A part i -mechanical properties and fracture

INTRODUCTORY CHAPTER
MECHANICAL PROPERTIES AND
FRACTURE
Prepared by
Prof. Dr. M.M. El-Gammal
Professor Emeritus of Material Sciences and
Engineering,
ex-Head of Marine Engineering Department,
Faculty of Engineering, Alexandria
University, Egypt

Fracture mechanics
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

 Now we have the formulae for Us & U (which are required to write down the Griffith’s condition):
dc
dU
dc
dUs 

2
2 c
4a
f
E
 

 LHS increases linearly with c, while RHS is constant.
 The ‘equal to’ (=) represents the bare minimum requirement (i.e. the critical condition) →
the minimum crack size, which will propagate with a ‘balance’ in energy (i.e. between
elastic energy released due to crack extension and the penalty in terms of the fracture
surface energy).
 The critical crack size (c*): (Note that ‘c’is half the crack length internal)
 A crack below this critical size will not propagate under a constant stress a.
 Weather a crack of size greater than or equal to c* will propagate will depend on the Inglis
condition being satisfied at the crack-tip (i.e. sufficient stress concentration should exist).
 This stress a now becomes the fracture stress (f)→ cracks of length c* will grow
(unstably) if the stress exceeds f (= a).
2
* 2
a
fE
c



 E
a
2
s c2
c
U 


 U
4
c
f





*
2
2 f
f
E
c




 According to Griffith’s criterion:
Understanding Griffith’s equation
*f
c
E2


 
At constant c (= c*)
when  exceeds f then specimen failsGriffith )1(c
E2
2*




f
Plane strain conditions
*2
2 c
4a
f
E
 

 
 
 
dc
dU
dc
dUs 

2
2 c
4a
f
E
 

At criticality (written with ‘*’) crack propagation just starts:
Putting the ‘*’around the variables: *2
2 c 4a fE        
2* *
c 2a fE  
Since, * is f : * 2
c 2f fE  
 This can be understood as follows: keep displacement imposed on the ends of the specimen
constant (& hence far field) and keep increasing ‘c’ till the crack beings to propagate (& hence
far field = f).
 Else, one can keep ‘c’ constant and increase displacements (leading to an increase in far field)
till crack propagation starts (i.e. c  c*,   f).
Which, can be written in two ways: *f
c
E2


 
*
2
2 f
f
E
c




  












E
a
f
22
c
c4UcrackaofonintroductitheonenergyinChange


c →
U→
0
*





 
cdc
Ud
*
c
0c
0
0
An alternate way of understanding the Griffith’s criterion (energy based), though personally I
prefer the previous method (discussed in the last few slides).
cγU f4
E
a
22
s
c
U


 This change in energy (U) should be negative with an increase in crack
length (or at worst equal to zero). I.e. (dU/dc) ≤ 0.
 At c* the slope of U vs c curve is zero [(dU/dc)c* = 0]. This is a point
of unstable equilibrium.
 With increasing stress the value of c* decreases (as expected→ more
elastic strain energy stored in the material).
Stable
cracks Unstable cracks
Equations for ready reference
Negative slope
Positive slope
c →
U→
*
1c *
2c

‘Modern’ Fracture Mechanics
 Stress Intensity Factor (K)
Material Parameter  Fracture Toughness (KC)
 Energy Release Rate (G)
Material Parameter  Critical Energy Release Rate (GC)
 J-integral (J)
Material Parameter: JC
 Crack Tip Opening Displacement (CTOD) ()
Material Parameter: C
[1] G.R. Irwin, “Fracture Dynamics”, in: “Fracture of Metals”, ASM, Cleaveland, OH, 1948, pp.147-166.
[2] G.R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate, J. Appl. Mech 24 (1957) 361-364.

