SMOOTH AND NON-SMOOTH SOLUTIONS FOR GEOMETRICALLY EXACT BEAMS SUBJECTED TO IMPACT

Abstract eng:
Shock and Impact problems have been the focus of research for decades in many engineering areas such as structural engineering, automobile industry, aerospace, multibody dynamics. In general, unlike classical dynamics, impact requires specials techniques to correctly model the underlying physics. The solutions for the problems of impact have been developed in different ways by many authors with each of them using different hypothesis. However, formulations of impact of beams have received limited attention. The solution methods for impact can be divided into two main groups: smooth and non-smooth approaches. In this paper we will develop both smooth and nonsmooth formulations for impact between projectiles and beams. The adopted beam kinematics is geometrically exact. In smooth formulations, the velocities and forces are continuous functions. The impact forces is determined by the corresponding deformations of the projectile, the deformation of beam is calculated with the applied impact force. This approach is appropriate when the projectile is deformable and it reflects the reality of many impact scenarios. When it comes to impact between rigid projectile and deformable beams, the problem becomes more complex and should be treated using non-smooth formulations where the velocities and the impact forces may have jumps, thus, they are discontinuous. Set-valued force laws are used to account for unilateral conditions on both displacement and velocities. The appropriate framework is differntial measure combined with convex analysis The coefficient of restitution plays also an important role. The integration is also split into smooth and non-smooth part. Though, having stable time integration schemes for geometrically exact beams is not a straightforward task. For both formulations, the use of a newly developed energy-momentum time integration scheme conserve perfectly the total energy and the momentum of the system during and after the impact. A range of numerical applications will be delivered to show excellent performances of the new formulations.

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