fm_lect1

8/11/2019 fm_lect1

1/35

ME293: Fracture Mechanics &

Applications

Course Instructor: R.Narasimhan

8/11/2019 fm_lect1

2/35

Course Contents

Introduction & Overview

Design against failure

Motivation for studying Fracture Mechanics

Evolution of Fracture Mechanics

Chap. 1: Energy Concepts in Fracture Mechanics

Atomistic view of fracture

Griffith energy balance & Irwin-Orowan extension

Energy release rate G; Compliance method

Crack growth stability and Resistance (R) curve

Chap. 2: Linear Elastic Fracture Mechanics

Field equations of elasticity

Stress / displacement fields near crack tip Williams eigenfunction expansion. Stress intensity factor (SIF) K and relation to applied load, fracture geometry

Relation between G and K

Fracture characterization by K Small scale yielding conditions

Irwins plastic zone correction; Dugdale model.

Fracture toughness Kc

8/11/2019 fm_lect1

3/35

Course Contents (continued )

Chap. 3: Analytical methods for determining SIF

Westergaard method

Principle of superposition Greens function method

Weight function method

Chap. 4: Nonlinear Fracture Mechanics J Integral

Plastic crack tip (HRR) fields

Ductile fracture criterion

J Integral Testing

J-controlled crack growth and stability

Engineering approach to Plastic Fracture

8/11/2019 fm_lect1

4/35

Course Contents (continued )

Chap.5 : Fatigue Failure

S-N Diagram and its limitations

Fatigue crack propagation Similitude concept; Empirical laws

Crack closure ; Fatigue Threshold

Variable amplitude loading ; Overload cycle

Damage Tolerant Design Methodology

Chap.6: Fracture of interfaces and thin films

Stresses and Failure modes in films.

Interface crack tip fields.

Crack kinking and crack deflection at interfaces

Crack channeling in thin films.

8/11/2019 fm_lect1

5/35

References

1. T.L.Anderson, Fracture Mechanics Fundamentals &Applications, CRC press, 3rd Edn., 2005.

2. M.F.Kanninen and C.H.Popelar, Advanced Fracture Mechanics,Oxford press, 1985.

3. D.Broek, Elementary Engineering Fracture Mechanics, MartinusNijhoff publishers, 1982.

4. Kare Hellan, Introduction to Fracture Mechanics, McGraw Hill,1984.

5. Fracture Journals : Engineering Fracture Mechanics (Elsevier);International Journal of Fracture (Springer); Fatigue and Fracture ofEngineering Materials and Structures (Blackwell).

Course Assessment Sessionals Assessment : 2 or 3 class tests / quizzes (50 marks -

tentative)

Final exam : (50 marks tentative)

8/11/2019 fm_lect1

6/35

Introduction & Overview


Failure of a structural component may occur in different ways

Example : Consider thin cantilever beam

Failure mechanism Failure Criterion

Yielding Yield

Strength

A

x y = =

max = c = Max.allowable displacement

Deflection above

certain limit

8/11/2019 fm_lect1

7/35


Failure Mechanism Failure Criterion

Fatigue

N : # cycles to failure

S : Stress amplitude

e : Endurance limit

Fracture

8/11/2019 fm_lect1

8/35

Fracture based design

Governing structural mechanics parameter in presence of crack (linear

elastic body) is Stress Intensity Factor or SIF K.

For cracked cantilever (small a/h) SIF can be approximatedas :

Crack propagation occurs when

K = KIc (Material property called

Fracture Toughness can be determined

from lab test)

max

max 2

1.12

6

K a

PL

where Bh

=

max : Maximum stress at crack location that would occur in absence of crack.More accurate estimate should include a factor f(a/h) in above equation.

(Units : MPam or N m-3/2)

8/11/2019 fm_lect1

9/35

Strength based vs Fracture based Design

For example of cracked

cantilever beam, maximum load

for given crack length a :

S Factor of safety

Traditional Strength of Materials

approach requires :

2

6 (1.12 )

IcKBhPLS a

Note eq.(1) is similar ro eq.(2)

when dN/dt = const and when

load P is cycled from 0 to max

value (or K: 0 Kmax)

(1)

(2)

8/11/2019 fm_lect1

12/35

Damage Tolerant Design Methodology (DTDM)

Initial flaw size ao is determined using

Non-Destructive Evaluation (NDE)

methods like Ultrasonics or Acoustic

Emission. Take ao = Min crack size that can be

detected by NDE method if no crack

is detected

In DTDM, one sets NDE-based

inspection intervals tI for the structuralcomponent from time t estimated forcrack length to increase from ao to ac by

integrating crack kinetic relations (1) or

(2).

