Page 80 - The Combined Finite-Discrete Element Method

POTENTIAL CONTACT FORCE IN 3D 63

To limit penetration, it is enough to select a penalty term to be proportional to the modulus
of elasticity, i.e.
p = αE (2.58)

In this way, contribution of the allowed penetration to the displacement field is limited to
1
d = u (2.59)
α
For α = 100, for example,
1
d = u (2.60)
100
i.e. the total local error in displacements will be less than 1%.
This is another advantage of the potential contact force approach. The error in the
displacements is easily controlled through setting penalty p as a function of E. In addition,
the error in displacements is reduced by reducing the size of finite element h. Thus, any
mesh refinements automatically reduce error introduced by contact approximation.

2.6.4 Contact damping

Any energy dissipation in contact is due to friction or plastic straining of surface asperities.
Plastic straining of surface asperities can be approximated by a viscous damping model.
Damping parameters for contact are defined in a similar way to the definition of penalty
parameters. For the situation in Figure 2.28, the frequency of the subsystem shown can
be approximated by
2 p
ω = (2.61)
h ρ

where ρ is the density and h is the size of the finite element, as shown in the figure. The
normal contact stress due to critical viscous damping is given by

σ c = 2ωd ˙ (2.62)

while in the general case of an underdamped system,

σ c = 2ωξd ˙ (2.63)

where ξ is the damping ratio. If ξ = 0, there is no energy dissipation. If ξ = 1, critical
damping is obtained.
After substituting the frequency from equation (2.62),
√
p/ρ
σ c = 4ξ d ˙ (2.64)
h
This damping is due to contact only. Physical interpretation of such damping is, for
instance, plastic deformation and/or breaking of surface asperities. This damping is not to
be confused with damping such as energy dissipation due to deformation of the discrete