## Abstract

In this paper, we investigate the formation of band-gaps and localization phenomena in an elastic strip nearly disintegrated by an array of transverse cracks. We analyse the eigenfrequencies of finite, strongly damaged, elongated solids with reference to the propagation bands of an infinite strip with a periodic damage. Subsequently, we determine analytically the band-gaps of the infinite strip by using a lower-dimensional model, represented by a periodically damaged beam in which the small ligaments between cracks are modelled as ‘elastic junctions’. The effective rotational and translational stiffnesses of the elastic junctions are obtained from an *ad hoc* asymptotic analysis. We show that, for a finite frequency range, the dispersion curves for the reduced beam model agree with the dispersion data determined numerically for the two-dimensional elastic strip. Exponential localization, boundary layers and standing waves in strongly damaged systems are discussed in detail.

## 1. Introduction

Localization around defects in solids is of high importance in mathematical models of elastic Bloch waves as well as in practical applications of engineering designs. Localization phenomena, in particular trapped modes appearing near defects, can occur in elastic structures with defects, cracks or discontinuities such as beams [1], plates [1–3] and microstructured media [4–7]. In addition, the dynamic response of elongated solids with a distribution of crack-like defects is used in the practical evaluation of properties of composite body armour as well as protection sheets and windscreens of armoured vehicles.

The earlier papers [8,9] present an efficient algorithm for the analysis of localized modes around crack-like defects distributed periodically in a bimaterial delaminating system. A special feature of the problem is the singular perturbation analysis in the region around a crack, and the reduction to a lower-dimensional approximation. Higher-order terms in the asymptotics are studied in [10], which allow for a higher accuracy in the description of the dispersion properties of Floquet–Bloch waves existing within the periodic system with longitudinal cracks. Boundary layers near the vertices of the cracks are analysed in problems involving unbounded domains, and full analytical solutions are derived by solving an equation of the Wiener–Hopf type. An alternative powerful approach used in [11] uses Stroh formalism and analysis of solutions to corresponding Riemann–Hilbert problems. Waves in non-periodic composites have received substantial attention in both mechanics and geophysics applications. In particular, an efficient self-consistent approach has been well-developed in Sabina & Willis [12–14]. The effect of perturbations in the periodic systems and subsequent localization is analysed in Guennaeu *et al*. [15], whereas rigorous spectral analysis of discrete systems is presented in Davies [16].

In this paper, we are interested in the dynamic response of nearly disintegrating systems. Examples include elongated solids containing transverse cracks that have grown to the extent that the solid is close to disintegration into several disjoint subsets. Such models also arise in the engineering designs of long bridges, pipelines and conveyors as well as in slender systems such as skyscrapers. More specifically, several bridges and viaducts are designed as series of simply supported spans, sustained by piers. In correspondence of each pier, the spans are connected only by the upper deck. Therefore, the junction at the pier behaves like a cracked section where the ligament is represented by the depth of the upper deck. An example of such structure is shown in figure 1.

We note that a static singularly perturbed problem for a disintegrating elongated elastic solid containing a transverse crack is studied in [17,18] for longitudinal and transverse loads, respectively. These works present the derivation of a lower-dimensional model and an effective junction condition, which serves as the condition of decay for the boundary layer occurring in the vicinity of the cracked region. A comparison with numerical results relevant to the calculation of the first eigenfrequency of a simply supported plate with a crack in the middle section shows that the asymptotic model in Gei *et al*. [18] behaves well for deep cracks, whereas the model proposed in Ostachowicz & Krawczuk [19] works better for small cracks.

The dynamic behaviour at high frequencies of a diffusively damaged structure may exhibit surprising features. Figure 2*a* includes an instance of a damaged bridge, which can be modelled as an elongated solid weakened by transverse cracks, as sketched in figure 2*b*. Figure 2*c* shows typical eigenmodes of the elongated solid. The eigenmodes at the top and bottom of the figure are localized near one end of the structure and are characterized by a different decaying rate, whereas the eigenmode at the middle of the figure presents a typical standing wave pattern. Standing waves and localization in strongly damaged systems possess new features and require a substantial additional effort with respect to static computations. In this paper, we identify and study standing waves leading to localization phenomena. We also show that damaged structures exhibit band-gaps, and we derive analytical estimates for the frequency ranges of such band-gaps.

