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Microstructure-Properties: IMicrostructure-Properties: ILecture 6A: FractureLecture 6A: Fracture

27-301Fall, 2007

Prof. A. D. Rollett

Microstructure Properties

Processing Performance

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ObjectiveObjective• The objective of this lecture is to relate the fracture resistance

of materials to their microstructure.• Both ceramics and metals exhibit strongly microstructure

dependent fracture resistance.• This section focuses on basic theory of brittle fracture and the

Griffith equation in particular:σbreak = Kc/√(πc).

• Part of the motivation for this lecture is to provide backgroundfor the 2nd Lab on the sensitivity of mechanical properties tomicrostructure. The fracture resistance of steels, for example,is very sensitive to the microstructure.

• The material property of interest is fracture toughness, whichhas dimensions of stress intensity and relates breakingstrength to flaw/crack size.

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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.• 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.

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RelevanceRelevanceFracture is more important than strength from an engineeringpoint of view! When we build something, we want it to resist theloads that we put on it and not break. Of course, objects cancome apart in way that is not necessarily disastrous just throughregular wear and tear. By fracture, we generally meanunanticipated (worse, unpredictable) breakage. Metallurgists arefond of quoting the Liberty ship experience because of itshistorical significance. Just consider the recent exposé of theFirestone tire problem, however, for a contemporary example ofthe seriousness of fracture in a composite system (primarilyrubber+steel). As an example of a polymer-ceramic composite,think of the inconvenience and pain of cracking or breaking atooth (never mind a bone).

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Definition of FractureDefinition of Fracture• Fracture is the propagation of a crack across a loaded section.• The material property that characterizes fracture resistance is

its toughness.• Toughness is related to the energy per unit crack advance.• From linear elastic fracture mechanics, the units of toughness

are MPa√m.• Just as for yield strength, toughness scales with elastic

modulus.• How do we relate a stimulus to a response, when the response

is breakage and the stimulus is stress?! The answer is thatfracture toughness is effectively a threshold property - belowthe critical stress intensity, nothing breaks but above it, fractureoccurs (or is highly likely to occur). Thus fracture toughness isakin to the critical resolved shear stress - the material is elasticbelow it and plastic, flowing above it.

• Fracture toughness is a non-linear property because it is athreshold value, rather than a proportional response to astimulus.

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Fracture CharacteristicsFracture Characteristics• Although shear stresses can generate fracture, 99% of

practical examples involve tensile stresses.• Tensile stresses are important because they can separate the

atoms and lead to crack formation.• Fracture can be either ductile or brittle.• Brittle fracture implies little energy consumed in the process,

e.g. breaking glass• Ductile fracture implies that much energy is expended in, e.g.

breaking a pure metal.• Ductility, or fracture toughness, is extremely important from an

engineering and materials selection perspective, because itensures that loads are re-distributed in a structure withoutfailure. If the load exceeds the yield in some location, thematerial is more likely to flow plastically, thus causing someother part of the structure to absorb the load.

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Concepts to be learnedConcepts to be learned• Cracks require energy to be supplied in order to propagate.• The energy consumed goes into - at a minimum - creating new

surface area. We can measure the energy consumed as thearea under the stress-strain curve - ∫ σ dε. The greatestamount of energy is consumed when plastic work must beperformed in the course of crack propagation. Microstructureplays the largest role here (and where the materials engineeracts).

• The energy supplied comes from the elastic energy stored in aloaded body. As the crack propagates, some part of the bodyis unloaded, thereby releasing energy. Geometry plays thelargest role here; e.g. stress concentration at the tip of a sharpcrack (which is what a mechanical engineer must avoid!).

• In addition to the trade-off between energy supplied andconsumed, there must be enough stress available to breakbonds at the crack tip. Generally speaking, however, thiscriterion is satisfied and the energy criterion above is thegoverning one.

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Fracture Depends on MicrostructureFracture Depends on Microstructure• The fracture properties (resistance to fracture) depend strongly

on microstructure.• If mechanisms exist for dissipating significant amounts of

energy during crack propagation, then a material will have hightoughness.

