A dynamic contact model for
adhesive spheres
Sebastiaan Krijt (Leiden Observatory)
with AGGM Tielens, Carsten Dominik, Carsten Guettler, Daniel
Heisselmann
Workshop on "Ice and Planet Formation"
Lund - (15-17/05/2013)
Particle 2
Particle 1
Material
Mass
Porosity
Material
Mass
Porosity
Velocity
Impact parameter
rotation
grains: A. Seizinger
Modelling approach
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Aggregate = collection of spherical micron-sized monomers, held
together by surface forces
Force laws governing normal motion (a), rolling (b), sliding (c), and
spinning (d) of monomers derived by Dominik & Tielens
(1995,1996,1997), based on static JKR-theory (Johnson et al 1971)
Normal force – (Static) JKR theory
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Material assumed perfectly elastic
Single contact described in terms of mutual approach (δ) and contact
radius (a)
F
Static theory assumes that for a given δ, the contact radius will adjust
itself to minimize the total elastic and surface energy
Results in unique analytic expressions relating a, δ and elastic
interparticle force F
Normal force – Dynamic theory
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Approach: drop perfectly elastic assumption, and assume material
responds visco-elastically, on a characteristic timescale T.
Results in dissipative stresses, when material is stressed on a similar
timescale. This has two effects:
1)
Answer comes from viscoelastic crack theory:
Effective surface is velocity-dependent, and energy is dissipated as
crack opens/closes. No unique relation between a and δ
Normal force – Dynamic theory
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Drop perfectly elastic assumption, and assume material responds
visco-elastically, on a characteristic timescale T.
Results in dissipative stresses, when material is stressed on a similar
timescale. This has two effects:
2)
Integrating over the dissipative stresses in the bulk yields a
dissipative force:
Normal collision - bouncing
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To follow a normal collision, both a(t) and δ(t) have to be integrated
numerically
For water-ice like microspheres, colliding at 8 m/s, we find:
Normal collision - sticking
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Decreasing the collision velocity to 2 m/s results in sticking:
Oscillation is damped, and JKR equilibrium retrieved after a couple of
oscillations
Critical sticking velocity
JKR sticking velocity
(Normal) Coefficient of restitution
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Collisional outcome captured by Coefficient of restitution (COR),
defined as the ratio of post- to pre-collision velocity
elastic
sticking
Comparison to experiments - results
Three fitting parameters; the
combined surface energy,
relaxation time, and yield
strength.
Increasing size
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Rolling friction
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Dominik and Tielens (1997) assume a constant critical rolling
displacement, which gives rise to an asymmetric pressure
distribution, and a rolling torque.
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In the visoelastic mode, the
effective surface energy will be
different on both sides of the
rolling contact.
The difference, and thus the
rolling torque, will be depend on
rolling velocity
How does this affect aggregate
restructuring?
Conclusions / Future work
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Viscoelastic particle response results in energy dissipation in the bulk,
and at the contact edge. Contact equations are derived
Resulting collision model, expanded to include plastic yield, succeeds
where purely elastic theory fails, fitting a large number of
experiments, varying in size, material, collision velocity, and
experimental setup
Rolling torque can be derived, which appears to depend on rolling
velocity
Next step: Implement dynamic normal & rolling force into N-body
code and simulate aggregate collisions/restructuring