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Microstructure-Properties: IMicrostructure-Properties: ILecture 6B:Lecture 6B:

Fracture Toughness:Fracture Toughness:how to use it, and measure ithow to use it, and measure it

27-301Fall, 2007

Prof. A. D. Rollett

MicrostructureProperties

ProcessingPerformance

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Stress vsGriffith

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ObjectiveObjective• The objective of this lecture is to build upon

the basic concepts of fracture toughnessand stress intensity introduced in part A.Realistic approaches to fracture toughnessare considered with information on how tomeasure toughness.

• Part of the motivation for this lecture is toprepare the class for a Lab on the sensitivityof mechanical properties to microstructure.

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Key PointsKey Points• The Griffith equation applies to technological materials.• Toughness scales with modulus, as does strength.• Toughness is highly dependent on material type: the most

important issue is the presence (toughness) or absence(brittleness) of plasticity.

• Plasticity makes a large contribution to the energy absorbed incrack propagation because plastic deformation at the crack tipblunts the tip (lower stress concentration) and substantiallyincreases the amount of work required per unit crack advance.

• Measurement of toughness uses many methods: two basicmethods measure critical stress intensity in plane strain, KIC,and the energy absorbed in impact (Charpy Test).

• Fractography, i.e. classification+quantification of the fracturesurfaces, is useful as a microstructural diagnostic fortoughness, in addition to the quantitative measures ofmechanical behavior.

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Toughnesses Toughnesses in Materialsin Materials• Before looking at the influence of microstructure on fracture

toughness, it is useful to review the range of toughnessesobserved in real materials.

• We find that to a first (crude!) approximation, toughnessscales with strength.

• An immediate refinement is to consider the bonding type inthe various classes of materials: metals tend to have simplerstructures with easier dislocation motion, i.e. more energyabsorbed in crack propagation. Ceramics have covalent orionic bonding with much higher resistance to dislocationmotion, especially at ambient conditions.

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Map ofMap oftoughness toughness vsvs..

strengthstrength

[Ashby]

Glass-likebrittleness

Tough

Design with carebelow this line!

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Measure-ment

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Use of the Griffith equationUse of the Griffith equation• The Griffith equation can be applied immediately to practical

problems.• Problem: estimate the strength of a brittle material (meaning

that we can ignore plastic yield) with properties,E = 100 GPa, γ = 1 J.m-2, and a crack length of 2.5 µm.The answer is σbreak = √(2Eγ/πc) = √(2.1011.1/π/2.5.10-6)= 160 MPa

• Now it is instructive to compare this result with that from thestress concentration equation, with the crack tip radius setequal to, say, 8a0:σbreak = √(Eγρ/4a0c) = √(Eγ8a0/4a0c) = √(2Eγ/c)

√(2.1011.1/2.5.10-6)= 283 MPa• So, we see that, even for a fairly sharp crack, the Griffith

(energy balance) equation sets the lower limit on fracturestrength.

!break

=2"E

#c

!break

="E#

4a0c

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Which equation controls?Which equation controls?

0

1

2

3

4

5

0 5 10 15 20

Fra

ctu

re S

tre

ng

th (

arb

itra

ry u

nits)

Tip Radius (multiples of a0)

!fr=!(2E"/"c)

(Griffith)

!fr=!(E"#/4a

0c)

(Stress concentration)Griffith eq.controls

Stress concentration equation controls

The paradox: although the Griffith equation (black line) appears to be anecessary but not sufficient condition for fracture because one also needs forthe stress at the crack tip to exceed the breaking stress (the red line), as amatter of practical experience, it does successfully predict when fracture willactually occur.

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Application to structural materialsApplication to structural materials• Notwithstanding the previous slides on energy balance

(Griffith) versus stress concentration, the experimental fact isthat the Griffith equation works well for many differentmaterials.

• It works well, not in its literal form with the surface energydetermining the energy consumed, but with an additionalenergy term that accounts for the effect of plasticity (crackbridging, phase transformation….). This was one of Orowan’s(many) contributions to the field.

!break =2(" surface +" plastic)E

#c

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ToughnessToughness• Recall that we define a stress intensity as K=σ√c.• Cracking is defined by K > Kc, where Kc is a critical stress

intensity or fracture toughness, and is a material property.

