Mechanical Properties of Ceramics - NONMET...1 Materials Science &Technology Materials Science II -...

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Materials Science & Technology

Materials Science II - 2010, Ceramic Materials, Chapter 6, Part 1

Mechanical Properties of Ceramics

Jakob Kübler

orMechanical Behavior of Brittle Materials

& Prof. L.J. Gauckler

1Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2010

Jakob Kübler Empa, Materials Science & Technology

Lab for High Performance Ceramics Überlandstrasse 129, CH-8600 Dübendorf

+41-44-823 [email protected]

& Prof. L.J. Gauckler ETH Zürich, Materials Department

References• Munz and Fett, Ceramics, Springer, 1999

• Barsoum, Fundamentals of Ceramics, IoP, 2003

• Askeland & Phulé: Science and Engineering of Materials, 2003

Lit t di• Literature, diverse

• ISO, CEN, ASTM standards


2

Aim of chapter & Learning targets 1. Introduction2. Stresses at a crack tip3. Griffith law4. KI and KIc

5 R

part

1C

rack

tip

th

lear

ning

ta

rget

s 1“Why mechanical testing …”

“Higher than you’d assume …”“Conditions for failure …”

“Stress intensity & critical stress intensity …”

g 2

“I i t h ”5. R-curve6. Properties7. Strength

8. Statistic9. Proof testing10. Fractography

part

2St

reng

tpa

rt 3

Stat

istic

s

lear

ning

targ

ets “Improving toughness …”

“Knowing what you measure …”“Just a value …”

lear

ning

ta

rget

s 3“Weibull, a name you’ll never should forget …”

“Make it or …”“Reading fracture surfaces …”


11. Thermal shock12. Slow crack growth13. SPT diagrams14. Creep15. Failure maps

part

4Ti

me&

Tem

p

lear

ning

ta

rget

s 4

“Temperature, time and geometry …” “After several years …”

“Combining strength, lifetime & statistics …”“Temperature makes it move …”

“Finding your way …”

part 5 - Case Study: Lifetime of All-Ceramic Dental Bridges

Introduction (1)

Deformation of …

Ceramics:

500

400

500

400

Al2O3 failure

C steel

Stress vs. strain diagram

Ceramics:Critical elasticity: ~ 0.01%Plasticity: ~0%

Metals:Critical elasticity: ~ 1 - 2 %Plasticity: up to 50 - 100 %

300

200

nom

inal

stre

ss F

/A (

MPa

)

300

200

elastic deformation

plastic deformation


PMMA*:Critical elasticity: ~ several %Plasticity: up to several 100 %

100

0

ΔL/L0 x100, %

0 5 10 15 20 25

100

0

nominal elongation ΔL/L0 x100, %

0 5 10 15 20 25

PMMA @ 122°C

* Polymethylmethacrylate (acrylic glass) Trade Name „Plexiglas“

3

Introduction (2)

Ceramic structure for optics Mounting of IR camera

5Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2010© Carl Zeiss Optronics GmbH & FCT Ingenieurkeramik

Why Si3N4 ?

Introduction (3)

What’s that?


4

LifetimePrediction

AdvancedNDE

ComponentTesting

Fabrication

......

MaterialDevelopment

Introduction (4)

Why do we need mechanical properties?

ine:

Sou

rce

Wik

iped

ia

Prediction

MaterialsTesting and

Charact.Structural

Design

StatisticalModeling

σ < 100 MPaT > 1100°C

Corrosion

Creep & FatigueTu

rbof

an E

ngi


σ < 250 MPaT > 900°C

σ < 350 MPaT > 800°C

Brittle fracture

Slow crack growth

Mechanical properties needed for the design of components.

Material properties→ Design: E, KIc , (σm), ...

Technological properties→ Comparison: Hardness, ...

Proof testingG d / N t d

Introduction (5)

Aim of mechanical testing

• temperature

→ Good / Not good

• micro structure• surface

• strength• elasticity• fracture

toughness• creep• scg


• environment

• time• static• cyclic

• hardness• thermal

shock resistance

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Introduction (6)

Vickers hardness

HV = 0.102 (2·F·sin[136/2]) d-2

= 1.891·F·d-2 [N mm-2]:

Technological testsF

[ ]

Exam

ple

650 HV 100 / 30

Duration 30 sLoad level 100 kp

Vickers procedureHardness value


• No materials constant• Comparison of materials under exactly defined conditions • Relative values or statements (e.g. „good“ or „not good“)

other technological tests:Charpy V-notch, Jominy test, wear tests, crash test, ….

Stresses at crack tip (1)

Determination of design relevantmechanical properties

Creep

Relation between creep rate and

Crack growth /Lifetime

static dynamic

Fracturetoughness

Relation between defect size and

strength

Strength

Relation between strength and


creep rate and load.

