R&DE (Engineers), DRDO
rd_mech@yahoo.co.in
Introduction to Fracture
Ramadas Chennamsetti
Introduction
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Yielding – Strength
Deflection – Stiffness
Buckling – critical load
Fatigue – Stress and Strain based
Vibration – Resonance
Impact – High strain rates
Fracture - ???
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Design of a component
Introduction
Design based on Strength of Materials /
Mechanics of Solids
Strength based design – Check for allowable
stress
Stiffness based design – Check for allowable
deformation / deflection
Presence of defects in the material – ideal
Imperfections – Higher factor of safety
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Introduction
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Fracture – separation of a body in response
to applied load
Crack
Fracture – Relieving
stress and shed excess
energy
Main focus – whether a
known crack is likely to
grow under a certain
given loading condition
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Crack tip
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
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
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R&DE (Engineers), DRDO
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
Irwin’s theory – application of FM to design
problems
Focus was on crack tip – not on the crack
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R&DE (Engineers), DRDO
Introduction
R&DE (Engineers), DRDO
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Crack Tip Opening Displacement (CTOD) –
1961 – Wells
J – Integral – 1968 – Rice
CTOD and J – integral – Ductile materials –
large plastic zone at tip
Modes of fracture
R&DE (Engineers), DRDO
mode I
KI
Opening
mode II
mode III
K II
K III
Sliding
Tearing
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Modes of Fracture –
Stresses at crack tip
R&DE (Engineers), DRDO
From Linear Elastic Fracture Mechanics
(LEFM) the stresses near the crack tip in
Mode I –
θ 3θ
y
σ
yy
σ xy
σ xx
r
θ
x
The displacements near the crack tip –
 θ
2θ 
cos
(
κ
−
1
+
2sin
)  Plane strain

u 
KI r
2
2
=


v 
θ
θ
  2G 2π sin (κ + 1 − 2 cos 2 )  Plane stress
 2
2 
Ramadas Chennamsetti
κ = 3 − 4υ
3 −υ
κ=
1+υ
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

1
−
sin
sin

2
2
σ x 
θ
θ 3θ 
  KI

cos  sin sin
σ y  =
2
2
2 
2πr
τ 
 xy 
1+ sin θ sin 3θ 

2
2 
Griffith’s theory
Theory of unstable crack growth
Theory establishes a relation – unstable
crack growth
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
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R&DE (Engineers), DRDO
Griffith’s theory
R&DE (Engineers), DRDO
Uo =
σ
βa
2a
2
2E
Vol
Ramadas Chennamsetti
σ
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An infinite body
Linear elastic – subjected
to stress σ
Initially there was no crack
Strain energy stored
σ
Griffith’s theory
Crack size = 2a – the crack surfaces –
traction free – no traction loads acting
Material above and below – stress free to
some extent
Stress relived portion – assuming triangular
distribution
Height of triangle = β
Griffith carried out extensive calculations
and experiments => β = π
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Griffith’s theory
Strain energy after introducing crack = Strain
energy before introducing crack – Strain
energy loss (relived)
1σ2
(Volume of ∆ le )
=> U = U o −
2 E
2
1σ 
1

