Basic Fracture Mechanic

8/11/2019 Basic Fracture Mechanic

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Lesson

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-

Assuring

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Lesson

5

-

Testing

Design

ConcePts

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PENTON

EDUCATION

DIVISION

PENTON

BUILDING

CLEVELAND,

OHIO

44113

A

ffiAS[C

COURSE

Eru

FRACTTI

RE

M

ECHAIUECS

Pkfr/,6ld

Prepared

bY

CARL

C.

OSGOOD

Member.

Technical

AdvisorY

Staff

Astro-Electronics

D

iv-

RCA

princeton,

N.

J.

for

MACHINE

DESIGN

Magazine

CONTENTS

'

t :rt

r

'

lt-_l__-:--tl

I-esson

I

-

VifLaL

rs rtautulc

lYlcuiiairiu)'

Lesson

2

-

Building

on

the F-undamentals

Lesson

3

-

Applying

Fracture

Mechanics

to Design

15

Rt

c

-{//-

'-ji

--/'

.

I

i

Copyrlqht

l97l

by

Th.

Prnton

h'rbtlthlng

Co'.

Clcvtlrn


2/8

.8

0,

basic

course

in

racture

mecha,nics

What

ls

Fracture

Mechanics?

tsuilding

on

the

Fundamentals

Applying

Fracture

Mechanics

to Design

Assuring

Gomponent

Ufe

I

esttng

Design

Concepts

Lesson

1

D

n

D

n

What

is

Fracture

Mechanics?

Fnacrtrnrs can

occur

in

pipelines, pressure

vessels,

ship

hulls,

and

aircraft

structures

at

stresses

below

the

yield

strength

of the

structural

materials.

Such

failures

have often

come

as

a surprise

to

both

user

and

designer

because

all

t h

e

conventional

strength-of-materials

rules

had

been

followed.

Oh-

viously,

something

was

overlooked-


3/8

sa

psetend

that

you're

the

captain

of a

Liberty ship,

pounding

in

a

heavy

sea

in

the

North

Atlantic

during

the winter

of

lg4l..

fhe

ship's so new

that

rust has

barely

had

time

to

set

in.

5uddenly,

you

feel

the

deck .trembte,

hear

the

snap

of

steel

cracking,

and

see the

bow

drift

away.

You've

iust

observed

what happens

when sq:called ductile

metals

don't

behave

in

a

ductile

mannsl-which

is

what

fracture

mechanics

is

all

about.

Here

is

the

first

part

of a

five-part course

that

will

explain

how

the

concept

of

fracture

mechanics evolved, how

to

recognize

designs

to

which

it

should

be

applied,

and

how to

put

the

concept

to

work.

Nomenclature

a

=

Half crack length,

in-

E

=

Modulus

of elasticity,

psi

G

=

Crack driving

force,

lb

K

=

Stress-intensity

factor,

psi-in-)l

Kl"

=

Critical

stress-intensity factor

for simple

cracking.

Psi-in-*Kr

=

Stress-conceatration

factor

(local

stress/gross

stress)

Q

=

Proportional constant

for

stress-lntensity-factor

equation

R

=

Resistance

to

crack

propagation,

lb

r

=

Polar

coordinate

(radius),

ln

T

=

Surface

tension,

ln

-Ib

t

=

Sheet

thict

ness,

ia-

U'=

PotenUal

energy, iL-Ib

x

=

Factor

of

lVestergaard

equetion

)

=

Factor

of

Westergaard

cquation

Zt

=

'rfr/estcrgaard

functloa

a

=

Angular

polar

coordinate, deg

p

=

Radius of crack 6p,

ln-

o

=

Tensile

stress,.

psi

r

=

Sbear stress,

Dsl

'Subscript

c

identifies

tle decrea-se

iu

fotential

eoergl'

due

to deformation;

o,

the

potential

energy

prior

to

inducing

a

crack;

and

T,

tle increase

in

potential

energy

due to

e

new

surface-

'

tPrime

indicates first

differeotial

lSubscript

x'

y,

and

ry

indicate

tbe

direction

or

plane Iu

which

a stress lies.

account

for minute

cracks in materials.