Take tI as fraction of t typically : tI = t/3 so

that there will be minimum of2 or 3 inspections before

crack becomes critical.

Adaptively change tbased on crack size found in

each inspection

Ref. Kanninen & Popelar

8/11/2019 fm_lect1

13/35

Application of Fracture Mechanics An Example

Leak-before-Break in pressure vessel / pipe.

Consider part-through surface crack

of length 2c and depth a on the inner

wall of a pressure vessel or pipe. It is

subjected to tensile circumferential

stress.

Considering the surface to be an

edge crack subjected to tensile stress

(K = 1.12 a) one can estimate

the critical crack depth ac

Next considering a crack through the

wall thickness of length 2c subjected to

tensile stress (K = c ) one can

estimate critical crack length 2cc.Thickness dependent Kc has to be

used in calculating cc.

Flaws A and B are benign while flaw

C is dangerous. Flaw B leads to leak-

before-break and can help indetecting the crack.

8/11/2019 fm_lect1

14/35


Man has always had an intuitiveknowledge of FractureMechanics.

Examples :

While cutting a tree we make aV-notch on the trunk with an axe(introduce a stress concn.) &

then pull it down with a rope.

Brick & mortar are weak undertensile loads. Hence bridges,windows, roof-spans, etc. built in

ancient times have arch shape causes compressive stressesto be transmitted.

Ref. Anderson, Fracture Mech.

Roman Bridge Design

8/11/2019 fm_lect1

15/35


Leonardo Da Vinci measured strength of wires & found : strength 1/(wirelength) - because longer wire larger sample volume higher probability ofwire having a flaw.

Due to industrial revolution many kinds of m/cs and structures were made withmetals several failures due to fatigue & fracture in bridges, boilers, ships,

locomotive wheels & axles, rails, etc., in 19th century.

Accelerated production of aircrafts & ships during World War II : Led to someserious failures.

Liberty ships : Cargo ships with all-welded hulls instead of riveted

construction used in earlier designs ; Sustained fractures (some broke

completely into two !) in cold waters of N.Atlantic.

Crack-like flaws present in welds had low KIc (especially at low temperatures)leading to catastrophic fracture .

Propagating cracks did not encounter any barriers in all-welded ships & were able

to traverse entire hull.

In riveted ships, cracks could not propagate across panels joined by rivets.

8/11/2019 fm_lect1

16/35


Comet Jet aircraft : Many

accidents in early 1950s in

Comet Jet aircrafts due to rapid

propagation of cracks initiated

due fatigue loading near anopening in fuselage.

Jetliner flying at high altitude is

like pressurized thin-walled

vessel (inside pressurized;

outside low atm pressure) with

fuselage under tensile stress.

Failure in Nuclear components :Boilers, piping.

Thermal shock in nuclear

reactor vessels.

Degraded nuclear plant piping

8/11/2019 fm_lect1

17/35


Development of new materials for high-tech applications has led to

fracture-related problems

Ceramics : In 80s and 90s ceramics were developed for structural

applications like in armours, high pressure jet engines, radomes etc.

High strength, good creep resistance, low thermal conductivity, etc.

Low KIc ~ 0.1 to 5 MPam compared to 50 to 100 MPam for metals. Considerable research for improving KIc

Fibre / particulate reinforcements. Sandwich (ceramic-metal) panels.

Transformation toughening (phase transformation).

Bulk Metallic glasses : Multi-component systems which solidify in

amorphous state.

Attractive mechanical properties (high strength / stiffness)

Can have low KIc - considerable research is ongoing to improve toughness.

8/11/2019 fm_lect1

18/35


Thin films / coatings - Thin ceramic,

metal, polymer coatings (thickness

varying from few nm to microns).

Applications like cutting tools, dental

implants, MEMS, optical systems, etc.

Prone to variety of failures due to

intrinsic / extrinsic stresses.

Intense research on failure of thin filmsand interfaces in 90s and 2000s.

Summary : Study of Fracture

Mechanics is important to preventfailures in critical engineering

components and to develop better

engineering materials.

