Introduction to FractureIntroduction Fracture mechanics approach – Implicit assumption – Crack...

R&DE (Engineers), DRDO

Introduction to Fracture

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Ramadas Chennamsetti

Introduction to Fracture


Introduction� Design of a component

� Yielding – Strength

� Deflection – Stiffness

� Buckling – critical load

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Ramadas Chennamsetti2

� Fatigue – Stress and Strain based

� Vibration – Resonance

� Impact – High strain rates

� Fracture - ???


Introduction � Design based on Strength of Materials /

Mechanics of Solids

� Strength based design – Check for allowable stress

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� Stiffness based design – Check for allowable deformation / deflection

� Presence of defects in the material – ideal

� Imperfections – Higher factor of safety


Introduction� Fracture – separation of a body in response

to applied load

CrackCrack tip Fracture – Relieving stress and shed excess

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stress and shed excess energy

Main focus – whether a known crack is likely to grow under a certain given loading condition


Introduction� Fracture mechanics approach – Implicit

assumption – Crack exists � Severity of the existing crack when loads

applied on the structure / component� Application of Fracture Mechanics (FM) to

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� Application of Fracture Mechanics (FM) to Crack growth – Fatigue failure

� 1920s – Griffith developed right ideas for growth of a crack – Working parameter

� Modern FM – 1948 – George Irwin –devised a working parameter


Introduction� Irwin’s work – Mainly for brittle materials –

Introduced Stress Intensity Factor (SIF) and Energy Release Rate (G) – Linear Elastic Fracture Mechanics (LEFM) – Plastic deformation negligible

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deformation negligible

� Irwin’s theory – application of FM to design problems

� Focus was on crack tip – not on the crack


Introduction� Crack Tip Opening Displacement (CTOD) –

1961 – Wells

� J – Integral – 1968 – Rice

� CTOD and J – integral – Ductile materials –

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� CTOD and J – integral – Ductile materials –large plastic zone at tip


Modes of fractureModes of Fracture –

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mode I

IKmode II

IIKmode III

IIIKOpening Sliding Tearing


Stresses at crack tip� From Linear Elastic Fracture Mechanics

(LEFM) the stresses near the crack tip in Mode I –

xxσσσσ

yyσσσσxyσσσσ

y

−

=

3sinsin

2

3sin

2sin1

cosθθ

θθ

θσσ

KI

x

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� The displacements near the crack tip –

θθθθr

xxσσσσ

x

+

=

2

3sin

2sin1

2

3sin

2sin

2cos

2θθ

θθθπτ

σr

KI

xy

y

2

2

cos ( 1 2sin )2 2

2 2 sin ( 1 2cos )2 2

Iu K r

v G

θ θκ

θ θπ κ

− + =

+ −

υκ 43−=Plane strain

Plane stressυυκ

+−=

1

3


Griffith’s theory � Theory of unstable crack growth

� Theory establishes a relation – unstable crack growth

� Basic underlying principle – Energy balance

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� Basic underlying principle – Energy balance

� Introducing a crack in a stressed / loaded component – release of strain energy

� Reduction in stiffness

� What happens to change in strain energy ??? or released strain energy


Griffith’s theory� An infinite body

� Linear elastic – subjected

to stress σ� Initially there was no crack βa

σ

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� Initially there was no crack

� Strain energy stored2a

βa

σ

VolE

Uo 2

2σ=


Griffith’s theory� Crack size = 2a – the crack surfaces –

traction free – no traction loads acting

� Material above and below – stress free to some extent

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� Stress relived portion – assuming triangular distribution

� Height of triangle = β� Griffith carried out extensive calculations

and experiments => β = π


Griffith’s theory� Strain energy after introducing crack = Strain

energy before introducing crack – Strain energy loss (relived)

( )o EUU ∆−==>

2

le of Volume2

1 σ

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( )

R

o

o

EE

taU

ataE

UUU

EUU

==∆=>

×=−=∆=>

∆−==>

22

2

22

12

2

1

le of Volume2

βσ

βσ

Energy released because of presence of crack


Griffith’s theory� When a body is cracked – breaking of

material bonds – Energy required for breaking bonds – Source of energy ???

