cracks€¦ · cracks: effective elastic properties - crack interactions – local fields (stress...
TRANSCRIPT
Cracks
versus
Rough Fractures
-Viscosity of
Tufts University
Mark Kachanov
• Fractures:
Rough surfaces with contacts
• Cracks:
Traction-free surfaces
Frequently confused
Fractures vs Cracks
• Fractures:
Rough surfaces with contacts
• Cracks:
Traction-free surfaces
Frequently confused
Fractures vs Cracks
Both determined by displacement discontinuities
But: are controlled by different microstructural parameters
Their compliances:
u
u
Root of the confusion?
Terminology:
words “Fracture” and “Crack”
are treated as synonyms
Clarification:
Fracture: Rough contacting surfaces
Crack: Traction-free surface
Root of the confusion?
Terminology:
words “Fracture” and “Crack”
are treated as synonyms
Clarification:
Fracture: Rough contacting surfaces
Crack: Traction-free surface
Outline:
Cracks
Fractures with contacts
Similarities and differences
Cracks:
Effective Elastic Properties
- Crack interactions – local fields (stress shielding and amplification)
- Large crack – “cloud” of microcracks
Review “Elastic solids with many cracks”
(Kachanov, in “Advances in applied mechanics”)
Other crack-related issues:
Cracks:
Effective Elastic Properties
--Crack interactions, effects on local fields
-- Large crack – “cloud” of microcracks
Review: Kachanov “Elastic solids with many cracks”
Other crack-related issues:
Circular (penny-shaped) cracks
V a
31 kaV
e
Crack density parameter (Bristow, 1960)
Individual crack contributions to compliance: proportional to 3a
Adequate for:
• circular cracks
(otherwise: adjustable parameter, no link to cracks)
• random orientations, isotropy (otherwise, must be tensor)
Circular (penny-shaped) cracks
V a
31 kaV
e
Crack density parameter (Bristow, 1960)
Individual crack contributions to compliance: proportional to 3a
Adequate for:
• circular cracks
(otherwise: adjustable parameter, no link to cracks)
• random orientations, isotropy (otherwise, must be tensor)
31 ka
Ve
Does not reflect
crack opening
(aspect ratio)
Crack density
31 ka
Ve
Crack compliance: Almost
independent of aspect ratio
if it is smaller than 0.1
crack radius - not volume
- is kept constant
Does not reflect
crack opening
(aspect ratio)
Crack density
“Crack porosity” : of no importance
(for elastic properties, wavespeeds)
provided aspect ratio < 0.1
Non-Interaction Approximation (“dilute” limit)
Each crack is placed into
No effect of neighbors
Individual compliance contributions summed up
V
Not necessarily!
Non-Interaction Approximation (“dilute” limit)
Each crack is placed into
No effect of neighbors
Individual compliance contributions summed up
V
k
εσ:Sε0
Overall strain
(per volume V)
Displacement jump across crack
Matrix compliance dS
VkS
unnu1
Not necessarily!
Non-Interaction Approximation (“dilute” limit)
Each crack is placed into
No effect of neighbors
Individual compliance contributions summed up
V
k
εσ:Sε0
Overall strain
(per volume V)
Displacement jump across crack
This representation is general
Cracks may be non-flat
Material - anisotropic
Matrix compliance dS
VkS
unnu1
Not necessarily!
Crack compliance contributions are summed up – effective
compliances (not stiffnesses!) are linear in crack density
k
εσ:Sε0
e
eG
G
0
000
245
51321
Random orientations (isotropy): Parallel cracks (TI):
(normal to
cracks)
eG
G
0
0
13
0
23
1161
eE
E
6
0
20
3
0
23
1321
eE
E
8.1
0
0200
245
3101161
Important:
formally, one can linearize, for small crack densities
e
eCeCE
E
1
1
1
0
stiffness linear in e
Important:
formally, one can linearize, for small crack densities
e
eCeCE
E
1
1
1
0
stiffness linear in e
However:
Linearization reduces range of
where non-interaction approximation remains accurate
e
Do not linearize !
e
?
1
00 G
Gor
E
EUpper curve remains accurate
at higher crack densities
References for these basic results:
Bristow (1960) Microcracks, and the static and dynamic elastic
constants, British Journal of Appl. Physics 11, 81-85
Walsh (1965) The effect of cracks on the compressibility of rocks,
Journal of Geophysical Research 70(2), 381-389.
Walsh (1965) The effect of cracks on uniaxial compression of rocks
Journal of Geophysical Research 70(2), 399-411.
Non-random orientations
Anisotropy?
More complex –
and realistic – cases
Non-random orientations
Complex crack shapes
Anisotropy?
Crack density
Parameters?
More complex –
and realistic – cases
Non-random orientations
Complex crack shapes
Anisotropic background (shale)
Anisotropy?
Crack density
Parameters?
More complex –
and realistic – cases
Non-random orientations
Complex crack shapes
Anisotropic background (shale)
Interactions
Intersections
Anisotropy?
shield amplify
Crack density
Parameters?
