Mechanical Properties of Ceramics

Materials Science & Technology

Materials Science II - 2009, Ceramic Materials, Chapter 6, Part 1

Mechanical Properties of Ceramics or Mechanical Behavior of Brittle Materials

Jakob Kübler & Prof. L.J. Gauckler Empa, Materials Science & Technology Lab for High Performance Ceramics Überlandstrasse 129, CH-8600 Dübendorf +41-44-823 4223 [email protected]

Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

ETH Zürich, Materials Department

1

References • Munz and Fett, Ceramics, Springer, 1999 • Barsoum, Fundamentals of Ceramics, IoP, 2003 • Askeland & Phulé: Science and Engineering of Materials, 2003 • Literature, diverse • ISO, CEN, ASTM standards


2

11. 12. 13. 14. 15.

Thermal shock Slow crack growth SPT diagrams Creep Failure maps

learning targets 1

8. Statistic 9. Proof testing 10. Fractography

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

learning targets 2

R-curve Properties Strength

“Weibull, a name you’ll never should forget …” “Make it or …” “Reading fracture surfaces …”

learning targets 3

part 2 Strength

5. 6. 7.

part 3 Statistics

Introduction “Why mechanical testing …” Stresses at a crack tip “Higher than you’d assume …” Griffith law “Conditions for failure …” KI and KIc “Stress intensity & critical stress intensity …”

“Temperature, time and geometry …” “After several years …” “Combining strength, lifetime & statistics …” “Temperature makes it move …” “Finding your way …”

learning targets 4

part 1 Crack tip

1. 2. 3. 4.

part 4 Time&Temp

Aim of chapter & Learning targets

part 5 - Case Study: Lifetime of All-Ceramic Dental Bridges Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

3

Introduction (1)

Stress vs. strain diagram 500 Al2O3 failure

Deformation of … C steel

nominal stress F/A (MPa)

400

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

plastic deformation 300

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

200 elastic deformation

PMMA*:

100

0

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

PMMA @ 122°C

0

5

10

15

20

nominal elongation ΔL/L0 x100, % Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

25

* Polymethylmethacrylate (acrylic glass) Trade Name „Plexiglas“

4

Introduction (2) Mounting of IR camera

Ceramic structure for optics

Why Si3N4 ? Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

© Carl Zeiss Optronics GmbH & FCT Ingenieurkeramik

5

Introduction (3)

What’s that?


6

Introduction (4) Component Testing

Turbofan Engine: Source Wikipedia

Why do we need mechanical properties?

Advanced NDE

......

Material Development

Lifetime Prediction

Materials Testing and Charact.

Fabrication

Statistical Modeling Structural Design

σ < 100 MPa T > 1100°C

Corrosion Creep & Fatigue

σ < 250 MPa T > 900°C

Brittle fracture

Mechanical properties needed for the design of components.

Slow crack growth σ < 350 MPa T > 800°C


7

Introduction (5)

Aim of mechanical testing

Material properties → Design: E, KIc , (σm), ... Technological properties → Comparison: Hardness, ... Proof testing → Good / Not good

• strength • elasticity • fracture toughness • creep • scg • hardness • thermal shock resistance


• micro structure • surface • temperature • environment

• time • static • cyclic 8

Introduction (6)

Technological tests F

Vickers hardness HV = 0.102 (2·F·sin[136/2]) d-2

Example:

= 1.891·F·d-2

[N mm-2]

650 HV 100 / 30 Duration 30 s Load level 100 kp Vickers procedure Hardness 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, …. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

9

Stresses at crack tip (1)

Determination of design relevant mechanical properties

Fracture toughness

Relation between defect size and strength.

Strength

Relation between strength and probability of failure.

Crack growth / Lifetime

static

dynamic

Creep

Relation between creep rate and load.

Relation between crack growth rate and load.


10

Stresses at crack tip (2) http://upload.wikimedia.org Aloha Airline Flight #243 after becoming a convertible on April 28, 1988 Source: ASTM Standardization News, Oct.1998

… famous failures … http://ngm.nationalgeographic.com

Titanic during sea trials, sunk on April 14, 1912 Source: http://www.dodger.com/titanic/titanic-history.htm Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

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

http://en.wikipedia.org/wiki/Liberty_ship 11

Stresses at crack tip (3) F

Stress in a component F

F

σ

2r

F W

σ = F / (W·h)

2r

F W/2

F W/2

σ = F / (W·h)

W/2

W/2

σ ≠ F / (W·h)

h = plate thickness


12


Stress concentration around a hole

X

X Strain distribution around a hole in a open hole tensile coupon by FEA Source: NPL, Teddington, UK

large stress gradient along x-x

■ 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. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

13


stress concentration factor Kt

Stress concentration factor for holes and notches

w 2r

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

b h

w

b

stress concentration factor Kt

2r/w

b/r=4 b/r=1 b/r=0.5

“.. the sharper the notch the higher Kt ..”

2r r/h

Predictions are in good agreement with numerous experimental observations. possible source: Pilkey, Walter D., Peterson's Stress Concentration Factors (2nd Edition), John Wiley & Sons, 1997 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

14


Radius of curvature at tip of a defect F

F

ρt

“round hole := special case of defect

ρt

2a 2a

F

F W/2

W/2

W/2

σm (1)

W/2

σm (2) σm (1) < σm (2)

… sharpness of defect (or crack) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

15

Stresses at crack tip (7) Concept of stress superelevation at tip of a defect

stress

σm

σ0 x1

x2

x1-x2 → σ = 0 Calculating σm is rather complicated and is a function of the type of - crack geometry - loading (uniaxial, biaxial, …) - sample size Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

σ0 σm a ρt

nominal stress stress at crack tip ½ crack width radius of curvature at crack tip

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.

Here only the final result:

σm

⎛ a ⎞⎟ ⎜ = σ 0 1+ 2 ⎜ ⎟ ρ t ⎝ ⎠

Question: Factor X for a very small hole? 16

Stresses at crack tip (8) For a >> ρt this reduces to:

σm = σ0 ⋅2⋅

= sharp crack

ρt

Stress concentration factor

σm a =2 Kt = σ0 ρt

70

60 50

σ max / σ 0

a

40

3 (hole)

30

20 10

0 1

10

100

log(a/p (a/ρt) log t) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

1000

Measure of how much an external stress is amplified (superelevated) at the crack tip. From this it follows that - large cracks and - sharp cracks are more deleterious than small, blunt cracks. 17

Griffith law (1)

Stress superelevation 1. All materials have defects.

(sometimes very small ones)

2. Small defects weaken brittle materials.

(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. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

18

Griffith law (2)

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

Portrait of Alan Arnold Griffith and 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. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

19

bird eyes view

Griffith law (3)

Definition of crack dimensions for rest of today’s lecture: Attention: In literature “c” and “a” are often used vise versa.

σ

σ 2a→∞

2a c

σ Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

c

σ e t a l p n i th 20

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.

2

3 relaxed volume

uniformly stressed solid

1

4 two surfaces created

Total energy change Utot of system upon introduction of a crack

U tot = U 0 + UVo − UV − crack + U S − crack 1 1 2 3 4

2

3

4

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


21

Total energy change of system upon introduction of a crack

Griffith law (5)

U tot = U 0 + UVo − UV − crack + U S − crack U - applied stress

U -released by crack

U - residual stress

U tot = U 0 +

U – needed to create 2 crack surfaces

2 V0 ⋅ σ app

2⋅ E

2 σ ⎡ π ⋅ c ⋅ t ⎤ app

−⎢ ⎣

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

with ε =

2

2

+ 2 ⋅ c ⋅t ⋅γ ⎥ ⎦ 2⋅ E

with: Vo σapp E c t 2 γ

volume applied stress (avg stress) Young’s modulus depth of crack width of crack two surfaces created intrinsic surface energy of material

σ app E


22

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

Griffith law (6)

2 ⋅γ ⋅ c ⋅t

U*

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

dU* / dc = 0

Elastic energy (release) term −

2 σ app ⎡π ⋅ c 2 ⋅ t ⎤

⎢ 2⋅ E ⎣

2

⎥ ⎦

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

σ f ⋅ π ⋅ ccrit = 2 ⋅ γ ⋅ E Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

23

Griffith law (7)

Griffith - equation

σ f = σ crit

2 ⋅γ ⋅ E = π ⋅ ccrit

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

Example MgO: (rock salt, CN 6)

c = 2·dei ~ 424 pm { dei= r0Mg2+ + r0O2- = 86 pm+126 pm} equilibrium interatomic distance E ~ 250 GPa 2 Barsoum γ ~ 1.15 J/m2 (100) Surface, air, RT 3 1 2 3

1

p82, p364, p103

c

σf, theor ~ 20.8 GPa Î 1/12 E c

this confirms with data given in 1st part of Materials Science II, Ceramic Materials

The theoretically possible tensile strength (~E/10) has never been observed … Ð Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

24

Griffith law (8) … in reality rather 1/100 to 1/1000 of E → Why? 1 nm

10 nm

100 nm 1 µm

25

1000

20

800

15

600

10

400

5

200

0

0

[GPa]

⎛ E ⋅ 2 ⋅γ ⎞ σf =⎜ ⎟ ⎝ π ⋅c ⎠

1

2

-1

σf,theoretical of MgO

0

1

2

3

E / σf

4

defect size [log(c)] … c in nm

σf,MgO polycrystall = 90 MPa → E / σf = 2’700 !! NIST - SCD Citation Number: Z00176 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

