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

Vibration of granular materials

• D. DUHAMEL

UR Navier

DES PONTS ET CHAUSSEES ECOLE NATIONALE

21 November 2007

Outline

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

T’

Problem of settlement of railway tracks

Ballast Layers Platform Ballast Layers Platform

Force Time Displacement

Settlement (≈ 10-6 mm) Deflexion (≈ 1 mm)

T’ 1 cycle

T

• 1. Different behaviors under vibration

Experimental device

Settlement at 320 and 400 km/h

0.5 1 1.5 2 x 10

5

−0.05 0.05 0.1 0.15 0.2 0.25 0.3

Nombre de cycles Tassement (mm) Blochet gauche − vréelle=320 km/h

1 vérin sollicité 3 vérins sollicités 0.5 1 1.5 2 x 10

5

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Nombre de cycles Tassement (mm) Blochet droit − vréelle=400 km/h

1 vérin sollicité 3 vérins sollicités

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

Experimental device Partially confined Fully confined

Fully confined samples

Partially confined samples

Most important parameters

• 2. Discrete element model

Dynamic equations Contact forces for spherical particules

∑ ∑

= + =

c ic t i i i c ic i i i

F R dt d I F g m dt r d m

2 2 2 2

ω

Dissipative terms

Equations

Load and boundary conditions

Preparation of samples

Influence of different parameters

Grains de ballast, r = 5.88 mm Steel balls, r = 2.25 mm Steel balls, r = 4.75 mm Glass balls, r = 3.5 mm

Comparison with experiments for static loads

Density versus number of cycles

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

partially confined fully confined

Ballast grains Steel balls Glass balls

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

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

Definition of residual displacement For a cyclic loading this quantity can be defined by

• 3. Computation of long term settlements

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)

Positions Coordination

Examples

Numerical simulation of settlement for cyclic loads

• Unilateral constitutive law in 3D :

– Isotropic and homogeneous medium – Small deformation – Stress tensor negative definite

• Linear continuous piecewise law

≤

j i ij n

n σ

ε ε σ ∂ ∂ = W ) (

• 4. Continuous model of granular media

Model of ballast with unilateral behavior

Example : Pressure on an embankment made of ballast

élastique linéaire unilatéral

Contrainte principale min.

déplacement vertical

Finite element model of a railway track

Vertical acceleration

Force signal by a bogie with two axles Two layers model

• ballast : 30 cm
• soil : infinite half-space

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