mmj1133-e fracture mechanicstaminmn/mmj1133_lecture slides-pdf/mmj1133-e...•fracture mechanics...

MMJ1133 – FATIGUE AND FRACTURE MECHANICS

E – ENGINEERING FRACTURE MECHANICS

FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM

E – ENGINEERING FRACTURE MECHANICS


WWII: Liberty ships


Reprinted w/ permission from R.W. Hertzberg, "Deformation and Fracture

Mechanics of Engineering Materials", (4th ed.) Fig. 7.1(b), p. 262, John Wiley

and Sons, Inc., 1996. (Orig. source: Earl R. Parker, "Behavior of Engineering

Structures", Nat. Acad. Sci., Nat. Res. Council, John Wiley and Sons, Inc., NY,

1957.)


Course Content:

A - INTRODUCTION

Mechanical failure modes; Review of load and stress analysis –

equilibrium equations, complex stresses, stress transformation,

Mohr’s circle, stress-strain relations, stress concentration; Fatigue

design methods; Design strategies; Design criteria.

B – MATERIALS ASPECTS OF FATIGUE AND FRACTURE

FRACTURE MECHANICS M.N.Tamin, CSMLab, UTM 3

Static fracture process; Fatigue fracture surfaces; Macroscopic features; Fracture mechanisms; Microscopic features.

C – FATIGUE: STRESS-LIFE APPROACH

Fatigue loading; Fatigue testing; S-N curve; Fatigue limit; Mean

stress effects; Factors affecting S-N behavior – microstructure, size

effect, surface finish, frequency.


D – FATIGUE: STRAIN-LIFE APPROACH

Stress-strain diagram; Strain-controlled test methods; Cyclic

stress-strain behavior; Strain-based approach to life estimation;

Strain-life fatigue properties; Mean stress effects; Effects of surface

finish.

E – LINEAR ELASTIC FRACTURE MECHANICS

Fundamentals of LEFM – loading modes, stress intensity factor, K;

Geometry correction factors; Superposition for Mode I; Crack-tip


Geometry correction factors; Superposition for Mode I; Crack-tip

plasticity; Fracture toughness, KIC ; Plane stress versus plane strain

fracture; Extension to elastic-plastic fracture.

F – FATIGUE CRACK PROPAGATION

Fatigue crack growth; Paris Law; da/dN-∆K; Crack growth test method; Threshold ∆Kth ; Mean stress effects; Crack growth life

integration.

.


A branch of mechanics that studies the relationships

between external loads applied to a deformable

body and the intensity of internal forces acting

within the body.

Mechanics of Materials


within the body.

The mechanics that describes the response of

materials to loading in the presence of crack or

crack-like defects.

Fracture Mechanics


• Introduction

Historical Review

Fracture mechanics approaches

• Linear Elastic Fracture Mechanics

Elastic stress field approach

Crack tip plasticity

Energy balance approach

LEFM testingA Short Course in

Fracture Mechanics


LEFM testing

• Elastic-Plastic Fracture Mechanics

J-integral

COD approach

• Fracture Mechanics Concept for Crack Growth

Fatigue crack growth

Dynamic crack growth and arrest

• Time-Dependent Fracture

• Fracture Mechanisms in Metals and Nonmetals

Fracture Mechanics

(Typical course content)


References:

• Anderson, T.L., Fracture Mechanics – Fundamentals and Applications, 3rd

edition, CRC Press, FL, USA, 2005.

• Broek, D., Elementary Engineering Fracture Mechanics, Kluwer Academic

Publishers, 1991.

•Atkins, A.G. and Mai, Y.W., Elastic and Plastic Fracture – Metals, Polymers,

ceramics, Composites, Biological Materials, Ellis Horwood Ltd., UK, 1985.


ceramics, Composites, Biological Materials, Ellis Horwood Ltd., UK, 1985.


Linear Elastic Fracture Mechanics (LEFM)

� Fracture mechanics within the confines of the theory of

linear elasticity.

� Analytical procedure that relates the stress magnitude and

distribution in the neighborhood of a crack to:


� the nominal applied stress

� crack geometry (size, shape) and orientation

� material properties

� An underlying principle is that unstable fracture occurs

when the stress-intensity factor at the crack tip reaches a

critical value.


