Contact Problems for Elasto-Plastic Impact in Multi-Body Systems

Abstract

Low to moderate speed collision between elastic-plastic bodies results in imperceptible permanent indentation of the contact surfaces if the bodies are hard. Nevertheless in these collisions the contact force-indentation relation is irreversible since internal energy gained from work done by the contact force during the compression is partially trapped in elastic waves, work done in plastic deformation and work done to overcome friction during sliding. These sources of kinetic energy loss in collisions depend on relative velocity between the bodies at the contact point, material properties, inertia properties related to the impact configuration and geometric constraints on the deformation field.

Colliding bodies that can be represented as elastic-plastic or visco-plastic solids have been analyzed using specific models of contact compliance to calculate the energy transformed into irrecoverable forms. These calculations show how elastic waves, plastic deformation and friction affect the energetic coefficient of restitution—a coefficient that is a measure of impact energy loss from internal sources. The calculations indicate that there is considerable difference in the sources of energy loss for 2D and 3D deformations. Also, that friction little affects the plastic energy losses unless the impact speed is large enough to cause uncontained plastic deformation beneath the contact area. Consequently, for low-speed impact $\rho v_3^2 \left( 0 \right)/Y < 10^{ - 3}$ , the energetic coefficient of friction e_*. is insensitive to friction.

In analysing impact of multi-body systems (e.g. mechanisms, kinematic chains or agglomerates of granules), it is crucial to employ these ideas by specifically modelling the contact compliance. During impact on a mechanism, the velocity changes in one area of contact induce small displacements that develop at other compliant contacts where the impacted body is supported or connected to other elements; the force that develops at these secondary contacts (as a consequence of relative displacement at each individual contact) is the means of transmitting the impact process through system. Although the global compliance of elements of the system may be small enough so that vibration energy is negligible, the dynamics of multi-body collisions requires consideration of local compliance in each contact regions.

Previously impact on a system of interconnected bodies has been analyzed as either a sequence of separate collisions or a set of simultaneous collisions. In general neither of these assumptions gives an accurate representation of the dynamic behaviour of multi-body systems. Rather, it is necessary to model the compliance of each contact and consider the contact forces which develop since it is these forces which prevent interpenetration or overlap. When applied to impact on multi-body systems such as mechanisms, the impulsemomentum methods used in rigid body dynamics give but one limit of the range of response - a range that depends on the distribution of local compliance at each contact between bodies in the system. An accurate analysis of multi-body system response to impact generally requires consideration of the time-dependent contact forces in a wave of reaction that propagates away from the initial site of impact.