G.V. Dreiden, K.R. Khusnutdinova*, A.M. Samsonov, I.V. Semenova. A.F. Ioffe Physical Technical Institute, St. Petersburg, , Russia ABSTRACT

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1 This preprint will appear in final form in Strain LONGITUDINAL STRAIN SOLITARY WAVE IN A TWO-LAYERED POLYMERIC BAR G.V. Dreiden, K.R. Khusnutdinova*, A.M. Samsonov, I.V. Semenova A.F. Ioffe Physical Technical Institute, St. Petersburg, , Russia *Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK ABSTRACT Both theoretical investigations and successful experimental research were performed recently, confirming the existence and demonstrating the main properties of bulk strain solitary waves in nonlinearly elastic solid wave guides. Our current research is devoted to non-linear wave processes in layered elastic wave guides with inhomogeneities modelling damage/delamination. Here, we present first experimental and theoretical results, demonstrating the change of the amplitude and width of a strain solitary wave propagating in an inhomogeneous two-layered wave guide made of PMMA. Parameters of such waves in layered structures may be of crucial importance for the assessment of their operational integrity and robustness. Keywords: Delamination, Holographic interferometry, Layered wave guides, Strain solitary waves.

2 Introduction Considerable attention has been paid to the development of various approaches to dynamic nondestructive testing of (small to large) laminated structural components in physics and engineering. Commonly used methods of nondestructive testing of adhesively bonded structures are based on ultrasonic technology and include normal and oblique ultrasonic scans, resonant ultrasonic spectroscopy and the Lamb wave NDT [1]. These methods allow one to detect voids, delamination, porosity, cracks, and poor adhesion in bonding layers. However, certain limitations of these methods (e.g., rapid decay of ultrasonic waves in solids), as well as recent theoretical and experimental developments in the area of non-linear waves stimulate both analytical and experimental research into the novel approaches. Combinations of different materials are widely used to improve mechanical properties of constructions. The major problem in the behaviour of laminated structures consists in a possibility of a sudden and irreversible delamination under dynamic loading. Integrity of such a structure is mostly defined by the quality of its interfaces. Delamination may be initiated by a considerable stress concentration near an interface, caused by an abrupt change in material properties and/or deformation under loading. It may result in a total failure of a structure with no visible signs up to its catastrophic manifestation, which necessitates development of advanced methods of non-destructive detection of delaminated areas. Various types of layered (or laminated) elastic structures are used in modern engineering. Layers can be oriented along or across a wave propagation direction, and bonded via various types of adhesive material (e.g., a glossy or rubber-like glue, etc.). There are various sources of nonlinearity in such structures, which may cause very unusual behaviour. Non-linear dynamics of condensed matter is often studied using lattice models (e.g., dynamics of molecular crystals, polymer chains, hydrogen-

3 bonded chains, etc., see [2] and references therein). In [3] a laminated composite was modelled as a modified Toda lattice with an external linear elastic term. It was shown that stable envelope solitons play central role in the dynamics of such a chain instead of Toda solitons, which may be of interest for applications as a way of avoiding the concentration of energy in a bulk solitary wave. Interestingly, in [4] the same Toda lattice has been related to a different composite, in which layers were assumed to be oriented across the direction of wave propagation. Coupled Klein-Gordon (ckg) chains and corresponding ckg equations were proposed as a model for long longitudinal waves in elastic bi-layers, where nonlinearity comes only from the bonding material [5]. Studies were devoted to general mathematical properties of the model, and special solutions exhibiting an energy exchange between the layers [6, 7]. Damage can also be studied using lattice models, where fracture is introduced as a vanishing link between respective elements (e.g., [8, 9, 10]). Lattice modelling of non-linear waves in a bi-layer with delamination has been attempted in [11]. The ckg model consisted of two one-dimensional periodic chains with linear links between elements and non-linear interactions between the chains (modeling the glue bond), in a situation when there is a delamination zone (which has been modeled as vanishing links between the chains). The results of this study have shown the sensitivity of a non-linear wave to the length of the delamination zone, and have indicated that bulk non-linear waves are potentially applicable to the detection of delamination zones in layered structures, inviting further detailed analytical and experimental studies. Such investigations are especially interesting due to the recent experiments showing that longitudinal strain bulk solitary waves can propagate for long distances without any significant decay [12]. The latter will be discussed in more detail in the next section. The mechanical properties of a layered wave guide depend, in particular, on the properties of a glue/ solid contact. Two different contact models were considered in a problem of wave propagation in a thin layer placed over an elastic half space in [13]. It was shown that, for a sliding contact, an equation governing the non-linear longitudinal strain wave may be reduced to the well-known