Scope of fracture mechanics



Basic loading of cracked bodies



Stress field ahead of crack tip

−

=2

3sin

2sin1

2cos

2

θθθσσ

r

ax

+

=2

3sin

2sin1

2cos

2

θθθσσ

r

ay

θθθ

rθ

Westergaard solution


=2

3cos

2sin

2cos

2

θθθστ

r

axy

KI is called stress

intensity factor (SIF)

( ) ( )termsorderhigherfr

Kij

Iij += θ

πσ

2


Stress intensity factor

The stress intensity factor,

KI describes the crack tip

stresses.τxy

σyy

σ

σ

( )θπ

σ ijI

ijr

fr

K

2lim

0

=→


2a

Crack

σxy

θr

σ

β- dimensionless parameter

KI has dimension of MPa√m

aK I βσ=

aK I πσ=

For infinite cracked plate


Stress field at notch tip

Compact tension C(T) specimen

σyy


von Mises


Crack-tip stress

r

K

r

K

Iy

Iy

πσ

θθθπ

σ

2

2

3sin

2sin1

2cos

2

=

+

=



Crack-tip plasticity

*2 p

IYS

r

K

πσ =

2

2

2

2

*

22

Ip

aKr

σσ

σπ==


2222 YSYS

pr σσπ==


Shape of plastic zone

( )

++

= θθ

σπθ 2

2

sin2

3cos1

4

1

YS

Iy

Kr


( ) ( ) ( )

++−

= θθν

σπθ 22

2

sin2

3cos121

4

1

YS

Iy

Kr

Plane strain


aYK I πσ=

Meaningful parameters are σ and a

Finite width correction

aK I πσ= Infinite cracked plate

σ


I

w

aY

πsec=

σ


Finite width correction for SIF

For 2a<<W,

aK I πσ=



SIF



Condition for fracture

Fracture occurs when the applied stress intensity factor, KI

reaches the value of the fracture toughness, KIC of the material

ICI KK ≥


σ

aa

aK I πσ1.1=

ICc Ka =πσ1.1

ICc Ka =πσ1.1

At fracture:


Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

0.5

) Mg alloys

Al alloys

Ti alloys

Steels

20

30

C-C(|| fibers)1

40

506070

100

Al/Al oxide(sf)2

Fracture toughness

represents the resistance of

materials to resist cracking.

Fracture Toughness


5

KIc

(MP

a ·

m0

.5

1Si crystal

Glass-soda

Concrete

Si carbide

PC

Glass6

0.5

0.7

2

4

3

10

<100>

<111>

Diamond

PVC

PP

Polyester

PS

PET

0.6

67

Al oxideSi nitride

C/C( fibers)1

Al/Al oxide(sf)2

Al oxid/SiC(w)3

Al oxid/ZrO2(p)4

Si nitr/SiC(w)5

Glass/SiC(w)6

Y2O3/ZrO2(p)4

Based on data in Table B5,

Callister 6e.

materials to resist cracking.

Fracture toughness values

are determined from

fracture toughness tests.


Effect of thickness


Ref. : T.L Anderson, CRC Press, 2005

Plane

stress

Plane

strain


Plane Stress versus Plane Strain



Plane stress versus plane strain fracture


Ref. : Atkins and Mai, 1985


Stress triaxiality at crack-tip


Ref. : T.L Anderson, CRC Press, 2005


Residual strength



K-controlled fracture• KI characterizes crack-tip

condition even though the 1/√ r

singularity does not apply to the

plastic zone.

• LEFM ceased to be valid when

the plastic zone size becomes

large relative to key dimensions



Case I - Maximum flaw size dictates the design

stress.

σdesign <Kc

Y πamax

σ

fracture

Design considerations


amaxno fracture

fracture

Case II - Design stress dictates the tolerable maximum

flaw size.

amax <1

πKc

Yσdesign

2


Energy release rate

Change in energy, dU due to

crack growth from a to a+da is

represented by the shaded

area.

drudU

da

∫= σ


drudU yy∫=0

σ


Energy release rate

drudU y

da

y∫=0

σ

r

ay

2σσ =

( ) ( )rdaaE

ur −−= 2122 σ

ν


( )daE

adU 2

2

1 νπσ

−=

( )22

1 νπσ

−=E

a

da

dU

Energy release rate, G


Energy release rate

( )22

1 νπσ

−==E

a

da

dUG Plane strain

E

aG

πσ 2

= Plane stress


( )22

1 ν−=E

KG

E

KG

2

=

Plane strain

Plane stress


Fracture Toughness Test

ASTM E399 Standard Test Method for Plane Strain Fracture Toughness of

Metallic Materials

Sample geometry




Direction of cut




Fatigue pre-cracking


∆=∆

W

af

WB

PK



Load-gage displacement

=W

af

WB

PK

Q

Q

a

Validity requirements

for KIC


Q

YS

Q

PP

KaB

W

a

10.1

5.2,

55.045.0

max

2

≤

≥

≤≤

σ

mmj1133-e fracture mechanicstaminmn/mmj1133_lecture slides-pdf/mmj1133-e...•fracture mechanics...

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