An exact solution for the impact law in thick elastic plates

We study the impact of a rigid sphere on a circular elastic plate whose thickness is not small with respect to its diameter, so that Kirchhoffs theory cannot be applied. For plate-like bodies of this kind it is convenient to apply a theory proposed by Levinson [J. Elasticity 7 (1985) 283], which is a compromise between Kirchhoff's solution and that obtained by the integration of Lame's equation of three-dimensional elasticity. The pressure distribution and the extent of the (circular) area of contact of the sphere on the plate-like body is mathematically described by Hertz's theory. By combining these two theories in a dynamical framework, we derive a non-linear ordinary differential equation able to describe the normal slow impact of a rigid sphere against an elastic plate-like body. (C) 2003 Elsevier Ltd. All rights reserved.