σbreak = Kc/√(πc)

• We can also define a toughness, Gc, which is given by

σbreak = √(EGc/πc)

and allows us to modify (increase) the apparent surface energyto account for plastic work at the crack tip.

• The toughness can be thought of as the combination ofsurface energy and plastic work done at the crack tip noted onthe previous slide: Gc = γsurface + γplastic

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Effect of plasticityEffect of plasticity• How important is the additional term?• In metals, very important: compared to typical

surface energies between 0.5 and 2 J.m-2, the plasticwork term ranges up to 103 J.m-2 . Therefore thesurface energy term can be neglected in most metalalloys.

• Again, we cannot use the Griffith equation in its basicform, even with the addition of the plastic work,however.

• The plasticity results in a plastic zone immediately infront of the crack tip. This is the zone within whichsignificant yielding has occurred. Remember that thestress concentration leads to locally higher stressesand so, only in the vicinity of the crack will the yieldstress be exceeded.

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Plastic ZonePlastic Zone• The plastic zone is a simple concept to visualize. Within a certain radius of

the crack tip, the yield stress is exceededand the materialhas deformed(consuming energythereby andcontributing totoughness). Clearlythe lower the yieldstrength, the largerthe plastic zone, rp.Actually the sizedepends on theratio of the appliedstress, σ, to theyield stress, σy :rp ∝ σ/σy

[Dowling]

rp

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Crack TipCrack Tip

[McClintock, Argon]

Different length scales at whichto view a crack tip

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Effective crack lengthEffective crack length• An important but slightly counter-intuitive idea is that the effective

crack length is longer than the actual value as a result of the plasticzone, i.e. ceffective

= cactual + rp.• Size of the plastic zone?

rp = K2/2πσ02 = σ2c/2σ0

2 ≡ σ2c/2σ2yield.

• Substituting this relationship into the standard Griffith equation, weobtain:

σbreak = Kc/√(πc),as σf ≡ σbreak = Kc/√(π{c+rp}) = Kc/√(πc{1 + Kc

2/2cπσ02}),

σ2 {1 + Kc

2/2cπσ02} = Kc

2/(πc), σ2 = Kc

2(1/(πc) - σ2 /2cπσ02},

πcσ2 = Kc2(1 - σ2 /2σ0

2),and re-arrange so that we obtain the following modified form:

!

Keffective

=" f #c

1$1

2

"

" yield

%

& ' '

(

) * *

2σ0 ≡ σyield

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Effective crack length, Effective crack length, contdcontd..• This second version is an empirical generalization of the first one: σf

is the fracture strength, σ is the operating stress in the material, andσyield is the yield stress of the material. KIc is the plane strain fracturetoughness (critical stress intensity). A, B and α are constants thatdepend on crack geometry (of order unity). In the next slides, B iswritten as a function of c/a, the ratio of the (elliptical) crack(semi-)length, a, to its depth, c.

• One can either calculate a fracture strength for a given set ofparameters, calculate a maximum operating stress similarly, or,determine whether the fracture toughness dictated by the quantitieson the RHS is higher than the actual fracture toughness of thematerial.

!

KIc =" f #$c

B % A"

" yield

&

' ( (

)

* + +

2

," f = KIc

B % A"

" yield

&

' ( (

)

* + +

2

#$c

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Example

Measure-ment

Fracto-graphy

ExampleExampleproblemproblem

[Courtney, p431]

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Measuring Fracture ToughnessMeasuring Fracture Toughness• How do we measure fracture toughness?• Two examples:

A - measure the critical stress intensity (KIc) in planestrain by measuring the stress required to propagatea sharp crack.B - measure the energy absorbed in a rapid fractureof a bar - the Charpy test.

• The first method measures a quantity correspondingto the values in the equations discussed (but a pre-existing crack is used).

• The second test is a more macroscopic test but itincludes the effect of crack nucleation (which may bedifficult enough to raise the effective toughness).

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Compact Tension testCompact Tension test• The load is increased until crack propagation starts: for a large enough

specimen, the stress intensity at this point is the critical stress intensity, KIC. Pis the load, t is the specimen thickness, b is the distance from the loadingpoint to the right-hand face, and Fp is a function of the crack geometry.

