Fracture Mechanics

1 Brittle Fracture
• Brittle fracture occurs in high-strength metals and alloys OR metals and
alloys with poor ductility and toughness.
• Any crack or imperfection limits the ability of a ceramic to withstand a
tensile stress.
• Brittle fractures often take place due to impact causes failure, rather than
overload.
• This is because a crack (sometimes called a Griffith flaw) concentrates and
magnifies the applied stress.
• The figure below shows a crack of length 𝑎 at the surface of a brittle
material.
• The radius 𝑟 of the crack is also shown

1.1 Characteristics of Brittle Fracture

• Little or no plastic deformation
o The metal does not deform permanently in the area of the fracture.
• Rapid crack propagation
o The crack happens quicker without any warning.
o Initiation of the crack normally occurs at small flaws, which results in
concentration of stress.
o The crack propagation speed can be as high as the speed of sound in
the metal.
o The crack can occur along specific crystallographic planes, often the
{100} planes, by cleavage.
o In some instances, the crack propagation may occur along the
intergranular path, which is along grain boundaries due to
segregation (preferential separation of different elements) or
inclusions (unwanted particles or impurities trapped in metal castings
during pouring or melting) weaken the grain boundaries.
• Smooth fracture surface/ flat fracture surface
o Normally, the fracture surface is flat and perpendicular to the applied
stress in a tensile test.
• Cleavage
o Type of trans granular fracture.
o Failure occurring by cleavage is identifiable when fractured grain is
flat and differently oriented.
 • Trans granular fracture:
o Crack travels through the grains of a material.
o The direction the crack path is dictated by different orientations of the
atoms in each grain.
o The crack path also follows a route of high-stress intensity.
• Inter granular fracture:
o Crack travels along the grain boundaries of a material.
o This may occur when the grain boundaries are weak.

1.2 Brittle fracture of ductile materials

• Materials which are normally ductile may fail in a brittle manner
o at low temperatures,
o in thick sections,
o at high strain rates such as impact,
o or when flaws play an important role.

1.3 Ductile to Brittle Transition Temperature (DBTT)

• The ductile to brittle transition temperature or DBTT is the temperature at
which the failure mode of a material changes from ductile to brittle
fracture.
• Example of DBTT: The Titanic
• This temperature may be defined by the average energy between the ductile
and brittle regions, at some specific absorbed energy, or by some
characteristic fracture appearance.
• BCC (Body-Centred Cubic) metals have transition temperatures, but most
FCC (Face-Centred Cubic) metals do not.

• 𝑇1 = 𝐹𝑇𝑃 (𝐹𝑟𝑎𝑐𝑡𝑢𝑟𝑒 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑃𝑙𝑎𝑠𝑡𝑖𝑐)

o Temperature at which the fracture is 100% ductile (Fibrous).
o Very conservative.
• 𝑇2 = 𝐹𝐴𝑇𝑇 (𝐹𝑟𝑎𝑐𝑡𝑢𝑟𝑒 𝐴𝑝𝑝𝑒𝑎𝑟𝑎𝑛𝑐𝑒 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)
o Temperature at which the fracture appears to be 50% Fibrous and
50% Cleavage.
• 𝑇3 = 𝐹𝐴𝑇𝑇 (𝐹𝑟𝑎𝑐𝑡𝑢𝑟𝑒 𝐴𝑝𝑝𝑒𝑎𝑟𝑎𝑛𝑐𝑒 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)
o Mean of upper and lower shelf value often approximated to 𝑇2
• 𝑇4 = 𝐷𝑇𝑇 (𝐷𝑢𝑐𝑡𝑖𝑙𝑖𝑡𝑦 𝑇𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)
o Arbitrary value of energy absorbed, i.e. 20 J
• 𝑇5 = 𝑁𝐷𝑇 (𝑁𝑖𝑙 𝐷𝑢𝑐𝑡𝑖𝑙𝑖𝑡𝑦 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)
o 100% Cleavage fracture
1.4 Notch Effects
• The presence of a notch will increase the ductile-brittle transition
temperature of a steel.
• The major effect of the triaxial stress state is to hinder the spread of the
plastic zone.
 Homework/Self Study
1 Stress Corrosion Cracking
• A failure mechanism that requires three simultaneous conditions to be
satisfied.
• These three conditions include,
o susceptible material exposed simultaneously
o to an aggressive corrosive environment,
o and a sustained tensile stress.
• Removal of any of these conditions, will eliminate stress corrosion cracking.
• The susceptibility of a material is a function of alloy composition and
microstructure.
• A sustained tensile stress can be an applied stress, or a residual stress
generated as a result of assembly or manufacturing processes.

