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DOKUZ EYLÜL UNIVERSITY
MECHANICAL ENGINEERING DEPARTMENT
INTRODUCTION TO FRACTURE MECHANICS
STRESS CONCENTRATIONS AND STRESS INTENSITY FACTOR
Prof. Dr. M. Evren Toygar
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Fracture
Seperation of a materials into at least two or more pieces due to stress,
at temperatures below the melting point is called as fracture. Steps of
fracture are:
Crack initiation
Crack propagations
Depending on the ability of material to undergo plastic deformation can
be classifies as:
- Ductile Materials - most metals (not too cold)
Extensive plastic deformation a head of the crack
If crack is stable, then it resists further extension unless applied stress
- Brittle Materials – ceramics, ice, glass, cold metals
Relativelly little plastic deformation
Crack is unstable propagates rapidly without increase in applied
stress
Prof.Dr. M.Evren TOYGAR
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Prof.Dr. M.Evren TOYGAR
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TENSILE TEST FOR DUCTILE - BRITTLE FRACTURE
Prof.Dr. M.Evren TOYGAR
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Brittle fracture
Crack propagation is fast
Propagates nearly perpendicular to
direction of applied stress
Occurs with Little or no plastic
deformation
many pieces and small deformation
occur during brittle failure
Prof.Dr. M.Evren TOYGAR
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Ductile Fracture
(Cup-and-cone fracture in Al)
The stress causing the fracture is (σ).
The stress causing the sliding (τ).
Occurs with plastic deformation
one piece and large deformation
occur during ductile failure
For plastic deformation it should
be σapp> σyield. However, even
when (σapp) is in the elastic region
the fracture can occur
Prof.Dr. M.Evren TOYGAR
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Ductile vs Brittle Failure
Very
Ductile
Moderately
DuctileBrittle
Fracture
behavior:
Large Moderate%Ra or %El Small
• Ductile fracture is
nearly always desirable!
Ductile:
warning before
fracture
Brittle:
No
warning
Prof.Dr. M.Evren TOYGAR
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Adapted from Fig. 8.3, Callister 7e.
cup-and-cone fracture brittle fracture
Ductile vs Brittle Failure
Prof.Dr. M.Evren TOYGAR
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Prof.Dr. M.Evren TOYGAR
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Stress Concentrations
Fracture strength of a brittle solid is related to cohesive forces
between atoms.
Theoretical strength: ~E/10
Experimental strength ~ E/100 - E/10,000
Difference due to:
Stress concentration at microscopic flaws
Stress amplified at tips of micro-cracks etc., called stress raisers
Prof.Dr. M.Evren TOYGAR
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Stress Concentrations
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Stress concentration at an elliptical hole for a = 3c.
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Crack perpendicular to applied stress: maximum stress near crack tip
Stress Concentration
ot
/
t
om Ka
21
2
Prof.Dr. M.Evren TOYGAR
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where
K t = stress concentration factor
t = radius of curvature of crack tip
σo = applied stress
σm = stress at crack tip
2/1
t0
mt
a2K
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Concentration of Stress at Crack Tip
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r/h
sharper fillet radius
increasing w/h
0 0.5 1.01.0
1.5
2.0
2.5
Stress Conc. Factor, Kt
max
o
=
r , fillet
radius
w
h
o
max
During engineering design avoid sharp corners
Prof.Dr. M.Evren TOYGAR
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Stress Intensity Approach and Crack
aKI
The unit of KI is MPam
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Crack deformation mode.
I
ij
II
ij0r
fr2
Klim
II
ij
IIII
ij0r
fr2
Klim
III
ij
IIIIII
ij0r
fr2
Klim
Cauchy stress around a crack tip and Stress Intensity Factors
K value can be evaluated using standard mathematical approaches,
Prof.Dr. M.Evren TOYGAR
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Stress intensity factor K can be described as fracture toughness KIC of
materials (material resistance to crack propagation) under conditions of
Brittle fracture
In the presence of sharp crack
Under critical tensile loading
cIC aK
LEFM- Linear Elastic Fracture Mechanics
KIC : is the critical stress intensity factor for plane strain condition in
mode I failure.
ac : is the critical crack length in an infinite plate
app : is the applied stress
Stress Intensity Factor K
Prof.Dr. M.Evren TOYGAR
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For a body subjected far field biaxial stress 0, with a double ended crack of
length 2a, the stress state is given by (this is mode-I loading):
Stresses near and tip of the crack
31
2 2 22
Ixx
KCos Sin S
rin
31
2 2 22
Iyy
KCos Sin S
rin
2
3
222
SinSinCos
r
KIxy
Note the inverse square root (of r) singularity at the crack tip. The intensity of the
singularity is captured by KI (the Stress Intensity Factor).
At = 0 and r → the stresses (xx & yy) should tend to 0( see equations ((1) & (2)). This
implies that the equations should be used only close to crack tip (with little errors) or
additional terms must be added to the equations.
At = 0 and r →0 the stresses goes to infinity.
(1)
(2)
(3)
Fig.1
1ij
r
Prof.Dr. M.Evren TOYGAR
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Stress field equation
2
3
21
22
SinSinCos
r
K Ixx
r
fK Ixx
2
)(→
aKI 0
‘Shape correction factor’ related to ‘Geometry’
Indicates mode I ‘loading’
Half the crack length (for a fully internal crack)
“KI (the Stress Intensity Factor) quantifies the magnitude of the effect of stress singularity at
the crack tip. K has units of [MPam].