ME 240: Introduction to Engineering Materials
Chapter 8. Failure 8.7
Plastic zone
What about ductile materials  consider y (i.e. y means direction not yield)

 Historically (in the ‘old times’ ~1910-20) fracture was studied using the Inglis and Griffith
criteria, wherein fracture stress (f) was calculated either using a global or local criterion.
 The birth of fracture mechanics (~1950+) led to the concepts of stress intensity factor (K)
and energy release rate (G) (due to Irwin and others). These concepts worked well in the
domain of ‘linear elasticity’ (LEFM); i.e. for brittle materials. Crack tip stresses were
characterized by the stress intensity factor.
 In the presence of crack tip plasticity some of these concepts were extended if the crack tip
plasticity was small (small scale yielding).
 In ductile materials with large plastic deformation at the crack tip, concept of J-integral was
evolved, wherein stress fields ahead of the crack tip are termed as ‘HRR’ fields and the J-
integral characterizes the field intensity of the crack tip.
 The concept of Crack Tip Opening Displacement (CTOD) was also proposed to characterize
cracks in ductile materials.
Fracture Mechanics

 G is defined as the total potential energy () decrease during unit crack extension (dc). ‘G’ is
also referred to as the crack extension force and is given by:
Concept of Energy Release Rate (G)
dc
d
G


The potential energy is a difficult quantity to visualize. In the absence of external
tractions (i.e. only displacement boundary conditions are imposed), the potential
energy is equal to the strain energy stored:  = Us.*
* It is better to understand the basics of fracture with fixed displacement boundary conditions (without any surface
tractions).
dc
dU
G s
 With displacement boundary conditions only.
 Crack growth occurs if G exceeds (or at least equal to) a critical value GC, which is the
fracture toughness of the material.
CGG  For perfectly brittle solids: GC = 2f (i.e. this is equivalent to Griffith’s criterion).
2 2
s
c
U a
E
 
As we have seen, Us is given by:

2
2 c a
G
E
 

‘G’has units of [J/m2] = [N/m]

 In spite of the fact that ‘G’ has a more direct physical interpretation for the crack growth
process, usually we work with ‘K’ as it is more amenable to theoretical computation.
 ‘K’ can be related to ‘G’ using the following equations:
Relation between K and G
2
2
2
:
.
:
(1 )
Planestress K GE
G E
Planestrain K





What does Proof Stress mean?
The proof stress of a material is defined
as the amount of stress it can endure until
it undergoes a relatively small amount of
plastic deformation. Specifically, proof
stress is the point at which the material
exhibits 0.2% of plastic deformation.
This type of stress is typically used in the
manufacturing industry to ensure that a
material is not stressed far beyond
its elastic limit. That is, a safe load is
applied such that the object is not
stressed to the point where it irreversibly
changes in size or shape. If a material is
allowed to exceed its proof stress, it can
deform permanently, and failure or
fracture can ultimately occur.

RESILIENCE
Resilience is the ability and the capacity
of a material to absorb energy when it is
deformed elastically and then, upon
unloading, to recover this amount of
energy. The maximum energy that can be
absorbed up to the elastic limit, without
creating a permanent deformation is
known as proof resilience. In the stress-
strain curve, it is given by the area under
the portion of a stress–strain curve (up
to yield point).
Under assumption of linear elasticity or up
to proportional limit, resilience can be
calculated by integrating the stress–
strain curve from zero to
the proportional limit.

TOUGHNESS
Toughness: How well the
material can resist fracturing
when force is applied.
Toughness requires strength
as well as ductility, which
allows a material to deform
before fracturing. Do you
consider silly putty to be tough
stuff? Under these terms,
believe it or not, it actually is
relatively tough, as it can
stretch and deform rather than
break.Alloy 2 is better than
alloy 1

Material KIC [MPam]**
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
* We have already noted that fracture toughness is a microstructure sensitive property and hence to get ‘true’value the
microstructure has to be specified.
** Note the strange units for fracture toughness!
[1] Fracture Mechanics, C.T. Sun & Z.-H. Jin, Academic Press, Oxford (2012).
Fracture Toughness* (KIC) for some typical materials [1]

 In ductile materials:
 Crack-tip stresses lead to plastic deformation at the crack-tip, which further leads to crack
tip blunting.
 Energy is consumed due to plastic deformation at the crack-tip (which comes from elastic
strain energy). This implies less energy is available for crack growth (& creation of new surfaces).
 Crack-tip blunting leads to a reduced stress amplification at the crack-tip. Blunting will
avoid ‘stress singularity’ at the crack tip and may lead to a maximum stress at a certain
distance from the crack-tip (as in the figure below).
 Crack-tip blunting will lead to an increased resistance to crack propagation (i.e. increased
fracture toughness).
Ductile fracture
* Note: For a material to be classified as ductile it need not
display large strain in a tensile test.

