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

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

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

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

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

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

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.

Mathematical Formulation of Directional Distribution Kanatani (1984, 1985)

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:

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

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

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

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

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

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

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

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

Sample Preparation

Localized Directional Distribution Function v directional porosity function

Directional porosity

Autocorrelation function Validation of the directional autocorrelation expression

Autocorrelation Function

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

Measured vs. Simulated ACF

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

Boundary Conditions p 1 p 2 h

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

Limitations Specific surface area

Flow Fields

increase in porosity Ottawa sand

Flow Fields silica sandOttawa sandglass beads

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

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

Moving Window Technique to Measure Heterogeneity

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)

Length Scale: 3-D Autocorrelation Function

Three-Dimensional Orientation of Aggregates

Aggregate Orientation

Damage Experiment 2 Replicates

Effect of Deformation on Void Content

Change in Void Measurements: Deformed Specimens

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

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

Model Parameters –  cohesion and adhesion

Model Parameters –  friction parameter

Model Parameters –  damage parameter

Model Parameters –  aggregate distribution

Experiments and Results “Compression” Gravel mixes

Compression Test Simulation Granite mixes

Compression Test Simulation Limestone mixes

Extension Test Simulation GravelGraniteLimestone

Lateral Strain Simulation GravelGraniteLimestone

Granite Limestone Gravel

Finite Element Simulation for Pavement Section Isotropic Anisotropic

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

Granite Limestone Gravel