Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering

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are prepared in order to minimize the likelihood of a fatigue failure. Shot peening is also

a process that is used very e¤ecti vely to enhance fatigue life of materials. Small metal

spheres are shot at the component. This leads to a residual compressive stress at the

surface similar to strengthening of inorganic glasses by tempering (Section 7-9).

EXAMPLE 7-11

Design of a Rotating Shaft

A solid shaft for a cement kiln produced from tool steel must be 240 cm long

and must survive continuous operation for one year with an applied load of

55,600 N. The shaft makes one revolution per minute during operation. Design

a shaft that will satisfy these requirements. The S-N curve for the tool steel is

shown in Figure 7-17.

SOLUTION

The fatigue life required for our design is the total number of cycles N that the

shaft will experience in one year:

N ¼ð1 cycle=minÞð60 min=hÞð24 h=d Þð365 d=yÞ

N ¼ 5:256  10

cycles=y

where y ¼ year, d ¼ day, and h ¼ hour.

From Figure 7-17, the applied stress therefore must be less than about

496 MPa. From Equation 7-13, the diameter of the shaft must be:

Gs ¼

16FL

¼ 5:09

496 MPa ¼

ð5:09Þð240 cmÞð55;600 NÞ

d ¼ 11:1cm

A shaft with a diameter of 11.1 cm should operate for one year under these

conditions. However, a signiﬁcant margin of safety might be incorporated in

the design. In additi on, we might consider producin g a shaft that would never

fail.

Let us assume the factor of safety to be 2 (i.e., we will assume that the

maximum allowed stress level will be 496/2 ¼ 248 MPa). The minimum diam-

eter required to prevent failure would now be:

248 MPa ¼

ð5:09Þð240 cmÞð55;600 NÞ

d ¼ 14 cm

Selection of a larger shaft reduces the stress level and makes fatigue less likely

to occur or delays the failure. Other considerations might, of course, be im-

portant. High temperatures and corrosive conditions are inherent in producing

cement. If the shaft is heated or attacked by the corrosive environment, fatigue

is accelerated. Thus, in the applications involving fatigue of components regu-

lar inspections of the components go a long way toward avoiding a catastro-

phic failure.

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior210

EXAMPLE 7-12 Do Materials have an Infinite Fatigue Life?

Typically, engineering materials such as alloys and composites are tested for

fatigue up to about 10

cycles (known as low-cycle fatigue). Following are the

data for an aluminum alloy designated as 7075T73 (the solid curve labeled as

A), used for making helicopter propellers. The solid circles are the data for an

aluminum matrix composite reinforced with SiC ﬁbers (labeled as the B-dashed

curve). (a) Based on these low-cycle fatigue data, which material appears to

have better fati gue resistance? (b) What appears to be the fatigue strength of

the aluminum matrix composites? (c) Can these data be used to predict the fa-

tigue life of these materials for longer tests involving fatigue cycles up to 10

SOLUTION

(a) From the low-cycle (i.e., up to 10

cycles) fatigue data, it is clear that the

aluminum alloy reinforced with SiC ﬁbers (material B) has a better fatigue

behavior at a low number of cycles.

(b) The fatigue curve for material B appears to show a plateau at around 10

cycles. This would suggest a fatigue strength of @300 MPa.

and 10

cycles, it appears that the

stress to failure for the SiC reinforced composite (material B) now is lower

and is approaching that for the alloy 7075T73 (material A). Thus, it prob-

ably would not be safe to assume that over a longer period of time (up to

cycles) that the SiC reinforced material will have a better fatigue life

than alloy 7075T73.

In fact, measurements conducted by Bathias and co-workers show that

maximum stress that can be supported for the SiC reinforced alloy decreases

even more to almost 200 MPa at @10

cycles.

Thus, an observation is that we must make fatigue measurements in the

giga-cycle range to see the fatigue behavior and not just extend what is seen

only in the low cycle fatigue regime. In fact, most materials may not have an

inﬁnite fatigue life, as is mostly suggested by the low-cycle fatigue data. These

giga-cycle fatigue tests can be done on modern fatigue machines that are driven

by piezoelectric materials at a frequency of @20 kHz, compared to 100 Hz.

