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| Fractal, or generally a self-similar (or self-affine) structure of infinite order, has been extensively used as a nonlinear mathematical tool to describe various irregular and complex phenomena in fracture mechanics. However, there are still critical issues remaining ambiguous, leading to inherent difficulties mainly resulted from the contradiction between the integral dimension immeasurability of a fractal and the integral dimension characteristic of a physical object in nature. Here we demonstrate that conceptually a physical object in nature can never be described as a fractal, rather a ubiquitiform (a terminology coined here for a finite order self-similar or self-affine structure), and show mathematically that a ubiquitiform must be of integral dimension in Euclidean space, which is radically different from a fractal in the sense of the Hausdorff measure and makes the fractal approximation of a ubiquitiform unavailable. Our result implies that a natural object is of ubiquitiform rather than fractal, and thus, instead of the existing fractal theory, a new type of ubiquitiform theory must be established in future. We anticipate this result to be a starting point for the coming universal ubiquitiform theory in fracture mechanics. Moreover, it can be expected that the coming ubiquitiform theory will be more easy-to-use in practice than the fractal one, since it can be constructed completely on the base of "classical" mathematics, and hence some intrinsic difficulties in applied fractal science can be avoided. |
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| Keywords:solid mechanics; ubiquitiform; fractal |
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