=> ∆ U = U o − U =
 2 × 2 a β at 
2 E 
2

2
2
σ βa t
=> ∆ U =
= ER
E
Energy released because of presence of crack
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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
Formation of crack => Generation of new
traction free surfaces
Energy used for breaking bonds stored as
surface energy on newly formed surfaces
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Griffith’s theory
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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
Increase in strain energy
Surface energy – crack growth
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Some external work is done
Griffith’s theory
R&DE (Engineers), DRDO
Assume a crack of ‘2a’ size exists
σ
Initial stress free area,
A1 = 2* ½ (2a*πa) = 2πa2
πa
∆a
2a
Crack grows by ∆a on both
sides
Stress free area after crack
growth,
A2 = 2*1/2 (2(a+∆a)*π(a+ ∆a)
σ
=> A2 = 2π(a+∆a)2
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π(a+∆a)
Griffith’s theory
R&DE (Engineers), DRDO
[
1σ2
=> ∆U =
∆V
2 E
1σ2
(4πa∆at ) = ∆ER
=> ∆U =
2 E
∆ER 2πσ 2 at
=>
=
- - - (1)
∆Ramadas
a
E
Chennamsetti
]
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Change in stress free area
2
2
=> ∆ A = A2 − A1 = 2π (a + ∆ a ) − a
=> ∆ A = 2π (2 a ∆ a ) = 4πa ∆ a
∆ V = 4πa ∆ at
Change in strain energy
Griffith’s theory
R&DE (Engineers), DRDO
Surface energy required to create new area
=> ∆Ws = γ s (4∆at )
dE R dWs
=
da
da
2πσ 2 at
=>
= 4γ s t
E
2
=> 2 Eγ s = σ πa
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∆Ws
=>
= 4γ s t - - - (2)
∆a
Unstable crack growth takes place, if
Griffith’s theory
R&DE (Engineers), DRDO
Expression
2Eγ s = σ πa
2
' a' existing crack length - Stress required
to grow crack
2Eγ s
=> σ =
πa
Given loading - Stress - maximumallowable
crack size - Damage tolerant design
ac =
2Eγ s
πσ
2
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2
c
Griffith’s theory
R&DE (Engineers), DRDO
=> ER = Ws
=>
πσ a t
2
E
2
= 4atγ s
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γs = Specific surface energy – Surface energy
per unit area of crack surface
Area of crack = 2(2a*t) = 4at
Surface energy stored => 4at γs
Energy balance
Griffith’s theory
R&DE (Engineers), DRDO
Plot ER and Ws wrt crack length ‘a’
2 2
πσ
a
t
ER
ER =
Ws
E
W
=
4
at
γ
s
s
E
R
Ws
ao
∆a
∆Ws
Crack grows by ∆a
∆ER
Strain energy
released = ∆ER
a
Ramadas Chennamsetti
Surface energy
required = ∆Ws
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Initial crack size = ao
Griffith’s theory
R&DE (Engineers), DRDO
Crack growth takes place, if
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
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∆E R ≥ ∆Ws
Griffith’s theory
Strain energy used to break the bonds –
stored as surface energy
Strain energy – source & surface energy –
sink – Irreversible thermodynamic process
Crack growth – energy conversion process
When a crack propagates – strain energy gets
reduced and surface energy increases for a
constant displacement case
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Griffith’s theory
R&DE (Engineers), DRDO
In the limiting case –
∆Ws
∆E R
=> lt
= lt
∆a → 0 ∆ a
∆a → 0 ∆ a
dE R dW s
=>
=
da
da
Check the slope of ER and WS – Satisfying above
condition – onset of crack growth
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=> ∆E R = ∆Ws
Griffith’s theory
R&DE (Engineers), DRDO
If strain energy release rate is higher than
required rate of surface energy – unstable
crack growth
Difference in energy rates => kinetic energy
Higher difference => Faster crack growth
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dER dWS
>
da
da
Griffith’s theory
R&DE (Engineers), DRDO
Strain energy release rate per unit increase
in area during crack growth => Griffiths = G
2
A = 2at
dER 1 dER πσ a
=> G =
=
=
dA 2t da
E
2
Units of G => N.m/m2 or J/m2
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dER
πσ a t
G=
, ER =
,
dA
E
=> dA = 2tda
2
Griffith’s theory
R&DE (Engineers), DRDO
Crack area (A) and crack surface area (As)
Crack area => Simply the
area of crack
A = 2a*t = 2at
2a
Crack surfaces = Two (top
and bottom)
Each => 2at
As = 2(2at) = 4at = 2A
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Crack surface area => Sum
of surface areas of all the
crack surfaces
Griffith’s theory
R&DE (Engineers), DRDO