The structure of the paper is as follows. In §2, we present the two-dimensional model of an elastic strip damaged at regular distances. We describe both a finite and an infinite periodic structure, and we derive numerically their dynamic responses. Section 3 is dedicated to the lower-dimensional model, which consists of a beam with periodic elastic connections that simulate the cracked sections. We revise the asymptotic method leading to the reduced model and then analyse the dispersion properties of the system obtained by means of the transfer matrix formalism. Simple analytical expressions for the frequency intervals of the propagation bands as a function of the damage parameters are provided. Finally, we determine the dynamic range of applicability of the reduced model by comparison with the two-dimensional model of §2. Final remarks in §4 conclude the paper.

## 2. Two-dimensional strip model

We consider an elastic strip with a diffuse damage, represented by transverse cracks distributed at equal distances in the direction of the strip length. For the strip with finite dimensions, we determine numerically eigenfrequencies and eigenmodes for different values of the strip length. Then, we show that the frequency response of the finite strip is connected with the dispersion properties of a periodic strip of infinite length.

### (a) Eigenfrequencies of the finite strip

A strip of finite dimensions is sketched in figure 3*a*. The length and the height of the strip is indicated by *L* and *h*, respectively. The distance between cracks is denoted by *l*. In the example of figure 3*a*, the strip is made of five identical units (or cells) of length *l*. One of these cells is drawn on an enlarged scale in figure 3*b*, where *ρ*_{ϵ} is the depth of the cracked section.

We assume that the strip is elastic and that the boundaries are traction-free. Accordingly, the time-harmonic governing equations for the strip are the following:
**x** is the position vector, **u** is the displacement vector, λ and *μ* are Lamé coefficients, *ρ* is the mass density, *ω* is the radian frequency and ** σ** is the traction vector associated with the unit outward normal

**n**. Furthermore,

*Ω*and ∂

*Ω*are the interior and the traction-free boundary of the strip, respectively, as indicated in figure 3

*a*.

By performing finite-element computations in COMSOL Multiphysics (v. 4.3) to solve numerically equations (2.1), we find the eigenfrequencies and eigenmodes of finite strips with different lengths. The values of the eigenfrequencies and the shapes of some eigenmodes are shown in figure 4, where the number of cells varies from 2 to 10. We point out that we have disregarded the eigenfrequencies corresponding to longitudinal motion, which are not relevant in this work. In addition, the crosses on the horizontal axis of figure 4 represent rigid-body motions, which are not of particular interest.

The eigenmodes corresponding to the eigenfrequencies indicated by grey dots exhibit localization, whereas those obtained from the eigenfrequencies coloured in black do not have decaying amplitudes. The latter eigenfrequencies increase in number as the number of cells is increased; however, they remain confined within specific ranges of frequency. In the following section, we identify such frequency ranges from the study of the dispersion properties of a periodic strip.

### (b) Dispersion properties of the periodic strip

We examine a strip of infinite length, consisting of a periodic array of cells, one of which is shown in figure 3*b*. The top and bottom boundaries of the cell and the sides of the crack are traction-free, whereas the vertical sides of the cell are subjected to Floquet–Bloch conditions. The equations of motion of the infinite periodic strip result to be
*Ω* is the interior domain of the cell, ∂*Ω* is its traction-free boundary, whereas ∂*Ω*_{p} indicates the ends of the cell where Floquet–Bloch conditions are imposed (figure 3*b*). Furthermore, *k* stands for the wavenumber.

We solve equations (2.2) numerically by developing a finite-element model in COMSOL Multiphysics. In particular, we determine the dispersion diagrams plotted in figure 5*a*,*b*, which refer to a larger and a smaller value of the cracked section depth *ρ*_{ϵ}, respectively. We stress that the numerical data shown in figure 5*a*,*b* are relative to transverse waves and are independent of the strip thickness. In fact, the finite-element code also provides the dispersion curves relevant to longitudinal waves; however, the latter are not reported in the figures for the sake of clarity.

From figure 5*a*,*b*, it is apparent that the damage generates bands of non-propagation (band-gaps), which are not present in the intact elastic system. The size of the band-gaps increases at a higher level of damage, namely as the depth of the cracked cross section *ρ*_{ϵ} is decreased.

It is interesting to observe that, though the strips considered in figure 5*a*,*b* are nearly disintegrating because *ρ*_{ϵ} is small, waves of high frequencies can still propagate in such structures. This is quite a counterintuitive result. Furthermore, by comparing figure 5*a*,*b,* we note that the main effect of reducing the depth of the cracked section is to drop the first dispersion curve, which becomes flatter as *ρ*_{ϵ} is decreased. On the other hand, the higher dispersion curves are not significantly modified by a change in *ρ*_{ϵ}.

