Introduction
Kinematics
Solutions
Rolling and Friction
in Discrete Element Simulations
Matthew R. Kuhn
Donald P. Shiley School of Engineering
University of Portland
EMI 2010 Conference
Boston, Massachusetts
June 2–4, 2011
National Science Foundation Grant No. NEESR-936408
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Classification of Rolling Resistance
Rolling Resistance — Two Categories
Velocity-preserving
(with contact torque)
M input > M output
θ̇ input = θ̇ output
Torque-preserving
(with contact creep)
M input = M output
θ̇ input > θ̇ output
Ẇ input = M input · θ̇ input
Ẇ output = M output · θ̇ output
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Classification of Rolling Resistance
Rolling Resistance — Two Categories
Velocity-preserving
(with contact torque)
M input > M output
θ̇ input = θ̇ output
Torque-preserving
(with contact creep)
M input = M output
θ̇ input > θ̇ output
Gears
Adhesive surfaces
Visco-elastic materials
Elasto-plastic materials
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Classification of Rolling Resistance
Rolling Resistance — Two Categories
Velocity-preserving
(with contact torque)
M input > M output
θ̇ input = θ̇ output
Torque-preserving
(with contact creep)
M input = M output
θ̇ input > θ̇ output
Gears
Adhesive surfaces
Visco-elastic materials
Elasto-plastic materials
Iwashita & Oda (1998)
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Classification of Rolling Resistance
Rolling Resistance — Two Categories
Velocity-preserving
(with contact torque)
M input > M output
θ̇ input = θ̇ output
Gears
Adhesive surfaces
Visco-elastic materials
Elasto-plastic materials
Torque-preserving
(with contact creep)
M input = M output
θ̇ input > θ̇ output
“Creep-friction”
Iwashita & Oda (1998)
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Creep-Friction: Definition
Creep-Friction
Rolling resistance and dissipation in the absence of a
contact moment
Contact mechanics are different than for Cattaneo-Mindlin
sliding
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Creep-Friction: Definition
Creep-Friction
Rolling resistance and dissipation in the absence of a
contact moment
Contact mechanics are different than for Cattaneo-Mindlin
sliding
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Creep-Friction vs. Cattaneo-Mindlin Friction
Cattaneo-Mindlin
Zero-Rolling Contact
Creep-Friction
Rolling Contact
Used in DEM?
Genealogy
Widespread
Cattaneo (1938)
Mindlin (1949)
Deresiewicz (1953)
Rare
Carter(1926)
Poritsky(1950)
Kalker (1967)
Original problem
Machinery sliding
(transient sliding)
Rail-wheel interaction
(steady-state rolling)
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Classification of rolling resistance
Creep-friction definition
Creep-friction vs. Cattaneo-Mindlin friction
Creep-Friction vs. Cattaneo-Mindlin Friction
Cattaneo-Mindlin
Zero-Rolling Contact
Creep-Friction
Rolling Contact
Hertz
Yes
Hertz
Yes
Slip & stick
within contact
area between
spheres
Symmetric
Non-symmetric
Reference frame
Lagrangian
Eulerian
Normal stress
Elastic-frictional?
Particle motions
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Eulerian kinematics
Transient & steady-state kinematics
Eulerian kinematics of a moving contact
Rigid displacement
Particle deformation
dudisp ≡ duq − dup
dudef ≡ duq,def − dup,def
Rolling
duroll ≡ 12 (duq +dup )
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Eulerian kinematics
Transient & steady-state kinematics
Eulerian kinematics of a moving contact
Rigid displacement
Particle deformation
dudisp ≡ duq − dup
dudef ≡ duq,def − dup,def
Rolling
duroll ≡ 12 (duq +dup )
Contact
area
Slip d s within the contact area:
q
∂u
∂up
d udef
disp
ds = du
−
−
dt
· d uroll −
∂x
∂x
∂t
Stick zone: d s = 0
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Eulerian kinematics
Transient & steady-state kinematics
Transient vs. Steady-State Conditions
Cattaneo-Mindlin problem — pure sliding
“Transient” conditions
q
∂up
∂u
d udef
disp
−
dt
· d uroll −
0 = du
−
∂x
∂x
∂t
Creep-friction problem — steady rolling
“Steady-state” creep-friction
q
∂u
d udef
∂up
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
General problem for DEM — transient sliding & rolling
q
∂u
∂up
d udef
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Eulerian kinematics
Transient & steady-state kinematics
Transient vs. Steady-State Conditions
Cattaneo-Mindlin problem — pure sliding
“Transient” conditions
q
∂up
∂u
d udef
disp
−
dt
· d uroll −
0 = du
−
∂x
∂x
∂t
Creep-friction problem — steady rolling
“Steady-state” creep-friction
q
∂u
d udef
∂up
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
General problem for DEM — transient sliding & rolling
q
∂u
∂up
d udef
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Current situation
Cattaneo-Mindlin sliding:
Solved!
