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

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2. The ability of a material to deform is critical. In ductile metals, the material near

the tip of the ﬂaw can deform, causing the tip of any crack to become blunt, reducing

the stress intensity factor, and preventing growth of the crack. Increasing the strength

of a given metal usually decreases ductility and gives a lower fracture toughness. (See

Table 7-1.) Brittle materials such as ceramics and many polymers have much lower

fracture toughness than metals (Figure 7-3).

3. Thicker, more rigid pieces of a given material have a lower fracture toughness

than thinner materials.

Figure 7-3 Fracture toughness versus strength of different engineered materials. (Reprinted by

permission of Waveland Press, Inc. from Courtney, T., Mechanical Behavior of Materials, 2/e,

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior190

4. Increasing the rate of application of the load, such as that encountered in an

impact test, typically reduces the fracture toughness of the material.

5. Increasing the temperature normally increases the fracture toughness, just as in

the impact test.

6. A small grain size normally improves fracture toughness, whereas more point

defects and dislocations reduce fracture toughness. Thus, a ﬁne-grained ceramic mate-

rial may provide improved resistance to crack growth.

7. In certain ceramic materials we can also take advantage of stress-induced trans-

formations that lead to compressive stresses that cause increased fracture toughness.

Fracture testing of ceramics cannot be performed easily using a sharp notch, since

formation of such a notch often causes the samples to break. We can use hardness

testing to gain a measure of the fracture toughness of many ceramics.

7-2 The Importance of Fracture Mechanics

The fracture mechanics approach allows us to design and select materials while taking

into account the inevitable presence of ﬂaws. There are three variables to consider: the

property of the material (K

or K

), the stress s that the material must withstand, and

the size of the ﬂaw a. If we know two of these variables, the third can be determined.

Selection of a Material If we know the maximum size a of ﬂaws in the material and

the magnitude of the applied stress, we can select a material that has a fracture tough-

ness K

or K

large enough to prevent the ﬂaw from growing.

Design of a Component If we know the maximum size of any ﬂaw and the material

(and therefore it’s K

or K

has already been selected), we can calculate the maximum

stress that the component can withstand. Then we can design the appropriate size of the

part to assure that the maximum stress is not exceeded.

Design of a Manufacturing or Testing Method If the material has been selected, the

applied stress is known, and the size of the component is ﬁxed, we can calculate the

maximum size of a ﬂaw that can be tolerated. A nondestructive testing technique that

detects any ﬂaw greater than this critical size can help assure that the part will function

safely. In addition, we ﬁnd that, by selecting the correct manufacturing process, we can

limit the size of the ﬂaws to be smaller than this critical size.

EXAMPLE 7-1

Design of a Nondestructive Test

A large steel plate used in a nuclear reactor has a plane strain fracture tough-

ness of 87.9 MPa

ﬃﬃﬃﬃ

and is exposed to a stress of 310 MPa during service.

Design a testing or inspection procedure capable of detecting a crack at the

edge of the plate before the crack is likely to grow at a catastrophic rate.

7-2 The Importance of Fracture Mechanics 191

SOLUTION

We need to determine the minimum size of a crack that will propagate in the

steel under these conditions. From Equation 7-1, assuming that f ¼ 1: 12:

¼ f s

ﬃﬃﬃﬃﬃﬃ

87:9 ¼ð1:12Þð310Þ

ﬃﬃﬃﬃﬃﬃ

a ¼ 2cm

A 2 cm deep crack on the edge should be relatively easy to detect. Often, cracks

of this size can be observed visually. A variety of other tests, such as dye pen-

etrant inspection, magnetic particle inspection, and eddy current inspection,

also detect cracks much smaller than this. If the growth rate of a crack is slow

and inspection is performed on a regular basis, a crack should be discovered

long before reaching this critical size.

Brittle Fracture Any crack or imperfection limits the ability of a ceramic to withstand

a tensile stress. This is because a crack (sometimes called a Gri‰th ﬂaw) concentrates

and magniﬁes the applied stress. Figure 7-4 shows a crack of length a at the surface

of a brittle material. The radius r of the crack is also shown. When a tensile stress s is

applied, the actual stress at the crack tip is:

actual

G 2s

ﬃﬃﬃﬃﬃﬃﬃ

a=r

ð7-3Þ

For very thin cracks (small r) or long cracks (large a), the ratio s

actual

=s becomes large,

or the stress is intensiﬁed. If the stress (s

actual

) at the crack tip exceeds the yield strength,

the crack grows and eventually causes failure, even though the nominal applied stress s

is small.

In a di¤erent approach, we recognize that an applied stress causes an elastic strain,

related to the modulus of elasticity, E, within the material. When a crack propagates,

this strain energy is released, reducing the overall energy. At the same time, however,

Figure 7-4

Schematic diagram of the Griffith flaw in a ceramic.

