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Fracture Mechanics
Brittle fracture
Fracture mechanics is used to formulate quantitatively
The degree of Safety of a structure against brittle fracture
The conditions necessary for crack initiation, propagation
and arrest
The residual life in a component subjected to
dynamic/fatigue loading
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Fracture mechanics identifies three primary factors that control the
susceptibility
of a structure to brittle failure.
1. Material Fracture Toughness. Material fracture toughness may be
defined as the ability to carry loads or deform plastically in the
presence of a notch. It may be described in terms of the critical
stress intensity factor, KIc, under a variety of conditions. (These
terms and conditions are fully discussed in the following chapters.)
2. Crack Size. Fractures initiate from discontinuities that can vary from
extremely small cracks to much larger weld or fatigue cracks.Furthermore,
although good fabrication practice and inspection can minimize the
size and
number of cracks, most complex mechanical components cannot be
fabricated without discontinuities of one type or another.3. Stress Level. For the most part, tensile stresses are necessary for
brittle
fracture to occur. These stresses are determined by a stress analysis
of the
particular component.
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Fracture at the Atomic level
Two atoms or a set of atoms are bonded
together by cohesive energy or bond energy.
Two atoms (or sets of atoms) are said to be
fractured if the bonds between the two atoms
(or sets of atoms) are broken by externally
applied tensile load
Theoretical Cohesive Stress
If a tensile force T is applied to separate the
two atoms, then bond or cohesive energy is
(2.1)
Where is the equilibrium spacingbetween two atoms.
Idealizing force-displacement relation as one
half of sine wave
(2.2)
ox
Tdx
x o
x
CT sin( )
+ +
xo
BondEnergy
CohesiveForce
EquilibriumDistance xo
Po
ten
tia
lE
nergy
Distance
Repulsion
Attraction
Tension
Compression
App
lie
dF
orce
k
BondEnergy
Distance
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Theoretical Cohesive Stress (Contd.)
Assuming that the origin is defined at and for small
displacement relationship is assumed to be linear such
that Hence force-displacement
relationship is given by
(2.2)
Bond stiffness k is given by
(2.3)
If there are n bonds acing per unit area and assuming
as gage length and multiplying eq. 2.3 by n then k
becomes youngs modulus and beecomes cohesive
stress such that
(2.4)
Or (2.5)
If is assumed to be approximately equal to the atomic
spacing
+ +
xo
BondEnergy
Cohesive
Force
EquilibriumDistance xo
Po
ten
tia
lE
nergy
Distance
Repulsion
Attraction
Tension
Compression
App
lie
dF
orce
k
BondEnergy
Distance
ox
xx
sin( )
C
xT T
CT
k
ox
ox
CT
C
c
o
Ex
c
E
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Fracture stress for realistic materialInglis (1913) analyzed for the flat plate with an
elliptical hole with major axis 2a and minor axis 2b,
subjected to far end stress The stress at the tip of
the major axis (point A) is given by
(2.8)
The ratio is defined as the stress
concentration factor,When a = b, it is a circular hole, then
When b is very very small, Inglis define radius of
curvature as
(2.9)
And the tip stress as
(2.10)
2a
2b
A
A
A
2a1
b
A
tktk 3.
2b
a
A
a1 a
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Fracture stress for realistic material (contd.)
When a >> b eq. 2.10 becomes
(2.11)
For a sharp crack, a >>> b, and stress at the crack tip tends to
Assuming that for a metal, plastic deformation is zero and the sharpest
crack may have root radius as atomic spacing then the stress is
given by
(2.12)
When far end stress reaches fracture stress , crack propagates and
the stress at A reaches cohesive stress then using eq. 2.7
(2.13)
This would
A
a2
0
ox
A
o
a2x
A C f
1/ 2
sf
E
4a
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Griffiths Energy balance approach
First documented paper on fracture
(1920)
Considered as father of FractureMechanics
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A A Griffith laid the foundations of modern fracture mechanics by
designing a criterion for fast fracture. He assumed that pre-
existing flaws propagate under the influence of an applied stressonly if the total energy of the system is thereby reduced. Thus,
Griffith's theory is notconcerned with crack tip processes or the
micromechanisms by which a crack advances.
Griffiths Energy balance approach (Contd.)
2a
X
Y
B
Griffith proposed that There is a simpleenergy balance consisting of the decrease
in potential energy with in the stressed
body due to crack extension and this
decrease is balanced by increase in surfaceenergy due to increased crack surface
Griffith theory establishes theoretical strength of
brittle material and relationship between fracture
strength and flaw size af
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2a
X
Y
B
Griffiths Energy balance approach (Contd.)
The initial strain energy for the uncracked plate
per thickness is
(2.14)
On creating a crack of size 2a, the tensile force
on an element ds on elliptic hole is relaxed
from to zero. The elastic strain energyreleased per unit width due to introduction of a
crack of length 2a is given by
(2.15)
2
iA
U dA2E
a1
a 20
U 4 dx v
dx
where displacement
v a sinE
usin g x a cos
2 2
a
aU
E
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Griffiths Energy balance approach (Contd.)
