Hypoplasticconstitutive modeling - TFE1€¦ · Constitutive modeling: disadvantages • Many...
Transcript of Hypoplasticconstitutive modeling - TFE1€¦ · Constitutive modeling: disadvantages • Many...
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MSM, CTW, UTwente, NL
Stefan Luding, [email protected]
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
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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
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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, [email protected]
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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.
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Bi-axial box (stress chains)
Bi-axial box (kinetic energy)
Bi-axial box (rotations)
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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
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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 =
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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
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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
…