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5. Crack Tip Stress Analysis

ME559Introduction to Fracture Mechanics

p y

Prof. M. Ramulu

© M. Ramulu 1

Outline

• Plane Problem &Airy’s Stress Functions

• Modes of Fracture

• Crack Tip Stress & Displacement fields

– Westergaard

M kh li h ili– Muskhelishvili

– Williams

• Stress Intensity Factor, K

• Relation between G & K

• Summary

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Crack Tip Stress Analysis• Fracture mechanics is based on the assumption that all 

engineering materials contain cracks from which failure starts.

• Require knowledge of the redistribution of stress caused by the cracks in conjunction with crack growth condition.

L di f k d b d i ll i d b• Loading of a cracked body is usually accompanied by inelastic deformation & other nonlinear effects near the crack tip, except ideally brittle materials. Assume these effects are small compared to crack size. Therefore linear theory is more than adequate to address stress distribution in the crack body.

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Crack Tip Stress Analysis: Stress Function Approach

• A “Valid” stress function is one that satisfies both the equilibrium and the compatibility conditions.  

• The “Airy Stress Function” automatically satisfies these conditions.

• To satisfy a real problem, the boundary conditions must match the problem.

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Crack Tip Stress Analysis: Stress Function Approach

• Plane problem

• Plane stress : 

x = x (x,y),  y = y(x,y),  xy = (x,y)  z = xz =  yz= 0Triaxial strains.Triaxial strains.

• Plane strain : 

u = u(x,y),  v = v(x,y), w = 0

z = xz = yz= 0; z= (x+y)

Triaxial stresses.

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Crack Tip Stress Analysis: Stress Function Approach

• Linear elasticity field equation

• 1) Equilibrium equations for plane problem.

2) Strain‐displacement relations.

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Crack Tip Stress Analysis: Stress Function

• 3) Stress‐strain relations (Hooks’ law).

4) Compatibility equations.

Combining the above lead to the compatibility equations :

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Crack Tip Stress Analysis: Airy’s Stress Function

• For a 2‐D elastic problem there exists a function (x,y) from which the stresses can be derived.

• The equilibrium and compatibility equations are q p y qautomatically satisfied if x,y has the biharmonic property

• The function is called the Airy stress function

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Crack Tip Stress Analysis: Airy’s Stress Function

• Airy’s stress function,  (or U)

• Where 1,2,3 are harmonic functions.

• Airy’s stress function,  or U; 

– satisfied 22 = 0 (Biharmonic equation). 

– This implies that the equilibrium and compatibility equations are automatically satisfied.

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Crack Tip Stress Analysis: Airy’s Stress Function

• In polar Coordinate;

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2D‐Crack Problems

• Stress Functions:

–Westergaard (1939)

–Williams (1952, 1957)

Muskhelishvili (1953)–Muskhelishvili (1953)

They all used Airy’s stress function.

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Crack Tip Stress Analysis: Westergaard’s Stress Function

• Westergaard stress function Z(z) expressed in complex variable form. 

Z( ) i d i d i i th k ti t• Z(z) is used in deriving the  crack tip stresses

• The Westergaard stress function is related to Airy stress function as follows

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Crack Tip Stress Analysis: Westergaard’s Stress Function

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Crack Tip Stress Analysis: Westergaard’s Stress Function

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Crack Tip Stress Analysis: Westergaard’s Stress Function

Stresses in terms of the Z(z) are

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Using these stresses and a polar coordinate system at the crack tip, the elastic stresses at the crack tip can be derived.

Crack Tip Stress Analysis: Basic Modes of Fracture

The three Modes of loading cracks are:

• Mode I: Opening

• Mode II: In-Plane Shear

Mode III: O t of Plane Shear

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• Mode III: Out-of-Plane Shear

Opening Mode is the predominant mode of fracture in real engineering problems

In-plane Mode is also called sliding mode

Out-of-plane Mode is also called tearing mode

Crack Tip Stress Analysis:CRACK TRIP STRESS AND DISPLACEMENT FIELDS

z

py

o

rx

z

py

o

rx

z

py

o

rx

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Crack Tip Stress Analysis:Mode I crack problem

y

r

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x2a

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Crack Tip Stress Analysis:Mode I crack problem

xx

yy

xy

iy

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xx

x

aa

rZ = x +iy

rei

Crack Tip Stress Analysis:Mode I crack problem

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Crack Tip Stress Analysis:Mode I crack problem

-

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Crack Tip Stress Analysis:Mode I crack problem

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Crack Tip Stress Analysis:Mode I crack problem

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Crack Tip Stress Analysis: Mode I crack problem (stress Intensity Factor , KI)

Mode I stress intensity factor, KI

• Stress analysis of a component containing crack(s) reduces to determination of KI at the tip of the crack(s).

• KI has units of psi (in)0.5 or MPa (m)0.5.

