ch2-LEFM- Fracture Mechanics

8/16/2019 ch2-LEFM- 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

dynamicfatigue loading


2/30

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!

"t may be described in terms of the critical stress intensity factor, #"c, under

a variety of conditions! $These terms and conditions are fully discussed in

the follo%ing chapters!&

'! Crack Size. Fractures initiate from discontinuities that can vary from

e(tremely small cracks to much larger %eld or fatigue cracks! Furthermore,

although good fabrication practice and inspection can minimi)e the si)e and number of cracks, most comple( mechanical components cannot be

fabricated %ithout discontinuities of one type or another!

*! 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!

+ther factors such as temperature, loading rate, stress concentrations,

residual stresses, etc!, influence these three primary factors!


3/30

Fracture at the tomic level

T%o atoms or a set of atoms are bonded

together by cohesive energy or bond energy! T%o atoms $or sets of atoms& are said to be

fractured if the bonds bet%een the t%o atoms

$or sets of atoms& are broken by e(ternally

applied tensile load

Theoretical -ohesive Stress

"f a tensile force .T/ is applied to separate the

t%o atoms, then bond or cohesive energy is

$'!0&

1here is the equilibrium spacing bet%een t%o atoms!

"deali)ing force2displacement relation as one

half of sine %ave

$'!'&

o(

Td(∞

φ = ∫

( o

(

-T sin$ &πλφ =

+ +

xo

BondEnergy

CohesiveForce

λ

EquilibriumDistance xo

Distance

Repulsion

Attraction

Tension

Compression k

BondEnergy

Distance


4/30

Theoretical -ohesive Stress $-ontd!&

ssuming that the origin is defined at and for small

displacement relationship is assumed to be linear such

that 3ence force2displacement

relationship is given by

$'!'&

Bond stiffness .k/ is given by

$'!*&

"f there are n bonds acing per unit area and assuming

as gage length and multiplying eq! '!* by n then .k/

becomes young/s modulus and beecomes cohesive

stress such that

$'!4&

+r $'!5&

"f is assumed to be appro(imately equal to the atomic

spacing

+ +

xo

BondEnergy

Cohesive

Force

λ


Distance

Repulsion

Attraction

Tension

Compression k

BondEnergy

Distance

o(

((

sin$ &πλπ

≈

λ

-

(T T

π≈

λ

-T

k

π

= λo(

o(

-T

-σ

c

o

6(λσ = π

c

6σ =

π

λ


5/30

Theoretical -ohesive Stress $-ontd!&

+ +

xo

BondEnergy

CohesiveForce

λ


Distance

Repulsion

Attraction

Tension

Compression k

BondEnergy

Distance

The surface energy can be estimated as

$'!7&

The surface energy per unit area is

equal to one half the fracture energy because t%o surfaces are created %hen a

material fractures! 8sing eq! '!4 in to

eq!'!7

$'!9&

( )(0's - -

:

sin d(λ

πλ∫ λ

γ ≈ σ = σπ

s

-

o

6(γ σ =


6/30

Fracture stress for realistic material"nglis $0;0*& analy)ed for the flat plate %ith an

elliptical hole %ith major a(is 'a and minor a(is 'b,

subjected to far end stress The stress at the tip of

the major a(is $point & is given by

$'!


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Fracture stress for realistic material $contd!&

1hen a >> b eq! '!0: becomes

$'!00&

For a sharp crack, a >>> b, and stress at the crack tip tends to

ssuming that for a metal, plastic deformation is )ero and the sharpest

crack may have root radius as atomic spacing then the stress is

given by

$'!0'&

1hen far end stress reaches fracture stress , crack propagates and

the stress at reaches cohesive stress then using eq! '!9

$'!0*&

This %ould

a'

σ = σ ÷ρ

:ρ = ∞

o(ρ =

o

a'( σ = σ ÷

-σ = σf σ = σ

0 '

sf

6

4a

γ σ = ÷


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?riffith/s 6nergy balance approach

•First documented paper on fracture

$0;':&

•-onsidered as father of FractureMechanics


9/30

?riffith laid the foundations of modern fracture mechanics by

designing a criterion for fast fracture! 3e assumed that pre2

e(isting fla%s propagate under the influence of an applied stressonly if the total energy of the system is thereby reduced! Thus,

?riffith@s theory is not concerned %ith crack tip processes or the

micromechanisms by %hich a crack advances!

?riffith/s 6nergy balance approach $-ontd!&

a

!

"

B

σ

σ

?riffith proposed that .There is a simpleenergy balance consisting of the decrease

in potential energy %ith in the stressed

body due to crack e(tension and this

decrease is balanced by increase in surfaceenergy due to increased crack surface/

?riffith theory establishes theoretical strength of

brittle material and relationship bet%een fracture

strength and fla% si)e .a/ f σ


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a

!

