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, propagation and arrest

• The residual life in a component subjected to dynamic/fatigue loading

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 particular component.

Other factors such as temperature, loading rate, stress concentrations, residual stresses, etc., influence these three primary factors.

Fracture at the Atomic level

Two atoms or a set of atoms are bonded together by cohesive energy or bond energy. Two atoms (or sets of atoms) are said to be fractured 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 the two atoms, then bond or cohesive energy is

(2.1)Where is the equilibrium spacing between two atoms.Idealizing force-displacement relation as one half of sine wave

(2.2)

ox

Tdx

xo

x

CT sin( )

+ +

xo

BondEnergy

CohesiveForce

EquilibriumDistance xo

PotentialE

nergy

Distance

Repulsion

Attraction

Tension

Compression

App

liedForce

k

BondEnergy

Distance

Theoretical Cohesive Stress (Contd.)

Assuming that the origin is defined at and for small displacement relationship is assumed to be linear such that Hence force-displacement relationship is given by

(2.2)

Bond stiffness ‘k’ is given by

(2.3)

If there are n bonds acing per unit area and assuming as gage length and multiplying eq. 2.3 by n then ‘k’ becomes young’s modulus and beecomes cohesive stress such that

(2.4)

Or (2.5)

If is assumed to be approximately equal to the atomic spacing

+ +

xo

BondEnergy

CohesiveForce

EquilibriumDistance xo

PotentialE

nergy

Distance

Repulsion

Attraction

Tension

Compression

App

liedForce

k

BondEnergy

Distance

ox

xx

sin( )

C

xT T

CT

k

ox

ox

CT

C

co

Ex

c

E

Theoretical Cohesive Stress (Contd.)

+ +

xo

BondEnergy

CohesiveForce

EquilibriumDistance xo

PotentialE

nergy

Distance

Repulsion

Attraction

Tension

Compression

App

liedForce

k

BondEnergy

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)

x12

s C C0

sin dx

s

C

o

Ex

Fracture stress for realistic materialInglis (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, thenWhen b is very very small, Inglis define radius of curvature as (2.9)

And the tip stress as

(2.10)

2a

2b

A

Ar

A

2a1

b

A

tk

tk 3.

2ba

r

A

a1 a r

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

a2

r

0r

oxr

Ao

a2

x

A C f

1/ 2

sf

E4a

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 (Contd.)

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

2a

X

Y

B

Griffith’s Energy balance approach (Contd.)

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

On creating a crack of size 2a, the tensile force on an element ds on elliptic hole is relaxed from to zero. The elastic strain energy released per unit width due to introduction of a crack 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

Griffith’s Energy balance approach (Contd.)

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 cracked plate is

(2.18)

wU Fdy,

where F= resultant force = area

=total relative displacement

p i a w U =U +U -U

s U = 4a

2 2 2

t sA

aU dA Fdy 4a

2E E

P1

P2

(a)

(a+da)

Load

,P

Displacement, v

Crack beginsto grow fromlength (a)

Crack islonger by anincrement (da)

2 2

aa

UE

v

Griffith’s Energy balance approach (Contd.)

Ene

rgy,

U

Cracklength, a

Surface

Energ

yU

=4a s

2 2

aa

UE

Elastic Strainenergy released

Total energy

Rates

,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

2Ea

1/ 2

sf 2

2Ea(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.

2s 0.54J / m .

1/ 2

s2E

MPa m.1/ 2

sc

2Ea

MPa m.

Modification for Ductile MaterialsFor 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

evaluate s (provided E is known) and if this is done, one finds the

value is many orders of magnitude higher than what is known to be the true value of the surface energy (which can be determined by other means). For these materials plastic deformation accompanies crack propagation even though fracture is macroscopically brittle; The released strain energy is then largely dissipated by producing localized plastic flow at the crack tip. Irwin and Orowan modified the Griffith theory and came out with an expression

Where prepresents energy expended in plastic work. Typically for

cleavage in metallic materials p=104 J/m2 and s=1 J/m2. Since p>>

s we have

 

           

1/ 2

s pf

2E( )

a

1/ 2

pf

2E

a

1/ 2f a

Strain Energy Release RateThe 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.

dUG

da

cG

cG G1/ 2

cf

1 EG

Y a

cG f

cG

cG cK2c cK EGcK

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 crack plane and parallel to the crack front (results from out-of-plane shear).

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;

xy, 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.)

CrackPlane

ThicknessB

ThicknessB

z z

z za

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.

In the former case, the overall stress state is reduced to the three components; x, y, xy, since; z, xz, yz= 0, while, in the latter

case, a normal stress, z, is induced which prevents the z

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

z = ν (x+y)

where νis Poisson's ratio.For plane problems, the equilibrium conditions are:

If is the Airy’s stress function satisfying the biharmonic compatibility Conditions

¶¶

¶¶

¶¶

¶¶

x xy y xy

x y y x0 0 ;

4

0

Then

For problems with crack tip Westergaard introduced Airy’s stress function as

Where Z is 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 iybg Re[ ] Im[ ] ; = +

And are 2nd and 1st integrals of Z(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

¶ ¶

¶ ¶¶ ¶

Opening mode analysis or Mode I

Consider an infinite plate a crack of length 2a subjected to a biaxialState of stress. Defining:

Boundary Conditions :• At infinity • On crack faces

x y xy| z | , 0

x xya x a;y 0 0

s

s

x

y

2a

s

2 2

zZ

z a

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

Zz a

z z a

b gb g2

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.

Za

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

Ix

Iy

Ixy

K 3cos 1 sin sin

2 2 22 r

K 3cos 1 sin sin

2 2 22 r

K 3sin cos cos

2 2 22 r

Iij ij I

KIn general f and K Y a

2 r where Y = configuration factor

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

The vertical displacements at any position along x-axis (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

22 2

v a x for plane stressE

(1 )v a x for plane strain

E

x

v

x

y

a

2a a2 2 2 2

a a0 0

2 2

1 U Fv

2For 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

2I

I

a (1 )

EThe strain energy release rate is given by G dU da

a (1 )aG = G =

E E

KG =

E

2 2I

I

K (1 ) G =

E

Sliding mode analysis or Mode 2

For problems with crack tip under shear loading, Airy’s stress function is taken as

Using Air’s definition for stressesII y Re[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

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 cos

2 2 2 22 r

K 3cos sin cos

2 2 22 r

K 3cos 1 sin sin

2 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

II o

2I

I

2 2I

I

K a

KG = for plane stress

E

K (1 ) G = for plane strain

E

Tearing mode analysis or Mode 3

In this case the crack is displaced along z-axis. Here the 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

0 x y

Strain displacement relationship is given by

w ww 0

x y

¶ ¶ ¶ ¶

¶¶ ¶ ¶

¶ ¶ ¶ ¶

xy yz

Z

if w is taken as

1w 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 ,

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 sin

G 2

u v 0

K a