1. Microstructure Analysis of Geomaterials: Directional Distribution Eyad Masad Department of Civil Engineering Texas A&M University International Workshop in Geomaterials September 25-27 Prague, Czech Republic

2. Soil Structure vs. Soil Fabric Mitchell (1993): Soil structure: combination of fabric (arrangement of particles) and interparticle bonding.

3. Applications Model microstructure parameters (anisotropy and heterogeneity). Model verification. Computer simulation of fluid flow, deformation at the microstructure level.

4. Anisotropy vs. Homogeneity A B Define G as a material property: Heterogeneity: Anisotropy : G (A1)  G(A2) A1 A2 G (A)  G(B)

5. Anisotropy vs. Homogeneity A B A1 A2 Assumptions: Aggregate material is isotropic Binder material in isotropic

6. Measurements Representative Elemental Volume l (min) l (max) n

7. Anisotropy within the RVE M ij : microstructure tensor E(l): probability density function l i denotes the unit normal of an elementary solid angle d .  represents the whole surface of a sphere representing the RVE, and d  = sin  d  d  for three dimensions, and d  = d  for two dimensions.

8. Mathematical Formulation of Directional Distribution Kanatani (1984, 1985)

9. Second Order Approximation of Directional Distribution 2 nd order directional distribution function of aggregate orientation: n(l): number of features oriented in the l-direction n a : average number of features Microstructure orientation tensor:

10. Parameters of Microstructure Distribution Tensor Tensor components: Second invariant of orientation tensor:

11. Microstructure Distribution Tensor Uniform Distribution  =1.0, Transverse Anisotropic Random Distribution  =0.0, Isotropic

12. Microstructure Quantities n+n+ n-n- Contact normal x1x1 x2x2 n+n+ n-n- Branch vector x1x1 x2x2 A B A, B : particle center n+n+ n-n- x x2x2 particle orientation

13. Correlation Function A B C  I = 1 (solids), I = 0 (voids)  M, N = number of points  i, j = distance between two points p p+h

14. Correlation Function i, j estimate of particle size S(i.j) n n2n2 slope = -s/4 estimate of pore size Two-point correlation function:  specific surface area  particle size  pore size

15. Normalized Correlation Function Normalize with respect to the solids ratio Use the spherical harmonic series with tensor notation

16. Quantifying parameters of directional distribution Average angle of inclination from the horizontal: Vector magnitude: q V.M. = 0 %>>>> random distribution q V.M. = 100% >>>> perfectly oriented distribution

17. Applications Ottawa Sand Glass Beads Silica Sand Quantifying the microstructure Low Angularity Smooth High Angularity Low Elongation Rounded High Elongation

18. Sample Preparation

20. Localized Directional Distribution Function v directional porosity function

21. Directional porosity

23. Autocorrelation function Validation of the directional autocorrelation expression

24. Autocorrelation Function

25. Simulation of Soil Microstructure Measure 3-D DirectionalACF Generate a 3-D Gaussian noise Filtering thresholding Compare ACF of the model with the actual ACF Control ACF Control the average porosity

26. Measured vs. Simulated ACF

27. Equations of Fluid Flow (two dimensional analysis) r Numerical solution of Navier-Stokes equation and the equation of continuity

28. Boundary Conditions p 1 p 2 h

29. Pressure difference maintained at inlet and outlet Periodic Boundary Conditions  u(x=0) = u(x=h)  v(x=0) = v(x=h)  u(y=0) = u(y=h)  v(y=0) = v(y=h) No slip: u s = 0, v s = 0

30. Limitations Specific surface area

31. Flow Fields

32. increase in porosity Ottawa sand

33. Flow Fields silica sandOttawa sandglass beads

34. Asphalt Mixes To quantify aggregates distribution 0 <  < 1 (= 0.5 for asphalt mixes)  

35. Aggregate Orientation in Asphalt Concrete Aggregate orientation exhibits transverse anisotropy (axisymmetry) with respect to the horizontal direction.

36. Moving Window Technique to Measure Heterogeneity

37. Length Scale: Autocorrelation Function r = (i 2 +j 2 ) 0.5 Two-point ACF is given as: Isotropic: S is independent on direction of i and j. Weak Homogeneity: S is not dependent on location (x,y)

38. Length Scale: 3-D Autocorrelation Function

39. Three-Dimensional Orientation of Aggregates

40. Aggregate Orientation

41. Damage Experiment 2 Replicates

42. Effect of Deformation on Void Content

43. Change in Void Measurements: Deformed Specimens

44. Damage Evolution Top Region Middle Region Bottom Region Strain: 0%Strain: 1%Strain: 2%Strain: 4% Strain: 8%

45. Extended Drucker-Prager Yield Surface Hardening/softening Shear, and stress path

46. Model Parameters –  cohesion and adhesion

47. Model Parameters –  friction parameter

48. Model Parameters –  damage parameter

49. Model Parameters –  aggregate distribution

51. Experiments and Results “Compression” Gravel mixes

52. Compression Test Simulation Granite mixes

53. Compression Test Simulation Limestone mixes

54. Extension Test Simulation GravelGraniteLimestone

55. Lateral Strain Simulation GravelGraniteLimestone

56. Granite Limestone Gravel

58. Finite Element Simulation for Pavement Section Isotropic Anisotropic

59. Effect of Anisotropy on Permanent Deformation a) Isotropic layer (  =0) b) Anisotropic layer (  =30 percent)

60. Granite Limestone Gravel