1.1 Characteristics of SCC

• Corrosive species
o Concentration of the key species may be very small.
o For example, the SCC susceptibility of copper alloys generally
increases as a function of zinc content.
o In addition, the SCC resistance of certain austenitic stainless steels
decreases as a result of grain boundary precipitation of chromium
carbides. This is due to the formation of a chromium-depleted zone
in the region immediately adjacent to the grain boundary that is
susceptible to corrosion in many environments.
• Metal-environment specific
o SCC will occur on specific metals in a specific environment.
o For example, carbon steel will experience SCC in seawater and
copper will not. Copper will experience SCC in ammonia and carbon
steel will not.
• Chemical composition influence
o Susceptibility of alloys is influenced by chemical composition.
o Susceptibility of copper to SCC in ammonia increases with zinc
composition but is eliminated with increase in nickel content.
• Microstructure influence
o Metals prone to sensitisation are susceptible to SCC.
o Stainless steel alloys are prone to sensitisation and will experience
SCC in most corrosive environments.
 o Sensitisation in stainless steels is caused by the depletion of
chromium at grain boundaries to less than 11.5%.
• Stress may be small or large
o Stress may be applied or residual.
o Strain rates as low as 10−7 /𝑠 can cause SCC

1.2 Mechanism of SCC

• The film-rupture metal-dissolution mechanism proposes that SCC occurs
in the following steps:
o Tensile stresses rupture any films on the surface, exposing fresh
metal to the corrosive species.
o Corrosive species dissolve the metal, forming a pit that acts as a
stress raiser and crack initiation site.
o Corrosion products form a protective film at the pit and prevent
further dissolution. exposing fresh metal to the corrosive species.
o Tensile stresses rupture the protective film, exposing fresh metal to the
corrosive species.
o Corrosive species dissolve the metal, propagating the crack.

1.3 Crack morphology for SCC

• SCC results in multi branched cracks.
• The primary crack, shown in the figure, is perpendicular to the applied
stress.
 • The crack can be inter granular or trans granular.

• Trans granular SCC occurs when applied stress is large and inter granular
SCC occurs when applied stress is low.
• At times the crack may progress so slow that the fracture surface appears to
have striations similar to those observed on fatigue fracture.

1.4 SCC Prevention

• Proper material selection
o SCC is metal-environment specific and material selection can go a long
way in eliminating SCC.
o Where practically possible, replacing metals with polymers can help
prevent SCC.
• Remove or avoid the corrosive environment
o Corrosion inhibitors may be used to alter corrosiveness of an
environment.
• Reduce stress
o Residual stress may be reduced by use of appropriate heat
treatment (normalizing, annealing, tempering and stress relieving)
• Avoid stagnant areas and crevices
o Stagnant areas and crevices promote formation of pits that act as
stress raisers and initiation sites for SCC.

2 Hydrogen Induced Cracking (HIC)

• Similarly, a failure mechanism that requires three simultaneous conditions to
be satisfied.
• These three conditions include,
o susceptible material exposed simultaneously
o to hydrogen,
o and a sustained tensile stress.
 • Removal of any of these conditions, will eliminate hydrogen induced
cracking.

2.1 Characteristics of HIC

• Hydrogen
o The concentration of which may be very small.
• Strength influence
o High strength increases susceptibility to HIC.
o Hardening of steels by heat treatment increases susceptibility.
o Strong metals are prone to HIC, while tough metals have better
resistance to HIC.
• Microstructure influence
o Rolled or elongated grain structures are susceptible to HIC.
o Austenitic steels and austenitic stainless steel are prone to HIC,
while ferritic versions of the alloys are not.
• Stress may be large or small

2.2 Sources of Hydrogen

• Casting
o When molten metal reacts with moisture, it forms an oxide and
hydrogen.

