Wave propagation in 1D elastic solids in presence of long-range central interactions Massimiliano Zingales n Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica, DISAG, Universit  a degli Studi di Palermo, Viale delle Scienze Ed.8, I-90128, Palermo, Italy article info Article history: Received 22 January 2009 Received in revised form 20 October 2010 Accepted 23 October 2010 Handling Editor: A.V. Metrikine Available online 6 May 2011 abstract In this paper wave propagation in non-local elastic solids is examined in the framework of the mechanically based non-local elasticity theory established by the author in previous papers. It is shown that such a model coincides with the well-known Kr ¨ oner–Eringen integral model of non-local elasticity in unbounded domains. The appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the Cauchy stress. In fact, the long-range forces between different volume elements are modelled, in the body domain, as central body forces applied to the interacting elements. It is shown that the shape change of travelling disturbances coalesces with those predicted by the non-local integral theory of elasticity in unbounded domains, but several differences arise in the case of bounded domains. The wave propagation problem has been formulated by means of the Hamiltonian functional of the proposed mechanically based model of non- local elasticity, introducing an additional term to the elastic potential energy that accounts for elastic long-range interactions. In this way, the wave equation may be obtained in a weak formulation and be further used to provide approximate analytical solutions to the governing equation in the context of standing wave analysis. An equivalent discrete point-spring model, similar to lattice-type networks, has also been introduced to show the mechanical equivalence of the non-local elastic model as well as to provide a mechanical scheme suitable for the numerical treatment of pressure waves travelling in non-local bounded domains. & 2011 Published by Elsevier Ltd. 1. Introduction The mechanics of elastic continuum has received enthusiastic support since its introduction at the beginning of the eighteenth century. Since then, very famous leading scientists have devoted strong research efforts to the formulation of elastic problems in several fields of physics and engineering, providing results in good agreement with experiments. Because a more precise description of the behaviour of solids was needed in the early forties, the classical continuum mechanics theory no longer seemed to be appropriate for describing experimental evidence. Discrepancies were observed in the presence of shear bands at the boundaries of tensile specimens, or in the presence of infinite stress in correspondence with crack tips, as predicted by continuum mechanics. Other drawbacks were observed in a dynamic context, for instance, the absence of dispersion in elastic waves travelling in 1D elastic solids. The main reason for the strong differences between the observed and predicted phenomena was found in the intrinsic size-independence of continuum mechanics. Since then, Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration 0022-460X/$ - see front matter & 2011 Published by Elsevier Ltd. doi:10.1016/j.jsv.2010.10.027 n Tel.: +39 091 656 8458. E-mail address: zingales@diseg.unipa.it Journal of Sound and Vibration 330 (2011) 3973–3989