Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media R. Kumar a, * , N. Sharma b , P. Ram b a Department of Mathematics, Kurukshetra University, Kurukshetra, India b Department of Mathematics, NIT, Kurukshetra 136119, India article info Available online 7 March 2008 Keywords: Orthotropic micropolar elastic solid Normal and transverse force stiffness Transverse couple stiffness Perfect bonding Amplitude ratios abstract This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coeffi- cients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction A micropolar elastic material solid differs from a classical elastic material in that each point has extra rotational degrees of freedom independent of translation, and that the material can transmit cou- ple stress as well as the usual force stress. Micropolar theory [1] is though to apply to structured materials with fibrous, lattice, or granular microstructure. Considered in [2] is the plane wave prop- agation in an infinite isotropic homogeneous micropolar elastic solid half-space and showed the existence of four basic waves (a longitudinal displacement wave, a longitudinal microrotational wave and two sets of two coupled waves) propagating with differ- ent velocities in an isotropic micropolar elastic solid. Discussed in [3,4] wave propagation in micropolar elastic solids. In many engineering problems, including the response of soils, geological materials and composites, the assumptions of an isotro- pic behavior may not reflect some significant features of the con- tinuum response. The formulation and solution of anisotropic problems in much more difficult and cumbersome than its isotro- pic counterpart. In the last years the elastodynamic response of anisotropic continuum has received the attention of several researchers. In particular, transversely isotropic and orthotropic materials, which may not be distinguished from each other in plane strain and plane stress cases, have been more regularly stud- ied. Static problems [5–7] have been considered for orthotropic micropolar elastic solids. Imperfect bonding considered in this study is to mean that the stress components are continuous and small displacement field is not. The small vector difference in the displacement is assumed to depend linearly on the traction vector. More precisely, jumps in the displacement components are assumed to be proportional (in terms of ‘‘spring-factor-type” interface parameters) to their respective interface components. The infinite values of interface parameters imply vanishing of displacement jumps and therefore correspond to perfect interface conditions. On the other hand, zero values of the interface parameters im- ply vanishing of the corresponding interface tractions which corre- sponds to complete debonding. Any finite positive values of the interface parameters define an imperfect interface. Such interface parameters may be present due to the presence of an interphase layer or perhaps interface bond deterioration caused by, for exam- ple, fatigue damage or environmental and chemical effects. The values of interface parameters depend upon the material proper- ties of the medium, i.e. microstructure as well as the bi-material properties. Significant work has been done to describe the physical conditions on the interface by different mechanical boundary con- ditions by different investigators. Notable among them are [8–16]. Recently various authors have used the imperfect conditions at an interface to study various types of problems [17–21]. Studied in [22–24] are various problems in orthotropic micropolar elastic medium. Discussed in [25] is the problem of moving load at bound- ary surface. Recently, wave propagation in an orthotropic micropo- lar elastic solid is considered in [26]. In what follows, two-dimensional governing equations in xy- plane for an orthotropic micropolar elastic solid are solved to show 0167-8442/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2008.02.007 * Corresponding author. E-mail address: rajneesh_kuk@rediffmail.com (R. Kumar). Theoretical and Applied Fracture Mechanics 49 (2008) 305–312 Contents lists available at ScienceDirect Theoretical and Applied Fracture Mechanics journal homepage: www.elsevier.com/locate/tafmec