The frac-

ture-mechanics

design

parameters

are:

a

tough-

ness

(enerry-absorption)

factor, applied stress,

size

of

crack, operating

temperature, and

t}te

"state

of

st1g5s"-qrfuigfi

is

often

far

more

complor t]an

that

usually

assumed.

This

concept

can be

applied

to

any

solid material,

although

most

testing

to

date

has

been concentratd

on

metals-

Reoson

for Doubt

Early

work on

the

effects

of notches

and holes

on

local

stresses accounted

for these

discontinui-

r.h

/r1

o

Jr r

3

in.

l'

L0.00t

\

Lo.

305

rl

|

2

4

5

810

20

40

60 80100

200

400

0istorxe

Aheod

ot

t{olch

Tip

lt0-3

in.)

.Fig-

l-Effect

of

crack-tip

radiu ,.

pr

on

stress eround

thc

crack

in err

infioite

plate

C-urvcs

iadicete

dret

sre$-concentration

fectors

K,

near

a

small

cra&

are

lerge,

but not as

lerge

es

drorc

dcrived

from

the

cooven-

tionel

gcooctric

apprech

1

rRclcrrocs

t.bglrtc{

rt

cr}d

oC

lgao-

._400

;

zoo

v,

o

G

t00

o$Q

, 60

6a9

o

o

J

2n

0

Gl0

0.r o.2

0.3

Tilckntg

(in)

t-1

Yield

Srenl

(ot

0-27J

tl

V

/

t,

rocl

n

:islon

C,R

-200

-r00

0

t00

200 300

400

Temperoture

(F)

Ei1,* z-C-omparison

of

coovenrionel

mcclreoice-l

propcr'

ricl

of a

tyiicat

low-alloy

steel

I'ith

is

resistrnce

to

cncking.

t5

.E

Es

ro

cv

9

t 5

Eto

Es5

lr,

0

?0

u

-Cl

ar-

lo

E-

0

200

0.4

E

_o

l5O

:o

vt=


4/8

ties

in

a

stress

field,

but

these

early

theories

were

unable

to

treat

fine

cracks-

For

instance,

ap.plying

conventional

stress-concentration

factors

to.

f

i

n

e

cracks

resulted

in

large,

unrealistic

values;

these

values

have.-

been

subs,equently

modified

by

frac-

ture-mechanics

concepi,

Fig-

1.

Also,

the

purely

geometric

approach

did

not

consider

strain'

load-

ing

rate,

and

toughness-

However,

these

theories

did

point

out

certain

inconsistencies

in

understanding

a

material's

me-

chanicalpropertiesthroughconventionalstrength-

of-materials

concepts-

In

conventional

design

theories,

elongation

is

often

considered

to

be

a

qualitative

measure

of

a

metal's

ability

to

yield

upon

loading,

thus

redistributing

the

sLresses

-

and

reducing

stress

concentrations

to

tolerable

values'

HowevJr,

evaluating

a

material's

fracture

resist-

ance

by

ils

ability

to

elongate

is

faulty

bgcause

elongation

is

not

directly

related

to

two

very

rm-

portint

design parameters: operating

temperature

Ld

section

thickness,

Fig.

2-

Simila.rly,

elongation

of

steel

remains

essentially

constant

with

temper-

ing

temperature,

Fig-

3;

ihis

fact

has

led

to

the

fal-se

conclusion

thal

tempering

to

the

maximum

Vield

strength

is

desirable

and

does

not

appreciably

reduce

toughness.

Actually,

using

yield

strength

as

a

guidJ

to

selectin

a

tempering

temperature

produles

the

minimum

notch

(crack)

strength'

Fie-

g.

