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 Kirchhoff's 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 Lamé 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.
An exact solution for the impact law in thick elastic plates
SBURLATI, ROBERTA
2004-01-01
Abstract
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 Kirchhoff's 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 Lamé 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.
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