Discrete Element Modeling of Soils as Granular Materials

Discrete Element Modeling of Soils as Granular Materials Matthew R. Kuhn Donald P. Shiley School of Engineering University of Portland National Science Foundation Grant No. NEESR-936408

Outline Discrete Element Method (DEM) 1 Discrete Element Method (DEM) 2 3

Contents Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM 1 Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM 2 3

DEM assembly of 6400 particles Background of DEM Coding and challenges Alternatives to DEM

Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM Background Granular media are modeled with individual particles Peter A. Cundall (1971) 1979 Geotechnique paper

DEM Algorithm Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM A finite difference (time-stepping) algorithm, in which Elemental particles (spheres, polyhedra, etc.) Interact in a pair-wise manner (contacts), such that the Imbalances in the forces on each particle Impel the particles to new positions With each time step Via Newton s equations of motion

DEM Software Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM Commercial software PFC3D Itasca, Inc. EDEM DEM Solutions Open source software Yade yet another dynamics engine ESyS LAMMPS OVAL

Coding Challenges Background of DEM Coding and challenges Alternatives to DEM Particle shapes Spheres = Easy Other shapes = Difficult Contact detection: an N 2 problem Contact force models Linear springs = Easy Hertz-Mindlin springs = Difficult Real (soil) particle interactions = Not yet attempted Problem types Element tests Field problems (realistic boundaries) Modeling real soil problems = Very difficult

Background of DEM Coding and challenges Alternatives to DEM DEM modeling disadvantages Shortcomings of DEM simulations: Realistic particle shapes and arrangements are difficult to create and to calibrate. Relative density is difficult to surmise. Roughness, texture, and sharp edges of particles are not modeled. Idealized contact models (Hertz-Mindlin, etc.) Particle breakage or chipping is (usually) disallowed.

Alternatives Discrete Element Method (DEM) Background of DEM Coding and challenges Alternatives to DEM Alternative computational methods: Discontinuous Deformation Analysis (DDA) G.-H. Shi and Y. Kishino [K][u]+[C][ u]+[m][ü] = 0 Contact Dynamics M. Jean and J.-J. Moreau Inequality constraints, non-linear programming Event-driven Models Instantaneous collision-based

Contents Discrete Element Method (DEM) Undrained loading & static liquefaction Cyclic liquefaction Severity measures 1 Discrete Element Method (DEM) 2 Undrained loading & static liquefaction Cyclic liquefaction Severity measures 3

DEM assembly of 6400 particles Undrained loading & static liquefaction Cyclic liquefaction Severity measures

Particle shape Discrete Element Method (DEM) Undrained loading & static liquefaction Cyclic liquefaction Severity measures

DEM model Discrete Element Method (DEM) Undrained loading & static liquefaction Cyclic liquefaction Severity measures Contact properties: Hertz-Mindlin (elastic-frictional) contact model E = 29 GPa, ν = 0.15 µ = 0.60 friction coefficient

Verifying the DEM model Undrained loading & static liquefaction Cyclic liquefaction Severity measures Undrained triaxial compression and extension tests 200 150 DEM, e = 0.734 Nevada Sand, e = 0.729 Nevada Sand, e = 0.734 Deviator stress, q, kpa 100 50 0-50 -100 0 50 100 Mean stress, p, kpa 150

Verifying the DEM model Undrained loading & static liquefaction Cyclic liquefaction Severity measures Undrained triaxial compression tests range of densities Deviator stress, q, kpa 150 100 50 Void ratio, e 0.789 0.766 0.734 0.641 0 0 50 Mean stress, p, kpa 100

Undrained simple-shear results Undrained loading & static liquefaction Cyclic liquefaction Severity measures Undrained simple-shear: Shear stress, τ, kpa 40 30 20 10 Void ratio, e = 0.734 Critical state line Phase transformation line Instability line 0 0 20 40 60 Mean stress, p, kpa 80

DEM modeling advantages Undrained loading & static liquefaction Cyclic liquefaction Severity measures Modeling soil behavior with DEM element simulations: Experiments can be initiated (or restarted) from the same assembly. Full stress and strain tensors can be measured. Arbitrary control of 6 stress or strain increments. Behavior simulated in the absence of shear bands.

Static liquefaction Undrained loading & static liquefaction Cyclic liquefaction Severity measures Stress path for inducing static liquefaction

Static liquefaction Undrained loading & static liquefaction Cyclic liquefaction Severity measures Drained shearing followed by undrained shearing: 50 Shear stress, τ, kpa 40 30 20 10 Void ratio, e = 0.734 0 0 20 40 60 Mean stress, p, kpa 80

Cyclic liquefaction Undrained loading & static liquefaction Cyclic liquefaction Severity measures Cyclic liquefaction simulations: Two loading cases Case I Uniform amplitude cyclic shearing Case II Erratic, seismic shearing Severity Measure for predicting initial liquefaction

DEM assembly of 6400 particles Undrained loading & static liquefaction Cyclic liquefaction Severity measures

Case I: Uniform cyclic shearing Undrained loading & static liquefaction Cyclic liquefaction Severity measures Uniform shearing amplitude: Shear strain, γ Time, t Shear stress, τ Time, t Control strain rate γ in a sawtooth pattern until the targeted shear stress τ is attained.