• Toughening mechanisms include plastic flow, crack bridging,phase transformation, fiber reinforcement.

• Defects such as voids and poorly bonded second phaseparticles decrease toughness by introducing nucleation sitesfor cracks. Residual stresses introduced by thermal cycling ordeformation tend to decrease toughness also because theymean that there must be a region with elevated tensile stress(balanced by compressive stresses elsewhere).

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Liberty ship, heart valve failuresLiberty ship, heart valve failures

This WW-2 era “Liberty”ship, with a welded steelhull, failed by brittlepropagation of a crack whilemoored in dock! Thefracture toughness of thesteel was too low for thedesign.

This is an artificial heartvalve. Failure has occurredin some types of valve, e.g.through breakage at theweld of the outlet strut tothe carrier ring in the Björk-Shiley Convexo-Concave(BSCC) mechanicalprosthetic heart valve

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DesignDesignThe importance of fracture mechanics in design of

structures is paramount. The example of theLiberty ship in the 1940’s is classic: many shipswere lost either at sea or even in dock because ofthe propagation of a crack around the welded steelhull. It turned out that failure could be associatedwith a Charpy test energy of less than 15 ft.lbs atthe temperature at which the accident happened.Thus was the link between toughness and designborn. The specific point is that engineering designrequires the designer to take account of all possiblemodes of fracture.

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Charpy Charpy Impact TestingImpact TestingCharpy Impact testing is a method ofassessing the fracture resistance ofmaterials at moderately high loadingrates.The Charpy test is the most widely used evaluationtechnique for measuring the toughness of materials; itutilizes impact loading conditions. Standard-sizedspecimens (10 mm sq x 60 mm long) containing a sharpnotch (2 mm deep with a .015 mm radius) to localize thestress, are hammer-impacted and the energy absorbedduring this fracture process is measured. As thependulum hammer has a fixed weight and drops thesame distance each time (see Figure) its kinetic energywhen it strikes the specimen is always the same. Part ofthis energy is consumed in breaking the specimen; theenergy remaining in the hammer causes the pendulum tocontinue its upward swing. By measuring the differencein the height of the upward swing after the pendulum hasfallen freely and after it has broken the sample, theenergy absorbed in breaking the sample may becalculated. This energy is the impact strength of thematerial and can be read directly from a dial gauge onthe machine.

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Griffith & glass Griffith & glass fibresfibres• A.A. Griffith is considered to have made the first

substantial scientific contribution to theunderstanding of brittle fracture (1920). Hemeasured the breaking strength of glass fibres ofvarying thicknesses and found that their strengthvaried in inverse proportion to their diameter, seefig. from Green’s book (next slide). He thenshowed that Inglis’s equation for the stress[concentration] at the root of an elliptical crack couldbe applied to the problem to rationalize the results.That is, by assuming that the largest flaw was oforder of the fibre diameter, he could demonstratethat his results were consistent with Inglis’s theory.

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Strength of Glass Strength of Glass FibresFibres

Figure 8.2 from Green’s book on CeramicMechanical Behavior, for strength of glass fibers, asdetermined by Griffith. Note the inverse relationshipbetween size and strength.

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Microstructure Effect on FractureMicrostructure Effect on Fracture• Whether or not a material fractures on loading

depends on a competition between flow and fracture.If flow [of dislocations] is easy then fracture will onlyoccur when necking (localization) happens. If flow isdifficult then fracture will relieve the loading instead.

• Microstructure: weakly bonded second phaseparticles tend to promote fracture by acting asinitiation sites for cracks.

• Fine grain size tends to inhibit fracture by providing ahigh density of crack arrest/deflection points. Also,even if a grain cracks, then the stress concentrationat the end of the crack decreases with decreasingcrack size [= grain size].

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Temperature EffectTemperature Effect• Temperature: temperature affects plasticity in many

materials. Higher temperatures promotedeformation whereas low temperatures promotefracture. In many materials, a ductile-to-brittletransition can be detected as you lower thetemperature, see fig 11.8 from Cottrell. The reasonfor this is rising yield stresses with decreasing Teventually mean that tensile stresses cause fracturebefore shear stress can cause deformation.

