Keywords: A. Elastic solids, Non-linear elasticity, Surface stress, Dynamics, C. Wave propagation

the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the ÿnite deformation are then described graphically. In particular, features which di er from those arising in the classical theory are highlighted.

Introduction

In a recent paper (Steigmann and Ogden, 1997a), henceforth referred to as (I), the authors have developed a static non-linear plane strain theory of an elastic solid coated with a thin ÿlm of elastic material that has both extensional and exural resistance. Applications of the theory were examined in (I), [START_REF] Ogden | The e ect of elastic surface coating on the ÿnite deformation and bifurcation of a pressurized circular annulus[END_REF][START_REF] Ogden | The e ect of elastic surface coating on the bending of an incompressible elastic block[END_REF] and [START_REF] Dryburgh | Bifurcation of an elastic surface-coated incompressible isotropic elastic block subject to bending[END_REF], and further theory was provided in Steigmann and Ogden (1997b). The corresponding three-dimensional theory is described in [START_REF] Steigmann | Elastic surface-substrate interactions[END_REF].

In the model for the surface coating, which extends that developed by Gurtin andMurdoch (1975, 1979) for membrane surfaces to allow for the inclusion of exural sti ness, the mechanical response of the coating is determined by the metric and curvature of the surface. Motivation for inclusion of exural sti ness in the considered model is discussed in detail in (I) and in [START_REF] Steigmann | Elastic surface-substrate interactions[END_REF], but it is appropriate to highlight certain key points here. Firstly, the signiÿcance of exural sti ness in respect of the mechanical stability of coatings produced by thin-ÿlm deposition processes has been elucidated by [START_REF] Gille | Buckling instability and adhesion of carbon layers[END_REF] and [START_REF] Yu | Buckle formation in vacuum-deposited thin ÿlms[END_REF]. In the absence of exural sti ness the coating cannot support compressive stresses. This fact e ectively rules out the pure membrane theory of Gurtin and Murdoch as a model for equilibrium states that support compression in the surface ÿlm. Indeed, according to the energy criterion of elastic stability, such states cannot be maintained in stable equilibrium, as was proved in (I). But, experiments indicate the presence of signiÿcant compressive stresses in the surface of certain thin-ÿlm=substrate systems (see, for example, [START_REF] Krulevitch | Stress and microstructure in LPCVD polycrystalline silicon ÿlms: experimental results and closed form modelling of stresses[END_REF], [START_REF] Kuba | Some theoretical problems of shot peening[END_REF] and the review by [START_REF] Nix | Mechanical properties of thin ÿlms[END_REF]). Thus, in order to accommodate such compressive stresses in equilibrium and to analyse exural wave propagation in compressed states it is essential to use a model which incorporates exural sti ness.

In the context of plane strain, [START_REF] Wu | The chemical potential for stress-driven surface di usion[END_REF] presented an interesting extension of the Gurtin-Murdoch theory to account for the e ects of material accretion at boundaries. His kinematical formulas are similar to those used here. He suppressed the dependence of the strain energy on local curvature, an e ect that we include, while in the present work the boundary ÿlm is considered to be a material surface in all conÿgurations and thus the e ects of accretion considered by Wu are not included. See also [START_REF] Wu | The e ect of surface stress on the stability of surfaces of stress solids[END_REF] and references to related work in these two papers.

In the present paper we extend the (plane strain) theory to allow for the dynamics of a substrate-coating structure, including the e ects of the rotatory inertia of the coating. The theory is then applied to a prototype problem in which inÿnitesimal surface waves of Rayleigh type are examined for a statically pre-stressed half-space of incompressible isotropic elastic material coated with a thin ÿlm on its plane boundary. This extends to the dynamic context the results obtained in (I) relating to the quasi-static bifurcation of a pre-stressed half-space. For classical isotropic elasticity, surface waves in a membrane-coated half-space with a residual surface tension were discussed by [START_REF] Murdoch | The propagation of surface waves in bodies with material boundaries[END_REF], and his results may be recovered by appropriate specialization of the theory discussed here.