Surface energy required for crack to grow
per unit area of extension – crack resistance
–R
∆WS = γ S (increase in total surface surface area )
Crack area => ∆A = 2∆at
dWS
=> R =
= 2γ S
dA
Brittle fracture – no plastic deformation – R = surface
energy – Elastic and plastic – R caters for surface
energy and energy Ramadas
for plastic
deformation at crack tip
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28
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=> ∆WS = γ S (2(2∆at ) ) = 4∆atγ S = 2γ S ∆A
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
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
29
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Griffith’s theory
R&DE (Engineers), DRDO
Onset of crack growth –
σ πa = 2 Eγ s
2
depends on loading or
a∝
1
σ
2
Loading decides the allowable
crack size
Bigger cracks – lower loads
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RHS – completely depends on material
properties – Constant
From the above - Maximum allowable crack size
Griffith’s theory
R&DE (Engineers), DRDO
Crack growth at different load / stress
conditions
1.8
1.6
1.4
Strain energy - High stress
1
0.8
Strain energy - Low stress
0.6
0.4
Surface energy
0.2
0
0
0.2
0.4
0.6
0.8
Ramadas Crack
Chennamsetti
size
1
1.231
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ER & Ws
1.2
Griffith’s theory
R&DE (Engineers), DRDO
Thin and thick brittle plates
Brittle – insignificant plastic zone at crack
tip - LEFM
Ductile materials – considerable plastic
deformation at crack tip – EPFM
Plane stress and planes strain – Poisson ratio
Allowable crack size of stress different
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Thin plate – Plane stress condition
Thick plate – Plane strain condition
Griffith’s theory
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Thin and thick plates – Griffith’s theory
σ πa = 2 Eγ s
Plane stress
 E 
σ πa = 2
γ
2  s
 1−υ 
Plane strain
2
2
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
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RHS higher in plane strain
Griffith’s theory
R&DE (Engineers), DRDO
Taking square root
 E 
=> σ c πac = 2
γ
2  s
 1−υ 
2
=> K IC = σ c
 E 
πac = 2
γ = Constant
2  s
 1−υ 
KI – Stress Intensity Factor (SIF)
Kc – Stress Concentration Factor (SCF)
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=> σ c
 E 
πac = 2
γ
2  s
 1−υ 
Griffith’s theory
R&DE (Engineers), DRDO
Critical strain energy release rate = GIC = R
Crack growth takes place, if
G=
πσ a
2
c
≥ GIC
πσ a
2
c
2
IC
K
GIC =
=
= 2γ S = R
E
E
=> K IC = GIC E = 2γ S E
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E
In the limiting case,
Griffith’s theory
R&DE (Engineers), DRDO
Typical values of critical stress intensity
factor – KIC
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Glass – 0.5 to 1 MPa m0.5
Alloy steel – 150 MPa m0.5
Aluminium alloy – 25 to 40 MPa m0.5
Griffith’s theory
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Plate under constant displacement
Po
A
P1
B
Plate stiffness reduced
OAB – Strain energy released
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O
C
Uo
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 – 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
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Fracture toughness – total potential
R&DE (Engineers), DRDO
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
Point of application of load – may or may not
move
Force moves => work is done by the force
Decrease in stiffness
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In this process
Conservation of energy
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Work performed per unit time = rate of
change of internal energy + plastic energy +
kinetic energy + surface energy (crack
•
•
•
•
•
formation)
W = UE +UP + K + Γ
Equation was wrt time. Crack grows with time – change
time to crack area
∂
∂ ∂A • ∂ Chain rule
=
=A
A = 2at
∂t ∂A ∂Ramadas
t
∂
A
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2a
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Crack grows slowly – inertia effects negligible => KE = 0
Conservation of energy
R&DE (Engineers), DRDO
Change variable to crack
dW • dU E • dU P • dΓ
A
=A
+A
+A
dA
dA
dA
dA
dW dU E dU P dΓ
=>
=
+
+
dA
dA
dA dA
dW dU E dU P dΓ
=>
−
=
+
dA
dA
dA dA
 dU E dW  dU P dΓ
=> −
−
+
=
dA dA
 dARamadasdA

