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
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.
Fracture at the Atomic level
Repulsion
Potential Energy
 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
Distance
Bond
Energy
Attraction
Equilibrium
Distance xo
Theoretical Cohesive Stress
Tension


k
Compression
   Tdx
(2.1)
xo
Where x
is the equilibrium spacing
o
between two atoms.
Idealizing force-displacement relation as one
half of sine wave
  T sin( )
C
Applied Force
If a tensile force ‘T’ is applied to separate the
two atoms, then bond or cohesive energy is
Bond
Energy
x

(2.2)
+
+
xo
Cohesive
Force
Distance
Theoretical Cohesive Stress (Contd.)
Assuming that the origin is defined at x o and for small
displacement relationship is assumed to be linear such
x
that sin( ) 
Hence force-displacement

relationship is given by
x

C
x

(2.2)
Potential Energy
TT
Repulsion
Distance
Bond
Energy
Attraction
Bond stiffness ‘k’ is given by
T
C
Tension
(2.3)

If there are n bonds acing per unit area and assuming x o
as gage length and multiplying eq. 2.3 by n x o then ‘k’
becomes young’s modulus and TC beecomes cohesive
stress C such that
E
c 
x o
Or
c 
E

Bond
Energy
Applied Force
k
Equilibrium
Distance xo

k
Compression
+
+
xo
(2.4)
(2.5)
If  is assumed to be approximately equal to the atomic
spacing
Cohesive
Force
Distance
Theoretical Cohesive Stress (Contd.)
The surface energy can be estimated as
    sin 

s
1
2
0
C
x


 dx  

(2.6)
C
Potential Energy
Repulsion
Distance
Bond
Energy
Attraction
Equilibrium
Distance xo
Tension
E
 
x
C
o
Bond
Energy
Applied Force
The surface energy per unit area
is
equal to one half the fracture energy
because two surfaces are created when a
material fractures. Using eq. 2.4 in to
eq.2.6

k
Compression
+
+
xo
s
(2.7)
Cohesive
Force
Distance
Fracture stress for realistic material

Inglis (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
2a
A    1  
b

(2.8)
The ratio  A  is defined as the stress
concentration factor, k t
When a = b, it is a circular hole, thenk t  3.
When b is very very small, Inglis define radius of
curvature as
b2
r
(2.9)
a
And the tip stress as

a 
A   1  a   

 r  

A
r
(2.10)
2b
2a
A
Fracture stress for realistic material (contd.)
When a >> b eq. 2.10 becomes
a
 A  2  
r
(2.11)
For a sharp crack, a >>> b,r  0 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 spacingr  x o then the stress is
given by
 a 
(2.12)
 A  2  
 xo 
When far end stress reaches fracture stress   f, crack propagates and
the stress at A reaches cohesive stress A  C
then using eq. 2.7
E
f   s 
 4a 
1/ 2
This would
(2.13)
Griffith’s Energy balance approach
•First documented paper on fracture
(1920)
•Considered as father of Fracture
Mechanics
Griffith’s Energy balance approach (Contd.)
A A Griffith laid the foundations of modern fracture mechanics by
designing a criterion for fast fracture. He assumed that preexisting flaws propagate under the influence of an applied stress
only if the total energy of the system is thereby reduced. Thus,
Griffith's theory is not concerned with crack tip processes or the
micromechanisms by which a crack advances.

Griffith proposed that ‘There is a simple
energy 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 surface
energy due to increased crack surface’
Griffith theory establishes theoretical strength of
brittle material and relationship between fracture
strength f and flaw size ‘a’
B
Y
X
2a

Griffith’s Energy balance approach (Contd.)

The initial strain energy for the uncracked plate
per thickness is
2
(2.14)
U i   dA
A 2E
On creating a crack of size 2a, the tensile force
on an element ds on elliptic hole is relaxed
from   dx to zero. The elastic strain energy
released per unit width due to introduction of a
crack of length 2a is given by
a
where displacement
U a  4  12  dx  v
0

v

a  sin 
usin g x  a  cos 
E
2 a 2
(2.15)
Ua 
E
B
Y
X
2a

Griffith’s Energy balance approach (Contd.)
External work = U w   Fdy,
(2.16)

B
where F= resultant force =   area
 =total relative displacement
Y
X
The potential or internal energy of the body is
U p =U i +U a -U w
2a