The limits of the band-gaps of figure 5*a* are represented by horizontal dashed lines in figure 4, in which the same value of *ρ*_{ϵ} has been taken into account. We observe that all the eigenfrequencies of the finite strips with different lengths indicated by black dots in figure 4, which are relevant to non-localized modes, lie within the propagation bands of the periodic structure (a similar conclusion was drawn in [20] for mono-coupled systems and in [21] for the analysis of a real bridge structure). On the contrary, the eigenmodes corresponding to the eigenfrequencies falling outside the propagation bands of the periodic structure, indicated by grey dots in figure 4, are localized near a boundary. Thus, we have shown that it is possible to identify the attainable ranges of the eigenfrequencies of the finite structure by studying the dispersion properties of the infinite periodic structure.

The periodic structure exhibits standing waves at the limits of the band-gaps, where the dispersion curves are flat (∂*ω*/∂*k*=0). The lowest eight modes relative to these standing waves, as in figure 5*b*, are plotted in figure 6. Modes (*a*), (*c*), (*e*) and (*g*) show a slope discontinuity at the cracked section, whereas modes (*b*), (*d*), (*f*) and (*h*) present a relative displacement in correspondence of the damaged section.

In concluding this section, we remark that we have solved numerically the two-dimensional problems defined by equations (2.1) and (2.2). In the following section, we investigate a lower-dimensional model that allows to derive analytically an efficient approximation of the dynamic properties of the two-dimensional model.

## 3. Asymptotic reduced beam model

Here, we study a reduced model, represented by a beam with cracked cross sections that are modelled as elastic junctions. The effective bending (rotational) and shear (translational) stiffnesses of the elastic junctions are denoted as *K*_{b} and *K*_{s}, respectively. The beam is of infinite length and the damaged cross sections are located at regular intervals. A periodic segment of the beam is sketched in figure 7.

### (a) Effective junction conditions

The notion of effective junction conditions was addressed in the book [22]. In the asymptotic models involving boundary layers near singularly perturbed boundaries that encompass connection of several bodies, these are equivalent to conditions of decay of boundary layers away from the relevant junction region. Asymptotic analysis relevant to our model engages effective bending and shear stiffnesses that were determined for the two-dimensional, nearly disintegrating strip shown in figure 8. This technique has been applied to asymptotic models of disintegrating solids in [17] for longitudinal loads and in [18] for flexural loads. In these papers, the attention was devoted to static problems, whereas here we describe the corresponding generalization to the time-harmonic regime.

Two classes of boundary conditions for a bending problem are analysed: symmetric rotations *q* at the boundaries (figure 8*a*) and antisymmetric displacements *p* at the two ends (figure 8*b*). For symmetry, only the half domain *ξ*_{1,2}=*x*_{1,2}/*ϵ*, the displacement vector *u*_{ϵ} admits the asymptotic approximation [23]
**u**^{(i)} (*i*=0,1,2,3) and **U** are functions of (*x*_{1},*ξ*_{2},*t*), whereas the boundary layer terms **L**_{ϵ} are functions of the scaled variables (*ξ*_{1},*ξ*_{2}) and decay away from the singularly perturbed boundary that also includes the junction region. The leading order term **u**^{(0)} has the form **U**, as discussed in [23]. Equation (3.2) has the structure of a beam equation of motion and requires four boundary conditions, which are determined from the analysis of the conditions of decay for the corresponding boundary layers. The conditions at the right end of the domain are
*x*_{1}=0, whereas the first derivative *φ* at the junction. By using the relationship between bending moment (per unit thickness) *M* and curvature *s*) *E* is Young's modulus, whereas *ν* is Poisson's ratio.

A similar procedure can be applied to the bending problem with antisymmetric displacements, from which the following expression of the translational stiffness (per unit thickness) is derived (see Gei *et al*. [18] for details):

The analysis of Gei *et al*. [18] has illustrated that the expressions (3.8) and (3.9) provided a sufficiently high accuracy for 0<*ρ*_{ϵ}/*h*≤0.35. In the following sections, we discuss equations (3.8) and (3.9) being applied to dynamic problems, in particular to describe the junction conditions of a periodic beam subjected to flexural waves.

