By Professor Isaak A. Kunin (auth.)
Crystals and polycrystals,composites and polymers, grids and multibar structures will be regarded as examples of media with microstructure. A attribute characteristic of all such versions is the life of scale parameters that are hooked up with micro geometry or long-range interacting forces. for this reason the corresponding conception needs to primarily be a nonlocal one. The ebook 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 idea of defects are thought of intimately. a lot consciousness is paid to approximate versions and proscribing tran sitions to classical elasticity. The booklet may be regarded as a revised and up-to-date version of the author's booklet below an analogous identify released in Russian in 1975. The frrst quantity offers a self-con tained thought of one-dimensional types. the idea of three-d versions may be thought of in a imminent quantity. the writer want 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
Sample text
The first problem for the homogeneous medium cannot be reduced to a integral equation with difference-type kernel because of the intrinsic inhomogeneity of the elastic bonds in the boundary layer. As will be seen later, in the nonlocal theory of elasticity this leads to the fact that the solution of the first problem for the homogeneous medium is much more difficult than is the second one. Developing further the analogy with the usual theory of elasticity, let us obtain the Green's formula in the nonlocal theory.
E. the conditions of translational invariance are fulfilled. In Chap. 4 will we consider some consequences of the breaking of translational invariance, within the framework of the energy method. 2), it is easy to see that for the chain with interaction of finite number of neighbors the function tP(k) is always bounded for real values of k and hence there exists a maximum frequency W max for nondecaying waves, which can be reached for the limiting values of k = ±1t/a as well as for the internal points.
Medium of Simple Structure SO Thus the chain plays the role of a filter of low frequencies: signals with frequencies (f) < (f)max will propagate and signals with higher frequencies will decay exponentially. 1), that (f)max appears also for the nonlocal model of the continuous medium under physically reasonable restrictions on P(x). The presence of the maximum (one or more) in the curve (f) = (f)(k) for real values of k, has as a consequence the existence of zones of frequencies in which to each frequency there correspond several nondecaying waves which propagate in one and the same direction.