Waves in lattices with imperfect junctions and localized defect modes

A correspondence between continuum periodic structures and discrete lattices is well known in the theory of elasticity. Frequently, lattice models are the result of the discretization of continuous mechanical problems. In this paper, we discuss the discretization of two-dimensional square thin-walled structures. We consider the case when thin-walled bridges have defects in the vicinity of junctions. At these points, the displacement satisfies an effective Robin-type boundary condition. We study a defect vibration mode localized in the neighbourhood of the damaged junction. We analyse dispersion diagrams that show the existence of standing waves in a structure with periodically distributed defects.


Introduction
The analysis of localized vibration modes within continuous and discrete lattice structures has a long history, starting with Maradudin et al. (1963). The application of defect vibration modes was found initially in solid-state physics (Maradudin 1965;Mead 1973Mead , 1996Mead & Parthan 1979;Mead & Yaman 1991) and many other publications. In the two-component chains, defect modes were discussed in Bacon et al. (1962). In the scattering of waves in solids, this subject was studied by Callaway (1964). There are more examples of localized vibration modes in theoretical and experimental physics, especially in photonic crystal fibres ( John 1987;Yablonovitch 1987Yablonovitch , 1993. One aspect of the technological applications involves models of acoustics and elasticity with photonic crystal models based on 'mass-spring' interaction lattices (Jensen 2003;Martinson & Movchan 2003;Movchan & Slepyan 2007). The passband Green functions for twodimensional lattices were discussed in the paper by Martin (2006).
It is well known that lattice structures possess phononic band gaps, i.e. intervals of frequencies for which there are no mechanical waves that could propagate through the structure. Neglecting the inertia of the thin-walled bridges in comparison with that of the masses at the junctions, we perform a discretization of the structure. Hence, we obtain a lattice with particles connected by springs. If the initial periodic structure has defects at the junctions such as cracks, at these points, the principal term of the displacement satisfies an effective Robin-type boundary condition characterized by a defect stiffness parameter (Zalipaev et al. 2007).
In the present paper, we study localized vibration modes for mechanical lattice structures of two types: a one-dimensional diatomic lattice and a twodimensional square lattice. The defects, periodically distributed in the lattice, change the nature of the band gaps. For these lattices, we analyse dispersion diagrams and explore possible localized standing waves. Furthermore, we explore the existence of a localized vibration mode for a one-dimensional lattice with a single defect. We study how the behaviour of this mode depends on the defect stiffness parameter.
In §2, we consider the transition from two-dimensional wave propagation through the square periodic structure in the anti-plane shear case to a discretized model of a two-dimensional lattice with masses connected by springs. In §3, we initially study phononic diagrams for a one-dimensional lattice with periodically distributed defects. Then, we discuss the existence of a localized vibration mode for a lattice with a single defect. In §4, we find the dispersion diagrams and analyse how standing waves depend on the defect stiffness parameter for a twodimensional lattice. Throughout this paper, the time dependence of the displacement of the masses is harmonic.