Figure 7-18

Low-cycle fatigue data for an

aluminum alloy and an

aluminum-alloy matrix

composite reinforced with SiC.

(Source: Ref. Bathias, C.,

Fatigue and Fracture in

Engineering Materials

Structures, Blackwell

Publishing, page 559–565,

Vol. 22 (1999).)

7-7 Results of the Fatigue Test 211

7-8 Application of Fatigue Testing

Material components are often subjected to loading conditions that do not give equal

stresses in tension and compression (Figure 7-19). For example, the maximum stress

during compression may be less than the maximum tensile stress. In other cases, the

loading may be between a maximum and a minimum tensile stress; here the S-N curve

is presented as the stress amplitude versus number of cycles to failure. Stress amplitude

) is deﬁned as half of the di¤erence between the maximum and minimum stresses;

Figure 7-19

Examples of stress cycles. (a) Equal

stress in tension and compression,

(b) greater tensile stress than

compressive stress, and (c) all of the

stress is tensile.

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior212

mean stress (s

) is deﬁned as the average between the maximum and minimum stresses:

max

 s

min

ð7-15Þ

max

þ s

min

ð7-16Þ

A compressive stress is considered a ‘‘negative’’ stress. Thus, if the maximum tensile stress

is 345 MPa and the minimum stress is a 69 MPa compressive stress, using Equa-

tions 7-15 and 7-16 the stress amplitude is 138 MPa and the mean stress is 207 MPa.

As the mean stress increases, the stress amplitude must decrease in order for the

material to withstand the applied stresses. This condition can be summarized by the

Goodman relationship:

¼ s

1 



ð7-17Þ

where s

is the desired fatigue strength for zero mean stress and s

is the tensile

strength of the material. Therefore, in a typical rotating cantilever beam fatigue test,

where the mean stress is zero, a relatively large stress amplitude can be tolerated with-

out fatigue. If, however, an airplane wing is loaded near its yield strength, vibrations of

even a small amplitude may cause a fatigue crack to initiate and grow.

Crack Growth Rate In many cases, a component may not be in danger of failure even

when a crack is present. To estimate when failure might occur, the rate of propagation

of a crack becomes important. Figure 7-20 shows the crack growth rate versus the

range of the stress-intensity factor DK, which characterizes crack geometry and the

stress amplitude. Below a threshold DK, a crack does not grow; for somewhat higher

Figure 7-20

Crack growth rate versus stress-

intensity factor range for a high-

strength steel. For this steel,

C ¼ 1:62  10

12

and n ¼ 3: 2 for

the units shown.

7-8 Application of Fatigue Testing 213

stress-intensities, cracks grow slowly; and at still higher stress-intensities , a crack grows

at a rate given by:

¼ CðDKÞ

ð7-18Þ

In this equation, C and n are empirical constants that depend upon the material. Fi-

nally, when DK is still higher, cracks grow in a rapid and unstable manner until fracture

occurs.

The rate of crack growth increases as a crack increases in size, as predicted from

the stress intensity factor (Equation 7-1):

DK ¼ K

max

 K

min

¼ fs

max

ﬃﬃﬃﬃﬃﬃ

 fs

min

ﬃﬃﬃﬃﬃﬃ

¼ fDs

ﬃﬃﬃﬃﬃﬃ

ð7-19Þ

If the cyclical stress Dsðs

max

 s

min

Þ is not changed, then as crack length a increases,

DK and the crack growth rate da=dN increase. In using this expression, however, one

should note that a crack will not propagate during compression. Therefore, if s

min

compressive, or less than zero, then s

min

should be set equal to zero.

Knowledge of crack growth rate is of assistance in designing components and in

nondestructive evaluation to determine if a crack poses imminent danger to the struc-

ture. One approach to this problem is to estimate the number of cycles required before

failure occurs. By rearranging Equation 7-18 and substituting for DK:

dN ¼

n=2

If we integrate this expression between the initial size of a crack and the crack size re-

quired for fracture to occur, we ﬁnd that

N ¼

2½ða

ð2nÞ=2

ða

ð2nÞ=2



ð2  nÞCf

n=2

ð7-20Þ

where a

is the initial ﬂaw size and a

is the ﬂaw size required for fracture. If we know

the material constants n and C in Equation 7-18, we can estimate the number of cycles

required for failure for a given cyclical stress (Example 7-13).