What happens to a ‘crack’ in a ductile material?Funda Check
 High magnitude of crack tip stresses can cause yielding at the crack tip (plastic
deformation).
 This leads to crack tip blunting, which reduces the stress amplification.
 There develops a zone ahead of the crack tip known as the process zone.
What else can happen at the crack tip due to high stresses?
Funda Check
 High magnitude of crack tip stresses can cause:
 phase transformation (tetragonal to monoclinic phase in Yttria stabilized Zirconia),

 We have seen that the crack tip gets blunted in ductile materials. As the stress intensity factor
approach assumes that the material in linear elastic, the solutions obtained are not applicable
to ductile materials.
 However, if the plastic deformation zone around the crack tip is smaller than the K-
dominance zone, then the LEFM approach can be used for ductile materials also (with
minimum errors). This is referred to as Small Scale Yielding (SSY).
 Irwin[1] estimated the size of the plastic deformation zone (near the crack tip):
(Assuming the plastic deformation to be small).
 If the yielding is large, then the applicability of LEFM breaks down. As a first approach
Irwin[1] suggested the use of ‘effective crack length’ instead of crack length, to obtain an
‘effective stress intensity factor’.
Effective crack length = crack length + ½(plastic zone size).
 Further Dugdale[2] presented a strip yielding zone model to determine the plastic zone size
(in thin cracked sheets).
Fracture Toughness in Ductile Materials
2
I
plastic
y
K
r


  
 
[1] G.R. Irwin, Proc. 7th Sagamore Ordnance Materials Conference, Syracuse Univ., 1960, p. IV-63.
[2] D.S. Dugdale, J. Mech. Phys. Solids 8 (1960) 100.

 The table below summarizes many criteria found in standard literature. Some details can be
found in other pages of the chapter.
Summary of Fracture Criteria
Criterion Comments
Fracture
occurs if
Relevant formulae
Inglis (f)
1913
Involves crack tip radius (& crack
length)
  f
Griffith (f)
1920
Involves crack length   f
Irwin [K]
Concept of stress intensity factor.
KI  KIC
(in mode I)
Irvin [G]
Energy release rate based. Same as
K based criterion for elastic bodies.
G  GC
Wells (, CTOD)
1961
Involves crack (tip) opening
displacement
  C
Rice [J-integral]
1968
Generalized energy release rate
concept. Applicable to non-linear
elastic and elastic-plastic materials.
J  JC
(crack initiation)
HRR fields.
ca
E
f
04

 
*f
c
E2


 
 cYKI 0
2
* *
I
y
K
E
 


  
 

Ductile – brittle transition
 Certain materials which are ductile at a given temperature (say room temperature), become
brittle at lower temperatures. The temperature at which this happens is terms as the Ductile
Brittle Transition Temperature (DBTT).
 As obvious, DBT can cause problems in components, which operate in ambient and low
temperature conditions.
 Typically the phenomena is reported in polycrystalline materials. Deformation should be
continuous across grain boundary in polycrystals for them to be ductile. This implies that
five independent slip systems should be operative (this is absent in HCP and ionic
materials).
 This phenomenon (ductile to brittle transition) is not observed in FCC metals (they remain
ductile to low temperatures).
 Common BCC metals become brittle at low temperatures (as noted before a decrease in
temperature can be visualized as an increase in strain rate, in terms of the effect on the
mechanical behaviour).
 As we have noted before a ductile material is one which yields before fracture (i.e. its yield
strength is lower in magnitude than its fracture strength).

Why do we need a large ductility (say more than 10% tensile elongation)
material, while ‘never’ actually in service component is going to see/need such
large plastic deformation (without the component being classified as ‘failed’).
Funda Check
 Let us take a gear wheel for an example. The matching tolerances between gears are so
small that this kind of plastic deformation is clearly not acceptable.
 In the case of the case carburized gear wheel, the surface is made hard and the interior is
kept ductile (and tough).
 The reason we need such high values of ductility is so that the crack tip gets blunted and the
crack tip stress values are reduced (thus avoiding crack propagation).

