l6b 24oct07 fracture

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Microstructure-Properties: I Microstructure-Properties: I

Lecture 6B: Lecture 6B: Fracture Toughness: Fracture Toughness:

how to use it, and measure it how to use it, and measure it

27-301

Fall, 2007

Prof. A. D. Rollett

MicrostructureProperties

ProcessingPerformance



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ObjectiveObjective

• The objective of this lecture is to build upon

the basic concepts of fracture toughness

and stress intensity introduced in part A.

Realistic approaches to fracture toughnessare considered with information on how to

measure toughness.

• Part of the motivation for this lecture is to

prepare the class for a Lab on the sensitivityof mechanical properties to microstructure.



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Key Points Key Points

• The Griffith equation applies to technological materials.

• Toughness scales with modulus, as does strength.

• Toughness is highly dependent on material type: the mostimportant 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, K IC ,

and the energy absorbed in impact (Charpy Test).• Fractography , i.e. classification+quantification of the fracture

surfaces, is useful as a microstructural diagnostic for toughness, in addition to the quantitative measures of mechanical behavior.



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ToughnessesToughnesses in Materialsin Materials

• Before looking at the influence of microstructure on fracture

toughness, it is useful to review the range of toughnesses

observed in real materials.

• We find that to a first (crude!) approximation, toughness

scales with strength.

• An immediate refinement is to consider the bonding type in

the various classes of materials: metals tend to have simpler

structures with easier dislocation motion, i.e. more energy

absorbed in crack propagation. Ceramics have covalent or

ionic bonding with much higher resistance to dislocation

motion, especially at ambient conditions.



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Use of the Griffith equationUse of the Griffith equation

• The Griffith equation can be applied immediately to practicalproblems.

• Problem: estimate the strength of a brittle material (meaningthat 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 σ break = √ (2E γ / 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:σ

break

= √ (E γρ /4a0

c) = √ (E γ 8a0

/4a0

c) = √ (2E γ /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

F r a c t u r e

S t r e n g

t h

( a r b i t r a r y

u n i t s )

Tip Radius (multiples of a0)

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

(Griffith)

!fr= ! (E"#/4a0c)(Stress concentration)

Griffith eq.controls

Stress concentrationequation controls

The paradox: although the Griffith equation (black line) appears to be a

necessary but not sufficient condition for fracture because one also needs for

the stress at the crack tip to exceed the breaking stress (the red line), as a

matter of practical experience, it does successfully predict when fracture will

actually occur.



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Application to structural materials Application to structural materials

• Notwithstanding the previous slides on energy balance

(Griffith) versus stress concentration, the experimental fact is

that the Griffith equation works well for many different

materials.

• It works well, not in its literal form with the surface energy

determining the energy consumed, but with an additional energy term that accounts for the effect of plasticity (crack

bridging, 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 > K c, where K c is a critical stress

intensity or fracture toughness, and is a material property .

σ break = K c / √ ( πc)

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

σ break = √ (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 of

surface energy and plastic work done at the crack tip noted onthe previous slide: Gc = γ surface + γ plastic



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Effect of plasticity Effect of plasticity

• How important is the additional term?

• In metals, very important: compared to typicalsurface energies between 0.5 and 2 J.m-2, the plasticwork term ranges up to 103 J.m-2 . Therefore the

surface 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 Zone Plastic Zone

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

the crack tip, the yield stress is exceeded

and the material

has deformed

(consuming energy

thereby and

contributing to

toughness). Clearly

the lower the yield

strength, the larger

the plastic zone, r p.

Actually the size

depends on theratio of the applied

stress, σ, to the

yield stress, σy :

r p ∝ σ/σy

[Dowling]

rp



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Crack TipCrack Tip

[McClintock, Argon]

Different length

scales at which

to view a crack tip



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Effective crack length Effective crack length

• An important but slightly counter-intuitive idea is that the effectivecrack length is longer than the actual value as a result of the plasticzone, i.e. ceffective

= cactual + rp.

• Size of the plastic zone? rp = K 2 /2 πσ 0

2 = σ 2 c/2σ 0 2 ≡ σ 2 c/2σ 2

yield .

• Substituting this relationship into the standard Griffith equation, we

obtain: σ break = K c / √ ( πc),as σ f ≡ σ break = K c / √ ( π {c+r p }) = K c / √ ( πc{1 + K c

2 /2c πσ 0 2 }),

σ 2 {1 + K c2 /2c πσ 0

2 } = K c2 /( πc),

σ 2 = K c2(1/( πc) - σ 2 /2c πσ 0

2 } , πcσ 2 = K c

2(1 - σ 2 /2 σ 02),

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

K effective

=

" f #c

1$1

2

"

" yield

%

& ' '

(

) * *

2σ 0 ≡ σ yield



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Effective crack length, Effective crack length, contd contd ..

• This second version is an empirical generalization of the first one: σ f is the fracture strength, σ is the operating stress in the material, andσ yield is the yield stress of the material. K Ic 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 of

parameters, 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.

K Ic =" f #$ c

B % A"

" yield

&

' ( (

)

* + +

2," f =K Ic

B % A"

" yield

&

' ( (

)

* + +

2

#$ c



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Example Example

problem problem

[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 (K Ic ) in plane

strain by measuring the stress required to propagate

a sharp crack.B - measure the energy absorbed in a rapid fracture

of a bar - the Charpy test.

• The first method measures a quantity corresponding

to the values in the equations discussed (but a pre-

existing crack is used).• The second test is a more macroscopic test but it

includes the effect of crack nucleation (which may be

difficult enough to raise the effective toughness).



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Compact Tension test Compact 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, K IC . P is the load, t is the specimen thickness, b is the distance from the loading

point to the right-hand face, and F p is a function of the crack geometry.

[Dowling]Fatigue

crack;

grown

before

fracture

expt.



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Charpy Test Charpy Test

• The Charpy test uses a

square bar with a small

notch in it.

• The further the pendulum

swings after breaking thespecimen, the less

energy was absorbed in

the impact, and vice

versa.

• Higher toughness results

in higher energy

absorbed.



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Fractography Fractography

• Fractography is the practice of characterizing

fracture surfaces.

• Surface preparation is not needed - one needs to

examine the surfaces as fractured, which means

that it should be done promptly so as to avoidchanges from oxidation, corrosion etc.

• The rough, irregular nature of fracture surfaces

means 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

Shear

Lips



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Grain scaleGrain scale

• These micrographs contrast the appearance of

ductile and brittle fractures at the microstructural

scale.

[Dowling]

Brittle (cleavage)

Ductile (tearing)



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Ductile fracture Ductile fracture

• In contrast to brittle

fracture, which is a

cleavage process

(and, in crystalline

materials typicallyfollows low index

planes), ductile

fracture only occurs

after much plastic

deformation.

Cup and cone fracture - each dimple is a void (which may or

may 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 in

crack propagation.

• Measurement methods contrasted between K IC and impact

testing (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 Case Failure Analysis of a Rocket Motor CaseA 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 of 150 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 relatesfracture toughness to yield strength and operating stress, we obtain:

Now we estimate the fracture stress iteratively by substituting values of K IC andthe crack depth, c , (not the half-length!) and assume the operating stress

value, 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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References References

• 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 Mechanicsof Engineering Materials, Wiley.

• N.E. Dowling (1998), Mechanical Behavior of Materials,Prentice Hall.

• D.J. Green (1998). An Introduction to the MechanicalProperties of Ceramics, Cambridge Univ. Press, NY.

• A.H. Cottrell (1964), The Mechanical Properties of Matter,Wiley, NY.

l6b 24oct07 fracture

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