4 Benjamin-Ono equation, while a full contact leads to an integro-differential equation which can be analyzed numerically. Interestingly, it was found that under certain conditions non-linear stresses in a contact zone can be sufficient for demolishing a thin layer glued to the half space, similar to the phenomenon observed in [14, 15, 16]. Our present study is devoted to a strain solitary wave propagating in a delaminated two-layered PMMA bar, where nonlinearly elastic layers are glued together by a thin layer of a cyanide glue. Understanding and modelling properties of the bonds constitute an integral part of the problem under consideration. Here, we approach this part of the problem by organizing a series of experiments for certain specific configurations of a wave guide. The experimental technique is based on laser generation and optical (by means of holographic interferometry) recording of the solitary wave, as described below. Theoretical approach and results of the previous research Nowadays a term soliton becomes associated not only with the well known localized traveling wave in fluid mechanics and optics, but also with their solid mechanics analogue the strain solitary wave in solids. Bulk strain soliton is a long non-linear localized traveling wave, which, under certain conditions, can be formed in a solid wave guide from an initial pulse. Being formed, the soliton propagates along the homogeneous wave guide with almost no change of amplitude, shape and velocity. The basic principle of strain soliton formation in a solid wave guide can be briefly described as follows. Bulk soliton generation in a nonlinearly elastic wave guide can be initiated by a short and strong wave of deformation (a weak shock wave, for instance), propagating along it. However, the gradient of a wave front may quickly increase yielding irreversible deformations, provided the non-

5 linear elasticity is not balanced with the wave dispersion in a solid wave guide (having small, but not infinitesimal cross section). Moreover, in absence of balance caused by various factors, e.g., by the nonlinearity of an opposite sign with respect to the strain sign in propagating wave, the initial pulse decays very rapidly. The balance of an elastic nonlinearity of a wave guide material and a dispersion in a wave guide results in the bulk soliton generation, resistive to main mechanisms of elastic energy decay, quite stable and with permanent shape when moving along a considerable distance. The theory of bulk strain solitons is well developed for rods, bars and plates, see, e.g. [20]. Our derivation of a leading order equation for longitudinal wave in a wave guide is based on the Murnaghan model of elasticity, an assumption that strains are small enough to be elastic, the planar cross section hypothesis and the approximate relations for the transversal displacements (V,W) via the linear longitudinal strain component as follows: V = yu ; W = zu, where is the Poisson ratio, which are similar to the Love hypothesis for a rod. Lower indices denote derivatives in space variable x, which is determined along the rod axis. The component U x =u of the gradient of the longitudinal displacement U, will be called here a strain for brevity. Description of a longitudinal strain (density) wave propagation in a nonlinearly elastic rod is based on a Doubly Dispersive Equation (DDE), see [20, 23], written in the following dimensional form for a component U x =u: x x 2a u c u u u c u tt xx = + tt xx ( 1 ) xx (1) Here t is time, a is the square side in a cross section, c and c 1 are the linear longitudinal and transversal sound velocities in a bar, respectively, is the material density, E + 2 l(1 2 ) + 4m (1 + ) (1 2 ) + 6n is the nonlinearity coefficient, depending upon the Young modulus E, and the Murnaghan moduli (l,m,n) of the 3d order. An explicit dependence of

