Continuous and discrete models for simulating granular assemblies Akke S.J. Suiker Delft University of Technology Faculty of Aerospace Engineering Chair of Engineering Mechanics January 1, 2019

Configuration of Lattice Graphical representation of 9-cell square lattice model Suiker, Metrikine, de Borst, Int. J. Sol. Struct, 38, 1563-1583, 2001 Suiker & de Borst, Phil. Trans. Roy. Soc. A., 363, 2543-2580, 2005 January 1, 2019

Equations of motion lattice January 1, 2019

Long-wave approximation of EOM (I.A. Kunin, 1983) Replace discrete kinematic d.o.f.’s by continuous field variables: Replace discrete d.o.f.’s of neighbouring cells by second-order Taylor approximations of continuous field variables: January 1, 2019

Equations of motion in long wave-approximation January 1, 2019

Equations of motion for Cosserat continuum (Cosserat E. , Cosserat F Equations of motion for Cosserat continuum (Cosserat E., Cosserat F., 1909; Günther, W., 1958; Schaefer, H., 1962; Mindlin, R.D., 1964; Eringen, A.C., 1968; Mühlhaus, H.-B., 1989; de Borst, R., 1991) The Cosserat continuum model is useful for studying: Localised failure problems, where rotation of grains is important High-frequency wave propagation, with deformation patterns of short wavelengths January 1, 2019

Mapping long-wave approximation on Cosserat model Relation between continuum material parameters and lattice parameters: Constraints that have to be satisfied to match the anisotropic lattice model with the isotropic Cosserat continuum model: January 1, 2019

Configuration reduced lattice Graphical representation of reduced 9-cell square lattice model January 1, 2019

Dispersion relations for plane harmonic waves Lattice Continuum Substitution into equations of motion yields: Dispersion relations: January 1, 2019

Direction of propagation (kx,kz) = (0,k) Dispersion curves for 9-cell square lattice and Cosserat continuum January 1, 2019

Second-gradient micro-polar model (microstructural approach) Constitutive coefficients are of the form (using ) Suiker, de Borst, Chang, Acta Mech., 149, 161-180, 2001 Suiker, de Borst, Chang, Acta Mech., 149, 181-200, 2001 Suiker & de Borst, Phil. Trans. Roy. Soc. A., 363, 2543-2580, 2005 January 1, 2019

Reduced forms of the second-gradient micro-polar model Linear elastic model, C(1) to C(6) = 0,: Second-gradient model, C(3) to C(6) = 0, (Chang & Gao, 1995): Cosserat model, C(1), C(2) and C(4) = 0, (Chang & Ma, 1992): January 1, 2019

Dispersion curves for various models Dispersion curves for compression wave, shear wave and micro-rotational wave January 1, 2019

Boundary value problem Layer of thickness H, consisting of equi-sized particles of diameter d - forced vibration under moving load Suiker, Metrikine, de Borst, J. Sound Vibr., 240, 1-18, 2001 Suiker, Metrikine, de Borst, J. Sound Vibr., 240, 19-39, 2001 January 1, 2019

Cells of square lattice Inner cell Boundary cell January 1, 2019

Response of layer to a moving load 4 boundary conditions (in Fourier domain): - boundary cells at top of layer subjected to moving load in z-direction and free of loading in x-direction - displacements at bottom of layer are zero (in x- and z-directions) Substituting harmonic displacements into these 4 boundary conditions gives: Solve above system, and transform solution to time domain by Inverse Fourier Transform (numerical). January 1, 2019

Displacement profile (H=300mm) (uz taken at 0.2H below layer surface) Case 1 Case 2 Velocity dependence Case 3 Harmonic load January 1, 2019

Model Configuration Cuboidal volume of randomly packed, equi-sized, cohesionless spheres (initial porosity is 0.382). Suiker & Fleck, J. Appl. Mech., 71, 350-358, 2004 January 1, 2019

Stress-strain Response at various Contact Friction Stress-strain response for various contact friction angles January 1, 2019

Effect of Contact Friction on Sample Strength Macroscopic friction angle versus contact friction angle January 1, 2019

Effect of Particle Redistribution Three different kinematic conditions: Particle sliding and particle rotation are allowed Particle sliding is allowed, particle rotation is prevented Particle sliding is allowed in correspondence with an affine deformation field, particle rotation is prevented. January 1, 2019

Stress-strain Responses left: Volumetric strain versus hydrostatic stress (volumetric deformation path ) right: Deviatoric strain versus deviatoric stress (deviatoric deformation path ) January 1, 2019

Collapse Contour in the Deviatoric Plane Left: Collapse contour for unconstrained and constrained particle rotation ( ) Right: Collapse contour for DEM model (unconstrained particle rotation) and various continuum models January 1, 2019

Points of discussion Higher-order continuum models approach discrete models accurately up to a certain wavelength of deformation Higher-order continuum models may be unstable for small wavelengths;  remedy: inclusion of higher-order time derivatives (and coupled time-space derivatives) Deformations with wavelengths < few times the particle diameter can not be decribed accurately with continuum models The number of constitutive coefficients increases drastically when continuum models are further kinematically enhanced (i.e., 4th-order, 6th-order etc.) January 1, 2019