mechanical properties of ceramics - eth z

08.02.2010

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Materials Science & Technology

Materials Science II - 2010, Ceramic Materials, Chapter 6, Part 4

Mechanical Properties of Ceramics

Jakob Kübler

orMechanical Behavior of Brittle Materials

& Prof L J Gauckler

1Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2010

Jakob Kübler Empa, 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

• Weibull: mathematical descriptionof failure / survival probability

What you already know and understand!

Repetition learning targets part 3

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−=−=

0

exp11VVPP

m

o

cSf σ

σσ

• Weibull parameters:- σ0 = strength @ 63% probability of failure- small m = large distributionlarge m = small distribution

• The effect of volume and surface area on acceptable stress level can be calculated

f(σc)

σc

1

1 2

2 1

mVV

σσ

⎛ ⎞= ⎜ ⎟⎝ ⎠


calculated.

• Proof testing will eliminate “bad” components. Lower end of distribution is cut off and new distribution isn’t a proper Weibull distribution anymore.

2 1Vσ ⎝ ⎠

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Reading fracture surfaces …

• Increasing level of information of a fracture by starting from its history.

• Fracture patterns will lead you to the origin zone

What you already know and understand!

Repetition learning targets part 3

• Fracture patterns will lead you to the origin zone.

• Macro- and micro-features point towards the origin.

• Fractography in combination with fracture mechanics:- develop materials- optimize procedures and processes- construct components- improve machining- design systems


g y

Guide for fractographer …• Get familiar with failure and its environment by naked eye and a map .. • Observe large markings and features with an optical microscope ..• Locate and understand small details with a SEM ..

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

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Thermal shock• All materials change their dimensions with temperature.

• If there is a temperature gradient in a material the resulting strain generate internal stresses.

• If the material is polycrystalline and multiphase we can get interphasestresses.

• If it is single phase but anisotropic we get intergrain stresses.

• These stresses can lead to cracking and as a result immediate fracture of whole specimen.


• Smaller thermal shocks may introduce damage by causing sub-critical extension of existing cracks. The material’s strength will be reduced.

• Thermal shock parameters (R, R´, R´´) are good for ranking materials’ resistance to complete failure or to major crack initiation.

sources: Fett & Munz and Oxford Ceramic Lectures

Thermal shock (2)

Thermal stressstationary thermal stresses, i.e. time-independent stress distribution

thermally inducedelongation

TTTl

lthermalthermal Δ⋅=−⋅=

Δ= ααε )( 01

0

0+εε

e.g.: αSi3N4 = 3·10-6 [K-1]coefficient of thermal expansion

Stress will develop when component is fixed dimensionally.T0 T1

Bar:- fixed length- bending prevented


)( 01 TTEEE elasticthermalthermal −⋅⋅=⋅=⋅= αεεσ

0=+ elasticthermal εε

failureif ⎯→⎯> elasticthermal εε

g p

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Thermal shock (3)

Plate:- free expansion in all directions- bending prevented

Thermal stressstationary thermal stresses, i.e. time-independent stress distribution

T0

T1

d

0=σ

z

max. stress@

z = d/2yx


2)TT(

)1(E 01

c−

⋅υ−α⋅

=σ

ν = Poisson’s ratio

yx σ=σ→={ }

Thermal shock (4)

A material initially at T1 has ("only") its surface temperature lowered (or raised) instantaneously to T2

Instantaneous surface temperature change= non-stationary stresses

= time-dependent stress distribution

Ttemperature lowered (or raised) instantaneously to T2

The stress in the top surface layer of a disc (2-D stress state) is:

)TT()1(

E21c −⋅

υ−α⋅

=σ RS

thermal shock resistance parameter

T1

α⋅υ−⋅σ

=E

)1(c


(the lower RS the larger the thermal shock sensitivity of a component)

low αhigh KIcandlow E

→high RS = good thermal shock resistance

for rapid surface temperature change which can occur without exceeding σc

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Thermal shock (5)

Heat flow

In some cases, thermal shock is a function of heat transfer ratetransfer rate.

