mechanical properties of ceramics - eth - nonmet · 2010-02-08 · prof. l.j. gauckler eth zürich,...

1

Materials Science & Technology

Materials Science II - 2010, Ceramic Materials, Chapter 6, Part 2

Mechanical Properties of Ceramics

Jakob Kübler

orMechanical Behavior of Brittle Materials

& Prof. L.J. Gauckler

1Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 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

• Design relevant mechanical properties (≠ properties by technological tests)Fracture toughness, strength, creep, subcritical crack growth, …

What you already know and understand!

Repetition learning targets part 1

• All materials exhibit a natural defect populationdue to production. Defects differ in size, form and orientation.

• Mechanical stress at crack tip is by factors larger than stress calculated from macroscopically available cross section and average applied stress.

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

ρσσ a210max

σmax stress at crack tipσ0 nominal stressρt radius of curvature at crack tip


• Brittle materials like ceramics can’t diminish stress superelevationat crack tip by plastic deformation.

⎠⎝ ρt pa ½ crack width

2

• Griffith’s basic idea: Balance energy consumed in forming new surfaces ascrack propagates against elastic energy released.

• Griffith’s law: Failure occurs whenrate at which energy is released is greater than rate at which it is consumed.(if defect related stress peak ≥ theoretical strength)

Repetition learning targets part 1

Valid as long as only factor keeping crack from extending is creation of new surfaces.

(if defect related stress peak ≥ theoretical strength)

• with help of Irwin’s correlation: Failure occurs if Stress Intensity Factor ≥ Critical Stress Intensity Factor

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

YcK I ⋅⋅= σ KIc ,σc fracture toughness, critical applied stressd th f k Y f t


• KIc is material specific and indicates how well it withstands the extension of a crack under stress. The higher KIc, the more difficult it is for a crack to advance.

• Y-factor predicts intensity and distribution of a stress field around a defect caused by an external load.

YcK cIc σ c, Y depth of crack, Y-factor

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

3

Definition of crack dimensions for today’s lecture≠ last weeks definition

(… just to stay flexible …)Attention:

In literature “c” and “a” are often used vise versa

2 c

a

σ

a

2 c → ∞

σare often used vise versa.


σ σ

R-curve behavior

KIR

KIR

Increasing resistance against crack propagation

Crack growth Δa

KIR= KIc

KIR

Why this increase ?


e.g. fracture toughness of a polycrystalline ceramic is significantly higher than that of a single crystals of the same composition, e.g. KIc of alumina

single-crystal ~ 2.2 MPa √mpolycrystal ~ 4 MPa √m

Crack extension isn’t characterized by a constant KIc anymore but by a KIR - Δa curve.

4

R-curve behavior (3)

Polycrystalline material:as crack deflects along weak grain

Why is KIR increasing?a) Crack deflection at grain boundaries

as crack deflects along weak grain boundaries, Ktip is reduced, because stress is no longer normal to crack plan

Barsoum, p380

crack plan

( ) apptip KK 23cos θ=


assuming Θavg = 45° → Ktip ~ 1.25 x single-crystal value

crack deflection accounts for some of the enhanced toughness, but not all


b) Crack bridging 1

Toughening results from bridging of

deflection of crack front along / around rod-shaped particles

Why is KIR increasing?

g g g gthe crack surfaces behind crack tip by a strong reinforcing phase e.g.

Bridging ligaments generate closure f k f hi h d K

• elongated grains• continuous fibers• whiskers• particles (metal …)ligament bridging mechanism with no

interfacial debonding

undeflected crack front


forces on crack face which reduce Ktip.

5


SiC whiskers in - glass- mullite- alumina

b) Crack bridging 3: example

lines: predictionpoints: experiments

Mineralogical name of only chemically stable intermediate phase in SiO2 - Al2O3 system. The natural mineral is rare, occurring on the Isle of Mull, west coast of Scotland.

What’s Mullite ?


Crack bridging and pullout can yield substantially increased fracture toughness.

Si3N4F. Monteverde, A. Bellosi, S. Guicciardi, © ISTEC-CNR


Fracture toughness of a “composite” due to elastic stretching of a partially debonded reinforcing phase at crack tip with no interfacial friction:

P. Becher, J.Am.Ceram.Soc.,

b) Crack bridging 2: amount of increase

⎟⎟⎞

⎜⎜⎛

⋅⋅⋅

+⋅= fcffI

EVrGEK

γσ 2

where: c, m, f , i composite, matrix, reinforcement, interfaceE, V, r Young’s modulus, volume fraction, radius of bridging ligament σ, G strength of reinforcement phase, toughness of unreinforced ligament γf/γi ratio of fracture energy of the bridging ligaments to that of

the reinforcement/matrix interface

74:255-269 (1991)⎟⎟⎠

⎜⎜⎝

+if

fmcIc EGEK

γσ

12


i.e. fracture toughness is increased for • “high” reinforcement content, • “weak” reinforcement (increasing Ec/Ef ratio) and • “weak” reinforcement / matrix interfaces (increasing γf/γi ratio)