• This also illustrates the essential aspect ofcompetition between fracture and plastic flow. Ifdislocation slip is easy, then even a artificially madecrack will blunt by plastic flow at its tip.

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Ductile-Brittle Transition Temperature (DBTT)Ductile-Brittle Transition Temperature (DBTT)

[Cottrell]

breaking stress (brittle)

yield stress (ductile)

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Ductile-Brittle Transition Temperature (DBTT)Ductile-Brittle Transition Temperature (DBTT)

• The basic explanation of the transition from ductile to brittlefailure as the temperature is decreased is that, as thetemperature is decreased, it becomes too difficult to movedislocations (as quantified by the critical resolved shear stress)relative to the stress required to propagate a crack (asquantified by the tensile breaking stress).

• Most materials exhibit a crss that increases rapidly withdecreasing temperature. This is generally caused by anincrease in the Peierls stress, i.e. the lattice friction (after theBritish physicist, Rudolf Peierls).

• The Peierls stress denotes the stress required to move adislocation from one atom position to the next (along the close-packed direction). In fcc metals, this stress is negligible. Inbcc metals at low temperatures, however, this stress is non-trivial. In most ceramics, the Peierls stress is so high (e.g.silicon) at room temperature that dislocations cannot bemoved. The Peierls stress is independent of other obstacles todislocation motion and depends only the atomic structure.

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Environment, LoadingEnvironment, Loading• Environment: certain interstitial elements are particularly

deleterious, e.g. hydrogen in steels. The exact causes are stillsubject to scientific debate but the presumption is that atomichydrogen lowers the cohesive energy between the planes thattend to cleave in brittle fracture. Hydrogen can be readilyintroduced in welding, e.g. by use of damp weld rods.Ammonia is notorious as a promoter of corrosion fatigue, e.g.cracking in brass. Similarly chloride ions (salt) in iron alloys(even stainless steel!).

• Type of loading: multiaxial stresses involving tension promotefracture whereas stresses involving compression promotedeformation, especially if deviatoric stresses are maximized.Monotonic loading is generally less severe than cyclic loading.Specimen design is also critical – notches promote fractureover deformation.

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Brittle Fracture: overviewBrittle Fracture: overview• What are we trying to establish about brittle fracture?• Real materials always have cracks (or potential

cracks): therefore one must always be concernedabout whether the cracks will grow catastrophically.

• Defects with high aspect ratios (cracks!) lead tostress concentrations which help cracks to grow.

• The most useful idea (Griffith) is that of an energybalance between elastic energy release rate andsurface energy consumption. In brittle solids, thisenergy barrier is easily overcome.

• We need to understand these basic concepts inorder to comprehend the role of microstructure!

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Pulling atoms apartPulling atoms apart• We will return to the basic concept of needing enough stress

at the tip of a crack in order to break bonds and propagate thecrack.

• For the moment, however, ignore the issue of stressconcentration in order to estimate the maximum possibletensile breaking stress.

• Remember that this is a necessary but not sufficient condition.• Step 1 = calculate the theoretical cohesive strength

Step 2 = calculate the effect of a crackFinish = relate fracture strength to modulus, crack geometry

[Courtney]

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Theoretical cohesive strength: Theoretical cohesive strength: stress stress basedbased

• The theoretical cohesive strength of a solid is veryhigh - between 1/4 and 1/50 of its Young’s modulus.

• Approximate theinteratomicforce-displacementcurve with a sine lawand equate the resultto the macroscopicelastic relationship:

σ = σcsin{πx/(λ/2)}.Force

Displacement

E

!/2

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Theoretical cohesive strength : Theoretical cohesive strength : stress stress basedbased

• Sine Curve: σ = σcsin{πx/(λ/2)} = σcsin{2πx/λ}

Small displacements: linearize: σ = σc2πx/λ

The slope near the origin: dσ/dx = σ/x = σc2π/λ

Compare with Hooke’s Law: strain = ∆l/l = x/a0

E = stress/strain = σ / (x/a0) = σ a0/x ∴ E/a0 = σ /xwhere a0 is the equilibrium distance between atoms.From equating the two expressions for the slope:

E/a0 = σ /x = σc2π/λ E/a0 = σc2π/λ

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Theoretical cohesive strength : Theoretical cohesive strength : stress stress basedbased

• Re-arranging to obtain the breaking stress, we obtain:

σc = Eλ/2πa0or,

2σc a0 /E = λ/π

• If we approximate the spacing for the sine curve as theinteratomic spacing, so that λ ≈ a0, then

σc = E/2π

which means that one estimate of the theoretical strength isabout 1/7 (simplified to 1/10 by some authors) of the Youngsmodulus.

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Theoretical cohesive strength : Theoretical cohesive strength : energy energy basedbased• An alternative approach is to consider the energy balance inherent in

this problem: we can estimate the work done to separate the twosurfaces from these equations, and equate it to the surface energies.Work to fracture the solid, based on the sine curve:

• By substituting the previously obtained relationship for l (to avoidmaking any assumptions about it) into the above relationship:

2σc a0 /E = λ/π, which gives: 2(σc)2 a0 /E = 2γ ,

we can obtain the following estimate of the fracture strength in termsof modulus, surface energy and bond length:

σbreak = √(Eγ/a0)• NO PRE-EXISTING CRACKS assumed for this derivation. The

resulting estimate gives similar values to the first one.

!

work done = "d#$ = " c0

% / 2

$ sin&x

% 2dx =" c

%

&= 2' surface

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Estimates of Fracture StrengthEstimates of Fracture Strength• Based on the observed maximum strengths

of whiskers and fibers, the ratio ofmodulus:strength approaches that of thetheoretical estimates.

• The last two materials are technologicalmaterials with useful, albeit small ductility.

732002.75Piano wire

642003.14Ausformedsteel

3349615.2Aluminawhisker

261666.5Siliconwhisker

2329513.1Ironwhisker

497.124.1Silica fiber

E/ σbreakE (GPa)σbreak(GPa)

Material

28.516Si28.5~45TiC

40.831MgO

6.33.7NaCl

37.222.5Ni24.812.1Cu17.77.8Au

√(Eγ/a0)(GPa)

E/10(GPa)

Material

• Evidently, the two different estimatesof fracture strength give similarresults.

• On this basis there is nothing tochoose between the stress-basedand the energy-based criteria.

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StressStressConcentrationConcentration

Stress concentration is an extremelyimportant concept. The local increase instress around a void in a materialconcentrates the stress next to it, farabove the level that you would computefrom the change in cross-sectional area.

Note the pattern of fringes in the upperright figure, shows the concentrationof stress around the tip of the crack.

Polarized light is shone throughthe specimen. Polycarbonate rotates theplane of polarization of light, so usingcrossed polars reveals contoursof maximum shear stress.

[Materials in Principle and Practice]

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Stress ConcentrationStress Concentration

2c

2b

[Courtney]

• Now we return to the basic concept of stress concentration atthe tip of a crack that is higher than the average stress in thebody. We can then ask what combination of stress and crackgeometry gives a high enough stress to break bonds andpropagate the crack.

• We employ a formula by Inglis for an elliptical crack of length cand thickness b:

σmaximum/σapplied = 1 + 2c/b

• The sharper thecrack, the greaterthe stressconcentration at thecrack tip, σmaximum.

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Fracture limited by stress concentrationFracture limited by stress concentration• Now notice that the radius of curvature, ρ, at the crack tip

(sharp end of the ellipse) is given by:

ρ = b2/c

• Combining the previous equation with this one and thensimplifying 1+2√(c/ρ) to 2√(c/ρ) because c»ρ,

σmaximum = 2σapplied√(c/ρ)

• The factor 2√(c/ρ) is sometimes referred to as a stressconcentration factor, k.

• The net result is that, with a crack present, less external stressis needed in order to reach the breaking stress of the material.

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Stress concentration + Breaking stressStress concentration + Breaking stress• What if we now assume that brittle fracture occurs when the

stress at the crack tip reaches the theoretical maximum tensilestress at which atomic bonds are ruptured (based on oursecond, energy-based criterion)?