"

B

σ

σ


The initial strain energy for the uncracked plate

per thickness is

$'!04&

+n creating a crack of si)e 'a, the tensile force

on an element ds on elliptic hole is rela(ed

from to )ero! The elastic strain energyreleased per unit %idth due to introduction of a

crack of length 'a is given by

$'!05&

'

i

8 d'6

∫ σ=

a0

a ':

8 4 d( v∫ = − σ× ×

d(σ×

%here displacement

v a sin6

σ= × θusin g ( a cos= × θ

' '

a

a8

6

πσ=


11/30


a

!

"

B

σ

σ

6(ternal %ork = $'!07&

The potential or internal energy of the body is

Aue to creation of ne% surface increase in

surface energy is

$'!09&

The total elastic energy of the cracked plate is

%8 Fdy,δ∫ =

%here F= resultant force = area

=total relative displacement

σ ×

δ

p i a %8 =8 8 28

s8 = 4aγ γ

' ' '

t s

a8 d Fdy 4a

'6 6 δ∫ ∫

σ πσ= + − + γ

#$

#

Displacement% v

Crack beginsto gro& 'rom

length (a)

Crack islonger by anincrement (da)

v

? iffi h/ 6 b l h $- d &


12/30


Cracklength% a

Elastic *trainenergy released

Total energy

#otential energyrelease rate + ,

*yr'ace energy-unitextension ,

Cracklength% a

ac

.nstable*table

(a)

(b)

(a) /ariation o' Energy &ith Crack length

(b) /ariation o' energy rates &ith crack length

The variation of %ith crack

e(tension should be minimum

Aenoting as during fracture

$'!0;& for plane stress

$'!':&

for plane strain

t8

'

t

s

d8 ' a

: 4 :da 6

πσ= ⇒ − + γ =

f σσ0 '

sf

'6

a

γ σ = ÷

π

0 '

sf '

'6

a$0 &

γ σ = ÷π − ν

The ?riffith theory is obeyed by

materials %hich fail in a completely

brittle elastic manner, e.g. glass,

mica, diamond and refractory

metals!

? iffi h/ 6 b l h $- d &


13/30


?riffith e(trapolated surface tension values of soda lime glass

from high temperature to obtain the value at room temperature as

8sing value of 6 = 7'?Ca,The value of as :!05 From the e(perimental study on spherical vessels he

calculated as :!'5 D :!'<

3o%ever, it is important to note that according to the ?riffiththeory, it is impossible to initiate brittle fracture unless pre2

e(isting defects are present, so that fracture is al%ays considered

to be propagation2 $rather than nucleation2& controlledE this is a

serious short2coming of the theory!

'

s :!54 m !γ =0 '

s'6γ ÷π MCa m!

0 '

sc

'6a

γ σ = ÷π MCa m!


14/30

Modification for Ductile Materials

For more ductile materials $e.g. metals and plastics& it is found that

the functional form of the ?riffith relationship is still obeyed, i.e.

. 3o%ever, the proportionality constant can be used to

evaluate γ s $provided 6 is kno%n& and if this is done, one finds thevalue is many orders of magnitude higher than %hat is kno%n to be

the true value of the surface energy $%hich can be determined by

other means&! For these materials plastic deformation accompanies

crack propagation even though fracture is macroscopically brittleE

The released strain energy is then largely dissipated by producing

locali)ed plastic flo% at the crack tip! "r%in and +ro%an modified

the ?riffith theory and came out %ith an e(pression

1here γ prepresents energy e(pended in plastic %ork! Typically for

cleavage in metallic materials γ p=0:4 m' and γ s=0 m'! Since γ p>>

γ s %e have

0 '

s pf

'6$ &a

γ + γ σ = ÷π

0 '

pf '6

aγ σ = ÷π

0 '

f aσ µ


15/30

Strain Energy Release RateThe strain energy release rate usually referred to

Gote that the strain energy release rate is respect to crack length and

most definitely not time! Fracture occurs %hen reaches a critical

value %hich is denoted !

t fracture %e have so that

+ne disadvantage of using is that in order to determine it is

necessary to kno% 6 as %ell as ! This can be a problem %ith somematerials, eg polymers and composites, %here varies %ith

composition and processing! "n practice, it is usually more

convenient to combine 6 and in a single fracture toughness

parameter %here ! Then can be simply determined

e(perimentally using procedures %hich are %ell established!

d8?

da=

c?

c? ?=0 '

cf

0 6?

