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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

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

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

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

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

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

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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.

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Mathematical Formulation of Directional Distribution Kanatani (1984, 1985)

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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:

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

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

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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

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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

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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

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

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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

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

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Sample Preparation

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Localized Directional Distribution Function v directional porosity function

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Directional porosity

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Autocorrelation function Validation of the directional autocorrelation expression

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Autocorrelation Function

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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

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Measured vs. Simulated ACF

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

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Boundary Conditions p 1 p 2 h

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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

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Limitations Specific surface area

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Flow Fields

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increase in porosity Ottawa sand

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Flow Fields silica sandOttawa sandglass beads

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

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

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Moving Window Technique to Measure Heterogeneity

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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)

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Length Scale: 3-D Autocorrelation Function

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Three-Dimensional Orientation of Aggregates

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Aggregate Orientation

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Damage Experiment 2 Replicates

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Effect of Deformation on Void Content

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Change in Void Measurements: Deformed Specimens

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

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

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Model Parameters – cohesion and adhesion

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Model Parameters – friction parameter

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Model Parameters – damage parameter

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Model Parameters – aggregate distribution

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Experiments and Results “Compression” Gravel mixes

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Compression Test Simulation Granite mixes

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Compression Test Simulation Limestone mixes

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Extension Test Simulation GravelGraniteLimestone

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Lateral Strain Simulation GravelGraniteLimestone

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Granite Limestone Gravel

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Finite Element Simulation for Pavement Section Isotropic Anisotropic

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

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Granite Limestone Gravel

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