Relation between crack growth rate

and load.

strength. probability of failure.

6


… famous failures …

Aloha Airline Flight #243after becoming a convertible on April 28, 1988Source: ASTM Standardization News, Oct.1998

http://upload.wikimedia.org

Liberty Ship "Valeri Chkolov"after breaking in two pieces on Dec. 1943Source: ASTM Standardization News, Oct.1998

http://ngm.nationalgeographic.com


http://en.wikipedia.org/wiki/Liberty_shipTitanic during sea trials, sunk on April 14, 1912Source: http://www.dodger.com/titanic/titanic-history.htm

FStress in a component


FF

F

W/2 W/2

2r σ

F

W/2 W/2

2r

F

W


σ = F / (W·h) σ = F / (W·h) σ ≠ F / (W·h)h = plate thickness

7


Stress concentration around a hole

X X

Strain distribution around h l i h l


large stress gradient along x-x

a hole in a open hole tensile coupon by FEA

Source: NPL, Teddington, UK

■ tension ■ compression

The mechanical stress around a hole is significantly higher than the stress simply calculated from the macroscopically available cross section and applied load.


“.. the smaller the hole the higher Kt ..”

Stress concentration factor for holes and notches

w2r co

ncen

trat

ion

fact

or K

t

g t

“.. the sharper the notch the higher Kt ”

stre

ss f

2r/w

w h

b

ss c

once

ntra

tion

fact

or K

t b/r=4

b/r=1


Predictions are in good agreement with numerous experimental observations.

notch the higher Kt ..

2r

b

stre

s

r/h

b/r=0.5

possible source: Pilkey, Walter D., Peterson's Stress Concentration Factors (2nd Edition), John Wiley & Sons, 1997

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Radius of curvature at tip of a defect FF “round hole := special

case of defect

F

W/2 W/2

2a

F

W/2 W/2

2a

ρt ρt


σm (1) σm (2)

σm (1) < σm (2)

… sharpness of defect (or crack)


σm

ss

Concept of stress superelevationat tip of a defect σ0 nominal stress

σm stress at crack tipa ½ crack widthρt radius of

curvature at crack tip

σ0

stre

s

The mechanical stress σm at a tip of a defect is by factors larger than

the stress simply calculated from the

macroscopically available cross section

and applied load.x1-x2 → σ = 0 x1 x2


⎟⎟⎠

⎞⎜⎜⎝

⎛+=

tm

aρ

σσ 210

Calculating σm is rather complicated and is a function of the type of - crack geometry - loading (uniaxial, biaxial, …)- sample size

Here only the final result:

Question: Factor X for a very small hole?

9

tm

aρ

σσ ⋅⋅= 20


For a >> ρt this reduces to:

Stress concentration factor

= sharp crack

Measure of how much an

factor

30

40

50

60

70

σm

ax /

σ0

3 (hole)


f mexternal stress is amplified (superelevated) at the crack tip.

From this it follows that large, sharp cracks are more deleterious than small, blunt cracks.

0

10

20

1 10 100 1000

log (a/pt)

σ

log (a/ρt)

Griffith law (1)

1. All materials have defects.(sometimes very small ones)

2. Small defects weaken brittle materials.( t l t d b G iffith i 1920)

Stress superelevation

(postulated by Griffith in 1920) 3. Ductile materials can diminish the stress superelevation.

(by plastic deformation at the crack tip and are therefore endangered far less than brittle materials)

4. All (not only) brittle materials exhibit a natural defect population.(due to production. Defects differ in size, form and orientation. Proposed by Griffith.)


5. Brittle materials (and all others) fail if nominal strength is overcome by a defect related stress peak.

Remark: This was examined on whiskers made of glass, on which fracture strains near the theoretical one (binding strength of atoms) have been measured.

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Griffith law (2)

Energy Criteria for Fracture(= condition for fast failure of brittle materials)

Portrait of Alan Arnold Griffithand the centre-cracked panel used to derive his fracture concept in 1920.

ESIS Newsletter nr. 40-2004


Griffith’s basic idea was to balance the energy consumed in forming new surfaces as a crack propagates against the elastic energy

released.The critical condition for fracture occurs when the rate at which energy is

released is greater than the rate at which it is consumed.

Definition of crack dimensionsfor rest of today’s lecture:

bird eyes view

Attention: In literature “c” and “a”

are often used vise versa.

Griffith law (3)

2 a

σ2 a → ∞

σ


c

σ

c

σ

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Griffith law (4)Remember:Griffith’s basic idea was to balance the energy consumed in forming new surfaces as a crack propagates against the elastic energy released.