6 the wave type on the sign of the nonlinearity coefficient and the analysis in [20,23] allows to obtain the solitary wave solution to (1), having a well known bell-shaped form: [ ] u = Acosh x± V( A) t LA ( ) 2 1 (2) along with the constraints for a solitary wave velocity V(A) and the pulse width L(A), under which it may exist: A 2 8a 3E 1+ 2 V = c + ; L = A 2(1 ) (3) It is easy to see that for >0 and sgn A sgn, one may have L 2 < 0 and no such wave exists, therefore < 0 provides a compression strain solitary wave propagation with A < 0 and vice versa. The equation (1) is proved not to be fully integrable [20], and for this reason we will call the solution in a form of the solitary wave (2) as a soliton only for brevity. The following features allowed us to certify the observed wave as the genuine strain (i.e., density) solitary wave: - while formed, the strain solitary wave keeps its shape permanent, when propagating; - no long wave of opposite sign is detected behind it; - both the amplitude A and pulse width L are proportional to the wave velocity, in contrast to that of any linear wave; - strain solitary wave is focused in a tapered wave guide. Previous theoretical and experimental research has been devoted to the generation and subsequent propagation of bulk strain solitary waves in various polymeric (made of polystyrene (PS) and plexiglas (PMMA)) wave guides, see [12, and references therein]. Strain solitons have been

7 successfully generated in rods, bars and plates. The soliton focusing in a gradually narrowing rod has also been observed. The process of soliton dissipation has been studied in long thin bars (over half a meter long and one square cm in cross section), and the possibility of solitons application to the measurement of the 3d-order elastic moduli of materials has been shown. Thus, experiments show that the wave is resistant to the usual causes of decay, is quite stable, has a permanent shape and can propagate for a considerable distance. Therefore, bulk solitary waves may transfer elastic energy for long distances with no significant losses. Thus, the previous research was mainly focused on fundamental aspects of the phenomenon of soliton formation and propagation in various homogeneous wave guides. The influence of certain inhomogeneities on wave parameters was also studied, mainly using numerical simulations (see [20] and references therein). Our present experimental and theoretical research is devoted to studying the behaviour of a soliton in a delaminated layered wave guide. Experimental approach and observed phenomenon The experimental technique used for the generation and observation of strain solitary waves in various wave guides is based on laser generation and subsequent optical (by means of holographic interferometry) recording of the waves under study. The apparatus consists of a channel to produce the strain wave in a solid from a weak shock wave in a water cell, which is induced via laser pulse evaporation of a metallic target, placed in water nearby the entrance cross section of the wave guide; a synchronizer; a holographic interferometer and a control unit for measuring the laser pulse energy (see Figure 1). Both the wave generation and recording were performed using Q-switched pulsed ruby lasers (20 ns, 0.5 J). The laser pulses were synchronized by means of the delayed-pulse generator allowing to vary the time interval between two pulses in a wide range and providing the accuracy of about 1 μs.

8 Figure 1. Experimental arrangement. Rb are pulsed ruby lasers used to generate and record waves; C is a water cell; WG is a wave guide; T is an aluminated film; PP is a photosensitive material; EM is an energy meter, BS is a beam splitter, W are wedges and M is a mirror To avoid unwanted interference of wave patterns under study with accompanying waves propagating in water, only a small section of the wave guide (50 mm long) is submerged into the water cell, allowing the shock wave, produced by laser evaporation of the foil target, to enter the wave guide. At longer distances, where the wave patterns are recorded, the wave guide is surrounded by air. The diameter of the area recorded on the interferogram equals to 50 mm. The current experimental set-up allows us to record a wave pattern inside and outside the optically transparent wave guide due to the wave induced density variations. The rotation of the wedge W (see Figure 1) placed in the object beam path after the water cell in a time moment between the hologram exposures results in the formation of the system of carrier fringes on the holographic interferogram. The number of fringes depends upon the angle to which the wedge is rotated and may be easily optimized for the desired experimental conditions. If an object did not undergo any changes between two hologram exposures the fringe pattern on the resulting holographic interferogram is formed as a system of straight equidistant fringes. Any density variations induced for example by the strain wave cause shifts of these carrier fringes on the interferogram in the corresponding area. As it was shown

