Euclidean solids

J) Euclidean solid. This type of solid cannot be deformed by stress, i.e. its elasticity coefficient is infinite. A Euclidean solid does not exist in reality, although its properties are approached by hard materials such as diamond. 

It is necessary here to point out that the rheology of all materials, especially the rheology of surfactant solutions, can depend strongly on the time-scale on which the material is studied. This can be demonstrated quite easily. For example, Newtonian liquid water behaves as an elastic solid if the stress is applied for nanoseconds or picoseconds on the other hand, granite, which can be regarded as an Euclidean solid to a first approximation, can flow like a liquid if a strong stress acts over a period of 10 -10 years. Therefore, it must be stated at this point that the description of the rheological effects of surfactant solutions is valid for observation times which are necessary for the measurements, i.e. several seconds up to several days, or weeks in some cases. 

The Tabor s condition (the Eq. (12.1) with c 3) in the Eq. (12.5) case is reached at 2.95, that is, at the greatest fractal dimension value for real solids [14]. This circumstance assumes two consequences. Firstly, the indicated above Tabor s condition is correct for Euclidean solids only. For fractal objects c < 3 and in the typical for solids range of d = 2.0 2.95 the value FZ/Oy varies within the limits of 1.5 3.0. Secondly, the value c = 3 reaching at d = 2.95 assumes the Eq. (12.5) higher precision in comparison with Eq. (12.4). The data of Fig. 12.1 are confirmed by this assumption, where the dependence of the experimental values (points) on d value is shown. They correspond well to the curve, calculated according to the Eq. 

From the Eq. (14.15) it follows, that polymers structure fractality < d) results to yield stress essential reduction. From the point of view of thermodynamics Oy indicated reduction is due to accumulation in sample of internal (latent) energy, the relative fraction of which is equal to about [41]. For Euclidean solids d = d) the Eq. (14.15) gives Hooke law. At the same time for the indicated materials extrudates strong increase in comparison with initial samples is due to E and d simultaneous growth. It is follows to note also, that the Eq. (14.15) can be used for description of polymers deformation on the elasticity part (at d = d) and on cold flow plateau (at d = d and elasticity modulus replacement on strain hardening modulus) [32]. 

Thus, the fractal analogues of the Hill and Marsch equations obtained above have shown that the microhardness of crosslinked epoxy polymers is defined only by their structure, characterised by its fractal dimension. Tabor s criterion is correct for Euclidean (or close to Euclidean) solids only. The degree of increase in crosslinking results in loosening of the loosely packed matrix and to a corresponding reduction of the microhardness of crosslinked epoxy polymers [60]. 

The BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is... 

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Figure 2.28 shows how the sum over all samples of the Euclidean distance, E between target output and actual output varies as a function of the number of epochs during a typical training session. Ed for the training set (solid line) falls continuously as training occurs, but this does not mean that we should train the network for as long as we can. 

In the simplest case, a system is called Euclidean or nonfractal if its topological dimension dt is identical to the fractal dimension df. This means dt = df = 1 for a curve, dt = df = 2 for a surface, and dt = df = 3 for a solid. The following relationship holds for the three expressions of dimensionality... 

The exponent df is denoted mass fractal dimension or simply fractal dimension. It characterizes the mass distribution in three dimensional space and can vary between lfractal analysis of furnace blacks was performed, e.g., by Herd et al. [108] or Gerspacher et al. [109, 110]. The solid volume Vp of primary aggregates is normally determined (ASTM 3849) from the cross-section area A and the perimeter P of the single carbon black aggregates by referring to a simple Euclidean relation [108] ... 

For a cube, a equals 6 and / is the side dimension of the cube. For Euclidean geometries, the term, fl is a constant (e.g., for spheres of radius I, a = 4tt), and d=2. The geometric surface area per gram of solid, Ageo (the specific surface area)... 

Surfaces of most materials, including natural and synthetic, porous and non-porous, and amorphous and crystalline, are fractal on a molecular scale. Mandelbrot defines that a fractal object has a dimension D which is greater than the geometric or physical dimension (0 for a set of disconnected points, 1 for a curve, 2 for a surface, and 3 for a solid volume), but less than or equal to the embedding dimension in an enclosed space (embedding Euclidean space dimension is usually 3). Various methods, each with its own advantages and disadvantages, are available to obtain... 

Euclidean geometry fails to describe disordered surfaces such as real solid surfaces. Fractal geometry, which has been developed to overcome this obstacle, covers surface, mass, and pore fractality. It has been pointed out that the diffusion process can be used to characterize the fractal dimension of a rough surface. The impedance response of a rough electrode could be used for the characterization of the roughness and. 

Solid surfaces can be of different kinds. They can be defined as Euclidean, nonporous, completely flat surfaces, with area, S, equal to the square of their linear dimension R [34] ... 

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

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