Fractal Dimensions in Elastic-Plastic Beam Dynamics

Calculations of two types of fractal dimension are reported, regarding the elastic-plastic response of a two-degree-of-freedom beam model to short pulse loading. The first is Mandelbrot’s (1982) self-similarity dimension, expressing independence of scale of a figure showing the final displacement as function of the force in the pulse loading; these calculations were made with light damping. These results are equivalent to a microscopic examination in which the magnification is increased by factors of 10²; 10⁴; and 10⁶. It is found that the proportion and distribution of negative final displacements remain nearly constant, independent of magnification. This illustrates the essentially unlimited sensitivity to the load parameter, and implies that the final displacement in this range of parameters is unpredictable. The second fractal number is the correlation dimension of Grassberger and Procaccia (1983), derived from plots of Poincare intersection points of solution trajectories computed for the undamped model. This fractional number for strongly chaotic cases underlies the random and discontinuous selection by the solution trajectory of the potential well leading to the final rest state, in the case of the lightly damped model.