Our results are discussed in relation to those for a thin, but ÿnite thickness, layer of material on a half-space (see, for example, [START_REF] Achenbach | Free waves in a plate supported by a semi-inÿnite continuum[END_REF], [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF] and [START_REF] Farnell | Elastic wave propagation in thin layers[END_REF] in the classical context and [START_REF] Ogden | On interfacial waves in pre-stressed layered incompressible elastic solids[END_REF] for the case of pre-stressed materials) and to those with the layer modelled using plate theory [START_REF] Achenbach | Free waves in a plate supported by a semi-inÿnite continuum[END_REF][START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF]. It is shown, in particular, that rotatory inertia, which is sometimes neglected in the classical theory [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF], can have a signiÿcant in uence on the surface wave behaviour even at relatively small values of the wave number.

In Section 2, we summarize the basic equations for the bulk solid=ÿlm combination and derive the equations governing the motion of the ÿlm. The role of the rotatory inertia term is discussed in some detail. We also derive the equations governing the coupling of mechanical power between bulk and ÿlm material and the associated energy ux. In Section 3, we obtain the (linearized) incremental equations governing small motions superimposed on a ÿnite static deformation. The equations are then applied, in Section 4, to the problem of inÿnitesimal surface waves propagating in a homogeneously pure strained incompressible isotropic elastic half-space with an elastic coating on its plane boundary. The two equations governing the motion of the ÿlm provide boundary conditions for the displacement in the bulk solid which, since the material is incompressible, is governed by a single scalar equation.

Surface wave solutions of harmonic type are used to obtain, on application of the boundary conditions, the secular equation for the speed of surface waves. Unlike in the situation for an uncoated half-space [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF], the waves here are dispersive and the wave speed depends on the deformation in the bulk material, the properties of the bulk and ÿlm materials, the wave number, the relative densities of the two materials and the local moment of inertia of the ÿlm cross-section. An explicit form of the secular equation is given in respect of an arbitrary incompressible, isotropic elastic substrate material and an arbitrary ÿlm material whose properties are described by a strain energy per unit reference length that depends only on the stretch and curvature of the ÿlm, as in (I). Additionally, we include in the properties of the ÿlm material both residual stress and residual moment. For a special form of energy function for the bulk material the secular equation reduces to a quartic, which contrasts with the cubic obtained for the uncoated case. Material properties for the coating material and a number of specializations of the secular equation are discussed in Section 5. In particular, the secular equation derived by [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF] for an uncoated half-space is recovered. Furthermore, the specialization of the secular equation for the undeformed conÿguration, with residual stress and moment incorporated, is given, and this, when the exural rigidity is omitted, can be shown to be equivalent to an equation obtained by [START_REF] Murdoch | The propagation of surface waves in bodies with material boundaries[END_REF] for a membrane ÿlm. A di erent specialization, again for the undeformed situation but with residual stress and moment excluded and rotatory inertia included, enables the results to be compared with those of [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF].

In Section 6, asymptotic results for large wave number are derived and it is shown that these have a signiÿcant in uence on the character of the wave speed throughout the range of wave numbers. It is also shown that when the wave speed is set to zero in the secular equation a bifurcation equation is obtained which generalizes that given in (I) for incompressible materials to the case in which residual stress and moment are included in the coating. Finally, in Section 7, numerical solutions of the secular equation are presented in graphical form to illustrate the dependence of the wave speed on the various material and geometrical parameters.

Basic equations

Kinematics

We consider plane motions of a (prismatic) body whose plane section is identiÿed in its reference conÿguration with a two-dimensional region . The motion is described by a mapping from to the two-dimensional region ˆ . A material particle is labelled by its position vector X in , and its position in ˆ is denoted by x. The motion is deÿned by the mapping E such that x = E(X; t); X ∈ ; t ∈ I;

(1

)
where t is the time and I some appropriate time interval. For each t ∈ I, E is invertible and is assumed to possess regularity properties necessary for the ensuing analysis. The deformation gradient tensor A is given by

A = Grad x; ( 2
)
where Grad is the (two-dimensional) gradient operator on . Let {e 1 ; e 2 } be a ÿxed orthonormal basis in the plane under consideration and let x i = x • e i ; X = X • e , with i; ∈ {1; 2}, denote the Cartesian coordinates of a particle in ˆ and , respectively. Then, in component form, Eq. ( 2) may be written as