More complex –
and realistic – cases
Effect on local fields, and on overall response?
Non-random orientations
Complex crack shapes
Anisotropic background (shale)
Interactions
Intersections
Anisotropy?
shield amplify
Crack density
Parameters?
More complex –
and realistic – cases
Effect on local fields, and on overall response?
Non-random orientations
Complex crack shapes
Anisotropic background (shale)
Interactions
Intersections
Fluid infill
Anisotropy?
shield amplify
Crack stiffening
Changes in anisotropy
Crack density
Parameters?
More complex –
and realistic – cases
Effect on local fields, and on overall response?
Non-Random orientations. Anisotropy
Flat cracks (of any shape)
V
k
εσ:Sε0
n
kSV
nbbn1
on crack constn
Non-Random orientations. Anisotropy
Flat cracks (of any shape)
V
k
εσ:Sε0
n
kSV
nbbn1
Crack compliance
tensor
Average
displacement jump Bσnuub
nnIBnnBB TN
Crack compliance
normal shear
on crack constn
circular crack:
Crack compliances, shear and normal (circular crack)
They are close
21 0TN BB
a
E
BT
00
20
23
132
Crack compliances, shear and normal (circular crack)
They are close
21 0TN BB
a
E
BT
00
20
23
132
σnnnnBBσnnBnbbn TNT :
Crack-generated strain:
relatively small
Crack compliances, shear and normal (circular crack)
They are close
21 0TN BB
a
E
BT
00
20
23
132
σnnnnBBσnnBnbbn TNT :
Crack-generated strain:
relatively small
Multiple cracks: 2-nd and 4-th rank tensors and
enters with small factor
nn nnnn
ka
V nnα
31
ka
V nnnnβ
31
Crack density tensor
symmetric 2-nd rank
jlikjlikjlikjlikijkl
ES
4
1
23
132
00
20
ijkl
2
0
In terms of
relatively small
4-th rank tensor
Its effect is relatively small
Extra compliances due to cracks (non-inter. approx):
ka
V nnα
31
ka
V nnnnβ
31
Crack density tensor
symmetric 2-nd rank
jlikjlikjlikjlikijkl
ES
4
1
23
132
00
20
ijkl
2
0
In terms of
relatively small
4-th rank tensor
Its effect is relatively small
Neglecting term: approximate orthotropy
Principal axes of orthotropy = principal axes of
β
α
Extra compliances due to cracks (non-inter. approx):
Overall elastic properties are orthotropic
for any orientation distribution of cracks
ortho. axes
Counter-intuitive
Implications for wavespeeds
Moreover: Orthotropy due to cracks is of simplified type:
1. Reduced number of independent constants:
From 9 (general ortho) to only 4
j
ji
i
ij
ij EEG
111Shear moduli are not independent
00331223112 EEEE
independent constants can be taken as 00321 ,,, EEEE
Implications for wavespeeds?
Moreover, orthotropy due to cracks is of simplified type:
1. Reduced number of independent constants:
From 9 (general ortho) to only 4
j
ji
i
ij
ij EEG
111Shear moduli are not independent
00331223112 EEEE
Independent constants can be taken as 00321 ,,, EEEE
(Implications for wavespeeds? )
2. Young’s modulus: Orientation dependence is ellipse
(not 4th order surface)
Elliptic cracks
Questions:
1. Can they be replaced by equivalent distribution of circular cracks?
2. If yes: what is the equivalent crack density ?
Results:
1. Yes, if: ellipse eccentricities are uncorrelated with orientations
2. Equiv. crack density: parameters S and PS2 must be matched
PS, - area & perimeter
References
Crack density tensor. Orthotropy:
Kachanov (1980) Continuum model of medium with cracks,
J. Eng-g Mechanics Division, 106, 1039-1051
Elliptic cracks (isotropic case, random orientations)
Budiansky and O’Connell (1976) Elastic moduli of cracked solid
Intern J. Solids & Structures 12
Elliptic cracks (anisotropy, non-random orientations & review):
Kachanov (1992) Effective elastic properties of cracked solids,
Applied Mech. Reviews, 45
Concept of and
Sayers, C. and Kachanov, M. (1995) Microcrack-induced elastic wave
anisotropy of brittle rocks, J. Geophys. Research, 100, 4149-4156
Concept of approximate elastic symmetry
Pioneering work: Fedorov (1968) Theory of elastic waves in crystals. Plenum Publ.