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior192

two new surfaces are created by the extension of the crack; this increases the energy

associated with the surfa ce. By balancing the strain energy and the surface energy, we

ﬁnd that the critical stress required to propagate the crack is given by the Gri‰th

equation,

critical

G 2s

ﬃﬃﬃﬃﬃﬃ

ð7-4Þ

where a is the length of a surface crack (or one-half the length of an internal crack) and

g is the surface energy (per unit area). Again, this equation shows that even small ﬂaws

severely limit the strength of the ceramic.

We also note that if we rearrange Equation 7-1, which described the stress intensity

factor K, we obtain:

s ¼

ﬃﬃﬃﬃﬃﬃ

ð7-5Þ

This equation is similar in form to Equation 7-4. Each of these equations points out the

dependence of the mechanical properties on the size of ﬂaws present in the ceramic.

Development of manufacturing processes (Chapter 15) to minimize the ﬂaw size be-

comes crucial in improving the strength of ceramics.

The ﬂaws are most important when tensile stresses act on the material. Com-

pressive stresses try to close rather than open a crack; consequently, ceramics often

have very good compressive strengths.

EXAMPLE 7-2 Properties of SiAlON Ceramics

Assume that an advanced ceramic, Sialon (acronym for silicon aluminum oxy-

nitride), has a tensile strength of 414 MPa. Let us assume that this value is for

a ﬂaw-free ceramic. (In practice, it is almost impossible to produce ﬂaw-free

ceramics.) A thin crack 0.025 cm deep is observed before a Sialon part is tested.

The part unexpectedly fails at a stress of 3.5 MPa by propagation of the crack.

Estimate the radius of the crack tip.

SOLUTION

The failure occurred because the 3.5 MPa applied stress, magniﬁed by the

stress concentration at the tip of the crack, produced an actual stress equal to

the tensile strength. From Equation 7-3:

actual

¼ 2s

ﬃﬃﬃﬃﬃﬃﬃ

a=r

414 MPa ¼ð2Þð3:5 MPaÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:025 cm=r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:025=r

¼ 59:1or0:025=r ¼ 3492:8

r ¼ 7:1  10

6

cm ¼ 710 A



The likelihood of our being able to measure a radius of curvature of this size by

any method of nondestructive testing is virtually zero. Therefore, although

Equation 7-3 may help illustrate the factors that inﬂuence how a crack prop-

agates in a brittle material, it does not help in predicting the strength of actual

ceramic parts.

7-2 The Importance of Fracture Mechanics 193

EXAMPLE 7-3 Design of a Ceramic Support

Design a supporting 7.5-cm-wide plate made of Sialon, which has a fracture

toughness of 9.9 MPa

ﬃﬃﬃﬃ

, that will withstand a tensile load of 177,920 N. The

part is to be nondestructively tested to assure that no ﬂaws are present that

might cause failure.

SOLUTION

Let’s assume that we have three nondestructive testing methods available to us:

X-ray radiography can detect ﬂaws larger than 0.05 cm, gamma-ray radiography

can detect ﬂaws larger than 0.02 cm, and ultrasonic inspection can detect ﬂaws

larger than 0.0125 cm. For these ﬂaw sizes, we must now calculate the minimum

thickness of the plate that will assure that ﬂaws of these sizes will not propagate.

From our fracture toughness equation, assuming that f ¼ 1:

max

ﬃﬃﬃﬃﬃﬃ

A ¼

ﬃﬃﬃﬃﬃﬃ

ð177;920 NÞð

ﬃﬃﬃ

Þð

ﬃﬃﬃ

9:9 MPa

ﬃﬃﬃﬃ

A ¼ 0:03

ﬃﬃﬃ

and thickness ¼



0:03 m

0:075 m



ﬃﬃﬃ

¼ 0:4

ﬃﬃﬃ

Smallest Minimum Minimum Maximum

Detectable Area Thickness Stress

NDT Method Crack (cm) (cm

) (cm) (MPa)

X-ray radiography 0.05 6.7 0.89 270

g-ray radiography 0.02 4.24 0.57 420

Ultrasonic inspection 0.0125 3.35 0.45 530

Our ability to detect ﬂaws, coupled with our ability to produce a ceramic

with ﬂaws smaller than our detection limit, signiﬁcantly a¤ects the maximum

stress than can be tolerated and, hence, the size of the part. In this example, the

part can be smaller if ultrasonic inspection is available.

The fracture toughness is also important. Had we used Si

,withafracture

toughness of 3.3 MPa

ﬃﬃﬃﬃ

instead of the Sialon, we could repeat the calculations

and show that, for ultrasonic testing, the minimum thickness is 1.4 cm and the

maximum stress is only 166 MPa.