Formulation of the problem
Consider a two-dimensional wave propagation through the square periodic structure shown in figure 1a. The axis x 3 is orthogonal to the plane of the figure with in-plane axes x 1 and x 2 . The displacement inside the walls of the structure (0, 0, u(x 1 , x 2 )) satisfies the wave equation where m is the shear modulus and r is the density. The solution must satisfy the quasi-periodicity condition uðx 1 C n 1 L; x 2 C n 2 LÞ Z expðiLðk 1 n 1 C k 2 n 2 ÞÞuðx 1 ; x 2 Þ; ð2:2Þ where L is the cell dimension; k Z ðk 1 ; k 2 Þ 2 D L is the Bloch vector (quasimomentum free parameter); n 1,2 2Z; and the domain D L is the square Let us assume that the periodic square structure is composed of two types of regions: the shaded regions in figure 1a having non-zero inertia, connected by thin-walled elements of negligible inertia.
We use a small parameter e, which is the ratio of the width to the length of a thin-walled bridge connecting the shaded regions in figure 1a. Hence, we obtain the following problem for a thin rectangle PZ fðx 1 ; x 2 Þ : 0! x 1 ! L; jx 2 j! eLg representing a thin-walled bridge in the undamaged structure: where q 1 and q 2 are displacements on the interface with the junction region. In the case of a damaged junction, as in figure 1b, the first boundary condition in (2.6) has to be replaced by vu vx 1 where r e is the length of the intact part of the junction.
(a ) Reduction to a lower dimensional model To the leading order, the solution of (2.4)-(2.6) is given by where Y 1 Zx 1 /(eL) and Y 2 Zx 2 /(eL) are the scaled variables and the principal term satisfies The boundary-layer term W decays exponentially away from the junctions. The term U satisfies the following problem: together with (2.8) gives a linear function for the solution u 0 . Thus, in this case, the principal term of the asymptotic approximation of the displacement within the thin-walled bridge is a linear function of the longitudinal coordinate. Hence, the thin-walled bridges in the periodic structure behave like springs connecting masses m located at the junction points. Thus, taking into account the approximations, we perform a discretization of the initial square periodic structure, and arrive at the equations of motion for a mass m in the twodimensional undefected lattice. The displacement u i, j of a mass located at the junction point with indices (i, j ) satisfies the equation where f is the magnitude of the tension force and L is the length of the thinwalled bridge. In addition, we must add the quasi-periodicity conditions for the displacement u i, j u iCn 1 ; j Cn 2 Z expðiLðk 1 n 1 C k 2 n 2 ÞÞu i;j ; ð2:11Þ where kZ ðk 1 ; k 2 Þ 2 D L is the Bloch vector. If the initial square periodic structure has defects at the junction points similar to cuts (figure 1b), the system of the equations of motion for the two-dimensional lattice, given by (2.10), must be modified. In this case, we have to take into consideration an effective boundary condition between the thin-walled bridge and the junction domain. The asymptotic solution to the problem was obtained by Zalipaev et al. (2007), and a summary is given in §2b.

(b ) Effective boundary conditions for a defected junction
Let LZ1 and consider a thin rectangular domain U e Z fx Z ðx 1 ; x 2 Þ 2 R 2 : 0! x 1 ! 1; jx 2 j! eg in dimensionless coordinates representing a thin-walled bridge (the width of the domain is much less than its length), where eO0 is a small non-dimensional parameter (figure 2). At the left-hand end of the domain, the displacement is specified on a smaller part of the boundary, whose length is 2r e . We denote the ratio of the two small parameters by C e Zr e /e. It is assumed that r e /e. The right-hand end of the domain U e is fixed. The remaining parts of the boundary (the upper and lower sides of U e ) are traction free. We assume that the domain is occupied by an elastic isotropic homogeneous material with the shear modulus m. The displacement field has the form ð0; 0; u e ðx 1 ; x 2 ÞÞ, where the function u e satisfies the following mixed boundary-value problem: where q and Q are assumed to be constant displacements. The leading term U(x) of the asymptotic expansion for the solution of the problem (2.12)-(2.16) satisfies, at x 1 Z0, one of the following boundary conditions (Zalipaev et al. 2007): where C is a positive constant independent of e.
aO 1: We are interested in case (ii) when aZ1. Now the effective junction conditions will be used in modelling defect modes within periodic structures.