25

Griffith law (9)

fac to r

sit y

KI … if

2 ⋅γ ⋅ E constant for a material

in t en

str es s

De fin

itio

n

:=

σ ⋅ π ⋅c

K Ic

KI ≥ KIc

Definition := critical stress intensity factor (or fracture toughness)

than fast fracture occurs

Remark: Valid as long as the only factor keeping the crack from extending is the creation of new surface. Only true, for extremely brittle systems such as inorganic glasses. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

26

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) 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) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

27

Irwin (1)

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

B

c W

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

σyy σxx

σxy

P

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

⎡ ⎛ θP ⎞ ⎧ ⎛ θ P ⎞ ⎛ 3θ P ⎞ ⎫ ⎤ ⎢cos ⎜ ⎟ ⋅ ⎨1 − sin ⎜ ⎟ sin ⎜ ⎟ ⎬⎥ 2 2 2 ⎝ ⎠ ⎩ ⎝ ⎠ ⎝ ⎠ ⎭⎥ ⎢ ⎡σ xx ⎤ KI ⎢ ⎢ ⎥ ⎛ θP ⎞ ⎧ ⎛ θ P ⎞ ⎛ 3θ P ⎞ ⎫⎥ ⎢cos ⎜ ⎟ ⋅ ⎨1 + sin ⎜ ⎟ sin ⎜ ⎬⎥ ⎟ ⎢σ yy ⎥ = 2π rp ⎢ ⎝ 2 ⎠ ⎩ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎭⎥ ⎢σ xy ⎥ ⎢ ⎥ ⎣ ⎦ PP 3 θ θ θ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ P ⎢ sin P ⋅ cos P ⋅ cos ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎥⎦ Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

28

Irwin (2) rP crack

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

θ

rp = „1.00“

point P crack -3, 0, 3, 6 x “nominal” stress Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

rp = „0.10“

σxx σyy σxy

rp = „0.01“

Which of the stresses is the most important one and why ? 29

Irwin (3)

Energy release rate G 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.

σ yy for Θ = 0 u y for Θ = π

2 Δx σ yy G = lim u y dx ∫0 Δx → 0 Δx 2

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 factor and energy release rate


K I2 G= E 30

Irwin (4) Irvin correlation

K I2 G= E

whereas is valid, too:

and if the load is constant:

combining the equations:

UM = UE =

K I2 = E

d[

G = dU M

σ 2 ⎡π ⋅ c 2 ⋅ t ⎤

⎢ 2⋅ E ⎣

σ 2 ⋅π ⋅ c2 ⋅ t 2⋅ E ⋅2 dc

dc

G energy release rate UM mechanical energy UE released energy

]

2

⎥ ⎦

and solving it:

K I = σ ⋅ c ⋅Y

Therefore, KI depends on • 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. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

31

Y-factor (1)

Y-factor Y depends on whether the flaw is intersected by the surface or is in the bulk of the component. For the surface connected flaw, the value at the centre is different from that where the flaw intersects the surface.

c

2c

2a

2a


32

Y-factor (2)

Crack (flaw) shape correction factors, Y

Attention: c and a have been changed to be in line with our definition.

CEN TS 843-6, 2004

This table should be used only as a guide if computing the stress or flaw 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 %. 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 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

33

Strength (1)

Strength of ceramics As the fracture toughness is a materials property …

K Ic = σ c ⋅ c ⋅ Y

[KIc ] = MPa

m

[σc] = MPa [c] = m [Y] = -

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

Î volume: pores, large grains, inclusions, … Î surface: cracks, pits, chipped edges, machining marks, … … is the (failure) strength “only” a characteristic of a sample, part, or component !

σc = Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

K Ic c ⋅Y 34

Strength (2)

Critical crack (defect) size With the critical stress intensity factor and the stress at failure it is possible to estimate the size of the defect which was responsible for the failure of a sample or component:

Conclusion • The fracture toughness KIc (= critical stress intensity factor) indicates how well a specific material under stress is able to withstand the extension of a crack (… design relevant property). • The higher the fracture toughness, the more difficult it is for a crack to advance in a material. • The strength at failure σc is a characteristic of a component. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.1, 2009

35

Learning targets part 1

What you should know and understand, now! Why mechanical testing … • Design relevant properties Fracture toughness, strength, creep, subcritical crack growth, … (technological tests = comparison of materials under exactly defined conditions) • Ceramics miss plasticity elasticity: ~ 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.

σmax stress at crack tip (max. stress) σ0 nominal stress ρt radius of curvature at crack tip a crack size • 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.

⎛ a ⎞⎟ ⎜ σ max= σ 0 1 + 2 ⎜ ρ ⎟⎠ ⎝


36


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. • Failure occurs in brittle materials if theoretical strength < defect related stress peak. σ, c applied stress, depth of crack σ ⋅ π ⋅ c ≥ 2 ⋅ γ ⋅ E 2, γ surfaces created, intrinsic surface energy of material E 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. 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. KIc ,σc fracture toughness, critical applied stress K Ic = σ c ⋅ c ⋅ Y c, Y depth of crack,Y-factor


37




Jakob Kübler & Prof. L.J. Gauckler Empa, Science & Technology Lab for High Performance Ceramics Überlandstrasse 129, CH-8600 Dübendorf +41-44-823 4223 [email protected]


1


Repetition learning targets part 1

What you already know and understand! • Design relevant mechanical properties (≠ properties by technological tests) Fracture toughness, strength, creep, subcritical crack growth, … • All materials exhibit a natural defect population due 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.

⎛ ⎜ ⎝

σ max= σ 0 ⎜1 + 2

a ⎞⎟ ρ ⎟⎠

σmax σ0 ρt a

stress at crack tip nominal stress radius of curvature at crack tip p ½ crack width

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


2

1

Repetition learning targets part 1 • Griffith’s basic idea: Balance energy consumed in forming new surfaces as crack propagates against elastic energy released.

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

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

σ ⋅ π ⋅c ≥ 2⋅γ ⋅ E

σ, c applied stress, depth of crack 2, γ surfaces created, intrinsic surface energy of material E Young’s modulus

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

K IcI = σ c ⋅ c ⋅ Y

KIc ,σc c, Y

fracture toughness, critical applied stress d th off crack, depth k Y-factor Yf 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. 3


11. 12. 13. 14. 15.


learning targets 1


“I “Improving i toughness t h …”” “Knowing what you measure …” “Just a value …”

learning g targets 2

R R-curve Properties Strength


learning targets 3

part 2 Strengtth

5. 5 6. 7.

part 3 Statistics



learning targets 4

part 1 Crack tip

1. 2. 3. 4.

part 4 Time&Temp



4

2

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. versa

σ

σ 2c→∞

2c a

a

σ

σ

5


R-curve behavior Increasing resistance against crack propagation KIR

KIR

KIR= KIc

Why this increase ?

Crack growth Δa

Crack extension isn’t characterized by a constant KIc anymore but by a KIR - Δa curve. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

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 √m polycrystal ~ 4 MPa √m 6

3

R-curve behavior (3)

Why is KIR increasing? a) Crack deflection at grain boundaries Polycrystalline material: as crack deflects along weak grain boundaries, Ktip is reduced, because stress is no longer normal to crack plan

(

crack plan

)

K tip = cos 3 θ2 K app Barsoum, p380

assuming Θavg = 45° → Ktip ~ 1.25 x single-crystal value crack deflection accounts for some of the enhanced toughness, but not all

7



Why is KIR increasing? b) Crack bridging 1 deflection of crack front along / around rod-shaped particles

g g results from bridging g g of Toughening the crack surfaces behind crack tip by a strong reinforcing phase e.g. undeflected crack front ligament bridging mechanism with no interfacial debonding


• • • •

elongated grains continuous fibers whiskers particles (metal …)

Bridging ligaments generate closure f forces on crackk face f which hi h reduce d Ktip.

8

4

R-curve behavior (5) b) Crack bridging 3: example SiC whiskers in - glass - mullite - alumina lines: prediction points: experiments

What’s Mullite ? 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.

Si3N4 F. Monteverde, A. Bellosi, S. Guicciardi, © ISTEC-CNR

Crack bridging and pullout can yield substantially increased fracture toughness. 9


R-curve behavior (6) b) Crack bridging 2: amount of increase Fracture toughness of a “composite” due to elastic stretching of a partially debonded reinforcing phase at crack tip with no interfacial friction:

⎛ r ⋅ V f Ec γ f ⎞ K IcI = Ec ⋅ Gm + σ 2f ⎜ ⋅ ⋅ ⎟ ⎜ 12 E f γ i ⎟ ⎝ ⎠ where: c, m, f , i E, V, r σ, G γf/γi

P. Becher, J.Am.Ceram.Soc., 74:255-269 (1991)

composite, matrix, reinforcement, interface Young’s modulus, volume fraction, radius of bridging ligament strength of reinforcement phase, toughness of unreinforced ligament ratio of fracture energy of the bridging ligaments to that of the reinforcement/matrix interface

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)


10

5

Data for calculation m: matrix ; f : whisker Em KIc Al2O3 Ew σw df lf whisker direction interface γf / γi

R-curve behavior (7) b) Crack bridging 4: amount of increase

400 GPa 3 MPa √m 580 GPa 8’400 MPa 1 μm 10 μm random-3D 1, 25, 125

Al2O3 & SiC-whisker composite

composiite KIc [MPa √ m]

(1 = super strong)

10 8 'super strong' interface 'strong' interface 'weak' interface

6 4 2 0 0.0


0.1

0.2

0.3

0.4

volume fraction of SiC-whisker

11

R-curve behavior (8) c) Transformation toughening … if tetragonal particles are fine enough, then upon cooling from Tprocess , they can be constrained from transformation by surrounding matrix.

original i i l metastable tetragonal zirconia particle

martensitically transformed zirconia 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.


