UNCERTAINTY QUANTIFICATION IN THE SIMPLIFIED PROBLEM OF CONTACT BETWEEN TWO ELASTIC BODIES

FERNANDO S. BUEZAS; MARTA B. ROSALES; RUBENS SAMPAIO; MARIO SANDOVAL

Maresias

Simposio; UNCERTAINTIES 2012; 2012

JBSMSE (ABCM) and IJUQ (Begell House)

A straightforward study of the uncertainty quantification in the contact problem between two elastic bodies that collide, is presented. An existing elastic model based on the contact Hertz theory is employed. It involves two parameters, one of them acounts for the material and geometric properties of the bodies and the other, is related to the Hertz model. The time of contact and the involved force of this direct impact are the quantities of interest. When the initial conditions and the mechanical and geometrical properties are known, the prediction of these quantities becomes a deterministic problem. However, if the data contains any uncertainty, a stochastic approach becomes appropriate. Based on the Principle of Maximum Entropy, and under certain restrictions on the parameter value, we derive a probability density function (PDF) for the stochastic parameter. In the case of the collision of two identical spheres, since the time of contact is governed by an algebraic relationship involving the above-mentioned parameters, the propagation of the uncertainty can be done symbolically and the confidence bands obtained straightforwardly. A study is carried out for various levels of dispersion in the input parameter. The influence of the impact velocity on the duration of the contact is evaluated statistically. On the other hand, the contact force is derived from the solution of a nonlinear ordinary differential equation. Given the PDF of the parameter, the problem is also tackled using Monte Carlo simulations. The PDF of the contact force is thus obtained and it is assessed for different times of contact. Additionally, the time of contact yields from this numerical study. This is useful to compare the two approaches resulting in an excellent agreement. The collision of two discs is also tackled. First, the small deformation problem within the Hertz theory is addressed with a Monte Carlo simulation. Finally, the collision of two discs undergoing large deformations is addressed using a finite element algorithm previously developed by the first author, combined with a Monte Carlo simulation. The differences between the two problems are shown.