Localization in a one-dimensional discrete system with imperfect junctions (a ) A periodic problem
Consider a one-dimensional lattice with a macro-cell consisting of two particles with masses m 1 and m 2 and with period 2L. Let the particles interact with their neighbours via thin strip connections. The displacements within thin ligaments could be modelled by a discrete one-dimensional harmonic oscillator equation (anti-plane shear case). This one-dimensional problem may be considered as a special case of wave propagation just in one direction for the two-dimensional problem described in (2.10) and (2.11). For a perfect connection between particles and strips, we have the following system of equations for the nth macrocell for the case of a time-harmonic dependence (Movchan & Slepyan 2007): n ; n are the displacements of the particles in the nth cell with masses m 1 and m 2 , respectively; u is the frequency of vibration of the particles; and f is the magnitude of the tension force in the connectors.
A defect in the continuous structure is a crack within the junction region, and it is modelled in the lattice approximation by a spring. In the lattice model, imperfect junctions are considered as additional spring connectors (figure 2) with the stiffness parameter c as in the effective boundary condition (2.18). Namely, it includes the Robin-type boundary condition modelling a spring connection well known in mechanics. Thus, the displacements u ð1Þ n and u ð2Þ n of the particles with masses m 1 and m 2 , respectively, satisfy the following system of equations: n represent the displacements of the points A 1 and A 2 , respectively. Moreover, for a periodic structure, we impose the quasiperiodicity conditions nCm Z expð2ikLmÞu ð2Þ n ; where UZ u 2 L=f and GZ cL=ðcLC f Þ.
The system (3.6) is similar to the one analysed in Movchan & Slepyan (2007), where GZ1. The transfer from perfect connections to imperfect ones as in the system (3.6) at the junction points A 1 and A 2 is obtained by the replacement of f/L in (3.1) by the parameter G.
We use the result of Movchan & Slepyan (2007) to write the dispersion equation as where rZm 1 /m 2 . This demonstrates that the dispersion relationship depends on the ratio of the masses, referenced to a value of m 1 and the degree of imperfection in the junction condition through the parameter c and hence G. Note that the dispersion relationship is separable with respect to r and G. The effect of the imperfect junction condition on the dispersion diagram is shown in figure 4 for m 1 Z1, LZ1 and rZ0.5. Frequencies within the band gap satisfy From figure 4, it is clearly seen that the width of the band gap is increasing and occurring at a higher frequency, with increasing G.

(b ) Localized mode for a single defect
Consider a non-periodic one-dimensional structure with masses m 1 and m 2 connected by springs. Let us assume that a defect is placed in the central nZ0 K1 , which satisfy the following equations of motion: where a 1 and a 2 are the displacements at the defect junctions A 1 and A 2 , respectively (figure 5), and a Z c L f : The system (3.11) may be written as follows: Using the Green function for the unperturbed system of equations (3.9) and (3.10) (see the case of GZ1 in Movchan & Slepyan (2007) and the case of arbitrary G in appendix A of the present paper) for the stopband one can represent (3.12) as follows: Without loss of generality, we set m 1 Cm 2 Z2, hence, Thus, all the Green matrix elements depend on U, c and r. The existence of the localized defect mode in the current problem directly follows from the existence of the solutions of (3.14), g K1 Z l i ðU; rÞ; i Z 1; 2; 3; ð3:15Þ where l i (U, r) are the eigenvalues of the matrix S(U, r). In figure 6, we illustrate the localized defect modes by showing a family of curves representing frequencies of the localized defect modes versus r for fixed values of cZ1.5, 3, 7 and LZ1. In figure 7, we show for a localized defect mode, r parameter dependence on c taken for the frequency U 0 (r), which corresponds to the case of the extremal localization (see formula (A 3) in appendix A) The result in figure 7 was obtained as the intersection of two surfaces u Z det SðU 0 ðrÞ; rÞK 1 C c L f I and uZ0. This intersection is shown in figure 8. The extremal localization for a defect mode was noted in the paper by Movchan & Slepyan (2007), where a similar result was obtained for the corresponding frequency inside the band gap.

