Fracture Mechanics:
fundamentals,
relation to plasticity,
potential
Henrik Stang
Technical University of Denmark
Department of Cvil Engineering
Outline
ƒ Fundamentals – historical overview
ƒ Fundamental properties of a fracture mechanical
analysis
ƒ Identification of material parameters
ƒ Application of fracture mechanics in analysis of
FRC disk
ƒ Outlook: service life analysis, understanding
materials
ƒ Acknowledgements
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LEFM historical overview
r
”square root r” singularity
σ
p
p
2a
Griffith (1920)
Griffith’s criterion for crack propagation:
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Irwin (1957)
σ2 =
2E 'γ
πa
3
The Fictitious Crack Model - FCM
σ w ( w)
Stress
Smooth crack closure
P
ft
σw (w)
w
σ w = σ w (w )
wc
GF =
wc
a
∫ σ (w )dw
lp
0
Cohesive law Stress – crack opening law
E , G F , ft
lc h
EGF
=
( ft ) 2
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x
P
Characteristic length
Hillerborg et al. 1976: NLFM for concrete,
the fictitious crack model.
4
LEFM/FCM
σ
2a
σ
LEFM: K IC = EGF
lch = 55 mm
GF = 14 J/m2, E = 31 GPa, ft = 2.8 MPa, a0 = 2 mm
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Basic properties of FM based analysis
-types of analysis
Analytical: Hinge Model
FEM: XFEM, embedded crack model
FEM: Interface model
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Basic properties of FM based analysis:
-full load-deformation history, size effect
80.00
Load (kN)
Modulus of rupture:
2.0
2.5
2.9
3.2
40.00
MPa
MPa
MPa
MPa
Beams:
4000x1200x150 mm
2000x600x150 mm
1000x300x150 mm
500x150x150 mm
Same cohesive law
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0.00
0.00
0.40
0.80 Deflection
(mm)
7
Parameter Identification
Eksperimental setup
Hinge model
and multilinear
cohesive law
Concrete
Mortar
Micro-mortar
‘Mixed Mode’ ?
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Application of FM in the analysis of FRC disk
w1*
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Application of FM in the analysis of FRC disk
Work equation
We =Wi
n
W (u ) = t disk ∑ D j (u )
i
Internal work = dissipation in cracks
j =1
w
G f ( w) = ∫ σ w (v) d v
Energy dissipated in a point on a crack
w* x
w( x) =
=Ω
l
Crack opening profiles are linear
0
l w( x )
D( w* ) = ∫
0
∫σ
w
(v) dv dx
Energy dissipation in each crack
0
w*
D( w* ) =
l
G f (Ω)dΩ = lF ( w* )
* ∫
w 0
n
Energy dissipation expressed as a function of
crack mouth opening displacement and crack
length only
j =1
Internal work depends on deformation
W (u ) = t disk ∑ l j F (w*j (u ))
i
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Outlook: Service life prediction
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Outlook: Service life prediction
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Outlook: Understanding materials
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Acknowledgements
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
ƒ
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Associate professor John Forbes Olesen
Associate professor Peter Noe Poulsen
Associate professor Leif Otto Nielsen
Ph.d. Lars Dick Nielsen
Ph.d. student Anders Ole Stubbe Solgaard
Ph.d. student Jon Spangenberg
Ph.d. student Jan Skocek
M.Sc. Mikkel Wyrtz
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