Mechanics of 3D Crack Growth in Compression

1
Mechanics of 3-D Crack Growth in Compression

• AV Dyskin

2
The Role of Internal Cracks
Perfect confinement
Uniform load
(after Peng Johnson, 1972)
3
Plan

• Conventional 2-D crack growth in uniaxial
  compression
• 3-D crack growth in uniaxial compression.
  Experiments and analysis
• Random stress fields as a mechanism of extensive
  crack growth in uniaxial compression
• Crack growth in biaxial compression
• Models of crack growth in compression
• Conclusions

4
2-D Crack. Fairhurst-Cook Model
?
?
F
?
F
?
The law of crack growth
5
Stable Crack Growth in Compression
K
I
?
4
?
3
K
?
2
Ic
?
l
6
3-D Crack growth in uniaxial compression.
Experiments and analysis

• 3-D vs. 2-D. Anticipated mechanism of crack
  growth
• Experiments
• Limits of growth of a single crack
• Analysis
• Crack interaction
• Initiation of large fractures

7
2-D vs. 3-D Crack Growth
?
3-D crack
K
III
K
II
b
c
a
K
Zone of tensile stress
II
Microcracks
Wing
8
Adams and Sines 3-D Test
Sample manufacturing
PMMA
Glue
Tensile stresses
Fish fin crack initiation experiments
artefact
Adams and Sines (1978)
9
UWA experiments
Polyester casting resin Polylite 61-209
Main technology
T-17ºC, E4GPa sc140 MPa KIc0.6 MPa m1/2
Controlling tests
Inclusion that models a 3-D crack. Diameters 10,
12, 15 mm
125
Teflon insert to ensure contact
30, 45
55
Glue
55
Second type 80 mm x 80 mm x 190 mm
10
Single 3-D Crack in Uniaxial Compression
11
Limits of Growth of a Single Crack
12
Size and position of initial crack
Crack near a free boundary
Large initial crack
13
Laser cracks in frozen PMMA and glass
Laser method of producing internal cracks,
Germanovich et al (1994), UO
PMMA
T-198ºC, E6.4GPa sc260 MPa KIc3.7 MPa m1/2
Bore-silicate glass 29 x 17 x 51 mm
E40GPa KIc1 MPa m1/2
Crack 1 cm
Crack 1 cm
14
Types of Growth of a Single Crack
Wing wrapping
In 3-D, a single crack cannot induce long wings
capable of producing fracture
Secondary cracks (wings)
Initial crack
10 and 12 mm initial crack
15 mm initial crack
15
Analysis
x2
Principal stresses (tensile)
Mechanism of wrapping
16
Crack from a spherical pore
17
Crack from a spherical pore. Other views
18
Multiple Cracks
2a
dgt4a
No interaction
19
Crack Interaction
2a
dlt4a
20
Analysis
d3a
d5a
21
Orientations of large fracture
22
Large crack initiation
23
Multicrack Interaction
Casting resin Laser cracks (UO experiments)
24
Mechanism of Extensive 3-D Crack Growth in
Uniaxial Compression
Dyskin (1998)

• Stress fluctuations in compressed rock samples
• Stress fluctuations as a driving force for cracks
  in compression
• Model of 3-D crack growth in uniaxial compression
• Mechanics of crack growth in uniaxial compression

25
Stress Fluctuations
s3
x2
P
x3
lts3gt-P
15
10
Compression
x3
Tension

• Randomly located wing cracks produce stress
  fluctuations
• In lateral directions stress fluctuations are
• self-equilibrating
• spatially random

x2
5
x1
0
s2
0
P
lts2gt0
26
Stress Fluctuations as a Driving Force for Cracks
in Compression
Newly developing crack location is determined by
peak stress
Pre-existing crack location is not correlated
with stress fluctuations
s
s
y
y
0
0
x
x
r
x
/
r
x
/
0
5
10
15
0
5
10
15
Average force acting on the crack is zero. Crack
cannot grow.
Average force acting on the crack is positive.
Crack will grow.
27
Crack growth in 3-D
Influence of stress field variations

• The location of the initiated crack is correlated
  with the stress field
• For Gaussian stress fluctuations the average
  stress distribution in the crack plane (x1, x2)

ltDs( )gt
x , x
1
2
x
2
Initial microcrack
x
1
B(x1,x2) is the correlation function of Ds
(eg, Feller, 1971)
28
Mechanism of Crack Growth due to Stress
Fluctuations
(
a)
(
b)
Future crack
Positive
Future crack
stress
fluctuation
(tension)
Negative
stress
fluctuation
Conditional average stress
(compression)
distribution (tension)
29
Model of 3-D Crack Growth
F

x
2
Grown crack
x
F
1
Condition of crack growth
The crack grows even if no external tensile load
is applied
30
Mechanics of Crack Growth and Fracture in
Uniaxial Compression
Extensive crack growth
Wing crack accumulation
Failure
31
Crack growth in Biaxial Compression

• Experiments on 3-D crack growth in biaxial
  compression
• Mechanism of extensive 3-D crack growth in
  biaxial compression. The role of intermediate
  principal stress
• Model of 3-D crack growth in biaxial compression

32
Experiments on Crack Growth in Biaxial Compression
Before testing
After testing
z
y
Initial crack inclined to yz plane at 30º
Casting resin, 100x100x100 mm, frozen. True
triaxial frame
Px0
33
Experiment 1
34
Experiment 2
35
Mechanism of Extensive Crack Growth. Role of
Intermediate Principal Stress
Uniaxial compression
Biaxial compression
Intermediate principal stress
36
Model of Crack Growth in Biaxial Compression

P


z

Extensive wing
F


growth
R

a
F

P


y
After testing

Model


Before testing



(Germanovich et al., 1996)
KIF(pR)-3/2 (Cherepanov, 1979)
Stable growth
37
Models of Crack Growth in Compression
KIF(pR)-3/2
Biaxial compression Initial crack
Uniaxial compression - Random stress fluctuations
Different mechanisms similar effects
38
Conclusions

• 3-D wing crack growth in uniaxial compression is
  limited, as opposite to 2-D case
• 3-D growth of wing cracks is limited by wing
  curving (wrapping)
• Large (macro) crack propagation can only result
  from a combined action of wing cracks
• In biaxial compression the wing crack is nearly
  planar and thus can grow significantly
• In 3-D, wing crack growth is controlled by
  intermediate principal stress, sII
• sII 0 ? limited growth
• sII sI ? extensive growth

39
Unresolved questions Splitting vs. shear failure
(after Paul, 1968)
Splitting
Shear fracture
Conventional interpretation the end effect
40
Failure modes under the same loading conditions
What is the mechanism of shear failure?
Cement/Sand Shear failure
Pure cement Splitting
41
Post-Peak Behaviour
1
0.8
Splitting
Shear failure
0.6
Pure cement
0.4
Cement/Silica Sand
Cement/Silica Flour
0.2
Cement/Garnet
0
e/e
0
0.5
1
1.5
2
peak
42
Influence of free surface
Fairhurst-Cook crack
Free surface
l/h
43
The Role of Lateral Pressure
? is friction angle
For proportional loading, pc?
44
Crack Coalescence
Initially collinear, mode I cracks avoid each
other (Melin, 1983)
a
b
Simultaneous growth of 2-D cracks in compression
Growth of initially collinear cracks in
compression (Horii Nemat-Nasser, 1986)
45
Non-interacting cracks
2a
dgt4a