DOKUZ EYLÜL UNIVERSITY MECHANICAL ...kisi.deu.edu.tr/evren.toygar/FRACTURE MECHANICS/fracture...For a body subjected far field biaxial stress 0, with a double ended crack of length

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DOKUZ EYLÜL UNIVERSITY

MECHANICAL ENGINEERING DEPARTMENT

INTRODUCTION TO FRACTURE MECHANICS

STRESS CONCENTRATIONS AND STRESS INTENSITY FACTOR

Prof. Dr. M. Evren Toygar

Fracture

Seperation of a materials into at least two or more pieces due to stress,

at temperatures below the melting point is called as fracture. Steps of

fracture are:

Crack initiation

Crack propagations

Depending on the ability of material to undergo plastic deformation can

be classifies as:

- Ductile Materials - most metals (not too cold)

Extensive plastic deformation a head of the crack

If crack is stable, then it resists further extension unless applied stress

- Brittle Materials – ceramics, ice, glass, cold metals

Relativelly little plastic deformation

Crack is unstable propagates rapidly without increase in applied

stress

Prof.Dr. M.Evren TOYGAR


TENSILE TEST FOR DUCTILE - BRITTLE FRACTURE


Brittle fracture

Crack propagation is fast

Propagates nearly perpendicular to

direction of applied stress

Occurs with Little or no plastic

deformation

many pieces and small deformation

occur during brittle failure


Ductile Fracture

(Cup-and-cone fracture in Al)

The stress causing the fracture is (σ).

The stress causing the sliding (τ).

Occurs with plastic deformation

one piece and large deformation

occur during ductile failure

For plastic deformation it should

be σapp> σyield. However, even

when (σapp) is in the elastic region

the fracture can occur


Ductile vs Brittle Failure

Very

Ductile

Moderately

DuctileBrittle

Fracture

behavior:

Large Moderate%Ra or %El Small

• Ductile fracture is

nearly always desirable!

Ductile:

warning before

fracture

Brittle:

No

warning


Adapted from Fig. 8.3, Callister 7e.

cup-and-cone fracture brittle fracture

Ductile vs Brittle Failure



Stress Concentrations

Fracture strength of a brittle solid is related to cohesive forces

between atoms.

Theoretical strength: ~E/10

Experimental strength ~ E/100 - E/10,000

Difference due to:

Stress concentration at microscopic flaws

Stress amplified at tips of micro-cracks etc., called stress raisers


Stress Concentrations


Stress concentration at an elliptical hole for a = 3c.


Crack perpendicular to applied stress: maximum stress near crack tip

Stress Concentration

ot

/

t

om Ka

21

2


where

K t = stress concentration factor

t = radius of curvature of crack tip

σo = applied stress

σm = stress at crack tip

2/1

t0

mt

a2K


Concentration of Stress at Crack Tip


r/h

sharper fillet radius

increasing w/h

0 0.5 1.01.0

1.5

2.0

2.5

Stress Conc. Factor, Kt

max

o

=

r , fillet

radius

w

h

o

max

During engineering design avoid sharp corners


Stress Intensity Approach and Crack

aKI

The unit of KI is MPam


Crack deformation mode.

I

ij

II

ij0r

fr2

Klim

II

ij

IIII

ij0r

fr2

Klim

III

ij

IIIIII

ij0r

fr2

Klim

Cauchy stress around a crack tip and Stress Intensity Factors

K value can be evaluated using standard mathematical approaches,


Stress intensity factor K can be described as fracture toughness KIC of

materials (material resistance to crack propagation) under conditions of

Brittle fracture

In the presence of sharp crack

Under critical tensile loading

cIC aK

LEFM- Linear Elastic Fracture Mechanics

KIC : is the critical stress intensity factor for plane strain condition in

mode I failure.

ac : is the critical crack length in an infinite plate

app : is the applied stress

Stress Intensity Factor K


For a body subjected far field biaxial stress 0, with a double ended crack of

length 2a, the stress state is given by (this is mode-I loading):

Stresses near and tip of the crack

31

2 2 22

Ixx

KCos Sin S

rin

31

2 2 22

Iyy

KCos Sin S

rin

2

3

222

SinSinCos

r

KIxy

Note the inverse square root (of r) singularity at the crack tip. The intensity of the

singularity is captured by KI (the Stress Intensity Factor).

At = 0 and r → the stresses (xx & yy) should tend to 0( see equations ((1) & (2)). This

implies that the equations should be used only close to crack tip (with little errors) or

additional terms must be added to the equations.

At = 0 and r →0 the stresses goes to infinity.

(1)

(2)

(3)

Fig.1

1ij

r


Stress field equation

2

3

21

22

SinSinCos

r

K Ixx

r

fK Ixx

2

)(→

aKI 0

‘Shape correction factor’ related to ‘Geometry’

Indicates mode I ‘loading’

Half the crack length (for a fully internal crack)

“KI (the Stress Intensity Factor) quantifies the magnitude of the effect of stress singularity at

the crack tip. K has units of [MPam].