)TT(A λ

In steady state, heat transfer rate Q from environment at T2 through the material to a heat sink at T1:

QxQ


x)TT(AQ 12 λ⋅−⋅−

=

A = surface areaλ = thermal conductivity of material

λ= FQ

λ⋅⋅−

=−A

xQTT 12

Thermal shock (6)

so, similarly to the “instantaneous” ΔT case:

⋅α⋅

=σ Fc

QEλυ−

σc )1(

λ⋅α⋅υ−⋅σ

=E

)1(Q cF s

'R= sR⋅λ=

Rs

thermal resistance parameterfor (constant) heat flow


high thermal conductivity λ high Rs’

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Thermal shock (7)

Rs R'sσc


Rs R'sσc σc

Important for thick sections subjected to constant heating or coolinge.g. in furnace design

Constant rate of heating (1)

The solution of the thermoelastic equations gives

Thermal shock (8)

dT/dt is the rate of change of temperature and θ is the thermal diffusivity

thermal diffusivity := thermal conductivity to volumetric heat capacity [m²/s]


λ thermal conductivityρ material’s densityCP heat capacity

low resulting stress if Θ is large large λlow ρ and low CP

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Constant rate of heating (2)

shape factor F

Thermal shock (9)

RS” := 3rd thermal shock parameter

maximum heating rate

F''RdtdT

s ⋅=


1. Heat sample up to temperature T1

2. Quench sample down to temperature T0 (≠ RT, 0°C, …..)

3. Measure failure strength of sample

Thermal shock (10)

Thermal shock testing

micro-cracks were produced @ ΔT = 300 K (or more)

additional damage produced@ ΔT 650 Kre

ngth

(M

Pa)


@ ΔT > 650 K

Str

ΔT = T1 – T0 [K]

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Subcritical crack growth (1)

Determination of design relevantstrength properties

☺ ☺Creep

Relation betweencreep rate and

load.

Crack growth /Lifetime

static dynamicRelation betweendefect size and

strength.

Relation between strength and probability of

Fracturetoughness Strength


Relation between crackgrowth speed and stress

intensity factor.

KIc

failure.

σ, σ0, m


"... after several years of service, sudden transverse rupture

t d i l t i t separated an insulator in two parts ... numerous insulators underwent a visual inspection, and longitudinal cracks were found in some of them..."

slow crack growth (scg)


Source: Fractography of Glasses andCeramics III, Woodtli et al, p 260, 1996

slow crack growth (scg)= time dependent failure= limited life time

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2ci

v=da/dtF

2cc aicrack size

ac

F

aiac


experimental finding:nIKA

dtdav ⋅==da


log v

III

crack velocity

“100%” vacuum

I

II

n)( IKf

dtdav ==


log KIKIc stress intensity

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-4

-5

Al2O3 - 998, water


-6

-7

-8

-9

-10

log

[ v (K

I)]

[m

/s]

static stress270 MPa (7 von 20)

250 MPa (20 von 20)

230 MPa (20 von 20)

PrivateCommunication

ith T F tt KfK


0.00-0.05-0.10-0.15-0.20log (KI / KIC)

-11

-12

n = 45.9 ; A* = 0.0229 m/s

c: | files | ukrw | v-k_0696.gfr

with T. Fett, KfK


Si-O-Si- + H2O → -Si-O-H + -Si-O-HSi

Si O

HO

H

H

O HHOH

H

Environment: H2O diffusion in crack and reaction at crack tip

Water induced brake up of bonds at crack tip in soda-lime glass.

Si OOH

H3

O+O HH

moving of free

Mass transport processes inside crack


mass flow (viscose liquid)

gmolecules (gas)

activated diffusion

diffusion (liquid)

adsorption reaction

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log vcrack velocity

[m/s]

• same material• same defect (same “a”)

nIKA ⋅=ν

( )• same stress

• different environment(= different n and A)

different crack velocity= different lifetime

10-6

10-9


log KI

KIstress intensity factor

Lifetime under static load σ

lg σ

Level of σabove which instantaneous

rupture after scg(= time delayed failure)

evaluate n if lifetime at different stress levels are measured

σc

n1

−

lif ti i t t ith

inert strength:determined with the help of a fast fracture test e.g. in vacuum

tB-trans

lg tB

σinstantaneous failure occurs

scg

tB

[ ]2nB −σ


lifetime in tests with σ(t) = const.

for n>10 equation simplifies to

[ ] ),,,()/(1 2Ic

ncn

cB KYAnfB

Bt =−= − with σσ

σσ

nIcn

nc

B KnYA

BB

t −−

−⋅=

⋅= 2

2

2

)2(2 with

σσ

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Lifetime with load ramps σevaluate n if strengths at different stress rates are measured

.

