Mechanics of Earth Crust with Fractal Structure

1. Mechanics of Earth Crust with Fractal Structure by Arcady Dyskin University of Western Australia

2. 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?

3. Example: Cantor set of pores sy0=1 sy Carpinteri's nominal stress x

4. 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

5. z sz L tzy tzx tyz txz txy sy tyx y sx l H H x Representative volume element Representative volume element (RVE) or representative elemental volume, VH Heterogeneous (discontinuous) material is modelled at amesoscaleby a continuum Macro Meso Modelling Micro H-Continuum H Interpretation

6. Multiscale continuum mechanics Material with multiscale microstructure Multiscale continuum modelling A set of continua … H1<<H2 <<H3 ... H3 H2 H1 Coarse Fine SCALE

7. 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. Self-similar mechanics H1 H3 H2 Material with self-similar microstructure Continuum models of different scales (H-continua) Scaling property of H-continuum

9. Self-similar elastic moduli General scaling Hooke's law 21 independent Engineering constants 21 independent? nij are bounded 6 independent? Scaling laws

10. Scaling laws for tensors Proposition: Scaling of elastic moduli and compliances Proof: 1. Power functions with different exponents are linearly independent 2. 3. Homogeneous system a=-b or Scaling is isotropic, prefactors can be anisotropic

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

12. 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 Rmax/Rmin (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. Differential self-consistent method Matrix Defects of one scale do not interact, Dp<<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

14. Contribution of the inhomogeneities at each step of the method is taken from the non-interacting approximation Scaling equations Voids and stiff inclusions

15. Isotropic distribution of elliptical cracks Scaling equations Scaling laws kais the aspect ratio; K(k) and E(k) are elliptical integrals of 1st and 2nd kind Fractal dimension, D=3.

16. x2 x1 Plane with two mutually orthogonal sets of cracks - total concentration l is the crack length w = w1 + w2 Orthotropic material, plane stress w1 w2

17. Plane with two mutually orthogonal sets of cracks (cont) Scaling laws Scaling equations Fractal dimension, D=2.

18. Average stress and strain Definition Scaling Scale G Scale H D3 is fractal dimension Fractal object

19. Fracturing of self-similar materials. General case Simplified fracture criterion: G(s11H, s12H, …, s33H)=scH Homogeneous function Scaling G(s11H, s12H, …, s33H)~HD-3, scH~Hac 1. ac <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. ac >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. ac =D-3: self-similar fracturing.

20. H 3 H2 H 1 Special case of self-similarity, scH=0 Weak plane • 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: • mH-DdH Number per unit volume of fractures of sizes greater than H Gutenberg-Richter law

21. Self-similar approximation Arbitrary function y lny y=f(x) y=xa y=f(x) Multiplications y=xa lnx x x0 lnx0

22. 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. 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