Chapter 10: Failure - FMAM Lab | Functional Materials & Applied Mechanics …fmam.cau.ac.kr/.../2017/11/2017-1-MSE-Lecture-Chapter-10.pdf · 2017-11-13 · Griffith (1920) used the

11 SCHOOL OF MECHANICAL ENGINEERINGLECTURER: PROF. SEUNGTAE CHOI

Chapter 10: Failure

ISSUES TO ADDRESS

Failure Modes:


Titanic on April 15, 1912

RMS Titanic was a British passenger liner that sank in the North Atlantic Ocean on 15 April 1912 after colliding with an iceberg during her maiden voyage from Southampton, UK to New York City, US. The sinking of Titanic caused the deaths of 1,502 people in one of the deadliest peacetime maritime disasters in history. (http://en.wikipedia.org/wiki/RMS_Titanic)


10.3 Ductile Fracture (Rupture)

Classification of fracture behavior

Callister& Rethwisch 9e.

AR EL


Moderately Ductile Fracture (Rupture)

Stages in the cup and cone fracture


Analysis ofMetallurgical Failures


10.4 Brittle Fracture

Brittle Fracture

I. Transgranular Fracture(Cleavage Fracture)

Callister & Rethwisch 9e.


II. Intergranular Fracture

II. Intergranular Fracture:



10.5 Principle of Fracture Mechanics

An Atomic View of Fracture

0

bx

E PdxBonding energy:

Interatomic force-displacement relation:

cxP P sin

For small displacements, force-displacement relationship is linear:

c c 0c

c

P xxP P kx, where k= E=

E

Surface energy:

s c c0

sc

0

1 xsin dx2

Ex


Stress Concentration Effect of Flows

Stress concentration around an elliptic hole by C. E. Inglis (1913)

The above equation show that as b 0 (the ellipse becomes a crack) a stresssingularity ( ~ 1/ r) develops at the crack tip.

0 0

0

0

x1

x2

2 2

2 2x y 1a b

2ba

MAXyy 0

0 0

0

a(A) 2 1b

a a2 1 2 1a

a2

Aba


Stress Analysis of Cracks

Three Modes of Fracture

Stress fields near a crack tip mI II III (m)I II III 2

ij ij ij ij m ijm 0

ij

I

II

IIIIij

K K Kf ( ) f ( ) f ( ) A r g ( )2 r 2 r 2 r

: stress tensorK : Mode I stress intensity factorK : Mode II stress intensity factorK : Mode III stress intensity factorf , fII III

ij ij, and f : dimensionless functions of


Stress Fields near a Crack Tip

11II

22

12

sin 2 2 cos 2 cos 3 2K sin 2 cos 2 cos 3 22 r

cos 2 1 sin 2 sin 3 2

I

II

III

K : Mode I stress intensity factorK : Mode II stress intensity factorK : Mode III stress intensity factor

11I

22

12

1 sin 2 sin 3 2K cos 2 1 sin 2 sin 3 22 r

sin 2 cos 3 2

Stress fields near a crack tip

31 III

32

sin 2Kcos 22 r


Design Criteria

Stress approach Fracture mechanicsapproach

: applied stress : ultimate stregthu

uI ICK K

: stress intensity factor (SIF) calculated value due to loading

: fracture toughness material property

I

IC

K

K

r

Iyy

K2 r


Examples of Stress Intensity Factors

A centered crack in an infinite plate:under uniform uniaxial stress

IK a

A penny shaped crack in an infinitedomain

2IaK


Elastic Energy

Strain energy density:

Internal energy of a deformable body:

Linear elastic materials (Hooke’s law):

or ( and : Lamé constant)

Strain energy density of linear elastic materials:

Internal energy of linear elastic materials:

ij ij0u d

ij ij0V V

U udV d dV

ij ij ij kk1

E E ij ij ij kk2

ij ij ij ij0

1u d2

ij ij ij ij0V V

1U d dV dV2


Conservation of Energy


Energy Balance During Crack Growth

The first law of thermodynamics (Law of conservation of energy)

The reduction of potential energy is equal to the energy dissipated in plasticwork and surface creation.For brittle materials, UP 0.

2a aa

Applied traction

E P

E

P

W U UW : Work done by the applied loadU : Elastic energyU : Plastic energy

: Surface energy

A At A t A

PE

U where =U WA A A

S2A A


Griffith Energy Balance

Griffith (1920) used the stress analysis of Inglis (1913) to show

Griffith fracture stress

Modified Griffith fracture stress

2a aa

2 2 2

0

S S

a B aE A E

4aB 2A

B1 2

Sf

2Ea

1 2P S

f2E( )

a P : plastic work per unit area of surface


Energy Release Rate

Energy release rate, G: A measure of theenergy available for an increment of crackextension (Irwin, 1948)

It is also called the crack extension force orthe crack driving force.