𝑴 + 𝑯𝟐 𝑶 → 𝑴𝑶 + 𝑯𝟐

o If the hydrogen does not escape before solidification, it will cause

HIC.
• Welding
o Moisture or grease on workpieces or the environment can lead to
hydrogen pick up.
o Fluxes or flux covered electrodes are typically hydroscopic and
increase opportunities for hydrogen pick up.
• Electroplating
o Electroplating from acidic solutions can generate hydrogen by the
following reaction,
𝟐𝑯+ + 𝟐𝒆− → 𝑯𝟐
o Corrosion in acidic solution and pickling with acid will also generate
hydrogen through the same reaction.
• Hydrogen related industries
o Some industries such as petrochemical inevitably produce hydrogen.
o Materials used in such environments must be able to resist HIC.
2.3 Mechanism of HIC
• Atomic hydrogen diffuses into the metal, and concentrates at a site such as
grain boundary, an inclusion, or the lattice. Hydrogen is very small and can
easily diffuse in metal.
• Atomic hydrogen combines to form molecular hydrogen.
• The formation of molecular hydrogen leads to an increase in internal
pressure.
• The internal pressure will generate enough stress to cause and propagate a
crack.

2.4 Crack morphology for HIC

• The crack morphology for HIC depends on where the hydrogen
concentrates.
• If the hydrogen concentrates at the grain boundaries, HIC will be inter
granular.
• If the hydrogen concentrates at the lattice, HIC will be trans granular OR
cleavage.
• If the hydrogen concentrates at dislocations, voids and pores, HIC will
result in a ductile fracture.
• HIC may also result in delamination and fisheye.

• Cracks due to HIC are typically stepwise.

• Crack orientation has no relation with direction of stress.

2.5 HIC Prevention

• Proper materials selection
• Control material hardness
o High strength materials are most prone to HIC
o If hardness is not a requirement, it must be kept as low as practically
possible.
• Reduce stress
• Remove or avoid the hydrogen source
o Use non-acidic pickling solutions
o Plate from non-acidic solutions
o Remove moisture from furnace and furnace charge
o Bake flux and flux covered electrodes
 1 Fracture Mechanics
• Fracture Mechanics is the discipline concerned with the behaviour of
materials containing cracks or other small flaws.
• The term “flaws” refers to small pores, holes, inclusions, or microcracks.
• It involves the evaluation of the strength of a structure or component in the
presence of a crack or flaw.

1.1 Theoretical vs Observed material strength

• Theoretical strength of a material is determined by,
o cohesive force required to break atomic bond.
o Work done to generate new crack surfaces.

• Cohesive strength
𝐸 𝐸𝛾
𝜎𝑐 ≈ =√
𝜋 𝑎0
o Most material,
𝐸
𝜎𝑐 ≈
10
where 𝑎0 is the atomic spacing and 𝛾 is the surface energy.

• Consider AISI 316 stainless steel (𝐸 = 164 𝐺𝑃𝑎)

𝐸
o Theoretical strength 𝜎𝑚𝑎𝑥 ≈ 10 = 16.4 𝐺𝑃𝑎
o Experimental value (UTS) = 0.62 − 0.795 𝐺𝑃𝑎

1.2 Stress concentration

• By analysing a plate containing an elliptical hole, Inglis was able to show
that the applied stress, 𝜎𝑎 was magnified at the ends of the major axis of the
ellipse, as seen in the figure below.
 𝜎𝑚𝑎𝑥 2𝑎 (1.1)
=1+
𝜎𝑎 𝑏

