1/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Introduction to

FRACTURE MECHANICS

2/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

ekspl =Rm/s < RH

This fundamental design formula is valid:1. Below load bearing capacity 2. Within elastic region

but it turned out to be unsatisfactory in two situations:

q=q(t) , P=P(t)

Because of material FATIGUE

Beacuse of material CRACKING

(Fatigue Mechanics)

(Fracture Mechanics)

When loading (and consequently – stress) is varying in time:

When geometry of a structure yields stress concentration

σ

t

Elasticity versus fracture

3/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

na

a s

CF

mr

r s

CF

s

mo

rno

a

s

C

s

Cra FF

Fr FaFa Fr

Reactive force (repulsion) Active force (attraction)

m > n (m10, n 5)

For s=so

so

Fr FaFa Fr

nmo

mnom

o

no

r

a

ss

s

s

C

C

1

Lennart-Jones model

4/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

m

mo

n

no

no

am

mo

mo

rn

no

no

amr

na

ra s

s

s

s

s

C

s

s

s

C

s

s

s

C

s

C

s

CFFF

s > so

no

amo

r

s

C

s

C

Fr FaFa FrF F

so

F

0

=s-so

F

s0

Lennart-Jones model

5/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

m

o

o

n

o

ono

a

m

o

n

ono

am

mo

n

no

no

a

s

s

s

s

s

C

s

s

s

s

s

C

s

s

s

s

s

CF

F

0

F=FR

=R

os

mn

no

a

mn

no

a

m

o

n

ono

a

s

C

s

C

sss

CF

111

1

1

1

1

1

1

1

0R

d

dF

011 11 mR

nRn

o

a mnd

dF

s

C

R

n

mmnR 111 1

1

nm

R n

m

=R

Lennart-Jones model

6/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

%1515,01215

10 5

1510

1

R

nm

m

nm

n

no

aR n

m

n

m

s

CF

m=10, n =5

1

1

nm

R n

m

no

ano

ano

aR s

C

s

C

s

CF 25,022

5

10

5

10 21510

10

510

5

=(101000)(FR experimental )

Reasons for discrepancy:

1. Extremely simplified two-atomic model

2. Defects of crystalline structure (theory of dislocation)

Lennart-Jones model

7/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

XV century

XIX century

X century

Fatigue Observed since prehistoric times: example – technology development in shipbuilding

8/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

A ship is working alternatively as double cantilever and simply supported beam

Deck under tension!Keel under compression!

Keel under tension!Deck under compression!

Until the middle of XX century the problem of ship cracking caused by fatigue remained unsolved.

Fatigue

9/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Transporter SS Schenctady, cracked in half when docked in the port on 16.01.1943, Portland, OR

10/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Fatigue appearance accelerated when steam railways were introduced in XIX century.

„The Rocket”, steam locomotive built byR. Stephenson, 1829

Fatigue

11/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

M

t

Wheel axis cross-section

A

A

A

A

12/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

A.Wöhler (1819-1914)

Wöhler stand for fatigue investigation

Wöhler diagram for high-cycle fatigue

Fatigue limit

Fatigue

13/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Stress approach

q[Pa]

q[Pa]

y

x

σy= 3q

G.Kirsch, 1898 – band of infinite width with circular hole

Cracking

14/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Stress approach

q[Pa]q[Pa]

y

x

C.E.Inglis, 1913 – A band of infinite width with elliptic hole

a

b

b

aqy

21

b 0 σ a b σ 3q

Independent of the hole half- radius size a !!!

Cracking

15/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

H.M.Westergaard, 1939, N.I.Muskhelischvili, 1943 – analysed 2D stress field at the tip of a sharp slit

y

...

2

3sin

2sin1

2cos

20

r

ay

A

r

aA 20

0r

aK 0r

K

r

aA

220

r

x

0

Ay

For

Singularity!

Stress intenstiy factor

a

Cracking

0For

16/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Three typical cases (modes) can be distinguished:

Stress intensity factors were calculated for different configuration of loading and specimen geometry (G.Sih)

Mode I - Tearing; crack surfaces separate perpendicularly to the crack front.Mode II – In-plane shear; crack surfaces slide perpendicularly to the crack front Mode III – Out-of-plane shear; crack surfaces slide parallely to the crack front .

KI KII KIII

Cracking

17/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

Design criteria are:

KI < KIc KII < KIIc KIII < KIIIc

where KIc , KIIc , KIIIc

are critical values of corresponding stress factors, being determined experimentally.

Cracking

18/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

2l

q

q

c c

What is the length of a central slit we can introduce to the structure shown without the

reduction of its load bearing capacity? (no interaction assumed)

For side crack

For central crack

cqK sideI 12,1

lqK centerI

sideIcenterI KK cqlq 12,1

cl 25,1

cl 212,1If e.g. c = 2 cm

2l 5 cm

Simple example of Fracture Mechanics application

2l ≤2c or 2l 2c ?

19/18M.Chrzanowski: Strength of Materials

SM2-12: Fracture

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