### (b) Dispersion in the asymptotic reduced model

We use the transfer matrix method to obtain the propagation and non-propagation zones for the periodic beam with elastic connections. The transfer matrix is a mathematical tool that can be efficiently implemented to analyse periodic media, both in electromagnetism [24] and in elasticity [25]. It allows to define the vector of generalized displacements and generalized forces at the end of a periodic cell in terms of the same vector at the beginning of the cell. Examples for mono-coupled elastic periodic structures can be found in [20,25–27]. In the present case, the structure is a bicoupled system, because there are 2 degrees of freedom, i.e. vertical displacement and rotation. The corresponding generalized forces are bending moment and shear. Therefore, the transfer matrix has dimensions 4×4 independently of the complexity of the periodic cell. Different instances of bicoupled periodic structures are investigated in [28]. In the case examined in this paper, the transfer matrix can be written in compact form as
**M**_{int} is the transfer matrix of an intact beam of period *l*:

where *β*=(*ρAω*^{2}/*EJ*)^{1/4}, *ω* is the radian frequency, whereas *A* and *J* are respectively the area and the second moment of inertia of the beam cross section. Furthermore, the matrix **I** appearing in equation (3.10) represents the 4×4 identity matrix, whereas **K** is the ‘stiffness matrix’ given by
**K** reflects that there are no discontinuities in the force and the bending moment (the last two rows of the matrix are zero), whereas the flexural displacements and their derivatives are discontinuous across the junction regions identified in figure 7.

By imposing Floquet–Bloch conditions, we find the dispersion relation, which is given by
*k* is the wavenumber.

If the stiffnesses *K*_{b} and *K*_{s} are chosen arbitrarily, the dynamic problem can be described by three non-dimensional parameters:
*ϕ* is a non-dimensional parameter related to frequency, which is henceforth referred to as ‘frequency parameter’. *κ*_{b} and *κ*_{s} represent the normalizations of the junction stiffnesses with respect to the flexural rigidity of the beam; they can be indicated as ‘effective damage parameters’.

The unimodular transfer matrix **M** is characterized by two independent invariants, which are expressed in terms of the frequency and effective damage parameters in equations (3.14) as follows
*I*_{1} and *I*_{2} by the following three curves
*f*_{1} and *f*_{2} represent straight lines, whereas *f*_{3} is a parabola. By introducing equations (3.15) into equations (3.16), we determine the boundaries of the propagation zones in the space defined by the three non-dimensional parameters (3.14), which is henceforth referred to as ‘physical space’.

By fixing the constitutive properties of the material (*E*, *ν*, *ρ*) and evaluating the stiffnesses *K*_{b} and *K*_{s} by means of equations (3.8) and (3.9), we reduce the non-dimensional parameters that fully characterize the problem to two: the frequency parameter *ϕ*, defined by equations (3.14a), and the ratio *ρ*_{ϵ}/*h*. As a consequence, the physical space reduces to a plane. The propagation zones for a particular choice of the material properties are shown in figure 9. In this diagram, the grey and white regions represent the pass–stop and the stop–stop zones, respectively. We observe that there are no pass–pass nor complex zones in the ranges of values considered.

We determine the dispersion curves from equation (3.13) for the cases *ρ*_{ϵ}=*h*/5 and *ρ*_{ϵ}=*h*/100, which are plotted in solid thick black lines in figure 10*a*,*b*. In figure 10*a,b*, the solid grey lines represent the dispersion curves of an intact beam, which can also be derived from equation (3.13) by taking

We note that waves of any frequency can travel in an intact beam (see also [29], §3). In such a case, the dispersion curves are straight lines in the (*ϕ*,*kl*) space. If the beam contains cracks, instead, non-propagation bands appear. The upper limit of the lowest stop–stop band is independent of the value of *ρ*_{ϵ} and coincides with the lowest value that the dispersion curves of an intact beam attain at *kl*=*π*; on the other hand, its lower limit decreases as the crack grows, as expected on physical ground. As already detailed in figure 5, it is evident that the ‘amount of damage’ strongly influences the acoustic pass-band, while it has a less relevant effect on the optical pass-bands.

### (c) Efficiency of the asymptotic approximation

The dynamic properties of the two-dimensional strip model, derived numerically in §2, can be predicted with good accuracy by the reduced beam model in a finite range of frequencies.

In figure 11*a*,*b*, we compare the analytical dispersion curves, obtained from the transfer matrix method applied to the periodic beam, with the numerical values provided by COMSOL Multiphysics for the periodic strip, for the cases *ρ*_{ϵ}/*h*=1/5 and *ρ*_{ϵ}/*h*=1/100.

From figure 11*a*,*b*, it can be seen that there is a very good agreement between the first three analytical dispersion curves and the numerical findings. At higher frequencies, the discrepancy between the two models increases. We stress the fact that two approximations are embedded into the damaged beam model: the reduced one-dimensional model with respect to the continuous two-dimensional one, and the asymptotic approximation for the effective junction conditions. Analogous computations for an intact beam show similar discrepancies between the continuous and the structural beam model. These computations, not reported here for brevity, also indicate that the validity frequency range of the one-dimensional model is wider as the slenderness of the unit cell is increased. These results show that the effective junction conditions are efficient within the frequency range where the beam model is valid.