Notch

strength

is

a

function

of

operating

temperature

and,

for

many

steels'

falls

rapidly

just

below

room

temperature'

Thus'

conventional

iesign

thinking

compounds

the

danger

of

abntpt'

brittle

fracturJ

of

a

material thought

to

be ductile

on

the

basis

of

its

room-temperature

eiongation'

Two

APProoches

The

techriolory

of

fracture

mechanics

is

pres-

.nUy

based

ott

t*o

approaches

that

are

some-

what

complementary

and

give

generally

similar

re-

sults.

one

"ppro"ih

grerr/

from

the

recognition

,trJ

bie"

d"ls''l

o

s-"

ptaiti

ei

tv

a

.'

lemPe-Ttqre-'st

The

other,

*or"

*iiiuc"r

ipproich'

was

derived

from

consideraiiont

ot

:lr

tgggt-f-teld

'and

'the

ptasticzol._1 hq 9.s--oJ-cg9


5/8

to

be

the most

fracture

resistant-

Results

of transition-temperature

tests

cannot

be

expressed

directly

in load-carrying

terms.

However,

a

fracture-analysis

diagram

(FAD)

--derived

from

laboratory

tests

and

service

failures-relates

ap-

plied

stiess,

defect

size,

and

temperature,

Fig. 5.

This

diagram

is

applicable

to the structural

grades

of

steel

(between

y2 and

2-in-

thick)

that

have

a

pronounced

ductile-to-brittle

transition

behavior.

Using

the FAD and

knowing

one

of

the

transition

temperatures

(most

commonly

NDT), the

com-

bination

of

stress

and defect

size

that will

cause

catastrophic

failure

can

be approximated

for

a

giv-

en

material.

Below

the

CAT

curve,

the

probability

of

crack

propagation

is small. Above the

CAT, the

constant

fiaw-size

cdrves show the

stress-tempera-

ture

combinations

that

will

intiate fracture

For

high-enerry

tear materials,

applying

the

FAD

:t'*

i?

::"#::';,,1'

liL,

ffff

'

:l

.

T,',flHH

tion

temperatures (preferably

NDT).

Then,

if

you

know

the

applied

stress,

you

can estimate

the

size

of

defect

required

to

start a brittle

fracture.

Con-

versely,

if

you

know

the operating

temperature

relative

to NDT and

the

size

of

a

defect

that

exists

in a

structure,

you

can

estimate the

stress t}at

will

start

a fracture-

Certain limirations

restrict the

general

use

of

the

FAD. Because

it

is

based

on

the stmctural

grades

of

steel

that

have an

abrupt ductiletcr.,b;ti,l;

X;:i.

tion, its use

should

be

restricted

to

these materials.

It is

not applicable

to

the ultrahigh-strength

steels

or to

aluminum

and

titanium alloys

because

t}ese

materials

do

not

have

this

abnrpt transition

be-

havior.

Stress-Analysis

Approach:

The

analytical

apr

proach

to

fracture

mechanics is

based

on

the

stress

field

in

the

vicinity

of

the

ever-present crack.

The

most

useful

way

of determining

this

stress

is

from

the

relationship

between

stress

ahd

changes

in

the

surface

and

potential

energiei

of

a.part-

This

approach

to

fracture

mechanics

assumes

that

a

part

contains

only

one

crack

with

one

plas-

tic

zone at its

tip.

There is

no

need

to consider

multiple

cracks

because

failure is

as

complete

when

a

part

fractures

into

two

pieces

as

it

is

into

three.

The

stress analysis for

fracture

mechanics

be.

Froclure

opPeoronce

Encrgy

T, Tz

Tr

T{

Temptrolrre

-+

ig.,{-Tnnsi

tion ternpcnnrrcs

and

correspondia.B

casrq.y

..vels

obrined from

Cherpy- V-notch

-

inpect

ierc

7r,

is

the trznsition

rcrnp&rnrrc derermincd

bi

a fixed

levei

of

impact cnergy

Er

(e

lcvel

rher can

often

bc estiaritJ

from

producer"

dea)-

The

fncrure-appc2nocc

tnnsidoa

tempcnture

(FATT),

Tz

(arbi,;erily

akea

*

507o

brittle

znd

50Vo

shczr

frzcture), ir

uscd

lo

comparc

thc

fr:cnrre

resisance

of

differcnt

mete.ie '

Ts

ic

the

aridpoint

tear-

pcrenrier-

with

corre-tpondiag

:"qtgy

lcvel

Ea. A[.*r

?r,

fracnrre

is entirdy

rhear.

r00

I

I

.o

Lz

o

a

st

,5

et

s

5

509

la

g

CD

gYs

rtt

=

l5O

C\

o

=

t00

E

o

c

O

;50

:s

c

o

o

b

c

o

fa,

250

200

4

34vs

r6vs

14

Ys

400

600

800

1,000

.

