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)
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