Mokashi Imrankhan Sherkhan

Guided by: M S Bobji

Seminar on

Importance of Fracture Importance of Fracture Mechanics in TribologyMechanics in Tribology

WearErosionSurface Formation

Fracture Mechanics approachFracture 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 PropagationModes of Crack Propagation

Energy ApproachEnergy Approach

+ +

xo

BondEnergy

CohesiveForce

l

EquilibriumDistance xo

Pot

ent

ialE

nerg

y

Distance

Repulsion

Attraction

Tension

Compression

App

lied

For

ce

k

BondEnergy

Distance

Griffith’s Energy balance approachGriffith’s Energy balance approach

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

Energy Approach Cont…Energy Approach Cont…

Surface energy Es = 2(2a B ᵞ)

ᵞ -Surface energy per unit areaB-Thickness of plate

Energy Approach Cont..Energy Approach Cont..

Total PE of Body with crack

U = UE – Uwork + US

Strain EnergySystem always try to minimize its PE

Thick plate is more Thick plate is more resistant to fracture or resistant to fracture or thin?thin?

J IntegralJ Integral J Integral over a closed loop = 0 Path independent

Displacement Displacement ApproachApproach

From Hooke’s law, displacement field can be obtained as

2I

2I

2(1 ) r 1u K cos sin

E 2 2 2 2

2(1 ) r 1v K sin cos

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

Displacement ApproachDisplacement Approach

Stress ApproachStress Approach

Stress Intensity FactorStress Intensity Factor

for infinite plate

for other geometry

can be obtained from : 1. handbook solution

2. approximate method

3. numerical method

aK

aK

Stress near crack tipStress near crack tip

]sinsin[cos2

3

21

22

r

Kx

]sinsin[cos2

3

21

22

r

Ky

2

3

222

coscossinr

Kxy

)( ijij fr

a

2

Plastic zone size

x

y

plane strain

plane stress

Plastic Zone ShapesPlastic Zone Shapes

Mixed LoadingMixed Loading

13

III

III2

II

II1

I

I

n

c

n

c

n

c K

K

K

K

K

K

where KIc, KIIc, KIIIc, ni (i = 1, 2, 3) – parameters that characterize material near the concentrator,

determined experimentally

in

c

in

c K

K

K

K

II

II

I

I = 1

in

c

in

c K

K

K

K

III

III

II

II = 1

Cont…Cont…

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

1III

III

II

II

I

I

iii n

c

n

c

n

c K

K

K

K

K

K(1)

FATIGUE FRACTUREFATIGUE FRACTURE

Every Coin has two Every Coin has two sides…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 Fracture Avoidance with Proper Use of MaterialProper 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 Fracture Avoidance with Proper Use of Material (cont’)Proper Use of Material (cont’)

• Roof spans and windows were arched to maintain compressive loading

REFERENCESREFERENCES

A book on “Fracture Mechanics” by Prashant Kumar“Introduction to Fracture Mechanics” book by C. H. Wang

Finite Element AnalysisFinite Element Analysis

Stress analysis near crack Quarter point element

J- integral