9 in [20] the bulk extension D of the wave guide caused by the longitudinal wave propagation may be estimated as follows: 2 D (1 2 ) u + O( ) (4) x u x The conservation of mass yields an equation for relative variation of density: 0 = 1+ (1 2 ) ; = (1 2 ) (5) u x u x which, on the other hand, is proportional to the refractive index variation: = n2 n1 n 1 1 (6) (this formula can be easily obtained following the Lorenz-Lorentz formula [24]), here n 1 and n 2 are refractive indices of the wave guide material before and during the deformation. The shift of carrier fringes on the interferogram caused by this density variation is due to the phase variation difference ( 1) of the recording light and can be written for our experimental configuration in the 2 following way: K 2 = 2 1 = ( n2(2h + 2h) 2hn0) 2hn1) (7) 0 where h is the bar thickness along the recording light path and h is the bar extension induced by the strain wave, n 0 is the refractive index of air. Using the Love hypothesis written above and after some transformations we finally obtain for the soliton amplitude A: K0 A = (8) h n 1)(1 ) ( 1 In our experiments the conventional holographic recording on a holographic film is followed by the digital recording of interferograms on the stage of reconstruction. The digital recording of holograms would simplify and improve the experimental procedure, and is highly desirable in our future studies. However, at present, this attractive technology is inapplicable to our experimental configuration due to insufficient resolution of the available digital cameras.

10 A 600 mm long PMMA bar with 10x10 mm square cross section has been used to observe solitons in a homogeneous wave guide. Special rod/bar combinations are used in the experiments with composite wave guides, as described below. The energy of the laser pulse generating the shock wave is controlled and kept constant, to ensure that variations of laser energy would not be a source of variations of soliton parameters. To correctly interpret the data obtained for a complex, inhomogeneous wave guide, it is necessary to know the properties of the wave under study in a homogeneous wave guide. The evolution of a soliton in lengthy wave guides made of PMMA was first reported in [12]. Interferograms recorded in different areas of the wave guide allowed us to visualize the processes of soliton formation and propagation along the wave guide. The typical interferogram of the strain soliton in the PMMA bar along with the descriptions is shown in Figure 2. The soliton looks like a rather extended symmetrical bell-shaped longitudinal wave, which is not followed by any tensile wave. The soliton propagates for a long distance showing very little decay. Figure 2. Holographic interferogram of a strain soliton in the PMMA bar at the distance mm from the input. The wave moves from left to right. Fringe shift, representing the wave, is shown below the interferogram

11 Table 1 summarizes the data obtained for solitons in lengthy PMMA bars: the fringe shift and soliton width (FWHM) measured on the interferograms and the corresponding soliton amplitude calculated using eq. (8). The fifth column represents the magnitudes of soliton dissipation decrement calculated via the conventional formula: = 1 1 log A x x A (9) where A 1 and A 2 are soliton amplitudes measured at the distances x 1 and x 2 respectively. Table 1. Variations of parameters of a soliton in a PMMA bar due to dissipation. Distance (mm) Fringe shift (K) Max strain amplitude, 10-4 Width (FWHM), (mm). 10-3, cm -1 e folding distance (mm) 1: : : : : : averaged An interval (x i -x j ), in which the decrement was calculated, is indicated in the column with numbers i-j. The corresponding e folding distance of the elastic solitary wave is of the order of meters, as shown in the last column. For comparison, the e folding distance of an ultrasonic wave in PMMA is equal to mm ([21]). The magnitude of dissipation decrement does not keep constant and changes (rises) considerably while soliton propagates along the bar. These variations are well above the measurement accuracy, and thus we can conclude that variations in values of may be due to the non-linear type of dissipation of non-linear waves in this material. Unfortunately neither theory nor experimental data are available in the literature on non-linear wave attenuation in polymers. The conventional values of available for linear waves in PMMA are much higher then those we have calculated for bulk solitons: lin =0.25 cm -1 vs soliton =0.009 cm -1.

