Mechanics Research Communications 47 (2013) 106–111 Contents lists available at SciVerse ScienceDirect Mechanics Research Communications jou rnal h om epa ge: www.elsevier.com/locate/mechrescom On the influence of internal degrees of freedom on dispersion in microstructured solids Kert Tamm ∗ , Tanel Peets Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia a r t i c l e i n f o Article history: Received 30 November 2011 Received in revised form 24 September 2012 Accepted 17 October 2012 Available online 26 October 2012 Keywords: Dispersion Microstructure Nonlinearity Pseudospectral method Solitary waves a b s t r a c t In the present paper a model describing wave propagation in the nonlinear dispersive media with microstructure is investigated. The model is based on the continuum approach following Mindlin’s and Eringer’s earlier theories which model a microstructure as a deformable cells in a macrostructure assuming that the deformation gradient is small. A generalized version of the Mindlin model called the Mindlin–Engelbrecht–Pastrone model (MEP) is used. The MEP model is solved numerically using the pseudospectral method and localized initial conditions together with periodic boundary conditions. The main focus of the study is on clarifying the influence of internal degrees of freedom of a microstructure on solutions. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction A model describing wave propagation in nonlinear dis- persive media with microstructure is studied. In this model the microelement is taken following Mindlin (1964) as a deformable cell with an additional assumption that the deforma- tion gradient is small. A generalized version of Mindlin model called Mindlin–Engelbrecht–Pastrone model (MEP) is used by Engelbrecht et al. (2006). While the MEP model has been well studied and generalized in the framework of dual internal variables (see, for example, Berezovski et al. (2011a) for the generalization and Berezovski et al. (2011b) for additional details on the concept of dual internal variables) there is still need for clarifications on how the inter- nal degrees of freedom can and do affect the macroscopic wave propagation. The concept that internal variables can affect the macroscopic behavior is well established in physics (see, for exam- ple, Brillouin (1953) and Eringen (1999)). In the present work the influence of the internal degrees of freedom is demonstrated in the MEP model on the macroscopic behaviour of the waves by inter- preting the higher order dispersion branch as manifestation of an internal degree of freedom. The main goals of the present study are: (i) to solve model equa- tions numerically under localized initial conditions and periodic ∗ Corresponding authors. E-mail addresses: kert@cens.ioc.ee (K. Tamm), tanelp@cens.ioc.ee (T. Peets). boundary conditions and (ii) to investigate the influence of the internal degrees of freedom on the solutions of the MEP. 2. Model equations In the present paper a model derived by Engelbrecht and Pastrone (2003), Engelbrecht et al. (2005, 2006), Janno and Engelbrecht (2005a,b) is applied to describe wave propagation in nonlinear dispersive media with microstructure. The model is based on Mindlin’s and Eringen’s earlier works (Mindlin (1964), Eringen (1966, 1999)). As mentioned in the introduction, in this model the microelement is taken as a deformable cell with an additional assumption that the deformation gradient is small, thus allowing one to express microdeformation in terms of macrodis- placement. Balance laws are formulated separately for the macro- and microscale and we deal with 1D case. In order to clarify the principal essence and the role of the parameters of the model, we repeat here briefly the basic steps of modeling. It should be noted that similar governing equations can be achieved also by starting with the lattice model (see, for example, Polyzos and Fotiadis (2011), Metrikine and Askes (2002), Metrikine (2006)) 2.1. Mindlin–Engelbrecht–Pastrone model In the 1D case the Lagrangian L is expressed as follows: L = K - W, K = 1 2 u 2 t + 1 2 Iϕ 2 t , W = W(u x , ϕ, ϕ x ). Here K is the kinetic energy, W is the free (potential) energy, I is the microinertia, ϕ is the 0093-6413/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2012.10.006