Tempering

Temperolure

(Fl

NoT+lzc

liq.

3--How

remnerlng

rcrnpccrure

affccs

ud

notch

strength

of

a

it"iot.ir

iii.t-t

Fis.

5-Fncnrre

analysis

diegram

(FAD).

Tt-.c

OIT

cutrve

is

the

critical

strcs-tempcrature

rcletionrhip

ftor

stopping

a

nrnning

craclc- Orher curves above

rre-

CrtT

cJrve represcnt

consnnt flew sizes and

indicete

combine-

tions

of

stress

and

tempcfilrure

that

rr.ill

cause

frecnrre.

1,200

elontedon

ield

ttrength

(ot

0.2%l

Froclure

Stress

NOT

NDI+60

Tenperolurr

(Fl


6/8

gins

wift

the

theory

of elasticity,

but

includes

the

efts:t

of

crack-tip

plasticity.

C;'acx-SrABrt.rry

(GnrrrrrH) TecHNreuei

Con-

sider

an

infinite

sheet

of elastic

material,

subject

to

uniform

biaxial

stress

q

at infinity,

into

which

a cr;rck'of

length

2a ii

introduced-

Potential

enerry

U of

the

system

is:

g'=Uo-Uq-FU1

(l)

where

Uo

is

the

potential

energy

prior

to

introduc-

ing

the

crack,

U.

is the

decrease

in

potential enerry

due

to

deformation

(strain

enerry

and

boundary

forct

work),

and U.

is

the

increase

in

surface

ener

ilJ

due to

the new

surface.

The

potential

ener-

W

r'f

deformation

is:

r o2o2t

tt

E

whr,re

t is

the

sheet

thickness

and

E

is

the

m@ulus

of

elasticily

of

the

material.

The

surface

energ/

is

;imply the

surface

tension

of

the

material,

T,

timci

the

new

crack

surface

area.

Ur

=

latT

Combining

Equations

1,2,

and 3

gives:

t&o2t

Ur=Uo-T*4atT

E

The

minimum

potentiat enerry

\*ith

respect

to

crack

size is

a stable

equilibrium

position,

whereas

the

ma:rimum

potential

energy

is

unstable'

Thus,

the

change

in

the

potential

enerry

with

respect

to

crack

sii-e

(the

first

derivative

of

Equation

4)

is

the erack

driving

force,

G, and

the

resistance,

R"

t&a

G=-=27-R

Cnncr-Tlp

Srness-FlElo

AppnoecH:

The

most

common

fracture

is

the

direct

opening

of a

crack'

where

thp

motion

of the

crack

face

is

normal

to

the

plane

of

the

face

(ModeJ)

,

Fig-

7-

The elastic

streis

field

near

the crack

till

can

be

described

by

the

theory

of

elasticity

(i.e.,

plane

stress

or

plane

strain).

The

analysis

of

the

simple

(Mode

I)

crack

is applicable

to

more

complex

cases (lvlodes

II

and

III)

where

shear

motion

occurs

between

the

crack

faces.

In

general,

a

plane

extensional

linear

elasticity

problem can

be

solved

by

finding

a stress

function

ihat

satisfies

a

suitable

biharmonic

equation

such

that

the

resulting

stresses

and/or

displacements

also

safisfy

the

given

boundary

conditions

of

a

problem.

Forthesimplecrack,theWestergaardfunction'

in

the

form

z

-

x

*

iy,

has

proved

(both

mathe'

matically

and

by

experience)

to

"fi "

this

apprcach

to

fracture

mechanics-

Using

this

function,

t-he

stresses

ctst

oyt

and

r'

are

expresed by:

c,

-

fReallZ

-

Y[Imag]Z'

(6)

.*

o,=

fReallZ*YUmag)Z'

';

--

-Y[Real]Z'

where

function

z

(z)

will

give

stresses

that'

auto'

matically

satisfy

the

elastic

theory.

