Nonlinear modulation of wave packets in a shallow shell on an elastic foundation

Wave Motion 34 (2001) 63–81

Nonlinear modulation of wave packets in a shallow shell on an elastic foundation B. Collet∗ , J. Pouget Laboratoire de Modélisation en Mécanique, Unité Mixte de Recherche (UMR) 7607, Université Pierre et Marie Curie/C.N.R.S, Case 162-4, Place Jussieu, 75252 Paris Cedex 05, France Received 6 December 1999; accepted 1 March 2000

Abstract Among a lot of fascinating nonlinear effects, there is a great deal of interest in self modulation of a plane wave, or modulational instability (MI), which occurs in nonlinear dispersive media. Qualitatively a MI is the tendency for amplitude of a modulated carrier wave to break into isolated structures or solitons. Energy originally resident in the carrier of wave packets or long pulse is gradually transferred by nonlinear interaction, in the medium, into the spectrum side bands. As the side-band energy grows in amplitude, the modulated wave breaks up into a series of localized objects or envelope solitons. Composite structures formed by a singly or doubly-curved shallow shell resting on a nonlinear elastic foundation are wave guides that enable one to focus a high energy density in order that nonlinearities be excited. This class of elastic structures is an interesting candidate for the real observation of two-dimensional localized modes in elastodynamics. The purpose of the present work is to study the influences of the geometric dispersion, the prestress on the shallow shell and the material nonlinearity of the elastic foundation on the vibrations modes of the structure. The basic equations which govern the dynamics of the elastic structure are deduced from a variational principle. The analysis is restricted to signals which consist of a slowly varying envelope in space and time modulating a harmonic carrier wave. In the limit of low amplitude the coupled equations are solved by means of a reductive perturbation method. It is shown that the complex amplitude of the envelope is governed by a two-dimensional nonlinear Schrödinger equation. This equation allows us to study the modulational instability conditions leading to different zones of instability. The mechanism of the self generated nonlinear waves in the elastic structure beyond the birth of modulational instability is numerically investigated on the original equations. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Modulational instability (MI); Nonlinear Schrödinger equation; Localized modes; Elastic shell

1. Introduction Propagation of nonlinear waves and formation of localized structures or modes in various fields of sciences have been the subjects of considerable interest for many years [1–5]. The introduction of the concept of spatio-temporal structures has played a fundamental role in wave motion in a variety of physical circumstances and becomes of practical importance in engineering physics. We intend, in the present work, to illustrate the formation of spatial and temporal structures in complex media by examining the wave propagation in elastic structures (e.g. beams, plates, shells, composite media, etc.). More precisely, such elastic structures possess some interesting effects: nonlinearities, dispersion, dissipation whose the influence is able to change the wave nature very drastically. ∗

Corresponding author. Fax: +33-1-44-25-52-59. E-mail addresses: [email protected] (B. Collet), [email protected] (J. Pouget). 0165-2125/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 0 1 ) 0 0 0 7 0 - 1