where 𝜎𝑚𝑎𝑥 = maximum stress at the end of the major axis

𝜎𝑎 = applied stress applied normal to the major axis

𝑎 = half major axis

𝑏 = half minor axis

• Since the radius of curvature, 𝜌 at the end of the ellipse is given by

𝑏2 (1.2)
𝜌=
𝑎

and combining equation (1.1) and (1.2),

(1.3)
𝑎
𝜎𝑚𝑎𝑥 = 𝜎𝑎 (1 + 2√ ) = 𝜎𝑎 𝑘𝑡
𝜌

In most cases 𝑎 ≫ 𝜌, therefore

𝑎 (1.4)
𝜎𝑚𝑎𝑥 ≈ 2𝜎𝑎 √
𝜌
• The stress concentration factor is,
𝜎𝑚𝑎𝑥 (1.4)
𝑘𝑡 =
𝜎𝑎
1.3 Griffith’s Theory of Brittle fracture
• Griffith's theory states that a crack will propagate when the applied stress
can generate elastic strain energy that is at least equal to the energy
required to create new crack surfaces.
• Assumptions:
o Material contains an elliptical flaw of length 2a
o Loaded perpendicular to the crack
o Wide sheet with negligible thickness (plane stress)

1.3.1 Energy balance

• The applied stress will cause elastic strain related to Young’s modulus 𝐸,
stored in the material.
• The elastic strain energy is released during crack propagation and used to
create new surfaces.
• The elastic strain energy released is,
𝜋𝑎2 𝜎 2
𝑈𝐸 = −
𝐸
• The surface energy is,
𝑈𝑠 = 4𝑎𝛾𝑠
𝛾𝑠 = 𝑆𝐸 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑟𝑒𝑎

• The stress required to create the new crack surface is,

2𝐸𝛾𝑠
𝜎𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = √
𝜋𝑎
 • Shortcomings
o The criterion does not consider the crack tip geometry and the stress
concentration.
o Only applicable to perfectly brittle materials under uniaxial stress
with cracks oriented perpendicular to the stress direction.
o Surface energy is not good and well-defined parameter for real
materials.
• Modified Griffith’s equation to account for ductility
o Irwin’s modification:

2𝐸(𝛾𝑠 + 𝛾𝑝 )
𝜎=√
𝜋𝑎

𝛾𝑝 = 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 𝑤𝑜𝑟𝑘

o Orowan’s modification:

𝐸𝐺𝑠
𝜎=√
𝜋𝑎

𝐺𝑠 = 𝐶𝑟𝑎𝑐𝑘 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒

1.4 Stress Intensity Factors (SIF)

• The SIF or K is useful way of describing the stress field around a flaw.
𝐾 = 𝛼𝜎√𝜋𝑎
𝛼 𝑖𝑠 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑎𝑐𝑡𝑜𝑟
• It is a function of;
o loading 𝜎,
o crack size 𝑎, and
o 𝛼 is dependent specimen and crack geometry.
• Dimension: 𝑀𝑃𝑎√𝑚

1.5 Crack deformation modes

• Dealing with the stress intensity factor (SIF) or 𝐾, a crack can be loaded in
one or some combination of the three modes.
o Mode 1 – Opening (most common)
o Mode 2 – Sliding
o Mode 3 – Tearing
1.6 Stress Intensity Modification
• The geometric factor (𝛼) varies:
o a plate of width 𝑤 loaded in tension with a centrally located crack of
length 2𝑎,
𝑤 𝜋𝑎 2
𝛼 = ( tan )
𝜋𝑎 𝑤

• a plate of infinite width in tension having a through-thickness crack,

𝛼 ≈ 1.0
• a plate of semi-infinite width in tension containing an edge crack of length
𝑎,
𝛼 ≈ 1.12

1.7 Stress Intensity Factor – Mode 1

• Mode 1 is opening of a crack.
𝐾𝐼 = 𝛼𝜎√𝜋𝑎

1.8 Fracture toughness

• Fracture toughness measures the ability of a material containing a flaw of
known size and geometry.
• Brittle fracture condition,
𝐾𝐼 ≥ 𝐾𝐼𝐶
• The critical value of stress intensity factor mode 1 (𝐾𝐼𝐶 ),
𝐾𝐼𝐶 = 𝛼𝜎√𝜋𝑎𝑐
Homework/Self Study
1. A structural plate component of an engineering design must support 207 MPa
in tension. If aluminium alloy 2024-T851 is used for this application, what is
the largest flaw size that this material can support? Use 𝛼 = 1 and 𝐾𝐼𝐶 of that
alloy is 26.4 𝑀𝑃𝑎√𝑚.
2. The critical stress intensity for a material for a component of a design is 22.5
𝑀𝑃𝑎√𝑚. What is the applied stress that will cause fracture if the component
contains an internal crack 12 mm long? Assume 𝛼 = 1

