Three-dimensional measurement, geometrical

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

One important application of matrix algebra is formulating the transformations of points or vectors which define a geometrical entity in space. In ordinary three-dimensional space that involves three axes, any point is located by means of three coordinates measured along these axes. Similarly... 

Here three constants appear Go is the equilibrium modulus of elasticity 0p is the characteristic relaxation time, and AG is the relaxation part of elastic modulus. There are six measured quantities (components of the dynamic modulus for three frequencies) for any curing time. It is essential that the relaxation characteristics are related to actual physical mechanisms the Go value reflects the existence of a three-dimensional network of permanent (chemical) bonds 0p and AG are related to the relaxation process due to the segmental flexibility of the polymer chains. According to the model, in-termolecular interactions are modelled by assuming the existence of a network of temporary bonds, which are sometimes interpreted as physical (or geometrical) long-chain entanglements. 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the ehaos assoeiated with motion on the attractor and some geometrical measme of the structural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the diseussion we consider three-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

The word shape is loaded with strong personal intuitions of what each of us thinks defines an object in three-dimensional (3D) space. Sometimes, we may view an object as a distribution of points in space and conceive its shape in terms of distances and angles, that is, local geometrical measures. More often, we associate the notion of shape with a solid object defined by an envelope surface. In this case, a precise geometrical characterization may lose its relevance, and it may be sufficient to characterize shape in a more qualitative fashion. Global topological measures are better suited here. Therefore, we... 

There are three general classes of descriptors topological, geometrical, and physicochemical. Topological descriptors are derived from the topological representation of the structure, the connection table. The geometrical descriptors are derived from the three dimensional model of the molecule. Physicochemical descriptors may be measured experimentally, calculated using a mathematical model, or represented by linearly correlated calculated descriptors. The descriptors that are currently available in ADAPT are as follows ... 

Blum, J.J. (1975) Slice synthesis of three dimensional work-of-fracture specimens, Eng. Fract. Mech. 7, 593. The slice model for determining Y, the geometric constant in measurements (cf. Munz et al.). 

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