C i f K i (i )0 5 1 0989* MP ( )0 5

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Conversion factor: Ksi (in)0.5 = 1.0989* MPa (m)0.5

• The difference between one cracked component and another lies in the magnitude of the stress field parameter KI. Higher K indicates higher stress field.

• In essence, KI serves as a scale factor to define the magnitude of the crack tip stress field.

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Crack Tip Stress Analysis:Mode I Crack Tip Elastic Stress Distribution

KI characterizes the magnitude (intensity) of the stresses in the vicinity of an ideally sharp crack tip in linear-elastic and isotropic material.

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KI is a measure for the stress singularity at the crack tip.

These equations are based on the theory of linear elasticity.

They describe the stress field near the crack tip.

Crack Tip Stress Analysis:Mode I crack problem

strain plane and stress plane

for different is and modulusshear is

2cos21

2sin

22

2sin21

2cos

22

2

2

Where

rKv

rKu

I

I

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Crack Tip Stress Analysis:Mode I crack problem

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Crack Tip Stress Analysis:Mode II crack problem

iy

XY

r

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x2a

Crack Tip Stress Analysis:Mode II crack problem

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Crack Tip Stress Analysis:Mode III crack problem

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2a

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Crack Tip Stress Analysis:Mode III crack problem

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Crack Tip Stress Analysis:Mode III crack problem

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Muskhelishvili’s stress function (1953)

• Suitable for solving boundary‐value problem.

• Utilize conformal‐mapping technique.

• Muskhelishvili’s stress function consists of two analytic functions (z) and X(z) which are selected to Airy’s stress function (x,y), throughfunction (x,y), through

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Muskhelishvili’s stress function (1953)

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Williams (1952, 1957)Using semi-inverse method Williams assumed an Airy stress function in polar coordinates of the form

Where the value(s) of are to be determined

y

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as part of the solution.

With this choice of stress function, the stresses can be expressed in polar coordinates

A Wedge of apex angle 2 with traction-free flanks.

Crack Tip Stress Analysis: Airy’s Stress Function

• Stresses in polar Coordinate;

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01111

2

2

22

2

2

2

22

2224

rrrrrrrr

Biharmonic Equation is :

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Williams (1952, 1957)Boundary Conditions:

Substituting the assumed airy stress function into

y

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the biharmonic equation ( ) yields:

Result in an ordinary differential equation and the general solution for f()

4

Williams (1952, 1957)The constants C1, C2, C3, and C4 are to be determined from the boundary conditions on the faces of the wedge. Using the boundary conditions in f() and its derivative :

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Williams (1952, 1957)The resulting equations, in matrix form

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Therefore,

The resulting eigenequation for the crack problem

Williams (1952, 1957)Let =n/2 and = in matrix form yields

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Williams (1952, 1957)General solution in terms of stress function for the Wedge shaped or single-ended crack with traction-free faces is:

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Williams (1952, 1957)Crack tip near-field equations

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Displacement Equation mode I and mode II

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Substituting rr, r,, r in Hooks law for strains and integrity them to obtain displacements

Displacement Equation mode I and mode II

Mode I

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Displacement Equation mode I and mode II

Mode II

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Crack tip elastic stress fields

Linear Elastic Fracture Mechanics (LEFM)

• Fracture  Modes • William’s Mode I and II Infinite Plate 

solution• Mode III solution• Truncation of Williams solution to 2‐terms• Introduction of T‐stressIntroduction of T stress

∑2

21

2 12 2

12

12

3

2 12 2

12

12

3

12 2

12

12

3

∞1

∑2

21

2 12 2

12

12

3

2 12 2

12

12

3

12 2

12

12

3

∞1  

22 1 2

1

21

∞

1,3,5,…

2 1 21

sin2

1

∞

2,4,..

 

1

12

21

232

21

232

2 232

12

400

1

12

22

232

2 232

21

232

22

000

 