Solution:

1) 𝐾𝐼𝐶 = 𝛼𝜎√𝜋𝑎𝑐

Given: 𝐾𝐼𝐶 = 26.4 𝑀𝑃𝑎√𝑚, 𝛼 = 1, 𝜎 = 207 𝑀𝑃𝑎

Unknown: The value of 𝑎𝑐 , which determines the crack size

26.4 𝑀𝑃𝑎√𝑚 = (1)(207 𝑀𝑃𝑎)√𝜋𝑎𝑐

26.4 𝑀𝑃𝑎√𝑚
√𝑎𝑐 =
(1)(207 𝑀𝑃𝑎)(√𝜋)
2
26.4 𝑀𝑃𝑎√𝑚
𝑎𝑐 = ( )
(1)(207 𝑀𝑃𝑎)(√𝜋)

𝑎𝑐 = 5.177 × 10−3 𝑚

2) 𝐾𝐼𝐶 = 𝛼𝜎√𝜋𝑎𝑐

Given: 𝐾𝐼𝐶 = 22.5 𝑀𝑃𝑎√𝑚, 𝑎 = 12 𝑚𝑚 = 0.012𝑚, 𝛼 = 1

Unknown: Applied stress, 𝜎 in 𝑀𝑃𝑎

22.5 𝑀𝑃𝑎√𝑚 = (1)𝜎√𝜋(0.012𝑚)

22.5 𝑀𝑃𝑎√𝑚
𝜎=
(1)(√𝜋(0.012𝑚))

𝜎 = 115.882 𝑀𝑃𝑎
Question & Answers
Q: Use a neat graph with complete details to explain the difference between “true
stress – true strain” and “engineering stress – strain”.

Fully describe Bridgman’s analysis by using sketches and the equation indicated
below and indicate on this same graph the application of his correction changes on
the “true stress – true strain”.
(𝜎𝑥 )𝑎𝑣𝑔
𝜎=
(1 + 2𝑅/𝑎)(ln (1 + 𝑎/2𝑅))

A:

Description of Bridgman analysis:

• Bridgman made a mathematical analysis which provide a correction

• to the average axial stress to compensate for the introduction of transverse
stresses.
• This analysis was based on the following assumptions:
o The contour of the neck is approximated by the arc of a circle
o The cross section of the necked region remains circular throughout the
test.
o The von Mises’ criterion for yielding applies
o The strains are constant over the cross section of the neck

• According to Bridgman’s analysis, the uniaxial flow stress corresponding to

that which would exist in tension test if necking had not introduced triaxial
stresses is
(𝜎𝑥 )𝑎𝑣𝑔
𝜎=
(1 + 2𝑅/𝑎)(ln (1 + 𝑎/2𝑅))

where (𝜎𝑥 )𝑎𝑣𝑔 is the measured stress in the axial direction.

 Q: Anisotropy of mechanical properties

A:

Anisotropy:

• Refers to when the property of a material varies or

• depends on the direction in which measurement is taken.
• Two types: Crystallographic anisotropy and mechanical fibering
• Crystallographic anisotropy:
o Refers to preferred orientation or texturing of grains in
polycrystalline materials due to severe plastic deformation.
o Has significant impact on yield strength and ductility in non-ferrous
alloys.
o Can result in defects in certain application such as deep drawing.
o Can be eliminated by recrystallization annealing.
• Mechanical fibering:
o Caused by alignment of impurities, inclusions, or voids during plastic
working of a metal
o thereby having properties that varies with relative orientation of the
applied stress to that of the defect.
o Important in wrought steel forging and plates.
o Has impact on ductility.

Q: Strain hardening

A:

• Strain hardening cause an increase in yield strength.

• This is because plastic deformation creates dislocations.
• Increasing dislocation-dislocation interactions that impede dislocation
motion thus increasing the strength of the material.