• Set the maximum local stress as follows:

σmaximum = σbreak = √(Eγ/a0).

• Combine with the Inglis equationand the net result is:

• This tells us that the breaking strength depends on both cracksize and the sharpness of the crack.

• Contrast this with the previous result for theoretical cohesivestrength that ignores the effect of stress concentration:σbreak = √(Eγ/a0).

!break

="E#

4a0c

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Elastic Energy Elastic Energy vsvs. Crack Surface. Crack Surface• Now we turn our attention to the role of energy, as opposed to just

stress concentration. The previous derivation looked for a localstress-based criterion for crack propagation. This turns out to be anecessary but not sufficient condition for a crack to grow. We willnow look for an energy-based criterion, as did Griffith.

• As a crack (drak green) advances, so a region of radius ~ cracklength unloads (light orange). The elastic energy of this region isnow available to do work, i.e. create new surface area.

• The elastic energy of the body goes down as the square of the cracklength whereas the energy consumed in creating new surface area ina longer crack is proportional (linear) to the crack length. Note thatthe relevant length, c, is the half-length of en elliptical crack.

c

dc

cσ

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Energy Balance: 2Energy Balance: 2

[Courtney]

low stress

high stress

• In order to compute the energy balance as thecrack lengthens we need, (a) the elastic energy released = πσ2c2t/Eand (b) the energy consumed = 4ctγ.

• To understand the balance point, consider theenergy of the system, UTOT, as a function of cracklength (similar to the nucleation criterion):

UTOT = -πσ2c2t/E + 4ctγ

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Griffith equationGriffith equation• To find the “peak of the curve”, apply standard calculus and

find the zero of the derivative.

• Notice that the “break-even point” for getting more elasticenergy back than you have to put into the crack surface isstress dependent: higher stress, shorter crack required. Alsothe failure stress is not dependent on the crack shape (i.e.sharpness of the tip).

• Contrast this with the result for theoretical cohesive strengththat ignores the effect of stress concentration:σbreak = √(Eγ/a0), and the result that includes stressconcentration:σbreak = √(Eγρ/4a0c).

!

d

dc"#$ 2

c2

E+ 4c%

&

' (

)

* + = 0; $

break=

2%E

#c

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Stress IntensityStress Intensity• Practical experience tells us that the Griffith approach is surprisingly

general. Materials show characteristic stress intensities at whichcracks will propagate. The presumption is that the former, stress-based criterion, is generally satisfied before (i.e. at lower stresses)the energy criterion is.

• Therefore 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.

σbreak = Kc/√(πc)

• Can also define a toughness, Gc, which is given by

σbreak = √(EGc/πc)

and allows us to modify (increase) the apparent surface energy toaccount for plastic work at the crack tip. It does, however, mean thatthere is no longer a direct link to the surface energy.

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Fracture Toughness ValuesFracture Toughness Values

0.0172Window glass0.0316Concrete0.06390Alumina0.1 - 0.34Epoxy (thermoset)

0.3 - 0.43Glassy thermoplastics(PMMA)

0.5 - 26-16Wood, along grain2 - 81-2.5thermoplastics1020Glass fiber composite8-200.2-0.7Wood, across grain7-3070Al 201420-110120Ti-6Al-4V100210Mild steel10-110210Tool steel

Toughness (kJ/m2)Young’s Modulus(GPa)

Material

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Summary (part A)Summary (part A)• Fundamental equations governing brittle fracture have been

reviewed.• Griffith demonstrated as a balance between the rate of [elastic]

energy release versus energy consumed in creating new cracksurface area.

• This approach leads to the concept of a critical stress intensity,written as KIC.where the “I” in the subscript “IC” means that thematerial is loaded in plane strain. Plane strain (zero strainalong one principal axis) gives the lowest measured toughnessand is therefore the most conservative estimate of toughness.

• Ductile-to-Brittle Temperature (DBTT) introduced ascharacteristic of most materials. Related to the relative ease ofplastic deformation versus brittle crack propagation.