4 2

Theoretical Results

Crack‐tip Stress fields T=0 T=+0.5 T=‐0.5

(a) Cartesian stresses in Cartesian and Polar coordinates

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.5

1.0

�ij 2�r

K1

�xy

�y

�x

�T�, 0.��0.4 �0.2 0.2 0.4 0.6 0.8 1.0

�1.0

�0.5

0.5

1.0

1.0

�ij 2�r

K11.0

(a) Cartesian stresses in Cartesian and Polar coordinates

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.5

1.0

�ij 2�r

K1

�xy

�y

�x

�T�, 0.5��0.4 �0.2 0.2 0.4 0.6 0.8 1.0

�1.0

�0.5

0.5

1.0

�ij 2�r

K11.0

(a) Cartesian stresses in Cartesian and Polar coordinates

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.5

1.0

�ij 2�r

K1

�xy

�y

�x

�T�, �0.5��0.2 0.2 0.4 0.6 0.8 1.0

�1.0

�0.5

0.5

1.0

�ij 2�r

K11.0

(b) Polar Stresses in Cartesian and Polar coordinates

(c) Principal stresses in Cartesian and Polar coordinates 

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

�r�

���

�rr

�T�, 0.��0.4 �0.2 0.2 0.4 0.6 0.8 1.0

�1.0

�0.5

0.5

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

1.0

1.2

�ij 2�r

K1

�3

�2

�1

�T�, 0.��0.5 0.5 1.0

�1.0

�0.5

0.5

1.0

(b) Polar Stresses in Cartesian and Polar coordinates

(c) Principal stresses in Cartesian and Polar coordinates 

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

1.0

�r�

���

�rr

�T�, 0.5��0.5 0.5 1.0

�1.0

�0.5

0.5

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

1.0

1.2

�ij 2�r

K1

�3

�2

�1

�T�, 0.5��0.5 0.5 1.0

�1.0

�0.5

0.5

1.0

(b) Polar Stresses in Cartesian and Polar coordinates

(c) Principal stresses in Cartesian and Polar coordinates 

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

1.0

�r�

���

�rr

�T�, �0.5��0.4 �0.2 0.2 0.4 0.6 0.8 1.0

�1.0

�0.5

0.5

30 ° 60 ° 90 ° 120 ° 150 ° 180 °�

0.2

0.4

0.6

0.8

1.0

1.2

�ij 2�r

K1

�3

�2

�1

�T�, �0.5��0.5 0.5 1.0

�1.0

�0.5

0.5

1.0

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Crack under Mode III action by torsion

A i l C

y

yz

y

x

yz

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Antiplane stress Components

Free

Free

x

r

z rz

y

z

Mode III – closely related to Torsion problem

equation harmonic has 02uzz

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)(ulet

0

0

.conditions

boundary under sloved be shouldIt

wzz

z

gr

uzz

z

Mode III – closely related to Torsion problem

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Mode III – closely selected to Torsion problem

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Mode III – closely selected to Torsion problem

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Stresses along the extended crack line and displacements along the crack sides will be of particular importance. We first rename the coefficients C11, C21, and D1 as follows:

© M. Ramulu 54

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22

1)(u :0xFor

2)0( :0xFor

:0 yII, Mode

yx

xK

ku

x

K

IIr

IIr

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2

2)( w:0xFor

2)0( :0xFor

:0 yIII, Mode

yz

xKu

x

K

IIIz

IIIz

Crack Tip Stress Analysis: Stress Intensity as a Similitude Parameter

Two cracked bodies of different size and shape, but of the same material with cracks of different size, the crack tip stress field is given by:

Ki characterizes the magnitude (intensity) of the stresses in the vicinity of an ideally sharp crack tip in linear-elastic and isotropic

iji

ij fr

K

2

© M. Ramulu 56

vicinity of an ideally sharp crack tip in linear-elastic and isotropic material. If the stress intensities for the two cases are equal, there is exact similitude ( ie.,two cracks respond in the same manner). Because of the similutude argument, any cracked body of this material will cause fracture at the same value of the stress intensity.

Restrictions:- Same thickness (plane stress or plane strain)

- No plastic deformation or negligible plastic deformation (Condition of the crack tip)

Relation Between Strain Energy Release Rate and Stress Intensity Factor

GUaa

lim

0

U F x V x x V x dxy yy

2

12

a a

2V

a

yy VG 02

lim 180

yy

a

0

K

2 r

K

2 xV

r

K I

2 2 21 2

22

sin cos

© M. Ramulu 57

a-x

Closure stress

yy

Closure Stress

Displacement

aaG

00 2

lim

Relation between G & K

Vr

K I

2 21

Va a a x

KI 2

12

at

2 2

G

a a a

a

a x

xdx

a

a

lim

0

0

1

4

K KI I

18

KI2

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GE

K I2

E

v

E Plane stress

E Plane strain

1 2

where

cossin2 and ; sinLet 2 adxax

Relation between G & K

rK

Where V is the crack opening displacement

COD

r

a x

K

E

I

2

1

4 2 2

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Relation between G & K

GEI

I2K

G II2K

Mode-I:

Mode-II: GEII

II

GIIIIII2K

2

E

v

E Plane stress

E Plane strain

1 2

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Mode II:

Mode-III:

Where

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Relation between G & K

K K K + KI II IIITotal

K K K + KI IA IB ICTotal Same mode of deformation

G G G GTotal

+ I II III

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However,

Summary

• Crack tip stress analysis using stress function approach was introduced.

– Airy’s stress function

• Westergaard

• Williams• Williams

• Muskhelishvili

• Crack tip stress state is characterized by a single parameter called stress intensity factor, K. Equal K guarantees similitude.

• Relation exists between energy release rate and the stress intensity factors for all modes of deformation 

© M. Ramulu 62