Vibration of granular materials D. DUHAMEL UR Navier DES PONTS ET - PDF document

21 November 2007 Vibration of granular materials D. DUHAMEL UR Navier DES PONTS ET CHAUSSEES ECOLE NATIONALE

Outline 1. Different behaviors under vibration 2. Discrete element model 3. Long term settlement 4. Continuous model

1. Different behaviors under vibration Problem of settlement of railway tracks Force T 1 cycle T’ Time Settlement ( ≈ 10 -6 mm) Deflexion ( ≈ 1 mm) T’ Displacement Ballast Ballast Layers Layers Platform Platform

Experimental device

Settlement at 320 and 400 km/h Blochet gauche − v réelle =320 km/h Blochet droit − v réelle =400 km/h 4.5 0.3 4 0.25 1 vérin sollicité 3.5 3 vérins sollicités 0.2 3 Tassement (mm) Tassement (mm) 0.15 2.5 1 vérin sollicité 3 vérins sollicités 2 0.1 1.5 0.05 1 0 0.5 0 −0.05 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Nombre de cycles 5 Nombre de cycles 5 x 10 x 10

Threshold of acceleration at 10 m/s² Threshold of speed at 360 km/h

Fully confined Experimental device Partially confined

Fully confined samples

Partially confined samples

Most important parameters

2. Discrete element model

Equations Contact forces for spherical particules Dynamic equations 2 i d r ∑ = + i i ic m m g F 2 dt c ω 2 i d ∑ = i i ic I R F t 2 dt c Dissipative terms

Load and boundary conditions

Preparation of samples

Influence of different parameters

Comparison with experiments for static loads Steel balls, r = 4.75 mm Steel balls, r = 2.25 mm Glass balls, r = 3.5 mm Grains de ballast, r = 5.88 mm

Density versus number of cycles

partially confined fully confined � The settlement increases with the acceleration � Movement of grains in the partially confined case � Critic acceleration at ~ 1.5g, in the partially confined case

Glass balls Ballast grains Steel balls

0.25 0.2 0.15 y(m) 0.1 0.05 0 0 0.05 0.1 0.15 x(m)