EXAMPLE 7-13

Design of a Fatigue-Resistant Plate

A high-strength steel plate (Figure 7-20), which has a plane strain fracture

toughness of 80 MPa

ﬃﬃﬃﬃ

, is alternately loaded in tension to 500 MPa and in

compression to 60 MPa. The plate is to survive for 10 years, with the stress

being applied at a frequency of once every 5 minutes. Design a manufacturing

and testing procedure that assures that the component will serve as intended.

SOLUTION

To design our manufacturing and testing capability, we must determine the

maximum size of any ﬂaws that might lead to failure within the 10-year period.

The critical crack size (a

), using the fracture toughness and the maximum

stress, is:

¼ fs

ﬃﬃﬃﬃﬃﬃﬃ

80 MPa

ﬃﬃﬃﬃ

¼ð1Þð500 MPaÞ

ﬃﬃﬃﬃﬃﬃﬃ

¼ 0:0081 m ¼ 8:1mm

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior214

The maximum stress is 500 MPa; however, the minimum stress is zero, not

60 MPa in compression, because cracks do not propagate in compression.

Thus, Ds is

Ds ¼ s

max

 s

min

¼ 500  0 ¼ 500 MPa

We need to determine the minimum number of cycles that the plate must

withstand:

N ¼ð1 cycle=5 minÞð60 min=hÞð24 h=dÞð365 d=yÞð10 yÞ

N ¼ 1;051;200 cycles

If we assume that f ¼ 1 for all crack lengths and note that C ¼ 1: 62  10

12

and n ¼ 3:2 in Equation 7-20, then

1;051;200 ¼

2½ð0:008Þ

ð23:2Þ=2

ða

ð23:2Þ=2



ð2  3:2Þð1:62  10

12

Þð1Þ

3:2

ð500Þ

3:2

3:2=2

1;051;200 ¼

2½18  a

0:6



ð1:2Þð1:62  10

12

Þð1Þð4:332  10

Þð6:244Þ

0:6

¼ 18 þ 2764 ¼ 2782

¼ 1:82  10

6

m ¼ 0: 00182 mm for surface ﬂaws

¼ 0:00364 mm for internal ﬂaws

The manufacturing process must produce surface ﬂaws smaller than 0.00182 mm

in length. We can conduct a similar calculation for specifying a limit on edge

cracks. In addition, nondestructive tests must be available to assure that cracks

exceeding this length are not present.

Effect of Temperature As the material’s temperature increases, both fatigue life and

endurance limit decrease. Furthermore, a cyclical temperature change encourages fail-

ure by thermal fatigue; when the material heats in a nonuniform manner, some parts of

the structure expand more than others. This nonuniform expansion introduces a stress

within the material, and when the structure later cools and contracts, stresses of the

opposite sign are imposed. As a consequence of the thermally induced stresses and

strains, fatigue may eventually occur. The frequency with which the stress is applied

also inﬂuences fatigue behavior. In particular, high-frequency stresses may cause poly-

mer materials to heat; at increased temperature, polymers fail more quickly. Chemical

e¤ects of temperature (e.g., oxidation) must also be considered.

7-9 Creep, Stress Rupture, and Stress Corrosion

If we apply stress to a material at an elevated tempe rature, the material may stretch and

eventually fail, even though the applied stress is less than the yield strength at that

temperature. A time dependent per manent deformation under a constant load or con-

stant stress and at high temperatures is known as creep. A large number of failures

occurring in components used at high temperatures can be attributed to creep or a

combination of creep and fatigue. Essentially, in creep the material begins to ﬂow

slowly. Di¤usion, dislocation glide or climb, or grain boundary sliding can contribute

7-9 Creep, Stress Rupture, and Stress Corrosion 215

to the creep of metallic materials. Polymeric materials also show creep. In ductile met-

als and alloys subjected to creep, fracture is accompanied by necking, void nucleation

and coalescence, or grain boundary sliding.

