failure - university of sheffieldashleycadby.staff.shef.ac.uk/materials/resources/failure.pdf ·...

$: Failure - University of Sheffieldashleycadby.staff.shef.ac.uk/Materials/Resources/Failure.pdf · ics of the various failure modes—i.e., fracture, fa- ... Ductile fracture is almost$
Failure

C h a p t e r 9 / Failure

An oil tanker that fractured in a brittle manner by crack propagation around its girth. (Pho-

tography by Neal Boenzi. Reprinted with permission from The New York Times.)

Why Study Failure?

The design of a component or structure often calls

upon the engineer to minimize the possibility of fail-

ure. Thus, it is important to understand the mechan-

ics of the various failure modes—i.e., fracture, fa-

tigue, and creep—and, in addition, be familiar with

appropriate design principles that may be employed

to prevent in-service failures. !For example, we dis-

cuss in Section 20.5 material selection and pro-

cessing issues relating to the fatigue of an automo-

bile valve spring."

234

of percent elongation (Equation 7.11) and percent reduction in area (Equation7.12). Furthermore, ductility is a function of temperature of the material, the strainrate, and the stress state. The disposition of normally ductile materials to fail in abrittle manner is discussed in Section 9.8.

Any fracture process involves two steps—crack formation and propagation—inresponse to an imposed stress. The mode of fracture is highly dependent on themechanism of crack propagation. Ductile fracture is characterized by extensiveplastic deformation in the vicinity of an advancing crack. Furthermore, the processproceeds relatively slowly as the crack length is extended. Such a crack is oftensaid to be stable. That is, it resists any further extension unless there is an increasein the applied stress. In addition, there will ordinarily be evidence of appreciablegross deformation at the fracture surfaces (e.g., twisting and tearing). On the otherhand, for brittle fracture, cracks may spread extremely rapidly, with very littleaccompanying plastic deformation. Such cracks may be said to be unstable, andcrack propagation, once started, will continue spontaneously without an increasein magnitude of the applied stress.

Ductile fracture is almost always preferred for two reasons. First, brittle fractureoccurs suddenly and catastrophically without any warning; this is a consequence ofthe spontaneous and rapid crack propagation. On the other hand, for ductile frac-ture, the presence of plastic deformation gives warning that fracture is imminent,allowing preventive measures to be taken. Second, more strain energy is requiredto induce ductile fracture inasmuch as ductilematerials are generally tougher.Underthe action of an applied tensile stress,most metal alloys are ductile, whereas ceramicsare notably brittle, and polymers may exhibit both types of fracture.

9.3 DUCTILE FRACTURE

Ductile fracture surfaces will have theirown distinctive features on bothmacroscopicand microscopic levels. Figure 9.1 shows schematic representations for two charac-teristic macroscopic fracture profiles. The configuration shown in Figure 9.1a isfound for extremely soft metals, such as pure gold and lead at room temperature,and other metals, polymers, and inorganic glasses at elevated temperatures. Thesehighly ductile materials neck down to a point fracture, showing virtually 100%reduction in area.

The most common type of tensile fracture profile for ductile metals is thatrepresented in Figure 9.1b, which fracture is preceded by only a moderate amount

236 ● Chapter 9 / Failure

(a) (b) (c)

FIGURE 9.1 (a) Highly ductile fracture inwhich the specimen necks down to a point.(b) Moderately ductile fracture after somenecking. (c) Brittle fracture without anyplastic deformation.

A is highly ductile B is moderately ductile C is Brittle

Ductile Fracture9.3 Ductile Fracture ● 237

of necking. The fracture process normally occurs in several stages (Figure 9.2).First, after necking begins, small cavities, or microvoids, form in the interior of thecross section, as indicated in Figure 9.2b. Next, as deformation continues, thesemicrovoids enlarge, come together, and coalesce to form an elliptical crack, whichhas its long axis perpendicular to the stress direction. The crack continues to growin a direction parallel to its major axis by this microvoid coalescence process (Figure9.2c). Finally, fracture ensues by the rapid propagation of a crack around the outerperimeter of the neck (Figure 9.2d), by shear deformation at an angle of about 45!with the tensile axis—this is the angle at which the shear stress is a maximum.Sometimes a fracture having this characteristic surface contour is termed a cup-and-cone fracture because one of the mating surfaces is in the form of a cup, theother like a cone. In this type of fractured specimen (Figure 9.3a), the central

