fractures/failures of materials … · lectures in fracture mechanics fractures/failures of...

Introduction Fracture Theory

Lectures in Fracture Mechanics

Fractures/Failures of Materials

K. Sakkaravarthi

Department of PhysicsNational Institute of Technology

Tiruchirappalli – 620 015Tamil Nadu

India

[email protected]

K. Sakkaravarthi Lectures in Fracture Mechanics 1/59


My sincere acknowledgments toElements of Fracture Mechanics, 1st Ed., P. Kumar,Tata McGraw-Hill Education, New Delhi (2009).

Mechanical Metallurgy, 3rd Ed., G.E. Dieter,Mc-Graw-Hill Book Company, New York (2004).

Many other free & copyright internet resources.



MotivationStudy on fracture mechanics was none/rare before World War II.

Cracks: Small, insignificant nuisances.Never be a threat to large structures (ships, aircraft, ...)

During & after WW II, many ships and aircraft failedsuddenly.

(Understood much later)Failures were caused by cracks in their metal structures.



SS Schenectady: 16 Jan. 1943Cost: USD 2,700,000

Cause of failure: Defective welding, “locked-in" stresses, sharpchanges in climate, or systemic design flaws. faulty workingpractices.Later research revealed that the failure was by brittle fracture.Caused by low-grade steel, highly brittle in cold weather.



A brittle fracture in an oil tanker.

1944: 2500 Liberty ships built/ 700 experienced severestructural failure/ 145 broke into 2 pieces:-(Reason: Flaws in welded joints & Low fracture toughness.It is easy to break large objects and very difficult tobreak a small ones!



Causes for Fracture/FailureThe usual causes of mechanical failures are:

Misuse or abuseManufacturing defectsAssembly errorsDesign errors or design deficienciesImproper material or poor selection of materialsImproper or inadequate maintenanceImproper heat treatmentsUnforeseen operating conditionsInadequate quality assuranceInadequate environmental protection/controlDegradation of (thermal, electrical, magnetic) propertiesCasting discontinuities, etc.



Mechanical FailureThe general types of mechanical failure include:

Failure by fracture due to static overload, the fracturebeing either brittle or ductile.Buckling in columns due to compressive overloading.Yield under static loading which then leads tomisalignment or overloading on other components.Failure due to impact loading or thermal shock.Failure by fatigue fracture.Creep failure due to low strain rate at high temperature.Failure due to the combined effects of stress and corrosion.Failure due to excessive wear.



Even though the causes of failure are known, prevention offailure is difficult to guarantee.

Our responsibility is to anticipate and prepare for possiblefailure; and in the event of failure, to assess its cause and thentake preventive measures.

Fracture Mechanics: courseNeed to minimize the possibility of failure, through a betterunderstanding of this complex subject “FractureMechanics".

Involve several principles & phenomena related tofracture/failure.

Necessary to understand different types of mechanicalfailure i.e. fracture, fatigue, creep, corrosion, wear, etc.



Outline of the CourseIt is important to understand the mechanisms forfailure, especially to prevent in-service failures due todesigning and usage problems.

This can be accomplished via Materials selection,Processing (strengthening), Design Safety(combination) and good knowledge in the utilization ofmaterials.



Failures in Structural elementsStructural elements may fail to perform their intended functionsin three general ways:(1) excessive elastic deformation,(2) excessive plastic deformation or yielding, and(3) fracture.

Excessive elastic deformationEx.: Too flexible machine shaft can cause rapid wear of bearing.(or) A sudden buckling type of failure may occur.

Failures due to excessive elastic deformation are controlled bythe modulus of elasticity, not by the strength of the material.

The most effective way to increase stiffness of a component is bytailoring the shape or dimensions.



Yielding or plastic deformationRender a component useless after a certain limit. Can becontrolled by the yield strength of the material.

At room temperature, continued loading over the yielding pointmay lead to strain hardening followed by fracture.