2uniformly stressed

solid

1 3 relaxed volume

Total energy change Utot of system upon introduction of a crack

solid 4 two surfaces created

crackScrackVVotot UUUUU −− +−+= 0


1 2 3 4

1 Free energy in absence of “external” stress2 Energy introduced into volume V0 by applying a stress (avg stress)3 Released energy in shaded volume of size π·c2·½ ·t (= Elastic energy term)4 Energy needed to create new surface area of size 2 · c · t (= Surface energy term)

γσπσ

⋅⋅⋅+⎥⎤

⎢⎡ ⋅⋅

−⋅

+= tctcVUU appapp 2

2220

0

crackScrackVVotot UUUUU −− +−+= 0

Griffith law (5)Total energy change of system upon introduction of a crack

U - residual stress

U - applied stress

U -released by crack

U – needed to create 2 crack surfaces

γ+⋅⎥

⎦⎢⎣⋅

+ tcEE

UUtot 22220

with:Vo volumeσapp applied stress (avg stress)E Y ’ d l

if the applied stress is increased and a crack is created (or growing)


Eappσ

ε =

E Young’s modulusc depth of crackt width of crack2 two surfaces createdγ intrinsic surface

energy of material

with

12

Critical crack length ccrit at which fast fracture will occur corresponds to U*maxccrit respectively equilibrium reached at

dU* / dc = 0 U*

tc ⋅⋅⋅γ2

Griffith law (6)

- differentiation of equation to c- equating it to zero - replacing σapp by σf - rearranging

leads to:

Ec γπσ 2

Elastic energy(release) term

⎥⎦

⎤⎢⎣

⎡ ⋅⋅⋅

−22

22 tcE

app πσ


Plot of equation terms - surface energy (top curve)- elastic energy (bottom curve)as a function of c and sum ofthe two curves ( U* )

Eccritf ⋅⋅=⋅⋅ γπσ 2

K ≥ Kcstress intensity

factor

criticalstress intensity

factor than fast fracture occurs

if

Griffith law (7)

With the help of the Griffith law it is possible to estimate the theoretical strength of a material

critcritf c

Eσσ ⋅

⋅⋅==

πγ2

Griffith - equation

theoretical strength of a materialif for the minimal possible defect size the distance c between 2 atoms is used and

the theoretical strength depends only on the forces between the atoms.

c = 2·dei ~ 424 pm { dei= r0Mg2+ + r0

O2- = 86 pm+126 pm} 1

E ~ 250 GPa 2

γ ~ 1.15 J/m2 (100) Surface, air, RT 3 Barsoum1p82, 2p364, 3p103

Example MgO:(rock salt, CN 6) equilibrium interatomic distance


σf, theor ~ 20.8 GPa 1/12 Ethis confirms with data given in 1st part of Materials Science II, Ceramic Materials

c

c

The theoretically possible tensile strength (~E/10) has never been observed …

13

20

25

800

1000

… in reality rather 1/100 to 1/1000 of E → Why?

[GPa]

1 nm 1 µm

Griffith law (8)

10 nm 100 nm

5

10

15

20

200

400

600

800

E / σf

21

2⎟⎠⎞

⎜⎝⎛

⋅⋅⋅

=c

Ef π

γσ


0-1 0 1 2 3 4

0

defect size [log(c)] … c in nmσf,theoretical of MgO

σf,MgO polycrystall = 90 MPa → E / σf = 2’700 !!NIST - SCD Citation Number: Z00176

Modes of failure

Mode I: opening mode (tensile load in a right angle to the crack)

Mode II: sliding mode (shear in the direction of the crack)


(shear in the direction of the crack)

Mode III: tearing mode (shear at a right angle of the direction of the crack)

Mode I is by far the most pertinent to crack propagation in brittle materials(… so this is the only one we are going to discuss)

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Irwin (1)

Stress field in accordance to Irwinat the tip of a crack with a coordinate system

Pσyy

Wc B

One-sided pre-chracked body= model for Griffith-concept

Stress at a point P in relation to its distance and direction from the crack tip leads to a stress matrix.

Pσxx

σxy

31 i iP P Pθ θ θ⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎨ ⎬⎢ ⎥


312 2 2

312 2 22

32 2 2

cos sin sin

cos sin sin

sin cos cos

P P P

xxI P P P

yy

xy PP P P

Kr

θ θ θ

σθ θ θσ

πσθ θ θ

⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ −⎨ ⎬⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥⎡ ⎤ ⎢ ⎥⎧ ⎫⎢ ⎥ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= ⋅ +⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎢ ⎥ ⎢ ⎥⎣ ⎦

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

rpP

Stress (elevation) in accordance to Irwin at an angel θ and at distance or rp

θ

Irwin (2)rP

1 00“ 0 10“


rp = „1.00“ rp = „0.10“

rp = „0.01“σxx

σyy

σxy

Which of the stresses is the most important one and why ?

point P

-3, 0, 3, 6 x “nominal” stresscrack

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Irwin (3)

The rate at which the energy is released while a crack is formed, can be obtained from the energy difference at the beginning and end of

the crack (= Δx), respectively.