To be plane strainPlane strain fracture toughness

Alloy
KIC
(MN/m3/2)
SCC
environment
KISCC
(MN/m3/2)
13Cr steel 60 3% NaCl 12
18Cr-8Ni 200 42% MgCl2 10
Cu-30Zn 200 NH4OH, pH7 1
Al-3Mg-7Zn 25 Aqueous halides 5
Ti-6Al-1V 60 0.6M KCl 20
http://en.wikipedia.org/wiki/Stress_corrosion_cracking

 Glass can be toughened without changing it composition by introducing compressive
residual surface stress. This is done as follows: surface of molten glass solidified by cold air
on the surface, followed by solidification of the bulk → the contraction of the bulk while
solidification, introduces residual compressive stresses on the surface → fracture strength
can be increased 2-3 times.
Case study: difference in fracture behaviour of ‘normal’ versus toughened glass
Fracture of normal
plate glass
Fracture of
toughened glass
 As more interfaces are created due to severe fragmentation of
toughened glass on impact, more energy is absorbed (in creating new
interfaces)  this leads to an increase in the toughness of glass.

Design using fracture mechanics
Example:
Compare the critical flaw sizes in the following metals subjected to
tensile stress 1500MPa and K = 1.12 a.
KIc (MPa.m1/2)
Al 250
Steel 50
Zirconia(ZrO2) 2
Toughened Zirconia 12
Critical flaw size (microns)
7000
280
0.45
16
Where Y = 1.12. Substitute values
SOLUTION

IMPACT TESTING
Tensile test vs. real life failures
Impact energy measured
or notch toughness

How do you practically make these fatigue measurements ?

Can make extrapolations
To obtain log A

FATIGUE
One of failure analysis goals = prediction of fatigue life of component
knowing service constraint and conducting Lab tests
Ignores crack initiation
and fracture times

Other effects:
a) Mean stress
b)stress concentrations
c) Surface treatments

Lead pipes deforming
under their own weight
CREEP
 = f (T, t, )
Time dependent and permanent deformation of materials
when subjected to load or stress (significant at T = 0.4Tm)

Effect of temperature and stress

Data extrapolation methods – e.g. prolonged exposures (years)
Perform creep tests in excess of T and shorter time but at same stress level

SHEAR FAILURES AND FRACTURES
SOURCE : https://www.sciencedirect.com/topics/engineering/shear-fracture
Shear Failure
Load applied to a body exceeds
maximum permissible value of
load that can be applied to a body.
The body undergoes a deformation
which consequently leads to
permanent deformation in the form
of a failure. Failure in a material
snatches the usability of material
for different purposes from the
material and makes it useless.

© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
MECHANICS OF MATERIALSFourth
Edition
Beer • Johnston • DeWolf
7 - 37
Yield Criteria for Ductile Materials Under Plane Stress
• Failure of a machine component
subjected to uniaxial stress is directly
predicted from an equivalent tensile test
• Failure of a machine component
subjected to plane stress cannot be
directly predicted from the uniaxial state
of stress in a tensile test specimen
• It is convenient to determine the
principal stresses and to base the failure
criteria on the corresponding biaxial
stress state
• Failure criteria are based on the
mechanism of failure. Allows
comparison of the failure conditions for
a uniaxial stress test and biaxial
component loading

Edition
7 - 38
Maximum shearing stress criteria:
Structural component is safe as long as the
maximum shearing stress is less than the
maximum shearing stress in a tensile test
specimen at yield, i.e.,
2
max
Y
Y

 
For a and b with the same sign,
22
or
2
max
Yba 
 
For a and b with opposite signs,
22
max
Yba 
 



Edition
7 - 39
Maximum distortion energy criteria:
distortion energy per unit volume is less
than that occurring in a tensile test specimen
at yield.
   
