Page 59 - The Combined Finite-Discrete Element Method

42 PROCESSING OF CONTACT INTERACTION

where df is the infinitesimal contact force due to infinitesimal overlap dA,defined by
overlapping points P c belonging to the contactor and P t belonging to the target.
Equation (2.17) can also be written as

df =−df t + df c (2.18)

where
df c =−gradϕ t (P t )dA c , dA c = dA
(2.19)
df t =−gradϕ c (P c )dA t , dA t = dA

In other words, the contact as described by (2.17) can be viewed as first the elemental
area of the contactor penetrating the target, and then the elemental area of the target
penetrating the contactor. Thus, for each of the discrete elements in contact, the contact
force is calculated as a gradient of the corresponding potential function. The field of
contact forces is therefore a conservative field for both the target penetrating contactor
and the contactor penetrating target.
If point P c of a contactor discrete element penetrates the target through any path defined
by end points A and B, the total work of the contact force due to the potential function
does not depend upon the path, but on the end points A and B only. If both points A and
B are on the boundary of the target discrete element, a contact-contact release situation
arises, i.e. point P c comes into contact with the target at point A of the target and the
contact is released through point B of the target. Preservation of the energy balance
requires that the total energy of the system before and after the contact is the same, i.e.
that no work is done by the contact force, which is equivalent to saying that for any
points A and B on the boundary of the target discrete element,

ϕ t (A) − ϕ t (B) = 0 (2.20)

i.e.
ϕ t (A) = ϕ t (B) (2.21)

The same is valid for the contactor discrete element, i.e. for any points A and B from the
boundary of the contactor discrete element

ϕ c (A) − ϕ c (B) = 0 (2.22)

i.e.
ϕ c (A) = ϕ c (B) (2.23)
Thus, provided that the potentials on the boundary of both the contactor and target discrete
elements are constant, the contact force as given by (2.17) preserves the energy balance
regardless of the geometry or shape of contactor and target discrete elements, the size of
the penalty term or the size of penetration (overlap) when in contact.
The total contact force is obtained by integration of (2.17) over the overlapping area S

f c = [gradϕ c − gradϕ t ]dA (2.24)
S=β t ∩β c