12

6


t → m in pure undoped ZrO2 during cooling is a reversible martensitic transformation, associated with a volume change (4–5%). Dopants (yttria, ceria, magnesia, i calcia l i etc.) t ) are usually ll added dd d tto stabilize t bili the th hi high h ttemperature t t and/or d/ c-phase in the sintered microstructure. Surface grinding induces the martensitic transformation, which in turn creates compressive surface layers and a concomitant increase in strength. Matensitically transformed zirconia particle

Metastable tetragonal zirconia particle 13


R-curve behavior (10) Shielding factor Ks

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

… if constrain t i iis llost, t ttransformation f ti iis iinduced d d ((volume l expansion i ~4% 4% → shear h 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

E Vf εT w Δa

Young’s modulus volume fraction of transformable phase transformation strain width of zone with transformed phase length of crack inside transformed zone

(Δa/w = ∞ → κ = -0.215 depends on shape of zone ahead of crack tip)

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


14

7

R-curve behavior (11) Toughened zirconia-containing ceramics PSZ: partially stabilized zirconia Cubic 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.

TZP: tetragonal zirconia polycristal 100% tetragonal phase and small amounts of yttria and other rare-earth additives. σb up to 2’000 MPa.

ZTC: zirconia-toughened ceramic Tetragonal or monoclinic zirconia particles finely dispersed in other ceramic matrices such as alumina, mullite, and spinel. 15


R-curve behavior (12) Increasing resistance against crack propagation by design, e.g.

compressive residual stresses in laminates Residual compressive stresses reduce actual stress in outer layer

L1

σLayer = σLoad - σCRes

L2

KIc = ( σ−|σC| ) • √a • Y

L1

σ+

σ-

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. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

16

8

R-curve behavior (13) Laminates (2): CTE mismatch to introduce residual stresses

CTE 10-6/ oC

6

4

2

+ X%TiN TiN Si3Si3N4 N4+30% Si3N4

Si3N4

0 0

200

400 600 Temperature °C C

800

1000

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

17


R-curve behavior (14) Laminates (3): Design

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

Si3N4

Si3N4 Si3N4 + 30 % TiN

150 μm 600 μm

1 mm

Si3N4 +TiN 10 µm

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


18

9

R-curve behavior (15) Laminates (4): Apparent Fracture Toughness = toughness you will measure but isn’t solely material related

a 0.5mm

Si3N4

Si3N4+TiN

Notch length a [mm]

• 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. 19


R-curve behavior (16) Laminates (5): improved design

Micro-layered laminates (with external tensile layers)


20

10

R-curve behavior (17) Laminates (6): further improved design

Micro-layered laminates (with external compressive tensile layers)

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

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

Kuebler J., et.al., KEM 333 (2007) 117-126 21


Properties (FT1)

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

SEPB

SEVNB

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). Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

22

11

Properties (FT2)

F

SEVNB (and SEPB, SENB) Single Edge V-Notched Beam (and Single Edge Precracked Beam, Single Edge Notched Beam)

(

)

K Ic = σ ⋅ a ⋅ Y =

Fmax S1 − S 2 3 ⋅ ΓM a W 2(1 − α ) 3 / 2

B W

with ΓM = 1.9887 − 1.326α −

(3.49 − 0.68α + 1.35α 2 )α (1 − α ) (1 + α ) 2

and

S2 W

α = aW

a S1

NC Si3N4

5 μm

23


Properties (FT3)

CN Chevron Notched Beam

F

not valid

d

K Ic=

Fmax B⋅ W

Ym'

S1 S2 ⎫ 2 ⎧ Y' m = ( 3.08 + 5.00a 0 + 8.33 a0 ) ⎨ 1 + 0.007 ----------2 ⎬ W ⎭ ⎩ Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

24

12

Properties (FT4)

SCF Surface Crack in Flexure

1 Knoop hardness indent 2 polished surface

a 2c improve visibility

Ymax : larger of Ys and Yd

25


Properties (FT5) Fracture toughness values measured with various test methods in comparison with SEVNB values

Why those differences? other methods, indiv. avg. SEVNB; G.P.Avg. G P Avg SEVNB; G.P.Std.Dev.

Alumina-999 6.0

H2O SEVNB-H (1 1/5)

SEVNB-N N2 (1/5)

SCF (10/5 5)

SCF (9/4))

SCF+halo o (10/5) 10/5) SCF-N2 (1

0.0

CN (8/5)

2.0

SEPB (26 6/5)

4.0 SEPB (25 5/4)

Frac. T Toughness [MPa √m]

8.0

Method (Participant / Number of Specimens) J. Kübler, ASTM STP 1409, 2002 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

26

13

Properties (FT6)

Development of Vickers indentation cracks

Vickers - IF Indentation Fracture

E K I c = 0.032H a⎛ ----⎞ ⎝H⎠

1 --2

⎛ c- ⎞ ⎝a ⎠

1 -2

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

F

27


Properties (FT7) Example: Micro-Hardness (… plasticity …)

plastic deformation !!

Si3N4 – 05

Optical Glass BK7, HV-1N Scanning Probe Microscope


28

14

Strength (1)

Determination of design relevant strength properties

☺ Fracture toughness



Strength


static

dynamic

Creep


Relation between crack growth speed and stress intensity factor.

KIc 29


Strength (2) Strength of ceramics; evolution


30

15

Strength (3)

Bend test Advantage • simple (fixation of sample, simulation of environment, …) • cheap (sample, (sample jig jig, …)) • universal (strength, fracture toughness, Young’s modulus, fatigue, …) • sensitive to surface defects Disadvantage • small volume tested • stress t gradient di t therefore th f nott valid by plastic deformation

σ B=

MB W 31


Strength (4)

3- vs. 4-pt-bending


32

16

Strength (5) Measurement uncertainty Example: 4-Pt-bending strength @ elevated temperature

σ (P, b, h, d ) :=

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 %

3⋅ P ⋅ d b ⋅ h2

((calculated from tolerances of test jjig) g)

2

Δy =

m ⎞ ⎛ d f ( x1... xn ) ⎟⎟ ⋅ Δxi + ∑ Δe j ∑ ⎜⎜ n =1 dxi m =1 ⎠ ⎝ n

Relevant factors, e.g.:

• σf ≥ 100 MPa / ≥1’000 MPa Δe1 > ± 2.7 % / < ± 0.3 % • Δl jig (T related) Δe2 ~ ± 3.0 % • chem. h reaction ti @ surface f Δ 3 ~ ± 5.0 Δe 50%

Relative measurement uncertainty: Δσ% ≤ 1.2 12%

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 ≥ 1000 MPa

± 1.2 % + 2.7 % + 3.0 % + 5.0 % = 11.9 % ± 1.2 % + 0.3 % + 3.0 % + 5.0 % = 9.5 % 33


Strength (6)

Scatter of mechanical strength f(σc)

Dispersion density of the strength measured on a series of components ∞

σc

σc1 σc2

∫ f (σ c ) dσ c = 1

σc

0

σ c2

P(σ c1 < σ c < σ c 2 ) = ∫ f (σ c ) dσ c σ c1

σc

F(σ c ) = ∫ f (σ c ) dσ c 0 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

34

17

Strength (7) Dispersion of the largest (failure relevant) defects and failure strengths

h(a)

1-F(σ ( c)

f(σc) H(a)

σc

a … large “largest” defect → low strength … … small “largest” defect → high strength … Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

35

Strength (8)

Strength of a

ceramic component …

largest, but not failure relevant defect … 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

σ component = f (σ , K Ic , ca,, σ ↑, ac⊕)

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

↑ direction ⊕ position

… therefore it’s difficult to predict the strength of a Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

failure relevant defect

component ! 36

18

Strength (9)

Sources for defects … … during fabrication of component: • always: powder agglomerates • friction when pressing • powder sedimentation when casting (slurry) • always: cracks and pores from sintering … during usage of component: • corrosion, pitting • subcritical crack growth, creep • friction, scratches • stress peaks (impact, …) (in ductile materials materials, e.g. e g in fcc-metals 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 37


Strength (10) Toughness ↔ defect size ↔ strength

~ 1 : 10’000 (!!) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

38

19

Strength (11) 1) increase σc by reducing ac, e.g. by improved processing

Two strategies to improve σc and KIc

2) increase KIc by increasing fracture energy, e.g. by crack bridging, transformation t toughening h i

100’000

1 log σ c = − ⋅ log ac + (log K Ic − log Y ) 2

σc (MPa)

10’000

1’000 1 000

1

10

100

1’000

critical defect size (μm) 39


Strength (12)

Summary • in ceramics strength controlling defects have a size of a few μm up to a few 100 μm • failure relevant is the largest volume or surface defect under stress

• identical components will not fail at one reproducible strength value (= strength value distribution) • When is the density of defects small enough so that we can be absolutely sure that no defect with a critical size is present ?

den nsity of defects

• ceramic materials don’t have a single strength value

Never … defect size

Statistical data is needed! The strength of ceramics is described by the Weibull statistics - see part 3. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.2, 2010

40

20


What you should know and understand, now! 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 … • 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) 41



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 - critical i i l stress iintensity i ffactor - 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


42

21

08.02.2010






1



What you already know and understand! • Fracture toughness can be enhanced by increasing energy required to extend crack. • C Ceramics i with ith R-curve R b h i behavior: - degradation in strength with increasing flaw size is less severe - reliability increases • Crack deflection, crack bridging, martensitic transformation are mechanisms that enhance KIc app. • Fracture toughness values measured with different test methods may differ. • Bend test:

- universal (e.g. strength, fracture toughness) - sensitive to surface defects - only small volume tested - value σ3Pt test > value σ4PT test - specimen sees stress gradient


2

1

08.02.2010

Repetition learning targets part 2 • All components have defects due to fabrication and usage. They have a size from a few μm up to a few 100 μm • Strength of a 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 • 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 b iincreasing i ffracture energy, e.g. crack kb bridging id i • Strength of ceramics must be described by statistics as identical components will not fail at one reproducible strength value.

