failure occurs by a static fracture

micro-scopiccyclicslip

cyclicmacro-crackgrowth

cyclicmicro-crackgrowth

micro-cracknucleation

Initiation phase Propagation phase Final failure

staticfracture

As noted above, micro-crack nucleation is in metallic materials due to cyclicslip, i.e. cyclic plastic flow on a microscopic scale, involving the back and forthmovement of dislocations.

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The physical nature of fatigue; cont.

We note that

• The final fatigue failure surface often (but not always) shows two regions

* one surface with so called striations, resulting from the cycling loading

* one rough surface, representing the final static fracture surface

schematically

crack front

micro-crack

macro-crack

finalfailuresurface

striations

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The physical nature of fatigue; cont.

• micro-cracks are typically initiated

* in shear bands (with extensive localized microscopic plastic flow)

* at the surface of a component or specimen, due to environment effects (oxidation), surface roughness and less restraint to plastic flow

• the distinction micro-crack/macro-crack and initiation phase/propagation phase is not sharp

It may e.g. be based on the size or detectability of the crack; 1mm (i.e. visible to the naked eye) could be a criterion

• The initiation phase is typically very much longer than the propagation phase for smooth laboratory specimens, but this is generally not true for notched specimens or real structures

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The physical nature of fatigue; cont.

• different factors may have very different effect on the initiation and the propagation phase, resp.

As an example, initiation is strongly affected by the surface conditions, while the propagation phase hardly depends on it at all

• there is often a large scatter in experimentally obtained lives, especially for lower loads/longer lives, since for this case the microstructure has a larger influence on the material behavior

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Fatigue design

As noted previously, fatigue design (i.e. designing against fatigue) is avery important topic, where life-, safety- and economic aspects are to beguaranteed and/or improved.

However, fatigue design is to its nature a complex and difficult topic.Furthermore, large uncertainties often exist regarding e.g.

• actual load spectrum

• the real material behavior

• residual stresses and surface conditions

• environment

which makes the task even more difficult.

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Stress concentrations

A stress concentration is simply a geometrical feature (hole, notch, etc), whichlocally increases the stresses. Since such stress raisers always will be presentin components and designs, we simply need to be able to handle their effecton e.g. the life (number of cycles to failure) of cyclically loaded components.

The probably most simple, and therefore the by far most discussed case, is acircular hole in a uniaxially loaded large flat plate. By using the stress functionapproach of 2-dimensional linear elastostatics,one finds for the tangential stresses at the hole the following formula

2cos21)( 0 ar

000

03 r

As can be seen, we geta 3-fold increase of thetangential stress at thehole!

a

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Stress concentrations; cont.

The nominal stress at the hole is defined as the stress that would be presentif no redistribution of stresses around the hole took place, i.e.

002

aww

nom

where w is the width of the plate.

nom0

The stress concentration factor isdefined as

nomtK max

which for the case above equals 3.

For a small hole in a large plate

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Stress concentrations; cont.

In the case that the stress concentration becomes sharper, a higher stressconcentration factor will be found. As an example, one may show (not to bedone here) that we for an elliptical hole in a large flat plate have the followingsituation.

As can be seen, we regain the previous result for the case of a circular hole(a=b). Furthermore, we see that the more elongated the hole will get, thehigher the stress concentration will be (as stated above).

Imagine what will happen for a crack!

0min

0max 21,

ba

KK tt

00max

b2 a2

min

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The stress concentration at the tip of a crack

As we saw previously, the stress concentration at the tip of an elliptical hole ina very large flat plate becomes higher as the hole becomes more elongated.In detail we have

0min

0max 21,

ba

KK tt

00max

b2 a2

min

Obviously, the stress concentration factor for a sharp crack becomes(in theory) infinite! This is of course not so in reality, since the material willyield plastically at the crack tip. However, surprisingly enough, the stresssolution based on linear elasticity has been shown to be extremely useful inpredicting e.g. static fracture and cyclic crack propagation!

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The stress concentration at the tip of a crack; cont.

Uni-axially loaded plateLet us now in some more detail look at the stress state at a crack located in avery large flat plate (for which we can assume that plane stress- or plane strainconditions prevail). More specifically, we will start by considering a uni-axialloading as illustrated below.

a2

0

0

y

x

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The stress concentration at the tip of a crack; cont.

Uni-axially loaded plate; cont.It can be shown that the stress state near the crack tip takes the following form

a2

'y

'x

r

aK

rK

rK

rK

I

Ixy

Iy

Ix

0

23

cos2

cos02

sin2

23

sin2

sin12

cos2

23

sin2

sin12

cos2

As can be seen, the stress has a so called singularity at the crack tip,where the intensity of the singularity is given by the KI parameter, which isreferred to as the stress intensity factor.

r/1

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Uni-axially loaded plate; cont.

• The stresses at the crack tip of a very large plate can thus be written

where the f's are coordinate functions

• If the size of the plate becomes "finite", the expression for the stress intensity factor will also depend on the relation between the crack size and the plate size and on the type of loading, which may be written

where W is the width of the plate; for details please see any formula table in fracture mechanics.

The stress concentration at the tip of a crack; cont.

),( rfK ijIij

)/(0 WafaK I

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Different Modus

A crack can in principle be loaded in three different ways, which we refer toas loading modus, see below. The stress field at the crack tip will in each ofthese cases have the same basic structure. The reason why we used the nameKI previously, was simply that the loading corresponded to a Mode I-loading!

Most dangerous!

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Fracture toughness

Even though, as was pointed out previously, the obtained stress field isclearly unphysical, it is still extremely useful in determining the risk ofstatic failure, or in calculating the propagation of a crack under cyclic loading!

Considering the former topic (static fracture), it has been shownthat materials have a fracture toughness KIC , such that they fail if KI exceedsthis value. Thus, the failure criterion for static failure becomes (in Mode I)

ICI KK

The value KIC is to be obtained under plane strain conditions. In order toensure that the actual loading condition is of this type, the followingrequirements are to be fulfilled (must be checked when solving problems!)

2

5.2

Y

ICK

aW

a

t

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Cyclic crack growth

If a crack is subjected to a varying load it may grow. We find experimentally

The linear region is described by the so called Paris law

nI

nII

KCdNda

KCdNda

KnCdNda

logloglogloglog

000

00

0

max,

min,max,

min,min,max,

II

III

IIII

KifK

KifKK

KifKKK

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Cyclic crack growth; damage tolerant approach

Let us assume that we need to design against fatigue in a structure where wehave cracks smaller than a certain size (found by e.g. non-destructive testing)

When designing this structure for infinite life, we require (for a certain safetyfactor s)

sKaK thinitialI /

When designing this structure for finite life, we require that an integration ofParis law from the known crack size, and for a given number of loadcycles/sequences (corresponding to e.g. an inspection interval), will give(for a certain safety factor s)

sKaK IcfinalI /