In rock mechanics:
Arts, Rasolofosaon and Zinzner (1991) Anisotropy due to defects in rocks,
in book Seismic Anisotropy (Soc. Of Explor. Geophys)
Approximate orthotropy, simplified orthotropy (cracks) and review:
Sevostianov and Kachanov (2007) On the concept of approximate
elastic symmetry and elliptic orthotropy, Int. J. Eng-g Sciences
Complex crack shapes
Results for circular cracks: often applied to realistic, complex shapes
Two possibilities, then:
1. Treat crack density as adjustable parameter (not always clearly said)
Link to microstructure is lost
2. Analyze quantitatively effects of complex, realistic shapes
Computational studies simple estimates
Work in progress
Complex crack shapes
Results for circular cracks: often applied to realistic, complex shapes
Two possibilities, then:
1. Treat crack density as adjustable parameter (not always clearly said)
Link to microstructure is lost
2. Analyze quantitatively
Computational studies, simple estimates
Work in progress
“Irregular” geometries
2. “Wavy” patterns
1. Flat cracks of “irregular” shapes
3. Intersections of flat cracks
Flat cracks, of “irregular” shapes
Hypothesis:
Multiple cracks can be replaced by equivalent set of circular cracks
if “irregularities” are random
Then: approximate orthotropy with reduced number of constants
Key fact: for circular crack: and are close
Irregular shape: is average over in-plane close to ?
Confirmed
Flat cracks of “irregular” shapes
Hypothesis:
Multiple cracks can be replaced by equivalent set of circular cracks
if “irregularities” are random
Then: approximate orthotropy with reduced number of constants
Key fact: for circular crack: and are close
Irregular shape: is average over in-plane close to ?
Confirmed
NB TB
TB NB
shapes for which closeness of and
confirmed computationally
NB TB
Equivalence to circular cracks
Which shape “details” are not important ?
Equivalent density of circular cracks?
- Moderate “roughness” of crack contours
- Sharpness of corner points
are not important
for effective properties
≈
≈
≈
≈
Towards “Database” of flat shapes:
Some results
on equivalence to circles
Very important factor: Contacts
between crack faces
Strong effect even if contacts are very small
They reduce crack compliance
a
r
c
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
0 = 0.10
0 = 0.25
0 = 0.50
Reff /a
Partial contacts reduce effective crack radii
reduce effective crack density a
r
c
ac
Off-center islands
Work in progress
Convex shapes: Radius of equivalent circle
Equivalence to circular crack: 1
1
ra
Multiple convex cracks:
Usual crack density retained
Its value explicitly found
r
a
centroid
rActually broader:
Any shape with r -
unique f-n of polar angle
Reference
Flat cracks, various “irregular” shapes:
Grechka and Kachanov (2006) The influence of crack shapes on
effective elasticity of fractured rocks, Geophysics 71
Non-Flat cracks
σ:HεnBnH
Flat crack x1
x2
x3
X Y
ZFrame 001 23 Jun 2006
Extra strain due to crack:
Non-Flat cracks
Hypothesis
Approximate by average over crack surface
S
nnnnnIn2
0
σ:HεnBnH
Flat crack x1
x2
x3
X Y
ZFrame 001 23 Jun 2006
H for circular crack
Extra strain due to crack:
Non-Flat cracks
Replace by large number of small tangent circles ?
average over crack surface
S
nnnnnIn2
0
σ:HεnBnH
Flat crack x1
x2
x3
X Y
ZFrame 001 23 Jun 2006
Then: Equivalence to set of circular cracks
Confirmed, except: high amplitude-to-wavelength ratio shapes
H for circular crack
Extra strain due to crack:
π/2 π/3 π/6 0 α
1.00
0.75
0.50
0.25
0.00
R
x2
x3
x1
x2
x1
π/2 π/3 π/6 0 α
-0.04
-0.03
-0.02
-0.01
0.00
π/2 π/3 π/6 0 α
π/2 π/3 π/6 0 α
0.20
0.15
0.10
0.05
0.00
Error total (Eucl.norm)
H3333
H1111
H1313
H1212
H1133
H1122
0.0
0.1
0.2
0.3
x1x2
x3
X Y
ZFrame 001 23 Jun 2006
π/2 π/3 π/6 0 α
1.00
0.75
0.50
0.25
0.00 π/2 π/3 π/6 0 α
-0.04
-0.03
-0.02
-0.01
0.00
π/2 π/3 π/6 0 α
π/2 π/3 π/6 0 α
0.20
0.15
0.10
0.05
0.00
Error total (Eucl.norm)
x1
x3
x2
H3333
H1212
H1122
H2222
H2323 H1313
x2
x1
0.0
0.1
0.2
0.3
Crack interactions: effect on overall properties
local fields: a different story
Closer look at the Non-Interaction Approximation (NIA)
Predicts: -orthotropy for any orientation distribution;
-reduced number of elastic constants (only 4)
NIA is often viewed as limited to small crack densities
Closer look at the Non-Interaction Approximation (NIA)
Predicts: -orthotropy for any orientation distribution;
-reduced number of elastic constants (only 4)
NIA is often viewed as limited to small crack densities
Importance of NIA is broader
Basic building block for approximate
schemes (differential, self-consistent, Mori-
Tanaka’s) that place non-interacting
cracks into some “effective environment”
Computational studies: NIA remains
accurate at substantial crack densities
Background considerations
Presence of crack does not affect
the average over volume stresses
ijij
ij
Note: -for each stress component !
-for cracks only (not for pores)
Amplification zones (near tips) are balanced by shielding zones
Zones contain singularity.