Thus,

oniy

fur.ction

z@)

that

satisfy

the

boundary

conditions

of

a crack

problem

must

be

found

Near

a crack

tip,

the

adjacent

crack

surfaces

are

:|-::::-f:::,

d :t".f :g

the

character

of

z(z)

in

that

vicinity.

Assuming

a coordinate

origin

at

the

right

end

oi a

crack

parallel

to

the

x-axis,

Fig.

8,

z

be'

comes:

;_

tG)

z*

where

f

(z)

must

approach

a

real

constant'at

the

origin.

-Tttus,

r3,

arrd

c,

appro?ch

zero

at

the

crack

siiriouc

(i,iie

su.-i"ii

;i

siress-frle)

-

However,

S

character

of

f

(z)

at any

distance

from

the

crack

tip

is

unspecified

and

can

be

adiusted

to

solve

-iny

configurations

of

simple

symmetrical

cracks

Therefore,

in

the crack

tiP

region:

(2)

(4)

(o

E

Equation

5

describes

the unstable

condltion

(un-

stairie

-irccause

this

is

a

riiaidtrrtiur

l*irtii

ior

a

craek

in

a

perfectly

brittle

material-

Because

plas-

ticity

and

other

effects

in

the

crack-tip

region

in-

crease

a

material's.

resistance

to

crack

extension,

R

=

2I

is

only

the approximate

resistance

to

crack

propagation.

For

ductile

materials'

t h e

additive

ternrs

are

large

compared

to

2T.

However,

frac-

ture

in

brittle

materials

can

be

assumed

to

be

an

enetry-nxts

exchange

where

the rate

of

elastic

enurry

available,

G,

opposes

the

material's

dissipa-

tion

rate,

R.

Rapid

extension

of

a crack

occurs

when

G

exceeds

R.

(As

a

rule

of

thumb,

tough

materials

have Gs

greater

than

50

ft-lb

per

in-:;

brittle

materials

have

Gs

smaller

than this

value.)

The relation

between crack

stobility

and

energy

rote

is

shown

in Fig.

6. The

available

energy

rate

increases

with

applied

stress. Therefore,

upon

load-

ing.

the

material

state

(brittle

or

ductile)

at any

time

is the

first

intersection

of

the

G

and

R

curves,

and

load

can

be

lncreased

until

the

instability

point

"

I

o,*o

where K1

is

a

real

constant.

If

polar

coordinates

(a

d)

are

fixed at

the

origin,

the stresses become:

(8)

t/

2rz

r,=

KI

.cos

t

(t

tiEz\

r

3t

\

-sin_.16_

I

(9)

22

c f r

3,

\

.

cos-. I I *sin-.

Jin-

I

2

\

2

2t

K1

ty=:

{

2rr

i(r

0

,tu=:

'

3in-

2rr

2

03c

cos-. cos-

aa

hr*


7/8

The

factor

K1,

the

crack-tip

stress-intensit5r

fac_

tor,

is

a

constant

in

that

vicinity

and

depen-ds

on

the

body

configuration

and

the

mode

of

loading.

(The

stress-intensity

factor

should

not

be

confused

with


factors,

Kr,

which

are

a

function

of

part

geometry.)

Because

stresses

in

Iinear

erasticity

depend

liae.

arly on

applied

load,

Equation

g

implies

that

stress-

intensity

factors

must

be,

in

part,

composed of

the

load-

Examining

the

dimensions

of Equation

g

thow-s

that

K, must

also

contain

length

paiameters

for

the

bodies

invorved,

incruding

J.".t

rength

cr-

Thus,

I(r=

e.

o,\l

From

Equation

L-the

enerry

criteria

for

crack

stability

in

brittle

materials-instability

occurs

at

a constant

stress-field

intensity

at

tjre

crack tip.