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Propagation of nonlinear waves including solitons and solitary waves has been extensively studied in numerous varieties of physical systems possessing a rich mathematical structure such as hydrodynamics, plasma physics, lattice dynamics, optical fibres, electrical transmission lines, Josephson junctions, magnetic films, surface acoustic waves and DNA just to quote some of them [6–17]. Nevertheless, wave problems related to elastic media are less explored because experimental demonstrations of solitons in elastic solids are scare and this seems to be an excellent reason for examining nonlinear wave propagation in elastic structures. Moreover, elastic structures, here shells, turn out to be an interesting candidate to design guided wave or active control devices using geometrical properties of the structure. It is worthwhile noting the observation of flexural wave packets with a soliton-shape envelope on a circular-cylindrical thin elastic shell [18]. In the present work, a particular attention is addressed to the influence of nonlinear and characteristic length of the structure (typically the shell thickness) on the plane wave propagation. Then we show the possible formation of localized modes on the shell. To be more specific, we are concerned with deformable structure consisting of an elastic thin shell perfectly bonded with a nonlinear foundation. The emphasis is placed on the formation of localized modes mediated by modulational instability of low-amplitude carrier wave [6,9,13,19]. Modulational instability is also named self-induced modulation of a plane wave, the instability occurs in nonlinear dispersive media. We want to study the conditions under which the modulational instability takes place and a question can be brought up, what is the nature of the wave thus produced? In fact, the answer is not obvious and it requires both analytical and numerical investigations for the physical problem in question. The elastic structure model is an elastic shallow shell of uniform thickness resting on a nonlinear elastic substrate [20–25]. In addition, the shell is subject to prestresses along its edges, in fact, the prestresses play an important role in the wave propagation and modulational instability mechanism. Since, shallow doubly-curved thin shells are considered the ratio of curvature radii is also an interesting parameter that can change the instability conditions. The equations for the elastic shell are obtained from the Hamiltonian principle and a set of nonlinear dynamical equations for the displacement field is thus deduced. Since we are concerned with low-amplitude wave, an analysis of the full nonlinear equations is done in the low-amplitude limit. In particular, by means of reductive perturbation method, we show that, in the low-amplitude limit, the wave modulation is governed by a bidimensional nonlinear Schrödinger equation (2D-NLSE) for the shell deflection. The model of a nonlinear Schrödinger type has an exceptional role in numerous physical systems because the nonlinear Schrödinger equation allows us to establish the modulational instability conditions. In the present situation — that is the elastic shell — the condition depends strongly on the material properties of the elastic shell and substrate, geometry of the shell (curvature) and prestresses. Nevertheless, the modulational instability does not give any information about the long-time evolution of the wave and the nature of the wave produced by the instability process. Accordingly, numerical simulations must be envisaged to ascertain the analytical conjecture and to characterize the qualitative nature of the localized modes emerging from the instability. The numerics exhibits rich and complex wave patterns on the elastic shell. The morphology of the localized waves is also studied according to the prestresses and ratio of curvature radii. The shell model is briefly presented in Section 2. The equations of motion of the elastic shell are deduced from a variational principle or Hamiltonian principle. The linear mode of propagation is examined in Section 3 leading to the dispersion spectra. The nonlinear analysis is presented in Section 4 as well as the derivation of the 2D nonlinear Schrödinger equation. In Section 5, we examine, in detail, the modulational instability on the basis of 2D-NLS equation, the zones of instability are obtained for different situations. The numerical simulations performed directly on the full equations of the elastic shallow shell are discussed in Section 6. The numerics displays rather complex wave patterns including localized modes. At length, by way of conclusion, the most pertinent results are summarized and some further extensions of the model are evoked in Section 7. 2. Model and basic equations Let us consider a homogeneous isotropic thin doubly-curved shallow shell resting on a nonlinear elastic foundation in a tri-orthogonal curvilinear coordinate system with x and y defining the shell neutral surface and z the normal

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Fig. 1. Geometry of the thin doubly-curved shallow shell on an elastic foundation.

direction as shown in Fig. 1. We restrict the analysis to shell having no twist (Rxy = ∞). In the natural state the shell has a uniform thickness h which is relatively thin in comparison to its constant radii of curvatures Rx and Ry (h/min Rx,y  1). Generic deflections (u, v, w) in three principal directions (x, y, z) are assumed to be small. Now we are interested in establishing the linear shell model undergoing the effects of external tangential initial loads and transverse force of the foundation. It is assumed that the general deformed configuration (referred to as the actual or final state) is obtained from the unstressed and unstrained state or natural state by passing through an intermediate equilibrium state, the state of initial stress. The displacements, associated with the initial stresses are noted u0 , v 0 and w0 . The current state of the elastic structure corresponds to small motions superimposed to the static deformations. The displacements u˜ and v˜ from intermediate configuration are tangent to the middle surface while w˜ is normal to the surface. Thus, the components of the field displacement in the final configuration from ˜ v = v 0 + v, ˜ w = w 0 + w. ˜ The application in mid-surface of the uniform static loads natural state are: u = u0 + u, N x0 ≷ 0 and N y0 ≷ 0 and N xy0 = 0 (N x0,y0 > 0, for traction forces per unit of length) leads to an intermediate prestressed state. Based on the linear equations of the perfect shell there exists a membrane state with an associated uniform transverse displacement w 0 solution to nonlinear equilibrium problem [26] N y0 N x0 + + c1 w 0 + c2 w 20 + c3 w30 = 0, Rx Ry

