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Steady-state and transient antiplane dynamic processes in a structured solids consisting of uniform periodic square-cell lattices connected by a lattice layer of different bond stiffnesses and point masses are analyzed. A semi-infinite lattice covered by a layer is also considered. Localization phenomena that are characterized by a waveguide-like propagation along the layer direction and exponential attenuation along its normal are studied. Waveguide pass-bands and attenuation factors are obtained analytically, while transient processes developed under the action of a monochromatic local source are numerically simulated. As a result, it is shown how a two-dimensional problem is transformed with time into a quasi-one-dimensional one and how a layer traps the source energy. Special attention is paid to revealing particularities of transient waves in cases where steady-state solutions are absent: resonant waves with frequencies demarcating pass-and stop-bands at the ends of the Brillouin zone and wave transition in the vicinities of transition points in dispersion curves. In the latter case, a simultaneous onset of different localization phenomena -a spatial star-like beaming and a one-dimensional waveguide-like localization -is shown.

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An axisymmetric contact-impact problem is considered for an elastic layer subjected by normal indentation of a rigid body. An exact analytical solution is obtained in the case1. Introduction. This paper is the continuation to the axisymmetric case of the plane problem analyzed in [9]. The introduction presented in [9] could be practically completely replicated herein, as well as sited works within [9]. With analytical approaches in mind, we refer review [1] reflecting the multitude of studies of a body's impact interaction with elastic and liquid media, while numerical approaches (primarily the method of finite elements) can be found in review [17]. A generalizing monograph in the field of contact interaction [3] is devoted to the development of analytical approaches to the solution of problems about the action of impact on an elastic medium. In the common case, the indentation problem is formulated as a unsteady-state mixed initially-boundary elasticity problem with an unknown (temporally varying) boundary, which must be determined in the course of the solution. The problem statement includes:· equations of dynamic deformation of the impacted solid; · the equation of the indenter motion; · the ratio presenting the resistive force (drag) as a function of contact stresses on a priori unknown contact surface; · the equation connecting the contact zone size with the indenter displacement, · the corresponding boundary and initial conditions; The overwhelming majority of publications (at least of those in which analytical methods are used) are devoted to the problem of impact by rigid or deformable indenter against a halfspace that precludes the possibility of analyzing the waves reflected from the boundaries of the impacted solid. Studies of indenter interaction with solids of finite size are much less represented. Positing such a problem appears topical in the practical aspect as well -in particular, in view of the wide use of laminate materials in modern aircraft and shipbuilding. It is noteworthy that scale effect is among the determinant qualitative factors for problems of stresses and fracture in impact interaction (see e.g. [10,11]): the structure element under impact loading is destroyed by stresses whose level is formed due to superposition of waves reflected from boundary surfaces. The classical Hertz theory of collision is known to be applicable in dynamics at large time values, i. е. after the wave processes have faded in the solid. The Saint -Venant wave theory of rod collision is well developed only for one-dimensional or quasi one-dimensional problems and does not take into account energy transfer in directions different from the impact one. The foresaid evaluates the motivation of the presented work devoted to the construction and investigation of more adequate models and methods for dynamic processes of indentation. This paper, in similar to [9], consists of two parts: in the first one a precise analytical solution of the problem is built of the indentation with a constant impact velocity of an smoot...

Transient wave processes in mass-spring lattices excited by point oscillating sources are studied. Dispersion properties of uniform periodic three-dimensional (3D) square-cell and two-dimensional (2D) hexagonal-cell lattices including revealed star-shaped localization phenomena are analysed. The resonant-like waves and localization-like patterns in non-uniform lattices possessing predetermined and randomly distributed defects are numerically examined in order to identify the sensitivity of star-shape forms to different types of defects. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.