(1)
),( rfK Ixx
Prof.Dr. M.Evren TOYGAR
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Stresses in the systems with crack
When the crack at the center of a plate the stress andstress intensity factors terms can be given as :
I
ijII
ij fr
K
2
ij
f
rafaKI /
: Shape correction factor
Prof.Dr. M.Evren TOYGAR
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Center crack in an infinite plate subjected to tension
Stress Intensity Factor Aprroach K :
aK
32
2.12152.1256.01
w
a
w
a
w
a
h
2a
b
P
P
asec where w = 2b
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Inclined crack in a plate
Mode I and Mode II calculations of a plate subjected to
tension. (Mixed Mode Loading)
Prof.Dr. M.Evren TOYGAR
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Types of Stress:
Plane stress problem : the stress in z direction becomes
zero.
z = xz = yz = 0,
Plane strain problem : the strain in z direction becomes
zero
xz = yz = 0 and
z = (x + y).
Prof.Dr. M.Evren TOYGAR
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Geometries of Fracture
Center crack and finite plate:
32
2.12152.1256.01
w
a
w
a
w
a
w
asec
aK Stress intensity factor
infinite plate:
Prof.Dr. M.Evren TOYGAR
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Single sided notch, subjected to stress
aK Stress intensity factor
if( a
432
w
a42.30
w
a74.21
w
a56.10
w
a23.012.1
w) semi infinite plate β=1.12
One-sided cracked plate
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Double-sided notch :
32
w
a46.15
w
a79.4
w
a43.012.1
aK Stress intensity factor
a) if( a w) semi infinite plate β=1.12
b)32
w
a46.15
w
a79.4
w
a43.012.1
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Stress intensity factors Y wrt a/w ratio
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Eliptic Crack
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Semi Eliptic Surface crack
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The Shape correction factor (β)
The geometry of the crack and its relation to the body will play an important role on its
effect on fracture.
The factor β depends on the geometry of the specimen with the crack.
β =1 for the center crack in an infinite body.
β =1.12 for a edge crack.
β = 2/ for a embedded penny shaped crack.
β = 0.713 for a surface half-penny crack.
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The crack tip fields consists of two parts:
(i) singular part (which blow up near the crack tip) and
(ii) the non-singular part.
The region near the crack tip, where the singular part can describe the stress fields is the K-
Dominance region. This is the region where the stress intensity factor can be used to
characterize the crack tip stress fields.
If in materials where the crack tip plastic zone is smaller than the concepts of linear elastic
fracture mechanics (LEFM) can be used with little error.
Stress Intensity Factor K
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One of the important goals of fracture mechanics is to derive a material parameter, which
characterizes cracks in a material. This will be akin to yield stress (y) in a uniaxial tension
test (i.e. y is the critical value of stress, which if exceeded ( y) then yielding occurs).
The criterion (critical condition) for fracture in mode-I can be written as:
Fracture Toughness (Irwins’s K- Based)
ICI KK Where, KIC is the critical value of stress intensity factor (K) and is known
as Fracture Toughness
KIC is a material property (like yield stress) and can be determined for different materials
using standard testing methods. KIC is a microstructure sensitive property.
The focus here is the ‘local’ crack tip region and not ‘global’, as in the case of Griffith’s
approach.
All the restrictions/assumptions on K will apply to KIC:
(i) material has a liner elastic behaviour (i.e. no plastic deformation or other non-linear
behaviour),
(ii) inverse square root singularity exists at crack tip (eq. (1)),
(iii) the K-dominance region characterizes the crack tip.
r
fK Ixx
2
)(
(1)
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Material KIC [MPam]
Cast Iron 33
Low carbon steel 77
Stainless steel 220
Al alloy 2024-T3 33
Al alloy 7075-T6 28
Ti-6Al-4V 55
Inconel 600 (Ni based alloy) 110
Fracture Toughness (KIC) for some typical materials [1]
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Is KIC really a material property like y?
KIC (in mode-I loading) (KIIC & KIIIC under other modes of loading) are the material property,
independent of the geometry of the specimen. In reality, KIC depends on the specimen
geometry and loading conditions.
The value KIC is especially sensitive to the thickness of the specimen. A thick specimen
represents a state that is closer to plane strain condition, which tends to suppress plastic
deformation and hence promotes crack growth (i.e. the experimentally determined value of
KIC will be lower for a body in plane strain condition). On the other hand, if the specimen is
thin (small value ‘t’ in the figure), plastic deformation can take place and hence the measured
KIC will be higher (in this case if the extent of plastic deformation is large then KI will no
longer be a parameter which characterizes the crack tip accurately).
KIC is the fracture toughness value determined from ‘plane strain tests’.
Prof.Dr. M.Evren TOYGAR
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First one has to decide if the material is brittle or ductile (i.e. if the crack tip is sharp or
blunted).
If the material is brittle one has to decide if the material is linear elastic or not.
For linear elastic materials we can apply the concept of K and use material property KC
(usually in mode-I these quantities become KI and KIC). We could also use G & GC.
If the material is ductile then we need to determine if the plastic zone is small compared to
the K dominance zone. If yes then we can continue to use the concepts of K and G.
If crack tip plasticity is large, then we have to use concepts like J-integral and CTOD. The
crack tip stress fields in this case is given by the HRR fields. (Noting that technically JC is for
crack nucleation).
Essentially there are two approaches: global (energy based) and local (stress based).
For linear elastic materials the energy and stress field approaches can be considered
equivalent.
Prof.Dr. M.Evren TOYGAR
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Sample Problem: 3 mm thick tension panel 10 cm wide containing an edge
crack of 1 mm yielded at a load of 150 kN. However, at a load of 120 kN,
another panel of same material cracked into two pieces when the crack was 5
mm long. With this information, calculate the yield stress and fracture
toughness of the material. SIF for the edge crack is rafaKI /12.1
Prof.Dr. M.Evren TOYGAR