(1)

where the coefficients cα (α = 1, 2, 3) are the elastic constants of the foundation (c1 > 0). Moreover, it is assumed that w0 satisfy the condition w0 / h  1. The Hamilton’s principle is now used to deduce the equations of motion in the actual configuration. The principle takes on the following form [27] Z t1 (K − U ) dt = 0, U = Us + Ub + Up + Uf , (2) δ t0

where the total strain energy U is the sum of the stretching membrane energy Us , the bending strain energy Ub , the prestress energy Up and the foundation energy Uf . We assume that the tangential inertia and rotatory inertia terms as well as the effects resulting from the transverse shear deformations are neglected [20,21,27,28]. The strain energy components are given, respectively, by   Z  w˜ w˜ 1 ˜ 2 − 2(1 + ν)[φ˜ xx φ˜ yy − (φ˜ xy )2 ]} − {(∇ 2 φ) φ˜ xx + φ˜ yy ds, (3) Us = Ry Rx S 2Eh

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Z

1 D {(∇ 2 w) ˜ 2 − 2(1 − ν)[w˜ xx w˜ yy − (w˜ xy )2 ]} ds, 2 S Z 1 {N 0x (w˜ x )2 + N 0y (w˜ y )2 } ds, Up = 2 S  Z  1 1 1 2 2 3 4 (c1 + 2c2 w0 + 3c3 w0 )w˜ + (c2 + 3c3 w0 )w˜ + c3 w˜ ds, Uf = 3 4 S 2 Ub =

(4) (5) (6)

where D = Eh3 /12(1 − ν 2 ) represents the flexural rigidity, E the Young’s modulus, ν the Poisson’s coefficient and ∇ 2 the Laplacian operator defined as (∂ 2 /∂x 2 + ∂ 2 /∂y 2 ). The kinetic energy of the vibrating structure is merely given by Z 1 (7) K = ρh (w˜ t )2 ds, 2 S where ρ is the mass density per unit volume. Using the variational principle, for any arbitrary of the transverse ˜ we arrive at the following set of equations of motion: displacement w˜ and the Airy stress function φ, D∇ 4 w˜ +

1 1 φ˜ yy + φ˜ xx − N x0 w˜ xx − N y0 w˜ yy + (c1 + 2c2 w0 + 3c3 w 20 )w˜ Rx Ry

+(c2 + 3c3 w0 )w˜ 2 + c3 w˜ 3 + ρhw˜ tt = 0,

(8)

1 1 1 4 w˜ yy − w˜ xx = 0, ∇ φ˜ − Eh Rx Ry

(9)

where the second equation is the compatibility equation. Variables and fields involved in the governing equations are, for sake of convenience, written in a nondimensional form as follows  1/4  1/2 y t D ρh w x , Y = , T = , Lc = , Tc = , W = , X= Lc Lc Tc c1 c1 wc Ry wc D Ry φ˜ w0 N 0x , Φ= , N= , φc = , Nc = (c1 D)1/2 , α= , W0 = wc φc Nc Rx L2c β=

N 0x N 0y

,

δ=

Eh , Ry c1

λ=

Nc , Ry c1 wc

µ=

c2 wc , c1

γ =

c3 wc2 , c1

(10)

where Lc and Tc are characteristic length and time of the model, wc a characteristic transverse deflection in the z-direction, φc a characteristic moment, Nc a characteristic load. The constant α is a form parameter which takes the values: α = −1 for a hyperbolic-paraboloidal or saddle-type shell, α = 0 for a cylindrical shell and α = 1 for a spherical shell. By substituting these appropriate dimensionless quantities into the Eqs. (1), (8) and (9) which govern the static deformed state and dynamic final state, one obtains γ W03 + µW02 + W0 + (α + β)λN = 0,

(11)

∇ 4 W + (ΦXX + αΦYY ) − (WXX + βWYY ) + (1 + 2µW0 + 3γ W02 )W +(µ + 3γ W0 )W 2 + γ W 3 + WTT = 0, ∇ 4 Φ − δ(WXX + αWYY ) = 0,

(12) (13)

where β is a load parameter, δ an elastic coupling parameter and λ a prestress coupling parameter. Also, µ and γ are quadratic and cubic nonlinear elastic coefficients of the foundation. By eliminating the stress function Φ between

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Eqs. (12) and (13) we obtain the following uncoupled motion equation for the deflection ∇ 4 {∇ 4 W − N(WXX + βWYY ) + (1 + 2µW0 + 3γ W02 )W + (µ + 3γ W0 )W 2 + γ W 3 + WTT } +δ(WXXXX + 2αWXXYY + α 2 WYYYY ) = 0.