lg σB

•σlg

n+11


and in its logarithmic form [ ]2)1(log

11log

11log −+

++

+= n

cB Bnnn

σσσ &

Lifetime under cyclic load (1)

Stress and stress intensity factor under cyclic load


y y

( ) )(tft am ⋅+= σσσ

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time till failure under static load

Correlation between static and cyclic load

time till failure under cyclic load

nnif Bt −− ⋅⋅= σσ 2 nn

lif Bt −− ⋅⋅= σσ 21

=

cstaticf Bt − σσ mcma

cyclicf Bng

t − σσσσ )/,(

dttfT

nT

m

a∫ ⎥⎦

⎤⎢⎣

⎡+=

0)(11

σσ


X

staticf

n

mmacyclicf t

ngt −− ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

σσ

σσ )/,(1

Porcelain


Example: Crack growth velocity under static and cyclic loads at RT

Zirconia


Why lower than prediction?… t m transformation of zirconia

• lifetime under cyclic loading usually shorter than under static loading

• lifetime is lowest for R=1

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6.40 540 MPa

Strength-Probability-Time diagramLink between: strength & probability & lifetime

example material: m = 15, n = 40 and B . σcn-2 = 107 (… ~alumina)

5.40

5.60

5.80

6.00

6.20

ln (s

tatis

che

Last

in M

Pa)

345 MPa

235 MPa


5.00

5.20

0 2 4 6 8 10 12 14 16 18 20 ln (Lebensdauer in s)

1 Min 1 Stunde 1 Woche 1 Jahr

0.99 0.90 0.50 0.10 0.01 Ausfallw.

lald-fik.wb1:graph-sb

Creep (1)

Determination of design relevantstrength properties

☺ ☺ ☺Creep

Relation betweencreep rate and

load.

Crack growth /Lifetime

static dynamicRelation betweendefect size and

strength.

Relation between strength and probability of

Fracturetoughness Strength


Relation between crackgrowth speed and stress

intensity factor.

KIc

failure.

σ, σ0, m n

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Creep (2)

Simple model by Nabarro-Herring and Coble for diffusional creep under load.


a) Vacancy concentration gradients that develop as a result of stress gradients. The vacancy concentrations are higher below the tensile surface. Curved arrows denote direction of vacancy fluxes.

b) Schematic of a grain of diameter d subjected simultaneously to a tensile and a compressive stress. Curved arrows denote direction of atomic fluxes.

c) Shape of grain after creep has occurred.

Creep (3)

Ashby and Verall model for grain boundary sliding (1973)

FF

FF


In this 2-D model with four hexagonal grains a structure is elongated by twisting and shifting the grains (without deforming them).

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Creep (4)

ε

εtl ti t i

Typical creep curve for a specific material at a defined temperature and load

tsp0 ε+ε+ε+ε=ε

t

εsεp

εο

εo elastic strainεp primary creepεs secondary creep

(stationary creep)εt tertiary creep


Creep (5)

Why creep should be measured in tension.


Time-dependent stress distribution in a bend bar.

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Creep (6)

High accuracy measurement of elongation(better than 1 µm at l0 = 25 mm up to 1’600°C)


NGK SN-88 (Specimen # 4)800

900

1000

3.2

3.6

4.0-50 0 50 100 150 200 250 300 350 400 450 500

Time [h]

Creep (7)

200

300

400

500

600

700

Elon

gatio

n [µ

m]

0.8

1.2

1.6

2.0

2.4

2.8

Elon

gatio

n [%

]

D2AD2B

200 MPa, 1375 °C


-100

0

100

-1.80E+05

0.00E+00

1.80E+05

3.60E+05

5.40E+05

7.20E+05

9.00E+05

1.08E+06

1.26E+06

1.44E+06

1.62E+06

1.80E+06

Time [s]

-0.4

0.0

0.4D2B

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Strain rates and LifetimeCreep (8)

εp primary strain rate (exponential orcreep law by Norton)

εs steady-state strain rate;generalized expression

·

'np tC ⋅=ε

generalized expressionεmin minimal strain rateσ stressµ shear modulusA, C constantsD diffusion coefficientG grain sizeT temperatureb Burgers vector

·

W.R. Cannon, T.G. Langdon, J.Mat.Sci., 18: 1-50 (1983)

np

s Gb

TkbDA

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⋅⋅⋅⋅

=μσμε&

mf Ct −⋅= minε&


gk Boltzmann constantm exponentn' time exponent (≥ 1)n stress exponentp grain size exponentt timetf lifetime (Monkman-Grant)

Stationary creep rate doesn’t decrease with increasing T since exponential dependence of

on T dominates D increases faster than 1/T.