7-3 Microstructural Features of Fracture in Metallic Materials

Ductile Fracture In metals that have good ductility and toughness, ductile fracture

normally occurs in a transgranular manner (through the grain s). Often, a considerable

amount of deformation—including necking—is observed in the failed component. The

deformation occurs before the ﬁnal fracture. Ductile fractures are usually caused by

simple overloads, or by applying too high a stress to the material.

In a simple tensile test, ductile fracture begins with the nucleation, growth, and

coalescence of microvoids at the center of the test bar. Microvoids form when a high

stress causes separation of the metal at grain boundaries or interfaces between the metal

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior194

and small impurity particles (inclusions). As the local stress increases, the microvoids

grow and coalesce into larger cavities. Eventually, the metal-to-metal contact area is

too small to support the load and fracture occurs.

Deformation by slip also contributes to the ductile fracture of a metal. We know

that slip occurs when the resolved shear stress reaches the critical resolved shear stress

and that the resolved shear stresses are highest at a 45



angle to the applied tensile stress

(Chapter 4, Schmid’s law).

These two aspects of ductile fracture give the failed surface characteristic features.

In thick metal sections, we expect to ﬁnd evidence of necking, with a signiﬁcant portion

of the fracture surface having a ﬂat face where microvoids ﬁrst nucleated and coalesced,

and a small shear lip, where the fracture surface is at a 45



angle to the applied stress.

The shear lip, indicating that slip occurred, gives the fractu re a cup and cone appear-

ance (see Figure 6-8). Simple macroscopic observation of this fracture may be su‰cient

to identify the ductile fracture mode.

Examination of the fracture surface at a high magniﬁcation—perhaps using a

scanning electron micros cope—reveals a dimpled surface. The dimples are traces of the

microvoids produced during fracture. Normally, these microvoids are round, or equi-

axed, when a normal tensile stress produces the failure [Figure 7-5(a)]. However, on the

shear lip, the dimples are oval-shaped, or elongated, with the ovals pointing toward the

origin of the fracture [Figure 7-5(b)].

In a thin plate, less necking is observed and the entire fracture surface may be a

shear face. Microscopic examination of the fracture surface shows elongated dimples

rather than equiaxed dimples, indicating a greater proportion of 45



slip than in thicker

metals.

Figure 7-5 Scanning electron micrographs of an annealed 1018 steel exhibiting ductile

fracture in a tensile test. (a) Equiaxed dimples at the flat center of the cup and cone, and

(b) elongated dimples at the shear lip (1250).

7-3 Microstructural Features of Fracture in Metallic Materials 195

EXAMPLE 7-4 Hoist Chain Failure Analysis

A chain used to hoist heavy loads failed. Examination of the failed link in-

dicates considerable deformation and necking prior to failure. List some of the

possible reasons for failure.

SOLUTION

This description suggests that the chain failed in a ductile manner by a simple

tensile overload. Two factors could be responsible for this failure:

1. The load exceeded the hoisting capacity of the chain. Thus, the stress

due to the load exceeded the yield strength of the chain, permitting failure.

Comparison of the load to the manufacturer’s speciﬁcations will indicate that

the chain was not intended for such a heavy load. This is the fault of the user!

2. The chain materia l was of the wrong composition or was improperly

heat-treated. Consequently, the yield strength was lower than intended by the

manufacturer and could not support the load.

Brittle Fracture Brittle fracture occurs in metals and alloys of high strength or those

with poor ductility and toughness. Furthermore, even metals that are normally ductile

may fail in a brittle manner at low temperatures, in thick sections, at high strain rates

(such as impact), or when ﬂaws play an important role. Brittle fractures are frequently

observed when impact, rather than overload, causes failure.

In brittle fracture, little or no plastic deformation is required. Initiation of the crack

normally occurs at small ﬂaws, which causes a concentra tion of stress. The crack may

move at a rate approaching the velocity of sound in the metal. Normally, the crack

propagates most easily along speciﬁc crystallographic planes, often the f100g planes,

by cleavage. In some cases, however, the crack may take an intergranular (along the

grain boundaries) path, particularly when segregation (preferential separation of di¤er-

ent elements) or inclusions weaken the grain boundaries.

Brittle fracture can be identiﬁed by observing the features on the failed surface.

Normally, the fracture surface is ﬂat and perpendicular to the applied stress in a tensile

test. If failure occurs by cleavage, each fractured grain is ﬂat and di¤erently oriented,

giving a crystalline or ‘‘rock candy’’ appearance to the fracture surface (Figure 7-6).

Figure 7-6

Scanning electron micrograph of a brittle

fracture surface of a quenched 1010 steel

(5000). (Courtesy of C.W. Ramsay.)

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior196

Often, the layman claims that the metal failed because it crystallized. Of course, we

know that the metallic material was crystalline to begin with and the surface appear-

ance is due to the cleavage faces.