(14)

We first note in the Eq. (14) the presence of nonlinearities and dispersion two necessary ingredients for the existence of localized modes. The dispersive effect is due to the geometry of the thin shallow-shell. The nonlinearity is the consequence of the nonlinear action of the foundation on the shell. We also note that initial static load N modifies, by means of W0 , the effective elastic coefficients of the foundation. In the particular case, where W0 = 0, for which α + β = 0 the Eq. (14) is reduced to a simple form ∇ 4 {∇ 4 W − N(WXX − αWYY ) + W + µW 2 + γ W 3 + WTT } + δ(WXXXX + 2αWXXYY + α 2 WYYYY ) = 0. (15)

3. Linear and nonlinear dispersion relations In this section, we derive the nonlinear dispersion relation [[1] (Chapter 17-4), [9]] corresponding to the motion Eq. (14). This relation represents the fundamental equation which, in the weak-amplitude limit, allows us to reduce Eq. (14) to two-dimensional nonlinear Schrödinger equation (2D-NLSE). In order to obtain the nonlinear dispersion relation, we consider plane-wave solutions of the form W = W (x, y, t) = A1 eiΘ +  2 A2 e2iΘ + c · c,

(16)

where A1 and A2 are the slowly varying amplitudes of the first and second harmonics, respectively. The second term is, however, introduced to take into account of second order nonlinearity of the elastic foundation, however, higher harmonics are neglected. The amplitudes A1 and A2 depend on the slow space and time variables X1 = x, Y1 = y, T1 = t. Here,   1 is a small dimensionless amplitude parameter. The phase is defined as Θ = kx + ly − ωt, where k and l are the components of the wave vector k = (k, l) and ω is the circular frequency of the carrier wave which varies rapidly. Here, c · c denotes the complex conjugate. On inserting (16) into (15), equating first and second harmonic terms and keeping terms up to order  2 , we arrive at D(ω, k, l, |A1 |2 ) = ω2 − (1 + 2µW0 + 3γ W02 ) − N (k 2 + βl 2 ) − (k 2 + l 2 )2 − δ(k 2 +αl 2 )2 (k 2 + l 2 )−2 −  2 |A1 |2 [−(µ + 3γ W0 )2 (12(k 2 + l 2 )4 −3(1 + 2µW0 + 3γ W02 )(k 2 + l 2 )2 − 3δ(k 2 + αl 2 )2 )−1 + 3γ ] = 0.

(17)

The first five terms in the right-hand side of Eq. (17) represent the linear contribution to the dispersion relation whereas the last term corresponds to the nonlinear contribution. The linear analysis of the model informs us about properties of the basic harmonic plane wave solutions. By neglecting the term of order O( 2 |A1 |2 ) in Eq. (17) we obtain the general linear dispersion equation ω2 = (1 + δ(cos2 θ + α sin2 θ)2 + 2µW0 + 3γ W02 ) + N (cos2 θ + β sin2 θ )κ 2 + κ 4 ,

(18)

where θ = (x, k) is the angle formed by the x-axis and the direction of the propagation of the carrier wave. In particular, the case where α + β = 0, i.e. in the absence of transverse displacement W0 = 0 in prestressed initial state, the linear dispersion relation is reduced to ω2 = (1 + δ(cos2 θ + α sin2 θ)2 ) + N(cos2 θ − α sin2 θ )κ 2 + κ 4 .

(19)

A simple observation of the curves plotted in Fig. 2 allows us to place the influence of the curvature ratio α and load parameter N on linear spectrum in evidence. In particular, we note the existence of a cut-off frequency at the origin for some values of α and N. We also observe a softening of the dispersion branches at a non zero κ for −2 < N < 0.

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Fig. 2. Linear dispersion curves: (ω vs. κ) with W0 = 0, δ = 0.1, θ = π/4: (a) N = 2; (b) N = 0; (c) N = −0.85; (d) N = −2.