Zones do not – for balance, they
have shapes of long “shadows”
Amplification zones (near tips) are balanced by shielding zones
Expect: For multiple cracks
(with uncorrelated mutual positions):
amplification and shielding effects on average, balance each other
excluded
Zones contain singularity.
Zones do not – for balance, they
have shapes of long “shadows”
Amplification zones (near tips) are balanced by shielding zones
Expect: For multiple cracks
(with uncorrelated mutual positions):
amplification and shielding effects on average, balance each other
Interactions do not violate NIA much
(although local fields may be strongly affected)
excluded
Zones contain singularity.
Zones do not – for balance, they
have shapes of long “shadows”
Finite Element Modeling
• areas of stress
shielding and
amplification
balance each other
• shielding (blue)
somewhat
dominates over
amplification
(warm)
xx (Pa)
Effective Stiffnesses
e = tr()
non-interaction
approximation
Non-Interaction Approximation has satisfactory accuracy
at least up to crack densities of the order of 0.15 (relatively high)
Intersecting cracks
Intersections affect local fields (near intersections)
Effect on the overall properties: Minimal
Intersections can be ignored
(unless we are close to the percolation point)
Cracks in Anisotropic background
Available results:
• 2-D problem: for any orientation distribution of cracks in orthotropic material
• 3-D problem: only for cracks in TI material, cracks parallel to isotropy plane
Major complication:
nnIBnnBB TN
Crack compliance
normal shear
Does not hold anymore
No such thing as or
Normal & shear modes are coupled
NB TB
Cracks in Anisotropic background
Further results
Most general case (3-D solid, arbitrary anisotropy, arbitrary orientations)
Approximation:
Cracksijklijkl
effijkl SSS 0
Assuming that cracks are placed in
the ‘best-fit’ isotropic matrix
If matrix anisotropy is not too strong (ratio of Young’s moduli < 1.5-1.6)
errors are small
Cracks with fluid infill
In each pore, fluid mass is assumed constant during deformation – not Biot theory
Relevant for: high frequencies/low permeabilities
Crack aspect ratio becomes important
It controls stiffening effect of the fluid
(For DRY cracks, it is not important, if small)
qVV POREPORE Linearly compressible fluid: fluid pressure reaction to load
load
highlow } fluid pressure response
Pressure polarization: fluid pressure in a cavity depends on
- Its aspect ratio
- Its orientation with respect to
Pressure polarization is coupled with effective elastic response
Stiffening effect of fluid for a given crack
Fluid pressure response to
load, in a given crack
controlled by dimensionless parameter
(Similar to Budiansky and O’Connell’s)
Fluid compressibility Crack aspect
ratio
00
20
213
14
E
This necessitates change of the crack density parameter !
The usual crack density parameter for circular cracks
assumes that crack contribution is proportional to
It is independent of aspect ratio , reflecting the fact that the
DRY crack compliance is almost independent of it
31 ka
Ve
3a
This necessitates change of the crack density parameter !
The usual crack density parameter for circular cracks
assumes that crack contribution is proportional to
For fluid-filled cracks: the aspect ratio becomes important: it determines the
change of volume for a given displacements of crack faces
It is independent of aspect ratio , reflecting the fact that the
DRY crack compliance is almost independent of it
stiffening effect of the fluid
Ignoring in crack density parameter would distort crack compliance
contributions; effective response would not be a unique function of crack density
31 ka
Ve
3a
ijkl
E
0
20
3
18
Extra compliances due to fluid-filled cracks:
In addition to the “dry” crack density tensor
ka
V nnα
31
the second crack density parameter emerges
Violation of orthotropy
αoftermsinDRYeApproximat
jlikjlikjlikjlikijklE
S
,
00
20
4
1
23
132
kkk where
ka
V
nnnnα
1
1 3
ka
Ve
1
1 3
Isotropic case
In addition to random orientations, isotropy requires:
aspect ratios should be uncorrelated with crack orientations
Second crack density parameter (scalar)
Effective compliances
inadequate, typically?
e
E
E
E
E
DRY 15
116 2000
e
G
G
G
G
DRY 45
116 000
When can we get away with one (conventional) crack density parameter?
Isotropic case: crack aspect ratios are uncorrelated with crack sizes
e
k
CorrectionRatioAspect
k
aV
a
Ve
3
3 1
1
1
1
1
Anisotropic case: aspect ratios are uncorrelated with
crack sizes
and crack orientations
Importance of shape “irregularities” for fluid-filled cracks
Shape changes leading to significant changes in crack volume become important
Piece of material fell off
Unimportant for dry crack
(loss of material in low stress zone)
Important for fluid-filled crack
(large volume change reduces
stiffening effect of the fluid)
Variation of Young’s modulus with direction
Parallel cracks
Aspect ratios 0.01; crack density 0.1
I: Dry
II: Intermediate
III: Incompressible fluid (or very small aspect ratios, or soft matrix)
Effect of fluid: illustration
References (fluid-filled cracks, non-Biot case, high frequencies)
Pioneering work:
Budiansky and O'Connell (1976) Elastic moduli of a cracked solid, Int. J.