However at elevated/higher temperatures, failure occurs in formof time-dependent yielding known as creep.



FractureA complete disruption of continuity of a component.It is a degradation process and inhomogeneous in time & space.Starts with initiation of a crack, its propagation & finallyfracture (coalescence of cracks).

Fracture of materials may occur in three ways:1. Ductile/brittle fracture occurs over short period of time,and distinguishable.

2. Fatigue/progressive failure is the mode in which mostmachine parts fail. It is caused by a critical localized tensilestress, occurs in parts which are subjected to fluctuating stress(random loading-unloading).

3. Delayed fracture Stress-rupture occurring when a metal isloaded at an elevated temperature for a long time.



Fractures...



Few Types of Fractures



Brittle & Ductile Fracture



Ductile FractureAssociated with overload of the structure or largediscontinuities.High plastic deformation & slow crack propagation.

Three steps:- Specimen forms neck and cavities within neck- Cavities form crack and crack propagates towards surface(perpendicular to stress)- Direction of crack changes to 450 resulting in cup-cone fracture

One piece & large deformationReasons: Error in design, incorrect selection of materials,improper manufacturing technique and/or handling.

Ductile metals experience observable plastic deformation priorto fracture.



Brittle FractureNo significant plastic deformation before fracture.

Common at high strain rates and low temperatures.

Three steps:1 Plastic deformation concentrates dislocations along slip

planes2 Microcracks nucleate due to shear stress where dislocations

are blocked3 Crack propagates to fracture

Due to defects like porosity, tears and cracks, corrosion damage.

Many pieces & small deformation



Some factors in Stress & StrainUniform across material?

No!Different components??Normal stress & shearing stress.Normal stress σ = p

A cos θ.Shearing stress σ = p

A sin θ.Strain: Average linear strain is the ratio of the change inlength to the original length of the same dimension!Average linear strain e = δ

La = ∆LLa

∴ Strain: Change in linear dimension divided by theinitial/instantaneous dimension.



Some factors in Stress & StrainUniform across material? No!

Different components??Normal stress & shearing stress.Normal stress σ = p



La = ∆LLa




Some factors in Stress & StrainUniform across material? No!Different components??

Normal stress & shearing stress.Normal stress σ = p



La = ∆LLa




Some factors in Stress & StrainUniform across material? No!Different components??Normal stress & shearing stress.

Normal stress σ = pA cos θ.

Shearing stress σ = pA sin θ.

Strain: Average linear strain is the ratio of the change inlength to the original length of the same dimension!Average linear strain e = δ

La = ∆LLa




Some factors in Stress & StrainUniform across material? No!Different components??Normal stress & shearing stress.Normal stress σ = p


A sin θ.

Strain: Average linear strain is the ratio of the change inlength to the original length of the same dimension!Average linear strain e = δ

La = ∆LLa




Some factors in Stress & StrainUniform across material? No!Different components??Normal stress & shearing stress.Normal stress σ = p



La = ∆LLa




Parameters used to analyze Fractures:-Energy Releas Rate (G):Applied to brittle or less ductile materials.Stress Intensity Factor (K):Developed for brittle or less ductile materials.J-Integral (J):To deal with ductile materials. Also, useful to brittlematerials.Crack Tip Opening Displacement (CTOD):For ductile materials and based on the displacement.



Difference: Brittle/Ductile Fracture in Engineering &Conventional Materials



Transition: Ductile ⇐⇒ Brittle FractureThree factors influencing the nature of fracture:1. Temperature: Decrease can change a ductile fracture canbecome brittle (ductile-to-brittle transition).(Reverse: Temperature increase ⇒ Brittle to Ductile.)2. Strain rate: Higher ⇒ Brittle (Lower ⇒ Ductile)3. State of stress: Plastic deformation order(tri-axial < bi-axial < uni-axial)



Modes of Cracks in MaterialsThree ideal cases of loading of a cracked body can beconsidered, which are called the modes of deformation:(i) Mode I: Opening mode(ii) Mode II: Sliding mode(iii) Mode III: Tearing mode

In general (for a crack in an arbitrarily shaped body, under anarbitrary loading), the mode is not pure (i.e. is mixed mode).