A material is considered failed by creep even if it has not actually fractured. When

a material does actually creep and then ultimately break the fracture is deﬁned as stress

rupture. Normally, ductile stress-rupture fractures include necking and the presence of

many cracks that did not have an opportunity to produce ﬁnal fracture. Furthermore,

grains near the fracture surface tend to be elongated. Ductile stress-rupture failures

generally occur at high creep rates and relatively low exposure temperatures and have

short rupture times. Brittle stress-rupture failures usually show little necking and occur

more often at smaller creep rates and high temperatures. Equiaxed grains are observed

near the fracture surface. Brittle failure typically occurs by formation of voids at the

intersection of three grain boundaries and precipitation of additional voids along grain

boundaries by di¤usion processes (Figure 7-21).

Stress-Corrosion Stress-corrosion is a phenomenon in which materials react with cor-

rosive chemicals in the environment. This leads to formation of cracks and lowering

of strength. Stress-corrosion can occur at stresses well below the yield strength of the

metallic, ceramic, or glassy material due to attack by a corrosive medium. In metallic

materials, deep, ﬁne corrosion cracks are produced, even though the metal as a whole

shows little uniform attack. The stresses can be either externally applied or stored re-

sidual stresses. Stress-corrosion failures are often identiﬁed by microstructural exami-

nation of the nearby metal. Ordinarily, extensive branching of the cracks along grain

boundaries is observed (Figure 7-22). The location at which cracks initiated may be

identiﬁed by the presence of a corrosion product.

Inorganic silicate glasses are especially prone to failure by reaction with water

vapor. It is well known that the strength of silica ﬁbers or silica glass products is very

high when these materials are protected from water vapor. As the ﬁbers or silica glass

components get exposed to water vapor, corrosion reactions begin leading to formation

of surface ﬂaws, which ultimately cause the cracks to grow when stress is applied.

Polymeric coatings are applied to optical ﬁbers to prevent them from reacting with

water vapor. For bulk glasses, special heat treatments such as tempering are used.

Tempering produces an overall compressive stress on the surface of glass. Thus, even if

the glass surface reacts with water vapor the cracks do not grow since the overall stress

at the surface is compressive. If we create a ﬂaw that will penetrate the compressive

Figure 7-21 Creep cavities formed at grain boundaries in an austentic stainless steel (500).

(From ASM Handbook, Vol. 7, (1972) ASM International, Materials Park, OH 44073.)

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior216

stress region on the surface, tempered glass will shatter. Tempered glass is used widely

in building and automotive applications.

Figure 7-22

Photomicrograph of a metal near a stress-

corrosion fracture, showing the many

intergranular cracks formed as a result of the

corrosion process (200). (From ASM

Handbook, Vol. 7, (1972) ASM International,

Materials Park, OH 44073.)

EXAMPLE 7-14

Failure Analysis of a Pipe

A titanium pipe used to transport a corrosive material at 400



C is found to fail

after several months. How would you determine the cause for the failure?

SOLUTION

Since a period of time at a high temperature was required before failure

occurred, we might ﬁrst suspect a creep or stress-corrosion mechanism for fail-

ure. Microscopic examination of the material near the fracture surface would

be advisable. If many tiny, branched cracks leading away from the surface are

noted, stress-corrosion is a strong possibility. However, if the grains near the

fracture surface are elongated, with many voids between the grains, creep is a

more likely culprit.

7-10 Evaluation of Creep Behavior

To determine the creep characteristics of a material, a constant stress is applied to a

heated specimen in a creep test . As soon as the stress is applied, the specimen stretches

elastically a small amount e

(Figure 7-23), depending on the applied stress and the

modulus of elasticity of the material at the high temperature. Creep testing can also be

conducted under a constant load and is important from an engineering design view -

point.