On the other hand, refering to Fig. 3 we note that for a saddle-type shell α = −1 the appearance of the dispersion curve is weakly influenced by the direction of propagation of the carrier wave, a slight translation of the curve is observed. The influence of the direction of propagation of the carrier wave on the spectrum are stronger in the cases of cylindrical α = 0 and spherical α = 1 shallow shells. Moreover, there exists a critical situation defined by (κ, N )κ=κcr ,N =Ncr ,α=α,δ= ˜ ˜ δ,θ= θ˜ = 0,

∂ω(κ, N ) = 0. ˜ =θ˜ ∂κ κ=κcr ,N =Ncr ,α=α,δ= ˜ δ,θ

(20)

The stability of the linear modes [30] is checked if ω is a real root to Eq. (17) for all wave number κ. Conversely, it is unstable if there is, at least, one value of κ for which the root ω becomes imaginary. In the latter case, the solution

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Fig. 3. Linear spectrum: (ω vs. κ) with W0 = 0; δ = 0.1, N = −0.85; (a) α = −1; (b) α = 0; (c) α = 1.

grows exponentially in time. The neutral stability curves for W0 = 0 are defined by N˜ = N˜ (κ) = −N = [(1 + δ(cos2 θ + α sin2 θ )2 )κ 2 + κ 4 ][(cos2 θ − α sin2 θ )κ 2 ]−1 .

(21)

The neutral stability curves reach a minimum N˜ cr at κcr as shown in Fig. 4. N˜ cr and κcr are the critical or buckling load [26,30] and the corresponding modulus of the wave vector. In this particular situation, we find N˜ cr = 2(1 + δ(cos2 θ + α sin2 θ)2 )1/2 (cos2 θ − α sin2 θ)−1 and κcr = (1 + δ(cos2 θ + α sin2 θ )2 )1/4 . It is also interesting to compute the components of the group velocity vector κ cos θ {N + 2κ 2 − 2δ(α − 1) sin2 θ (cos2 θ + α sin2 θ )κ −2 }, ω κ sin θ {βN + 2κ 2 + 2δ(α − 1) cos2 θ (cos2 θ + α sin2 θ )κ −2 }. = ω

Vgk = Vgl

(22)

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Fig. 4. Neutral stability curves (N˜ vs. κ) for W0 = 0, δ = 0.1 and θ = π /4 in the vicinity of κcr : (a) α = −1; (b) α = 0.

We deduce from Eqs. (20) and (21) that Vgk and Vgl are, respectively, zero for θ = (π/2) + nπ (n = 0, 1, 2, . . . ) or N = −2κ 2 + 2δ(α − 1) sin2 θ(cos2 θ + α sin2 θ )κ −2 and θ = mπ (m = 0, 1, 2, . . . ) or N = −β −1 (2κ 2 + 2δ(α − 1) cos2 θ(cos2 θ + α sin2 θ)κ −2 ). In the particular situation of the isotropic prestressed shallow spherical shell (β = 1, α = 1) the components Vgk and Vlk are equal for θ = (π/4) + n(π/2). 4. Derivation of the 2D nonlinear Schrödinger equation Now, to reduce Eq. (16) to 2D-NLSE we can use the multiple scale technique [19] or consider the nonlinear dispersion relation [9]. We have verified that both approaches give the same results. Here, we adopt the second method. Namely we consider slow modulation in space and time of a carrier wave with given components of the wave vector k and l, the expansion of the nonlinear dispersion relation Eq. (16) around the carrier wave parameters (ωc , kc , lc ,  2 |A1 |2 = 0) yields       2  ∂ω ∂ω 1 2 2 2 ∂ ω + (l − lc ) + (k − kc ) ω(k, l,  |A1 | ) = ωc + (k − kc ) ∂k c ∂l c 2 ∂ 2k c  2   2  ∂ ω ∂ ω ∂ω +2(k − kc )(l − lc ) + (l − lc )2  2 |A1 |2 + · · · , (23) + 2 2 ∂k∂l c ∂ l c ∂ |A1 |2 where the subscript ‘c’ denotes evaluation at ωc , k = kc , l = lc ,  2 |A1 |2 = 0. The frequency ωc is the carrier frequency provided by the linear dispersion relation Eq. (17). We set the following operators k − kc = i∂/∂X1 , l −lc = i∂/∂Y1 , ω−ωc = iω∂/∂T1 . Here X1 , Y1 , T1 represent the slow variables. Therefore Eq. (17) is an operator equation for the amplitude function A = A1 (X1 , Y1 , T1 ) written in the Fourier space [29]. Going back to the real variables, we find the following partial derivative equation i(AT1 + (Vgk )c AX1 + (Vgl )c AY1 ) +  2 P1 AX1 X1 +  2 P2 AY1 Y1 +  2 P3 AX1 Y1 +  2 Q|A|2 A = 0, where higher terms have been neglected and we have set    2    1 ∂ 2ω 1 ∂ 2ω ∂ ω , P = , P = , P1 = 2 3 2 2 2 ∂ k c 2 ∂ l c ∂k∂l c

