Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
A new critical velocity
in the dynamic fracture
of brittle amorphous materials
J. Scheibert1,2,3, C. Guerra1, F. Célarié1, D. Dalmas2, D. Bonamy1
1. Grp. Complex Systems & Fracture, SPCSI, CEA Saclay, France
2. SVI, UMR CNRS/Saint-Gobain, Aubervilliers, France
3. PGP, University of Oslo, Norway
Dynamic: crack tip velocity v ~ speed of sound cR
Brittle: material is elastic up to fracture
crack propagation = bond breaking
� silica glass and polymethylmethacrylate (PMMA)
Amorphous: no long range cristalline order
heterogeneities: Glass ~ 0.1nm ; PMMA ~ 10nm
In the following: only linear elastic materials
Dynamic fracture of brittle amorphous materials
v
Equation of motion (Freund, 1990)
E
KvAGvAv
2
)()()( ==Γ
cR : Rayleigh speed (predicted limiting crack speed)
−≈
Rc
vvA 1)(
K = quasi-static stress intensity factor !
Energy balance between: - energy release rate G = K²/E
- dynamic fracture energy Γ(v)
- kinetic energy (velocity v)
Linear Elastic Fracture Mechanics (LEFM, Freund 1990)
� single smooth crack in linear elastic medium
Micro-branching: a first critical velocity
Micro-branching occurs over vb ~ 0.4 cR
Sharon et al
PRL 1995
Crack propagation
100µm
Equation of motion valid only for v < vb
PMMA Glass
vb vb
Dynamic fracture energy under vb ?
Derived from
Dissipating mechanisms at low velocities ?
E
KvAv
2
)()( =Γ
Γ(v=0)
(Sharon and Fineberg, Nature 1999)
Experimental setup: Wedge Splitting test
Hole: adjustable stored energy U0
Quasi-static loading
Rollers: no friction
Measurements:
1. Load when crack initiates (3-20 kN)
2. Velocity profile
3. Fractography
Load Wedge
Jaw
Roller Groove
Seed crack
Hole
Sample
(PMMA)
Support
140mm
125mm
Thickness: 15mm
Decelerating configuration
E
KvAv
2
)()( =Γ
R
25 lines
350µsU
UmesRmes
R >> Rmes :
Umes varies linearly with crack length
NR
RUU mes
mes =
Velocity measurement
90 lines
E
KvAv
2
)()( =Γ
Stress intensity factor: finite elements calculations
Ingredients: linear elasticity
plane stress
elastostatics
θθθθw
F
Hyp : constant wedge position
during crack propagation
E
KvAv
2
)()( =Γ
Dynamic fracture energy vs velocity
E
K
c
v
R
2
1
−≈Γ
v from measurements
K from FEM
Complete Γ curve from v=0 to vb
vb ~ 350m/s (micro-branching)
New critical velocity: va = 160m/s ~ 0.2 cR
Dynamic fracture energy vs stress intensity factor
E
K
c
v
R
2
1
−≈Γ
2 linear regimes:a = 0.77
b = 0.48
Origin of the transition at va ? � fractography analysis
va
vb
R
Ra
cva
cv
vv−−
−=<Γ
1
1
)( α
R
Ra
cvb
cv
vv−−
−=>Γ
1
1
)( β
2 critical velocities:
(1 – a) cR = 0.23 cR ~ va
(1 – b) cR = 0.52 cR ~ effective limiting crack speed
Fractography: parabolic marks
velocity threshold at va
�Same transition as on Γ
v < vavb < vva < v < vb
A
B
C
500µm
Simple model (Smekal 1953, Ravi-Chandar 1997)
Nucleation center +
Nucleation distance d
vmaincrack = vmicrocrack
� intersection = parabola
Micro-cracking damage
d
For v > va, the crack goes faster
because many micro-cracks
open simultaneously
Γ
vva
Conclusions
• New dynamic transition in the brittle fracture of amorphous solids…
…at a well-defined critical velocity va ~ 0.2cR
• Transition corresponds to the onset of micro-cracking damage
� Brittle to quasi-brittle transition
• Generic transition:
Microcracks in many brittle amorphous materials
(polymers, glass, rocks,…)
Ref: J. Scheibert, et al., arXiv:0906.2877