V.M. Sglavo – 2018 V.M. Sglavo – CerMatEng - UNITN 2018 1. Energetic approach (Griffith): − dU M dc = G crack resistance force dU S dc = R 0 crack extension force dU dc = 0 ➔ G=R 0 Complex geometries!! R 0 = adhesion work for unità area = 2 γ material property strong material? brittle-ductile fracture? safety criterion (failure or not?) Aim: to define intrinsic material properties Fracture mechanics principles Fracture of Brittle Solids, 2nd ed., B. Lawn, 1998 - Ch. 2 An Introduction to the Mechanical Properties of Ceramics, D.J. Green, 1998 – Ch. 8
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V.M. Sglavo –CerMatEng-UNITN 2018 · 2018-12-04 · V.M. Sglavo –2018 V.M. Sglavo –CerMatEng-UNITN 2018 Mechanical approach: F1 F2 F3 Fi in anypoint: stress and strain (displacement)
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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