Creep-friction rolling:
Steady-state rolling of two cylinders: solved, Carter (1926)
Steady-state rolling of two spheres: no exact solution!
Only analytical approximations of the form
v disp
|Q|
ξss ≡ roll = Fss G, µ, a,
µP
v
Transient rolling & sliding: no solutions. Only numerical
approximations.
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Current situation
Cattaneo-Mindlin sliding:
Solved!
Creep-friction rolling:
Steady-state rolling of two cylinders: solved, Carter (1926)
Steady-state rolling of two spheres: no exact solution!
Only analytical approximations of the form
v disp
|Q|
ξss ≡ roll = Fss G, µ, a,
µP
v
Transient rolling & sliding: no solutions. Only numerical
approximations.
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Current situation
Cattaneo-Mindlin sliding:
Solved!
Creep-friction rolling:
Steady-state rolling of two cylinders: solved, Carter (1926)
Steady-state rolling of two spheres: no exact solution!
Only analytical approximations of the form
v disp
|Q|
ξss ≡ roll = Fss G, µ, a,
µP
v
Transient rolling & sliding: no solutions. Only numerical
approximations.
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Recent approximation of the transient problem
Dahlberg & Alfredsson (2009) approximation:
Non-steady state (transient) rolling of cylinders
Start with the general 2-D problem
q
∂u
∂up
d udef
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
Express in an approximate 1-D scalar form
|Q| du roll disp
disp
du
= CC-M (dQ) + Fss
du
µP du disp
Invert the Cattaneo-Mindlin contact compliance “CC-M ” to
find force increment:
|Q| du roll
−1
disp
dQ = CC-M du
1 − Fss
µP du disp
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Recent approximation of the transient problem
Dahlberg & Alfredsson (2009) approximation:
Non-steady state (transient) rolling of cylinders
Start with the general 2-D problem
q
∂u
∂up
d udef
disp
0 = du
−
−
dt
· d uroll −
∂x
∂x
∂t
Express in an approximate 1-D scalar form
|Q| du roll disp
disp
du
= CC-M (dQ) + Fss
du
µP du disp
Invert the Cattaneo-Mindlin contact compliance “CC-M ” to
find force increment:
|Q| du roll
−1
disp
dQ = CC-M du
1 − Fss
µP du disp
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Transient rolling of spheres
3-D Transient rolling of spheres. Reduce to a 2-D form:
|Q| Q
−1 −1
disp
disp
roll
d Q = CC-M d u
⇒ CC-M d u
− Fss
|d u |
µP |Q|
(1)
using Kalker’s (2000) approximation of the steady-state
creepage:
"
#
|Q| 1/3
3µP
1− 1−
Fss =
µP
Ga2 C11
In a displacement-driven DEM code, compute the force
increment d Q by using a standard Cattaneo-Mindlin algorithm
−1
for CC-M
. But use (1) instead of d udisp .
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Tangential force, Q/µP
Effect of rolling upon tangential force
1
duroll
=0
10
dudisp
100
200
0.5
500
0
0
1
3
4
2(1 − ν)R
Sliding movement, dudisp ×
(2 − ν)f a2
Kuhn — June 4, 2011
2
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Effect of rolling in DEM simulation
Drained triaxial compression of 4096 spheres
Stress ratio, q/p′
1.5
1
0.5
0
0
0.05
0.1
0.15
Axial strain,
Kuhn — June 4, 2011
0.2
0.25
ǫ
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
Conclusions
Conclusions . . .
Creep-friction: a type of rolling resistance without rotational
“springs”.
A new approximate solution to the creep-friction problem
for 3D transient rolling & sliding of spheres.
Unless the rolling rate is large (relative to sliding rate), the
effect of creep-friction is small.
An example simulation with a dense packing of spheres
reveals a minimal effect of creep-friction.
Questions?
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf
Introduction
Kinematics
Solutions
Current situation
Dahlberg & Alfredsson
New 3D model
J. J. Kalker 1990.
Three-dimensional elastic bodies in rolling contact.
Kluwer Academic Publishers.
J. J. Kalker 2000.
Rolling contact phenomena — linear elasticity.
Rolling Contact Phenomena, Jacobsen, B. & Kalker, J.J.
(eds.), 1–84.
J. Dahlberg & B. Alfredsson 2009.
Transient rolling of cylindrical contacts with constant and
linearly increasing applied slip.
Wear, 266(1-2):316–326.
Kuhn — June 4, 2011
http:// faculty.up.edu / kuhn / papers / EMI2011.pdf