Another common fracture feature is the Chevron pattern (Figure 7-7), produced by

separate crack fronts propagating at di¤erent levels in the material. A radiating pattern

of surface markings, or ridges, fans away from the origin of the crack (Figure 7-8). The

Chevron pattern is visible with the naked eye or a magnifying glass and helps us iden-

tify both the brittle nature of the failure process as well as the origin of the failure.

Figure 7-7

The Chevron pattern in a 1.25-cm-

diameter quenched 4340 steel. The

steel failed in a brittle manner by an

impact blow.

Figure 7-8

The Chevron pattern forms as the crack

propagates from the origin at different levels. The

pattern points back to the origin.

EXAMPLE 7-5

Automobile Axle Failure Analysis

An engineer investigating the cause of an automobile accident ﬁnds that the

right rear wheel has broken o¤ at the axle. The axle is bent. The fracture sur-

face reveals a Chevron pattern pointing toward the surface of the axle. Suggest

a possible cause for the fracture.

SOLUTION

The evidence suggests that the axle did not break prior to the accident. The

deformed axle means that the wheel was still attached when the load was

applied. The Chevron pattern indicates that the wheel was subjected to an

intense impact blow, which was transmitted to the axle, causing failure. The

preliminary evidence suggests that the driver lost control and crashed, and the

force of the crash caused the axle to break. Further examination of the fracture

surface, microstructure, composition, and properties may verify that the axle

was manufactured properly.

7-3 Microstructural Features of Fracture in Metallic Materials 197

7-4 Microstructural Features of Fracture in Ceramics, Glasses, and Composites

In ceramic materials, the ionic or covalent bonds permit little or no slip (Chapter 4).

Consequently, failure is a result of brittle fracture. Most crystalline ceramics fail by

cleavage along widely spaced, closely packed planes. The fracture surface typically is

smooth, and frequently no characteristic surface features point to the origin of the

fracture [Figure 7-9(a)].

Glasses also fracture in a brittle manner. Frequently, a conchoidal fracture sur-

face is observed. This surface contains a very smooth mirror zone near the origin of

the fracture, with tear lines comprising the remainder of the surface [Figure 7-9(b)]. The

tear lines point back to the mirror zone and the origin of the crack, much like the

chevron pattern in metals.

Polymers can fail by either a ductile or a brittle mechanism. Below the glass tem-

perature ðT

Þ, thermoplastic polymers fail in a brittle manner—much like a glass.

Likewise, the hard thermoset polymers, whose structure consists of inter-connected long

chains of molecules, fail by a brittle mechanism. Some plastics whose structure consists

of tangled but not chemically cross-linked chains, however, fail in a ductile manner

above the glass temperature, giving evidence of extensive deformation and even necking

prior to failure. The ductile behavior is a result of sliding of the polymer chains, which

is not possible in glassy or thermosetting polymers. Thermosetting polymers have a

rigid, three-dimensional cross-linked structure (Chapter 16).

Fracture in ﬁber-reinforced composite materials is more complex. Typically, these

composites contain strong, brittle ﬁbers surrounded by a soft, ductile matrix, as in

boron-reinforced aluminum. When a tensile stress is applied along the ﬁbers, the soft

aluminum deforms in a ductile manner, with void formation and coalescence eventually

producing a dimpled fracture surface. As the aluminum deforms, the load is no longer

transmitted e¤ectively to the ﬁbers; the ﬁbers break in a brittle manner until there are

too few of them left intact to support the ﬁnal load.

Fracturing is more common if the bonding between the ﬁbers and matrix is poor.

Voids can then form between the ﬁbers and the matrix, causing pull-out. Voids can also

form between layers of the matrix if composite tapes or sheets are not properly bonded,

causing delamination (Figure 7-10). Delamination, in this context, means the layers of

di¤erent materials in a composite begin to come apart.

Figure 7-9 Scanning electron micrographs of fracture surfaces in ceramics. (a) The fracture

surface of Al

, showing the cleavage faces (1250), and (b) the fracture surface of glass,

showing the mirror zone (top) and tear lines characteristic of conchoidal fracture (300).

C HA P T E R 7 Fracture Mechanics, Fatigue, and Creep Behavior198

Figure 7-10

Fiber-reinforced composites can fail by

several mechanisms. (a) Due to weak bonding

between the matrix and fibers, fibers can pull

out of the matrix, creating voids. (b) If the

individual layers of the matrix are poorly

bonded, the matrix may delaminate, creating

voids. (c) Interactions of an advancing crack

with the materials microstructure. (This

article was published in Materials Today,

Vol. 10, No. 9, Sharvan Kumar and William

A. Curtin, ‘‘Crack interaction with

microstructure,’’ pp. 34–43, Copyright

Elsevier (2007).)

7-4 Microstructural Features of Fracture in Ceramics, Glasses, and Composites 199

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