(a) (b) (c)

(d) (e)

FibrousShear

FIGURE 9.2 Stages in the cup-and-conefracture. (a) Initial necking. (b) Smallcavity formation. (c) Coalescence ofcavities to form a crack. (d) Crackpropagation. (e) Final shear fracture at a45! angle relative to the tensile direction.(From K. M. Ralls, T. H. Courtney, andJ. Wulff, Introduction to Materials Scienceand Engineering, p. 468. Copyright 1976 by John Wiley & Sons, New York.Reprinted by permission of John Wiley &Sons, Inc.)

FIGURE 9.3 (a) Cup-and-cone fracture in aluminum. (b) Brittle fracture in amild steel.

(a) (b)

a.) Initial Necking b.) Small cavity formation c.) Coalescence of cavities to form crack d.) Crack propagation e.) Final shear fracture at a 45 degree angle

Scanning Electron Micrograph of Ductile Fracture Surface

Brittle Fracture

9.5b Principles of Fracture Mechanics (Concise Version) ● 239

STRESS CONCENTRATION

The measured fracture strengths for most brittle materials are significantly lowerthan those predicted by theoretical calculations based on atomic bonding energies.This discrepancy is explained by the presence of very small, microscopic flaws orcracks that always exist under normal conditions at the surface and within theinterior of a body of material. These flaws are a detriment to the fracture strengthbecause an applied stressmay be amplified or concentrated at the tip, the magnitude

(a)

(b)

FIGURE 9.5 (a) Photograph showing V-shaped ‘‘chevron’’ markings characteristicof brittle fracture. Arrows indicate origin of crack. Approximately actual size.(From R. W. Hertzberg, Deformation and Fracture Mechanics of EngineeringMaterials, 3rd edition. Copyright 1989 by John Wiley & Sons, New York.Reprinted by permission of John Wiley & Sons, Inc. Photograph courtesy ofRoger Slutter, Lehigh University.) (b) Photograph of a brittle fracture surfaceshowing radial fan-shaped ridges. Arrow indicates origin of crack. Approximately2!. (Reproduced with permission from D. J. Wulpi, Understanding HowComponents Fail, American Society for Metals, Materials Park, OH, 1985.)

314 • Chapter 9 / Failure

Brittle fracture of normally ductile materials, such as that shown in the chapter-opening Figure b (the oil barge), has demonstrated the need for a better understand-ing of the mechanisms of fracture. Extensive research endeavors over the lastcentury have led to the evolution of the field of fracture mechanics. This allowsquantification of the relationships between material properties, stress level, the pres-ence of crack-producing flaws, and crack propagation mechanisms. Design engineersare now better equipped to anticipate, and thus prevent, structural failures. The pres-ent discussion centers on some of the fundamental principles of the mechanics offracture.

9.5 PRINCIPLES OF FRACTURE MECHANICS1

1A more detailed discussion of the principles of fracture mechanics may be found in Section M.4 of the MechanicalEngineering Online Module.

fracture mechanics

intergranularfracture

(a) (b)

In some alloys, crack propagation is along grain boundaries (Figure 9.7a); this frac-ture is termed intergranular. Figure 9.7b is a scanning electron micrograph showing atypical intergranular fracture, in which the three-dimensional nature of the grains maybe seen.This type of fracture normally results subsequent to the occurrence of processesthat weaken or embrittle grain boundary regions.

Grains

SEM Micrograph

Path of crack propagation

Figure 9.6 (a) Schematic cross-section profile showing crack propagation through the interior of grains fortransgranular fracture. (b) Scanning electron fractograph of ductile cast iron showing a transgranular fracturesurface. Magnification unknown.[Figure (b) from V. J. Colangelo and F. A. Heiser, Analysis of Metallurgical Failures, 2nd edition. Copyright © 1987 by John Wiley& Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.]

c09Failure.qxd 6/18/11 4:53 AM Page 314

of the length of an internal crack. For a relatively long microcrack that has a small tipradius of curvature, the factor (a/!t)1/2 may be very large.This will yield a value of "m thatis many times the value of "0.

Sometimes the ratio "m/"0 is denoted the stress concentration factor Kt:

(9.2)

which is simply a measure of the degree to which an external stress is amplified at thetip of a crack.

