slide 1av dyskin, geomechanics group. uwa. australia mechanics of earth crust with fractal structure...

AV Dyskin, Geomechanics Group. UWA. Australia

Mechanics of Earth Crust with Fractal

Structure

by

Arcady DyskinUniversity of Western Australia


Fractal modelling Highly irregular objects, power scaling law Highly discontinuous objects, no

conventional properties like stress and strain can be defined

Questions» How can continuum mechanics be reconciled

with fractal modelling?» What kind of real systems would allow fractal

modelling?


Example: Cantor set of pores

0 0.2 0.4 0.6 0.8 10

2

4

6

8

2( )t

C( )t

tx

y

y0=1

Carpinteri's nominal stress


Plan

Self-similar mechanics Scaling of tensors Scaling of elastic moduli for materials with

cracks Average stress and strain characteristics Fracturing of self-similar materials Self-similar approximation Conclusions


Representative volume element

l H L Heterogeneous (discontinuous) material is modelled at a mesoscale by a continuum

Representative volume element (RVE) or

representative elemental volume, VH

L

H

l

Macro

Micro

Meso

H

z

x

y

z

y

x

zy

zx

yx

yz

xy

xz

H

Modelling

Interpretation

H-Continuum


Multiscale continuum mechanics

Coarse FineSCALE

Material with multiscale microstructure

Multiscale continuum modelling

H3 H2 H1

A set of continua … H1<<H2 <<H3 ...


Materials with self-similar microstructure. General

propertySelf-similar structures (i.e., the structures which look the same at any scale) have no characteristic length.Then, according to dimensional analysis, any function of length, f(H), satisfies:

11)(

)(

H

Hf

Hf

Hf

This implies the power law:

AHHf )(


Self-similar mechanics

Material with self-similar microstructure

Continuum models of different scales(H-continua)

H3 H2H1

Scaling property of H-continuum HHE )(


Self-similar elastic moduli

Hooke's law

1

2

3

23

13

12

11 12 13 16

21 22 23 26

61 62 63 66

1

2

3

23

13

12

S S S S

S S S S

S S S S

SE Eij

ij

i jii

, 1

Engineering constants

ij are bounded

2,1,,ji

i HSconstHE ijiiiji

Scaling laws

ijHSij

General scaling

21 independent

21 independent?

6 independent?


Scaling laws for tensors......

............ ,,... lmnijk HTHTTrrrT lmnijklmnnlimilijk

Proposition: ...,or0Either ...... ijkconstHTT ijkijk

1. Power functions with different exponents are linearly independent

2.

3. Homogeneous system

Proof:

pqCHC q

p

q

qq ,1,00

1

0... ... lmnnlimil Trrr

Scaling of elastic moduli and compliances

61,,)(,)( jiHaHAHcHC ijijijij

3,2,1,,,,)(,)( lkjiHaHAHcHC ijklijklijklijkl

Scaling is isotropic, prefactors can be anisotropic

or=-


Self-similar distributions of inhomogeneities

Range of self-similarity

maxmin RRR Lower cutoff(microscale)

Upper cutoff(macroscale)

mR

wRf )(

t

R

R

dRRfR v)(

max

min

3

w0 as Rmax/Rmin

4whenln

4when14)(),(v

443

mnw

mnRm

wdRRfRnRR

mmnR

R

Concentration of inhomogeneities ranging between R and nR, n>1

Normalisation:

minmax ,)(

max

min

RRNdRRf

R

R

- the total number of inhomogeneities per unit volume

Dimensionless concentration is constant


Self-similar crack distribution

Concentration of cracks is the same at every scale

4)(

R

wRf

Property: Probability that in a vicinity of a crack of size R there are inhomogeneities of sizes less than R/n, n>1:

13

)(~)( 33 nw

dRRfRnP

R

nR

- asymptotically negligible

as Rmax/Rmin (w0)

Wide distribution of sizes (Salganik, 1973)•Interaction between cracks of similar sizes can be neglected; •Interaction is important only between cracks of very different sizes

minmaxln

v

RRw t


Differential self-consistent method

Defects of one scale do not interact, p<<1.Successive application of solution for one defect in effective matrix.