H a

σ = ÷π c? f σ

c?

c? c# '

c c# 6?=c#


16/30


17/30

I"G6J 6IST"- FJ-T8J6 M6-3G"-S $I6FM&


18/30

I"G6J 6IST"- FJ-T8J6 M6-3G"-S $I6FM&For I6FM the structure obeys 3ooke/s la% 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 characteri)ed by stress intensity factor # %hich is related to both

the stress and the si)e of the fla%! The analytic development of the stress intensity

factor is described for a number of common specimen and crack geometries belo%!

The three modes of fracture

Mode ! Opening mod e" %here the crack surfaces separate symmetrically %ith

respect to the plane occupied by the crack prior to the deformation $results from

normal stresses perpendicular to the crack plane&E

Mode ! Sliding mod e" %here the crack surfaces glide over one another in

opposite directions but in the same plane $results from in2plane shear&E and

Mode ! Tearing mod e" %here the crack surfaces are displaced in the crack plane and parallel to the crack front $results from out2of2plane shear&!


19/30

"n the 0;5:s "r%in K9L and co%orkers introduced the concept of stress intensity

factor, %hich defines the stress field around the crack tip, taking into account

crack length, applied stress σ and shape factor H$ %hich accounts for finite si)e

of the component and local geometric features&! The #iry stress function."n stress analysis each point, (,y,), of a stressed solid undergoes the stressesE

σ( σy, σ), τ(y, τ(),τy)! 1ith reference to figure '!*, %hen a body is loaded andthese loads are %ithin the same plane, say the (2y plane, t%o different loading

conditions are possible

I"G6J 6IST"- FJ-T8J6 M6-3G"-S $-ontd!&

Crack#lane

ThicknessB

ThicknessB

σ

σ

σ

σ

σ0 σ0

σ0 σ0a

#lane *tress #lane *train

y

!

σ

σ

σ

σyy

0! plane stress (PSS), %hen the

thickness of the body is

comparable to the si)e of the

plastic )one and a free

contraction of lateral surfacesoccurs, and,

'! plane strain (PSN), %hen the

specimen is thick enough to

avoid contraction in the

thickness )2direction!


20/30

"n the former case, the overall stress state is reduced to the three

componentsE σ(, σy, τ(y, sinceE σ), τ(), τy)= :, %hile, in the latter

case, a normal stress, σ), is induced %hich prevents the )displacement, ε) = % = :! 3ence, from 3ooke@s la%

σ) = ν $σ(σy&%here ν is Coisson@s ratio!For plane problems, the equilibrium conditions are

"f φ is the iry/s stress function satisfying the biharmoniccompatibility -onditions

∂∂

+ ∂

∂ =

∂

∂ +

∂

∂ =

σ τ σ τ x xy y xy

x y y x: : E

∇ = 4

: φ


21/30

Then

For problems %ith crack tip 1estergaard introduced iry/s stress

function as

1here Z is an analytic comple( function

' ' '

( y (y' ', ,

y ( (y

∂ φ ∂ φ ∂ φσ = σ = τ = −

∂ ∂ ∂

JeK L y "mKNLN−=

φ = +

Z z z y z z x iy= +JeK L "mK L E =

nd are 'nd and 0st integrals of Z(z)

Then the stresses are given by

N,N= −

'@

( '

'@

y '

'@

(y

@

JeKNL y "mKN Ly

JeKNL y "mKN L(

y "mKN L(y

%here N =dN d)

∂ φσ = = −

∂

∂ φσ = = +∂∂ φ

τ = = −∂

+ i d l i M d "


22/30

+pening mode analysis or Mode "

-onsider an infinite plate a crack of length 2a subjected to a bia(ial

State of stress! Aefining

Boundary -onditions • t infinity

• +n crack faces

( ) ( y (yO ) O , := ∞ σ = σ = σ τ =

( ) ( (ya ( aEy : :− < < = σ = τ =

( )' ' )N

) aσ=

−

By replacing ) by z+a , origin shifted to crack tip!

Z z a

z z a=

−

+

σ

b '

nd %hen Oz|: at the vicinity of the crack tip


23/30

nd %hen O z|: at the vicinity of the crack tip

# " must be real and a constant at the crack tip! This is due to a

Singularity given by

The parameter # " is called thestress intensity factor for opening

mode "!

