Fracture Mechanics

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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.
• 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.
• 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.
• Other factors such as temperature, loading rate,
  stress concentrations, residual stresses, etc.,
  influence these three primary factors.

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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 spacing between
  two atoms.
• Idealizing force-displacement relation as one
  half of sine wave
• 
  (2.2)

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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

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Theoretical Cohesive Stress (Contd.)
The surface energy can be estimated as


(2.6) 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

(2.7)
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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


(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)

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Fracture stress for realistic material (contd.)
When a gtgt b eq. 2.10 becomes

(2.11) For a
sharp crack, a gtgtgt 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)
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Griffiths Energy balance approach

• First documented paper on fracture (1920)
• Considered as father of Fracture Mechanics

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Griffiths Energy balance approach (Contd.)
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
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 and flaw size a
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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 energy released
per unit width due to introduction of a crack of
length 2a is given by

(2.15)
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Griffiths Energy balance approach (Contd.)
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 cracked
plate is


(2.18)
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Griffiths Energy balance approach (Contd.)
The variation of with crack extension
should be minimum Denoting as
during fracture
(2.19)
for plane stress
(2.20)
for plane strain
The Griffith theory is obeyed by materials which
fail in a completely brittle elastic manner, e.g.
glass, mica, diamond and refractory metals.
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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 0.15 From the experimental
study on spherical vessels he calculated
as 0.25 0.28 However, it is important
to note that according to the Griffith theory, 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.
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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
crack plane and parallel to the crack front
(results from out-of-plane shear).
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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
s 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 sx sy, sz, 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.
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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
By replacing z by za , origin shifted to crack
tip.
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And when z?0 at the vicinity of the crack tip
KI must be real and a constant at the crack tip.
This is due to a Singularity given by
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
Further Simplification gives
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Sliding mode analysis or Mode 2
For problems with crack tip under shear loading,
Airys stress function is taken as
Using Airs definition for stresses
Using a Westergaard stress function of the form
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• Boundary Conditions
• At infinity
• On crack faces

With usual simplification would give the stresses
as
Displacement components are given by
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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
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Using Westergaard stress functionas
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Example 1. Suppose that you are asked to select
a material for some design and available
materials are AB. The material must not fracture
and deform plastically. The material Properties
are given as follows
dys (MPa) KIC (MPavm)
Material A 345 55
Material B 482 66
Select the suitable material according to the two
design approaches