Introduction Fracture Mechanics 2

7/24/2019 Introduction Fracture Mechanics 2

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Fracture Mechanics

Brittle fractureFracture mechanics is used to formulate quantitatively

The degree of Safety of a structure against brittle fracture

The conditions necessary for crack initiation, propagationand arrest

The residual life in a component subjected todynamic/fatigue loading


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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. (Theseterms 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

particular component.


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Fracture at the Atomic level

Two atoms or a set of atoms are bondedtogether by cohesive energy or bond energy. Two atoms (or sets of atoms) are said to befractured 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 thetwo atoms, then bond or cohesive energy is

(2.1)Where is the equilibrium spacing

between two atoms.Idealizing force-displacement relation as onehalf of sine wave

(2.2)

ox

Tdx

xo

x

CT sin( )

+ +

xo

BondEnergy

CohesiveForce

EquilibriumDistance xo

Po

ten

tia

lE

nergy

Distance

Repulsion

Attraction

Tension

Compression

App

lie

dF

orce

k

BondEnergy

Distance


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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 suchthat Hence force-displacementrelationship is given by

(2.2)

Bond stiffness k is given by

(2.3)

If there are n bonds acing per unit area and assumingas gage length and multiplying eq. 2.3 by n then k

becomes youngs modulus and beecomes cohesivestress such that

(2.4)

Or (2.5)

If is assumed to be approximately equal to the atomicspacing

+ +

xo

Bond

Energy

CohesiveForce


P o

t e n

t i a

l E

n e r g y

Distance

Repulsion

Attraction

Tension

Compression

A p

p l i

e d F

o r c e

k

BondEnergy

Distance

ox

xx

sin( )

C

xT T

CT

k

o

x

ox

CT

C

co

E

x

c

E


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Theoretical Cohesive Stress (Contd.)

+ +

xo

BondEnergy

CohesiveForce


P o

t e n

t i a

l E

n e r g y

Distance

Repulsion

Attraction

Tension

Compression

A p p

l i e

d F

o r c e

k

BondEnergy

Distance

The surface energy can be estimated as

(2.6)

The surface energy per unit area isequal to one half the fracture energybecause two surfaces are created when amaterial fractures. Using eq. 2.4 in toeq.2.6

(2.7)

x12s C C

0

sin dx

s

C

o

E

x


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Fracture stress for realistic materialInglis (1913) analyzed for the flat plate with anelliptical hole with major axis 2a and minor axis 2b,subjected to far end stress The stress at the tip ofthe major axis (point A) is given by

(2.8)

The ratio is defined as the stressconcentration factor,

When a = b, it is a circular hole, thenWhen b is very very small, Inglis define radius ofcurvature as

(2.9)

And the tip stress as

(2.10)

2a

2b

A

A 2a1 b

A

tk

tk 3.

2b

a

A

a1 a


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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 sharpestcrack may have root radius as atomic spacing then the stress isgiven by

(2.12)

When far end stress reaches fracture stress , crack propagates andthe stress at A reaches cohesive stress then using eq. 2.7

(2.13)

This would

A

a2

0

ox

A

o

a2

x

A C f

1/ 2

sf

E

4a


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Griffiths Energy balance approach

First documented paper on fracture(1920)

Considered as father of FractureMechanics


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A A Griffith laid the foundations of modern fracture mechanics bydesigning a criterion for fast fracture. He assumed that pre-existing flaws propagate under the influence of an applied stressonly if the total energy of the system is thereby reduced. Thus,

Griffith's theory is not concerned with crack tip processes or themicromechanisms by which a crack advances.

Griffiths Energy balance approach (Contd.)