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41
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•
Conservation of energy
R&DE (Engineers), DRDO
Ideal brittle material => Negligible plastic
deformation => Up = 0
 dU E dW  d Γ
=> −
−
=
dA  dA
 dA
π = U E +V = U E −W
dπ dΓ
−
=
dA
dA
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Total potential
Surface energy
R&DE (Engineers), DRDO
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
∆Γ
dΓ
=>
= 2γ s =>
= 2γ s
∆A
dA
It was shown that, GI = 2γs
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∆Γ = γ s ∆As = γ s (4∆at ) = 2γ s ∆A
G – total potential
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Surface energy and total potential
Impending of crack growth
GI ≥ R
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dπ d Γ
−
=
dA dA
dπ
=> −
= 2γ s = R = GI
dA
G – total potential
R&DE (Engineers), DRDO
Relation between GI and π
=> ∆Wext − dU E = − dπ = G I dA
=> ∆Wext = dU E + G I dA
External work done on the body => increase in
elastic strain energy
and formation of new surfaces
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dπ
=> G I = −
dA
=> − dπ = GdA
dπ = dU E − ∆Wext
G – total potential
R&DE (Engineers), DRDO
No movement of external forces, Wext = 0
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
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dU E
GI = −
dA
Compliance approach
R&DE (Engineers), DRDO
Compliance => Inverse of stiffness
P = ku => u = CP
Fracture toughness – compliance approach
Constant load
Constant displacement
Increase in compliance with crack length –
loss of stiffness
Calculation of Fracture toughness, GI
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1
C= ,
k
Constant load
R&DE (Engineers), DRDO
Consider a body with some initial crack of
length ‘a’- Load applied => Po
B
a
D
π 1 = U1 + V1
a+da
u1
A
Write total potentials when
crack sizes are ‘a’ and ‘a+da’
C
u2
E
u
1
1
=> Po u1 − Pou1 = − Po u1
2
2
π 2 = U 2 + V2
π1 = - Area ABF, π2 = - Area ADF
∆π = π2 - π1
1
1
=> Po u2 − Po u2 = − Po u2
2
2 48
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Po
P
F
Constant load
R&DE (Engineers), DRDO
1
1
π 1 = − Pou1 , π 2 = − Pou2
2
2
1
1
=> ∆π = π 2 − π 1 = − Pou2 + Pou1
2
2
1
1
=> ∆π = Po (u1 − u2 ) = − Po ∆u
2
2
Strain energy release rate, GI
Po du
dπ 1 d
(Pou ) =
GI = −
=
dA
2 dA
2 dA
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Potentials,
Constant load
R&DE (Engineers), DRDO
Use compliance relation
du
dC
=>
= Po
dA
dA
2
1 du Po dC
GI = Po
=
2 dA 2 dA
2
o
P dC
GI =
2 dA
Experimental measurements required to estimate
change in compliance with crack area
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u = CPo
Constant displacement
R&DE (Engineers), DRDO
Constant displacement – Fixed grip
condition
2a
Since the grip is fixed no external work is
done by the forces, Wext = 0
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2(a+∆a)
Constant displacement
R&DE (Engineers), DRDO
Load-displacement curve
P
P1
D
A
a
P2 E
B
a+da
C
O
uo
u
1
∆π = (P2 − P1 ) uo => − ve
2
1
=> ∆π = ∆Puo
2
dπ
1 dP
=> GI = −
= − uo
2 dA
dA
dC
dP
uo = CP => P
= −C
dA
dA52
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Change in potential,
Constant displacement
R&DE (Engineers), DRDO
This gives,
dP
P dC
=−
dA
C dA
1 dP
1  P dC 
GI = − u o
= − uo  −