A = A i e i ⊗ e ; A i = x i; ≡ 9x i 9X : (3)
We also note the (two-dimensional) polar decomposition

A = RU; ( 4
)
where R is proper orthogonal and U is positive deÿnite and symmetric. The right stretch tensor U has the spectral decomposition

U = 1 u (1) ⊗ u (1) + 2 u (2) ⊗ u (2) ; (5)
where 1 and 2 are the principal stretches of the deformation and {u (1) ; u (2) } the associated (orthonormal) principal directions of U. Since we are considering plane deformations it is implicit that the stretch 3 normal to the considered plane is set to unity. Then, for an isochoric deformation, we have

1 2 = 1: (6)
Let S be the arclength parameter describing the boundary 9 of in the usual sense and let N be the rightward unit normal to 9 . We write X(S) as the parametrization of points on 9 , as discussed in (I).

Let a subset P of 9 be coated with a thin elastic ÿlm that deforms as a material curve. The image of P on 9 ˆ , the boundary of ˆ , is denoted by P, and we write r(S; t) = E(X(S); t)

for those values of S for which X(S) is on P. The unit tangent to P is T(S) = X (S), where the prime indicates di erentiation with respect to S. Di erentiation of Eq. ( 7) with respect to S yields r (S; t) = A(X(S); t)T(S) ≡ (S; t) (S; t);

where

(S; t) = |A(X(S); t)X (S)| (9)
is the stretch of P induced by the motion E, (S; t) is the unit tangent to P at the point with arclength station S on P at time t, and the prime represents 9=9S. Let s, which depends on S and t, be the arclength parameter for P; we then have s (S; t) = (S; t). Let Â(S; t) be the angle that the tangent E to P makes with the axis e 1 measured in the counterclockwise sense. Then, we have (S; t) = cos Â(S; t)e 1 + sin Â(S; t)e 2 :

(10)

We denote by (S; t) = k × (S; t) the leftward unit normal to P, where k = e 1 × e 2 is the unit normal to the plane of and ˆ . Then, (S; t) = Ä(S; t) (S; t); Ä(S; t) ≡ Â (S; t);

with the curvature of P being -1 Ä.

Constitutive laws

Let W (A) be the strain energy of the bulk material per unit area of . Since the material is assumed to be incompressible, with the constraint

det A = 1 in (12)
holding, the (plane) nominal stress tensor S is given by

S = 9W 9A -pA -1 ; (13)
p being a Lagrange multiplier arising from constraint (12). There will also be a stress normal to the considered plane necessary to maintain the plane strain condition, but we do not require an explicit expression for it here.

For an isotropic material the corresponding Cauchy stress A = AS has principal components

1 = 1 9W 9 1 -p; 2 = 2 9W 9 2 -p; (14)
where W ( 1 ; 2 ) = W ( 2 ; 1 ), with 1 2 = 1. Following Steigmann and Ogden (1997a,b), we take the strain energy of the coating, per unit length of P, to depend only on and Ä. We write this as U ( ; Ä), and use the notations deÿned by

F = 9U 9 ≡ U ; M = 9U 9Ä ≡ U Ä ; (15)
which represent the tangential force and moment on P, respectively. The force at a point on P consists of both the tangential component F and a normal component and may be written as

F = F + G ; (16)
as described in (I), where G, the transverse shear force on the ÿlm, is not determined by a constitutive equation (it is a Lagrange multiplier). In this model the response of the coating is determined solely by the geometry of the surface (in this plane strain situation actually the curve corresponding to the coating). An alternative approach, which we do not consider here, would be to assign an independent director ÿeld to the surface which would not be directly related to the ÿlm geometry but would require an additional constitutive law for G.

In Steigmann and Ogden (1997b) it was proved that a necessary condition for a (static) deformed conÿguration of the ÿlm-substrate structure to be an energy minimizer is that the Hessian matrix

U U Ä U Ä U ÄÄ (17)
is positive semideÿnite. Moreover, it was established in (I) that for a membrane ÿlm (with no bending sti ness) a necessary condition is that the force F is non-compressive, that is

U ¿ 0: (18)

Equations of motion

In the absence of body forces the equation of motion of the bulk material in may be written in the standard form Div S = x ;tt ;

( 19) where is the mass per unit area of , Div is the divergence operator in and ;t signiÿes the material time derivative. Boundary conditions were discussed in (I) for static problems, and such conditions can be carried over to the present context as needed, but here we shall not consider conditions on 9 other than on the part P, which are provided by the following derivation of the equations of motion of P.