2a

X

Y

B

Griffith proposed that There is a simple

energy balance consisting of the decreasein potential energy with in the stressed

body due to crack extension and thisdecrease is balanced by increase in surface

energy due to increased crack surface

Griffith theory establishes theoretical strength ofbrittle material and relationship between fracture

strength and flaw size af


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2a

X

Y

B


The initial strain energy for the uncracked plateper thickness is(2.14)

On creating a crack of size 2a, the tensile forceon an element ds on elliptic hole is relaxedfrom to zero. The elastic strain energyreleased per unit width due to introduction of acrack of length 2a is given by

(2.15)

2

iA

U dA2E

a1

a 20

U 4 dx v

dx

where displacement

v a sinE

usin g x a cos 2 2

a

aU

E


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2a

X

Y

B

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 crackedplate is

(2.18)

wU Fdy,

where F= resultant force = area

=total relative displacement

p i a wU =U +U -U

sU = 4a

2 2 2

t sA

aU dA Fdy 4a

2E E

P1

P2

(a)

(a+d

a)

Loa

d,

P

Displacement, v

Crack beginsto grow from

length (a)

Crack is

longer by anincrement (da)

2 2

a aU E

v


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Energy

,U

Cracklength, a

Surfa

ceEn

ergy

U =

4as

2 2

a

aU

E

Elastic Strainenergy released

Total energy

Rate

s,

G,

s

Potential energyrelease rate G =

Syrface energy/unitextension =

U

a

Cracklength, 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

2t

s

dU 2 a0 4 0

da E

f1/ 2

sf

2E

a

1/ 2

sf 2

2E

a(1 )

The Griffith theory is obeyed bymaterials which fail in a completely

brittle elastic manner, e.g. glass,mica, diamond and refractorymetals.


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Griffith extrapolated surface tension values of soda lime glassfrom high temperature to obtain the value at room temperature as

Using value of E = 62GPa,The value of as 0.15From 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 consideredto be propagation- (rather than nucleation-) controlled; this is aserious short-coming of the theory.

2s 0.54J / m .

1/ 2

s2E

MPa m.1/ 2

sc

2Ea MPa m.


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Modification for Ductile Materials

For more ductile materials (e.g. metals and plastics) it is found that

the functional form of the Griffith relationship is still obeyed,i.e.. However, the proportionality constant can be used to

evaluates (provided E is known) and if this is done, one finds thevalue is many orders of magnitude higher than what is known to be

the true value of the surface energy (which can be determined byother means). For these materials plastic deformation accompaniescrack propagation even though fracture is macroscopically brittle;

The released strain energy is then largely dissipated by producinglocalized plastic flow at the crack tip. Irwin and Orowan modifiedthe Griffith theory and came out with an expression

Where prepresents energy expended in plastic work. Typically forcleavage in metallic materials p=10

4 J/m2 and s=1 J/m2. Since p>>

s

we have

1/ 2

s p

f

2E( )

a

1/ 2

pf

2Ea

1/ 2f a


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Strain Energy Release RateThe strain energy release rate usually referred to

Note that the strain energy release rate is respect to crack length andmost definitely not time. Fracture occurs when reaches a criticalvalue which is denoted .At fracture we have so that

One disadvantage of using is that in order to determine it isnecessary to know E as well as . This can be a problem with some

materials, eg polymers and composites, where varies withcomposition and processing. In practice, it is usually moreconvenient to combine E and in a single fracture toughness

parameter where . Then can be simply determinedexperimentally using procedures which are well established.

dUG

da

cG

cG G1/ 2

cf

1 EG

Y a

cG f

cG

cG cK2

c cK EGcK


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LINEAR ELASTIC FRACTURE MECHANICS (LEFM)For LEFM the structure obeys Hookes law and global behavior is linearand if any local small scale crack tip plasticity is ignored

The fundamental principle of fracture mechanics is that the stress field around acrack tip being characterized by stress intensity factor K which is related to boththe 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


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In the 1950s Irwin [7] and coworkers introduced the concept of stress

intensity factor, which defines the stress field around the crack tip, takinginto account crack length, applied stress and shape factor Y( whichaccounts 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 bodyis 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.)

CrackPlane

ThicknessB

ThicknessB

z z

z za

Plane StressPlane Strain

y

X

yy

1.plane stress (PSS), when thethickness 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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In the former case, the overall stress state is reduced to the

three components; x, y, xy, since; z, xz, yz= 0, while, inthe latter case, a normal stress, z, is induced whichprevents the z

displacement, z

= w = 0. Hence, from Hooke's law:

z = (x+y)where is Poisson's ratio.