(1)

),( rfK Ixx


Stresses in the systems with crack

When the crack at the center of a plate the stress andstress intensity factors terms can be given as :

I

ijII

ij fr

K

2

ij

f

rafaKI /

: Shape correction factor


Center crack in an infinite plate subjected to tension

Stress Intensity Factor Aprroach K :

aK

32

2.12152.1256.01

w

a

w

a

w

a

h

2a

b

P

P

asec where w = 2b


Inclined crack in a plate

Mode I and Mode II calculations of a plate subjected to

tension. (Mixed Mode Loading)


Types of Stress:

Plane stress problem : the stress in z direction becomes

zero.

z = xz = yz = 0,

Plane strain problem : the strain in z direction becomes

zero

xz = yz = 0 and

z = (x + y).


Geometries of Fracture

Center crack and finite plate:

32

2.12152.1256.01

w

a

w

a

w

a

w

asec

aK Stress intensity factor

infinite plate:


Single sided notch, subjected to stress


if( a

432

w

a42.30

w

a74.21

w

a56.10

w

a23.012.1

w) semi infinite plate β=1.12

One-sided cracked plate


Double-sided notch :

32

w

a46.15

w

a79.4

w

a43.012.1


a) if( a w) semi infinite plate β=1.12

b)32

w

a46.15

w

a79.4

w

a43.012.1


Stress intensity factors Y wrt a/w ratio


Eliptic Crack


Semi Eliptic Surface crack



The Shape correction factor (β)

The geometry of the crack and its relation to the body will play an important role on its

effect on fracture.

The factor β depends on the geometry of the specimen with the crack.

β =1 for the center crack in an infinite body.

β =1.12 for a edge crack.

β = 2/ for a embedded penny shaped crack.

β = 0.713 for a surface half-penny crack.


The crack tip fields consists of two parts:

(i) singular part (which blow up near the crack tip) and

(ii) the non-singular part.

The region near the crack tip, where the singular part can describe the stress fields is the K-

Dominance region. This is the region where the stress intensity factor can be used to

characterize the crack tip stress fields.

If in materials where the crack tip plastic zone is smaller than the concepts of linear elastic

fracture mechanics (LEFM) can be used with little error.

Stress Intensity Factor K


One of the important goals of fracture mechanics is to derive a material parameter, which

characterizes cracks in a material. This will be akin to yield stress (y) in a uniaxial tension

test (i.e. y is the critical value of stress, which if exceeded ( y) then yielding occurs).

The criterion (critical condition) for fracture in mode-I can be written as:

Fracture Toughness (Irwins’s K- Based)

ICI KK Where, KIC is the critical value of stress intensity factor (K) and is known

as Fracture Toughness

KIC is a material property (like yield stress) and can be determined for different materials

using standard testing methods. KIC is a microstructure sensitive property.

The focus here is the ‘local’ crack tip region and not ‘global’, as in the case of Griffith’s

approach.

All the restrictions/assumptions on K will apply to KIC:

(i) material has a liner elastic behaviour (i.e. no plastic deformation or other non-linear

behaviour),

(ii) inverse square root singularity exists at crack tip (eq. (1)),

(iii) the K-dominance region characterizes the crack tip.

r

fK Ixx

2

)(

(1)


Material KIC [MPam]

Cast Iron 33

Low carbon steel 77

Stainless steel 220

Al alloy 2024-T3 33

Al alloy 7075-T6 28

Ti-6Al-4V 55

Inconel 600 (Ni based alloy) 110

Fracture Toughness (KIC) for some typical materials [1]


Is KIC really a material property like y?

KIC (in mode-I loading) (KIIC & KIIIC under other modes of loading) are the material property,

independent of the geometry of the specimen. In reality, KIC depends on the specimen

geometry and loading conditions.

The value KIC is especially sensitive to the thickness of the specimen. A thick specimen

represents a state that is closer to plane strain condition, which tends to suppress plastic

deformation and hence promotes crack growth (i.e. the experimentally determined value of

KIC will be lower for a body in plane strain condition). On the other hand, if the specimen is

thin (small value ‘t’ in the figure), plastic deformation can take place and hence the measured

KIC will be higher (in this case if the extent of plastic deformation is large then KI will no

longer be a parameter which characterizes the crack tip accurately).

KIC is the fracture toughness value determined from ‘plane strain tests’.


First one has to decide if the material is brittle or ductile (i.e. if the crack tip is sharp or

blunted).

If the material is brittle one has to decide if the material is linear elastic or not.

For linear elastic materials we can apply the concept of K and use material property KC

(usually in mode-I these quantities become KI and KIC). We could also use G & GC.

If the material is ductile then we need to determine if the plastic zone is small compared to

the K dominance zone. If yes then we can continue to use the concepts of K and G.

If crack tip plasticity is large, then we have to use concepts like J-integral and CTOD. The

crack tip stress fields in this case is given by the HRR fields. (Noting that technically JC is for

crack nucleation).

Essentially there are two approaches: global (energy based) and local (stress based).

For linear elastic materials the energy and stress field approaches can be considered

equivalent.


Sample Problem: 3 mm thick tension panel 10 cm wide containing an edge

crack of 1 mm yielded at a load of 150 kN. However, at a load of 120 kN,

another panel of same material cracked into two pieces when the crack was 5

mm long. With this information, calculate the yield stress and fracture

toughness of the material. SIF for the edge crack is rafaKI /12.1


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DOKUZ EYLÜL UNIVERSITY MECHANICAL ...kisi.deu.edu.tr/evren.toygar/FRACTURE MECHANICS/fracture...For a body subjected far field biaxial stress 0, with a double ended crack of length