 Q=−

∂ω 2 ∂ |A1 |2

(24)

 . c

(25)

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These coefficients represent the group velocity dispersion and nonlinearity. On considering a frame moving with the wave and using the transformations ξ = X1 − (Vgk )c T1 , ζ = Y1 − (Vgl )c T1 , τ = T1 , we can write Eq. (24) governing the amplitude modulation of the signal in the standard form of the 2D nonlinear Schrödinger equation iAτ + P1 Aξ ξ + P2 Aζ ζ + P3 Aξ ζ + Q|A|2 A = 0,

(26)

where we have P1 =

P2 =

P3 =

Q=

1 {N + (6 cos2 θ + 2 sin2 θ)κc2 + 2δ(α − 1)[3 cos2 θ cos 2θ 2ωc +α sin2 θ(5 cos2 θ − sin2 θ)]κc−2 − (Vgk )2c },

(27)

1 {βN + (2 cos2 θ + 6 sin2 θ)κc2 + 2δ(α − 1)[3α sin2 θ cos 2θ − cos2 θ (5 sin2 θ 2ωc − cos2 θ)]κc−2 − (Vgl )2c },

(28)

4κc2 cos θ sin θ {1 − δ(α − 1)[ cos2 θ (cos2 θ + (α − 1) sin2 θ ) ωc − sin2 θ(α sin2 θ − (α − 1) cos2 θ)]κc−4 − (Vgk )c (Vgl )c (4 cos θ sin θ )−1 κc−2 },

(29)

1 {2(µ + 3γ W02 )[12κc8 − 3δ(cos2 θ + α sin2 θ )κc4 − 3(1 + 2µW0 + 3γ W02 )κc2 ]−1 − 3γ }. 2ωc

(30)

It is worthwhile noting that for a given propagation direction of the carrier wave θ , load parameters β and N, form parameter α, coupling parameter δ the sign of dispersion coefficients Pa (a = 1, 2, 3) can be changed by varying the wave number κc . It is the same for θ, β, N, α, δ, transverse displacement in static initial state W0 and elastic coefficients of the foundation µ and γ fixed, the sign of the nonlinear coefficient Q depends on κc . We point out that in the case where α + β 6= 0, i.e W0 6= 0 the coefficient Q can change of sign for a nonlinear elastic foundation purely cubic (µ = 0, γ 6= 0). In particular, P3 is equal to zero for θ = n(π/2) (n integer), i.e for carrier wave traveling along the x or y direction. We also remark that for spherical shallow shell (α − 1 = 0) the coefficients Pa are reduced to 1 1 {N + (6 cos2 θ + 2 sin2 θ)κc2 − (Vgk )2c }, P2 = {βN + (2 cos2 θ + 6 sin2 θ )κc2 − (Vgl )2c }, 2ωc 2ωc 4κ 2 cos θ sin θ {1 − (Vgk )c (Vgl )c (4 cos θ sin θ )−1 κc−2 }, (31) P3 = c ωc P1 =

For this particular geometry, an isotropic loading β = 1, and for a carrier wave vector such as θ = (π/4) + n(π/2)(n = 0, 1, 2, . . . ), the dispersion coefficients P1 and P2 are equal. Further step in the simplificity consists of introducing new spatial coordinates χ = cos υξ + sin υζ,

ψ = − sin υξ + cos υζ,

(32)

where υ=

P3 π 1 arctan +n 2 P2 − P1 2

υ = sin(P3 )(π/4) + nπ/2

(n = 0, 1, 2, . . . ) if (n = 0, 1, 2 . . . ) if

P2 6= P1 ,

P2 = P1 .

(33) (34)

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Fig. 5. The coefficients of the nonlinear Schrödinger equation as fonction of the carrier wave number according to the curvature ratio parameter α: (a) nonlinear coefficient Q; (b) dispersive coefficient R; (c) dispersive coefficient S.

Eq. (26) is then transformed into the classical 2D-NLSE iAτ + R Aχ χ + S Aψψ + Q|A|2 A = 0,

(35)

where R = P1 cos2 υ + P2 sin2 υ + P3 cos υ sin υ,

(36)

S = P1 sin2 υ + P2 cos2 υ − P3 cos υ sin υ.