12 Therefore the solitary wave propagating in a uniform homogeneous polymeric bar remains amazingly resistant to the decay in comparison with any conventional elastic bulk wave in polymers. We can suggest the long non-linear solitary strain waves as a useful tool for nondestructive testing of inhomogeneous (in particular, layered) wave guides. Indeed, since in the absence of inhomogeneities the wave does not change much as it propagates for considerably long distances, then any observed variations of parameters of the wave can be almost directly attributed to the influence of inhomogeneities. Soliton in a two-layered PMMA bar: results and discussion Complex PMMA wave guides, used in our experiments, are shown schematically in Figure 3, a-d. In all cases the soliton is first generated in the uniform part (in a rod, 50 mm long), and then propagates into either a uniform bar (Fig 3, a) or a layered bar, glued to the rod by a thin layer of a cyanide glue. Thus, the soliton enters a layered part (a bar, consisting of two layers being glued together (Fig. 3, b), clenched together (Fig. 3, c) or totally separated (Fig. 3, d)) through a transverse bonding layer. (a)

13 (b) (c) (d) Figure 3. Schematics of layered wave guides Figures 4 a,b show holographic interferograms of soliton propagation (transmission and reflection, respectively) in two structures: two PMMA rods glued together (Figure 4, a) and, for comparison, a PMMA rod glued to a quartz rod (Figure 4, b). In both cases the second rod has a bigger diameter,

14 allowing us to record the entire wave front in the first case, and to confirm that no elastic energy is transmitted through the interface in the second case. (a) (b) Figure 4. Strain soliton behaviour at the transversal interface of two rods glued together by a thin layer of a cyanide glue. (a) PMMA/PMMA interface, (b) PMMA/quartz interface In the first case, the bonding layer seems to be invisible for the bulk strain soliton which enters the second PMMA rod as if there was no interface (interference fringes on both sides of the glue layer are disturbed to a similar magnitude, no noticeable discontinuity is observed). This allows us to treat the interface as an ideal one (perfect contact). In the second case, the soliton is completely reflected from the interface (interference fringes in the quartz rod remain undisturbed), which can be

15 explained by a considerable difference in acoustic resistances of PMMA ( kg/(m 2 s)) and quartz ( kg/(m 2 s)) [22]. Hundreds of shots performed on these two structures showed that the transverse layer of a cyanide glue is resistant to loads induced by a head-on impact of the solitary wave with parameters given above (see Table 1). There were no signs of fracture or delamination caused by the propagating wave. Figures 5-8 show examples of the recorded interferograms of the strain soliton at one and the same distance from the input ( mm), but in 4 different situations. Figure 5 shows the soliton in the homogeneous bar (sketched in Figure 3, a). Figure 6 shows the soliton in the 2-layered glued bar (sketched in Figure 3, b) (the horizontal interface is of the same type as the vertical interface described above, the experiment again allows us to treat it as a perfect interface). In Figure 7 the soliton propagates in a 2-layered bar, where layers are not glued, but clenched together (sketched in Figure 3, c), the initial uniform part (rod) is glued to both layers symmetrically. In Figure 8 a 2- layered bar is similar to that in Figure 7, but the separation of layers is provided by thin (1 mm in diameter) wires placed between the layers in the transversal direction (sketched in Figure 3, d). Figure 5. Soliton in the homogeneous bar

16 Figure 6. Soliton in the 2-layered glued bar Figure 7. Soliton in the 2-layered bar with layers clenched together Figure 8. Soliton in the 2-layered bar with completely separated layers