)exp(0 TRQDD⋅

−⋅=

Creep (9)

Creep curves are depending on temperature, stress and grain size

increasing

cree

p

increasing load and temperature

increasing grain size


time

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-14 /s] Activation Energy Q

Al2O3 ; grain size 4.75 µm ; ALF

Creep (10)

-6 /s] Stress-Exponent n

(Sumitomo AKP-53; starting particle size 0.2 µm; sintering @ 1650°C for 1.5 h)

12.5 MPa 20 MPa 30 MPa

-22

-20

-18

-16

Ln S

train

Rat

e [1

/

7.2E-05 7.4E-05 7.6E-05 7.8E-05 8.0E-05 1/RT

- Q [KJ/mol]465

575619

602

631

-10

-9

-8

-7

Log

Stra

in R

ate

[1

1.0 1.2 1.4 1.6 1.8 2.0 Log Load [MPa]

n=1.60

n=1.58n=1.35


12.5 MPa 20 MPa 30 MPa50 MPa 75 MPa1245°C 1278°C 1329°C 1378°C

n → 1 with increasing temperature activation energy -Q decreases with increasing load

Creep (11)

Grain Size Exponent pAl2O3 - Sumitomo AKP-53; starting particle size 0.2 µm, 1329°C


sintering1400 °C, 1.5 h

sintering1650 °C, 2 h

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Creep (12)

Mechanisms at creep failure

I Crack tip is blunted by creep deformation Afterwards crack propagates


I Crack tip is blunted by creep deformation. Afterwards crack propagates.II Pores form at crack tip and coalesce with crack. Afterwards crack extends.III Pores, created by creep deformation, join to a crack. *IV Failure by oxidation. First an oxid layer is developing in which cracks

generate and extend into the bulk material. *

* no pre-existing crack is needed

Failure maps (1)

Lifetime of silicon nitride calculated for an elastic stress in the outer fiber of a bend bar.


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Failure maps (2)


First complete failure map for silicon nitride by Quinn (1986/1990).(Norton, NC 132, MgO doped)

What you should know and understand, now! Learning targets part 4

• Ceramic materials are susceptible to thermal shock,

they fail if exposed to too fast temperature changes (ΔT/Δ t) and locally to too large temperature gradients (ΔT/ Δ x).

• Low CTE, high KIc, and low E = high RS (= good thermal shock resistance)

• Slow Crack Growth (scg): time dependent failure → limited life time

• Crack velocity is influence by humidity → brake up of bonds at crack tip e.g. in soda-lime glass by water (Si-O-Si- +H2O → -SiOH + -SiOH).

• SCG parameters can be measured in accordance to state of load(static, dynamic, cyclic, or a combination thereof).


• Correlation between largest failure relevant defect, failure strength and lifetime.

• Link between strength, probability of failure and lifetime: Strength-Probability-Time diagram

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Learning targets part 4

• Creep: Boundary (diffusional) or lattice (dislocation) mechanism

• Diffusional creep: Free surfaces and grain boundaries work as source and assembly point for voids and atoms. Voids diffuse from surfaces under tension to surfaces under compression and matter flows in reverse direction.p

• Grain boundary sliding: Structure elongates by shifting & twisting of grains (without deforming them)

• Stages of creep: - primary- secondary (steady-state) - tertiary


• Creep should be measured in tensionand not in bending.

• Stain rate is increasing with - increasing load & temperature and - decreasing grain size.

ned!

Brittle vs. tough / Failure stress / Crack resistanceStress elevation at crack tipGriffith’s law : correlation between failure stress and critical flaw size

Summary:What you must know and understand!

you

shou

ld h

ave

lear

n

R-curve behavior: Process zone / Crack deflection & bridging / Transformation toughening Sub-critical crack growth : Environment / Life prediction / Fatigue static & dynamicWeibull statistic : Distribution of strength & Life time / m, σ0

Influence of surface area & volume on probability of survivalProof testing


This

y Proof-testingDeformation & failure under load at elevated temperatures: creep / Norton law / Monkman-Grant relationThermal shock behavior: thermally introduced stresses

Those laws & equations should be known by heart!

mechanical properties of ceramics - eth z

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