Due to creation of new surface increase in
surface energy is
(2.17)
U  = 4a s
2 2
P1
Load, P
The total elastic energy of the cracked
plate is
2
2a 2
U t   dA 
  Fdy  4a s
A 2E

E
 a
Ua 
E
P2
Crack begins
to grow from
length (a)
Crack is
longer by an
increment (da)
)
(a
)
da
+
(a
v
(2.18)
Displacement, v
Denoting  as f during fracture
2E s 
f  

 a 
Energy, U
Griffith’s Energy balance approach (Contd.)
The variation of U t with crack
extension should be minimum
=
2
U
dU t
2 a
y
erg
n
0
 4 s  0
eE
ac
f
r
da
E
Su
s
4a
(a)
Crack
length, a
1/ 2
Total energy
(2.19)
for plane stress
Stable
Unstable
1/ 2
The Griffith theory is obeyed by
materials which fail in a completely
brittle elastic manner, e.g. glass,
mica, diamond and refractory
metals.
 2a 2
E
Potential energy  ¶U 
release rate G =  ¶a 
s
(2.20)
for plane strain
Ua 
Rates, G,
 2E s 
f  
2 
 a(1   ) 
Elastic Strain
energy released
Syrface energy/unit
extension =
(b)
Crack
length, a
ac
(a) Variation of Energy with Crack length
(b) Variation of energy rates with crack length
Griffith’s Energy balance approach (Contd.)
Griffith extrapolated surface tension values of soda lime glass
from high temperature to obtain the value at room temperature as
 s  0.54J / m2 . Using value of E = 62GPa,The value of  2E  as 0.15
  
From
the
experimental
study
on
spherical
vessels he
MPa m.
calculated  a   2E  as 0.25 – 0.28 MPa m.
1/ 2
s
1/ 2
s
c


However, it is important to note that according to the Griffith
theory, it is impossible to initiate brittle fracture unless preexisting 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.
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.
f  a1/ 2 . However, the proportionality constant can be used to
evaluate s (provided E is known) and if this is done, one finds the
value 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
 2E(  s   p ) 
f  

a


1/ 2
Where prepresents energy expended in plastic work. Typically for
cleavage in metallic materials p=104 J/m2 and s=1 J/m2. Since p>>
s we have
1/ 2
 2E p 
f  

 a 
Strain Energy Release Rate
The strain energy release rate usually referred to
G
dU
da
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 G c.
At fracture we have G  G c so that
1/ 2
1 EG c 
f  

Y  a 
One disadvantage of using G c is that in order to determine f it is
necessary to know E as well as G.c This can be a problem with some
materials, eg polymers and composites, where varies with
composition and processing. In practice, it is usually more
convenient to combine E andG c in a single fracture toughness K c
parameter Kcwhere Kc2  EG. cThen can be simply determined
experimentally using procedures which are well established.
LINEAR ELASTIC FRACTURE MECHANICS (LEFM)
For LEFM the structure obeys Hooke’s 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 (Contd.)
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; x y, z, txy, txz,tyz. 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:

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 surfaces
occurs, and,
2. plane strain (PSN), when
the specimen is thick enough
to avoid contraction in the
thickness z-direction.
Thickness
B
Thickness
B

y

yy

z
z
z
z
Crack
Plane
X
a


Plane Stress

Plane Strain
In the former case, the overall stress state is reduced to the
three components; x, y, txy, since; z, txz, tyz= 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 = ν (x+y)
where ν is Poisson's ratio.
For plane problems, the equilibrium conditions are:
¶ y ¶t xy
¶ x ¶t xy
¶x

¶y
0 ;
¶y

¶x
0
If  is the Airy’s stress function satisfying the biharmonic
compatibility Conditions    0
4
¶2
¶ 2
¶ 2
x  2 , y  2 , t xy  
¶y
¶x
¶xy
Then
For problems with crack tip Westergaard introduced Airy’s stress
function as