Energy release rate G

dxux

G yx yy

x∫

Δ= Δ

→Δ0

0 22lim

σ 0=Θforyyσπ=Θforyu

uy half of the opening of the crack

for purely brittle solids = 2 γ !!


Irwin showed that this simple relation exists between the

stress intensity factorand energy release rate

EKG I

2

=

Irwin (4)

whereas isvalid, too: dc

dUG M=G energy release rateUM mechanical energyUE released energy

Irvin correlation

EKG I

2

=

⎤⎡ 22and if the load

is constant:

combining theequations: YcK I ⋅⋅= σdc

Etcd

EK I

]22

[22

2⋅⋅

⋅⋅⋅

=

πσ

⎥⎦

⎤⎢⎣

⎡ ⋅⋅⋅

==22

22 tcE

UU EMπσ

andsolving it:


• externally applied stress • size of the crack • Y-factor defined by geometry of crack

Y-factor predicts intensity and distribution of a stress field around a defect in the material of a component caused by an external load.

Therefore, KI depends on

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Crack (flaw) shape correction factorsY depends on whether the crack is intersected by the surface or is in the bulk of the

component. For the surface connected crack, the value at the centre is different from that where the crack intersects the surface.

Irwin (5)

CEN TS 843-6, 2004 → c and a have changed to be in line with definition.


Factors for intermediate shapes of a flaw can be computed using the equations given by Newman, C.R., Raju, I.S., Eng. Fract. Mech. 1981, 15 [1-2], 185-92

This table should be used only as a guide if computing the stress or crack size at failure, respectively. If you don’t have Y for your specific problem chose 1.5 and

as 1.1 ≤ Y ≤ 2.0 in the worst case you’re of by less than 30 %.

YcK cIc ⋅⋅= σAs the fracture toughness is a materials

property …

Strength of ceramics[ ]IcK MPa m=[σc] = MPa

[Y] = -[c] = m

Irwin & Griffith

… is the (failure) strength “only” a characteristic of a sample,

part or component ! YcKIc

c⋅

=σ

… and as no material and component is free of defects due to the purity and homogeneity of raw materials, processing, machining, handling, etc. …

volume: pores, large grains, inclusions, … surface: cracks, pits, chipped edges, machining marks, …


part, or component ! Yc ⋅Conclusion• KIc indicates how well a specific material under stress is able to withstand the

extension of a crack (… design relevant property). • The higher the KIc, the more difficult it is for a crack to advance in a material. • The strength at failure σc is a characteristic of a component.

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What you should know and understand, now! Learning targets part 1

Why mechanical testing …• Design relevant properties

Fracture toughness, strength, creep, subcritical crack growth, …(technological tests = comparison of materials under exactly defined conditions)(technological tests comparison of materials under exactly defined conditions)

• Ceramics miss plasticityelasticity: ~ 0.01 % plasticity: ~ 0 %

Stresses at a crack tip are higher than you’d assume …• Mechanical stress at crack tip is by factors larger than stress simply calculated from

macroscopically available cross section and average applied stress.

⎟⎞

⎜⎛ a σmax stress at crack tip (max. stress)


• For a given load, as crack grows and bonds are sequentially ruptured, σtip moves up the stress versus displacement curve towards σmax. Catastrophic failure occurs when σtip ≈ σmax.

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ρσσ a210max

σ0 nominal stressρt radius of curvature at crack tipa crack size

Griffith law explains conditions for failure …• All materials exhibit a natural defect population due to production. Defects differ in

size, form and orientation. • Brittle materials can’t diminish stress superelevation at crack tip by plastic

deformation.• Small defects weaken brittle materials.

Learning targets part 1

• Failure occurs in brittle materials if theoretical strength < defect related stress peak.

Ec ⋅⋅≥⋅⋅ γπσ 2σ, c applied stress, depth of crack2, γ surfaces created, intrinsic surface energy of materialE Young’s modulus

Failure occurs if stress intensity ≥ critical stress intensity … • KIc indicates how well a material under stress withstands the extension of a crack.


c

• The higher KIc, the more difficult it is for a crack to advance in a material. • KIc is material specific.• Y-factor predicts intensity and distribution of a stress field around a defect caused by

an external load.

YcK cIc ⋅⋅= σ KIc ,σc fracture toughness, critical applied stressc, Y depth of crack,Y-factor

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