Solids & Structures, 12
Modified crack density parameter, general anisotropic case
(plus pores, not only cracks):
Shafiro and Kachanov (1997) Materials with fluid-filled pores of various
shapes, Int. J. Solids & Structures 34
.
Its limitations: -isotropic case, identical aspect ratios of cracks
-crack density parameter not modified
Wavy and curved cracks:
Mear, M, Sevostianov, I and Kachanov, M (2007) Elastic compliances of
non-flat cracks, Intern. J. Solids & Structures 44, 6412-6427
Intersecting cracks:
Grechka, V and Kachanov, M (2006) Effective elasticity of rocks with closely
spaced and intersecting cracks, Geophysics 71, D85-91
Overview of these topics:
Tutorial: Grechka, V. and Kachanov, M (2006) Effective elasticity of fractured
rocks: a snapshot of work in progress, Geophysics 71, W45-58
2-D Anisotropic material with arbitrarily oriented cracks
Mauge, C and Kachanov, M (1994) Effective elastic properties of an anisotropic
material with arbitrarily oriented cracks, J. Mech& Physics Solids 42, 561-584
3-D Anisotropic TI material, with cracks parallel to isotropy plane
Levin, V and Markov, M (2005) Elastic properties of inhomogeneous
transversely isotropic rocks, Intern. J. Solids & Structures 42, 393-408
Rough fractures with contacts
similarities and differences with cracks
Contacts between fracture faces are common
Rock
Contacts between fracture faces are common
Large Scale:
Ceramics Rock
1 m
Contacts between fracture faces are common
Large Scale: Small Scale:
Ceramics Rock
• Fractures:
Rough contacting surfaces with contacts
• Cracks:
Finite traction-free surfaces
Both determined by displacement discontinuities
But: are controlled by different microstructural parameters
Frequently confused
Their compliances:
Fractures vs cracks: Similarities and differences?
u
u
Strong stiffening effect of contacts:
even if they are small
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
0 = 0.10
0 = 0.25
0 = 0.50
Reff / a
(a)
Reduces “effective” radius
ac
a c
Sharp drop
when contact forms
Stiffness and conductance – in terms of relevant microstructural features
Stiffness and conductance – in terms of relevant microstructural features
Incremental
(linear elastic) response
Low stresses - wave propagation, etc
Stiffness and conductance – in terms of relevant microstructural features
What are they? Incremental
(linear elastic) response
Low stresses - wave propagation, etc
Single elliptical contact
general shape for
locally-smooth
Hertzian contacts
Single elliptical contact
general shape for
locally-smooth
Hertzian contacts
Single elliptical contact
• Incremental normal compliance of the contact
• Incremental shear compliance of the contact, along one of ellipse’s axes:
Shape factor
Shape factor
general shape for
locally-smooth
Hertzian contacts
Elliptical contact:
Effect of aspect ratio of ellipse
• Long narrow contacts are stiffer than circular contacts of the same area
• At g > 0.4, replacement by a circle of the same area is accurate
Shear anisotropy: mild Ratio of shear shape factors
along ellipse axes Normal and shear shape factors
Fracture with multiple contacts:
Normal incremental stiffness (non-interacting contacts)
• Individual stiffnesses are summed up (parallel springs)
• Individual contacts share the same displacement (non-bending plates approx).
Fracture with multiple contacts:
Normal incremental stiffness (non-interacting contacts)
• Individual stiffnesses are summed up (parallel springs)
• Individual contacts share the same displacement (non-bending plates approx).
• Normal incremental stiffness in the non-interaction approximation:
controlled by microstructural
parameter
Number of contacts per unit area
Large number of small contacts is stiffer than small number of large ones (same total area)
Fracture with multiple non-interacting contacts:
Shear to normal stiffness ratio
The ratio controls deviations from orthotropy for multiple rough fractures
somewhat larger deviations than for traction-free cracks
Fracture with multiple non-interacting contacts:
Shear to normal stiffness ratio
Effect of contact interactions
via cross-property connections
using results for conductance across rough surfaces
Compliance-resistance connection
(Barber, 2003; Sevostianov & Kachanov, 2004)
dP
dw
ER
212
For any rough interface with multiple contacts
Material resistivity
Young’s
modulus
Interface Compliance Interface Resistance
Transfer of results between two fields:
11. Contacts of complex shapes:
Rigid indenters conductance
2. Interactions of multiple contacts
Conductance elastic contacts
(Fabrikant, 1989)
(Greenwood, 1966)
ji ijin
lnaEZ
11
2
1122
2
Approximate solution for multiple contacts
(Greenwood, 1966, in the context of conductivity)
Reformulated for elasticity:
Normal compliance of
rough interface non-interaction
term interaction effect – typically dominant
Controlled by the double sum
Interactions between contacts are very strong:
decrease as with distance between contacts
negligible only at spacing two orders of magnitude larger than contact sizes
1r
Interactions between contacts are very strong:
decrease as with distance between contacts
negligible only at spacing two orders of magnitude larger than contact sizes
Compare: For inhomogeneities (inclusions, cracks) interactions are much
weaker: decrease as with distance
1r
3r
(negligible at distances of the order of inhomogeneity size)
One consequence
of mechanics of interactions between contacts (the double sum):
Closely-packed cluster is almost equivalent to a single spot – its envelope
Mechanics of contacts vs Mechanics of traction-free cracks:
Similarities and differences
Compliance of each crack
depends on its size and shape
Compliance of each fracture
depends on statistics of contacts
Insensitivity to near-edge microgeometry: Similar
Near-tip geometries for cracks (sharp or blunted)
Hertzian vs welded for contacts
≈
≈
Exact microgeometry of contact: does not matter for the incremental compliance
Ultrasonics: quality of welded spots cannot be accessed via wavespeeds
Exact microgeometry of contact: does not matter for the incremental compliance
For incremental compliances only!