222
2222
00
6
1
6
1
Ybbaa
YYbbaa
Yd
GG
uu






Edition
7 - 40
Fracture Criteria for Brittle Materials Under Plane Stress
Maximum normal stress criteria:
maximum normal stress is less than the
ultimate strength of a tensile test specimen.
Ub
Ua




Brittle materials fail suddenly through rupture
or fracture in a tensile test. The failure
condition is characterized by the ultimate
strength U.

Edition
7 - 41
Transformation of Plane Strain
• Plane strain - deformations of the material
take place in parallel planes and are the
same in each of those planes.
• Example: Consider a long bar subjected
to uniformly distributed transverse loads.
State of plane strain exists in any
transverse section not located too close to
the ends of the bar.
• Plane strain occurs in a plate subjected
along its edges to a uniformly distributed
load and restrained from expanding or
contracting laterally by smooth, rigid and
fixed supports
 0
:strainofcomponents
x  zyzxzxyy 

Edition
7 - 42
Mohr’s Circle for Plane Strain
• The equations for the transformation of
plane strain are of the same form as the
equations for the transformation of plane
stress - Mohr’s circle techniques apply.
• Abscissa for the center C and radius R ,
22
222 










 



xyyxyx
ave R


• Principal axes of strain and principal strains,
RR aveave
yx
xy
p







minmax
2tan
  22
max 2 xyyxR  
• Maximum in-plane shearing strain,

Edition
7 - 43
Measurements of Strain: Strain Rosette
• Strain gages indicate normal strain through
changes in resistance.
 yxOBxy   2
• With a 45o rosette, x and y are measured
directly. xy is obtained indirectly with,
333
2
3
2
3
222
2
2
2
2
111
2
1
2
1
cossinsincos
cossinsincos
cossinsincos



xyyx
xyyx
xyyx



• Normal and shearing strains may be
obtained from normal strains in any three
directions,

(a) Von Mises yield criterion (b) Tresca yield criterion
NOTE:
- LEFM can be applied in cases if the external applied stress is less than 0.5*Yield
Stress,
- GYFM can be applied for cases of loading if the external applied stress is greater
than 0.5*Yield Stress.
- Plane stress plastic zone sizes are larger than plane strain plastic zone size.
- Tresca plastic zones are larger than von Mises plastic zones.

Theoretical Cohesive Stress
+ +
xo
Bond
Energy
Cohesive
Force
l
Equilibrium
Distance xo
PotentialEnergy
Distance
Repulsion
Attraction
Tension
Compression
AppliedForce
k
Bond
Energy
Distance
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)
 x1
2
s C C
0
sin dx
l

l
l
    

s
C
o
E
x

 

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
a
2
 
     
0  
ox 
A
o
a
2
x
 
    
 
A C  
f  
1/2
s
f
E
4a
    
 

Griffith’s Energy balance approach
•First documented paper on fracture
(1920)
•Considered as father of Fracture
Mechanics

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’s Energy balance approach
2a
X
Y
B


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’f

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)
2a
2b
A

A


A
2a
1
b
     
 
A
 tk
tk 3.
2
b
a
 
A
a
1 a
  
        

Griffith’s Energy balance approach (Contd.)
Energy,U
Crack
length, a
Surface Energy
U 
=
4a s
2 2
a
a
U
E


Elastic Strain
energy released
Total energy
Rates,G,s
Potential energy
release rate G =
Syrface energy/unit
extension =
U
a
 
 
 
Crack
length, 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 0
da E

     
f
1/2
s
f
2E
a
    
 
1/ 2
s
f 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.

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
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.
2
s 0.54J / m . 
1/2
s2E 
 
 
MPa m.
1/2
s
c
2E
a
 
   
 
MPa m.

Strain Energy Release Rate
The 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 some
materials, 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.
dU
G
da

cG
cG G
1/2
c
f
1 EG
Y a
    
 
cG f
cG
cG cK
2
c cK EGcK

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

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, 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.)
Crack
Plane
Thickness
B
Thickness
B




z z
z z
a
Plane Stress Plane 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 surfaces
occurs, and,
2. plane strain (PSN), when
the specimen is thick enough
to avoid contraction in the
thickness z-direction.