3


11. 12. 13. 14. 15.


learning targets 1


“I “Improving i toughness t h …”” “Knowing what you measure …” “Just a value …”

learning g targets 2

R R-curve Properties Strength


learning targets 3

part 2 Strengtth

5. 5 6. 7.

part 3 Statistics



learning targets 4

part 1 Crack tip

1. 2. 3. 4.

part 4 Time&Temp



4

2

08.02.2010

Weibull statistic (1)

Waloddi Weibull: 1887-1979 Swedish engineer famous for his pioneering work on reliability, providing a statistical treatment of - fatigue, - strength and - lifetime in engineering design design. The widely-usable, reliable and user-friendly Weibull distribution is named after him.

Today used for many other time-dependant fault mechanisms, too. 5



Model of the chain with the weakest link (1) • a chain is only as strong as it‘s weakest link • if the strength of the links is distributed evenly then the probability of survival of a chain with length L is defined as :

Ps(L)

σ0

1

m=1

Pf

σ

0

1

m=∞ Pf 0


12

6

08.02.2010

• “tough” ceramic components: m = 10-40

• small m: wide distribution, large spread Î unreliable material • “bad” ceramic components: m = 1-10

1 large m “reliable” reliable

0 stress

1

probability of faillure

• large m: narrow distribution, small spread Î reliable material

probability of failure


small m “unreliable”

0 stress 13


Weibull statistic (10) Calculation of m und σ0 with defined volume, e.g. test bar:

⎛ ⎛ σ ⎞m ⎞ Pf = 1 − expp⎜ − ⎜⎜ ⎟⎟ ⎟ ⎜ ⎝σo ⎠ ⎟ ⎝ ⎠

make a graph with the left term on the y-axis and the right term as x-axis and insert measured values · slope of straight line Æ m · ln (ln(1/(1-Pf))) = 0 Æ σ0 = 0.632

rearrange and take twice the logarithm:

⎛ ⎛ 1 y = ln ⎜ ln⎜ ⎜ ⎜ 1 − Pf ⎝ ⎝

⎞⎞ ⎟⎟ ⎟⎟ ⎠⎠

0

m

y = m⋅x +C σ0 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

x = ln σ 14

7

08.02.2010


Determination of Weibull-parameter • • • •

conduct measurements classify failures (>30 per class) rank results assign relative frequency P fi =

n − 0 .5 or N

n N +1

63.2% combined confidence intervals

• draw Weibull-diagram • calculate σ 0 and m • calculate confidence intervals

15



Test N No.

Failurre strength (MPa)

Rankk

Failurre strength (MPa) Failure probability (Pf)

Example: ground glass rods

1

178

1

178 0.1

2

276

2

210 0.2

3

262

3

235 0.3

4

296

4

248 0.4

5

210

5

262 0.5 05

6

248

6

276 0.6

7

235

7

296 0.7

8

318

8

318 0.8

9

345

9

345 0.9

⎛ ⎛ 1 y = ln ⎜ ln⎜ ⎜ ⎜ 1 − Pf ⎝ ⎝

m=4.76

σ0 Pf =


⎞⎞ ⎟⎟ ⎟⎟ ⎠⎠

n N +1

σ 0 = exp(

C ) = 285 MPa m 16

8

08.02.2010


Confidence Interval Interval for which it can be stated with a given confidence level that it contains at least a specified portion of the population of results (= measure of uncertainty of parameters).

“Cook book” • •

• •

d t determine i required i d confidence fid llevel,l 1 - α (common practice: 90 % → α = 0.1) for a given number of test-pieces N → upper confidence interval limit factor tu @ α/2 → lower confidence interval limit factor tl @ (1 - α/2) tu and tl are determined from tables (e.g. EN 843-5) upper & lower values of σˆ 0 :

upper limits of confidence interval:

⎛ t ⎞ Cu = σˆ 0 exp⎜ − u ⎟ ⎝ mˆ ⎠

lower limits of confidence interval:

⎛ t ⎞ Cl = σˆ 0 exp⎜ − l ⎟ ⎝ mˆ ⎠

^ = maximum likelihood estimate of Weibull characteristic strength of test piece

“Cook book” for confidence interval for m is identical 17


P


• used next to the “naturally” “ present defects (mainly volume) a 2nd defect population (surface) leads to lower failure stresses


probability of failure Pf

• typical value: m = 10 • probability of failure for components can be calculated without knowing defect density • weakest components (or sample) determine widely the slope of Weibull line • statistically relevant Weibull parameters require ≥ 30 experimentally measured values

tensile failure stress of glass fibres

used

new

failure stress

18

9

08.02.2010

Proof testing (1) • proof test stress lower than the design stress and higher than the expected stress in use is applied to components • this will eliminate “bad” components (samples) • the lower end of the distribution is therefore cut off and the new distribution isn’t a proper Weibull distribution anymore

probability of failure Pf

… assuring that no component fails while in use …

before proof test after proof test

Proof test stress

failure stress

Components should be used after proof testing @ σ < σP it is possible to calculate the failure rate if σ < σP but there is %-wise only a small improvement in failure 19


Failure strength (MP Pa) Failure probability Pf = n/(N+1)

1

178 0.1

2

210 0.2

3

235 0.3

4

248 0.4

5

262 0.5

6

276 0.6

7

296 0.7

8

318 0.8

9

345 0.9

Proof stress 240 MPa P

Rank

Proof testing (2)


G(σc) is not a Weibull distribution.

20

10

08.02.2010

Proof testing (3) distribution of stress for F and G after proof testing G

What does it mean ? ln σP f(σc) g(σc)

⎛ ⎛σ F (σ c ) = 1 − exp⎜ − ⎜⎜ c ⎜ σo ⎝ ⎝

⎞ ⎟ ⎟ ⎠

⎛ ⎛σ G (σ c ) = 1 − exp⎜ − ⎜⎜ c ⎜ σo ⎝ ⎝

⎞ ⎛σ ⎞ ⎟ +⎜ p ⎟ ⎟ ⎜σ ⎟ ⎠ ⎝ o⎠

m

m

⎞ ⎟ ⎟ ⎠ m

⎞ ⎟ ⎟ ⎠

σc σP 21



Influence of volume … and

⎛ ⎛ σ ⎞m V ⎞ P2 S = exp⎜ − ⎜⎜ 2 ⎟⎟ 2 ⎟ ⎜ ⎝ σ 0 ⎠ V0 ⎟ ⎝ ⎠

… the influence of volume V2 for a stress σ2 can be calculated from volume V1 and stress σ1 if m is know: 1

Pf )

⎛ ⎛ σ ⎞m V ⎞ P1S = exp⎜ − ⎜⎜ 1 ⎟⎟ 1 ⎟ ⎜ ⎝ σ 0 ⎠ V0 ⎟ ⎝ ⎠

σ 1 ⎛ V2 ⎞ m =⎜ ⎟ σ 2 ⎝ V1 ⎠

example: 3 x 4 x 45 mm bend bar: 3-pt BT V2 eff ≈ 1 mm3 4-pt BT - 40 / 20 mm V1 eff ≈ 11 mm3 tensile - 3x4x20 mm V3 eff ≈ 240 mm3 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

→ 1.27 x σ1 → σ1 → 0.73 x σ1 22

11

08.02.2010


Influence of tested volume or surface

23


Fractography (1) … why ? “Improvement through feedback.” (… cause of failure) Question (reason)

Answer for … (application)

Where did it break from?

Engineering

Did it crack k suddenly dd l or slowly? l l ?

E i Engineering i

Why did it break form here?

QA, process monitoring

Nature of fracture source?

Material development, QA

Stress at fracture?

Design

Environment or fatigue?

Engineering

Good test?

Material evaluation

Whose fault?

Commercial, legal

… skill seldom taught academically – poor ability to interpret reasons for failure – leads to negative impression of value of ceramic components (liability !) – leads to wrong conclusions concerning causes of failure (materials versus manner of use/abuse) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

24

12

08.02.2010

Fractography (2)

roman temple rebuild after earth quake

shell like chip

example glass

Garni, Armenia © Kübler, 2003

Am. Ceram. Soc. Bulletin Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

25

Fractography (3)

… old but not well known science First mention of ceramic fractures by E. Bourry in: A treatise on ceramic industries (first English editon 1901)

“… observation of the structure or homogeneity should consist of the examination of a fracture, either by the naked eye or by a magnifying glass.“ “… it will be advisable to note: (a) appearance of the fracture, whether granulated, rough or smooth, or with a conchoidal surface. (b) size of the grains……. (c) homogeneity…, whether there are any planes of cleavage or scaling, and whether these are numerous and pronounced”

Guide for hobby astronomer … • .. get familiar with the firmament simply by the naked eye and a map .. • .. observe satellites and stars with a simple field glass .. • .. locate and enjoy details of far away stars and galaxies with a telescope .. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

26

13

08.02.2010

Fractography (4)

Flow diagram Objective:

Stage 1:

Action:

Location of origin

Deduction:

Collect and clean fragments

History of fracture

Visual inspection p

Primary fracture face

Binocular macroscope inspection

Identify features and locate origin

Result:

increasing level of information

Tentative classification of origin More?