Characteristics of CracksBased on its connection with the external free surface:(i) Completely internal,(ii) Internal cracks with connections to the outer surfaces,(iii) Surface cracks.

Cracks with some contact with external surfaces areexposed to outer media and hence may be prone tooxidation and corrosion (cracking). (More details later)

Crack length (depends on the type of crack (i, ii or iii).Crack tip radius (the sharper the crack, the moredeleterious it is). Crack tip radius depends in the type ofloading and the ductility of the material.Crack orientation with respect to geometry and loading.



Early Studies of Fracture: For crack growth1. C.E. Inglis: Stress based criterion (local) (1913)2. A.A. Griffith: Energy based criterion (global) (1920)

3. G. Irwin: Stress intensity & Energy release rate (1948)4. Wells: Crack Tip Opening Displacement (1961)5. Rice: J-Integral (1968)



Cohesive Strength of Materials* Every material is held together by strong atomic bonds.* The attractive and repulsive force acting between two atomslead to cohesive force.* This varies with respect to the separation between these atoms* Consider a crack in the material, we need to knowHow much stress must be added to break the bond?

Hooke’s lawTensile modulus E = Tensile Stress

Strain = σx/a0

.Shear modulus G = Shear Stress

Strain = τx/a0

.



σc: maximum stress required to separate two atoms.Cohesive Strength of Materials...1. Based on the displacement* Now, actual stress σ = σc sin(2πx/λ).* For small values (sin θ = θ) ⇒ σ = 2πxσc

λ* Small displacements/strain: dσ/dx = 2πσc

λ* From the Hooke’s law: σ = Eε⇒ Ex/a0.* Change in σ: dσ/dx = E/a0.

Equating the above two dσ/dx⇒ σc = Eλ2πa0

: Cohesive Strength* Cohesive Strength: σc = E

π ∵ a0 = λ/2



σc = σmax

Cohesive Strength of Materials...2. Based on Work on fracture:Work =

∫ λ/20 σc sin(2πx/λ)⇒ −σcλ

2π cos(2πx/λ)|λ/20 ⇒ σcλπ

* Let σcλπ = 2γs. (γs: surface energy)

* Using the previous equations σc = Eλ2πa0

and σcλπ = 2γs: we get

Cohesive strength σc =√

Eγsa0

.



Difference in Cohesive Strength: Stress Concentration* For most of the materials, a huge difference between thetheoretical and experimental cohesive strength.

* 1913: Inglis tried to solve/identify the reason/relation byintroducing “stress concentration factor"



Summary of Inglis formulationForce applied to ends of an elastic plate would producelocally increased tensile stresses at the tip of a crack.Stress concentration may exceed the elastic limit of thematerial and lead to the propagation of the crack.Increase in the length exaggerates the crack further, suchthat it would continue to spread.Small crack tip radii increase the stress at the crack tip.The shape of the crack was important rather than thesize/scale in determining the stress concentration.

Problems with Inglis formulationThe stress concentration factor (K = σmax

σo) has no

dependence on the crack size/scale, depends ratio only c/ρ.Can not explain the crack propagation. Formation of alocal failure/crack does not ensure the crack propagation,because it requires more energy over the critical limit.No clarity to design flawless materials, through themaximum stress a material can withstand with given a flawsize distribution.



Inglis vs GriffithInitially he tried to understand crack propagation in brittlematerials (wherein the cracks are sharp and there is verylittle crack-tip plasticity). This is the domain of LinearElastic Fracture Mechanics (LEFM).For a crack to propagate the necessary global (Griffith)criterion and the sufficient local (Inglis) criterion have to besatisfied.The kind of loading/stresses also important. Tensilestresses tend to open up cracks, while compressive stressestend to close cracks.