The

stress

fields

surrounding

tjre

tips

of

simpie

cracks

have

the

same

distribution

and

airrer

galy

in

their

intensity

from

one

case

to

another.

tLri,

unstable crack

grofih

takes

place

when

intensity

Kr reaches

a

critical

value,

Kr".

This value

is

a

material property

and

reflects

a

material's

abifiry

to

withstand

a

given

stress

at a

crack-tip.

This

value

can

be

applied

to

cracks

where

a

small

zone

of plasticity

is

present.

For

a

given

material,

plasticity

always

disturbs

the

elasti.lfiua

equation

(Equation

9)

in

the

same.way,

and

con-

sequently,

failure

occurs

at

an

..apparent"

field

intensit5r,

K1.

Also,

K,

=

Kr" can

-be

used

as

a

"brittle

fracture"

criterion

for

ductile

materials,

if

the

material's

resistance

curve

is

shaped

like

the

one

shown

in

Fig.

6.

This

curve

is iuitable

for

most high-strength

materials,

but

it

is

often iaap

propriatc

for

low

strength

and

Ngh-toughness

ma-

terials-

A summary

of

the stress-intenslry

factors

is

given

in

Table

l.

FrxprNc

Srnrss-h.rreNslTy

Fectons:

Tbe

strex

intensity

factbr,

K,

can

be

ob^iair.ed

for;a:-;-;=

;;.-i

shapes

from

tJre

theory

of

elasticity.

The

-o"i

O-

rect

method

is

to

compute

K

from

the

westergaard

O2

Hotf

Croct

trngth

Fig-

fi-{reck

ertension

cnergy

G

and

_R

vcrsus

cnck

tcngth

_foi

a?i

iJeally

Vhen

stress

is

low,

G


8/8

sress

function,

z,

which

must

be

estimated,

on

die

basis

of

experience.

For

specialists

in

the

ileor|

of

elasticity,

guessing

tl,e

functions

is

noc

difficult.

But

for

the

less experienced,

techniques

for

finding

K

for_

two

general

c"ses are

presented

below.

Case

l:

For

the

Griffith

shape

(an

infinite

sheet

with

an

edge

crack

on

the

x-axis,

extending

-a I x

S

o, gdth uniform

bia-.cial

stress,

o, at

in-

nrury)

the

stress

function

is:

oz

L_

(ll)

vv4

where

z

=

x

+

iy,

with

the

origin at

the

center

of

the

crack-

Boundary

stresses,

from

Equation

6

Srei o,

=

s,

=

ol

Txt

=

0, aS

lZl

*

co.

On the

CraCk

surface:

o,

=

r,,

=

0.

If

the

right-hand crack

lip

is

moved

to

the

origin,

z

become

s z

*

a-

o(z

*

o)

(z

* 2a)-n

z*

K,

becomes:

j(12)

Kr

=

o(rra)'A

(13)

which

is

the

constant

of

proportionality

for

Equa-

tion

10.

Case

2:

For

an

infinite sheet having

a

centrally

located

crack

with

a

pair

of

equal

and

opposite

wedge

forces,

P

(force

per

unit

thickness),

at

the

center

of the

crack

prylng

it open,

the

stress

func-

tion

is:

Pa

L-

Toble

I

-stress-lntensity

Foctors

for

Tensile Looding

B:slc Equation

Ky2

-

Qo%a

Modlbdng

Faitor

Q.

Crack

tlrough

infinite

Platej

Iuternal

circular

crack:

Internal

elliptical

cradc

Long

surface

crack (shallow):

Eliptical

surface

craclc

frf

g-J

o

I

l[(62-a\7c21

a

a

o

a

o

I

4lrrz

L/Q2

t.2

1.2162

sinzd dd

L_

From

rz ffi

Whea

this

is

compared

with

Equations

7

and

the

orign

is

relocatd at the crack

becomes:

where

d

is

the

angular

polar

coordinate

Cir-cular

cracls

c

=

crack

radius

Eliptical

cracls:

a

=

Ienglh

of semiminor

axis

c

=

lengtb

of semimajor

axis

Questions

l.

