Seminar on
Mokashi Imrankhan Sherkhan
Guided by: M S Bobji
Importance of Fracture Mechanics in
Tribology
Wear
 Erosion
 Surface Formation

Fracture Mechanics approach
Energy Based
: Energy release rate(G) and J
integral
Stress Based
: Stress intensity factor (K)
Displacement Based : Crack tip opening
displacement (CTOD)
Modes of Crack Propagation
Energy Approach
Potential Energy
Repulsion
Distance
Bond
Energy
Attraction
Equilibrium
Distance xo
Tension
Applied Force
Bond
Energy
l
k
Compression
+
+
xo
Cohesive
Force
Distance
Griffith’s Energy balance approach
•First documented paper on fracture
(1920) Considered as father of
Fracture Mechanics
Energy Approach Cont…
Surface energy Es = 2(2a B ᵞ)
ᵞ -Surface energy per unit area
B-Thickness of plate
Energy Approach Cont..
Total PE of Body with crack
U = UE – Uwork + US
Strain Energy
System always try to minimize its PE
Thick plate is more resistant
to fracture or thin?
J Integral
 J Integral over a closed loop = 0
 Path independent
Displacement Approach
Displacement Approach
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  
Stress Approach
Stress Intensity Factor
K   a
for
K    a
for
 can be obtained from :
infinite plate
other geometry
1. handbook solution
2. approximate method
3. numerical method
Stress near crack tip


3
cos [1  sin sin ]
2
2
2
2r
 x
K
 y
K
 xy
K


3
cos [1  sin sin ]
2
2
2
2r


3
sin cos cos
2
2
2
2r
 ij
 a
f ij ( )
2r
Plastic zone size
Plastic Zone Shapes
y
plane strain
x
plane stress
Mixed Loading
 KI 


 K Ic 
n1
 K II 

 
 K IIc 
n2
 K III 

 
 K IIIc 
n3
1
where KIc, KIIc, KIIIc, ni (i = 1, 2, 3) – parameters that characterize material near the concentrator,
determined experimentally
 KI 


 K Ic 
ni
 K II  = 1

 
 K IIc 
ni
ni
 K II 


 K IIc 
ni
 K III 
 = 1
 
 K IIIc 
Cont…
 KI

 K Ic
ni

K
   II

 K IIc
ni

K
   III

 K IIIc
ni

  1

(1)
Diagrams of the deformed body limiting-equilibrium state the conditions of mixed
(I+II), (I+III) fracture mechanisms
Curve 1 – according to formula (1)
when KIII = 0 and ni = 4, curve 2
according formula (1) when KIII = 0
and ni = 2
Curves 1 and 2 are plotted according
to formula (1) when ni = 4 and ni =2
correspondingly
FATIGUE FRACTURE
Every Coin has two sides…

Useful Fracture
 Metal cutting
 Rock cutting in Mining
 “Biting” of candies

Unwanted Fracture
 Fracture of Liberty ships in World war II
 Fracture of wheels, axles and rails
Fracture Avoidance with Proper
Use of Material
Pyramid of Egypt
•
•
Schematic Roman Bridge Design
The primary construction material prior to 19th were timber, brick and
mortar
Arch shape producing compressive stress  stone have high
compressive strength
Fracture Avoidance with Proper Use
of Material (cont’)
•
Roof spans and windows were arched to maintain compressive loading
REFERENCES
A
book on “Fracture Mechanics” by Prashant
Kumar
“Introduction to Fracture Mechanics” book by C. H.
Wang
Finite Element Analysis

Stress analysis near crack
 Quarter point element

J- integral