First, we discuss the balance of linear momentum for the ÿlm. Consider an arbitrary section [S 1 ; S 2 ] ⊂ P, where S measures arclength along P and S 2 ¿ S 1 . Let F 1 and F 2 , respectively be the forces applied to the ends S 1 and S 2 of the section, where F 2 = F(S 2 ; t) and F(S; t), given by Eq. ( 16), is the force exerted by the material in (S; S 2 ] on that in [S 1 ; S]. The balance of linear momentum for the ÿlm is assumed to have the form

F 1 + F 2 + B = d dt S2 S1 0 r ;t dS; ( 20
)
where 0 is the mass density of the ÿlm per unit reference length and

B = - S2 S1 S T N dS; ( 21
)
the force exerted by the substrate on the ÿlm, is equal and opposite to that transmitted to the substrate by the ÿlm.

If we assume that the integrands in Eqs. ( 20) and ( 21) are bounded, that F(S; t) is a continuous function of S, and then let the length of the interval approach zero, we obtain F(S 1 ; t) + F 1 = 0.

This allows us to write Eq. ( 20) as

S2 S1 (F -S T N -0 r ;tt ) dS = 0; (22)
and the arbitrariness of the interval then leads to the local equation of motion

F = S T N + 0 r ;tt : (23)
We also introduce a moment-of-momentum balance for the ÿlm. This is assumed to have the form

(M 1 + M 2 )k + r 1 × F 1 + r 2 × F 2 + S2 S1 r × (-S T N) dS = d dt S2 S1 0 r × r ;t dS + k d dt S2 S1 IÂ ;t dS; ( 24
)
where M 1 and M 2 are the moments applied to the ends of the interval, k = × is the ÿxed unit normal to the plane of motion, and I (S) is the mass moment of inertia per unit reference length. The ÿnal integral is the rotatory inertia term and is discussed below. We assume that M 2 = M (S 2 ; t), where M (S; t)k is the moment exerted by the material in (S; S 2 ] on that in [S 1 ; S]. This is the value at (S; t) of constitutive function (15) 2 . By invoking appropriate boundedness and continuity assumptions, we obtain M (S 1 ; t) + M 1 = 0, and with our previous results the leading terms on the left-hand side of Eq. ( 24) may then be combined into the expression

k S2 S1 M dS + [r × F] S2 S1 : (25)
In Eq. ( 25), the second term may be written as the integral of r × F + r × F in which r and F are replaced by Eqs. ( 8) and ( 23), respectively, and F is given by Eq. ( 16).

The resulting form of the global balance law is equivalent to the local equation

M + G = IÂ ;tt ; (26)
which e ectively determines the otherwise arbitrary function G(S; t) in terms of the motion. Thus, the equations governing the motion of the ÿlm are Eqs. ( 23) and ( 26), which are to be taken in conjunction with Eqs. ( 15) and ( 16). Essentially, Eq. ( 26) provides an expression for G while Eq. ( 23) couples the ÿlm behaviour with that of the bulk material and can be regarded as a boundary condition for the solution of Eq. ( 19) with Eqs. ( 2), ( 12) and ( 13) in .

We close this subsection with some remarks on the form of the rotatory inertia term in Eq. ( 24). This form is essentially that proposed by [START_REF] Tadjbakhsh | The variational theory of the plane motion of the extensible elastica[END_REF] in the context of a theory for the planar motions of an extensible rod, which also serves as a model for the present study of plane motions of a thin ÿlm (or plate). Tadjbakhsh regarded the rod as a one-dimensional continuum and speciÿed a constitutive framework for the total mechanical energy density consisting of kinetic energy and strain energy jointly. The model equations were then obtained by a formal application of Hamilton's principle. The constitutive hypotheses for the energy density were tailored a posteriori so that the resulting model had the same structure as in the elementary dynamical theory of rods. Precisely the same form of the theory, again for plane motions of extensible rods, was discussed by Antman (1995, Chapter 4), who restricted the actual two-dimensional motion of a thin body in accordance with the Bernoulli-Euler hypothesis. In these developments, as in this work, the inertia coe cient I is independent of the motion.

