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21 – Fracture and Fatigue Revision

EG2101 / EG2401 March 2015 Dr Rob Thornton Lecturer in Mechanics of Materials

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Fracture concepts

• Fracture:

– Initiation and propagation of cracks within a material

– Structure no longer sustains any applied loading

– Often occurs at nominal stresses/strains below those material is expected to sustain (why?)

• Design against fracture:

– Crack-free materials are difficult to produce

– Therefore, we design for non-propagation or controlled propagation of cracks

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Fracture mechanisms (1)

• Brittle (fast) fracture:

– Failure occurs without plastic deformation

– Crystalline materials split along defined planes

• Ductile fracture:

– Failure occurs following necking or shearing

– Substantial plastic deformation

Transgranular (Cleavage)

Intergranular

Rupture

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Fracture mechanisms (2)

• Type of fracture dependent on:

– Material bonding and structure

– Type of stress applied (e.g. tensile, shear, torsion)

– Rate of stress application

– Temperature (creep)

– Operating environment (corrosion)

– Component geometry (stress concentrators)

– Internal flaws (e.g. vacancies, precipitates, cracks etc.)

– External flaws (e.g. cracks, oxides etc.)

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Brittle fracture by cleavage

• Occurs in materials where the yield strength is high relative to the bonding strength: – e.g. ceramics, glasses

• As p.d. does not occur, stress at sharp crack tips becomes extremely high: – Sufficient to break

interatomic bonds

• Brittle or fast fracture can propagate at the speed of sound

Crack propagation by cleavage Jones and Ashby, (2011), Engineering Materials 1

So what do brittle fractures look like?

c

6

Brittle fracture – Zn at -120°C

• Very ductile at room temperature Not so much at -120°C when Zn has gone through its ductile-brittle transition phase

• Smooth surfaces are characteristic of brittle fractures

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Brittle fracture – Al-Li 8090

• Aluminium alloys are usually ductile – but not when inclusions form along grain boundaries… this is an example of an intergranular fracture

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Ductile fracture by tearing

• In ductile materials, yield strength is low relative to bonding strength, hence p.d. occurs near crack tip: – e.g. metals, polymers

• Within the plastic zone, voids nucleate and grow near defects

• Voids grow, coalesce (merge); allowing crack to propagate

Nucleation

Growth

Coalescence

Crack propagation by ductile tearing Jones and Ashby, (2011), Engineering Materials 1

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Ductile fracture – (Almost) pure Cu

• Cu is a very ductile material – and can display stable necking if processed correctly

• Ductile tearing can occur as voids nucleate and grow around precipitates or inclusions and then coalesce

• Rough surfaces are characteristic of ductile fracture

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Fracture mechanics approaches

• Energy balance criterion (Griffith, brittle materials):

– An existing crack will propagate if crack growth releases more stored energy than is absorbed by the creation of the new crack surface

– 𝑈 = 𝑈𝑒 + 𝑈𝑠 ⇒d𝑈

d𝑎> 0 for crack stability

• Stress intensity factors (Irwin, ductile materials):

– The elastic stress distribution in a loaded material is distorted near crack tips and can be characterised if we assume standard crack geometries

– 𝜎𝑖𝑗 𝑟, 𝜃 ≈𝐾

2𝜋𝑟𝑓𝑖𝑗 𝜃 where 𝐾 = 𝑌𝜎 𝜋𝑎

Objective: To determine the critical size of defect necessary for fast fracture to occur

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Energy balance criterion (1)

• Assume a plate contains an edge-crack (grossly enlarged!)

– Under an applied stress, crack grows by δa

• For crack growth, work must be done (energy must be input):

– δW represents work required to enlarge crack by δa

• Crack growth releases elastic energy but requires energy for crack surface:

– ⇒ δ𝑊 = δ𝑈𝑒 + δ𝑈𝑠

σ

σ

a δa

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Energy balance criterion (2)

• ⇒ δ𝑊 = δ𝑈𝑒 + δ𝑈𝑠

• Energy absorbed by new crack tip:

– δ𝑈𝑠 = 𝐺𝑐𝑡δ𝑎

• Gc is a material property:

– Toughness or critical strain energy release rate

• ‘Tough’ materials have high Gc:

– Copper, Gc ≈ 106 Jm-2

– Glass, Gc ≈ 10 Jm-2

• In tough materials it is difficult for cracks to propagate

σ

σ

a δa

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U

a

Energy balance criterion (6)

• 𝜎 𝜋𝑎𝑐 = 𝐸𝐺𝑐 or

𝜎𝑐 𝜋𝑎 = 𝐸𝐺𝑐

• Total energy, 𝑈 = 𝑈𝑒 + 𝑈𝑠

• Critical crack length ac:

– Unstable equilibrium

• If a < ac:

– Growth requires additional energy (stress)

• If a > ac:

– Growth reduces energy

– Spontaneous and catastrophic

ac

d𝑈

d𝑎= 0

𝑈𝑠 = 𝐺𝑐𝑡𝑎

𝑈𝑒 = −𝜎2

2𝐸

𝜋𝑎2𝑡

2

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Stress intensity factor (1)

