Time-harmonic wave propagation in a pre-stressed compressible elastic bi-material laminate Priza Kayestha a , Anil C. Wijeyewickrema a, * , Kikuo Kishimoto b a Department of Civil Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan b Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1, Meguro-ku, Tokyo 152-8552, Japan article info Article history: Received 10 March 2009 Accepted 28 August 2009 Available online 8 October 2009 Keywords: Compressible Dispersion curves Elastic waves Pre-stress Wave propagation abstract The dispersive behaviour of time-harmonic waves propagating along a principal direction in a perfectly bonded pre-stressed compressible elastic bi-material laminate is considered. The dispersion relation which relates wave speed and wavenumber is obtained by formulating the incremental boundary value problem and the use of the propagator matrix technique. At the low wavenumber limit, depending on the pre-stress, both the fundamental mode and the next lowest mode may have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region, an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and higher modes tend to phase speeds of the surface wave, the interfacial wave or the limiting phase speed of the composite. For numerical examples, either a two-parameter compressible neo-Hookean material or a two-parameter compressible Varga material is assumed. Ó 2009 Elsevier Masson SAS. All rights reserved. 1. Introduction Materials that can undergo considerable deformation before failure have been widely used as seismic isolators in bridges, high- rise buildings, nuclear power plants and other critical infrastruc- ture as a passive protection measure against earthquakes (Chris- topoulos and Filiatrault, 2006). Extensive studies in the field of pre- stressed elastic media have been carried out, see for example, Fu and Ogden (2002). In most instances a particular wave propagation problem in pre-stressed media was first solved for incompressible elastic media, prior to considering the same problem for compressible elastic media. Wave propagation in a pre-stressed layer has been studied by many researchers. Ogden and Roxburgh (1993) studied the vibra- tion and stability of a pre-stressed incompressible elastic plate which is finite in all directions, and the compressible counterpart was reported by Roxburgh and Ogden (1994). An asymptotic analysis of the dispersion relation for waves propagating in a pre- stressed incompressible elastic plate was carried out by Rogerson and Fu (1995). The corresponding problem for a nearly incom- pressible elastic layer was solved by Sandiford and Rogerson (2000) and for a compressible elastic layer by Nolde et al. (2004). In a recent paper Wijeyewickrema et al. (2008) investigated time- harmonic waves propagating in a pre-stressed compressible elastic layer with constrained boundaries. While most of the studies have been on wave propagation along a principal direction there have been a few papers on wave prop- agation along a non-principal direction. See Rogerson and Sandi- ford (1999) for wave propagation in an incompressible layer and Destrade et al. (2005) for wave propagation in an incompressible half-space. The dispersive behaviour of waves in pre-stressed elastic lami- nated composites has also been studied previously. Wave propa- gation in a pre-stressed symmetric elastic layered composite was considered by Rogerson and Sandiford (1997, 2000a) for incom- pressible perfectly bonded material and the corresponding prob- lems with an imperfect interface was solved by Leungvichcharoen and Wijeyewickrema (2003) and Leungvichcharoen et al. (2004). The pre-stressed imperfectly bonded compressible symmetric elastic layered composite was investigated by Wijeyewickrema and Leungvichcharoen (2009). The propagation of harmonic waves in a pre-stressed incompressible elastic bi-material laminate was investigated in detail by Rogerson and Sandiford (2000b). Recent studies in the area of wave propagation through inho- mogeneous nonlinear elastic media have also been reported for non- laminated geometries. Parnell (2007) considered small amplitude elastic waves propagating in a one-dimensional pre-stressed composite bar consisting of two distinct elastic phases that are periodically distributed while Bertoldi and Boyce (2008) studied phononic band gaps of periodic elastomeric materials with either * Corresponding author. Tel.: þ81 3 5734 2595; fax: þ81 3 5734 3478. E-mail address: wijeyewickrema.a.aa@m.titech.ac.jp (A.C. Wijeyewickrema). Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.08.005 European Journal of Mechanics A/Solids 29 (2010) 143–151