6


b) Crack bridging 4: amount of increase

Al2O3 & SiC-whisker composite

Data for calculationm: matrix ; f : whiskerEm 400 GPaKIc Al2O3 3 MPa √mEw 580 GPaσw 8’400 MPadf 1 μmlf 10 μmwhisker direction random-3Dinterface γf / γi 1, 25, 125

4

6

8

10

ite K

Ic [

MPa

√m

]'super strong' interface'strong' interface'weak' interface

γf γi(1 = super strong)


0

2

4

0.0 0.1 0.2 0.3 0.4

volume fraction of SiC-whisker

com

posi

c) Transformation toughening

… if tetragonal particles are fine enough, then upon cooling from Tprocess , they can be constrained from transformation by surrounding matrix.

i i l


martensitically transformed zirconia particle

original metastable tetragonalzirconia particle

compressive stress field around crack tip


… very large toughness due to stress-induced transformation of metastable phase (tetragonal → monoclinic Zr) in vicinity of propagating crack.

zirconia particle

7

t → m in pure undoped ZrO2 during cooling is a reversible martensitic transformation, associated with a volume change (4–5%). Dopants (yttria, ceria,

i l i t ) ll dd d t t bili th hi h t t t d/


Surface grinding induces the martensitic transformation, which in turn creates compressive surface layers and a concomitant increase in strength.

Matensitically transformed

magnesia, calcia etc.) are usually added to stabilize the high temperature t and/or c-phase in the sintered microstructure.


Metastable tetragonal zirconia particle

zirconia particle

if t i i l t t f ti i i d d ( l i 4% h

Shielding factor Ks

R.M.Mc. Meeking and A.G. Evans, J.Amer.Ceram.Soc., 63:242-246 (1982)


… if constrain is lost, transformation is induced (volume expansion ~4% → shear strain up ~7%). Approaching crack front (= free surface) triggers transformation,

which in turn places zone ahead of crack tip in compression.

To extend crack into compressive zone extra energy is required → KIc and σ ↑

κ dimensionless constant (Δa/w = ∞ → κ = -0.215 depends on shape of zoneahead of crack tip)


E Young’s modulusVf volume fraction of transformable phaseεT transformation strainw width of zone with transformed phaseΔa length of crack inside transformed zone

w = 5 μm ; E = 210 GPa ; Vf =0.92 ; εt = -0.07

8


PSZ: partially stabilized zirconiaCubic phase is less than totally stabilized by the addition of MgO, CaO, or Y2O3. Heat treatment needed to keep precipitates small enough so that they do not spontaneously transform within the cubic zirconia matrix.

Toughened zirconia-containing ceramics

TZP: tetragonal zirconia polycristal100% tetragonal phase and small amounts of yttria and other rare-earth additives. σb up to 2’000 MPa.


ZTC: zirconia-toughened ceramicTetragonal or monoclinic zirconia particles finely dispersed in other ceramic matrices such as alumina, mullite, and spinel.

Residual compressive stresses


Increasing resistance against crack propagation by design, e.g.

compressive residual stresses in laminates

L1

= σLoad - σCResσLayer

KIc = ( σ−|σC| ) • √a • Y

reduce actual stress in outer layer

L1

L2

L1

σσ


How can compressive stresses be introduced into surfaces,e.g. in glass and ceramics?Glass: Rapid cooling of outer surfaces.Ceramic: CTE gradient from surface to core.

σ-σ+

9

4

6

C


Laminates (2): CTE mismatch to introduce residual stresses

0

2

0 200 400 600 800 1000Temperature °C

Si3N4 + X% TiNSi3N4

CTE

10-6

/ o C

Si3N4+30% TiN

Si3N4


Temperature C

• TiN particle addition in Si3N4 increases the CTE.• Si3N4 gives layers under compressive residual stress.• Si3N4 +TiN gives layer under tensile residual stress.


Laminates (3): Design

• Strong boundary layer interfaces. • External layers under compressive stress.

Si3N4 + 30 % TiNSi3N4

1 mm

150 μm

600 μm

Si3N4


1 mm

10 µmSi3N4 +TiN

Remark: “Joining” temperature ~1’100°C

10


Laminates (4): Apparent Fracture Toughness = toughness you will measure but isn’t solely material related

Si3N4 Si3N4+TiN

0.5mma


• KIc-app increases with notch length towards interface in compressive layer• KIc-app decreases in tensile layer.• KIc-app more than three times KIc of Si3N4.