Non-linearities are different:
Hertzian contacts – non-linear; welded – linear
Ultrasonics: quality of welded spots cannot be accessed via wavespeeds
Interactions between contacts within one fracture
vs Interactions between cracks/fractures
Between cracks/fractures: Interactions relatively weak, decrease as r-3
Contacts within rough fracture: Very strong interactions, decrease as r -1
Applicability of the non-interaction approximation:
For Cracks:
reasonable accuracy up to moderate crack densities
(compliances, not stiffnesses linear in crack density)
For Contacts:
very limited applicability, at spacings > 102 contact sizes
Interactions between contacts within one fracture
vs Interactions between cracks/fractures
Between cracks/fractures: Interactions relatively weak, decrease as r-3
Contacts within rough fracture: Very strong interactions, decrease as r -1
Applicability of the non-interaction approximation:
For Cracks:
reasonable accuracy up to moderate crack densities
(compliances, not stiffnesses linear in crack density)
For Contacts:
very limited applicability, at spacings > 102 contact sizes
Interactions between contacts within one fracture
vs Interactions between cracks/fractures
Between cracks/fractures: Interactions relatively weak, decrease as r-3
Contacts within rough fracture: Very strong interactions, decrease as r -1
Applicability of the non-interaction approximation:
For Cracks:
reasonable accuracy up to moderate crack densities
(compliances, not stiffnesses linear in crack density)
For Contacts:
very limited applicability, at spacings > 102 contact sizes
Complex Contact shapes - by interaction mechanics
ijij fl
1 abg
Arbitrary complex shape: break into elementary squares
Complex Contact shapes - by interaction mechanics
ijij fl
1 abg
Arbitrary complex shape: break into elementary squares
Collective effect: Greenwood’s sum:
ji ijin
lnaEZ
11
2
1122
2
non-interaction term interaction effect
Overall compliance
Squares: replace by circles (shape effect is weak)
Complex Contact shapes - by interaction mechanics
ijij fl
1 abg
Arbitrary complex shape: break into elementary squares
Collective effect: Greenwood’s sum:
ji ijin
lnaEZ
11
2
1122
2
non-interaction term interaction effect
Overall compliance
Suggested by Boyer (2001) for electric conductance
Squares: replace by circles (shape effect is weak)
Can be applied to stiffness
Conclusion:
Much can be learned by transfer of knowledge between two fields
Conductance Stiffness
The END
Viscosity of Suspensions
containing diverse particles
Effective viscosity?
ijij e 2
? stress
deviator
strain rate
deviator
Preferential (non-random)
orientations of particles
Anisotropic viscosity?
klijklij e 2
Viscosity tensor
Does the anisotropy actually exist?
If orientations are originally random (isotropy)
they gradually become non-random (aligned with flow)
Does the anisotropy actually exist?
If orientations are originally random (isotropy)
they gradually become non-random (aligned with flow)
Hypothesized by Jeffery (1922)
Experimentally observed by Taylor (1923) and others
Does the anisotropy actually exist?
Viscosity gradually becomes anisotropic
Available results:
Spherical particles (isotropy)
Available results:
Spherical particles (isotropy)
Classical formula (Einstein, 1911): 2510 Non-interaction approximation Volume
fraction
Available results:
Spherical particles (isotropy)
Classical formula (Einstein, 1911): 2510
Quadratic term correction (Batchelor, 1972):
Non-interaction approximation Volume
fraction
267 .
1
10
100
0.0 0.2 0.4 0.6
Lewis and Nielsen
Bachelor
Einstein
Shapiro and Probstein
0
Poor agreement with experimental data
(except for the initial slope)
Serious problem with Einstein’s formula
Violates rigorous bound
2
51
0
Serious problem with Einstein’s formula
Violates rigorous bound
rigid particles
incompressible fluid
12
51
0
Hashin & Shtrikman, 1963 for elasticity
Reformulated for viscosity
2
51
0
1
10
100
0.0 0.2 0.4 0.6
Lewis and Nielsen
Bachelor
Lower bound
Einstein
Shapiro and Probstein
0
Einstein’s formula: Root of the problem:
Incorrect formulation
of the non-interaction approximation
2510
Non-interaction approximation has two dual formulations:
1. Summation of viscosity contributions of particles
2. Summation of fluidity contributions
251
10
Einstein
2510
Non-interaction approximation has two dual formulations:
1. Summation of viscosity contributions of particles
2. Summation of fluidity contributions
251
10
Einstein
They coincide in the limit
But: the second one does not violate the bound !