Griffith’s Theory: MotivationThe crack length does not appear ‘independently’ (onlyalong with the crack tip radius) in Inglis’s formula.(What is the role of crack length alone?)We can feel that longer crack must be more dangerous.Assumption in Inglis’s approach: sufficient energy isavailable in the elastic body to do work to propagate thecrack.(‘What if there is insufficient energy?’)(‘What if there is no crack in the body?’).Also, intuitively we can understand that the energy (whichis the elastic energy stored in the body) should be availablein the proximity of the crack tip (i.e. energy available faraway from the crack tip is of no use!).



Griffith’s Theory...The development of linear-elastic fracture mechanics (LEFM)started with Griffith work on glass!

Fundamentally, the Griffith theory considers the energychanges associated with incremental crack growth.He used an energy balance approach to predict thefracture stress of glass in 1921.When a stressed plate of an elastic material containingcracks, the potential energy per unit thickness (∆U)decreases and the surface energy per unit thickness (Us)increases during crack growth.Then, the total potential energy of the stressed solid bodyis related to the release of stored energy and the work doneby the external loads. The “surface energy" arises from anonequilibrium configuration of the nearest neighbor atomsat any surface in a solid.



Griffith’s CriterionKeeping some of these factors in view, Griffith proposed twoconditions for crack propagation:

1 Bonds at the crack tip must be stressed to thepoint of failure (as in Inglis’s criterion).

2 The amount of strain energy released (by the ‘slight’unloading of the body due to crack extension) must begreater than or equal to the surface energy of crackfaces.The second condition can be written as:

dUsda≥ dUγ

da.

Here Us: strain energyUγ : surface energy (Energy per unit area: [J/m2])a: crack lengthda: (‘infinitesimal’) increase in the length of the crack



Griffiths Theorem* A large or an infinite brittle plate containing one centerthrough-thickness crack of length 2a with two crack tips* A remote and uniform tensile load perpendicular to the crackplane.* The stored elastic strain energy is released within a cylindricalvolume of material of length B.



Griffiths Theorem...* Elastic strain energy=released elastic strain energy density ×cylindrical volume

We = −2πa2B ×∫σdε

* By using the Hook’s law σ = E′εWe = −2πa2B ×

∫E′εdε⇒ −2πa2B × E′ε2/2.

∴We = −πa2B × σ2/E′.

Here E: Modulus of elasticity (MPa), ε: Elastic strain,σ: Applied remote stress (MPa), a: radius of the crack (mm),v: Poisson’s ratio, B: Plate thickness (mm),4aB = 2× 2aB: Total surface crack area (mm2),E′ = E/(1− v2) for plane-strain conditions.

Reason for E′ is to control either plane stress or plane-strain.



Griffiths Theorem...Elastic surface energy for creating new crack surfaces duringcrack growth (from two crack tips) Ws = 2× 2aBγs.γs: surface energy (J/mm2).

For any elastically stressed solid body, Griffith energybalance contains (i) the decrease in potential energy(due to the release of stored elastic energy and the workdone by external loads) and (ii) the increase in surfaceenergy resulting from the growing crack.

So, the total elastic energy is the total potential energy!W = Ws +We = 4aBγs − πa2Bσ2/E′.

Total potential energy per unit thicknessW = 4aγs − πa2σ2/E′ ⇒ U = Us + Ue.Us: Elastic surface energy per unit thickness (J/mm)Ue: Released elastic energy per unit thickness (J/mm)



Griffiths Theorem...The Griffith’s energy criterion for crack growth:

Ue ≥ Us when dUda = 0.

This gives the energy balance 4γsE′ = 2πaσ2.

From the above condition, we can easily findthe applied stress (σ),the crack length (a) orthe strain energy release rate (GI) for brittle materials.