17 The analysis of the series of recorded interferograms allowed us to register the typical variation of parameters of a soliton during its propagation in the above mentioned complex layered wave guides. Table 2 shows the averaged values of these parameters for the four specified configurations of a wave guide. Table 2. Averaged values of soliton parameters in layered bars. Wave guide Strain amplitude, 10-4 Width (FWHM), mm Homogeneous bar layer glued bar layer bar with clenched layers layer bar with separated layers The solitons in the homogeneous and glued bars are identical (within the experimental error), which means that a thin bonding layer of a cyanide glue between the layers does not introduce any noticeable inhomogeneity, and may be treated as a perfect interface. On the contrary, when the layers are just clenched, the soliton is slightly amplified, its width decreases. When the layers are well separated, the amplitude of the soliton rises more noticeably, and its width decreases further. This effect is similar to the behaviour of a soliton in a wave guide with an abrupt change of the cross section, recently observed in numerical simulations [20]. Three consecutive graphs of a soliton propagating along a rod with the abruptly changing cross section are shown in Figure 9 as a result of numerical simulations for the model equation. In Fig.9 the strain is dimensionless, and the distance along Ox axis is designated in arbitrary units for comparison. An initial bell-shaped strain wave U = Acosh 2 k( x Vt) with the unit amplitude, A = 1 (part I), propagates through an abrupt two-fold decrease of the cross section area (which was smoothened in numerical simulations). This has lead to a 19% amplification of the main soliton and to a formation of the second soliton with much lower

18 amplitude (part II). The velocity of the soliton is proportional to its amplitude, and that is why the second soliton is left behind the first one as they become well separated (part III). Note that the behaviour of a soliton in a gradually narrowing rod is completely different: a 47% increase of the amplitude and no generation of the second pulse were reported in numerical and analytical studies of the problem in [18, 20]. Figure 9. Numerical simulations of the evolution of a soliton due to its transition from the thick to the thin part of the rod (see [20])

19 The transformation of the strain solitary wave observed in our experiments can be modelled theoretically, by considering a two-layered bar and assuming that there is a perfect interface when x < 0 and complete debonding when x > 0, where the axis Ox is directed along the bar. This problem can be solved analytically using a combination of the method of matched asymptotic expansions with the method of multiple scales, and the subsequent use of results from the theory of integrable systems. The solution is rather technical and long, and will be published elsewhere [23]. Nevertheless, it is interesting to compare here our first experimental results with the theoretical predictions of this study. The analysis mentioned above leads, in particular, to the following analytical formula for the ratio C A T of the amplitude of the transmitted solitary wave (in the delaminated area) to the amplitude of the incident solitary wave (in the glued area): 2 8 A C T = 1+ 1, (10) 4 where = and is the ratio of the height to the half-width of a single layer. In 2 4(1 + ) A our experiment this ratio was equal to 1, which yields C T The same coefficient in our experimental study turned out to be: (clenched layers) 1.10 C (separated layers) Thus, first results are very encouraging, and we are now looking to expand and refine this study in order to improve the accuracy of both our measurements and estimates, and to check some other rather interesting theoretical predictions [23]. A Exp

20 Conclusions The first results of the experimental observation of the behaviour of a bulk solitary wave in layered and delaminated wave guides allow us to conclude that the amplification of a soliton in a delaminated two-layered wave guide of the type considered in this paper is detectable (ca %). This experimental result is in a very good agreement with the analytical formula derived by considering a two-layered bar under assumptions that there is a perfect interface when x < 0 and complete debonding when x > 0 (the axis Ox is directed along the bar). The results of our study may find applications in non-destructive recognition of certain delamination areas. This idea looks attractive due to the low decay rate of a bulk solitary wave compared to that of other known elastic waves (at least, in the experiments with PMMA rods). Since the agreement between the experimentally measured amplification of a soliton with that predicted by our theoretical model is very good, we are now looking to expand and refine both our experimental and theoretical studies into this topic. Acknowledgments The support of our joint research in the framework of the Research agreement between the A.F. Ioffe Physical Technical Institute and Loughborough University, as a part of the project on Mathematical modelling of non-linear waves in layered elastic wave guides with inhomogeneities (EPSRC grant EP/D035570/1) is gratefully acknowledged. We also thank the Department of Mathematical Sciences at Loughborough University for partial financial support of the experimental study. References 1. Maeva E., Severina I., Bondarenko S., Chapman G., O'Neill B., Severin F., Maev R., (2004), Acoustical methods for the investigation of adhesively bonded structures: A review. Can. J. Phys./ Rev. Can. Phys. 82, 12,

