Singular fractal functions and mesoscopic effects in mechanics

Abstract

Many natural phenomena are characterised by step-like mesostructural changes and from the macroscopic point of view look not completely smooth. The phenomenological description of such processes is difficult enough because the conventional equations in mechanics deal usually with sufficiently smooth functions. The possibilities of conventional mechanical theories can be extended to include these not completely smooth processes by introducing into constitutive equations singular fractal functions, for example, not absolutely continuous functions [B. Bernstein, M. Karamolengos and T. Erber, Sneaky plasticity and mesoscopic fractals, Chaos, Solitons & Fractals, 3, 269–277 (1993)]. The most essential property of such functions is that their integrals and derivatives are not constrained by the usual theorem of calculus. We discuss two simple examples which illustrate the consequences of such procedure: elastic material with progressive damage and stick-slip behaviour in friction.