Page 57 - The Combined Finite-Discrete Element Method

40 PROCESSING OF CONTACT INTERACTION

Problems of contact interaction in the context of the combined finite-discrete element
method are even more important, due to the fact that in this method, the problem of contact
interaction and handling of contacts also defines the ‘constitutive’ behaviour of the system,
because of the presence of large numbers of separate bodies. Thus, algorithms employed
must pay special attention to contact kinematics in terms of the realistic distribution of
contact forces, energy balance and robustness.
One of the early contact interaction algorithms that took into account energy balance
considerations is the contact interaction algorithm, based on the so-called concept of a
contact element, in which the penalty parameters are given physical meaning through the
concept of a contact layer, which is in a way a crude approximation of surface roughness.
This early algorithm was originally developed to handle 2D problems, and was difficult
to implement in 3D problems where, for instance, a surface normal may not be defined
at all surface points. In addition, if overlap of discrete elements in contact exceeds the
contact layer, the energy balance is not preserved, which is also the case if new surfaces
are created due to a fracture process. The contact forces resulting from the algorithm are
concentrated, which greatly influences stress and strain fields close to the boundary, and
may considerably influence the results of fracture and fragmentation analysis. For this
reason, this and many other contact algorithms of mainly historical importance are not
considered in this book.
The latest generation of contact interaction algorithms makes use of finite element
discretisations of discrete elements, and combines this with the so-called potential contact
force concept. These algorithms assume discretisation of individual discrete elements
into finite elements, thus imposing no additional database requirements on handling the
geometry of individual discrete elements. They also yield realistic distribution of contact
forces over finite contact areas resulting from the overlap of discrete elements that are
in contact. Thus, numerical distortion of local strain fields close to the boundary due to
contact is much reduced–an important aspect when, for instance, the fracture of brittle
material is analysed.

2.2 THE PENALTY FUNCTION METHOD

The penalty function method in its classical form assumes that two bodies in contact
penetrate each other, and this penetration results in a contact force. The standard contact
functional for the penalty function method takes the form

1 T

U c = p(r t − r c ) (r t − r c )d (2.15)
2
c
where p is the penalty term, while r t and r c are position vectors of the points on the
overlapping boundaries of the target and contactor bodies, respectively. In the limit for
infinite penalty terms, no penetration would occur, i.e.

lim U c = 0 (2.16)
p→∞