2a
X
Y
B
External work = (2.16)
The potential or internal energy of the body is
Due to creation of new surface increase in
surface energy is
(2.17)
The total elastic energy of the crackedplate is
(2.18)
wU Fdy,
where F= resultant force = area
=total relative displacement
p i a wU =U +U -U
sU = 4a
2 2 2
t sA
aU dA Fdy 4a
2E E
P1
P2
(a)
(a+d
a)
L
oa
d,
P
Displacement, v
Crack beginsto grow from
length (a)
Crack islonger by an
increment (da)
2 2
a
aU
E
v
G iffi h E b l h (C d )
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Griffiths Energy balance approach (Contd.)
Energy
,U
Cracklength, a
Surfa
ceEnerg
yU
= 4a
s
2 2
a
aU
E
Elastic Strainenergy released
Total energy
Ra
tes,
G,
s
Potential energyrelease rate G =
Syrface energy/unitextension =
U
a
Cracklength, a
ac
UnstableStable
(a)
(b)
(a) Variation of Energy with Crack length
(b) Variation of energy rates with crack length
The variation of with crack
extension should be minimum
Denoting as during fracture
(2.19)for plane stress
(2.20)
for plane strain
tU
2
t
s
dU 2 a
0 4 0da E
f1/ 2
sf
2E
a
1/ 2
sf 2
2E
a(1 )
The Griffith theory is obeyed by
materials which fail in a completely
brittle elastic manner, e.g. glass,
mica, diamond and refractory
metals.
G iffith E b l h (C td )
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Griffiths Energy balance approach (Contd.)
Griffith extrapolated surface tension values of soda lime glass
from high temperature to obtain the value at room temperature as
Using value of E = 62GPa,The value of as 0.15From the experimental study on spherical vessels he
calculated as 0.250.28
However, it is important to note that according to the Griffiththeory, it is impossible to initiate brittle fracture unless pre-
existing defects are present, so that fracture is always considered
to be propagation- (rather than nucleation-) controlled; this is a
serious short-coming of the theory.
2
s 0.54J / m . 1/ 2
s2E
MPa m.1/ 2
sc
2Ea
MPa m.
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Modification for Ductile Materials
For more ductile materials (e.g.metals and plastics) it is found that
the functional form of the Griffith relationship is still obeyed, i.e.
. However, the proportionality constant can be used to
evaluate s (provided E is known) and if this is done, one finds thevalue is many orders of magnitude higher than what is known to be
the true value of the surface energy (which can be determined by
other means). For these materials plastic deformation accompanies
crack propagation even though fracture is macroscopically brittle;The released strain energy is then largely dissipated by producing
localized plastic flow at the crack tip. Irwin and Orowan modified
the Griffith theory and came out with an expression
Where prepresents energy expended in plastic work. Typically for
cleavage in metallic materials p=104J/m2 and s=1 J/m
2. Since p>>
swe have
1/ 2
s pf
2E( )a
1/ 2
pf 2E
a
1/ 2
f a
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Strain Energy Release RateThe strain energy release rate usually referred to
Note that the strain energy release rate is respect to crack length and
most definitely not time. Fracture occurs when reaches a critical
value which is denoted .
At fracture we have so that
One disadvantage of using is that in order to determine it is
necessary to know E as well as . This can be a problem with somematerials, eg polymers and composites, where varies with
composition and processing. In practice, it is usually more
convenient to combine E and in a single fracture toughness
parameter where . Then can be simply determined
experimentally using procedures which are well established.
dUG
da
cG
cG G1/ 2
cf
1 EG
Y a
cG f
cG
cG cK2
c cK EGcK
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LINEAR ELASTIC FRACTURE MECHANICS (LEFM)
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LINEAR ELASTIC FRACTURE MECHANICS (LEFM)For LEFM the structure obeys Hookes law and global behavior is linear
and if any local small scale crack tip plasticity is ignored
The fundamental principle of fracture mechanics is that the stress field around a
crack tip being characterized by stress intensity factor K which is related to both
the stress and the size of the flaw. The analytic development of the stress
intensity factor is described for a number of common specimen and crack
geometries below.
The three modes of fracture
Mode I - Opening mode: where the crack surfaces separate symmetrically
with respect to the plane occupied by the crack prior to the deformation(results from normal stresses perpendicular to the crack plane);
Mode II - Sliding mode: where the crack surfaces glide over one another in
opposite directions but in the same plane (results from in-plane shear); and
Mode III - Tearing mode: where the crack surfaces are displaced in the
LINEAR ELASTIC FRACTURE MECHANICS (C d )
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In the 1950s Irwin [7] and coworkers introduced the concept of stress
intensity factor, which defines the stress field around the crack tip, taking
into account crack length, applied stress and shape factor Y( which
accounts for finite size of the component and local geometric features).The Airy stress function.