An alternative viewpoint in which the inertia coe cients are taken to depend on the strain was advanced recently by [START_REF] Hilgers | Kinetic energy of highly elastic membranes[END_REF] and [START_REF] Hilgers | Dynamics of elastic sheets with bending sti ness[END_REF]. The associated equations of motion, again obtained from Hamilton's principle, involve the strain-dependent part of the inertia in the form of a distributed couple. A similar set of dynamical equations may be obtained by specializing the Cosserat theory of directed surfaces so that the director ÿeld is constrained to be aligned with the surface normal but permitted to change length, this being intended to account for the thinning or thickening that accompanies surface strain (Naghdi, 1982, Section 6). In yet another treatment of the dynamics of cylindrical motions of shells, Libai and Simmonds (1998, Chapter 4) require the exact expression for the through-thickness integral of the moment of momentum of a two-dimensional body to be expressible in the form I!, where I is the (ÿxed) mass moment of inertia as in the present theory and ! is a spin that depends on the actual position and velocity ÿelds in the considered thin body. This spin is not in general equal to  ;t . In the associated mechanical power identity the moment on the cross-section is power-conjugate to an angle ÿ obtained by integrating ! with respect to time. While such a formulation has the advantage of delivering an exact consequence of the moment-of-momentum balance for the thin body, it su ers from the drawback that the angle ÿ cannot be related to the geometry of the curve used to model the body in the absence of information about the actual position and velocity ÿelds, information that is usually unavailable.

Regardless of the particular formulation chosen, there remains an open question as to the relationship, if any, of the dynamics predicted by such theories to those predicted by conventional non-linear elasticity theory. Comparisons made in the context of classical linear theory without initial stress or rotatory inertia (see, for example, [START_REF] Bogy | Re ection coe cient for plane waves in a uid incident on a layered elastic half-space[END_REF] indicate that solutions of the present model are in accord with those of elasticity theory if the characteristic wavelength of the motion greatly exceeds the ÿlm thickness. Thus, we expect our results to be quantitatively meaningful in this limit, even if the rotatory inertia is suppressed. Moreover, the problems we consider are based on equations of motion linearized about a ÿxed conÿguration in which the associated rotatory inertia term involves a constant mass moment of inertia, equal to I in the present formulation or a function of the underlying stretch of the ÿlm in the case of a strain-dependent inertia coe cient, the di erence being equivalent to an adjustment of the ÿlm thickness that has no e ect on the qualitative behaviour of the solutions.

Mechanical power and energy ux

The equations derived in Section 2.3 lead to a mechanical power balance which in turn may be used to obtain expressions for the uxes of energy in the ÿlm and substrate. This is constructed by scalar multiplying Eq. ( 23) by r ;t , Eq. ( 26) by  ;t and integrating the resulting equalities over a part, say Q, of the ÿlm. To this expression we add the integral of the scalar product of Eq. ( 19) with x ;t over an arbitrary simply connected region of the substrate, and then use Green's theorem and integration by parts to reduce the result to

[F • r ;t ] 9Q + [MÂ ;t ] 9Q - 9 E • N dS = [K S ;t + tr(SA ;t )] dA + Q (K F ;t + F • r ;t + MÂ ;t -GÂ ;t + Sr ;t • N) dS: (27)
Here, the notation [ • ] 9Q is used to denote the di erence between the values of the enclosed quantity at the endpoints of Q, while K S and K F , respectively are the kinetic energy densities of the substrate and ÿlm, deÿned by

2K S = |x ;t | 2 ; 2K F = 0 |r ;t | 2 + I (Â ;t ) 2 ; (28)
and

E = -Sx ;t ( 29
)
is the energy ux in the substrate. To the second integrand on the right-hand side of Eq. ( 27) we apply Eq. ( 16) and obtain

