The multiindentation problem of rigid bodies of
arbitrary shapes on a viscoelastic halfspace.
The contact between a tyre and a road is mainly modeled in the frame of the elastic theory.
Generally, the viscoelastic effects have not been taken into account for computing the pressure
distribution. But in fact the dynamic modulus of
rubber is frequency dependent and the viscoelastic
part can be more important than the elastic modulus. The experimental studies
of dynamical tyre/road contact show that the contact
areas obtained during rolling were smaller than in static
conditions, that is mainly explained by the dynamical properties of tyre compounds, like the viscoelastic
behaviour of the rubber. So
the aim of our work is to propose a new numerical method for problems of
multiindentation on a viscoelastic halfspace with
asperities of arbitrary shapes for an increasing and then decreasing vertical
loading.
The contact problem between rigid indenters (micro asperities of the
road) of arbitrary shapes and a viscoelastic
halfspace (tyre) is considered. Under the action of
a normal force the penetration of the indenters changes and a few contact areas
appeared.
The contact between a rigid indenter of arbitrary shape and
a viscoelastic halfspace 
We wish to find the relations which link the
pressure distribution, the resultant forces on the indenters and the
penetration on the assumption that the surfaces are frictionless. For indenters
of arbitrary shapes the problem may be solved numerically
by using the Matrix Inversion Method (MIM), extended to viscoelastic
cases. In this method the boundary conditions are
satisfied exactly at specified "matching points" (the midpoints of
the boundary elements). It can be validated by
comparing the numerical results to the analytic solutions in cases of a
spherical asperity (loading and unloading) and a conical asperity (loading
only). The pressure distribution and the contact area are
found for a single spherical, conical and cylindrical indenter.


the
conical indenter 
the
sphericalended cylindrical indenter 
the
cylindrical indenter 
When the
problem involves a large number of points the MIM can
become very timeconsuming. Here the problem is solved
using an alternative scheme, called the Twoscale Iterative Method (TIM). In
this method the Local Matrix Inversion Method is used
at the microscale for each contact area to compute the pressure distribution
taking into account interacting effect (the forces on the other contact areas
which can be calculated at the macroscale) between indenters. Two algorithms
were proposed. The first algorithm takes into account the distribution of
forces on the other contact areas and the second is the approximation of the
first algorithm and takes into account the resultant forces on the other
contact areas. The method was implemented for a simple
configuration of seven spherical indenters, seven sphericalended cylindrical
indenters and seven flatended cylindrical indenters as well as for a more
complex configuration of twelve randomly positioned indenters of arbitrary
shapes. For all these cases the TIM permits to find
the pressure distribution and the contact forces versus the penetration. It can be validated by comparing the numerical results to the
numerical results obtained with the MIM. The TIM was also
validated by comparing to the Finite Elements Method (FEM) and to the
measured data.
The
indentation of a viscoelastic halfspace by seven
rigid sphericalended cylindrical indenters (Algorithm 2) 
For a simple configuration of seven sphericalended cylindrical indenters one can see the difference between the forces for
central and peripheral indenters. The peripheral indenters deform the viscoelastic halfspace under the central indenter and the
contact force for the central indenter is less than for the peripheral
indenters and the peripheral contact areas are larger
than the central contact area.
The
indentation of a viscoelastic halfspace by seven
rigid sphericalended cylindrical indenters (Algorithm 1) 

Algorithm 1 catches better the influence of the central indenter on the
peripheral indenters. Under the influence of the central indenter
the maximum value of pressure shifts outside and no more in the tip of the
peripheral indenter. Thus, one can see the asymmetry of the distribution of the
pressure for peripheral indenters.
Now we consider the more general case of twelve indenters of arbitrary
shapes. This configuration is composed of three rigid sphericalended
cylindrical indenters, of three flatended cylindrical indenters, of three
conical indenters and of three cylindrical indenters (finite cylindrical shape
with its curved face). This case is more complex since the indenting geometry doesn't have an axisymmetric
profile and the positions of indenters are arbitrarily.
General case: three sphericalended cylindrical indenters,
three flatended cylindrical indenters, three conical indenters, three
cylindrical indenters (Algorithm 1) 






The maximum pressure for flatended cylindrical indenters is
concentrated at the boundaries of each contact area. The maximum pressure for
cylindrical indenters is concentrated at the boundaries of the contact area which are perpendicular to the axis y which is the axis
of symmetry of the cylinder. The maximum pressures for cylindrical and
flatended cylindrical indenters are comparable and the distributions are not
symmetric. The maximum pressure for sphericalended cylindrical indenters is
less than for the others indenters. The absolute maximum pressure is on the
conical indenters, but the contact areas are the smallest.
The last example of twelve randomly positioned indenters of arbitrary
shapes shows that the proposed method could be probably applied to practical
contact problems with a large number of asperities, such as for tyreroad contact computations.