Surface crack

Channeling

Substrate cracking

(crack penetration)

Crack deflection& debonding

Spalling

Failure mechanism

Ref. Hutchinson / Suo: Adv

in Appl Mech. V.29, 1992

8/11/2019 fm_lect1

19/35

Evolution of Fracture Mechanics

In this course we will focus on Applied Mechanics

8/11/2019 fm_lect1

20/35

Griffith Energy Approach

Theoretical cohesive strength of materials (peak stress required to

break atomic bonds) is very high: c ~ E/6 to E/3 much higherthan observed fracture strengths.

Griffith performed tensile tests with glass fibres of different thickness

& observed failure stress f with thickness :

Note:

f fbulk-glass for large thickness

c for very thin fibres( Glass / graphite are usefulas reinforcements in

polymer-matrix composites).

Griffith postulated above behavior was caused by presence of cracks whose

size with fibre thickness.

8/11/2019 fm_lect1

21/35


Griffith postulated :Rate of decrease of = Energy required

PE wrt crack extension to create new surfaces

Griffith considered an infinite body containing a crack of length 2asubjected to remote tensile stress

Using elasticity solution by Inglis for a

narrow elliptical void in an infinite plate

subjected to remote stress , Griffithdetermined the potential energy (PE) of

cracked body as:

2 2

oa B

E =

2 sA

=

(plane stress)

8/11/2019 fm_lect1

22/35


Problems with Griffith theory :

Applies to crack of length 2a in infinite body.

Plastic yielding will occur near crack tip (due to large stresses).

Inglis solution may not apply.

Additional energy expended in plastic dissipation as crack grows.

Theory cannot be directly used in engineering applications because G is

not convenient parameter to compute for practical configurations.

G - Energy Release Rate

8/11/2019 fm_lect1

23/35

Linear Elastic Fracture Mechanics (LEFM)

George Irwin extended Griffith hypothesis & made it a sound

engineering discipline.

He replaced (2s) by (2s + p) where : p is additional plastic workdissipated during creation of unit crack area A.

For structural metals Irwin & Orowan estimated p >> 2s(Griffith Irwin Orowan Theory of Fracture)

Using analysis of crack tip fields by Williams (based on an eigenfunction

approach) and Westergaard (based on complex variable method), Irwin

identified a new fracture characterizing parameter called Stress Intensity

Factor or SIF K.

8/11/2019 fm_lect1

24/35

Modes of Fracture

Y

Y

Y

Z

ZZ

Mode I : Tensile or

Opening Mode

Crack faces opensymmetrically wrt X-Y &

X-Z planes

Mode II : In-plane

Sliding Mode

Crack faces slide

symmetrically wrt X-Yplane &

anti-symmetrically wrt

X-Z plane

Mode III : Out-of-plane

Sliding Mode

Crack faces slide anti-

symmetrically wrt both X-Yand X-Z planes.

Arbitrarily oriented & loaded crack : Superpose response in the three modes.

8/11/2019 fm_lect1

25/35

Linear Elastic Crack Tip Fields

Linear elastic analysis shows that the stress / displacement fields near the tip

for all three modes is of following form.

K : KI / KII / KIII : SIF

for modes I, II , III . f(), g(,) : Universalfunctions : depend only

on modes I, II, III &

plane strain / plane

stress.

Note :

Stresses / strains are singular as 1/r as crack tip is approached ( r 0 ).

SIF KI / KII / KIII uniquely characterizes near-tip field (at given (r, ) from tip)irrespective of fracture geometry Concept of similitude.

SIF depends on applied load, crack length and other geometricalparameters.

8/11/2019 fm_lect1

26/35

Importance of SIF K

If similitude applies, crack propagation can be postulated to occur

when :

K (,a) = Kc

KI = (a) for Griffith problem (crack of length 2a in infinite platesubjected to remote tensile stress ).

Irwin showed SIF K is related to energy release rate by :

G = KI

2 / E for Mode I

8/11/2019 fm_lect1

27/35

Nonlinear Fracture Mechanics

LEFM predicts infinite stresses at crack tip Yielding will occur near cracktip. Rice proposed concept of Small Scale Yielding (or SSY) to extend

application of LEFM theory (SIF-based) to ductile materials which show

yielding at crack tip.

Need alternate parameters to characterize fracture initiation.