Required stress σ =√

2γsE′

πa .

Crack length a = 2γsE′

πσ2 .

Strain energy release rate GI = 2γs ⇒ πaσ2

E′ .

Rearrange GI ⇒ σ√πa =

√GIE′.

σ√πa⇒ KI (stress intensity factor).



Griffiths TheoremRelationship between applied nominal stress and crack length.

When the stress becomes energetically favourable for a crack togrow.



Griffiths Theorem...For a loaded brittle body: The only contributors to energychanges are the energy of the new fracture surfaces (twosurfaces per crack tip) and the change in potential energyin the body.

The surface energy term (Us) represents energy absorbed incrack growth, while the some stored strain energy (Ue) isreleased as the crack extends (due to unloading of regionsadjacent to the new fracture surfaces).

Surface energy has a constant value per unit area (or unitlength for a unit thickness of body) and is therefore a linearfunction of crack length, while the stored strain energyreleased in crack growth ∝ square of crack length (a2).



Griffiths Theorem: Alternate Algorithm

(a) Unstretched plate (b) Stretched plate(c) Stretched plate with a crack

* Energy release =volume of triangle× strain energy density,ER = 2

(122a(2λa)B

)× σ2

2E ⇒2λa2Bσ2

E

* For a thin plate (λ = π/2) ⇒ ER = πa2Bσ2

E .* Energy required to create two new surfacesES = 2× 2aBγ ⇒ 4aBγ.



Griffiths Theorem: Alternate Algorithm...

Energy vs Crack length (critical dERdac

= dESdac

)



Griffiths Theorem: Alternate Algorithm...

Critical Energy dERda ≥

dESda ⇒

2πacBσ2

E ≥ 4Bγ.

Critical crack length ac ≥ 2Eγπσ2 . (Safer crack length a < 2Eγ

πσ2 )

Strain energy release rate GI = 2γ ⇒ πaσ2

E .

Required stress σc ≥√

2Eγπa .

For thick plates E −→ E/(1− v2)

∴ Required stress σc ≥√

2Eγπa(1−v2)

.



Limitations in Griffiths TheoryGriffith’s investigation was based on glass materials.

If crack length is greater than the thickness of thespecimen???Griffith’s theory cannot be applied:-(The highest fracture strengths were found with thesmallest-diameter fibers, since on the average these fiberswould have the shortest microcracks.Other factors also can affect the strength:method of preparation, temperature of the melt, andamount and rate of drawing from the melt.Recent results: No dependence of strength on diameterwhen different-size glass fibers are prepared under nearlyidentical conditions.The strength of glass fibers is extremely sensitive to surfacedefects.



Limitations in Griffiths TheoryGriffith’s investigation was based on glass materials.If crack length is greater than the thickness of thespecimen???Griffith’s theory cannot be applied:-(The highest fracture strengths were found with thesmallest-diameter fibers, since on the average these fiberswould have the shortest microcracks.Other factors also can affect the strength:method of preparation, temperature of the melt, andamount and rate of drawing from the melt.Recent results: No dependence of strength on diameterwhen different-size glass fibers are prepared under nearlyidentical conditions.The strength of glass fibers is extremely sensitive to surfacedefects.



Modified Griffiths TheoryWhat will be the effect in other materials?

Joffe effect: The fracture strength of NaCl crystals could begreatly increased when the test was carried out under water.This shows the healing of surface cracks by the solution ofthe salt crystal in the water. The fracture behavior of otherionic crystals has been shown to depend on theenvironment in contact with the surface.However, the Joffe effect in these crystals cannot always beexplained simply by surface dissolution.Metals show evidence of a thin layer of plastically deformedmetal when the fracture surface is examined by XRD.Irwin & Orowan: modified Griffith theorem

Critical stress σc ≥√

2E(γs+γp)πa .