� Loss / Deficiency in strain energy => caters for formation of crack

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for formation of crack

� Formation of crack => Generation of new traction free surfaces

� Energy used for breaking bonds stored as surface energy on newly formed surfaces


Griffith’s theory� Two key aspects in crack growth

� How much energy is released – strain energy –when crack advances

� Minimum energy required for the crack advance in forming two new surfaces

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in forming two new surfaces

� Some external work is done � Increase in strain energy

� Surface energy – crack growth


Griffith’s theory� Assume a crack of ‘2a’ size exists

σ

π(a+∆a)

Initial stress free area,

A1 = 2* ½ (2a*πa) = 2πa2

Crack grows by ∆a on both

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2a

πa

σ

∆a

Crack grows by ∆a on both sides

Stress free area after crack growth,

A2 = 2*1/2 (2(a+∆a)*π(a+ ∆a)

=> A2 = 2π(a+∆a)2


Griffith’s theory� Change in stress free area

( )[ ]( )

ataV

aaaaA

aaaAAA

∆=∆∆=∆=∆=>

−∆+=−=∆=>

πππ

π

4

422

2 2212

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� Change in strain energy ataV ∆=∆ π4

( )

(1)--- 2

42

12

1

2

2

2

E

at

a

E

EataE

U

VE

U

R

R

πσ

πσ

σ

=∆

∆=>

∆=∆=∆=>

∆=∆=>


Griffith’s theory� Surface energy required to create new area

( )

ta

W

atW

ss

ss

γ

γ

(2) --- 4

4

=∆

∆=>

∆=∆=>

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Ramadas Chennamsetti18aE

tE

at

da

dW

da

dE

s

s

sR

πσγ

γπσ

2

2

2

42

if place, esgrowth takcrack Unstable

==>

==>

=


Griffith’s theory� Expression

2

crack grow to

required Stress -length crack existing a''

2

γ

πσγ s aE =

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2

2

2

design tolerant Damage - sizecrack

allowable maximum - Stress - loadingGiven

2

πσγ

πγσ

sc

sc

Ea

a

E

=

==>


Griffith’s theory� γs = Specific surface energy – Surface energy

per unit area of crack surface

� Area of crack = 2(2a*t) = 4at

� Surface energy stored => 4at γs

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� Surface energy stored => 4at γs

� Energy balance

s

sR

atE

ta

WE

γπσ4

22

==>

==>


Griffith’s theory� Plot ER and Ws wrt crack length ‘a’

ss

R

atWE

taE

γ

πσ

4

22

=

=

ER

Ws

ER

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ssER

Ws

a∆a

ao

∆ER

∆Ws

Initial crack size = ao

Crack grows by ∆a

Strain energy released = ∆ERSurface energy required = ∆Ws


Griffith’s theory� Crack growth takes place, if

� Not satisfying above inequality – crack

sR WE ∆≥∆

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� Not satisfying above inequality – crack remains dormant

� Body acts as a strain energy reservoir

� Surface energy required can be obtained from external sources as well – increase in applied stress – No change in ∆Ws


Griffith’s theory� Strain energy used to break the bonds –

stored as surface energy

� Strain energy – source & surface energy –sink – Irreversible thermodynamic process

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� Crack growth – energy conversion process

� When a crack propagates – strain energy gets reduced and surface energy increases for a constant displacement case


Griffith’s theory� In the limiting case –

Wlt

Elt

WE

sR

sR

∆=∆=>

∆=∆=>

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da

dW

da

dEa

Wlt

a

Elt

sR

s

a

R

a

==>

∆∆=

∆∆=>

→∆→∆ 00

Check the slope of ER and WS – Satisfying above condition – onset of crack growth


Griffith’s theory� If strain energy release rate is higher than

required rate of surface energy – unstable crack growth

dWdE SR >

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� Difference in energy rates => kinetic energy

� Higher difference => Faster crack growth

da

dW

da

dE SR >


Griffith’s theory� Strain energy release rate per unit increase

in area during crack growth => Griffiths = G

atAE

taE

dA

dEG R

R22

2 , ,πσ ===

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E

a

da

dE

tdA

dEG

tdadAEdA

RR2

2

1

2

πσ====>

==>

Units of G => N.m/m2 or J/m2


Griffith’s theory� Crack area (A) and crack surface area (As)Crack area => Simply the area of crack