N

Y

Stage 2:

SEM inspection. O i i size, Origin i fracture mechanics

M h i l Mechanical nature of origin

Mechanical circumstances of fracture More?

N

Y

Stage 3:

EDX analysis. Origin chemical inhomogenity

Chemical nature of origin

Chemical causes of failure

Overall conclusions

Report

27


Fractography (5)

Bifurcated compression curl

s

s

s

s

s

Medium stored energy test piece; primary fracture in centre with compression curl; secondary fractures caused by impact between test piece and jig parts

Fracture patterns in four-point flexural strength test pieces

High stored energy fracture with ith m multiple ltiple cracking near the origin; cracks bifurcate shortly after initiation; fracture origin may be lost in fragmentation

“low energy” e.g. porcelain

Low to medium stored energy fracture; primary failure close to loading rod; secondary break due to impact with jig parts

“high energy” e.g. silicon nitride

Low to medium energy fracture outside the loading span; usually due to larger than normal fracture origin

“medium energy” e.g. fine grained alumina

s : secondary failure • often due to shock wave • 5% increased load @ roller • “long” fracture piece hits jig

Four-point bend test piece, tensile face on lower side


28

14

08.02.2010

Fractography (6)

Fracture patterns in ring-on-ring test pieces 2

1 likely origin zone 2 primary crack 29


Fractography (7)

Macro-features in flexural test bars

Origin inside body

Origin at or close to surface

Ridge and compressive curl Hackle

Mist (when visible)

Mirror

Origin inside, but to one side Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

Origin 30

15

08.02.2010

Fractography (8)

Microscopic: ‘fracture lines’ – fine hackle Features: near fracture origins

Fracture lines from an extended origin such as a machining flaw

Fracture lines from a pore associated with an agglomerate

Twist due to two parts of crack meeting

Fracture lines from a large surface connected pore

Fracture initiating from both sides of origin in different planes and joining

31


Fractography (9)

Example 1: High purity alumina bend bar 2

1

1

3

6

4

2

1 mm

Optical fractography showing: 1 matched fracture surfaces of a flexural strength test bar 2 mirror region 3 compression side marked by compression curl 4 hackel (appears laterally only) 5 large internal pore 6 tail (wake hackle) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

5

0,5 mm

tensile surfaces together

32

16

08.02.2010

Fractography (10)

Example 2: Failure from agglomerate intersected by machining the surface 4

5

2

1

3

1

0 2 mm 0,2

1 tensile surface 2 directions of failure

10 µm

4 extended void 5 agglomerate

(“rising sun” – use light to illuminate topography)

3 origin region

33


Fractography (11)

Example 3: Chemical inhomogeneity in silicon nitride

(a)

(b)

Macro:

Meso: 1

1 0,5 mm

50 µm 3

2 (c)

(d)

Mi Micro:

BS: 1

1 10 µm

1 tensile surface 2 secondary electron image Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

4

10 µm

3 backscattered electron image 4 high ytterbium (= sintering additive) concentration around pore (EDX)

34

17

08.02.2010

Fractography (12)

Example 4: Fracture toughness calculated with natural flaw

Al2O3 σC = 292 MPa 2a ~ 2c ~ 160 µm Æc/a ~ 1 ÆY ~ 1.13 .. go and calculate KIc !!!

KIc ~ 3.0 MPa √m KIc measured in VAMAS / ESIS round robin 3.6 MPa √m

possible reasons: → effective elliptical flaw size is larger … → granulate / effective defect isn’t a sharp crack …

Küb @ Fractography of Glasses and Ceramics III Alfred University, NY, USA, 29.061995 35


Fractography (13) Macro

Example 5: TZP bend bar failing from large pore 0,1 mm

0,5 mm

Meso

σc = 728 MPa

a ~ 35 µm 2c ~ 140 µm

c/a ~ 2 Ycent = 1.59 > Ysurf = 1.24

Æ KIc ~ 6.8 MPa√m

Measured: KIc = 4.7 MPa√m

possible reasons: → effective elliptical flaw size is smaller … → pore isn’t a sharp crack … Kübler Empa-HPC, ETHZ MW-II Ceramics-6.3, 2010

0,05 mm

Micro Küb @ FAC 2001, Stara Lesná, Slovakiá 36

18

08.02.2010


What you should know and understand, now! Weibull, a name you’ll never should forget … • Weibull: mathematical description of failure / survival probability

⎛ ⎛ σ −σ c Pf = 1 − PS = 1 − exp⎜ − ⎜⎜ ⎜ σo ⎝ ⎝

• Weibull parameter m describes the width of the distribution: - small m = large distribution - large m = small distribution

m ⎞ V ⎞⎟ ⎟ ⎟ V ⎟ ⎠ 0⎠

f(σc) σc

• If you talk from “characteristic strength” σ0 already 2/3 of your components failed! • The effect of volume and/or surface area on the acceptable stress t level l l can b be calculated. l l t d (If you wantt tto hid hide the th poor quality of your material use 3-point bend test to get failure stress values.)

1

σ 1 ⎛ V2 ⎞ m =⎜ ⎟ σ 2 ⎝ V1 ⎠

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

37



Reading fracture surfaces … • Increasing the level of information of a fracture by starting from its history. • Fracture patterns will lead you to the origin zone. • Macro- and micro-features point towards the origin. • Fracture mechanics and fractography combined are strong tools to - develop materials - optimize procedures and processes - construct components - improve machining - design systems

Guide for fractographer … • .. get familiar with the failure and its “environment” simply by the naked eye and a map .. • .. observe large markings and features with a simple optical microscope .. • .. locate and understand small details with a SEM ..


38

19







1


What you already know and understand! • Weibull: mathematical description of failure / survival probability • Weibull parameters: - σ0 = strength @ 63% probability of failure - small m = large distribution large m = small distribution • The effect of volume and surface area on acceptable stress level can be calculated.

⎛ ⎛ σ −σ c Pf = 1 − PS = 1 − exp⎜ − ⎜⎜ ⎜ σo ⎝ ⎝

m ⎞ V ⎞⎟ ⎟ ⎟ V ⎟ ⎠ 0⎠

f(σc) σc

σ 1 ⎛ V2 ⎞ =⎜ ⎟ σ 2 ⎝ V1 ⎠

1 m

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


2


What you already know and understand! Reading fracture surfaces … • Increasing level of information of a fracture by starting from its history. • 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

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 .. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

3

11. 12. 13. 14. 15.


learning targets 1


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

learning targets 2

R-curve Properties Strength


learning targets 3

part 2 Strength

5. 6. 7.

part 3 Statistics



learning targets 4

part 1 Crack tip

1. 2. 3. 4.

part 4 Time&Temp



4

Thermal shock (1) 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).

α : coefficient of thermal expansion

critical for “bad heat conductors”

thermally induced elongation

e.g.: αSi3N4 = 3·10-6 [K-1]

ε thermal

Δlthermal = = α ⋅ (T1 − T0 ) = α ⋅ ΔT l0

dimensionally fixed components:

ε thermal + ε elastic = 0 if ε thermal > ε elastic ⎯ ⎯→ failure

Stresses:

no stress will develop if component isn’t fixed dimensionally

σ thermal = E ⋅ ε thermal = E ⋅ ε elastic = E ⋅ α ⋅ (T1 − T0 )


… keep in mind the the size effect: - small components = small σthermal - large components = large σthermal

5

Thermal shock (2) 1. heat sample up to temperature T1 2. quench sample down to temperature T0 (≠ RT, 0°C, …..)

Strength (MPa)

3. measure failure strength of sample

micro-cracks were produced @ ΔT = 300 K (or more) additional damage produced @ ΔT > 650 K

ΔT = T1 – T0 [K] Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

6

Thermal shock (3)

Thermal shock resistance R RS

onal i s n e m 3-di

(1)

:= sensibility of a component against a rapid change of temperature (the lower RS the larger the thermal shock sensitivity of a component)

for an infinite large heat transfer:

σ c ⋅ (1 −ν ) RS = α ⋅E

ν = Poisson’s ratio

This equation shows that RS is noting else than the acceptable temperature difference in K(elvin)

low α high KIc and low E


= high RS (= good thermal shock resistance)

7

Thermal shock (4)

Thermal shock resistance R’ and R”

(2)

for constant heat transfer the coefficient thermal conductivity λ of the material is added:

λ ⋅ σ c ⋅ (1 − ν ) R' S = = λ ⋅ RS α ⋅E

if the surface is heated by a constant rate the density ρ and the heat capacity Cp of the material are added:

λ ⋅ σ c ⋅ (1 − ν ) λ ⋅ RS R'S R' 'S = = = α ⋅ E ⋅ ρ ⋅ Cp ρ ⋅ Cp ρ ⋅ Cp high thermal conductivity λ

high Rs’

low density ρ plus low heat capacity Cp

high Rs’’


8

Subcritical crack growth (1)




☺ Strength



static

dynamic

Creep



KIc Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

σ, σ0, m 9

Subcritical crack growth (2) "... after several years of service, sudden transverse rupture 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) = time dependent failure = limited life time Source: Fractography of Glasses and Ceramics III, Woodtli et al, p 260, 1996 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

10


F

v=da/dt

2ci ai ac

F 2cc

experimental finding:


ai

crack size

ac

da da n v= = A⋅ KI dt 11


log v crack velocity “100%” vacuum

I n

thre


II

technical v acuum !!!

env iron me nt

III

da v= = f (K I ) dt

KIc

log KI stress intensity

sho ld

12


Si

O

Si

Si O

H3 O+

O

H

O H H

H

O H H

OHOH

Environment: H2O diffusion in crack and reaction at crack tip Si-O-Si- + H2O → -Si-O-H + -Si-O-H Water induced brake up of bonds at crack tip in soda-lime glass.