Notch length a [mm]


Laminates (5): improved designMicro-layered laminates

(with external tensile layers)


11


Laminates (6): further improved designMicro-layered laminates

(with external compressive tensile layers)


Suggested design and KIc app behaviour of micro-laminate design with layers of five different compositions.

KIc app as function of crack length of the 2nd micro-laminate design with external compressive layers superimposed onto WFA model.

Kuebler J., et.al., KEM 333 (2007) 117-126

Properties (FT1)

Test methods for determination of fracture toughness KIc (KIc → resistance displayed by a material to propagation of crack through it)

SEVNBSEPB

CNB IF


SCF IS

Fracture toughness should be qualified with the conditions under which the test is performed (e.g. method, test conditions, crack

size, geometry, stress field, crack velocity).

12

Properties (FT2)FSEVNB (and SEPB, SENB)

Single Edge V-Notched Beam(and Single Edge Precracked Beam, Single Edge Notched Beam)

( )with

aW

SSWB

FYaK M

Ic −Γ⋅−

=⋅⋅=α

σ

2

2/321max

)1(23

S2

S1

W a

NCSi3N4

Wa

and

M

=

+−+−

−−=Γ

α

αααααα 2

2

)1()1()35.168.049.3(326.19887.1

23Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 20105 μm

Properties (FT3)

CNChevron Notched Beam F

not validd

not valid

F


Y'm 3.08 5.00a0 8.33a02+ +( ) 1 0.007

S1 S2

W2-----------+⎩ ⎭⎨ ⎬⎧ ⎫=

'maxmIc Y

WBFK⋅

=

13

Properties (FT4)

1 Knoop hardness indent 2 polished surface

SCFSurface Crack in Flexure

improve visibility

2c

a


Ymax : larger of Ys and Yd

Properties (FT5)

8.0

√m] other methods, indiv. avg.

SEVNB; G P AvgAlumina-999

Why those differences?

Fracture toughness values measured with various test methods in comparison with SEVNB values

4.0

6.0

Tou

ghne

ss [

MPa

√ SEVNB; G.P.Avg.SEVNB; G.P.Std.Dev.

5/4)

6/5) 5)) o ) 10/5

)

N2 5) H2O 1/5)


0.0

2.0

Method (Participant / Number of Specimens)

Frac

. T

SEPB

(25

SEPB

(26

SCF

(10/

5

CN

(8/5

)

SCF

(9/4

)

SCF+

halo

(10/

5SC

F-N

2 (1

SEV

NB

-N (1/

SEV

NB

-H (1

J. Kübler, ASTM STP 1409, 2002

14

Properties (FT6)Development of Vickers indentation cracks

��

Vickers - IFIndentation Fracture

KIc 0.032H a EH----⎝ ⎠

⎛ ⎞

12---

ca--⎝ ⎠

⎛ ⎞

12---

=��

��

��

H⎝ ⎠ a⎝ ⎠H = F/2a (hardness)only valid if c/a > 2.5

F


��

��

plastic deformation !!

Properties (FT7)

Example: Micro-Hardness (… plasticity …)

Si3N4 – 05


Scanning Probe Microscope

Optical Glass BK7, HV-1N

15

Strength (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.

Strength (2)

Strength of ceramics; evolution


16

Strength (3)

Advantage• simple (fixation of sample,

simulation of environment, …)cheap (sample jig )

Bend test

• cheap (sample, jig, …)• universal (strength, fracture

toughness, Young’s modulus, fatigue, …)

• sensitive to surface defects

Disadvantage• small volume tested

t di t th f t


• stress gradient therefore not valid by plastic deformation

WM B

B=σ

Strength (4)

3- vs. 4-pt-bending


17

Strength (5)

P: load at failure r ΔP = 0.2 %b, h: size (3x4 mm) r Δb, r Δh = 0.07 %d: o/i rolls (10 mm) r Δd = 1.0 %

(calculated from tolerances of test jig)

Example: 4-Pt-bending strength @ elevated temperature

Measurement uncertainty

( ) 2

3:,,,hb

dPdhbP⋅

⋅⋅=σ

( j g)

Relative measurement uncertainty:% 1 2 %

Relevant factors, e.g.:• σf ≥ 100 MPa / ≥1’000 MPa Δe1 > ± 2.7 % / < ± 0.3 %• Δl jig (T related) Δe2 ~ ± 3.0 %

h ti @ f Δ 5 0 %

∑ Δ+∑ Δ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

==

m

mj

n

ninxx

i

exfdxdy

11

2

)...1(


Δσ% ≤ 1.2 % • chem. reaction @ surface Δe3 ~ ± 5.0 %Remark:• Considering uncertainty of TC ± 2.0 °C, registration equipment• Not considered: test speed variation, surface quality, rel. humidity