Dislocation Climb High temperatures permit dislocations in a metallic material to

climb. In climb, atoms move either to or from the dislocation line by di¤usion, causing

the dislocation to move in a direction that is perpendicular, not parallel, to the slip

plane (Figure 7-24). The dislocation escapes from lattice imperfections, continues to

slip, and causes additional deformation of the specimen even at low applied stresses.

7-10 Evaluation of Creep Behavior 217

Creep Rate and Rupture Times During the creep test, strain or elongation is measured

as a function of time and plotted to give the creep curve (Figure 7-23). In the ﬁrst stage

of creep of metals, many dislocations climb away from obstacles, slip, and contribute

to deformation. Eventually, the rate at which dislocations climb away from obstacles

equals the rate at which dislocations are blocked by other imperfections. This leads to

second-stage, or steady-state, creep. The slope of the steady-state portion of the creep

curve is the creep rate:

Creep rate ¼

D strain

D time

ð7-21Þ

Eventually, during third-stage creep, necking begins, the stress increases, and the speci-

men deforms at an accelerated rate until failure occurs. The time required for failure

to occur is the rupture time. Either a higher stress or a higher temperature reduces the

rupture time and increases the creep rate (Figure 7-25).

The combined inﬂuence of applied stress and temperature on the creep rate and

rupture time (t

) follows an Arrhenius relationship:

Creep rate ¼ Cs

exp 



ð7-22Þ

¼ Ks

exp



ð7-23Þ

where R is the gas constant, T is the temperature in Kelvin, C, K, n, and m are con-

stants for the material, Q

is the activation energy for creep, and Q

is the activation

energy for rupture. In particular, Q

is related to the activation energy for self-di¤usion

Figure 7-23

A typical creep curve showing the

strain produced as a function of

time for a constant stress and

temperature.

Figure 7-24

Dislocations can climb (a) when

atoms leave the dislocation line to

create interstitials or to fill vacancies

or (b) when atoms are attached to the

dislocation line by creating vacancies

or eliminating interstitials.

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior218

when dislocation climb is important. Relative creep resistance of three magnesium

alloys, namely AZ91D, AS21, and MRI153M is shown in Figure 7-25(b).

In creep of polycrystalline ceramics, other factors—including grain boundary slid-

ing and nucleation of microcracks—are particularly important. Often, a noncrystalline

or glassy material is present at the grain boundaries; the activation energy required for

the glass to deform is low, leading to high creep rates compared with completely crys-

talline ceramics. For the same reason, creep occurs at a rapid rate in ceramic glasses

and amorphous polymers.

The stress exponent ðnÞ and the creep activation energy ðQ

Þ encountered in Equa-

tion 7-22 for Mg alloys MRI 151, MRI 153, and As21 are shown in Table 7-2.

In Mg alloys containing Al, an intermetallic compound Mg

can form at grain

boundaries, and this causes the creep resistance to be lowered. The creep resistance is

enhanced by adding small concentrations of alkaline earth metals, such as Ca or Mg.

These metals react with Al preferentially and form other intermetallics (i.e., Al

Ca and

Mg). These intermetallics have melting temperatures greater than 1000



C. This

translates into enhanced creep resistance, as grain boundary sliding is suppressed.

Figure 7-25 (a) The effect of temperature or applied stress on the creep curve. (b) Relative

creep resistance of Mg alloys AZ91D, AS21, and MRI153M. (Ref. A.M. Russell and K.L. Lee

in Structure-Property Relations in Nonferrous Metals, Publ. Wiley, (2005), page 176.)

TABLE 7-2 9 Creep exponent (n) and activation energy (Q

) for

some Mg Alloys

Alloy

Stress Exponent (n)

T F 135˚ C, stress

85–110 MPa

Activation energy (Q

)

kJ/mol (90 MPa,

130–150˚ C)

MRI 151 7.0 175

MRI 153 7.6 181

AS21 19.5 166

(Source: A.M. Russell and K.L. Lee in Structure-Property Relations in

Nonferrous Metals, Publ. Wiley, (2005), page 176.)

7-10 Evaluation of Creep Behavior 219

Askeland D.R., Fulay P.P. Essentials of Materials Science &amp; Engineering

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Askeland D.R., Fulay P.P. Essentials of Materials Science & Engineering