Contact interaction law based on the Hertz theory for multibody applications

Resumo

The normal component of the contact interaction between two bodies arises from the fact that their
material boundaries cannot go through each other. One way to deal with this non-penetration geometric constraint
is to allow its violation on a controlled manner by adding an increasing magnitude force pair to the system, as long
as the constraint violation increases. Methods that use this approach are usually referred as penalty-based methods.
The simplest penalty method imposes a force proportional to the normal indentation. The proportionality constant
acts as a stiffness parameter. An alternative interface law originates from Hertz’s theory of nonconformal contact
between elastic bodies. According to this classic theory, the contact force is proportional to the normal indentation
raised to the power 3/2. The proportionality coefficient is not a constant, it has a dependency on the local curvatures
of the contacting surfaces. Ultimately, the proportionality constant depends on the configuration of the system at
a given time. In this work, we implement a penalty method based on Hertz force law for imposing contact between
rigid bodies. The nonlinear dynamical equations for multibody dynamics are solved numerically. In the required
linearization process, the dependency of the stiffness parameter on the degrees of freedom of the system is
neglected. The results of the present method for a set of simple examples are presented. We verified that the

considered simplification has a negative effect on the solution process if the stiffness varies throughout a time-
step. The effect may be diminished using a large quantity of time-steps.