J Integral (extension of Griffith energy release to nonlinear elastic solids).

Crack tip opening displacement (CTOD).

8/11/2019 fm_lect1

28/35

Nonlinear Fracture Mechanics

J-integral plays dual role during monotonic loading of a stationary

crack in a plastic solid :

Energy release rate.

Characterizing (amplitude) parameter of crack tip fields.

Crack tip stress / strain / displacement fields are described by the

Hutchinson / Rice / Rosengren (HRR) solution for plastic solids

obeying a power-law relation ( ~ n) Stresses behave as ~ r -1/(n+1) ; Strains behave as ~ r -n/(n+1) as r 0

Experimental methods needed to determine ductile fracture

toughness Jc

- Multiple Specimen Testing; Single Specimen Testing; ASTM standardtest procedure

Handbook style approach (EPRI method) to estimate J for standard

lab specimens & some structural configurations

8/11/2019 fm_lect1

29/35

Dynamic Fracture Mechanics

Inertial effects important in fractureanalysis.

Stationary cracks subjected to dynamicloading

SIF K(t) higher than static case due tostress wave reflection & interaction effects.

Fracture criterion :

Dependence of Kdc

on influenced bylocal failure mechanism :

CCP subjected to tensile pulse

Normalized SIF vs normalized time

)K(K)t(K dcI =

K

8/11/2019 fm_lect1

30/35

Dynamic Fracture Mechanics

Rapidly propagating cracks

Stress distribution near crack tip depends on

crack speed v

Energy release rate G and K are related by

similar expressions as stationary cracks :

Generally KI0 as vcR & so also does G.

Crack propagation criterion :

( ) ( , ) 02I

ij ijK t v as r r

=

2I1

2

K)v(fE

1G

=

( ) ( )I pc

K t K v=

Crack Arrest Toughness

Kpc vs crack speed for AISI

4340 steel

8/11/2019 fm_lect1

31/35

Brittle Fracture Mechanism in metals

Transgranular cleavage

Crack grows by cleavage of weak

crystallographic planes with littleplastic deformation (generally in

BCC & HCP metals due to limited

number of slip systems for plastic

flow).

Especially important in steels at low

temperature.

Cleavage cracks are initiated due tofracture of small grain boundary

carbides or inclusions within grains.


8/11/2019 fm_lect1

32/35

Cleavage Fracture Mechanism in steels

Cleavage of individual grains in apolycrystalline alloy gives a facetedappearance.

Change in crack plane at grain boundaries leads to river pattern-like features withineach facet.

By following river-patterns one can locate

the cleavage-triggering particle.

Cleavage fracture is statistical in natureand depends on probability of sampling avolume containing a cleavage-triggeringparticle leads to scatter in fracture

toughness data.

Mechanism is stress-controlled andcharacterized by low fracture energy.


Origin of river-patterns

8/11/2019 fm_lect1

33/35

Ductile or Fibrous Fracture Mechanism

Has 3 distinct stages :

Void nucleation (due to fracture ordebonding at inclusions / impurities)

Void growth (influenced by plastic

deformation of matrix and by highhydrostatic stress).

Coalescence of growing voids

Direct impingement (due to necking

of ligaments connecting grownvoids).

Due to void sheeting coalescenceof small voids in ligaments bridgingtwo large voids.

Crack growth is generally stable andshows high fracture energy.

Steels show brittle-ductile transition astemperature increases above NDT.


8/11/2019 fm_lect1

34/35

Inter-granular crack growth

Under following conditions cracks can

form & propagate along grain

boundaries :

Precipitation of a brittle phase (due to

improper tempering) at grain

boundaries.

Hydrogen & liquid metal embrittlement.

Environmental assisted cracking &

intergranular corrosion.

Grain boundary cavitation & cracking

at high temperatures


8/11/2019 fm_lect1

35/35

Current Fracture Mechanics Research An Example

Bulk Metallic Glasses (BMGs) are new materials being developed exhibitinhomogeneous plastic deformation by shear banding; Fracture occurs due to

cracking inside shear bands in ductile BMGs & by cavitation in brittle BMGs

Finite element modeling of shear

banding near notch tip Tandaiya

et.al., J.Mech.Phys.Solids, 2009

MD simulation of crack growth by cavitation

in a brittle BMG Falk , PRB, 1999

fm_lect1

Documents