### (d) Standing waves. Analytical estimates of the band-gap boundaries

Here, we give simple analytical expressions for the limits of the band-gaps. These can be obtained by computing the eigenfrequencies of simple beam models, whose boundary conditions can be deduced from the standing waves reported in figure 6.

The standing waves obtained for *kl*=*π* at the lower limits of the stop–stop zones (figure 6*a*,*e*) are identical to the eigenmodes of a simple beam simply supported at one end and with a rotational spring (of stiffness 2*K*_{b}) at the other end, that is sketched in figure 12*a*. In figure 12*a*, the dashed lines represent the first two eigenmodes. The normalized eigenfrequencies of this simple beam can be calculated from the following implicit equation
*ϕ* is proportional to the square root of the frequency, and it is a non-dimensional quantity; therefore, we denote *ϕ* as ‘normalized eigenfrequency’.

On the other hand, the standing waves computed for *kl*=*π* at the upper limits of the stop–stop zones (figure 6*b*,*f*) have the same shapes of the eigenmodes of a simple beam with a guided support at one end and a translational spring (of stiffness 2*K*_{s}) at the other end, shown in figure 12*b*. The normalized eigenfrequencies of this simple beam can be determined from the equation
*b*.

The standing waves at *kl*=0, determined at the beginnings of the stop–stop zones (figure 6*c*,*g*) and at the ends of the stop–stop zones (figure 6*d*,*h*), resemble the eigenmodes of the simple beams drawn in figure 12*c* and *d*, respectively. The corresponding normalized eigenfrequencies can be found from the equations

From the above considerations, it can be concluded that the limits of the stop–stop zones can be evaluated analytically from the eigenfrequencies of simple structures. This approach is easier than solving the dispersion relation of a periodic structure. The normalized eigenfrequencies of the beams sketched in figure 12*a*–*d* are plotted in figures 9 and 10 in dashed, dotted, solid and dot-dashed black lines. Especially, figure 9 shows that the limits of the stop–stop zones coincide with the solutions of equations (3.17)–(3.20).

The transcendental equations (3.17)–(3.20) can be expanded in Taylor series for *ϕ*=0 in order to find an approximation to their exact solution. Accordingly, the first solution of equations (3.17)–(3.20), which depends on either *κ*_{b} or *κ*_{s}, is approximated with high accuracy by, respectively:
*a,b*, the solid grey lines represent the exact solutions obtained from equations (3.17)–(3.20), showing the excellent agreement with the approximated solutions (3.21). We point out that we have considered a wider range of values for *κ*_{s}, because *κ*_{s} is generally larger than *κ*_{b} for a given *ρ*_{ϵ}. All functions increase monotonically with the stiffness on which they depend.

## 4. Conclusion

In this paper, we have examined the propagation of transverse waves in a two-dimensional elastic strip with periodically distributed transverse cracks. Numerical simulations concerning infinite periodic strips have shown that band-gaps arise as a consequence of the cracks present in the structure, and the limits of the band-gaps depend on the depth of the cracked sections. If, instead, the strips are of finite length, the eigenfrequencies of non-localized modes fall within well identified frequency intervals, coinciding with the pass-band of the periodic structure. The number of eigenfrequencies of the finite structure in each pass-band is shown to increase linearly with the number of cells composing the system.

In order to predict the positions and the sizes of the band-gaps for a strip with a periodic damage, we have developed a lower-dimensional periodic beam model, in which the cracked sections are represented by elastic junctions with a bending and a shear spring. The effective rotational and translational stiffnesses are derived by means of an asymptotic analysis. A comparison with the numerical findings obtained from the two-dimensional model has shown that the lowest band-gaps of the strip can be determined with a high level of accuracy from the dispersion curves of the periodic beam.

The limits of the band-gaps coincide with the eigenfrequencies of simple beams, whose boundary conditions have been deduced from the shapes of the standing waves of the elastic strip. This result is very important in practice, because the determination of the eigenfrequencies of simple beams is more straightforward than the solution of the dispersion relation of a periodic structure.

The results of this work can be used to design systems with filtering properties, and to detect and possibly estimate quantitatively the presence of cracks inside structural and mechanical elements by means of non-destructive techniques.

## Funding statement

G.C. and M.B. wish to acknowledge the financial support of the Regione Autonoma della Sardegna (LR7 2010, grant ‘M4’ CRP-27585). M.B. and A.B.M. acknowledge the financial support of the European Community's Seven Framework Programme under contract numbers PIEF-GA-2011-302357-DYNAMETA and PIAPP-GA-284544-PARM-2.

- Received February 18, 2014.
- Accepted March 31, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.