By Professor Isaak A. Kunin (auth.)
Crystals and polycrystals,composites and polymers, grids and multibar platforms will be regarded as examples of media with microstructure. A attribute function of all such versions is the lifestyles of scale parameters that are attached with micro geometry or long-range interacting forces. for this reason the corresponding concept needs to basically be a nonlocal one. The publication is dedicated to a scientific research of results of microstructure, internal levels of freedom and nonlocality in elastic media. The propagation of linear and nonlinear waves in dispersive media, static difficulties, and the speculation of defects are thought of intimately. a lot awareness is paid to approximate versions and restricting tran sitions to classical elasticity. The ebook may be regarded as a revised and up to date variation of the author's ebook below an analogous name released in Russian in 1975. The frrst quantity provides a self-con tained concept of one-dimensional types. the speculation of third-dimensional versions should be thought of in a coming near near quantity. the writer wish to thank H. Lotsch and H. Zorsky who learn the manuscript and provided many suggestions.
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Additional info for Elastic Media with Microstructure I: One-Dimensional Models
8). It follows that the elastic energy ~ is to be invariant under the transformation u(x) - u(x) + uo, where Uo = constant, since this transformation does not change distances between the particles of the medium. The direct application of this condition for the unbounded medium meets with some difficulties, because the energy ~ is not defined a priori on displacements, not vanishing as Ix I - 00. Therefore, let us first consider the case of a bounded medium. 15) or taking into account the arbitrariness of u(x) and Uo I ~(x, x') dx' = 0 .
Imk (wzO) j L O 0 WL::L If/a k 0.... 10. Trajectories of roots 2. 6). The three topologically different diagrams of the trajectories on the evolvent of the cylinder are shown in Fig. 11 (the dispersion curves are shown there too). Imh 1-2 rw-o) 0 .... ll. 8 Kernel of Operator tPOI in the Complex Region Such diagrams characterize qualitatively different chains, and we will call them the phase diagrams. 1) splits into three subregions with the same phase portraits. The last ones are determined by the set of singular points of the trajectories and their nature.
Elastic Media with Microstructure I: One-Dimensional Models by Professor Isaak A. Kunin (auth.)