Cracks€¦ · Cracks: Effective Elastic Properties - Crack interactions – local fields (stress shielding and amplification) - Large crack – “cloud” of microcracks Review

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!





V

k

εσ:Sε0

Overall strain

(per volume V)

Displacement jump across crack

Matrix compliance dS

VkS

unnu1

Not necessarily!





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


Complex crack shapes

Anisotropy?

Crack density

Parameters?

More complex –




Anisotropic background (shale)

Anisotropy?

Crack density

Parameters?

More complex –





Interactions

Intersections

Anisotropy?

shield amplify

Crack density

Parameters?

More complex –


Effect on local fields, and on overall response?




Interactions

Intersections

Anisotropy?

shield amplify

Crack density

Parameters?

More complex –






Interactions

Intersections

Fluid infill

Anisotropy?

shield amplify

Crack stiffening

Changes in anisotropy

Crack density

Parameters?

More complex –



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


They are close

21 0TN BB

a

E

BT

00

20

23

132

σnnnnBBσnnBnbbn TNT :

Crack-generated strain:

relatively small


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


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


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


H for circular 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


Then: Equivalence to set of circular cracks

Confirmed, except: high amplitude-to-wavelength ratio shapes

H for circular 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


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”


Expect: For multiple cracks

(with uncorrelated mutual positions):

amplification and shielding effects on average, balance each other

excluded





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




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


Large Scale:

Ceramics Rock

1 m


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


Incremental

(linear elastic) response

Low stresses - wave propagation, etc


What are they? Incremental

(linear elastic) response

Low stresses - wave propagation, etc

Single elliptical contact

general shape for

locally-smooth

Hertzian contacts


general shape for

locally-smooth

Hertzian contacts


• 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






For Cracks:



For Contacts:







For Cracks:



For Contacts:


Complex Contact shapes - by interaction mechanics

ijij fl

1 abg

Arbitrary complex shape: break into elementary squares


ijij fl

1 abg


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)


ijij fl

1 abg


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)


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


Viscosity gradually becomes anisotropic

Available results:

Spherical particles (isotropy)

Available results:


Classical formula (Einstein, 1911): 2510 Non-interaction approximation Volume

fraction

Available results:


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


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


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


Its reduction?



Its reduction?

“Slippery” coating of proppant? Viscosity

reduction

Viscosity as function of concentration/ shape

of proppant?



Its reduction?


reduction

spherical:

up to 2.5 times

reduction


of proppant?



Its reduction?


reduction Shape of proppant?

spherical:

up to 2.5 times

reduction


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.


*

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


This would distort contributions of diverse particles


Change of fluidity

due to particles

Volume fraction

is not a correct concentration parameter !

m

mijklmijkl LV

Vf

1


This would distort contributions of diverse particles


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.


n na

V

31 Proper concentration

parameter




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


parameter

Thin Platelets

aspect ratios < 0.1

Further comment on:

Legitimacy of volume fraction

as concentration parameter




What if:

we DO use volume fraction as concentration parameter ?

n na

V


parameter

Ellipsoids: Arbitrary mixture of



Ellipsoids: Arbitrary mixture of



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



Summary



Mixtures of diverse shapes (including anisotropic ones)

can be analyzed in straightforward way

Summary



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


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:



p Ku

1D’Arcy law

permeability tensor

Fissure contribution to

permeability: proportional to 3h K: in terms of k

hAV

nn 31

However:



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


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


“Irregular” crack geometries

Rough contacting plates, mechanics of multiple contacts

“Irregular” Morphology of Rocks- further work



Fracture of rock under compression

Brittle-ductile transition

at hundreds of MPa

Viscosity of fluid suspensions

ijij e 2

? stress

deviator

strain rate

deviator


1. Mixtures of diverse shapes (platelets, needles, …)

2. Slippery particles

The End

thank you for attention

Cracks€¦ · Cracks: Effective Elastic Properties - Crack interactions – local fields (stress shielding and amplification) - Large crack – “cloud” of microcracks Review

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