F • r ;t + MÂ ;t -GÂ ;t = F • r ;t + G( • r ;t -Â ;t ) + MÂ ;t ; (30)
wherein the term in parentheses vanishes identically by virtue of the relation

r ;t = ;t + Â ;t ; (31)
which follows by di erentiating Eq. ( 8). Then, on recalling Eqs. ( 11) and ( 15), we see that the right-hand side of Eq. ( 30) reduces to F ;t + MÂ ;t = U ;t . We also have tr(SA ;t ) = W ;t , and, if is chosen such that Q ⊂ 9 we may use x ;t | Q = r ;t to write Eq. ( 27) in the form

[F • r ;t ] 9Q + [MÂ ;t ] 9Q - 9 \Q E • N dS = E S ;t dA + Q E F ;t dS; (32)
where

E S = W + K S ; E F = U + K F (33)
are the mechanical energy densities of the substrate and ÿlm, respectively. This is the mechanical power identity for coupled ÿlm=substrate motions.

The conventional power identity for the substrate material follows from Eq. ( 19) by taking the region to be contained entirely within its interior. This yields

E S ;t dA + 9 E • N dS = 0; ( 34
)
which is equivalent to the local conservation law

E S ;t + Div E = 0: (35)
A similar balance law for the ÿlm may be derived directly from the di erential equations that hold on Q, or, alternatively, by using Eq. ( 35) in Eq. ( 32) to obtain

Q E F ;t dS = Q E • N dS + [F • r ;t ] 9Q + [MÂ ;t ] 9Q : (36)
This in turn is equivalent to the local equation

E F ;t + E = E • N; (37)
where

E = -F • r ;t -MÂ ;t ( 38
)
is the energy ux in the ÿlm. From Eq. ( 37) it is evident that the substrate supplies energy to the ÿlm through the ux E • N.

Incremental motions

Henceforth, we shall be concerned with a small incremental motion superimposed on an initial quasi-static ÿnite deformation. The equations governing this initial deformation are given by Eqs. ( 2), ( 12), ( 13) and ( 19) for the bulk material with the time dependence omitted and by Eqs. ( 15), ( 16), ( 25) and ( 26) for the coating. Let a superimposed dot denote an increment in the quantity concerned. In particular, ẋ, Ṡ; ṗ are increments in x, S and p, respectively. The incremental version of the equation of motion ( 19) is then Div Ṡ = ẋ;tt :

(39) Let u(x; t) denote the incremental displacement, identiÿed with ẋ(X; t) through the change of variable x = E(X), which deÿnes the initial ÿnite deformation. With (x; t) regarded as independent variables, Eq. ( 39) may be expressed in the form div = u ;tt ;

where div is the divergence operator in ˆ and = A Ṡ. The (linearized) incremental form of the constitutive law (13) may be written as

= A 0 + p -ṗI; ( 41
)
where I is the (two-dimensional) identity tensor, =grad u, with grad being the gradient operator in ˆ , is the displacement gradient having components

ij = u i; j ≡ 9u i 9x j (42)
and A 0 is the (fourth-order) tensor of instantaneous elastic moduli. Explicit expressions for the components of A 0 are given in, for example, [START_REF] Ogden | Non-linear Elastic Deformations[END_REF] and are not listed here. Note, however, that A 0 and p depend on the form of the initial ÿnite deformation.

The incremental form of incompressibility condition (12), duly linearized, is

div u = 0: (43)
On taking the increments of Eqs. ( 22) and ( 24) and updating from the variable S to s, noting that = ds=dS, we obtain Ḟ (s; t) = T n + -1 0 w ;tt (s; t) (44) and Ṁ (s; t)

+ -1 ˙ G + Ġ = -1 I Â;tt (s; t); (45)
where the prime now signiÿes partial di erentiation with respect to s, w = u for u evaluated on P, and n is the rightward unit normal to 9 ˆ , use having been made of Nanson's formula in the form A T n = N. It is straightforward to verify that the form of the inertia term in the last equation is appropriate whether or not I is strain dependent due to the fact that the linearization is carried out with respect to a ÿxed conÿguration of the ÿlm. In Eqs. ( 44) and ( 45) we require the incremental counterparts of Eqs. ( 15) and ( 16), namely

