International Journal of Non-Linear Mechanics 99 (2018) 281–287 Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm Two-dimensional waves in extended square lattice A.V. Porubov a, b, *, A.M. Krivtsov a, b , A.E. Osokina b a Institute for Problems in Mechanical Engineering, Bolshoy 61, V.O., St. Petersburg 199178, Russia b Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya st., 29, St. Petersburg 195251, Russia article info Keywords: Square lattice Continuum limit Non-linear wave Instability abstract We consider a two-dimensional square lattice model extended by additional not closed neighboring interactions. We assume the elastic forces between the masses in the lattice to be nonlinearly dependent on the spring elongations. First, we use an analysis of the linearized discrete equations to reveal the influence of additional interactions on the properties of the dispersion relation for longitudinal and shear plane waves. Then we develop an asymptotic procedure to obtain continuum two-dimensional non-linear equations to study the transverse instability of weakly non-linear localized plane longitudinal and shear waves. We find that the additional interactions used in the model may affect the sign of the amplitude of the plane strain waves (existence of compression (minus sign) or tensile (plus sign) plane waves) and their transverse stability. © 2017 Elsevier Ltd. All rights reserved. 0. Introduction The study of the discrete model with non-neighboring interactions between the particles in the lattice has attracted considerable interest due to the dispersion of waves propagating in such a system [1–8]. In particular, this model is important for the study of the influence of the microstructure of materials. Dynamic processes in one-dimensional lattices have been investigated more extensively [1,3,9], while two- dimensional lattices are mainly considered in the linearized case [3,6,7]. Some two-dimensional processes can be modeled in the one-dimensional approximation, like plane waves propagation, while the study of their transverse instability requires two-dimensional consideration. Also some physical phenomena cannot be modeled in the one-dimensional case, in particular, for a negative Poisson ratio or auxetic behavior [10–13]. The structural features of the lattice are usually taken into ac- count [10,14,15] to describe a negative Poisson ratio. In [11] it was obtained that a negative Poisson ratio is observed for some directions in many cubic metals due to their crystalline lattice features. It is also known that anisotropic systems like cubic ones are typically nonauxetic or partially auxetic [16]. The relationships for an anisotropic Poisson ratio in some materials may be found in [17,18]. There is a procedure for comparing the continuum limits of 2D discrete models with the 2D limit of the continuum cubic crystal model [15] to establish a connec- tion between the rigidities of the lattice model and the cubic elastic constants. It turns out that these relationships hold only for the Cauchy * Corresponding author at: Institute for Problems in Mechanical Engineering, Bolshoy 61, V.O., St. Petersburg 199178, Russia. E-mail address: alexey.porubov@gmail.com (A.V. Porubov). condition [19]. It applies to materials with a cubic symmetry where only central interactions are taken into account; however, deviations from the conditions may be considerable, e.g., for cubic metals [20]. However, it was found in [21] that the Cauchy relations do not hold for positive temperatures. Comparison with the 2D model, e.g., the auxetic properties of 2D media, were studied in [22]. Dynamic processes in lattices have been studied using both discrete and continuum modeling [1,9]. In the linear case, both discrete and continuum equations can be solved analytically. However, only a few discrete non-linear equations, such as the Toda lattice equation or the Ablowitz–Ladik equation, possess exact solutions [23]. That is why an approach based on the continuum limit of the original discrete equation is needed to obtain the governing non-linear continuum equations. The familiar acoustic branch continuum limit [1,9] requires the long wavelength approximation and corresponds to the discrete model only for small wave numbers. The mechanical properties and stability of lattices depend on their structure and particle interaction [19,24,25]. Discrete and continuum models both possess analytical solutions in the linear case, which allows complex analysis of the mechanical phenomena from micro- and macroscopic points of view [26]. This analysis becomes crucial for nano-objects where the discreteness of the atomic structure cannot be neglected [27]. Nonlinearity is essential for a description of thermo- mechanical effects [28] including peculiarities such as negative thermal expansion [29]. https://doi.org/10.1016/j.ijnonlinmec.2017.12.008 Received 10 July 2017; Received in revised form 10 December 2017; Accepted 10 December 2017 Available online 14 December 2017 0020-7462/© 2017 Elsevier Ltd. All rights reserved.