Physics of inhomogeneous inorganic materials

Evaluating a fractal curve's approximate length by walking a compass defines a compass exponent. Long ago, I showed that for a self-similar curve (e.g., a model of coastline), the compass exponent coincides with all the other forms of the fractal dimension, e.g., the similarity, box or mass dimensions. Now walk a compass along a self-affine curve, such as a scalar Brownian record B(t). It will be shown that a full description in terms of fractal dimension is complex. Each version of dimension has a local and a global value, separated by a crossover. First finding: the basic methods of evaluating the global fractal dimension yield 1: globally, a self-affine fractal behaves as it if were not fractal. Second finding: the box and mass dimensions are 1.5, but the compass dimension is D = 2. More generally, for a fractional Bownian record BH(t), (e.g., a model of vertical cuts of relief), the global fractal dimensions are 1, several local fractal dimensions are 2-H, and the compass dimension is 1/H. This 1/H is the fractal dimension of a self-similar fractal trail, whose definition was already implicit in the definition of the record of BH(t).

http://iopscience.iop.org/article/10.1088/0031-8949/32/4/001/pdf

Self-Affine Fractals and Fractal Dimension

The mechanical and hydraulic behavior of discontinuities in rock, such as joints and faults, depends strongly on the topography of the contacting surfaces and the degree of correlation between them. Understanding this behavior over the scales of interest in the earth requires knowledge of how topography or roughness varies with surface size. Using two surface profilers, each sensitive to a particular scale of topographic features, we have studied the topography of various natural rock surfaces from wavelengths less than 20 microns to nearly 1 meter. The surfaces studied included fresh natural joints (mode I cracks) in both crystalline and sedimentary rocks, a frictional wear surface formed by glaciation, and a bedding plane surface. There is remarkable similarity among these surfaces. Each surface has a “red noise” power spectrum over the entire frequency band studied, with the power falling off on average between 2 and 3 orders of magnitude per decade increase in spatial frequency. This implies a strong increase in rms height with surface size, which has little tendency to level off for wavelengths up to 1 meter. These observations can be interpreted using a fractal model of topography. In this model the scaling of the surface roughness is described by the fractal dimension D. The topography of these natural rock surfaces cannot be described by a single fractal dimension, for this parameter was found to vary significantly with the frequency band considered. This observed inhomogeneity in the scaling parameter implies that extrapolation of roughness to other bands of interest should be done with care. Study of the increase in rms height with profile length for two extreme cases from our data provides an idea of the expected variation in mechanical and hydraulic properties for natural discontinuities in rock. This indicates that in addition to the scaling of topography, the degree of correlation between the contacting surfaces is important to quantify.

Broad bandwidth study of the topography of natural rock surfaces

Semisolid metal (SSM) processingis a relatively new technology for metal forming. Different from the conventional metal forming technologies which use either solid metals (solid state processing) or liquid metals (casting) as starting materials, SSM processing deals with semisolid slurries, in which non-dendritic solid particles are dispersed in a liquid matrix. Semisolid metal slurries exhibit distinctive rheological characteristics: the steady state behaviour is pseudoplastic (or shear thinning), while the transient state behaviour is thixotropic. All the currently available technologies for SSM processing have been developed based on those unique rheological properties, which in turn originate from their non-dendritic microstructures. Year 2001 marks the 30th anniversary of the concept of SSM processing. Today, SSM processing has established itself as a scientifically sound and commercially viable technology for production of metallic components with high integrity, improved mechanical properti...

Semisolid metal processing

From the Publisher:
Why do similar patterns and forms appear in nature in settings that seem to bear no relation to one another? The Windblown ripples of desert sand follow a sinuous course that resembles the stripes of a zebra or a marine fish In the trellis-like shells of microscopic sea creatures we see the same geometry as in the bubble walls of foam, Forks of lightning mirror the branches of a river network or a tree
This book explains why these are not coincidences Nature commonly weaves its tapestry without any master plan or blueprint Instead, these designs build themselves by self-organization The interactions between the component parts -- whether they be grains of sand, molecules or living cells -- give rise to spontaneous patterns that are at the same time complex and beautiful Many of these patterns are universal, recurring again and again in the natural order: spirals, spots, stripes, branches, honeycombs Philip Ball conducts a profusely illustrated tour of this gallery, and reveals the secrets of how nature's patterns are made

The Self-Made Tapestry: Pattern Formation in Nature

When a piece of metal is fractured either by tensile or impact loading (pulling or hitting), the fracture surface that is formed is rough and irregular. Its shape is affected by the metal's microstructure (such as grains, inclusions and precipitates, whose characteristic length is large relative to the atomic scale), as well as by ‘macrostructural’ influences (such as the size, the shape of the specimen, and the notch from which the fracture begins). However, repeated observation at various magnifications also reveals a variety of additional structures that fall between the ‘micro’ and the ‘macro’ and have not yet been described satisfactorily in a systematic manner. The experiments reported here reveal the existence of broad and clearly distinct zone of intermediate scales in which the structure is modelled very well by a fractal surface. A new method, slit island analysis, is introduced to estimate the basic quantity called the fractal dimension, D. The estimate is shown to agree with the value obtained by fracture profile analysis, a spectral method. Finally, D is shown to be a measure of toughness in metals.

Alvin J. Paullay

Papers

Fractal character of fracture surfaces of metals