0
1
10
100
0.0 0.2 0.4 0.6
Lewis and Nielsen
OUR
Bachelor and Green
lower bound
Einstein
Shapiro and Probstein
0
Accurate at concentrations up to 30%
2510
Note on Batchelor’s quadratic correction:
Einstein 267 .
2510
Note on Batchelor’s quadratic correction:
251
10
Einstein
2256 .
It is close to a quadratic term of Taylor’s expansion
of the proper non-interaction approximation
267 .
Analogy with elasticity of cracked solids:
eCeCE
E
1
1
1
0
stiffnesses linear in e
(“first-order” Hudson’s theory) Summation of compliance
contributions of cracks
Analogy with elasticity of cracked solids:
eCeCE
E
1
1
1
0
stiffnesses linear in e
(“first-order” Hudson’s theory) Summation of compliance
contributions of cracks
However, computational studies show:
The linearization drastically reduces the range of where the
non-interaction approximation remains accurate
e
No point in linearizing
e
?
1
00 G
Gor
E
E
The upper curve remains accurate
at much higher crack densities
Extend to:
Non-spherical shapes
Anisotropic cases
Extend to:
Non-spherical shapes
Anisotropic cases
Fluidity
contribution
of a particle
by means of:
Fluidity contribution tensor of a particle
Change in strain rate (average over V )
due to presence of particle of volume
klijklij LV
Ve
V
L-tensor
shape-dependent
transfer of results from elasticity
(via the correspondence principle)
L-tensor:
Effective elastic properties
Volume V with one inhomogeneity: strain per V under applied stress
matrix compliance
ijklijklij ΔεσSε 0
V
Effective elastic properties
Volume V with one inhomogeneity: strain per V under applied stress
matrix compliance
extra strain due to inclusion:
linear function of applied stress (linear elasticity)
klijklij σHε
ijklijklij ΔεσSε 0
compliance contribution tensor
of the inclusion
V
rigidK,G
000
2
HL
Transfer of results from elasticity:
Fluidity contribution
tensor of a particle
Compliance contribution
tensor of an inclusion
Components of fluidity contribution tensor of a
spheroidal particle as a function of its aspect ratio
(spheroid diameter kept constant)
*
1
ijkl
n n
ij ijkl kl
n
f
e V LV
1 44 2 4 43
Multiple particles
diverse shapes and orientations
Change of deformation
rate due to particles Change of fluidity
due to particles
Ellipsoids: explicit results
arbitrary mixture of diverse aspect ratios
“Irregular” shapes (some results available)
Particle shapes
Limiting case of ellipsoids: Thin Platelets
aspect ratios < 0.1
In this limit, results are independent of aspect ratios.
Volume fraction is irrelevant
distorts contributions of diverse particles
Aspect ratios 0.1 and 0.01: Contributions almost the same
Cannot be used as concentration parameter
Thin Platelets
aspect ratios < 0.1
n na
V
31
radii of platelets
1691
0
eff
Proper concentration
parameter
Isotropic case (random orientations)
Effective
viscosity
Parallel platelets - anisotropic viscosity
Parallel platelets - anisotropic viscosity
Change of fluidities due to particles ( -axis normal to platelets)
3x
27
562222011110 ff
27
3233330 f
16
912120 f
27
4011220 f
27
162233011330 ff
3
22323013130 ff
concentration parameter
n na
V
31
Platelets – arbitrary orientation distribution
Concentration parameter:
symmetric 2nd rank tensor
similar to:
crack density tensor in solid mechanics
n
na
Vmmα
31
platelets
radii normals
to platelets
Platelets – arbitrary orientation distribution
Concentration parameter:
symmetric 2nd rank tensor
similar to:
crack density tensor in solid mechanics
n
na
Vmmα
31
platelets
radii normals
to platelets
Effective viscosities:
explicitly given in terms of components
for any orientation distribution
ij
Suspensions used in hydro-fracking
Effective viscosity?
Its reduction?
Suspensions used in hydro-fracking
Effective viscosity?
Its reduction?
“Slippery” coating of proppant? Viscosity
reduction
Viscosity as function of concentration/ shape
of proppant?
Suspensions used in hydro-fracking
Effective viscosity?
Its reduction?
“Slippery” coating of proppant? Viscosity
reduction
spherical:
up to 2.5 times
reduction
Viscosity as function of concentration/ shape
of proppant?
Suspensions used in hydro-fracking
Effective viscosity?
Its reduction?
“Slippery” coating of proppant? Viscosity
reduction Shape of proppant?
spherical:
up to 2.5 times
reduction
Viscosity as function of concentration/ shape
of proppant?
Unimportant shape factors
Roughness of boundaries
≈
Corner points
sharp or blunted
≈
≈
≈
Convexity / concavity
vs
Concave shapes produce (much) stronger effect
same volume
Important shape factor:
Thin Platelets
aspect ratios < 0.1
In this limit, results are independent of aspect ratios.