Mass transport processes inside crack moving of free molecules (gas) mass flow (viscose liquid) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

activated diffusion adsorption reaction

diffusion (liquid) 13


log v crack velocity [m/s]

ν = A⋅ K In • same material • same defect (same “a”) • same stress • different environment (= different n and A)

10-6

different crack velocity = different lifetime

10-9

log KI KI Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

stress intensity 14

Lifetime under static load σ evaluate n if lifetime at different stress levels are measured

−

lg σ Level of σ above which instantaneous failure occurs

1 n

σc

rupture after scg (= time delayed failure)

σ

tB-trans

lg tB

tB

[

n−2 lifetime in tests with t = Bσ c 1 − (σ / σ ) n − 2 B c σ(t) = const. σn

for n>10 equation simplifies to


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

scg

tB =

B ⋅ σ cn − 2

σn

]

with B = f ( n, A, Y , K Ic )

with

B=

2 2−n K Ic A ⋅ Y 2 ( n − 2) 15

. Lifetime with load ramps σ evaluate n if strengths at different stress rates are measured

lg σB 1 1+ n

•

lgσ

and in its logarithmic form


log σ B =

[

1 1 log σ& + log (n + 1) Bσ cn − 2 n +1 n +1

]

16

Lifetime under cyclic load (1)

Stress and stress intensity factor under cyclic load

σ (t ) = σ m + σ a ⋅ f (t )


17


Correlation between static and cyclic load time till failure under static load

t f − static = B ⋅ σ c

n−2

⋅σ

−n

time till failure under cyclic load

t f − cyclic =

1 g ( n, σ a / σ m )

B ⋅σ c

n−2

⋅σ m

−n

n

⎤ 1 ⎡ σa f (t )⎥ dt = ∫ ⎢1 + T 0⎣ σm ⎦ T

=

t f − cyclic

⎛ σ ⎜ = g (n, σ a / σ m ) ⎜⎝ σ m 1

n

⎞ ⎟ t f − static ⎟ ⎠ X


18


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

Porcelain

pred ictio n

• lifetime under cyclic loading usually shorter than under static loading • lifetime is lowest for R=1 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

tcy

cl

from

tsta

t

Why lower than prediction? …t

m transformation of zirconia 19

Strength-Probability-Time diagram Link between: strength & probability & lifetime example material: m = 15, n = 40 and B . σcn-2 = 107 (… ~alumina) 6.40 ln (statische Last in MPa)

540 MPa

6.20 6.00 345 MPa

5.80 5.60

235 MPa

5.40 5.20

1 Min

1 Stunde

1 Woche

1 Jahr

5.00 0

2

4

6

8 10 12 14 ln (Lebensdauer in s)

16

18

20

lald-fik.wb1:graph-sb

Ausfallw. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

0.99

0.90

0.50

0.10

0.01 20

Creep (1)




☺

☺ Strength



static

dynamic

Creep



KIc Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

σ, σ0, m

n 21

Creep (2) Simple model by Nabarro-Herring and Coble for diffusional creep under load.

Free surfaces and grain boundaries work as source and assembly point for voids and atoms. The concentration of voids near surfaces under tension (C+) is bigger than in the core (C0) → voids diffuse from surfaces under tension to surfaces under compression → matter flows in reverse direction Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

22

Creep (3) Ashby and Verall model for grain boundary sliding (1973) F

F

F

F

In this 2-D model with four hexagonal grains a structure is elongated by twisting and shifting the grains (without deforming them). Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

23

Creep (4) Typical creep curve for a specific material at a defined temperature and load

ε

ε = ε 0 + εp + ε s + ε t

εt εs εp εο


t

εo elastic strain εp primary creep εs secondary creep (stationary creep) εt tertiary creep

24

Creep (5)

Why creep should be measured in tension.

Time-dependent stress distribution in a bend bar.


25

Creep (6)

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


26

Creep (7) Time [h] 0

50

100

150

200

250

300

350

1000

450

500 4.0

900

NGK SN-88

3.6

800

(Specimen # 4)

3.2

700 Elongation [µm]

400

600

2.8

200 MPa, 1375 °C

2.4

500

2.0

400

1.6

300

1.2

D2A D2B

200 100 0

Elongation [%]

-50

0.8 0.4 0.0

-100 -0.4 -1.80E 0.00E+ 1.80E+ 3.60E+ 5.40E+ 7.20E+ 9.00E+ 1.08E+ 1.26E+ 1.44E+ 1.62E+ 1.80E+ +05 00 05 05 05 05 05 06 06 06 06 06 Time [s]


27

Creep (8)

stationary Strain Rate ; generalized expression εp

ε p = C ⋅ t n' A⋅ D ⋅ μ ⋅b ⎛ b ⎞ ε&s = ⎜ ⎟ k ⋅T ⎝G⎠

p

⎛σ ⎞ ⎜⎜ ⎟⎟ ⎝μ⎠

n

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

−m t f = C ⋅ ε&min Stationary creep rate doesn’t decrease with increasing T since exponential dependence of

D = D0 ⋅ exp( − on T dominates

Q ) R ⋅T

D increases faster than 1/T.


ε· s ε· min σ µ A, C D G T b k m n' n p t tf

primary strain rate (exponential or creep law by Norton) stationary strain rate minimal strain rate stress shear modulus constants diffusion coefficient grain size temperature Burgers vector Boltzmann constant exponent time exponent (≥ 1) stress exponent grain size exponent time lifetime (Monkman-Grant) 28

Creep (9)

Creep curves are depending on temperature, stress and grain size

creep

increasing load and temperature

increasing grain size

time


29

Creep (10)

Al2O3 ; grain size 4.75 µm ; ALF

Stress-Exponent n

-6

n=1.35 n=1.58

-7 -8

n=1.60

-9

-10 1.0

1.2

1.4 1.6 1.8 Log Load [MPa]

2.0

1245°C 1278°C 1329°C 1378°C

n → 1 with increasing temperature


Ln Strain Rate [1/s]

Log Strain Rate [1/s]

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

Activation Energy Q

-14

465

-16

- Q [KJ/mol]

-18

602 575 619

-20

-22 7.2E-05

631

7.4E-05

7.6E-05 7.8E-05 1/RT

12.5 MPa 20 MPa 50 MPa 75 MPa

8.0E-05

30 MPa

activation energy -Q decreases with increasing load 30

Creep (11)

Grain Size Exponent p Al2O3 - Sumitomo AKP-53; starting particle size 0.2 µm, 1329°C

sintering 1400 °C, 1.5 h


sintering 1650 °C, 2 h

31

Creep (12)

Mechanisms at creep failure

I II III IV

Crack tip is blunted by creep deformation. Afterwards crack propagates. Pores form at crack tip and coalesce with crack. Afterwards crack extends. Pores, created by creep deformation, join to a crack. * 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 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

32

Failure maps (1) Lifetime of silicon nitride calculated for an elastic stress in the outer fiber of a bend bar.


33

Failure maps (2)

First complete failure map for silicon nitride by Quinn (1986/1990). (Norton, NC 132, MgO doped) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

34


What you should know and understand, now! • 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) • Small component = small σthermal // Large component = large σthermal • 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 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

35

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. • 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 tension and not in bending. • Stain rate is increasing with - increasing load & temperature and - decreasing grain size. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.4, 2009

36

Summary: What you must know and understand! This you should have learned!

Brittle vs. tough / Failure stress / Crack resistance Stress elevation at crack tip Griffith’s law

: correlation between failure stress and critical flaw size

R-curve behavior: Process zone / Crack deflection & bridging / Transformation toughening Sub-critical crack growth Fatigue static & dynamic Weibull statistic

: Environment / Life prediction /

: Distribution of strength & Life time / m, σ0

Influence of surface area & volume on probability of survival Proof-testing Deformation & failure under load at elevated temperatures: creep / Norton law / Monkman-Grant relation Thermal shock behavior: thermally introduced stresses Those laws & equations should be known by heart!


37



Mechanical Properties of Ceramics or Mechanical Beha Behavior ior of Brittle Materials




1


What you already know and understand! • 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). • Small component = small σthermal // Large component = large σthermal • Low CTE, high KIc, and low E = high RS (= 1st thermal shock parameter) 2nd thermal shock parameter → constant heat transfer 3rd thermal shock parameter → surface heated by constant rate • Slow Crack Growth (scg): time dependent failure → limited life time • Crack velocity y is influence byy humidity y → brake up p of bonds at crack tip p 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, (static, dynamic cyclic, cyclic or a combination thereof) thereof). • Correlation between largest failure relevant defect, failure strength and lifetime. • Link between strength strength, probability of failure and lifetime: Strength-Probability-Time diagram Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

2

Repetition learning targets part 4 • Creep: Boundary (diffusion) 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. • Grain boundary sliding: Structure elongates by shifting & twisting of grains • Stages of creep: - primary - secondary (steady state) - tertiary • Creep should be measured in tension and not in bending. • St Stain i rate t is i iincreasing i with ith - increasing load & temperature and - decreasing grain size.