What

parameter distinguishes

tough

materials

from brittle

materials?

2.

On

a

conventional

stress-strain

plot

showing

both

elastic

and

plastic

aclion, which

of

the

following

is

true?

a-

BrittJe

materials

can.

exist

at

any

yield

point

level;

b.

Brittle materials can exist at

any ultimate

. fracture

lqvel-

'

3. Distinguish between

the

stress{oncentration

factor,

Kr,

and

the

stress-intensity

factors, K,

Ro

K1,

K1", etc.

'

RrrrrExcEs

L Y.

\FcEr rnd

S

-kres-"Qrttlcrl

Apprrts.l

of

frcturr

Xcclralc+"

rrprbt troEB srrapostuEo

oD

ltrctur.

Toutlacsr

TsdLDa r-Dd Itl Appllcrttoor,

ASn{,

STP No. 3EL 19Ea

2-

J. L

qhrn;oD,

r.

rud

\tr.

F.

Bmr.l.

Jn-"Protltg

tB ftrctutt

X.cb.lDlcr,"

llacEtxt

DErcx,

ldlr. 5,

19?0.

3.

E- t. Wcucl,

j

rt

-"f,seJrrccrlaf

Mctbodr tor tbc D.:taE

a d.

Sehcuor ol {rtcrlrlr

Atallst

l:leturc."

W*tlaglottsr

Res-

Irb..,

AD

tO10O5, Ju,nr

2{,

196{L

{.

J.

L SbrDloq,

Jr.-"tr:rstun

t{ccbrllcs," }[IcErNr

DtE

to}r,

EcpL

2a. 196?.

5. P.

C. Prdr

r, d

C.

C. Sif"Strcss

Alrtfie

ol Craclr." ttlB

Totglts4tt.

?cttl*g

aad,

Itt Lr?ltcotlo t,

A.gT a,

gIP.

38L

Answers

'u5.(ttl8uOD

x

(actog

:Jo

suolsuaup'-ar.fl

r{1it'r

'uopern8lJuoc

ued

e

Jo

cllsFaloererp

s

leq]

q15m1 E

Jo

puE

ssarts

p

Jo

slJnpord are

faqa

'dp

4oere

B

lE

ssarls

aql aqFosap

pue

'stoleey

d11sua1u1-ssa4s

arp

sX

aql

?oor

rlllou

aql

le

ssa4s

aql

JaAo

ssaJls

uoll3as ssor8

'lEulruou

aql

'sasruqs

oAU

Jo

oltu

ssaluolsuaulp

aql

sl

,X

'g

('araq pa:ou8J

s1

fltult

leuoluodo:d

pue

luld

plal/(

uaaryrlaq

uoll

-Jul?slp

,(uV)

';(11ec1rse1d

ur:ogap

?ou

ll /v\

ielral

-etu

anllrq

e

'uorllur;ap

.,(g

'arul

sl

q

luauralels

'Z

..'urf,

Jad 8Ja

ro

r'ul

.Iad

q1-11

u1 ,t11ecrd.(1 'earE

tlun

rad

IloA\

'I

)

fl

P

(r4)

and

8,

tip, Kr

(1s)

stress field

dimin-

and running cracks

K1

-

(ra)'A

In

this situation,

the crack:tip

ishes

as

the

crack

gets

longer,

are

sometimes

self-arresting.

Stress-intensity

factors

can

also

be

computed

from


factors

as

the

notch

radius,

p,

appnlaches

zro.

The

maxirnum

stress

occurs

on

the

notch

and is

proportional

to

the

stress

field

around

iL

Thus:

K1

=

cc.".rpll

(16)

where

C

is a

constanl

For an

ellipticat

hole,

taking

the

limit

as

p

-+

0

gives

C

-

ox/2.

Thus:

.

tta

K1

=

lim-

---

'

glarr

p+0

2

This

may

be used

to

I

crack

pta

(

l7)

lvlodeetermine

K,

for

any

Next lesson

further

of

fracture

mechanics.

I

I

explains the

fundamentals

Basic Fracture Mechanic

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Transcript of Basic Fracture Mechanic