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.
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
 

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
2
2 2
v a x for plane stress
E
(1 )
v a x for plane strain
E

 
  
  x
v
x
y
   
a
2a a
2 2 2 2
a a
0 0
2 2
1
U Fv
2
For 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
I
I
a (1 )
E
The strain energy release rate is given by G dU da
a (1 )a
G = G =
E E
K
G =
E
  

   
2 2
I
I
K (1 )
G =
E
 

Plane strain or plane stress
In general, the conditions ahead of a crack tip are neither plane stress
nor plane strain. There are limiting cases where a tw-dimensional
assumptions are valid, or at least provides a good approximation.
The nature of the plastic zone that is formed ahead of a crack tip
plays a very important role in the determination of the type of failure
that occurs. Since the plastic region is larger in PSS than in PSN,
plane stress failure will, in general, be ductile, while, on the other
hand, plane strain fracture will be brittle, even in a material that is
generally ductile. This phenomenon explains the peculiar thickness
effect, observed in all fracture experiments, that thin samples exhibit
a higher value of fracture toughness than thicker samples made of the
same material and operating at the same temperature. From this it can
be surmised that the plane stress fracture toughness is related
to both metallurgical parameters and specimen geometry while the
plane strain fracture toughness depends more on metallurgical factors
than on the others.

Effect of plate thickness on fracture toughness
PSS is for thin plates
PSN is for thick plates

Due to presence of crack
tip, stress in a direction to
normal to crack plane yy
will be large near the crack
tip. This stress would in turn
tries to contract in x and z
direction. But the material
surrounding it will
constraint it, inducing
stresses in x and z direction,
there by a triaxial state of
stress exists near the crack
tip. This leads to plane
strain condition at interior.
At the plate surface zz is
zero and zz is maximum.
This leads to plane stress
condition at exterior.

The state of stress is also dependent on size of plate thickness.
If the plastic zone size is small compared to the plate thickness,
plane strain condition exists.
If the plastic zone size is larger than the plate thickness, plane
stress condition prevails.
As the loading is increased, plastic zone size also increases
leading to plane stress conditions.

Limits of LEFM = Linear Elastic Fracture Mechanics
As per ASTM standard LEFM is applicable for components of size
2
2
2
2.5
or B 2.5
or b 2.5
 
    
 
    
 
    
I
ys
I
ys
I
ys
K
a
K
t
K
W
As per ASTM standard fracture toughness testing can be done on
Specimens of size
2
, , 2.5
 
    
Ic
ys
K
a B W

SHEAR STRENGTH AND
TESTING IN METALS
In order to understand
shear strength,
understanding different
theories of failure are
important.
Generally, for brittle
materials, the cause of
failure is due to a force in
tension.
For ductile materials, the
cause of failure is often
due to shearing forces.

Maximum shear stress theory
This theory states that material failure will take place when the maximum
shear stress brought on by combined stresses will equal or be greater than
the obtained shear stress value at yield in a uniaxial tensile test
For a uniaxial test, the principal stresses are:
σ1=Sy
σ2= 0
σ3= 0
where σ1, σ2, σ3 = maximum normal stress a body can withstand at a certain
point;
Sy = yield strength.
The shear strength at yielding is:
Ssy=(σ1−σ2)/2 =Sy/2
Or
Ssy=(σ1−σ3)/2 =Sy/2
For a homogenous, isotropic, ductile material with two or three-dimensional
static stress, to identify and compute for σ1, σ2, σ3 and the maximum shear
stress, 𝜏max:
𝜏max =(σ1−σ2)/2

SHEAR TEST PERFORMANCE
SHEAR TEST
FACILITY AND
SET UP

Still water bending
moment
Before a ship even goes out to
sea, some stress distribution
profile exists within the
structure. The Figure shows
how the summation of
buoyancy and weight
distribution curves of an
idealized
rectangular barge lead to shear
force and bending moment
distribution diagrams.
Stresses apparent in the still
water condition generally,
become extreme only in cases
where concentrated loads
are applied to the structure,
which can be the case when
the holds of a cargo vessel are
selectively filled.

End of PART I
TO BE FOLLOWED
TO PART II

A part i -mechanical properties and fracture

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