Coordination pour N=80 Coordination pour N=90 Coordination pour N=70 1.5 1 5 . 5 1 . 5 7 2 5 0.18 1.75 1 . 7 5 . . 3.4 1 3.4 2.5 3.4 1.75 5 2 1 2.25 1.75 5 1.5 . 2 1 5 2 2 0.16 2 1 . 1.5 2 . 1 . 0.18 7 5 2 2 1.75 1.5 1 . 2 5 2.75 2 2 2 2 2 2.5 . 2 2.25 3.2 5 3.2 3.2 2.75 2.5 0.16 1.5 1.75 2 1.75 3 1 2 2.5 . 7 . 2 2 5 1.25 2 0.14 5 0.16 5 5 1.75 2 3 2 3 3 . 1 2 . 1 1.25 1.5 . 2.75 1.75 2 7 2.25 0.14 5 1.75 2.25 2.75 2 1.25 2.25 1.5 2 2 2 . 5 2 2 2 2.25 3 2.5 2 0.14 0.12 3 3 3 2.25 2.8 2.8 1.75 1.75 2.8 1.5 2.5 2 2 2 2 5 1 . 2.5 0.12 2 2 2 1 . . 2 . 5 2 1 . 5 1.75 2.25 2 2.75 5 5 2 2.75 2 . 7 5 2 2.6 1.5 2.6 0.12 2.6 2.25 3 2.75 3 3 0.1 3 3.25 2.25 1.75 2.75 5 2.5 7 2 . 5 2.75 . 2 . 2.75 2 2.5 5 3 2.75 0.1 2 3 2.4 2.4 2.75 2.4 2.5 2.5 2.5 2.75 0.1 2 2 . 3 3 5 2 . 7 5 2.25 . 2.5 0.08 2 2.5 3 2 2 . 7 5 5 2 . 2.75 7 5 . 3 2 . 7 5 3.25 2.2 0.08 7 2.5 3 . 2 5 3 2.2 2 2.2 2.75 . 1 5 3 3 5 . 3 0.08 7 2 2.75 2.75 5 5 3 . 3 5 7 2.75 2.75 3 . 2.75 5 2 2 2 3 3 0.06 2.75 . 5 2 3 . . 2 3 7 2 2.75 2 5 3 5 2 0.06 5 7 2.75 3 2 . 5 3 . 0.06 2 3 2 2 5 . 2.5 2.5 . 5 2 2 2.25 2.5 . 2 2.75 2 2 1.8 2 1.8 2 . 5 1.8 0.04 . 2 7 5 2.75 5 3 5 2 2 . 7 0.04 . 2 2.5 5 2.75 7 2.25 5 . 2 5 . 7 0.04 2 7 3 5 . 2.5 3 3 . 3 3 3 3 2 1.6 3.25 2.75 2.75 3 1.6 3 1.6 3 3.25 3 2.5 3.25 2 2.75 0.02 3 3 2.75 . 3 7 3 0.02 5 2.75 0.02 3.25 . 2.75 2 2.75 2.75 3 . 5 2.5 2 1.4 2 . 2.5 3 1.4 5 1.4 . 2.75 2 3 5 5 3 5 2 . 5 7 2 . 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Densité pour N=70 Densité pour N=90 Densité pour N=80 0.8 0.8 0.5 0.8 0 0.55 . 6 0.5 0.5 5 0.5 0 . 6 0.55 0.65 0.55 0.6 0.6 0.16 0.65 0.18 0.7 0.7 0.8 0.18 0.7 0.8 0.7 0.7 0.75 0.7 0.75 0.75 0.7 0 0.75 0.55 0.75 0.7 0.6 . 8 0.5 0.55 0.6 0.16 0.5 0.14 0.65 0.65 0.16 0 0.5 0.55 0.6 0.6 0.7 0.75 0.75 . 0 0.65 8 0.8 . 7 0.55 0.6 0.6 0.8 0.8 0.65 0.75 0.6 0.7 0.65 0 0.6 0.65 0.14 0.6 . 0.14 0.12 0.8 0.5 7 6 5 . 0.55 0.8 0 0.65 0.65 0 0.5 . 0.55 8 8 0.6 0.5 8 0.5 0.65 0.7 0.8 . . 0 0.7 0 . 7 0 5 0.12 0.75 0.12 0 . 5 0.6 0.55 0 . 0.6 0.75 0.65 5 0.8 0 . 6 0.5 0.75 0.1 0.8 0.8 0 . 7 0.5 0.7 5 0 . 8 0.75 0.55 7 0.8 0.7 0 0.8 0.4 0.1 . 0.75 . 5 0.4 0 7 0.1 7 0.75 0 . 8 0.4 8 0.08 . 0 . 0 0.65 0.8 0.7 7 0.7 0.8 0.65 0.6 . 0.08 0 0 0.08 0.8 0.3 0.3 0.75 . 0.8 6 0.75 0.8 0.65 5 0.06 0.6 0.3 0.06 0 0.06 0.75 0.2 . 7 0.2 0.7 5 0.04 0.2 0.04 0.7 0.8 0.7 0.75 0.04 8 0.8 . 0 0.1 0 . 0.1 8 0.02 0.02 0.65 0.75 0.1 0.7 0.02 0.8 0 . 5 0.8 8 . 6 0 0 . 7 5 0 75 0.8 0.8 0 0.6 0.7 0 0.75 . 0.75 7 0 5 0.8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.8 0.75 0.75 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

3. Computation of long term settlements Definition of residual displacement For a cyclic loading this quantity can be defined by

The parameters of the model are obtained by minimizing the quantity After k cycles of calculation using the molecular dynamic procedure, a linear estimate of the residual displacements is obtained by The logarithmic estimate is obtained by

For linear estimations, the error is given by For logarithmic estimations, the error is given by

The function to minimize is Example: obtained by a DM computation and an extrapolation with (ki = 20, k = 20, and h= 20)

Coordination Examples Positions

Numerical simulation of settlement for cyclic loads

4. Continuous model of granular media • Unilateral constitutive law in 3D : – Isotropic and homogeneous medium – Small deformation – Stress tensor negative definite ∂ = W σ ≤ σ ε ( ) ij n n 0 ∂ ε i j • Linear continuous piecewise law

Model of ballast with unilateral behavior Example : Pressure on an embankment made of ballast élastique linéaire unilatéral principale min. Contrainte déplacement vertical

Finite element model of a railway track

Vertical acceleration Two layers model • ballast : 30 cm • soil : infinite half-space Force signal by a bogie with two axles

Conclusion � Influence of the shape of the grains – Ball versus polygons – Dynamic versus static behavior � Problem a large number of cyclic loads – Long term procedure � Mechanisms of settlement � Coupling granular with continuous media for structural computations