Propagation of uncertainties and multimodality in the impact problem of two elastic bodies

Abstract

An uncertainty quantification study is carried out for the problem of the frontal collision of two elastic bodies. The time of contact and the resultant force function involved during the collision are the quantities of interest. If the initial conditions and the mechanical and geometrical properties were known, the response prediction would be deterministic. However, if the data contains any uncertainty, a stochastic approach becomes appropriate. Based on the Principle of Maximum Entropy (PME), and under certain restrictions on the parameter values, we derive the probability density function (PDF) for each of the stochastic parameters to construct a probabilistic model. Two cases are dealt with: one of a collision involving two spheres and another of a collision of two discs. In the first case, a parameter involving geometry and material properties is assumed stochastic. Since a functional relationship exists, the propagation of the uncertainty of the time of contact can be done symbolically. However, the interaction force function can only be computed from the solution of a nonlinear ordinary differential equation. Given the PDF of the parameter, the problem of uncertainty propagation is tackled using Monte Carlo simulations. The comparison of both approaches yields an excellent agreement. With respect to the collision of two discs, first the small deformation problem, within the Hertz theory, is addressed with a Monte Carlo method. When the discs undergo large deformations, the problem is approximated using the equations of Finite Elasticity discretized by the finite element method (FEM) and combined with Monte Carlo simulations. In a first illustration, the modulus of elasticity is assumed stochastic with a gamma PDF. Further, the disc collision problem is analyzed when two parameters are stochastic: the modulus of elasticity and the Poisson's ratio. It is shown that under certain dispersion ranges, the PDF of the interaction force function undergoes a qualitatively change exhibiting bimodality.