21 2. Braun, O.M., Kivshar, Y.S., (2004), The Frenkel-Kontorova model. Concepts, methods, and applications. Springer, 472 pp. 3. Yagil D., Kawahara T., (2001), Strongly non-linear envelope soliton in a lattice model for periodic structure. Wave Motion, 34, LeVeque R., Yong D.H., (2003), Solitary waves in layered non-linear media. SIAM J. Appl. Math., 63, Khusnutdinova, K.R., (2003), Non-linear waves in a bi-layer and coupled Klein-Gordon equations. Solid Mech. and Appl., 113, Khusnutdinova, K.R., (2007), Coupled Klein-Gordon equations and energy exchange in twocomponent systems. Eur. Phys. J. - Special Topics, 147, 1, S.D. Griffiths, R.H.J. Grimshaw, K.R. Khusnutdinova, (2006), Modulatiional instability of two pairs of counter-propagating waves and energy exchange in a two-component system. Physica D, 214, Smith, E., (1977), The effect of the discreteness of the atomic structure on cleavage crack extension: use of a simple one-dimensional model. II. Mater. Sci. Eng., 30, Ginzburg, V.V. and Manevitch, L.I., (1993), The extended Frenkel-Kontorova model and its application to the problems of brittle fracture and adhesive failure. Int. J. Fract. 64, Slepyan, L.I., (2002), Models and phenomena in fracture mechanics. Springer, 576 pp. 11. Khusnutdinova K., Silberschmidt V., (2003), Lattice modelling of non-linear waves in a bi-layer with delamination. Proc. Estonian Acad. Sci, 51, 1, Semenova I.V., Dreiden G.V., Samsonov A.M., (2005), On non-linear wave dissipation in polymers. Proc. SPIE, 5880, Porubov A.V., Samsonov A.M., (1995), Long non-linear strain waves in layered elastic halfspace. Intern. Journ. Nonlin. Mech., 30, 6, 861.

22 14. Dyakonov K.V., Ilisavsky Y.V., Yahkind E.Z., (1988), Non-linear effects into surface acoustic waves propagation in LiNbO3 at T = K, Sov. Tech. Phys. Lett., 14, 12, Dyakonov K.V., Ilisavsky Y.V., Yahkind E.Z., (1988), The influence of sound on superconductive lead flims. Sov. Tech. Phys. Lett., 14, 12, Cho Y., Miagawa N., (1993), Surface acoustic soliton propagating on the metallic grating wave guide. Appl. Phys. Lett., 63, Dreiden G.V., Porubov A.V., Samsonov A.M., Semenova I.V., Sokurinskaya E.V., (1995), Experiments in the propagation of longitudinal strain solitons in a nonlinearly elastic rod. Sov. Tech. Phys. Lett., 21, 6, Samsonov A.M., Dreiden G.V., Porubov A.V., Semenova I.V., (1998), Longitudinal strain soliton focusing in a narrowing nonlinearly elastic rod. Physical Review B, 57, 10, Semenova I.V., Dreiden G.V., Samsonov A.M., (2003), A novel approach to determination of the 3d order elastic moduli of isotropic materials. Proc. SPIE, 5144, Samsonov A.M., (2001), Strain solitons in solids and how to construct them. Chapman&Hall/CRC Press, 248 pp. 21. Physical data, (1991), Grigoriev I.S., Melikhov E.Z., eds., Moscow, Energoatomizdat (in Russian). 22. Shutilov V.A. (1980), Foundations of physics of ultrasound, Leningrad university press (in Russian). 23. Khusnutdinova K.R., Samsonov A.M. (2008), Fission of a longitudinal strain solitary wave in a delaminated layered elastic bar. Physical Review E; 77, Born M., Wolf E. (1964) Principles of optics. Pergamon Press.