A = 2a*t = 2at

Crack surface area => Sum

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2a

Crack surface area => Sum of surface areas of all the crack surfaces

Crack surfaces = Two (top and bottom)

Each => 2at

As = 2(2at) = 4at = 2A


Griffith’s theory� Surface energy required for crack to grow

per unit area of extension – crack resistance – R

( )SS

atA

W γ2 areaCrack

area surface surface in total increase

∆=∆=>=∆

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( )

SS

SSSS

dA

dWR

AatatW

atA

γ

γγγ

2

24)2(2

2 areaCrack

===>

∆=∆=∆=∆=>∆=∆=>

Brittle fracture – no plastic deformation – R = surface energy – Elastic and plastic – R caters for surface energy and energy for plastic deformation at crack tip


Griffith’s theory� Crack start propagating if, G > = R � Strain energy release rate of a crack must be

greater than the crack resistance for the crack to grow

� Crack propagation occurs – G is sufficient to

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� Crack propagation occurs – G is sufficient to provide all the energy that is required for the crack formation

� Energy release rate is more than crack resistance, crack acquires KE – the growth speed may be faster than speed of a supersonic flight


Griffith’s theory� Onset of crack growth –

� RHS – completely depends on material

sEa γπσ 22 =

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� RHS – completely depends on material properties – Constant

� From the above -

2

1

σ∝a

Maximum allowable crack size depends on loading or

Loading decides the allowable crack size

Bigger cracks – lower loads


Griffith’s theory� Crack growth at different load / stress

conditions

1.4

1.6

1.8

Strain energy - High stress

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0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

ER & Ws

Crack size

Surface energy

Strain energy - Low stress


Griffith’s theory� Thin and thick brittle plates

� Thin plate – Plane stress condition � Thick plate – Plane strain condition

� Brittle – insignificant plastic zone at crack tip - LEFM

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tip - LEFM� Ductile materials – considerable plastic

deformation at crack tip – EPFM � Plane stress and planes strain – Poisson ratio� Allowable crack size of stress different


Griffith’s theory� Thin and thick plates – Griffith’s theory

strain Plane 1

2

stress Plane 2

22

2

s

s

Ea

Ea

γυ

πσ

γπσ

−=

=

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� RHS higher in plane strain � For a given load – higher crack size in thick

plate than in thin � For a given crack size – higher load in thick

plate than in thin plate

1 2 sγυ

πσ

−

=


Griffith’s theory� Taking square root

2

1

22

2

==>

−==> sc

Ea

Eac

γπσ

γυ

πσ

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Constant 1

2

1

2

2

2

=

−===>

−==>

sccIC

scc

EaK

Ea

γυ

πσ

γυ

πσ

KI – Stress Intensity Factor (SIF)

Kc – Stress Concentration Factor (SCF)


Griffith’s theory� Critical strain energy release rate = GIC = R

� Crack growth takes place, if

GE

aG IC

cπσ 2

≥=

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EEGK

RE

K

E

aG

E

SICIC

SICc

IC

γ

γπσ

2

2

case, limiting In the22

===>

====


Griffith’s theory� Typical values of critical stress intensity

factor –KIC

� Glass – 0.5 to 1 MPa m0.5

� Alloy steel – 150 MPa m0.5

� Aluminium alloy – 25 to 40 MPa m0.5

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� Aluminium alloy – 25 to 40 MPa m0.5


Griffith’s theory� Plate under constant displacement

APo

P

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� Plate stiffness reduced

� OAB – Strain energy released

O

B

CUo

P1


Griffith’s theory� No crack => More slope – high stiffness –

more force required to pull

� Presence of crack => lower slope – reduced stiffness – less force required to pull same length – u

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length – uo

� Difference in strain energy – Area OAB used to form new surface / break the bonds

� Major portion of strain energy release takes place at above and below crack


Fracture toughness – total potential

� A body with a crack subjected to external loading – work done on the body

� Utilization of this energy� Increase in strain energy � Utilization of energy to create two new surfaces

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� Utilization of energy to create two new surfaces

� In this process� Point of application of load – may or may not

move� Force moves => work is done by the force� Decrease in stiffness


Conservation of energy� Work performed per unit time = rate of

change of internal energy + plastic energy + kinetic energy + surface energy (crack formation) •••••