• From our fast fracture condition, 𝜎 𝜋𝑎 = 𝐸𝐺𝑐, we can also see that: – A critical combination of stress and crack length exist

when fast fracture will commence

• Defining LHS as the stress intensity factor:

– 𝐾 = 𝜎 𝜋𝑎 (MN m-3/2)

• Defining RHS (only material properties) as the fracture toughness:

– 𝐾𝑐 = 𝐸𝐺𝑐 (MN m-3/2)

• Fast fracture occurs when:

– 𝐾 = 𝐾𝑐

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Stress intensity factor (2)

• Strictly the result 𝐾 = 𝜎 𝜋𝑎 is only valid for wide plates (thin, semi-finite materials)

• A correction factor must be applied:

– For an edge-crack in a semi-finite plate (W >> a) 𝐾 = 1.12𝜎 𝜋𝑎

– In general, for other geometries:

𝐾 = 𝑌𝜎 𝜋𝑎

• Values for Y can be found in data books

σ

σ

a

W

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Stress intensity factor (3)

σ

a

σy

Failure by yielding

Failure by fast fracture

σc

𝜎𝑐 =𝐾𝑐

𝜋𝑎

ay

𝜎𝑦 =𝐾𝑐𝜋𝑎𝑦

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What is fatigue?

• Fatigue: – Slow crack growth at loads less than that described by

the fast fracture criterion, 𝐾 = 𝐾𝑐 – Occurs due to cyclic loading

• Why is understanding fatigue important? – Estimated that 75% of all failures in engineering

components due to fatigue – e.g. De Havilland Comet failures

• Design against fatigue: – Minimise both initial size and rate of growth of cracks – Use of S-N curves to ensure stress cycling does not

exceed the fatigue limit of the material

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σy

σa

σa

εa

εa

Fatigue loading (2)

Linear-elastic Linear-elastic / yielding

Δεpl

Δεel

Δσ

Δεtot

Δσ

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101 102 103 104 105 106

∆𝜀𝑡𝑜𝑡

2 (

log

scal

e)

2𝑁𝑓 (log scale)

Fatigue loading (4)

• Fatigue curve results from elastic and plastic strain amplitudes: ∆𝜀𝑡𝑜𝑡

2≈

𝜎𝑓′ 2𝑁𝑓𝑏

𝐸+ 𝜀𝑓′ 2𝑁𝑓

𝑐

• Constants b and c determined from fitting test data; typically:

– -0.12 < b < -0.05

– -0.7 < c < -0.5

𝜎𝑓′

𝐸

𝜀𝑓′

𝜎𝑚, 𝜀𝑚 = 0

True fracture strain

True fracture stress

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Fatigue failure surfaces

• Characteristics:

– No necking prior to failure elastic strain

– Flat fracture surface

– ‘Beach’ marks visible

– Brittle or ductile fracture follows fatigue surface

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Fatigue failure – rotating steel shaft

• Initiation at stress concentrating feature

• Beach marks formed during each loading cycle

• Fracture occurs once component can no longer sustain applied stress

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Fatigue of cracked components (3)

Δσ

Δσ

a δ

Initial crack

Tension Crack widens by δ

Unloading Crack grows by ≈ δ

Tension Crack grows by da/dN

≈ δ

Fatigue crack growth Jones and Ashby, (2011), Engineering Materials 1

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104 105 106 107 108 109

500

400

300

200

100

Stress-cycle (S-N) curves

σa / MPa

Nf

Fatigue limit

0.4%C steel

2000 series Al-Cu

Wöhler curve: Stress amplitude (S) against logarithmic cycles to failure (Nf)

N

Endurance limit for N cycles

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Fatigue of cracked components (5)

• Crack growth per cycle (da/dN):

– Steady-state crack growth rate, d𝑎

d𝑁= 𝐴 ∆𝐾 𝑚

• If initial (a0) and failure lengths (af) are known:

𝑁𝑓 = d𝑁𝑁𝑓

0

⇒ 𝑁𝑓 = d𝑎

𝐴 ∆𝐾 𝑚

𝑎𝑓𝑎0

log

d𝑎

d𝑁

log ∆𝐾 K0 Kmax = Kc

Fast fracture

Thre

sho

ld

Paris’ Law

d𝑎

d𝑁= 𝐴 ∆𝐾 𝑚

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Variable loading

• Typically, Δσ changes during life of components (e.g. due to wind loading, haulage loads, fuel load):

– How do we calculate total cycles to failure?

• Miner’s Rule:

– The sum of the fractions of the cycles to fracture under each loading regime equals 1

𝑁𝑖

𝑁𝑓𝑖𝑖 = 1 =

𝑁1

𝑁𝑓1+

𝑁2

𝑁𝑓2+

𝑁3

𝑁𝑓3…

– Procedure: 1) For each stress range, divide number of cycles by calculated cycles to failure 2) Add fractions until the sum exceeds 1 – at which point fracture should have occurred

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Exam reminder

• Two and a half hours:

– 6 questions in two sections; answer 4 in total

– Part A – Answer one question out of two

– Part B – Answer three questions out of four

• Questions in Part A do not follow style of previous years’ examples:

– New lecturer, new content – new example slides!

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Good luck!