Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

Abstract

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the “fixed points” of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C ⁰-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.