Bloch-Floquet waves and standing modes in two-dimensional lattices with imperfect junctions
Consider a two-dimensional square lattice with a macro-cell consisting of four particles with period 2L. Let the first particle with mass m 1 ( jZ0) of the macrocell interact with its neighbours and particles with mass m 2 ( jZ1, 2, 3) via spring connections (figure 9). Similar to (3.6), within such a macro-cell, the particle displacements u ðj Þ n 1 ;n 2 , jZ0, 1, 2, 3, satisfy the following system of equations of motion: The particle displacements u ðj Þ n 1 ;n 2 , jZ0, 1, 2, 3, satisfy the quasi-periodicity conditions u ðj Þ nCN ;mCM Z expð2iLðk 1 N C k 2 M ÞÞu ðj Þ n;m ; where k Z ðk 1 ; k 2 Þ 2 D 2L is the Bloch vector (quasi-momentum); N ; M 2 Z; and the domain D 2L is the square

4G
KGð1 C e 2ik 1 L Þ KGð1 C e 2ik 2 L Þ 0 As before, the dispersion equation of the lattice det½UM KsðkÞ Z 0; gives four non-negative roots U 1;2;3;4 ðkÞ. They are shown as the dispersion diagram in figures 10 and 11 for the parameters m 1 Z1, m 2 Z2, LZ1, GZ0.5 and 0.1, respectively. Making the parameter G smaller, we make the band gap wider. Moreover, the new low-frequency band gap appears. The decrease of G results in the appearance of flat parts of the dispersion curves representing standing waves. The dispersion surfaces as a function of the Bloch vector around the contour ABCA and G are shown in figure 12, for m 1 Z1, m 2 Z2 and LZ1. It shows the presence of standing waves. The eigenvectors indicate that the lowest frequency of standing waves ( ffiffiffiffi U p z0:6) corresponds to the relatively large amplitude oscillations of the mass m 1 when compared with the other masses. For small values of G, the spring constant is also small compared with the force per unit length in the connecting links.
In figures 13 and 14, we show the displacement within the lattice fragment for m 1 Z1, m 2 Z2, LZ1 and GZ0.1 for the first and the second eigenvectors, respectively, with k 1 Z0, k 2 Z0 (see points P and Q in figure 11). In the case of the point P, for small G, the amplitude of vibration of particle 0 is much larger compared with amplitudes for other particles. This is a localized vibration mode with respect to the macro-cell. A simple asymptotic approximation for the first non-zero eigenfrequency can be obtained by assuming that all the particles, except for the particle 0, within the macro-cell are stationary. This situation may be modelled as follows: KUm 1 u Z Gða 1 C a 2 C a 3 C a 4 K4uÞ; where u is the displacement of the particle 0. Eliminating a i , iZ1, 2, 3, 4, and using the asymptotic approximation U asymp 1 for G/1, we obtain KU asymp 1 m 1 u Z 4uG: In figure 15, we show that the difference U 1 ðGÞKU asymp 1 ðGÞ tends to 0 as G/0. In the second case, we have only particles 1 and 2 vibrating, whereas the other particles 0 and 3 of the macro-cell are stationary. This situation may be modelled as follows: where u is the displacement of the particle 1 or 2 as the behaviour of both the particles is identical. Hence, KU 2 m 2 u ZK2uK2Gu: It is worth noting that, for both the cases of the vibrating standing waves, the centre of mass of four particles of the macro-cell is zero.

Conclusion
In this paper, we analyse vibrations of two types of mechanical lattices with defects: one-dimensional and two-dimensional discrete structures possessing band gaps and the localized oscillation modes. The existence of a localized defect mode for a one-dimensional lattice, having a single imperfect junction in the central macro-cell, was considered. We have derived the representation for the lattice Green function for frequencies within the stopband. The presence of a crack in the junction region leads to a localized defect mode. We have introduced the notion of extremal localization for a lattice with an imperfect junction, and we have studied the corresponding vibration modes.
For a two-dimensional lattice, we presented the analysis of dispersion surfaces and showed the existence of standing waves within a periodic system of defects. It is shown that the increase in the crack length leads to a formation of a lowfrequency stopband on the dispersion diagram. We also note that a standing wave of a low frequency appears within periodic structures with cracked junctions.
The financial support via the grant no. EP/D035082/1 from the UK Engineering and Physical Scientific Research Council is gratefully acknowledged. We would like to thank the referees for their valuable comments and suggestions.