Ḟ = U ˙ + U Ä Ä; Ṁ = U Ä ˙ + U ÄÄ Ä (46)
and

Ḟ = Ḟ + Ġ + F ˙ + G ˙ ; (47)
which, in turn, require the expressions

-1 ˙ = • w ; Ä = • w -Ä • w ; (48) ˙ = ( • w ) ; ˙ = -( • w ) ; (49)
these being derived by making use of the incremental versions of Eqs. ( 8), ( 10) and (11), as described in (I). We also note that

 = • w ; w = -1 ˙ + ( • w ) : (50)
4. Application to a coated half-plane

We now consider a half-plane subject to a pure homogeneous strain with stretches 1 ; 2 satisfying Eq. ( 6), and we take the coordinate axes to coincide with the principal axes of strain. In the deformed conÿguration the half-plane corresponds to x 2 ¡ 0 and the surface coating is on the boundary x 2 = 0. The coating stretch is then equal to the stretch 1 , and hence, by Eq. ( 6), we have 2 = -1 .

We may therefore regard W as a function of deÿned through Ŵ ( ) = W ( ; -1 );

(51) from which, with the help of Eq. ( 14), we deduce that

1 -2 = Ŵ ( ): (52)
For the superimposed incremental motion the incompressibility condition (43) enables the components u 1 ; u 2 of the displacement to be expressed in the form

u 1 = ; 2 ; u 2 = -; 1 ; (53)
where (x 1 ; x 2 ; t) is a scalar function.

Following, for example, [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF] we may use Eq. ( 53) together with Eqs. ( 41) and (42) in Eq. ( 40) to eliminate ṗ and obtain an equation for for the region x 2 ¡ 0. This is

; 1111 + 2ÿ ; 1122 + ; 2222 = ( ; 11tt + ; 22tt ); (54)
where the material parameters ; ÿ; (constants here) are given by

= 4 = 5 Ŵ ( ) 4 -1 ; 2ÿ + 2 = 2 Ŵ ( ): (55)
We note that the strong ellipticity inequalities in this case require ¿ 0; ÿ¿ -2 ;

(56) as in [START_REF] Ogden | Non-linear Elastic Deformations[END_REF], for example. Since, in its deformed conÿguration, the ÿlm on x 2 = 0 is straight, we have Ä = 0. Also, U ( ; Ä) is constant with respect to arclength so that M = 0, and, from Eq. ( 26) specialized to the static case, we have G = 0, while F = U ( ; 0) with = -e 1 , and = -e 2 . From Eq. ( 23) we also deduce that 2 = 0, so that Eq. ( 52) is modiÿed accordingly. We note, however, that Eq. ( 23) may be modiÿed to accommodate non-zero 2 if required. From Eqs. ( 46) and ( 47) we then have

Ḟ = (U ˙ + U Ä Ä) + U ˙ + Ġ ; (57)
with the derivatives of U evaluated for Ä = 0, together with ˙ = -w 1 ; Ä = -w 2 ; ˙ = w 2 e 2 ;

(58) and hence

Ḟ = (U w 1 + U Ä w 2 )e 1 + (U w 2 -Ġ)e 2 : (59)
Eq. ( 44) now yields the two equations

U w 1 + U Ä w 2 = 21 + -1 0 w 1;tt ; (60)
U w 2 -Ġ = 22 + -1 0 w 2;tt ;

(61) while, from Eq. ( 45), with the help of Eqs. (46) 2 , ( 58) and ( 50), we obtain Ġ = U Ä w 1 + U ÄÄ w 2 --1 Iw 2;tt ;