  Re[Z]  y Im[Z]
Where Z is an analytic complex function


bg
Z z  Re[ z ]  y Im[ z ] ; z = x + iy
And Z, Z are 2nd and 1st integrals of Z(z)
Then the stresses are given by
¶2
x  2  Re[Z]  y Im[Z' ]
¶y
¶2
y  2  Re[Z]  y Im[Z' ]
¶x
¶2
t xy 
  y Im[Z' ]
¶xy
where Z' = dZ dz
Opening mode analysis or Mode I
Consider an infinite plate a crack of length 2a subjected to a biaxial
s
State of stress. Defining:
Z
z
z
2
 a2 
s
y
x
Boundary Conditions :
• At infinity | z |   x  y  , txy  0
• On crack faces
 a  x  a;y  0 x  txy  0
2a
s
By replacing z by z+a , origin shifted to crack tip.
Z
b g
zb
z  2a g
 za
And when |z|0 at the vicinity of the crack tip
Z
a
KI

2az
2 z
K I   a
KI must be real and a constant at the crack tip. This is due to a
1
Singularity given by
z
The parameter KI is called the
stress intensity factor for opening
mode I.
Since origin is shifted to crack
tip, it is easier to use polar
Coordinates, Using
z  ei
Further Simplification gives:
KI
 

3 
cos   1  sin   sin   
2 r
2
 2   2 
KI
 

3 
y 
cos   1  sin   sin   
2 r
2
 2   2 
KI
 

3 
t xy 
sin   cos   cos   
2 r
2 2
 2 
KI
In general ij 
f ij    and K I  Y a
2r
where Y = configuration factor
x 
From Hooke’s law, displacement field can be obtained as
u
2(1  )
r
  1
 
KI
cos   
 sin 2   
E
2
2 2
 2 
v
2(1  )
r
  1
 
KI
sin   
 cos2   
E
2  2   2
 2 
where u, v = displacements in x, y directions
  (3  4) for plane stress problems
3 
  
 for plane strain problems
 1  
The vertical displacements at any position along x-axis (  0 is
given by
y

2
2
v
a
x
E
(1  2 )
v
E

a
for plane stress
v
2
x
2

x
for plane strain
x
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
1
U a  Fv
2
For plane stress
For plane strain
a

(1   2 )
2
2
Ua  4  
a  x dx
Ua  4  

E
E
0
0
2 a 2
2 a 2 (1   2 )
E
E
The strain energy release rate is given by G  dU a da
a
2 a
GI =
E
K 2I
GI =
E
a
2 (1   2 )a
GI =
E
K 2I (1  2 )
GI =
E
2
 x 2 dx
Sliding mode analysis or Mode 2
For problems with crack tip under shear loading, Airy’s stress
function is taken as
t

II   y Re[Z]
0
Using Air’s definition for stresses
¶2
 x  2  2 Im[Z]  y Re[Z' ]
¶y
¶2
y  2   y Re[Z' ]
¶x
¶2
t xy  
 Re[Z]  y Im[Z' ]
¶xy
Using a Westergaard stress function of the form
Z
t0 z
z
2
 a2 
y
2a
t0
Boundary Conditions :
• At infinity | z |   x  y  0, txy  t0
• On crack faces
 a  x  a;y  0 x  txy  0
With usual simplification would give the stresses as
x 
K II
 
 

3  
cos   cos    2  cos   cos    
2r
 2    2 
2
 2  
K II
  
3  


y 
cos   sin   cos   
2r
2 2
 2 
K II
 

3 
t xy 
cos   1  sin   sin   
2r
2
 2   2 
Displacement components are given by
u
K II
E
r

(1  )sin      2  cos   
2
2
v
K II
E
r

(1  )cos      2  cos   
2
2
K II  to a
K 2I
GI =
E
K 2I (1   2 )
GI =
E
for plane stress
for plane strain
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
e x  e y   xy   yx  0
¶w
¶w
and  yz   zy 
¶x
¶y
the equilibrium equation is written as
¶t xz ¶t yz

0
¶x
¶y
Strain displacement relationship is given by
 xy   yx 
¶2 w ¶2 w
2



w0
2
2
¶x
¶y
if w is taken as

1
w  Im[ Z]
G
Then
txy  Im[Z]; t yz  Re[Z]
Using Westergaard stress functionas
Z
t0 z
z
2
 a2 
where t0 is the applied boundary shear stress
with the boundary conditions
on the crack face  a  x  a; y  0  z  t yz  t xy  0
on the boundary x  y  , t yz  t0
The stresses are given by
K III

sin  
2 r
2
K

t yz  III cos  
2 r
2
 x   y  t xy  0
t xz 
and displacements are given by
K III 2r

sin  
G 
2
uv0
w 
K III  to a