1

MSM, CTW, UTwente, NL

Stefan Luding, s.luding@utwente.nl

Hypoplastic constitutive modeling

Stefan Luding

MSM, CTW, UTwente, NL

Thanks to:

K. Nuebel (Zueblin, Stuttgart)

J. Tejchman (Gdansk)

Andres A. Pena O. (Stuttgart)

DFG, FOM

Granular Materials

Real:

• sand, soil, rock,

• grain, rice, lentils,

• powder, pills, granulate,

• … and many others …

Model Granular Materials

• steel/aluminum spheres

• spheres with dissipation/friction/cohesion

Why ?

• `Many particle’ simulations work

for small systems only (104- 106)

• Industrial scale applications rely on FEM

• FEM relies on continuum mechanics

+ constitutive relations

• `Micro-macro’ can provide those !

• Homogenization/Averaging

2

Single particle

Contacts

Many particle

simulation

Continuum Theory

Contents E

x p e

r i

m e

n t s

• Introduction

• Results DEM

• Micro-Macro transition

• Anisotropy

• Outlook

Equations of motion

2

2

ii i

d rm

dtf=�

�

Overlap ( ) ( )1

2i j i j

d d r r nδ = + − − ⋅� � �

�( )i j

ij

i j

r rn n

r r

−= =

−

� ��

� �

c w

ii i ic wf f f m g= + +∑ ∑

�� �

Forces and torques:

i

i i

dI

dtt

ω=

���

cc

i ici r ft = ×∑�� ��

Normal

Contacts

Many particle

simulation

Discrete particle model

Material behavior of granular media

Non-Newtonian Flow behavior under slow shear

inhomogeneity rotations

Yield-surface

Particle Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft

Stefan Luding, s.luding@tnw.tudelft.nl

3

Biaxial box set-up

• Top wall: strain controlled

• Right wall: stress controlled

• Evolution with time … ?

( )0 ff

( ) 1 cos2

z zz t z tω

−= + +

const.p =

εzz=9.1%εzz=4.2%εzz=1.1%εzz=0.0%

Simulation results

Bi-axial compression with px=const.

4

Bi-axial box (stress chains)

Bi-axial box (kinetic energy)

Bi-axial box (rotations)

5

Constitutive modeling: disadvantages

• Many constitutive models on the market

• Typically a large number of paramters

• Calibration with experiment is difficult

• Little microscopic insight/motivation

• Limited range of applicability

Constitutive models:

• Gradient continua

• Micro-polar continua

• Elasto-(visco)-plastic models

• Hyper-, hypo-plastic, -elastic models

• Challenges:

• Cohesion, creep, cyclic loading, ageing, …

Basics: Hypoplasticity

Assumptions:- state of the material is described by: stress and void ratio- simple incremental formulation for all states, and

also for loading and unloading

Strain increment ∆ε, Stress increment ∆σ, constitutive functions L, N of stress and void ratio e

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Basics: Hypoplasticity

Hypoplasticity … and the critical state

Void ratio:e=Vvoid/Vsolid

e=(1/ν ) – 1

Volume fraction:ν= Vsolid /(Vvoid+Vsolid)

Hypoplasticity … and its limit states

Void ratios: ei, ec, ed

Limit void ratios: ei0, ec0, ed0

Hardness: hs, power n

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Hypoplasticity

Density factor – Pressure factor

Hypoplasticity: parameter calibration

Critical friction angle: φc

- Angle of repose of a sandpile- Shear- or tri-axial tests

Herle & Gudehus 1999

8

Micro-macro transition

friction vs. void ratio

Hypoplasticity: parameter calibration

Limit void ratios: ei0, ec0, ed0

Minimum density: ei0 Prepare smoothly

Critical density: ec0 Shear tests …

Maximum density: ed0 Cyclic shear …

Biaxial box set-up

• Top wall: strain controlled

• Right wall: stress controlled

• Evolution with time … ?

( )0 ff

( ) 1 cos2

z zz t z tω

−= + +

const.p =

9

Preparationppp=2=2=2ppp=20=20=20ppp=200=200=200ppp=2000=2000=2000ppp=20000=20000=20000

Bi-axial compression with px=const.

Results for friction µ=0.5 and different px and kc=0

Pressure dependence

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Micro-macro transition

… void ratio vs. pressure …

Density vs. isotropic stress (=pressure)

p0/k ≈ 1.2 and ν0 ≈ 0.84( )

4 3

0

0

p

pν ν= −

3 4

0

0

p

pν ν

= −

( )

3 4

0 00 0

11

1

e p

e pν ν

⇒ ≈ −

−

0

exp 1

n n

s s

e p p

e h h

= − ≈ −

Micro-macro transition

void ratio vs. pressure

Hypoplasticity: parameters 2D

Hardness: hs, n ≈ 0.66-0.75

Minimum density: ei0 ≈ 0.244

Critical density: ec0 ≈ 0.222

Maximum density: ed0 ≈ 0.196

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Bi-axial: px=200 – varying friction

Hypoplasticity: parameter calibration

Exponents: α, β

Hypoplasticity parameters

8 Parameters

Critical friction angle: φc

Limit void ratios: ei0, ec0, ed0

Hardness: hs

Exponents: α, β, n

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Hypoplasticity

Material φc [o] hs

[MPa] n ed0

ec0

ei0 α β

Hochstetten gravel

Hochstetten sand

Hostun sand

Karlsruhe sand

Lausitz sand

Toyura sand

36

33

32

30

33

30

32000

1500

1000

5800

1600

2600

0.18

0.28

0.29

0.28

0.19

0.27

0.26

0.55

0.61

0.53

0.44

0.61

0.45

0.95

0.96

0.84

0.85

0.98

0.50

1.05

1.09

1.0

1.0

1.1

0.10

0.25

0.13

0.13

0.25

0.18

1.9

1.0

2.0

1.0

1.0

1.1

+ successful tool – few parameters

- microscopic foundations ?

- extensions & parameter identification

Continuum Theory

deformation - rotations

0.5

0.6

0.7

0.8

0.9

1.0

-0.1 0 0.10.5

0.6

0.7

0.8

0.9

1.0

u/h

e

cyclic deformations - creep

Hypoplastic FEM model

Hypoplasticity – Limits

Micro-polar and Gradient extensions

Cyclic loading

Anisotropy

…