For plane problems, the equilibrium conditions are:

If is the Airys stress function satisfying the biharmoniccompatibility Conditions

x xy y xy

x y y x0 0;


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Then

For problems with crack tip Westergaard introduced Airys stressfunction as

WhereZis an analytic complex function

2 2 2

x y xy2 2, ,

y x xy

Re[ ] y Im[Z]Z

Z z z y z z x iy Re[ ] Im[ ] ; = +

And are 2nd and 1st integrals ofZ(z)

Then the stresses are given by

Z,Z

2'

x 2

2

'y 2

2'

xy

'

Re[Z] y Im[Z ]y

Re[Z] y Im[Z ]x

y Im[Z ]xy

where Z =dZ dz


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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:

Boundary Conditions : At infinity On crack faces

x y xy

| z | , 0

x xya x a;y 0 0

x

y

2a

2 2z

Zz a

By replacing z byz+a , origin shifted to crack tip.

Zz a

z z a

b2

d h | h i i i f h k i


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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 thestress intensity factor for opening

mode I.

Z a

az

K

z

K a

I

I

2 2

1z

Since origin is shifted to cracktip, it is easier to use polar

Coordinates, Using

Further Simplification gives:

z ei


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Ix

Iy

Ixy

K 3cos 1 sin sin

2 2 22 r

K 3cos 1 sin sin2 2 22 r

K 3sin cos cos

2 2 22 r

Iij ij IKIn general f and K Y a2 r

where Y = configuration factor

From Hookes law, displacement field can be obtained as

2I

2

I

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 problems1

Th ti l di l t t iti l i ( i


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The vertical displacements at any position along x-axis ( isgiven by

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

work done by force acting on the crack face while relaxing thestress to zero

2 2

22 2

v a x for plane stress

E(1 )

v a x for plane strainE

x

vx

y

a

2a a2 2 2 2

a a0 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

2II

a (1 )

EThe strain energy release rate is given by G dU da

a (1 )aG = G =

E E

KG =E

2 2II

K (1 ) G =E


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Sliding mode analysis or Mode 2

For problems with crack tip under shear loading, Airys stressfunction is taken as

Using Airs definition for stresses

II yRe[Z]

2'

x 2

2'

y 2

2'

xy

2 Im[Z] y Re[Z ]y

y Re[Z ]x

Re[Z] y Im[Z ]xy

y

2a

0

0

Using a Westergaard stress function of the form

0

2 2

zZ

z a


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Boundary Conditions : At infinity On crack faces

x y xy 0| z | 0,

x xya x a;y 0 0 With usual simplification would give the stresses as

IIx

IIy

IIxy

K 3cos cos 2 cos cos2 2 2 22 r

K 3cos sin cos

2 2 22 rK 3

cos 1 sin sin2 2 22 r

Displacement components are given by

II

II

K ru (1 )sin 2 cos

E 2 2

K rv (1 )cos 2 cos

E 2 2

K


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II o

2I

I

2 2I

I

K a

KG = for plane stress

EK (1 )

G = for plane strainE

Tearing mode analysis or Mode 3

In this case the crack is displaced along z-axis. Herethe displacements u and v are set to zero and hence

x y xy yx

xy yx yz zy

yzxz

2 22

2 2

0

w w and

x y

the equilibrium equation is written as

0x y

Strain displacement relationship is given by

w ww 0x y


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xy yz

Z

if w is taken as

1

w Im[ ]GThen

Im[Z ]; Re[Z ]

Using Westergaard stress functionas

0

2 2

0

z yz xy

yz 0

zZ

z a

where is the applied boundary shear stress

with the boundary conditions

on the crack face a x a;y 0 0

on the boundary x y ,


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IIIxz

IIIyz

x y xy

III

III o

The stresses are given by

Ksin

22 r

Kcos

22 r0

and displacements are given by

K 2rw sinG 2

u v 0

K a

Introduction Fracture Mechanics 2

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