Note that stress amplification is not restricted to these microscopic defects; it mayoccur at macroscopic internal discontinuities (e.g., voids or inclusions), sharp corners,scratches, and notches.

Furthermore, the effect of a stress raiser is more significant in brittle than in ductilematerials. For a ductile metal, plastic deformation ensues when the maximum stressexceeds the yield strength. This leads to a more uniform distribution of stress in thevicinity of the stress raiser and to the development of a maximum stress concentrationfactor less than the theoretical value. Such yielding and stress redistribution do not occurto any appreciable extent around flaws and discontinuities in brittle materials; therefore,essentially the theoretical stress concentration will result.

Using principles of fracture mechanics, it is possible to show that the critical stress "c

required for crack propagation in a brittle material is described by the expression

(9.3)

where E is the modulus of elasticity, #s is the specific surface energy, and a is one-half thelength of an internal crack.

sc ! a 2Egs

p ab 1/2

Kt !sm

s0! 2 a a

rtb 1/2

316 • Chapter 9 / Failure

(a)

Figure 9.8 (a) The geometry of surface and internal cracks. (b) Schematic stress profile along the line X–X" in(a), demonstrating stress amplification at crack tip positions.

!t

"0

"0

X'X

x'x2a

a

Position along X–X'

"0

"m

Stre

ss

x'x

(b)

Critical stress forcrack propagation ina brittle material


A relatively large plate of a glass is subjected to a tensile stress of 40 MPa. If the specific surface energy and modulus of elasticity for this glass are 0.3 J/m2 and 69 GPa, respectively, determine the maximum length of a surface flaw that is possible without fracture.

Fracture Toughness

Plane strain fracture toughness318 • Chapter 9 / Failure

Figure 9.10 The threemodes of crack surfacedisplacement. (a) Mode I,opening or tensile mode;(b) mode II, sliding mode; and(c) mode III, tearing mode.

(a) (b) (c)

Figure 9.9 Schematicrepresentations of (a) aninterior crack in a plate ofinfinite width, and (b) anedge crack in a plate ofsemi-infinite width.

2a a

2Two other crack displacement modes, denoted II and III and illustrated in Figures 9.10b and 9.10c, are also possible;however, mode I is most commonly encountered.

(a) (b)

Plane strain fracturetoughness for mode Icrack surfacedisplacement

represented in Figure 9.9a, there is no strain component perpendicular to the front andback faces. The Kc value for this thick-specimen situation is known as the plane strainfracture toughness, KIc; it is also defined by

(9.5)

KIc is the fracture toughness cited for most situations. The I (i.e., Roman numeral “one”)subscript for KIc denotes that the plane strain fracture toughness is for mode I crack dis-placement, as illustrated in Figure 9.10a.2

Brittle materials, for which appreciable plastic deformation is not possible in frontof an advancing crack, have low KIc values and are vulnerable to catastrophic failure. Onthe other hand, KIc values are relatively large for ductile materials. Fracture mechanicsis especially useful in predicting catastrophic failure in materials having intermediateductilities. Plane strain fracture toughness values for a number of different materials arepresented in Table 9.1 (and Figure 1.6); a more extensive list of KIc values is given inTable B.5, Appendix B.

The plane strain fracture toughness KIc is a fundamental material property thatdepends on many factors, the most influential of which are temperature, strain rate, andmicrostructure. The magnitude of KIc decreases with increasing strain rate and decreasing

KIc ! Ys1p a

plane strain fracturetoughness


tests, on the other hand, are more qualitative and are of little use for designpurposes. Impact energies are of interest mainly in a relative sense and for makingcomparisons—absolute values are of little significance. Attempts have been madeto correlate plane strain fracture toughnesses and CVN energies, with only limitedsuccess. Plane strain fracture toughness tests are not as simple to perform as impacttests; furthermore, equipment and specimens are more expensive.


FIGURE 9.19 (a)Specimen used for

Charpy and Izodimpact tests. (b) A

schematic drawing ofan impact testing

apparatus. The hammeris released from fixed

height h and strikes thespecimen; the energy

expended in fracture isreflected in the

difference between hand the swing height h!.