Case I: Uniform cyclic shearing Undrained loading & static liquefaction Cyclic liquefaction Severity measures Conditions: τ = ±16 kpa, p o = 80 kpa Shear stress, τ/po Mean stress, p/p o Shear strain, γ, percent 0 0.2 0.4 0.6 0.8 1-1.5-1 -0.5 0 0.5 1 0.2 0.1 0-0.1-0.2 1.5 0.2 0.1 0-0.1-0.2 Number of cycles 6 4 2 0 1 0.8 0.6 0.4 0.2 0 Pore pressure ratio, r u 6 4 2 0 1.5 1 0.5 0-0.5-1 -1.5 Shear strain, γ, percent

Case I: Uniform cyclic shearing Undrained loading & static liquefaction Cyclic liquefaction Severity measures Liquefaction curves 0.3 Cyclic stress ratio, τ/po 0.25 0.2 0.15 0.1 0.05 Void ratio, e 0.734 0.641 0 1 10 Cycles to failure 100

Case II: Seismic shearing Undrained loading & static liquefaction Cyclic liquefaction Severity measures Select 24 sequences of seismic loading (Dr. Steven L. Kramer) 1 Earthquake ground accelerations from PEER data base Acceleration, a/g 0.1 0.05 0-0.05-0.1-0.15 Time, t Landers, 1992 M = 7.3 MCF000 2 Create CSR, cyclic shear record (Dr. Kramer) Shear stress, τ/po 0.2 0.1 0-0.1-0.2 Time, t 3 Scale the CSR to prolong pre-liquefaction Scaled, α τ/po 0.2 0.1 0-0.1-0.2 Time, t SHAKE91 Scale factor: α = 0.531

Case II: Seismic shearing Undrained loading & static liquefaction Cyclic liquefaction Severity measures Landers 1992 CSR record, scaling factor α = 0.531 Undrained cyclic simple-shear e = 0.641 po = 80 kpa Landers MCF000 α = 0.531 Shear stress, τ/po 0.1 0-0.1 Mean stress, p/po 0 0.2 0.4 0.6 0.8 1 Shear strain, γ, percent -0.02 0 0.02 0.1 0-0.1 60 60 Time, sec 40 20 40 20 0-0.1 0 0.1 Shear stress, τ/po 1 0.8 0.6 0.4 0.2 0 Pore pressure ratio, ru -0.02 0 0.02 Shear strain, γ, percent 0

Undrained loading & static liquefaction Cyclic liquefaction Severity measures Severity Measures for cyclic loading Ranking the severities, 1/α, of 24 stress records, as surmised from DEM simulations

Severity Measures Undrained loading & static liquefaction Cyclic liquefaction Severity measures Severity Measure : a scalar predictor of initial liquefaction computed from a cyclic stress (or strain) record Shear stress, τ/po 0.2 0.1 0-0.1-0.2 Time, t = Scalar value at initial liquefaction

Severity Measures Undrained loading & static liquefaction Cyclic liquefaction Severity measures Shear stress, τ/po 0.2 0.1 0-0.1-0.2 Time, t Possible Severity Measures for the 24 stress records: Maximum shear stress, τ/p o max Energy demand, τ dε plastic Strain path, dε 2 Stress path, τ dτ p o p Use DEM results to test the efficiency and sufficiency of each Severity Measure.

Severity Measures Undrained loading & static liquefaction Cyclic liquefaction Severity measures Efficiencies of four Severity Measures: 24 cyclic stress records

Severity Measures Undrained loading & static liquefaction Cyclic liquefaction Severity measures Sufficiency of the Maximum Shear Stress as a Severity Measure: τ/p o max

Severity Measures Undrained loading & static liquefaction Cyclic liquefaction Severity measures Sufficiency of a stress path scalar as a Severity Measure: τ dτ p o p

Contents Discrete Element Method (DEM) Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects 1 Discrete Element Method (DEM) 2 3 Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects

Granular Mechanics Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands and non-classical continua Shear bands have a characteristic thickness Granular materials have an inherent length scale This scale is not acessible via classical continuum mechanics

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands in DEM simulation free deformation

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands in DEM simulation free deformation Band Displacement

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands in DEM simulation free deformation Band Displacement Shear strain 0.10

Non-classical continuum models Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Continuum models with inherent length scale: 1 Cosserat / micropolar continua 2 Strain gradient-dependent material models τ = f (ǫ, ǫ) Simple material τ = f (ǫ, ǫ, ǫ, ( ǫ),...) Gradient-dependent material 3 Non-local material material models

Strain gradient-dependent materials Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Does stress depend upon the spatial gradients of strain?

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands in DEM simulation free deformation Band Displacement

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Constrained deformation using body forces

Effect of the first strain gradient Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects An increasing first gradient, γ, has a softening effect.

Effect of the second strain gradient Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects 0.6 0.4 0.2 0 0 0.02 n γ D 2 50/γ 0 0 2-0.0088 4-0.035 6-0.079 8 0.04 0.06 An increasing second gradient, γ, has a hardening effect.

Shear bands Discrete Element Method (DEM) Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Shear bands and gradient-dependent behavior: Persistent bands develop near the peak stress state Shear strain γ is non-uniform within a shear band Shear stress depends upon γ, γ, and γ In incremental form, dτ = f(dγ, dγ, dγ ) Shear stress is constant within a shear band: dτ/dx 2 = 0 Can an incremental model explain the profile of strain within a shear band?

Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects Solution of the incremental model: 65 60 55 50 45 40 35 30 0 0.02 Shear band 15 Model Gradient-dependent 1 Model model 2 10 5 0-5 -10-15 0.04 0.06

Questions? Discrete Element Method (DEM) Shear bands and non-classical continua Strain gradient-dependent materials DEM measurement of strain gradient effects