3

11. 12. 13. 14. 15.


learning ttargets 1

8. Statistic 8 St ti ti 9. Proof testing 10. Fractography

“Improving toughness …” “K “Knowing i what h t you measure …”” “Just a value …”

learn ning targe ets 2

R-curve P Properties ti Strength


learning targets 3

parrt 2 Stren ngth

5. 6 6. 7.

part 3 Statistics S s

Introduction “Why Why mechanical testing …” ” Stresses at a crack tip “Higher than you’d assume …” Griffith law “Conditions for failure …” KI and KIc “Stress intensit intensity & critical stress intensit intensity …” ”


learniing targetts 4

part 1 Crack tip C

1. 1 2. 3. 4 4.

part 4 Temp Time&T



4

Acknowledgements This lecture is based on / a compilation of • A.R. Studart, F. Filser, P. Kocher, H. Lüthy, L.J. Gauckler dental materials 23 (2007) 115-123 115 123 and dental materials 23 (2007) 177-185 177 185 • A.R. Studart, F. Filser, P. Kocher, L.J. Gauckler dental materials 23 (2007) 106-114 and Biomaterials 28 (2007) 2695-2705 • I. I Sailer, Sailer A A. Fehér Fehér, F F. Filser Filser, L L.J. J Gauckler Gauckler, H H. Lüthy Lüthy, Ch.H. Ch H F F. Hämmerle The International Journal of Prosthodontics, 20 (2007) 151-156 • presentation Lifetime of All-Ceramic Dental Bridges @ ETH Materials Day 2005 F 2005, F. Filser • presentation Load Bearing Capacity and Reliability of All-Ceramic Four-Unit Posterior Bridges @ CICC-4 (2005), F. Filser • lecture Lifetime Prediction of Dental Ceramics under Cyclic Fatigue @ ETHZMW-II-2007, F. Filser Special p thanks go g to • University of Zürich, Center for Dental & Oral Medicine M. Schumacher, O. Loeffel, C. Löscher • Degudent, g , Dentsply p y Ceramco,, Ivoclar and VITA Zahnfabrik for supporting pp g those studies. Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

5

Outline • Introduction - Dental bridges in time - Why ceramics - Mechanical short & longtime properties - Aim Ai & Obj Objectives ti off study t d • Production process, various • Experimental & Results - Mechanical short-time properties - Lifetime prediction - influence of water - subcritical crack growth & fatigue • Lifetime diagrams • Summary, Conclusion & Outlook


6

Introduction

D t lb Dental bridges id in i time ti Etruscan prosthesis

Metal-porcelain bridge

All-ceramic bridge

Filser , PhD thesis, ETHZ

200 BC


1960s

1999

7

Crown (frame & veneer)

Abutment Screw Jaw-bone

Implant

http://bpi-implants.com


8

Introduction

• Chemically Ch i ll iinertt • Low allergenic potential • Pink esthetic of gums - Biocompatibility - Low deposition p of p plaque q Metallic framework

• White esthetic of tooth - natural look - imitation i it ti off nature t - translucent

Ceramic framework

Why ceramics ? 5-unit Bridge in the Molar Region after3 years in service (2002) (Courtesyof A. Fehérand I. Sailer, University of Zurich) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

9

Introduction AB cross section ti

1

2

Filser et al. PhD thesis ETHZ (2001)

typical failure

situation

3

800 N = 80 kkg Î A

B

1

2

3

Ch i fforces Chewing Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

High-strength High strength materials are required i d! Î The h “f “failure l lload” d” was set to 800 N N, 900 N or 250 N / teeth depending on the time in the long-time research project a specific part has been conducted. 10

Introduction Number off failed sam mples

40

Mechanical strength g of ceramics and metals

Ceramic Metal

30 20 10

Probab bility of failu ure (%)

100

12 2

11 6

11 0

10 4

98

92

86

80

74

68

0

Strength (MPa) Ceramic

80

Metal Improved Improved ceramic ceramic

60

Strength of traditional ceramics had to be improved !

40 20 0 60

80

100 Strength (MPa)


120 11

Introduction

Fatigue and subcritical crack growth Mastication: Cyclic stresses

100 Stress on the bridge

80 60 40 20

Improved ceramic

80 60 40 20 0

0 60 300

Probabilitty of failure e (%)

Probabilitty of failure e (%)

100

+ Water

80 400

100 500 Stress (MPa)

120 600

1.E+00

1.E+10

Time

Fatigue mechanisms decrease the strength over time ! Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

12

Introduction

New materials T Toughn ness, KIC (MP Pa m1/22)

10 Advanced ceramics

ZrO2 (Y,Mg--PSZ) (Y,Mg

Glass-infiltrated Glassporous aluminas

8

6

Glass-ceramics Glass& reinforced porcelains p

Al2O3-ZrO2-Glass Inceram--Zirconia) (Inceram Al2O3-Glass (Inceram Inceram--Alumina)

Nano-ZrO2 Nano(Y--TZP) (Y

4

2

Li2O.2SiO2 (Empress 2)

Dicor MGC MK II

0

Al2O3

Conventional porcelains

0

Omega

200

Empress 1

400

600

800

1000

Strength, σc (MPa) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

13

Aim & Ojectives Systematically evaluate the lifetime under cyclic conditions for selected, representative dental ceramics.

Glass--infiltrated Glass porous aluminas

With variables effect of water and effect of coating (porcelain) Establish guidelines for materials selection, fabrication process and bridge design Framework: Inceram Zirconia (VITA) Al2O3-ZrO2-Glass

Nano--ZrO2 (Y Nano (Y--TZP)

10 10μm

Veneer: Glass A Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

Framework: Framework: Zirconia (Cercon, degudent) LiSilicate (Ivoclar, Empress II)

500 nm

Veneer: Glass Z

Li2O.2SiO2

2 μm

Veneer: Apatite 14

Introduction

Empress-2 Process

Pre-operative view shows the presence of extensive infiltrated composites.

Teeth are prepared.

The four waxed copings are sprued and then invested. invested

The frameworks after being pressed d iin the h pressing i furnace.

View of the finished frameworks k on the h master model. d l

The crowns and veneer are again placed on the master model.

IvoclarVivadent: “i, Vol. 7, Nr. 1, 2005 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

15

Introduction

Empress-2 Process

Facial view of the four restorations.

Labial view of the four restorations.

IvoclarVivadent: “i, Vol. 7, Nr. 1, 2005 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

Facial view of the four restorations following immediate operative placement

16

Introduction

InCeram-Zirconia (Alumina) Process

model made of gypsum

crown caps after f grinding i di

digitizing

crown caps fitted fitt d on the th gypsum model A. David, SpectrumInternational •IDS 2003, p. 5-7


grinding

l iinfiltration fil i glass 17

Introduction

InCeram-Zirconia (Alumina) Process

infiltrating the porous alumina

overworking the excess infiltration glass

veneering g the framework with VITADUR® ALPHA

A. David, SpectrumInternational •IDS 2003, p. 5-7 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

situation after cementation of the 3-unit bridge

18

Introduction

Process of Machining in the Soft Sintered State

D

Cercon® (Degudent) system W

M

F

S A

M C A

F. Filser et al., ETH Hochschulverlag, ISBN 3-7281-2635-7, 1998, p. 165-189 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

19

Strength

Load Bearing Experimental Setup e g 3-unit e.g. 3 unit bridges load teflon disk fframeworkk (dental bridge) steel post elastic rubber bb hose die holder

adapted from H. Lüthy et al., Dental Materials, 21 (2005) p930 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

20

Strength

Load Bearing g Capacity p y and Reliability of 4-Unit Frameworks

H. Lüthy et al. Dental Materials, 21 (2005) p930 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

21

Strength

Mean Load Bearing Capacity of 3 and 3d 4-unit 4 it Frameworks F k


22

Strength

First conclusions • Cercon Zirconia is the superior material • in-vitro load bearing capacity and reliability of Cercon Zirconia 3-unit 3 unit bridges are superior • 7 mm2 connector area is sufficient for 3-unit Cercon bridges g • 7 mm2 connector area is NOT sufficient for Cercon 4-unit bridges g • 4-unit bridges need larger connector areas, or less span width or stronger materials

And what about lifetime ? Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

23

Theory Lifetime

Subcritical crack growth σc< σc σc = Zr

ac/2

Y ac

O

ac/2

ai/2

How fast does the crack propagate t under d subcritical conditions ?

K IC

Griffith‘s law

Zr

σσc < σc

v = f (K I )

?

Flaw size (μm)) 65

60

Zr Zr H O Zr H O

200

H O H O

Zr

Inert strength

O

Stress (MPa)

Flaw w size (μm)

125

80

500

H H O

Zr

Zr

100

50

75

60

20

0

1

2

Time (hours) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

3

4

Failure !

40

0

Stress applied

0

1

2

3

4

Time (hours) 24

Theory Lifetime

Lifetime prediction v = f (K I )

?

