Page 53 - The Combined Finite-Discrete Element Method

36 PROCESSING OF CONTACT INTERACTION

Contact interaction between neighbouring discrete elements occurs through solid sur-
faces which are in general irregular and, as a consequence, the contact pressure between
two solids is actually transferred through a set of points at which surfaces touch. At
small normal pressure, surfaces only touch at a few points, and with increasing normal
pressure, elastic and plastic deformation of individual surface asperities occurs, resulting
in an increase in the real contact area.
Theoretical and micro-mechanical models of contact take into account this complex
phenomenon, and are usually based on assumptions such as shape, distribution and defor-
mation of individual asperities, etc.
In the computational literature, theoretical assumptions about contact are simplified
by employing variational formulation of contact combined with the most simple contact
law that defines contact pressure as a function of approach, with tangential resistance to
motion being a function of normal pressure and/or slip condition. This is usually done
using variational formulation.
The variational formulation of a boundary value problem with contact is equivalent to
the problem of making a functional stationary subject to the contact constraints over
boundaries of the domain :
C(u) = 0 (2.1)

Variational formulation of contact problems involves an additional functional due to con-
tact, through which no penetration conditions are enforced. Among the classic approaches
employed are:

• The least square method: to enforce contact constraints on the boundary , functional

T
C (u)C(u)d (2.2)

is introduced. This functional is always positive except when contact constraints are
satisfied exactly. Minimisation of this functional produces

C(u) = 0 (2.3)

• The Lagrangian multipliers method: to enforce contact constraints, the additional
functional

T
λ C(u)d (2.4)

is added to the original functional

T
˜
(u, λ) = (u) + λ C(u)d (2.5)

where λ is a set of independent functions defined on . These functions are known
as Lagrangian multipliers. A stationary point is found by employing the variation of
this functional

T
˜
δ (u, λ) = δ (u) + δ λ C(u)d = 0 (2.6)