V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

1.Energeticapproach(Griffith):

€

−dUMdc = G

crackresistanceforce

€

dUSdc = R0

crackextensionforce

€

dUdc

= 0 ➔ G=R0 Complexgeometries!!

R0 =adhesion workforunitàarea=2γ

material property

strong material?

brittle-ductile fracture?safety criterion (failure or not?)

Aim:todefine intrinsic material properties

Fracture mechanics principlesFracture of Brittle Solids, 2nd ed., B. Lawn, 1998 - Ch. 2An Introduction to the MechanicalProperties of Ceramics, D.J. Green, 1998 – Ch. 8

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

Mechanicalapproach:

F1

F2

F3

Fi

in any point:stress and strain (displacement)

1.straincompatibility

�

2∂2ε12

∂x1∂x2

=∂2ε11

∂x22 +

∂2ε22

∂x12

2.stressequilibrium

�

∂σ11

∂x1

+∂σ12

∂x2

=∂σ22

∂x2

+∂σ12

∂x1

= 0

3.Hooke’s law

�

ε11 =1E

(σ11 −νσ22)

�

ε22 =1E

(σ22 −νσ11)

�

2ε12 =2(1+ν)σ12

E

(2D; no body forces)

4.boundary conditions

ρ >0

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

€

σ ij = K fij (θ)2π r

€

ui = K gi (θ)2E

r2π

K=stressintensity factor=ψ σa c0.5

external load

shape factor (system geometry)

cracklength,c

when ρ è0….

K = lim

ρ→0

π ρ2 σ C

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

¢ K=driving forceforfracture

Fracturecriteria:

• G ≥ Gc = R0 fractureenergy• K ≥ Kc = T fracturetoughness

G =R0

€

⇒K2

E=

KC2

E material property

€

G = −dUM

dc=

K2

E

Young’s modulus

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

σ

σ

c

“applied load”=stressintensity factor:K=σ ψ c0.5=driving forceforfailure:G=K2/E

material “strength”=fracture toughness:KC=fracture energy:GC

critical condition: K=KC i.e.G=GC

nominalappliedstress

➔ failurestress=sf =Kc/(y c0.5)

lengthofthemostcriticaldefect(size,location,orientation…)

c

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

Shape factor

2c

y = 2 wc

tan p c2 wæ

è ç

ö

ø ÷

é

ë ê

ù

û ú 0.5

2 c

2 w

y = p

y =1.12 p

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

“penny”crack “half-penny”crack

y = 2p

y = 2.24p

2 c 2 c

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

SENB(singleedgenotchedbeam)test

€

K = ψ3P Ddh2 c

€

ψ = 1.99− 2.47ch +12.97 c

h$

% & '

( ) 2

− 23.17 ch$

% & '

( ) 3

+ 24.80 ch$

% & '

( )

4

KC measurement

if D=L/2(3-pointbending)andL/h=8

€

ψ = 1.93−3.07ch +14.53c

h$

% & '

( )

2

−25.11ch$

% & '

( ) 3

+ 25.80 ch$

% & '

( )

4

Mechanical properties of ceramics, J.B. Watchman, J. Wiley, 1996

main issue: production of controlled sharp crack

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

DT(doubletorsion)

DCB(doublecantileverbeam)

K =P cd h1.5!

"#

$

%& 3.47+ 2.43

hc

'

()

*

+,

K = P D 3(1+υ)L h 3 hn

!

"##

$

%&&

0.5

Mechanical properties of ceramics, J.B. Watchman, J. Wiley, 1996

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

Norms:UNICEN/TS14425-1/3/5,UNIENISO18756,ASTMC1421

surface crack - indentation(SCF)

through-thickness notch(SENB)

Chevron notch(CN)

2c

Si3N4

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

Fracture toughness /fracture energy

energydissipatingprocesses

Fracture of brittle solids, 2° ed., B.R. Lawn, Cambridge Univ. Press, 1993

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

100

10

1

0.1

0.01

Young’s modulus, E (GPa)

Frac

ture

toug

hnes

s, K

1c (

MP

a.m

1/2 )

0.001 0.01 0.1 1 10 100 1000

0.01

0.1

1

10

100

1000

Polymers and elastomers

Metals

Technicalceramics

Composites

Natural materials

Lead alloys

W alloys

SteelsTi alloys

Mg alloys

CFRPGFRP

Al alloys

Rigid polymer foams

Flexible polymer foams

Ni alloysCu alloys

Zinc alloys

Ps

PTFE

PC

Cork

Wood

Butyl rubber

Silicone elastomers

Concrete

Al2 O3

SiCSi3 N4

Fracture toughness–Modulus

B4 C

PP

EVAPolyurethane

Leather

Non-technical ceramics

Cast irons

WC

Soda glass

Silica glass

Silicon

Stone

Brick

ABS

Epoxies

Ionomers

MFA, 07

Toughness Gc = (K1c)2 /E kJ/m2

Foams

0.001

Figure 8.8 A chart of fracture toughness Klc and modulus E. The contours show thetoughness, Gc.

Yield strength or elastic limit, σy (MPa)

Fra

ctur

e to

ughn

ess,

K1c

(M

Pa.

m1/

2 )

0.1 1 10 100 1000

0.01

0.1

1

10

100

1000100 10

1

0.1

0.01

1000Transition cracklength, ccrit, mm

Non-technical ceramics

Foams

Polymers and elastomers

Metals

Technicalceramics

CompositesLead alloys

W alloys

Stainless steels

Ti alloys

Mg alloys

CFRP

GFRP

Al alloys

Rigid polymer foams

Flexible polymer foams

Ni alloys

Cu alloys

Zinc alloys

PMMA

Cork

Wood

Butyl rubber

Silicone elastomers

Concrete

Al2 O3

SiCSi3 N4

Fracture toughness–Strength

B4 C

Neoprene

Isoprene

Leather

Castirons

WC

Soda glass

Silica glassSilicon

Stone

Brick

ABS

Epoxies

Ionomers

Low alloy steels

Carbon steels

Polyurethane

PAPC

PEPTFE

PS

PP

Phenolic

MFA, 07

Figure 8.9 A chart of fracture toughness K1c and yield strength σy. The contours show thetransition crack size, ccrit.

Ch08-H8391.qxd 1/17/07 10:45 AM Page 173

tough (ductile)

brittle

V.M.Sglavo– 2018

V.M. Sglavo – CerMatEng - UNITN 2018

What does occur first?Plasticdeformation orbrittle failure?

- material

- volume under (tensile) stress

- stress state (1D vs. 2D vs. 3D)