Γ+++= KUUW

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Γ+++= KUUW PE

Crack grows slowly – inertia effects negligible => KE = 0

Equation was wrt time. Crack grows with time – change time to crack area

AA

t

A

At ∂∂=

∂∂

∂∂=

∂∂ •

Chain rule

A = 2at 2a


Conservation of energy� Change variable to crack

ddUdUdWdA

dA

dA

dUA

dA

dUA

dA

dWA

PE

PE

Γ++==>

Γ++=••••

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Ramadas Chennamsetti41dA

d

dA

dU

dA

dW

dA

dUdA

d

dA

dU

dA

dU

dA

dWdA

d

dA

dU

dA

dU

dA

dW

PE

PE

PE

Γ+=

−−=>

Γ+=−=>

Γ++==>


Conservation of energy� Ideal brittle material => Negligible plastic

deformation => Up = 0

dA

d

dA

dW

dA

dU E Γ=

−−=>

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� Total potential

dAdAdA

dA

d

dA

d

WUVU EE

Γ=−

−=+=π

π


Surface energy� Surface energy per unit area => γs

� Total surface area, As = 2*(2at) = 4at = 2A

� Change in surface energy to form new crack of length ∆a

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of length ∆a

( )

ss

ssss

dA

d

A

AatA

γγ

γγγ

2

2

24

=Γ=>=∆∆Γ=>

∆=∆=∆=∆Γ

It was shown that, GI = 2γs


G – total potential� Surface energy and total potential

GRd

dA

d

dA

d

===−=>

Γ=−

γπ

π

2

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Is GRdA

d ===−=> γπ2

Impending of crack growth

RGI ≥


G – total potential� Relation between GI and π

GdAddA

dGI

=−=>

−==>

π

π

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dAGdUW

dAGddUW

WdUd

GdAd

IEext

IEext

extE

+=∆=>=−=−∆=>

∆−==−=>

ππ

π

External work done on the body => increase in elastic strain energy and formation of new surfaces


G – total potential� No movement of external forces, Wext = 0

dA

dUG E

I −=

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When there is no external work done, energy required for crack growth is obtained from the strain energy stored in the body

Conservation of energy => Decrease in strain energy = increase in surface energy


Compliance approach� Compliance => Inverse of stiffness

� Fracture toughness – compliance approach

CPukuPk

C ==>== ,1

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� Constant load� Constant displacement

� Increase in compliance with crack length –loss of stiffness

� Calculation of Fracture toughness, GI


Constant load� Consider a body with some initial crack of

length ‘a’- Load applied => Po

B D

P

Po

Write total potentials when crack sizes are ‘a’ and ‘a+da’

F

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A C Eu

u1 u2

a a+da

222

222

111

111

2

1

2

1

2

1

2

1

uPuPuP

VU

uPuPuP

VU

ooo

ooo

−=−=>

+=

−=−=>

+=

π

π

π1 = - Area ABF, π2 = - Area ADF

∆π = π2 - π1


Constant load� Potentials,

uPuP

uPuP

oo

oo

+−=−=∆=>

−=−=

2

1

2

1

2

1 ,

2

1

1212

2211

πππ

ππ

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( ) uPuuP

uPuP

oo

oo

∆−=−=∆=>

+−=−=∆=>

2

1

2

122

21

1212

π

πππ

Strain energy release rate, GI

( )dA

duPuP

dA

d

dA

dG o

oI 22

1 ==−= π


Constant load� Use compliance relation

dCP

du

CPu o

==>

=

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dA

dCP

dA

duPG

dA

dCP

dA

du

ooI

o

22

1 2

==

==>

dA

dCPG o

I 2

2

=

Experimental measurements required to estimate change in compliance with crack area


Constant displacement� Constant displacement – Fixed grip

condition

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2a 2(a+∆a)