2 dA
2  C dA 
P dC
GI =
2 dA
GI same in both constant load and displacement
conditions
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2
Constant load & displacement
R&DE (Engineers), DRDO
Comparison
P
P
P = const
B
Po
a
D
P1
D
u = const
A
a
P2 E
a+da
B
a+da
A
C
u2
E
U2 > U1 Increase in strain
energy
Caters for crack growth
C
u
O
uo
u
No external work done
U2 < U1 Decrease in strain
energy – used for crack
growth
54
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u1
Energy release rate
R&DE (Engineers), DRDO
P
D
Linear
B
-π
P Nonlinear
B
D
-π
U
U
C
A
dπ
J =−
dA
u A
Area ‘ABD’
C
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u
For linear elastic,
G=J
55
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Energy release rate referred as rate of strain
energy flux flowing towards a crack tip as
the crack extends
‘J’ integral –
Linear and Nonlinear
contour integral
Measurement of ‘GIC’
R&DE (Engineers), DRDO
Expression for GIC – Critical SERR
2
P dC
GIC =
2 dA
A1
A2
k1 k2 k3
u1
u2
A3
A4
dC/dA
C
k4
u3
u4 u
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A
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P
Po
Parameter to be measured
experimentally => dC/dA
Measurement of ‘GIC’
R&DE (Engineers), DRDO
Size of crack => ao, corresponding area = Ao
Load required to initiate crack growth, Pc
2
c
Ao
If GI is expressed in terms of stiffness,
2
c
2
o
P dk
GIC = −
2k dA Ao
Ramadas Chennamsetti
dk/dA is -ve
57
rd_mech@yahoo.co.in
P dC
GIC =
2 dA
Griffith’s theory
R&DE (Engineers), DRDO
Crack in stiff and flexible components
Same force applied on stiff and flexible
components having same size of crack
Energy stored
Flexible component stores
1 2
1 P
U = P C=
2
2 K
1
(= C )
=> U ∝
K
More store of energy –
availability of energy for
crack propagation
Stiff bodies do not store
more energy – less release
Ramadas Chennamsetti
58
rd_mech@yahoo.co.in
2
more energy than stiffer
component
Surface energy
R&DE (Engineers), DRDO
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
Ramadas Chennamsetti
59
rd_mech@yahoo.co.in
Surface energy per unit area – R – material
property
In Fracture Mechanics – R => total energy
consumed in propagation of crack
In ductile materials
Surface energy
R&DE (Engineers), DRDO
Crack growth accompanied with plastic
deformation
G = 2 (γ s + γ p ) = 2 w f = R
γp => Plastic work per unit area of surface created,
Most of metals – surface energy is much
smaller than (1/1000th) of the total energy
consumed in crack propagation
Ramadas Chennamsetti
60
rd_mech@yahoo.co.in
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
energy – Higher crack resistance - Plane
stress
Smaller plastic zone – less dissipation of
energy – Lower crack resistance – Plane
strain
Ramadas Chennamsetti
61
rd_mech@yahoo.co.in
R&DE (Engineers), DRDO
Crack in a pipe
R&DE (Engineers), DRDO
Eg. Longitudinal crack of 10 mm is allowed
in a pipe. What is the allowable pressure?
P
P
2r
KIC = 93 MPa m0.5
2a
Allowable stress = 878 MPa
Ramadas Chennamsetti
62
rd_mech@yahoo.co.in
2a
Crack in a pipe
R&DE (Engineers), DRDO
pmax r
σ=
=> Hoop stress
t
K IC = σ πac => Critical SIF
=> K IC
pmax r
=
πac
t
Ramadas Chennamsetti
63
rd_mech@yahoo.co.in
Hoop stress causes crack opening – mode – I
Considering this as an infinite plate with a
crack of length 2a
Crack in a pipe
R&DE (Engineers), DRDO
Maximum pressure
=> pmax
t
= 742.03 ×10
r
6
Design based on Fracture
Ramadas Chennamsetti
64
rd_mech@yahoo.co.in
=> pmax
pmax r
K IC =
πac
t
6
t K IC
t 93
93×
×10
=
=
r πac r π × 5 ×10 −3
Crack in a pipe
R&DE (Engineers), DRDO
pmax r
6
σ allowble = Y =
= 878×10
t
6 t
=> pmax = 878×10
r
If design is based on fracture criteria, it
does not fail in yielding
Ramadas Chennamsetti
65
rd_mech@yahoo.co.in
Same maximum pressure causes Hoop stress –
Check for yield failure
Ramadas Chennamsetti
66
rd_mech@yahoo.co.in
R&DE (Engineers), DRDO