Introduction

Local deformations in the contact region must be accounted for in the analysis in order to accurately predict the contact force history. The indentation, defined as the difference between the displacement of the projectile and that of the back face of the laminate, can be of the same order as or larger than the overall displacement of the laminate. One could consider the projectile and the structure as two solids in contact and then analyze the impact problem as a dynamic contact problem. However, this approach is computationally expensive and cannot describe the effect of permanent deformation and local damage on the unloading process. The unloading part of the indentation process can be modeled only using experimentally determined contact laws. To predict the contact force history and the overall deformation of the target, a detailed model of the contact region is not necessary. A simple relationship between the contact force and the indentation, called the contact law, has been used by Timoshenko (1913) to study the impact of a beam by a steel sphere. This approach has been used extensively since then and is commonly used for the analysis of impact on composite materials.

Although the impact event is a highly dynamic event in which many vibration modes of the target are excited, statically determined contact laws can be used in the impact dynamics analysis of low-velocity impacts because strain rate and wave propagation effects are negligible with commonly used material systems. Hunter (1957) calculated the energy lost by elastic waves during the impact of a sphere on an elastic half-space.