Since the grip is fixed no external work is done by the forces, Wext = 0


Constant displacement� Load-displacement curve

P1 AD

P

a

Change in potential,

( )uPP o −=>−=∆ ve 2

112π

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uo

P2

O

B

C

E

u

a

a+da

dA

dPC

dA

dCPCPu

dA

dPu

dA

dG

Pu

o

oI

o

−==>=

−=−==>

∆=∆=>

2

12

1

π

π


Constant displacement� This gives,

−−=−=

−=

dCPu

dPuG

dA

dC

C

P

dA

dP

ooI

11

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−−=−=dAC

udA

uG ooI 22

dA

dCPGI 2

2

=

GI same in both constant load and displacement conditions


Constant load & displacement

� Comparison

B D

P

Po

a a+da

P1

P2

A

B

D

E

P

a

P = const u = const

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A C Eu

u1 u2uoO

B

Cu

a+da

U2 > U1 Increase in strain energy

Caters for crack growth

No external work done U2 < U1 Decrease in strain energy – used for crack growth


Energy release rate� Energy release rate referred as rate of strain

energy flux flowing towards a crack tip as the crack extends

� Linear and Nonlinear‘J’ integral –contour integral

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u

P Linear

U

- π

A

B

C

D

dA

dJ

π−=P

A

B

Cu

- π

U

Nonlinear

D

Area ‘ABD’

For linear elastic, G = J


Measurement of ‘GIC ’� Expression for GIC – Critical SERR

dA

dCPGIC 2

2

= Parameter to be measured experimentally => dC/dA

P dC/dA

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Po

P

uu1 u2 u3 u4

A1 A2 A3 A4

k1 k2 k3 k4

C

AAo

dC/dA


Measurement of ‘GIC ’� Size of crack => ao, corresponding area = Ao

� Load required to initiate crack growth, Pc

cIC dA

dCPG

2

2

=

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oAIC dA

G2

=

If GI is expressed in terms of stiffness,

oAo

cIC dA

dk

k

PG

2

2

2−= dk/dA is -ve


Griffith’s theory� Crack in stiff and flexible components

� Same force applied on stiff and flexible components having same size of crack

� Energy stored Flexible component stores

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� Energy stored

( )CK

U

K

PCPU

=∝=>

==

12

1

2

1 22

Flexible component stores more energy than stiffer component

More store of energy –availability of energy for crack propagation

Stiff bodies do not store more energy – less release


Surface energy� Surface energy per unit area – R – material

property� In Fracture Mechanics – R => total energy

consumed in propagation of crack� In ductile materials

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� In ductile materials � Plastic deformation takes place – energy

required to deform � Energy required for a crack to grow is much

larger than the surface energy

� Significant plastic deformation


Surface energy� Crack growth accompanied with plastic

deformation

( ) RwG fps ==+= 22 γγ

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� Most of metals – surface energy is much smaller than (1/1000th) of the total energy consumed in crack propagation

γp => Plastic work per unit area of surface created,

γs => Surface energy per unit area of surface created


Surface energy� Crack resistance (R) – depends on material

and geometry – thickness � Plastic zone size at the crack tip depends on

thickness � Bigger plastic zone – more dissipation of

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� Bigger plastic zone – more dissipation of energy – Higher crack resistance - Plane stress

� Smaller plastic zone – less dissipation of energy – Lower crack resistance – Plane strain


Crack in a pipe� Eg. Longitudinal crack of 10 mm is allowed

in a pipe. What is the allowable pressure?

P P

2a

2r

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2a

2a

KIC = 93 MPa m0.5

Allowable stress = 878 MPa


Crack in a pipe� Hoop stress causes crack opening – mode – I

� Considering this as an infinite plate with a crack of length 2a

rpσ max stress Hoop =>=

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cIC

cIC

at

rpK

aK

t

rp

π

πσ

σ

max

max

SIF Critical

stress Hoop

==>

=>=

=>=


Crack in a pipe� Maximum pressure

tKtp

at

rpK

IC

cIC

6

max

1093×===>

= π

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r

tp

r

t

a

K

r

tp

c

IC

6max

3max

1003.742

105

1093

×==>

×××===>

−ππ

Design based on Fracture


Crack in a pipe� Same maximum pressure causes Hoop stress –

Check for yield failure

t

rpYallowble

6max 10878×===σ

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r

tp

tallowble

6max 10878×==>

If design is based on fracture criteria, it does not fail in yielding


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Introduction to FractureIntroduction Fracture mechanics approach – Implicit assumption – Crack...

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Transcript of Introduction to FractureIntroduction Fracture mechanics approach – Implicit assumption – Crack...