Keywords: vibration, fractal, modal analysis.

Continuum mechanics continues to be the dominant methodology underpinning models for the prediction of material deformation and vibration. This is despite the increasing interest in material behaviour at meso- and micro-scales. The fundamental idea underpinning continuum mechanics is the concept of the continuum where quantities are assumed to be defined at arbitrarily small length scales. The principal link between the material and spatial volume is the physical quantity density. In continuum mechanics density is defined at a point by considering the ratio of mass over volume in the limit as the volume shrinks to zero. It is recognised that this limit breaks down at molecular length scales. Physics at molecular length scales is well described using molecular dynamics, where molecules can be approximated by spherical shapes and forces between the molecules are assumed to be known. Moving up to the meso-scale presents something of a dilemma where it is not practicable to apply molecular dynamics and continuum mechanics only provides limited accuracy. In addition to this the complex heterogeneous structures present with modern composites and cellular materials necessitate the development of complex material models.

In order to address the problem of material structure some researchers have considered the application of fractals. The vast majority of existing current approaches can be viewed as indirect in the sense that they involve the use of fractal quantities, obtained from the representative fractal, in a continuum type model. The most commonly applied quantity is the fractal dimension, with many researchers claiming that the dimension is a material property. An example of this is in the area of fracture mechanics, with many researchers suggesting a relationship between fracture toughness and fractal dimension [1,2,3,4].

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating rods. The fractals considered are necessarily deterministic and relatively simple in form to facilitate the solution methodology. Although, as a result, the fractals are not representative of realistic physical systems the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.

1

Mandelbrot B.B., Passoja D.E., Paullay A.J., "Fractal Character of Fracture Surfaces of Metals", Nature, Vol. 308, pp. 721-722, 1984. doi:10.1038/308721a0

2

Shek C.H., Lin G.M., Lai J.K.L., Tang Z.F., "Fractal fracture and Transformation Toughening in CuNiAl single crystal", Metallurgical and materials transactions A, 28A, pp. 1337-1340, 1997. doi:10.1007/s11661-997-0269-1

3

Carpinteri A., Chiaia B., "Crack-resistant behaviour as a consequence of self-similar fracture topologies", International Journal of Fracture, Vol. 76(4), pp. 327-340, 1996.

4

Mecholsky Jr. J.J., "Estimating theoretical strength of brittle materials using fractal geometry", Materials Letters, Vol. 60(20), pp. 2485-2488, 2006. doi:10.1016/j.matlet.2006.01.054

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