Specimen placementsfor both Charpy and

Izod tests are alsoshown. (Figure (b)

adapted from H. W.Hayden, W. G. Moffatt,

and J. Wulff, TheStructure and Properties

of Materials, Vol. III,Mechanical Behavior, p.

13. Copyright 1965by John Wiley & Sons,New York. Reprintedby permission of John

Wiley & Sons, Inc.)

Cyclic Stress

Also indicated in Figure 9.23b are several parameters used to characterize thefluctuating stress cycle. The stress amplitude alternates about a mean stress !m ,defined as the average of the maximum and minimum stresses in the cycle, or

!m "!max # !min

2(9.21)

Furthermore, the range of stress !r is just the difference between !max and !min ,namely,

!r " !max $ !min (9.22)


0

!min

!max

Time

+

!

Stre

ssTe

nsio

nCo

mpr

essi

on

(a)

0

!min

!max

Time

+

!

Stre

ssTe

nsio

nCo

mpr

essi

on

(b)

!m

!a

!r

Time

+

!

Stre

ssTe

nsio

nCo

mpr

essi

on

(c)

FIGURE 9.23 Variationof stress with time thataccounts for fatiguefailures. (a) Reversedstress cycle, in which thestress alternates from amaximum tensile stress(#) to a maximumcompressive stress ($) ofequal magnitude. (b)Repeated stress cycle, inwhich maximum andminimum stresses areasymmetrical relative tothe zero stress level;mean stress !m , range ofstress !r , and stressamplitude !a areindicated. (c) Randomstress cycle.

Two distinct types of S–N behavior are observed, which are representedschematically in Figure 9.25. As these plots indicate, the higher the magnitudeof the stress, the smaller the number of cycles the material is capable of sustainingbefore failure. For some ferrous (iron base) and titanium alloys, the S–N curve(Figure 9.25a) becomes horizontal at higher N values; or, there is a limitingstress level, called the fatigue limit (also sometimes the endurance limit), belowwhich fatigue failure will not occur. This fatigue limit represents the largestvalue of fluctuating stress that will not cause failure for essentially an infinitenumber of cycles. For many steels, fatigue limits range between 35 and 60% ofthe tensile strength.

Most nonferrous alloys (e.g., aluminum, copper, magnesium) do not have afatigue limit, in that the S–N curve continues its downward trend at increasinglygreater N values (Figure 9.25b). Thus, fatigue will ultimately occur regardless ofthe magnitude of the stress. For these materials, the fatigue response is specifiedas fatigue strength, which is defined as the stress level at which failure will occur


Cycles to failure, N(logarithmic scale)

Stre

ss a

mpl

itude

, S

103 104 105 106 107 108 109 1010

Fatiguelimit

(a)

Cycles to failure, N(logarithmic scale)

Fatigue lifeat stress S1

Stre

ss a

mpl

itude

, S

103 104 107 N1 108 109 1010

Fatigue strengthat N1 cycles

S1

(b)

FIGURE 9.25 Stressamplitude (S) versus

logarithm of thenumber of cycles to

fatigue failure (N) for(a) a material that

displays a fatigue limit,and (b) a material that

does not display afatigue limit.

Crack Propogation

firms that the cause of failure was fatigue. Nevertheless, the absence of either orboth does not exclude fatigue as the cause of failure.

One final comment regarding fatigue failure surfaces:Beachmarks and striationswill not appear on that region over which the rapid failure occurs. Rather, the rapidfailure may be either ductile or brittle; evidence of plastic deformation will bepresent for ductile, and absent for brittle, failure. This region of failure may benoted in Figure 9.32.


FIGURE 9.31 Transmissionelectron fractograph showingfatigue striations inaluminum. Magnificationunknown. (From V. J.Colangelo and F. A. Heiser,Analysis of MetallurgicalFailures, 2nd edition.Copyright 1987 by JohnWiley & Sons, New York.Reprinted by permission ofJohn Wiley & Sons, Inc.)

FIGURE 9.32 Fatiguefailure surface. A crackformed at the top edge.The smooth region alsonear the top correspondsto the area over which thecrack propagated slowly.Rapid failure occurredover the area having adull and fibrous texture(the largest area).Approximately 0.5 !.(Reproduced bypermission from MetalsHandbook: Fractographyand Atlas of Fractographs,Vol. 9, 8th edition, H. E.Boyer, Editor, AmericanSociety for Metals, 1974.)

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