[Dowling]Fatigue crack;grownbeforefractureexpt.

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Charpy TestCharpy Test• The Charpy test uses a

square bar with a smallnotch in it.

• The further the pendulumswings after breaking thespecimen, the lessenergy was absorbed inthe impact, and viceversa.

• Higher toughness resultsin higher energyabsorbed.

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FractographyFractography• Fractography is the practice of characterizing

fracture surfaces.• Surface preparation is not needed - one needs to

examine the surfaces as fractured, which meansthat it should be done promptly so as to avoidchanges from oxidation, corrosion etc.

• The rough, irregular nature of fracture surfacesmeans that optical microscopy is of little use.

• Scanning electron microscopy is most useful infractography.

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Sample scaleSample scale• Example of high strength steel from a compact

tension test.

Crack tip

Crackpropagation

ShearLips

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Grain scaleGrain scale• These micrographs contrast the appearance of

ductile and brittle fractures at the microstructuralscale.

[Dowling]

Brittle (cleavage)

Ductile (tearing)

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Ductile fractureDuctile fracture

• In contrast to brittlefracture, which is acleavage process(and, in crystallinematerials typicallyfollows low indexplanes), ductilefracture only occursafter much plasticdeformation.

Cup and cone fracture - eachdimple is a void (which may ormay not have a particle in it)

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Summary (part B)Summary (part B)• The Griffith equation has been extended to technological

materials.• Toughness scales with modulus, as does strength.• Toughness is highly dependent on material type: the most

important issue is the presence (toughness) or absence(brittleness) of plasticity.

• Plasticity makes a large contribution to the energy absorbed incrack propagation.

• Measurement methods contrasted between KIC and impacttesting (Charpy).

• Fractography introduced as a diagnostic for toughness, inaddition to the quantitative measures.

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Case Study:Case Study:Failure Analysis of a Rocket Motor CaseFailure Analysis of a Rocket Motor Case

A rocket motor case was made of a material that had a yield strength of 215 ksi (=1485 MPa) and a KIC of 53 ksi(in)1/2 (= 58 MPa.m3/2) and it failed at a stress of150 ksi. Examination of the failed component showed that there was anelliptical surface crack with a depth of 0.039 inches (= 0.99 mm) and a lengthof 1.72 in (= 43.7 mm). Could this flaw have been responsible for the failure?

Answer:The value of f(c/a) (=B) for this flaw is 1.38. Rearranging the equation that relates

fracture toughness to yield strength and operating stress, we obtain:

Now we estimate the fracture stress iteratively by substituting values of KIC andthe crack depth, c, (not the half-length!) and assume the operating stressvalue, σ, of 150 ksi, in order to estimate the RHS; then we compare the valueon the RHS with the known fracture stress on the LHS. The answer turns outto be 156 ksi, which is not far off the actual fracture stress of 150 ksi.Substituting 156 ksi as the operating stress value, σ, into the RHS produces156 ksi as the computed fracture stress. At this point the iteration hasconverged well enough for our purposes. The close agreement between theactual and the computed fracture stresses suggests that the flaw was verylikely to have been the cause of the failure.

Source: Courtney: Mechanical Behavior of Materials, Ch. 9.

!

" fracture =f c a( ) # 0.212" " y( )

2

1.20$cK IC =

1.38 # 0.212" " y( )2

1.20$cK IC

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ReferencesReferences• Materials Principles & Practice, Butterworth Heinemann,

Edited by C. Newey & G. Weaver.• G.E. Dieter, Mechanical Metallurgy, McGrawHill, 3rd Ed.• Courtney, T. H. (2000). Mechanical Behavior of Materials.

Boston, McGraw-Hill.• R.W. Hertzberg (1976), Deformation and Fracture Mechanics

of Engineering Materials, Wiley.• N.E. Dowling (1998), Mechanical Behavior of Materials,

Prentice Hall.• D.J. Green (1998). An Introduction to the Mechanical

Properties of Ceramics, Cambridge Univ. Press, NY.• A.H. Cottrell (1964), The Mechanical Properties of Matter,

Wiley, NY.