Bending strength "Real" relative measurement uncertainty ≥ 100 MPa ± 1.2 % + 2.7 % + 3.0 % + 5.0 % = 11.9 %

≥ 1000 MPa ± 1.2 % + 0.3 % + 3.0 % + 5.0 % = 9.5 %

Strength (6)

Scatter of mechanical strength

f(σc)

Dispersion density of the strength measured on a series of components

σc1 σcσc2σc ( )∫ =

∞

01cc σσ df


( )∫=<<2

121 )(

c

cccccc

σ

σσσσσσ dfP

( )∫=c

ccc

σ

σσσ0

)( dfF

18

Strength (7)

Dispersion of the largest (failure relevant) defects and failure strengths

h(a) 1-F(σc)h(a)

H(a)

f(σc)( c)


… large “largest” defect→ low strength …

a σc

… small “largest” defect→ high strength …

Strength (8)

Strength of a ceramic component …

… is defined by a combination of• critical stress intensity factor• size of critical defect• position of critical defect• stress and stress direction the crack sees

failure relevant defect

largest,but not failure relevant defect

A large number of small defects present in a component are loaded too, but aren’t responsible for catastrophic failure


stress and stress direction the crack sees

… therefore it’s difficult to predict the strength of a component !

↑ direction⊕ position),,,,( ⊕↑= ccKf Iccomponent σσσ a, a

19

Strength (9)

… during fabrication of component:• always: powder agglomerates• friction when pressing• powder sedimentation when casting (slurry)

Sources for defects …

• always: cracks and pores from sintering

… during usage of component:• corrosion, pitting• subcritical crack growth, creep • friction, scratches• stress peaks (impact, …)

(in ductile materials e g in fcc-metals stress peaks can be reduced


(in ductile materials, e.g. in fcc metals, stress peaks can be reduced by plastic deformation at RT due to 5 independent plains for sliding)

→ ceramic materials are very brittle - they fail without warning even at elevated temperatures (KIc is between 1 MPa √m and 20 MPa √m)

→ increase of toughness in ceramics has to happen in a different way than over sliding and plastic deformation

Strength (10)

Toughness ↔ defect size ↔ strength


~ 1 : 10’000 (!!)

20

Strength (11)

1) increase σc by reducing ac, e.g. by improved processing

Two strategiesto improve σc and KIc 2) increase KIc by increasing

fracture energy, e.g. by crack bridging, transformation t h itoughening

σ c(M

Pa)

100’000

10’000

1’000

)log(loglog21log YKa Iccc −+⋅−=σ


σ

1 10 100 1’000

critical defect size (μm)

1 000

Strength (12)

• in ceramics strength controlling defects have a size of a few μm up to a few 100 μm

• failure relevant is the largest volumeor surface defect under stress

Summary

or surface defect under stress

• ceramic materials don’t have a single strength value

• identical components will not fail at onereproducible strength value (= strength value distribution)

• When is the density of defects smallNever …

nsity

of d

efec

ts


• When is the density of defects small enough so that we can be absolutely sure that no defect with a critical size is present ?

Statistical data is needed!The strength of ceramics is described by the Weibull statistics - see part 3.

defect size

den

21

What you should know and understand, now! Learning targets part 2

Improving toughness …• Fracture toughness is related to the work required to extend a crack and is

determined by the details of the crack propagation process. It can be enhanced by increasing the energy required to extend the crack.

• Ceramics with R-curve behavior:- degradation in strength with increasing flaw size is less severe- reliability increases (some recent evidence shows that thermal shock resistance increases)

• Only for the fracture of the most brittle solids is the fracture toughness simply related to surface energy.

• Crack deflection, crack bridging, martensitic transformation (next to others) (and design) are mechanisms that enhance KIc app.

Know what you measure


Know what you measure …• Fracture toughness values measured with different test methods may differ.• Bend test: - universal (e.g. strength, fracture toughness)

- sensitive to surface defects- only a small volume is tested- value σ3Pt test > value σ4PT test- specimen sees stress gradient (not valid by plastic deformation)

Learning targets part 2

Strength is “just a value” …• All components have defects due to fabrication and usage• The strength controlling defects in ceramic components have a size of a few μm up

to a few 100 μm• The strength of a component is defined by a combination of

i i l i i f- critical stress intensity factor - size of critical defect- position of critical defect- stress and stress direction the crack sees

• Identical components will not fail at one reproducible strength value = strength value distribution

• Ceramic materials fail without warning even at elevated temperatures KIc is between 1 MPa √m and 20 MPa √m


• The aim is always to improve both- σc by reducing ac , e.g. by improved processing- KIc by increasing fracture energy, e.g. crack bridging, transformation toughening …

• The strength of ceramics must be described by statistics

mechanical properties of ceramics - eth - nonmet · 2010-02-08 · prof. l.j. gauckler eth zürich,...

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