Page 74 - The Combined Finite-Discrete Element Method

POTENTIAL CONTACT FORCE IN 3D 57

are the coordinates of the four nodes of the tetrahedron finite element (Figure 2.22). Using
the centroid, the tetrahedron is subdivided into four sub-tetrahedra:

tetrahedron 1 − 2 − 3 − 5
tetrahedron 2 − 4 − 3 − 5
(2.35)
tetrahedron 3 − 4 − 1 − 5
tetrahedron 4 − 2 − 1 − 5

For each point p of the sub-tetrahedron i − j − k − l, the potential ϕ is defined as

V i−j−k−p
ϕ(p) = k (2.36)
4V i−j−k−l
where k is the penalty parameter, V i−j−k−l is the volume of the tetrahedron i − j − k − l,
while V i−j−k−p is the volume of the sub-tetrahedron i − j − k − p,i.e.the tetrahedron
with one of the nodes being replaced by the point p.

2.6.1 Evaluation of contact force
Penetration of any elemental area dV of a contactor into the target results in an infinites-
imal contact force, given by
df =−gradφ t (P t )dV (2.37)
where df is the infinitesimal contact force due to infinitesimal overlap dV defined by
overlapping points P c belonging to the contactor and P t belonging to the target.
Since the target and contactor are relative attributes, to retain symmetry, a reverse
situation of the target penetrating the contactor produces force:

df = gradφ c (P c )dV (2.38)

The total infinitesimal contact force is given by

df = [gradφ c (P c ) − gradφ t (P t )]dV (2.39)

The total contact force exerted by the target onto the contactor is obtained by integration
over the overlap volume:

f = [gradϕ c − gradϕ t ]dV (2.40)
V =β t ∩β c
which can also be written as an integral over the surface S of the overlapping volume:

f = n (ϕ c − ϕ t )dS (2.41)
S β t ∩β c
where n is the outward unit normal to the surface of the overlapping volume.