Three-dimensional body

Neuhauser D, Baer M, Judson R S and Kouri D J 1989 Time-dependent three-dimensional body frame quantal wavepacket treatment of the atomic hydrogen + molecular hydrogen exchange reaction on the Liu-Siegbahn-Truhlar-Horowitz (LSTH) surfaced. Chem. Phys. 90 5882... 

A three-dimensional body limited by two curvilinear surfaces is called a shell if a distance called a thickness of the shell between the afore mentioned surfaces is small enough. We assume that the thickness is the constant 2h > 0. The surface equidistant from the surfaces is called a mid-surface. Thus, a shell can be uniquely defined introducing a mid-surface, a thickness and a boundary contour. 

In the book, two- and three-dimensional bodies, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, thermoelastic, elastoplastic. The book gives a new outlook on the crack problem, displays new methods of studying the problems and proposes new models for cracks in elastic and nonelastic bodies satisfying physically suitable nonpenetration conditions between crack faces. 

Here is a point inside the volume V. Hence, it is impossible to distinguish between the field caused by a volume distribution of masses and the field generated by masses on the equipotential surface S, provided that the condition (4.6) is met and the observation point is located outside S. As a rule, a three-dimensional body and... 

Next, suppose that the gravitational field is caused by masses in a three-dimensional body. In principle, the field can be calculated from Equation (4.12) directly using known algorithms of integration, but even with fast computers a numerical integration over the volume for many observation points requires a lot of time. For... 

By analogy with the previous case, let us represent the three-dimensional body as a system of elementary layers located in horizontal planes whose thickness is much smaller than the distances from them to the observation points. In such a case, every layer can be replaced by a horizontal plane of finite dimensions with the surface density... 

Correspondingly, determination of the vertical component of the field due to masses in a three-dimensional body may consist of calculating a set of solid angles. 

The general case of a three-dimensional body enclosed by a surface 5 will be treated using vector analysis. The time change in the amount M is given by the relationship... 

The difference is of smaller order than the error in either solution and Eq. (3-35) is exact to 0(Re) (P3). In fact, the Re term in (Cq/Cdsi — 1) can be deduced from the Stokes drag alone for any three-dimensional body symmetrical about a plane normal to the direction of motion (C6). 

Different eorrelations are required for three-dimensional bodies (spheres, disks, and spheroids) than for the two-dimensional shapes (cylinders and wedges). For three-dimensional shapes transfer in the aft region is correlated by... 

Harper and Chang (H4) generalized the analysis for any three-dimensional body and defined a lift tensor related to the translational resistances in Stokes flow. Lin et al (L3) extended Saffman s treatment to give the velocity and pressure fields around a neutrally buoyant sphere, and also calculated the first correction term for the angular velocity, obtaining... 

Material bodies scattered throughout Plane P have neither zero thickness nor are they infinitely small. They have, instead, a very small, finite thickness, and this is formed by superimposed atoms or molecules. This thickness is not suspected by the Plat Men [or inhabitants of a wholly imaginary, two-dimensional Flat Land ], for whom there is only a bed of atoms and for whom groupings of atoms constituting the molecules are all formed within Plane P. This hypothesis does not, however, deny the possibility, if one prefers, of considering some identical bodies to be like slices [tranches] within the three-dimensional body of our Universe.. . . ... 

Here, the first term on the right-hand side gives the net diffusive inflow of species A into the volume element. We have assumed that the diffusive process follows Fick s law and that the diffusion coefficient does not vary with position. The spatial derivative term V2a is the Laplacian operator, defined for a general three-dimensional body in x, y, z coordinates by... 

Equation 4.40 gives the solution for one-dimensional diffusion from a point source on an infinite line, an infinite thin line source on an infinite plane, and a thin planar source in an infinite three-dimensional body (summarized in Table 5.1). Corresponding solutions for two- and three-dimensional diffusion can easily be obtained by using products of the one-dimensional solution. For example, a solution for three-dimensional diffusion from a point source is obtained in the form... 

A soil is a naturally occurring three-dimensional body with morphology and piopeities resulting fioni effects of climate, floia and fauna, paient rock materials, topography, and time. A soil occupies a portion of the land surface, is mappable and is composed of horizons that parallel the land surface. A vertical section downward through all the horizons of the soil is called a soil profile. See Fig. 1. 

If a three-dimensional body such as a tetrahedron is substituted for the two-dimensional triangle, the branching sites he on the surface of an imaginary sphere. This accounts for the shell-like structure of dendrimers. 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

Other Methods for Particle Shape Determination Gotoh and Finney [50] proposed a mathematical method for expressing a single, three-dimensional body... 

Beddow [42] showed how a number of particle silhouette shapes could be analyzed and reproduced by Fourier transforms. Gotoh and Finney [52] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid having the same volume, surface area and projected area as the original body. 

Transient heat transfer for a three-dimensional body formed by the intersection of three one-dimensional bodies 1, 2, and 3 is given by... 

In the second formal step of the process, the nuclear arrangement relaxes to a nearby minimum of the potential energy surface of the new electronic state. The three-dimensional body of the electron distribution "follows" the nuclear rearrangement, hence the shape of the electron distribution changes in this step too. This change is called the shape change due to relaxation. 

One technique which is applicable for surfaces that are not everywhere differentiable is also suitable for the shape characterization of dot representations of molecular surfaces such as the Connolly surfaces [87], which are not only nondifferentiable, but are not even continuous. The method of 1-hulls [351] is based on a generalization of the concept of convex hull. The convex hull of a set A is the smallest convex set that contains A. Consider a three-dimensional body T. The T-hull of a point set A is the intersection of all rotated and translated versions of T which contain A. The T-hull method is suitable for shape comparisons with a common reference shape, chosen as that of the body T. Alternatively, when the shapes of two molecules, T and A are compared, one molecular body can be chosen as T and the T-hull of the other molecular body A provides a direct shape comparison [351]. 

The image is somewhat distorted in some areas such as the hips, suggesting cloth draped over a three-dimensional body form. 

The years between 1860 and 1880 were characterized by a strong dispute about the concept of -> molecular structure, arising from the studies on substances showing optical isomerism and the studies of Kekule (1861-1867) on the structure of benzene. The concept of the molecule as a three-dimensional body was first proposed by Butlerov (1861-1865), Wislicenus (1869-1873), Van t Hoff (1874-1875) and Le Bel (1874). The publication in French of the revised edition of La chimie dans I espace by Van t Hoff in 1875 is considered a milestone in the three-dimensional conception of the chemical structures. 

Mezey, P.G. (1991c). The Degree of Similarity of Three-Dimensional Bodies Application to Molecular Shape Analysis. J.Math.Chem., 7, 39-49. 

Keuhauser. D.. Baer, M., Judson. R.S. and Kouri. D.J. (1989) Time-dependent three-dimensional body frame tjnantal wave packet treatment of the H + H2 exchange reaction on the Liu-Si< gbahn-Truhlar-Horovi itz (LSTH) surface,. 7. Chem. Phys. 90, 5882-5884. 

The quartz crystal microbalance (QCM) is a well-known tool to measure film thicknesses in the nanometer range [1-3]. It is difficult to imagine a device which is simpler than a quartz crystal resonator, and simphcity is one of the principal advantages of the QCM. A QCM is a disk of crystalline quartz. The disk displays acoustic resonances like any other three-dimensional body. As a resonator, it distinguishes itself from other resonators by a number of features ... 

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