1. Slide 1AV Dyskin, Geomechanics Group. UWA. Australia Mechanics of Earth Crust with Fractal Structure by Arcady Dyskin University of Western Australia

2. Slide 2AV Dyskin, Geomechanics Group. UWA. Australia Fractal modelling l Highly irregular objects, power scaling law l Highly discontinuous objects, no conventional properties like stress and strain can be defined l Questions »How can continuum mechanics be reconciled with fractal modelling? »What kind of real systems would allow fractal modelling?

3. Slide 3AV Dyskin, Geomechanics Group. UWA. Australia Example: Cantor set of pores x yy  y 0 =1 Carpinteri's nominal stress

4. Slide 4AV Dyskin, Geomechanics Group. UWA. Australia Plan l Self-similar mechanics l Scaling of tensors l Scaling of elastic moduli for materials with cracks l Average stress and strain characteristics l Fracturing of self-similar materials l Self-similar approximation l Conclusions

5. Slide 5AV Dyskin, Geomechanics Group. UWA. Australia Representative volume element Heterogeneous (discontinuous) material is modelled at a mesoscale by a continuum Representative volume element (RVE) or representative elemental volume, V H L H l Macro Micro Meso H zz x y z yy xx  zy  zx  yx  yz  xy  xz H Modelling Interpretation H-Continuum

6. Slide 6AV Dyskin, Geomechanics Group. UWA. Australia Multiscale continuum mechanics Coarse Fine SCALE Material with multiscale microstructure Multiscale continuum modelling H3H3 H2H2 H1H1 A set of continua … H 1 <<H 2 <<H 3...

7. Slide 7AV Dyskin, Geomechanics Group. UWA. Australia Materials with self-similar microstructure. General property Self-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: This implies the power law:

8. Slide 8AV Dyskin, Geomechanics Group. UWA. Australia Self-similar mechanics Material with self-similar microstructure Continuum models of different scales (H-continua) H3H3 H2H2 H1H1 Scaling property of H-continuum

9. Slide 9AV Dyskin, Geomechanics Group. UWA. Australia Self-similar elastic moduli Hooke's law Engineering constants ij are bounded Scaling laws General scaling 21 independent 21 independent? 6 independent?

10. Slide 10AV Dyskin, Geomechanics Group. UWA. Australia Scaling laws for tensors Proposition: 1. Power functions with different exponents are linearly independent 2. 3. Homogeneous system Proof: Scaling of elastic moduli and compliances Scaling is isotropic, prefactors can be anisotropic or  =- 

11. Slide 11AV Dyskin, Geomechanics Group. UWA. Australia Self-similar distributions of inhomogeneities Range of self-similarity Lower cutoff (microscale) Upper cutoff (macroscale) w  0 as R max /R min  Concentration of inhomogeneities ranging between R and nR, n>1 Normalisation: - the total number of inhomogeneities per unit volume Dimensionless concentration is constant

12. Slide 12AV Dyskin, Geomechanics Group. UWA. Australia Self-similar crack distribution Concentration of cracks is the same at every scale Property: Probability that in a vicinity of a crack of size R there are inhomogeneities of sizes less than R/n, n>1: - asymptotically negligible as R max /R min  (w  0) 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

13. Slide 13AV Dyskin, Geomechanics Group. UWA. Australia 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 1 Material with smallest defects, non-interacting Step 2 Effective matrix with larger defects, non-interacting Step 3 Effective matrix with next larger defects, non-interacting Matrix

14. Slide 14AV Dyskin, Geomechanics Group. UWA. Australia Scaling equations Contribution of the inhomogeneities at each step of the method is taken from the non-interacting approximation Voids and stiff inclusions

15. Slide 15AV Dyskin, Geomechanics Group. UWA. Australia Isotropic distribution of elliptical cracks Fractal dimension, D=3. Scaling equations k a is the aspect ratio; K(k) and E(k) are elliptical integrals of 1 st and 2 nd kind Scaling laws

16. Slide 16AV Dyskin, Geomechanics Group. UWA. Australia Plane with two mutually orthogonal sets of cracks x2x2 x1x1 Orthotropic material, plane stress 22 11  1  2 l is the crack length - total concentration

17. Slide 17AV Dyskin, Geomechanics Group. UWA. Australia Plane with two mutually orthogonal sets of cracks (cont) Scaling equations Fractal dimension, D=2. Scaling laws

18. Slide 18AV Dyskin, Geomechanics Group. UWA. Australia Average stress and strain Definition Scaling Scale H Scale G D  3 is fractal dimension Fractal object

19. Slide 19AV Dyskin, Geomechanics Group. UWA. Australia Fracturing of self-similar materials. General case Simplified fracture criterion: G(  11 H,  12 H, …,  33 H )=  c H Homogeneous function Scaling G(  11 H,  12 H, …,  33 H )~H D-3,  c H ~H  c 1.  c <D-3: as H  0, the stresses increase stronger than the strength. Defects are formed at the smallest scale: damage accumulation, possibly plastic-type behaviour 2.  c >D-3 : as H  0, 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.

20. Slide 20AV Dyskin, Geomechanics Group. UWA. Australia Special case of self-similarity,  c H =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 : M H =cH -m, m>1 Number per unit volume of volume elements of sizes ( H, H+dH) in which the fracture criterion is satisfied:  H -D dH H 1 H2H2 H 3 Number per unit volume of fractures of sizes greater than H Gutenberg-Richter law Weak plane

21. Slide 21AV Dyskin, Geomechanics Group. UWA. Australia Self-similar approximation Arbitrary function Multiplications x y y=xy=x y=f(x)y=f(x) x0x0 lnx lny y=xy=x y=f(x)y=f(x) lnx 0

22. Slide 22AV Dyskin, Geomechanics Group. UWA. Australia Summation Necessary condition of fractal modelling Power functions with different exponents are linearly independent If an object allows fractal modelling, its additive characteristics must have the same logarithmic derivatives at the point of approximation

23. Slide 23AV Dyskin, Geomechanics Group. UWA. Australia Conclusions l Materials with self-similar structure can be modelled by a sequence of continua l 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 l Stress distributions can be characterised by point-wise averages. They scale with the exponent 3-D l For fracturing related to self-similar distributions of pre- existing weak planes, the number of fractures obeys Gutenberg-Richter law l Not all systems exhibiting power dependencies allow self- similar approximation: the necessary condition (summation property) must be tested