Effective medium

Step 1Material with

smallest defects, non-interacting

Step 2Effective matrix

with larger defects, non-interacting

Step 3Effective matrix with next larger defects,

non-interacting

Matrix


Scaling equations

61,,,, 661211 jiaaawSa ijij

61,,,, 661211 jicccwc ijij

61,,)(,)( jiHaHAHcHC ijijijij

Contribution of the inhomogeneities at each step of the method is taken from the non-interacting approximation

Voids and stiff inclusions


Isotropic distribution of elliptical cracks

Fractal dimension, D=3.

213120

3145

8

213

3212

CCC

CCCw

Scaling equations

22

2

3

222

22

2

22

22

1

1,)(

)()(

)()(

aa

aa

a

a

a

kkkE

kC

kKkkEkk

kkC

kKkkEk

kkC

ka is the aspect ratio; K(k) and E(k) are elliptical integrals of 1st and 2nd kind

Scaling laws

wkE

k

eHE

a

)(9

8

,0


Plane with two mutually orthogonal sets of cracks

126621

12

2221122

2121111

A

AA

AA

x2

x1

Orthotropic material, plane stress

3)(

llf

minmaxln llt

2

1

1 2l is the crack length

max

min

)(2

l

l

t dllfl - total concentration


Plane with two mutually orthogonal sets of cracks

(cont)

221166221

22116611266

22116622222

22116611111

24

24

24

24

aaaa

aaaaa

aaaaa

aaaaa

Scaling equations

Fractal dimension, D=2. 0

2

2

12

1

21166

2

1

21122

21

a

aa

aa

HaA ijij

Scaling laws


Average stress and strain

HH

HH

V

Hij

HV

Hij

V

Hij

HV

Hij

dVV

dVV

x

x

x

x

)(1

lim

)(1

lim

3

3

DGij

Hij

DGij

Hij

GH

GH

Definition

Scaling

Scale H Scale G

D3 is fractal dimension

GHCHC Gijklijkl )(

H1

H2

H3

Fractal object


Fracturing of self-similar materials. General case

Simplified fracture criterion: G(11H, 12

H, …, 33H)=c

H

Homogeneous function

Scaling G(11H, 12

H, …, 33H)~HD-3, c

H~Hc

1. c <D-3: as H0, the stresses increase stronger than the strength. Defects are formed at the smallest scale: damage accumulation, possibly plastic-type behaviour2. c >D-3 : as H0, the stress fluctuations increase weaker than the strength. Defects are formed at the largest scale: a large fracture, brittle behaviour.3. c =D-3 : self-similar fracturing.


Special case of self-similarity, cH =0

•Fracturing is related to sliding over pre-determined weak planes resisted by cohesionless friction•Number per unit volume of weak planes

of sizes greater than H: MH=cH-m, m>1

•Number per unit volume of volume

elements of sizes (H, H+dH) in which

the fracture criterion is satisfied: H-DdH

H1

H2

H3

4

4

mDH HmD

cN

Number per unit volume of fractures of sizes greater than H

Gutenberg-Richter law

Weak plane


Self-similar approximationArbitrary function ),0(,0)( 1 Cfxfy

)(

)(,)()(

0

00

00 xf

xxf

x

xxfxf

Multiplications

2121

)()()( 21

xfxfxf 11

)()(1

xfxf

x

yy=x

y=f(x)

x0

lnx

lny

y=x

y=f(x)

lnx0


Summation

Necessary condition of fractal modelling

20201

021

0201

0121

)()(

)(

)()(

)(

21 )()()(

xfxf

xf

xfxf

xf

xfxfxf

)(

)(

)(

)(or

02

02

01

0121 xf

xf

xf

xf

Power functions with different exponents are linearly independent

212121 0,,,21 CCCCxxCxC

If an object allows fractal modelling, its additive characteristics must have the same logarithmic derivatives at the point of approximation


Conclusions Materials with self-similar structure can be modelled by a

sequence of continua In passing from one continuum to another, tensorial

properties and integral state variables (average stress and strain) scale by power laws. The scaling is always isotropic only pre-factors account for anisotropy

Stress distributions can be characterised by point-wise averages. They scale with the exponent 3-D

For fracturing related to self-similar distributions of pre-existing weak planes, the number of fractures obeys Gutenberg-Richter law

Not all systems exhibiting power dependencies allow self-similar approximation: the necessary condition (summation property) must be tested

slide 1av dyskin, geomechanics group. uwa. australia mechanics of earth crust with fractal structure...

Documents