From geometric fractals to physical fractals: Concepts, models, and methods of fractional-order mechanics

Since the era of Newton and Leibniz, the concept of the material element in continuum mechanics has played a fundamental role. When the research perspective expands from traditional engineering materials to living organisms, the classic concept of the material element faces new challenges due to the multi-scale and multi-level characteristics of living organisms. This article focuses on the intersection of fractals and fractional calculus, analyzes the abstraction process of fractal snowflakes and physical fractal spaces from biological multi-level structures, and reviews the exploration process of fractional-order mechanics. This article starts from two classic problems, namely the "Mandelbrot Question" and the "L’Hospital Question", and presents the following exploration paths: (a) from geometric fractals to physical fractals, (b) from kinematics to dynamics, and (c) from fractal dimension to fractional order. The specific content includes: (a) fractal snowflakes abstracted from living organisms and their growth kinematics, (b) fractional-order operators abstracted from living organisms and their theoretical construction; (c) applications of fractional-order operator theory. On this basis, this article delineates the evolutionary trajectory of the material element concept: from the classical element, the representative volume element (RVE), the lattice unit cell, and the mechanical metamaterial unit cell to the fractal cell, and discusses the multiple origins of fractional orders and the spatiotemporal correlation between fractals and fractional calculus. This article concludes that mechanics in the physical fractal spaces exhibits features that are non-integer-order, non-rational, non-elementary, and non-local, offering novel perspectives for broadening the boundaries of mechanics.