γp: plastic work per unit area of surface created (γp >> γs)Irvin: crack-extension force or strain-energy release rate



Modified Griffiths TheoryWhat will be the effect in other materials?Joffe effect: The fracture strength of NaCl crystals could begreatly increased when the test was carried out under water.This shows the healing of surface cracks by the solution ofthe salt crystal in the water. The fracture behavior of otherionic crystals has been shown to depend on theenvironment in contact with the surface.However, the Joffe effect in these crystals cannot always beexplained simply by surface dissolution.

Metals show evidence of a thin layer of plastically deformedmetal when the fracture surface is examined by XRD.Irwin & Orowan: modified Griffith theorem


2E(γs+γp)πa .




Modified Griffiths TheoryWhat will be the effect in other materials?Joffe effect: The fracture strength of NaCl crystals could begreatly increased when the test was carried out under water.This shows the healing of surface cracks by the solution ofthe salt crystal in the water. The fracture behavior of otherionic crystals has been shown to depend on theenvironment in contact with the surface.However, the Joffe effect in these crystals cannot always beexplained simply by surface dissolution.Metals show evidence of a thin layer of plastically deformedmetal when the fracture surface is examined by XRD.Irwin & Orowan: modified Griffith theorem


2E(γs+γp)πa .




Problem: 1A large and wide brittle plate containing a single-edge crack (a)fractures at a tensile stress of 4MPa. The critical strain energyrelease rate (Gc) and the modulus of elasticity (E) are 4 J/m2

and 65,000 MPa, respectively. Assume plane stress conditionand include the thickness B = 3mm in all calculations. (a) Plotthe theoretical total surface energy (Us), the released strainenergy (Ue), and the total potential energy change (W).Interpret the energy profiles. Determine (b) the critical cracklength and (ac) the maximum potential energy change (Wmax)(d) stability nature of the crack? (e) What is the critical stressintensity factor for this brittle plate?

Given parameters:Stress σ = 4 MPa,Modulus of Elasticity E = 65, 000 MPa (plane stress)Critical energy release rate GI = 4 J/m2,Thickness B = 3mm



Metallographic Aspects of Fracture* Difficulty in Griffith theory :(

* Use of microscopes in search for cracks in metals.* However, with the electron microscope, there is no reliableevidence for Griffith cracks in metals in the unstressed condition.

* The microcracks can produce by plastic deformation!

* Metallographic evidence for the formation of microcracks atnon-metallic inclusions in steel as a result of plastic deformationhas existed for a number of years.* These microcracks do not necessarily produce brittle fracture.But, contribute to the ductile fracture strength.* The fact vacuum-melted steel proved this idea.* It shows a reduction in the fracture anisotropy supports theidea of microcracks being formed at second-phase particles.



Metallographic Aspects of Fracture* Difficulty in Griffith theory :(

* Use of microscopes in search for cracks in metals.* However, with the electron microscope, there is no reliableevidence for Griffith cracks in metals in the unstressed condition.* The microcracks can produce by plastic deformation!

* Metallographic evidence for the formation of microcracks atnon-metallic inclusions in steel as a result of plastic deformationhas existed for a number of years.* These microcracks do not necessarily produce brittle fracture.But, contribute to the ductile fracture strength.* The fact vacuum-melted steel proved this idea.* It shows a reduction in the fracture anisotropy supports theidea of microcracks being formed at second-phase particles.



Metallographic Aspects of Fracture...* Low-1956: Correlation between plastic deformation,microcracks, and brittle fracture.* Test:- Mild steel with given grain size at -196oC.Brittle fracture occurs at stress required.* Microcracks only one or two grains long were observed.* More detailed studies with tensile tests on mild steel atcarefully controlled subzero temperatures.

Microcracks produced in iron by tensile deformation at -140°CK. Sakkaravarthi Lectures in Fracture Mechanics 49/59


Metallographic Aspects of Fracture...