Volume fraction is irrelevant
*
1
ijkl
n n
ij ijkl kl
n
f
e V LV
1 44 2 4 43
ijkljkiljlikijkl ff
f 2
0
Multiple particles
Effective fluidity
Change of deformation
rate due to particles
Change of fluidity
due to particles
Covers arbitrary mixture of shapes
(generally anisotropic)
Cannot be expressed in terms of volume fraction !
Exception: (1) identical shapes, plus (2) random orientations (isotropy)
m
mijklmijkl LV
Vf
1
Change of fluidity
due to particles
m
mijklmijkl LV
Vf
1
Cannot be expressed in terms of volume fraction !
This would distort contributions of diverse particles
Exception: (1) identical shapes, plus (2) random orientations (isotropy)
Change of fluidity
due to particles
Volume fraction
is not a correct concentration parameter !
m
mijklmijkl LV
Vf
1
Cannot be expressed in terms of volume fraction !
This would distort contributions of diverse particles
Exception: (1) identical shapes, plus (2) random orientations (isotropy)
Change of fluidity
due to particles
Ellipsoids
arbitrary mixture of
- diverse aspect ratios
- diverse orientations
m
mijklmijkl LV
Vf
1Change of fluidity
due to particles
However:
It cannot be expressed in terms of any concentration parameter
Does not exist!
is explicitly calculated
Platelets (aspect ratios < 0.1)
Effective viscosity is (almost) independent of aspect ratios.
Volume fraction is irrelevant
n na
V
31 Proper concentration
parameter
Platelets (aspect ratios < 0.1)
Effective viscosity is (almost) independent of aspect ratios.
Volume fraction is irrelevant
What if:
we DO use volume fraction as concentration parameter ?
Effective viscosity is not a unique function of it
-Double the number of platelets. Viscosity will change significantly
-Double the thickness. Almost no change
n na
V
31 Proper concentration
parameter
Thin Platelets
aspect ratios < 0.1
Further comment on:
Legitimacy of volume fraction
as concentration parameter
Platelets (aspect ratios < 0.1)
Effective viscosity is (almost) independent of aspect ratios.
Volume fraction is irrelevant
What if:
we DO use volume fraction as concentration parameter ?
n na
V
31 Proper concentration
parameter
Ellipsoids: Arbitrary mixture of
- diverse aspect ratios
- diverse orientations
Ellipsoids: Arbitrary mixture of
- diverse aspect ratios
- diverse orientations
m
mijklmijkl LV
Vf
1
Change of fluidity due to particles
is explicitly calculated
Summary
Non-interaction approximation – if formulated properly –
remains accurate up to substantial concentrations
Summary
Non-interaction approximation – if formulated properly –
remains accurate up to substantial concentrations
Summary
Non-interaction approximation – if formulated properly –
remains accurate up to substantial concentrations
Mixtures of diverse shapes (including anisotropic ones)
can be analyzed in straightforward way
Summary
Non-interaction approximation – if formulated properly –
remains accurate up to substantial concentrations
Mixtures of diverse shapes (including anisotropic ones)
can be analyzed in straightforward way
Volume fraction is not a proper concentration parameter
(except for cases of -identical shapes -isotropy)
Topics for Future Research
Geomechanics
Topics for Future Research
Fluid permeability elasticity correlation ?
Frequent statement: There is a correlation
Can estimate permeability from wavespeeds
However:
The two properties are controlled by
very different microstructural parameters
p Ku
1D’Arcy law
permeability tensor
However:
The two properties are controlled by
very different microstructural parameters
p Ku
1D’Arcy law
permeability tensor
Fissure contribution to
permeability: proportional to 3h K: in terms of k
hAV
nn 31
However:
The two properties are controlled by
very different microstructural parameters
p Ku
1D’Arcy law
permeability tensor
Fissure contribution to
permeability: proportional to 3h K: in terms of k
hAV
nn 31
in Elasticity: kA
Vnn 231
Microstructural parameters are different
Anisotropy orientation different
Correlation cannot be established
except some special cases
Topics for Future Research
Wavespeed patterns in cracked rocks
their dependence on fluid saturation
Young’s modulus: Directional Variation
I: Dry
II: Intermediate
III: Incompressible fluid (or very small aspect ratios, or soft rock)
Effect of fluid
parallel cracks
Implications for wavespeeds, extraction of information from them?
“Irregular” Morphology of Rocks- further work
Topics for Future Research
“Irregular” crack geometries
Rough contacting plates, mechanics of multiple contacts
“Irregular” Morphology of Rocks- further work
Topics for Future Research
Topics for Future Research
Fracture of rock under compression
Brittle-ductile transition
at hundreds of MPa
Viscosity of fluid suspensions
ijij e 2
? stress
deviator
strain rate
deviator
Topics for Future Research
1. Mixtures of diverse shapes (platelets, needles, …)
2. Slippery particles
The End
thank you for attention