Mechanics Research Communications 38 (2011) 512–517 Contents lists available at ScienceDirect Mechanics Research Communications jo ur nal homep age : www.elsevier.com/locate/mechrescom On the strong ellipticity for orthotropic micropolar elastic bodies in a plane strain state Francesca Passarella, Vincenzo Tibullo ∗ , Vittorio Zampoli Dipartimento di Ingegneria Elettronica e Ingegneria Informatica, Università di Salerno, Italy a r t i c l e i n f o Article history: Received 21 May 2011 Received in revised form 29 June 2011 Available online 12 July 2011 Keywords: Strong ellipticity Orthotropic micropolar materials Progressive plane waves a b s t r a c t In the present paper we consider an orthotropic micropolar elastic material subject to a state of plane strain. In this context, we establish necessary and sufficient conditions for the strong ellipticity of constitu- tive coefficients. Furthermore, we study existence of progressive plane waves under the strong ellipticity conditions previously determined. Finally, we detail the results obtained for a specific class of materials related to tetragonal systems. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction During the past years, several theories concerning continuum mechanics characterized by a complex microstructure have been object of intensive study. The reason of such a kind of investiga- tion has to be researched in particular in modern technological and industrial necessities, as well as in biological and medical recent developments, which led to increasingly refined mathemat- ical models for elastic bodies. We remind that classical elasticity associates three degrees of freedom to material points of the considered body: for this reason, its mechanical properties are expressed in terms of displacement vectors. On the other hand, if we consider the points as particles, taking into account also their intrinsic rotation, then we approach the micropolar elasticity theory (Eringen, 1966, 1967, 1968, 1999) that provides a more complex mathematical model able to manage materials which exhibit microrotation effects and which can sup- port body and surface couples. Consequently, in micropolar theory, all the mechanical quantities are written in terms of displace- ment and microrotation vectors. In this context, important results have been obtained in Ies ¸ an (1969, 1970, 1971), Chandrasekharaiah (1986), Ciarletta (1991), and Ciarletta and Ies ¸ an (1993). More- over, Ies ¸ an (1974a) considers the operator of micropolar elasticity, proving its positive definiteness for the first boundary value problem. Parfitt and Eringen (1969) discuss plane waves propaga- tion in an infinite isotropic homogeneous micropolar elastic ∗ Corresponding author. E-mail addresses: vtibullo@unisa.it, enzotib@gmail.com (V. Tibullo). half-space; the authors show that there exist four waves in a micropolar elastic material, two of which disappear below a critical frequency dependent upon the properties of the medium. On the other side, in many engineering frameworks (such as the behavior of soil, geological or composite materials), an assumption of isotropy can be inadequate to realistically reproduce features of the considered medium: elastodynamic response of anisotropic continua has thus received the attention of several researchers. For example, orthotropic materials have been intensively studied by Ies ¸ an: he writes the constitutive equations for an orthotropic micropolar continuum in Ies ¸ an (1974b), while in Ies ¸ an (1973) he considers the static problem of plane micropolar strain, deriv- ing uniqueness and existence theorems. Moreover, transversely isotropic and rhombic systems modeled as Mindlin-type plates have been investigated in Passarella and Zampoli (2009a, 2009b) and Passarella et al. (2010). In particular, in the last two works the study of spatial behavior of solutions has been performed under the strong ellipticity condition on the elasticity tensor. For trans- versely isotropic and rhombic materials, strong ellipticity has been object of intensive study, see e.g. Merodio and Ogden (2003), Chirit ¸˘ a (2006), and Chirit ¸˘ a et al. (2007). In Chirit ¸˘ a and Danescu (2008) Chirit ¸˘ a and Danescu study the strong ellipticity conditions for all crystal classes of tetragonal systems in a linearly elastic mate- rial. The condition of strong ellipticity shows its applicability, for example, in the study of the uniqueness of solutions or in wave propagation, as can be seen in Gurtin (1972). The ellipticity analy- sis has important applications in several contexts. For example, in plane strain for fiber reinforced elastic materials, this analysis has been related to fiber instabilities, see Merodio and Ogden (2005, 2006). 0093-6413/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2011.06.006