Z a

az

K

z

K a

I

I

= =

=

σ

π

σ π

' '

0

z

Since origin is shifted to crack

tip, it is easier to use polar

-oordinates, 8sing

Further Simplification gives

z ei= θ

# *θ θ θ


24/30

"(

"y

"(y

# *cos 0 sin sin

' ' '' r

# *cos 0 sin sin

' ' '' r

# *sin cos cos' ' '' r

θ θ θ σ = − ÷ ÷ ÷ π θ θ θ σ = + ÷ ÷ ÷ π

θ θ θ τ = ÷ ÷ ÷ π

( )"ij ij "#

"n general f and # H a' r

%here H = configuration factor

σ = θ = σ ππ

From 3ooke/s la%, displacement field can be obtained as

'

"

'"

'$0 & r 0u # cos sin

6 ' ' ' '

'$0 & r 0v # sin cos6 ' ' ' '

− ν θ κ − θ = + ÷ ÷ π

− ν θ κ − θ = + ÷ ÷ π

%here u, v = displacements in (, y directions

$* 4 & for plane stress problems

* for plane strain problems0

κ = − ν

− ν κ = ÷+ ν

The vertical displacements at any position along ( a(is


25/30

The vertical displacements at any position along (2a(is

$θ = 0) is given by

The strain energy required for creation of crack is given by the

%ork done by force acting on the crack face %hile rela(ing the

stress σ to )ero

( )

( )

' '

'' '

v a ( for plane stress6

$0 &v a ( for plane strain

6

σ= −

σ − ν= − x

v

x

y

( ) ( )

a

'a a' ' ' '

a a: :

' '

0 8 Fv

'For plane stress For plane strain

$0 &8 4 a ( d( 8 4 a ( d(

6 6

a

6

∫ ∫

=

σ σ − ν= σ× − = σ× −

πσ ' ' '

a

' ' '

" "

'

""

a $0 &

6

The strain energy release rate is given by ? d8 da

a $0 &a? = ? =

6 6

# ? =6

πσ − ν

=

πσ πσ − ν

' '

"" # $0 & ? =

6− ν

Slidi d l i M d '


26/30

Sliding mode analysis or Mode '

For problems %ith crack tip under shear loading, iry/s stress

function is taken as

8sing ir/s definition for stresses

"" yJeKNL

−

φ = −

'@

( '

'@

y '

'@

(y

' "mKNL y JeKN Ly

y JeKN L(

JeKNL y "mKN L(y

∂ φσ = = +

∂

∂ φσ = = −

∂

∂ φτ = − = −

∂

8sing a 1estergaard stress function of the form

( ):

' '

)N

) a

τ=

−


27/30

Boundary -onditions • t infinity• +n crack faces

( ) ( y (y :O ) O :,= ∞ σ = σ = τ = τ

( ) ( (ya ( aEy : :

− < < = σ = τ =1ith usual simplification %ould give the stresses as

""(

""y

""(y

# *cos cos ' cos cos

' ' ' '' r

# *cos sin cos' ' '' r

# *cos 0 sin sin

' ' '' r

θ θ θ θ σ = + ÷ ÷ ÷ ÷ ÷ π

θ θ θ σ = ÷ ÷ ÷ π θ θ θ τ = − ÷ ÷ ÷ π

Aisplacement components are given by

( )[ ]

( )[ ]

""

""

# r u $0 &sin ' cos

6 ' '

# r v $0 &cos ' cos

6 ' '

θ = + ν κ + + θ ÷π

θ = + ν κ − + θ ÷

π

# a= τ π


28/30

"" o

'

""

' '

"

"

# a

# ? = for plane stress

6

# $0 & ? = for plane strain

6

= τ π

− ν

Tearing mode analysis or Mode *

"n this case the crack is displaced along )2a(is! 3ere

the displacements u and v are set to )ero and hence

( y (y y(

(y y( y) )y

y)()

' ''

' '

:

% % and

( y

the equilibrium equation is %ritten as

:( y

Strain displacement relationship is given by

% %

% :( y

ε = ε = γ = γ =

∂ ∂γ = γ = γ = γ =

∂ ∂

∂τ∂τ+ =

∂ ∂

∂ ∂

+ = ∇ =∂ ∂


29/30

(y y)

N

if % is taken as

0% "mK L

?

Then

"mKN LE JeKN L

−=

′ ′τ = τ =

8sing 1estergaard stress functionas

( )

( )

:

' '

:

) y) (y

y) :

)N) a

%here is the applied boundary shear stress

%ith the boundary conditions

on the crack face a ( aEy : :

on the boundary ( y ,

τ=−

τ

− < < = σ = τ = τ =

= = ∞ τ = τ


30/30

"""()

"""y)

( y (y

"""

""" o

The stresses are given by

# sin

'' r #

cos'' r

:

and displacements are given by# 'r

% sin? '

u v :

# a

θ τ = ÷π

θ τ = ÷π

σ = σ = τ =

θ = ÷π = =

= τ π

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ch2-LEFM- Fracture Mechanics