(37)

Eq. (35) has been extensively studied especially in plasma physics, hydrodynamics, solid state physics and optics [7–11] (Figs. 5 and 6).

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Fig. 6. The same curves as in the previous picture but for α = −1 and various values of the load N.

5. Modulational instability of a uniform wave train Now, we focus our attention on the discussion of the modulational instability due to small perturbations in two space dimensions. From Eq. (35) we can easily find a solution of constant amplitude which is oscillatory in time of the form A = A0 exp i(QA20 τ ) + c · c,

(38)

where A0 is a real amplitude. We suppose that there is one of small sideband disturbances superimposed to a uniform wave train. According to [6,9,19], the perturbed wave train can take the form A = [A0 + εa(χ, ψ, τ )] exp i(QA20 τ + εb(χ , ψ, τ )) + c · c,

(39)

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where the real functions a and b are the amplitude and phase, respectively, induced by the disturbances ε  1 the small parameter. On inserting Eq. (39) into Eq. (35) we obtain at O(ε) a set of linear equations aτ + RA0 bχχ + SA0 bψψ = 0,

(40)

−1 bτ − RA−1 0 aχχ − SA0 aψψ − 2QA0 a = 0.

(41)

In order to know under which conditions the disturbances become unstable, i.e. grow exponentially with time, we look for a solution to Eqs. (40) and (41) as (a, b) = (a0 , b0 )exp i(rχ + sψ − $ τ ),

(42)

Fig. 7. Sketch of the regions of modulational instability in the perturbation wave vector plane (r, s): (a) R > 0, S > 0 and Q > 0 or R < 0, S < 0 and Q < 0, the region is bounded by an ellipse; (b) R < 0, S > 0 and Q < 0 or R > 0, S < 0 and Q > 0, the region is defined by the zone between a hyperbola and its asymptotes; (c) R < 0, S > 0 and Q > 0 or R > 0, S < 0 and Q < 0, the region is bounded by a hyperbola and its asymptotes.

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where a0 and b0 are real amplitudes, r and s correspond to the wave vector components of the disturbance and $ is an eigenvalue or frequency to be determined. On substituting Eq. (42) into Eqs. (40) and (41), the condition for a nontrivial solution leads to the eigenvalue or dispersion relation $ 2 = (Rr2 + Ss2 )(Rr2 + Ss2 − 2QA20 ).

(43)

If the right-hand side of Eq. (43) is negative then the angular frequency $ is imaginary ($ = iσ , with σ > 0) and the disturbance is growing. The parameter σ which is similar to the inverse of time is called, here, the amplitude or growth rate of the disturbance. The maximum growth rate is given by σmax = |Q|A20

occurring at

QA20 − Rr2 − Ss2 = 0.

(44)

Now, it is obvious that the instability criterion depends on the model parameters and the characteristics of the carrier wave. The analysis of the growth rate σ allows us to distinguish three regions of instability in the wave vector space (r, s). It is clear that the discussion of the instability regions is cumbersome since it depends on the signs of R, S and Q which are themselves varying as the carrier wave vector (θ, κc ), the ratio of curvature of the shallow shell α, the coupling parameter δ, the prestresse parameters (β, N ), the initial transverse displacement W0 and the elastic coefficients of the foundation (µ, γ ) (Fig. 7).