Metallographic Aspects of Fracture...* Temperature dependence of yield stress, fracture stress, andductility and microcrack development.



Metallographic Aspects of Fracture...* Region A (near room temperature): a tensile specimen failswith a ductile cup-and-cone fracture. The reduction of area atfracture is of the order of 50 to 60 per cent.

* Region B: The fracture is still ductile, but the outer rim of thefracture contains cleavage facets.* A transition from ductile to brittle fracture occurs at theductility transition temperature Td.

* The existence of a transition temperature is indicated by thedrop in the reduction of area at the fracture to practically zero.* Accompanying this is a large decrease in the fracture stress.



Metallographic Aspects of Fracture...* Region C: The percentage of grains containing microcracksincreases rapidly just below Td. However, microcracks are foundabove Td too, but their role is very less in developing the crack.

* The ductility transition occurs when the conditions aresuitable for the growth of microcracks into propagatingfractures.

* The initiation of microcrack contribution is not a sufficientcriterion for brittle fracture.* Microcracks occur only in regions which have undergonediscontinuous yielding as a result of being loaded through theupper yield point.

* As the temperature drops in region C, eventually the fracturestress drops to a value equal to the lower yield stress.



Metallographic Aspects of Fracture...* Region D: The lower yield stress and fracture stress arepractically identical.* Fracture occurs at a value equal to the lower yield stress afterthe material has undergone some discontinuous yielding. Thefracture stress increases because the yield stress is increasingwith decreasing temperature.* The number of unsuccessful microcracks increases=> yield strength increases: discontinuous yield.

* Region E: Cleavage fracture occurs abruptly before there istime for discontinuous yielding.* Fracture occurs as it undergo discontinuous yielding.* At very low temperatures (region F), fracture is initiated bymechanical twins.* Mechanical twins are observed at temperatures as high as Td,but it is only in region F that they appear to be the source ofinitiation of fracture.



Dislocation Theory of FractureZener: High stresses at the head of a dislocation pile-up &produce fracture.At some critical value of stress the dislocations at the headof the pile-up are pushed close together & coalesce into anembryonic crack or cavity dislocation.Stroh: Condition for the formation of a cleavage crack(After analyzing the stresses & Griffith criterion) when ndislocations piled up under the action of a resolved shearstress τs:

nbτs = 12γb: Burgers vector(represents the magnitude and direction of the latticedistortion resulting from a dislocation in a crystal lattice).The length slip plane L = nbG

π(1−v)τs

Combining the above two eqns. τ2sL = 12γG

π(1−v) .



Dislocation Theory of Fracture...Consider a specimen of grain size D,shear stress τs = σ/2 & length D/2.

Fracture stress σf = 4[

6γGπ(1−v)

]1/2D−1/2 ⇒ KD−1/2.

Petch: From experimental data for iron & steel⇒ σf = σi +KD−1/2.

Yield-strength equation: σ0 = σi +KyD−1/2.

σi: Frictional stress resisting the motion of an unlockeddislocation.Ky: Localized stress needed to unlock dislocations held upat a grain boundary so that yielding can be transmitted tothe next grain by the propagation.



Dislocation Theory of Fracture...Cottrell and Fetch: The growth of a microcrack into aself-propagating fracture is a more difficult step than thenucleation of microcracks from glide dislocations (manynon-propagating microcracks are observed).

Crack nucleation by dislocation coalescence should dependonly on the shear stress(not the hydrostatic component of stress).

Many situations where fracture is strongly influenced bythe hydrostatic component of stress.

Griffith-type criterion: For the propagation of microcracks,the stress normal to the crack would be an important factor.Cottrell:The stress required to propagate a microcrack σ = 2γ

nbK. Sakkaravarthi Lectures in Fracture Mechanics 58/59


From the past few lectures, we have discussed severalimportant parameters & principles involved in definingthe fracture/failure of materials.

Thank You!


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