Flaw size (μm) 65

60

75

125

80

Stress (M MPa)

Fracture strength

60 40 20 0

De Aza et al., Biomaterials 23, 937 (2002)

v = A.K I n

σc =

K IC Y ac

Empirical relation Griffith‘s law


Failure

Stress applied

tf 0

1

2

3

4

Time (hours)

⎛ 2 2− n ⎞ n −2 − n n −2 −n ⎟ t f = ⎜⎜ K σ σ = B σ σ IC c c 2 ⎟ ⎝ AY (n − 2 ) ⎠

For constant stress σ ! 25

Theory Lifetime

Lifetime prediction v = f (K I )

? Stresss (MPa)

80

Inert strength

60

failure (= fracture)

40

Stress applied

20 0

tf 0

1

2

σc =

K IC Y ac

Empirical relation Griffith‘s law


4

5

6

7

8

Time (hours)

De Aza et al., Biomaterials 23, 937 (2002)

v = A.K I n

3

t f = Bσ cn − 2σ − n

tf =

1

h (n, σ m

σa)

Constant stress σ Bσ

n −2 c

σ

−n a

Any given stress variation i ti 26

Theory Lifetime

Scheme of the ageing process a) Nucleation on a particular grain at the surface, leading to o microcracking c oc ac g and a d stresses s esses to o the e neighbours e g bou s b) Growth of the transformed zone, leading to extensive microcracking and surface roughening. c) Grain pull-out induced by wear

water induced

t →m

Chevalier, Leading Opinion: What future for zirconia as a biomaterial? Biomaterials 27 (2006) 535–543

transformation (degradation of strength)

σ Monoclinic Tetragonal

σ (c) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

27

Experimental

Experimental Approach Sample preparation

Set-up fatigue g machine

100

Strength and lifetime measurements

80 60 40 Zirconia

20

10

100

1000

10000 100000

Crack velocity, v (m/cycle)

Number of cycles

10

-6

10

-7

Li2O.2SiO O 2SiO2

10

-8

ZrO2

10

-9

10

-10

10

-11

10

-12

10

-13

80 60 40 20 0 200

0 1

Failure probability (%)

Failure probablity (%)

100

Zirconia 400

600

800

1000

1200

Stress (MPa)

Al2O3-ZrO2-Glass

1

Subcritical crack growth curves 2

3

4

5

6

Stresses on posterior bridges

7

1/2

Stress intensity factor factor, KI,max (MPa.m (MPa m )

Lifetime diagrams Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

28

Experimental

Framework

Sample preparation Framework + veneer

Apatite Li O.2SiO 2 2 (veneer) ((framework))

1 mm

inert strength + lifetime in water X coated + uncoated X 3 materials + TZP rods = 240 samples (effective 232)


29

Experimental

Detection of sample breakage

Aqueous environment


30

Failure prob bability (%)

100 80

Time (min)

σmax

100

peak stress for accelerated fatigue g tests

60 40 20 0 200

Zirconia

Stress (MPa) Failure prob bablity (%)

Experimental

600

800

1000

1200

30

Accelerated Fatigue

10 Hz

650 80

σmax

60 0 40

Zirconia

20 650 0

400

Applied stress5 0.5

1

10

0.1 100 Time (s)1000

Stress (MPa)

Number of cycles

Mechanical strength

Fatigue


0.2 10000 100000

31

Results

Determination of SCG behavior e.g. TZP Inert strenght (MPa) 900

1000 1100 1200 1300

1

90 TZP

lnln(1/(1-F))

0

70 50

-1

30

-2

10

6.9

7.0

7.1

-1

30

-2

10

-6

7.2

-4

-2

0

2

4

6

ln(Lifetime)

lnσc

-8

Crack velocity v (m/cycle)

10

Air n = 79.8

-9

10

-10

10

n = 66.5

-11

10

0.4 Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

90 70 50

-4 6.8

50

0

-4 6.7

10

TZP

-3

6.6

1

1

-3

6.5

0.1

2

Failure probability (%))

800

lnln(1/(1-F))

700

Failure e probability (%)

2

Lifetime (min)

0.5

0.6 ΔKI / KIC

0.7

0.8

Describes the properties p p of the material ! 32

Results

Air

-8

10

n = 79.8 -9

10

-10

10

n = 66.5

-11

10

0.4

0.5

06 0.6

0.7

ΔKI / KIC

Materials‘ p properties p

08 0.8

Maximu um applied stre ess (MPa)

Crack velocity v (m/cy ycle)

Lifetime prediction of dental materials

Inert strength (MPa)

1000

1350

Air

600

Water ae

2

100

6 mm

2

8 mm

2

Stress at the connector of a 3-unit bridge loaded at 900 N

10 mm

Connector area

10 0 10

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

Number of cycles

20 years with ith 1’400 cycles/day Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

Predict lifetime for any bridge ! 33

Results

Fracture mode Imbeni et al, Nature Materials, 4, 229 (2005)

Enamel

Arrested crack

Dentin

Enamel Dentin


34

Results Fatigue and subcritical crack growth

Veneer 10Ca2+ + 6PO43- + 2OH 1 mm

Ca10((PO4)6((OH))2 Enamel: Apatite


35

Results Fatigue and subcritical crack growth

Framework

-6

Crac ck velocity y, v (m/cy ycle)

10

Human dentin (Nalla et al, 2003)

-7

10

Li 2 O.2SiO 2 Al 2 O3-ZrO2-Glass Nano-ZrO 2

-8

10

-9

10 0

Enamel: Apatite

-10

10

-11

10

-12

10

-13

10

1

2

3

4

5

6

7

Dentin: ~ 75 % Apatite + 25 % collagen

1/2

Stress intensity factor, KI,max (MPa.m ) Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

36

Results

Lifetime diagram

20 years with 1’400 cycles/day and Loadmax = 130 N

Probability of fa ailure (%)

100 Stress on the bridge

80

Initial strength

60 40 20

Improved ceramic

0 60 300

80 400

100 500

120 600

Stress (MPa)


37

Results

Lifetime diagram g : Veneer Glass Z

Apatite

40

100

20

40

100

300

Veneer strength, σc (MPa)

20

10

Crack size (μm)

1 bite


10

Crack size (μm)


10

> 20 years

20

40

100

300

Stress on veneer surface, σ (MPa)


4 3 2.5

4 3 2.5

Crack size (μm)

Glass A

300


P 2

1.7 mm

1.7 mm

2

4 3 2.5

2

1.7 mm

P P 4 3.5

3

2.6 mm

3.5

3.1 mm

4 3.5

3

2.6 mm

3.5

3.1 mm

4 3.5

3

2.6 mm

P P P 4

4

4

3.5

3.1 mm

1400 cycles/day Loadmax = 130 N Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

38

Results

Lifetime diagram g : Framework Nano--ZrO2 Nano

Li2O.2SiO2

1 bite

40

100

300

Stress on framework, σ (MPa)

Framework strength,, σc (MPa)

100

20

10

Crack size (μm)

40

Fra amework strength h, σc (MPa)

20

> 20 years

10

Crack size (μm)

Fra amework strength h, σc (MPa)

10

20

40

100

300


Crack size ((μm)

Al2O3-ZrO2-Glass

300


P 4 3 2.5

2

1.7 mm

4 3 2.5

2.8 mm

4

2

1.7 mm

4 3 2.5

2.8 mm

4

2

1.7 mm

P P 4

3.5

3

3.5

3

3.5

3

2.8 mm

P P P 4

3.5

3.3 mm

4

3.5

3.3 mm

4

3.5

3.3 mm

1400 cycles/day Loadmax = 130 N Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

39

Summary Framework:

Framework:

Al2O3-ZrO2-Glass

Framework: Li2O.2SiO2

Nano--ZrO2 Nano

Veneer: Glass Z

Veneer: Glass A P

Veneer: Apatite

P 4 3 2.5

P P

2

1.7 mm

Max ∅ (≥ 4 mm)

P P P

1400 cycles/day Loadmax = 130 N

P 4 3 2.5

P P

P P P

2

1.7 mm

Max ∅ (≥ 4 mm) Max ∅ (≥ 4 mm)

P P

P P P

Veneer and not framework dictates what’s feasible …..


40

Courtesy: University of Zürich, Center for Dental & Oral Medicine Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

41

Conclusions Nano-ZrO2 ceramics and ZrO2-based ceramic composites p can withstand the high g stresses applied on posterior bridges during mastication

-9 -4 1 5 10 15 20 25 30 34

Long-term failure due to fatigue and subcritical crack growth can be avoided through proper bridge design

2

1.7 mm

Max ∅ (> 4 mm) Max ∅ (> 4 mm)

4

3

Connector diameter (mm) 2.5 2 1.7

20

40

Crack size (μm)

10

Framework strength, σc (MPa)

Lifetime diagrams are suitable tools for the selection of new materials and fabrication technologies for dental restorations

4 3 2.5

100 300


Apatite-based materials should be avoided in the connector region of posterior bridges d due e to the pronounced subcritical crack growth Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

Ca10(PO4)6(OH)2 10Ca2+ + 6PO43- + 2OH 42

Outlook Etruscian prosthesis

200 BC

Metalporcelain bridge

1960s

2005

Ceramic high strength, non veneer bridge ?

Natural tooth genesis mediated by stem cells ?

20 20yy

2 2zzz

different materials strength not anymore the issue treatment methods shifting Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

All-ceramic bridge

- wishful - 1st steps successful - lots of challenges ahead 43

This you u should d have le earned!

Summary: What you know and understand, understand now! 9 9 9 9 9 9 9 9 9 9 9 9

Hook s law … Hook’s Stress … Griffith’s law … Weibull statistic … Influence of surface, volume, microstructure … R-curve R curve … Environment … Sub-critical crack growth … St ti & d Static dynamic i ffatigue ti … Proof-testing … Deformation & failure @ elevated temperatures … Thermal shock …

Those relations, laws & equations you known by heart, now! Kübler Empa-HPC, ETHZ MW-II Ceramics-6.5, 2010

44