In stress analysis each point, x,y,z, of a stressed solid undergoes the
stresses; xy, z, xy, xz,yz. With reference to figure 2.3, when a body
is loaded and these loads are within the same plane, say the x-y plane,
two different loading conditions are possible:
LINEAR ELASTIC FRACTURE MECHANICS (Contd.)
CrackPlane
ThicknessB
ThicknessB
z z
z za
Plane StressPlane Strain
y
X
yy
1.plane stress (PSS), when the
thickness of the body is
comparable to the size of the
plastic zone and a free
contraction of lateral surfacesoccurs, and,2.plane strain (PSN), when
the specimen is thick enough
to avoid contraction in the
thickness z-direction.
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In the former case, the overall stress state is reduced to the
three components; x, y, xy, since; z, xz, yz= 0, while, in
the latter case, a normal stress, z, is induced which
prevents the z
displacement, ez= w = 0. Hence, from Hooke's law:
z= (x+y)where is Poisson's ratio.
For plane problems, the equilibrium conditions are:
Ifis the Airys stress function satisfying the biharmonic
compatibility Conditions
x xy y xy
x y y x0 0;
4
0
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Then
For problems with crack tip Westergaard introduced Airys stress
function as
WhereZis an analytic complex function
2 2 2
x y xy2 2, ,
y x xy
Re[ ] y Im[Z]Z
Z z z y z z x iy Re[ ] Im[ ] ; = +
And are 2nd and 1st integrals ofZ(z)
Then the stresses are given byZ,Z
2'
x 2
2'
y 2
2'
xy
'
Re[Z] y Im[Z ]y
Re[Z] y Im[Z ]x
y Im[Z ]xy
where Z =dZ dz
O i d l i M d I
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Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a subjected to a biaxial
State of stress. Defining:
Boundary Conditions :At infinity
On crack faces
x y xy| z | , 0
x xya x a;y 0 0
s
s
x
y
2a
s
2 2zZ
z a
By replacing z byz+a, origin shifted to crack tip.
Zz a
z z a
b 2
And when |z|0 at the vicinity of the crack tip
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And when |z|0 at the vicinity of the crack tip
KImust be real and a constant at the crack tip. This is due to a
Singularity given by
The parameter KI is called thestress intensi ty factor for opening
mode I.
Z a
az
K
z
K a
I
I
2 2
1
z
Since origin is shifted to crack
tip, it is easier to use polar
Coordinates, Using
Further Simplification gives:
z ei
K 3
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Ix
Iy
Ixy
K 3cos 1 sin sin
2 2 22 r
K 3cos 1 sin sin
2 2 22 r
K 3sin cos cos2 2 22 r
Iij ij IK
In general f and K Y a2 r
where Y = configuration factor
From Hookes law, displacement field can be obtained as
2
I
2I
2(1 ) r 1u K cos sin
E 2 2 2 2
2(1 ) r 1v K sin cosE 2 2 2 2
where u, v = displacements in x, y directions
(3 4 ) for plane stress problems
3 for plane strain problems1
The vertical displacements at any position along x-axis ( 0 is
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The vertical displacements at any position along x-axis ( 0 is
given by
The strain energy required for creation of crack is given by the
work done by force acting on the crack face while relaxing the
stress to zero
2 2
22 2
v a x for plane stressE
(1 )v a x for plane strain
E
x
v
x
y
a
2a a2 2 2 2
a a0 0
2 2
1 U Fv
2For plane stress For plane strain
(1 )U 4 a x dx U 4 a x dx
E E
a
E
2 2 2
a
2 2 2
I I
2
II
a (1 )
E
The strain energy release rate is given by G dU da
a (1 )aG = G =
E E
KG =E
2 2
II K (1 ) G =
E
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Boundary Conditions :
At infinity
On crack faces
x y xy 0| z | 0,
x xya x a;y 0 0
With usual simplification would give the stresses as
IIx
IIy
IIxy
K 3cos cos 2 cos cos
2 2 2 22 r
K 3cos sin cos2 2 22 r
K 3cos 1 sin sin
2 2 22 r
Displacement components are given by
II
II
K ru (1 )sin 2 cos
E 2 2
K rv (1 )cos 2 cos
E 2 2
K a
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II o
2
II
2 2
I
I
K a
KG = for plane stress
E
K (1 )G = for plane strain
E
Tearing mode analysis or Mode 3
In this case the crack is displaced along z-axis. Here
the displacements u and v are set to zero and hence
x y xy yx
xy yx yz zy
yzxz
2 22
2 2
0
w w and
x y
the equilibrium equation is written as
0x y
Strain displacement relationship is given by
w w
w 0x y
e e
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xy yz
Z
if w is taken as
1w Im[ ]
G
Then
Im[Z ]; Re[Z ]
Using Westergaard stress functionas
0
2 2
0
z yz xy
yz 0
zZz a
where is the applied boundary shear stress
with the boundary conditions
on the crack face a x a;y 0 0
on the boundary x y ,
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IIIxz
IIIyz
x y xy
III
III o
The stresses are given by
Ksin
22 rK
cos22 r
0
and displacements are given byK 2r
w sinG 2
u v 0
K a