From the Griffith energy balance, the crackextension occurs when G reaches a criticalvalue, i.e.,

where Gc is a measure of the fracturetoughness of the material.

A

2

c Sa 2 ,

A E A


10.6 Fracture Toughness Testing

Standard: ASTM E399, D5045Specimen: Compact Tension (CT) Calculation

KIC : Fracture toughnessPQ : Critical load

f(x) = 8.34 at x = 0.459.66 at x = 0.5011.35 at x = 0.55

x = a/W, (0.45 < x < 0.55)

QIC 1/2

PK f x

BW

2

IC

yield

KB, a 2.5


Plastic Effect on Crack Tip

Plastic zone size

Yielding zone size:

yr pr r

• Ductile fracture : sufficient plastic deformation before fracture• Brittle fracture : small plastic deformation before fracture

2

Iy

y

K1r2

2

Ip y

y

K1r 2r for plane stress

2

Ip y

y

K1r 2r for plane strain3


3D Aspects of Plastic Zone


Variation of KC with Specimen Thickness

Planestress

Transitionregion

Planestrain

Specimen thickness

CK

ICK


Fracture Toughness Ranges


ASM Handbook

Fracture Mechanicsof Ceramics

Ceram. Eng. Sci. Proc.

KIC(MPa

m0.5 )


Impact Testing

Impact loading:

final height initial height


The Structure and Properties ofMaterials Mechanical Behavior


Influence of Temperature on Impact Energy

Ductile to Brittle Transition Temperature (DBTT)...


T

y E


Liberty Ship duringWorldWar II


Liberty Ship duringWorldWar II

John P. Gaines

Empire Duke


Patterning by Controlled CrackingK. H. Nam, I. H. Park, & S. H. Ko, Patterning by controlled cracking, Nature, Vol. 485, pp. 221-224, 2012.


Fragmentation of Ice in the Arctic Ocean


10.7 Cyclic Stresses Fatigue

Fatigue = failure under applied cyclic stress.

Stress varies with time.

Key points: Fatigue...




Fatigue limit, Sfat: For somematerials,


10.8 The S N Curve

Sfat



10.9 Crack Initiation and Propagation

Rate of Fatigue Crack Growth: Crack grows incrementally

Failed rotating shaft

a~

Callister & Rethwisch 9e.Understanding How Components Fail

mKdNda )(


1. Impose compressive surface stresses(to suppress surface cracks from growing)

10.10 Factors That Affect Fatigue Life


bad

bad

better

better


m

m

m


10.12 Generalized Creep Behavior

Creep phenomenon: Timedependent and permanentdeformation of materials whensubjected to constant load orstress.

e t)

e

t


Creep Fracture Mechanism

Schematic drawing of three fracture mechanisms in a high temperature creepregime

[Abe et al., Creep-Resistant Steels, 2008]


10.13 Stress and Temperature Effects

Occurs at elevated temperature, T > 0.4 Tm (in K)


elastic

primary

secondary

tertiary


Secondary Creep

Strain rate is constant at a given (T, s)

Strain rate increases with increasing T,

1020

40

100200

10 2 10 1 1Steady state creep rate (%/1000hr)es

Stress(MPa) 427 C

538 C

649 C

Callister & Rethwisch 4e.Metals

Handbook: Properties and Selection:Stainless Steels, Tool Materials, andSpecial Purpose Metals

RTQK cn

s exp2


Arrhenius Equation

The Arrhenius equation (Svante Arrhenius, 1889)

kTAEaREBkB

TkEAk

RTEAk

B

Ba expor exp


10.14 Data Extrapolation Methods

The Larson–Miller parameter is a means of predicting the lifetime of materialvs. time and temperature using a correlative approach based on the Arrheniusrate equation.Larsen Miller parameter PLM is used to represent creep stress rupture data.

rLM

c

c

c

cns

tCTP

lAtBT

RQ

RTQA

tl

RTQA

tl

RTQK

log

lnB whereln

lnln

exp

exp2


Prediction of Creep Rupture Lifetime

T s

LMr PtT )log20(

tr

310x24)log20)(K 1073( rt

trCallister & Rethwisch 9e.

Trans. ASME 74

T tr


SUMMARY

Engineering materials not as strong as predicted by theoryFlaws act as stress concentrators that cause failure at stresses lower thantheoretical values.Sharp corners produce large stress concentrations and premature failure.Failure type depends on T and :

Documents

Chapter 10: Failure - FMAM Lab | Functional Materials & Applied Mechanics …fmam.cau.ac.kr/.../2017/11/2017-1-MSE-Lecture-Chapter-10.pdf · 2017-11-13 · Griffith (1920) used the