6. Numerical simulations The analytical study tells us that, in the low-amplitude limit, the wave propagation can be approximately described by a 2D-NLS equation. However, the long-time evolution of the wave cannot be predicted from the 2D-NLS model. In this section, we intend to examine the role played by the modulational instability in the response of the elastic shell to an initial homogeneous wave with low amplitude. Moreover, we want to characterize the qualitative nature of the localized waves that appear in the long-time evolution and cannot be described by 2D-NLS equation. We undertake this task by means of numerical simulations performed directly on the full equations of motion of the elastic shell (see Eqs. (11) and (14)). The numerical scheme is provided by a simple explicit finite difference algorithm applied to the original equations of motion of the shell. In order to materialize the numerical scheme, a square grid made of 89 × 89 nodes is considered, in addition, periodic boundary conditions on the left and right sides and on the lower and upper boundaries of the domain are used. At the initial time, the shell is subject to a flexural harmonic wave travelling along the first diagonal of the square θ = (π/4). The harmonic plane wave is characterized by the amplitude of the initial deflection W and its wave number κc connected with the circular frequency through the dispersion relation (17). A first simulation is done by taking, W = 0.12, N = −0.85, six periods along the diagonal is considered leading to κc = 0.8078 and the corresponding frequency is ωc = 0.9933. The other parameters of the model are δ = 0.1, µ = −2, γ = 1 and we take β = −α so that the predeformation is zero. We consider a saddle-type shell for which the ratio of curvature radii is α = −1. In order to force the instability, a small (≈10−3 ) random noise is added to the initial velocity and removed afterward. The numerical results are shown in Fig. 8 where the shaded-contour plots represent the propagation of the flexural wave on the elastic shallow shell. The initial condition is given in Fig. 8a, this is the harmonic carrier wave. After a short period of time, we can see small perturbations that are taking place along the direction perpendicular to the first diagonal as depicted in Fig. 8b. In particular, we observe stretched structures along the perpendicular direction. After a rather long lapse of time, a localized wave is then produced. The resulting pattern is shown in Fig. 8c and we can see very clearly a disk-shaped structure. This localized object is moving along the first diagonal of the square and it looks as stable as a 2D-pulse soliton. We let the structure run over a long time scale and we recover the identical structure located at a different point of the shell as in Fig. 8d. A second series of numerical simulations is considered for a larger prestress, we take N = −1.6, the other parameters are kept unchanged. We use the same initial condition as in the previous simulation and it is still

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Fig. 8. Shaded-contour plots for the deflection W for α = −1 and N = −0.85: (a) initial condition; (b) the birth of the localized structures at time T = 75; (c) the moving 2D-pulse at time t = 1650; (d) the 2D-pulse after a long-time t = 3640.

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Fig. 8. (Continued).

given in Fig. 8a. After a very short lapse of time (T = 20), we observe the birth of instability characterized by small disk-shaped structures along the second diagonal of the square as in Fig. 9a. By continuing the numerical simulations a little bit longer, we can notice that the instability growth leads to fewer localized structures but with larger amplitudes (see Fig. 9b). After a rather long time, the structures merge and form a “zig-zag” pattern stacked in the direction perpendicular to the first diagonal as shown in Fig. 9c. The structures thus produced seem to be stationary and they are very robust (see Fig. 9d obtained at T = 3000).

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Fig. 9. Shaded-contour for the deflection W for α = −1 and N = −1.6: (a) the birth of instability at time t = 20; (b) the beginning of the localized objects formation at time t = 30; (c) formation of zig-zag structures at time t = 1500; (d) the same type of structure at time t = 3000.

7. Closing remarks In this paper, we have examined the formation of localized waves and modes mediated by the modulational instability of a plane wave with low amplitude. Here, we have a good illustration of the nonlinear wave propagation

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Fig. 9. (Continued).

on an elastic structure, more precisely, a nonlinear elastic substrate coated with a thin doubly-curved shallow shell. Furthermore, the model possesses the necessary nonlinearity (due to the action of the foundation on the elastic shell) and dispersion effects in order to expect solitons or solitary waves. The numerics shows that the localized mode can be identified to a 2D pulse soliton, the latter seems to be very stable and robust. An important point of the work is that, in the low amplitude limit, the rather complicated equation of motion for the shell deflection, is

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approximated by 2D-NLS equation. The 2D-NLS equation is, in fact, an universal model to predict the modulational instability (MI). Here, we have examined, in detail, the influence of the material properties of the elastic shell and parameters foundation, the geometric (ratio of curvature of radii), the prestresses as well as the carrier wave parameters (wave-vector) on the existence of the window of modulational instability. The numerical simulations have been done for two values of the prestresses N leading then to different morphologies of localized structures. For the first numerical simulation, we have obtained a moving 2D pulse soliton, whereas, for the second numerical simulation, for a prestress closer to the buckling load, we have observed the formation of “zig-zag” stationary structures. It turns out that the prestress parameter plays a crucial role in the structure formation, this problem should be well understood in a further work. In the present paper, we have restricted ourselves to the saddle-type shell (α = −1), it will be interesting to study the wave propagation on other types of shallow shells, for instance, spherical or cylindrical shells. This will be presented in further works.

Acknowledgements This work has been performed in the framework of the TMR European Contract number FMRX-CT-960062 entitled “Spatio-temporal instabilities in deformation and fracture mechanics, material science and nonlinear physics aspects”. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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