I being a constant. Substitution of this into Eq. ( 61) then yields

U w 2 -U Ä w 1 -U ÄÄ w 2 = 22 + -1 0 w 2;tt --1 Iw 2;tt : (62)
Note that Eq. ( 60), appropriately specialized and without the U Ä term, is similar to an equation used by [START_REF] Murdoch | The propagation of surface waves in bodies with material boundaries[END_REF] in the context of waves propagating on a linearly isotropic elastic half-space with a residual surface tension. Eq. ( 62), on the other hand, has no counterpart in the Murdoch theory. In isolation from the half-plane, Eq. ( 62) is the equation of motion of a thin rod accounting for rotatory inertia [START_REF] Antman | Nonlinear Problems of Elasticity[END_REF]. In the theory describing waves propagating in a thin layer on a half-space in the classical context it is sometimes assumed that the rotatory inertia term (involving I ) is negligible compared with the term in 0 , at least in the low-frequency regime (see, for example, [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF]. As we show in Section 5, however, it is in general necessary to include the term in I . Without the I term and with U = U Ä = 0, a version of Eq. ( 62) appropriate for the classical theory was given in [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF], as was the counterpart of Eq. ( 60).

From the specialization of Eq. ( 41) to the present circumstances (in which 2 = 0), we have 21 = ( ; 22 -; 11 );

(63) and, after di erentiation of 22 with respect to x 1 followed by use of the ÿrst component of Eq. ( 40) to eliminate ṗ,

-22; 1 = (2ÿ + ) ; 112 + ; 222 -; 2tt : (64)
Details of the derivation of Eq. ( 64) may be found in, for example, [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF].

By taking s = -x 1 we may also write respectively, each of which is evaluated on x 2 = 0 and for Ä = 0. Eqs. ( 66) and ( 67) provide boundary conditions on x 2 = 0 for the solution of Eq. ( 54).

w 1 (s; t) = ; 2 (x 1 ; 0; t); w 2 (s; t) = -; 1 (x 1 ; 0; t); (65

Surface waves

Surface waves on a pre-stressed half-space of incompressible isotropic elastic material without a surface coating were examined by [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF]. Here we evaluate the e ect that the surface coating has on such waves.

We seek solutions of the form = Ae ik(x1-ct)+skx2 ;

(68) where c (¿ 0) is the wave speed, k (¿ 0) is the wave number while s is to have positive real part so that → 0 as x 2 → -∞ and is determined by substitution of Eq. ( 68) into Eq. ( 54). We distinguish between this s and the arclength parameter s used for P used earlier, which does not feature in the remainder of the paper. Let s 1 ; s 2 be the two solutions with positive real part. Then, the general solution of Eq. ( 54) of the given form with the desired properties is written as = (Ae s1kx2 + Be s2kx2 )e ik(x1-ct) ;

(69) as in [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF], and s 1 ; s 2 are such that

s 2 1 + s 2 2 = (2ÿ -c 2 )= ; s 2 1 s 2 2 = ( -c 2 )= : (70)
We recall from Dowaikh and Ogden (1990) that 0 6 c 2 6 ;

but, as discussed in [START_REF] Ogden | On interfacial waves in pre-stressed layered incompressible elastic solids[END_REF], the upper limit is replaced by a lower limit if 2ÿ ¡ in order to ensure that the decay condition is satisÿed (this is made explicit below). Substitution of Eq. ( 69) into boundary conditions Eq. (66) and Eq. ( 67) for x 2 = 0 yields (a + bs 1 -s 2 1 )A + (a + bs 2 -s 2 2 )B = 0;

(u + vs 1 -s 3 1 )A + (u + vs 2 -s 3 2 )B = 0; (72)
where the parameters a; b; u; v are deÿned by

a = (k 2 U Ä -)= ; b = k( -1 0 c 2 -U )= ; v = (2ÿ -a -c 2 )= ; u = ( k 3 U ÄÄ + kU --1 Ik 3 c 2 -k -1 0 c 2 )= : (73)
Note that if Eq. ( 23) is modiÿed to allow for a normal stress on the boundary x 2 = 0 then 2 = 0 and 2 enters the coe cients (73) only through the parameter a and as an additional term 2 = in a. Thus, the e ect of such a normal stress is similar to the e ect of the term k 2 U Ä in a and could therefore be accommodated by choosing U so that U Ä does not vanish.

For a non-trivial solution for (A; B) to exist the determinant of coe cients in Eq. ( 72) must vanish. After removal of the factor (s 1 -s 2 ), whose vanishing is accounted for as a special case of the remaining equation, we obtain bu -av -u(s 1 +s 2 )+a(s 2 1 +s 2 2 +s 1 s 2 )-vs 1 s 2 + bs 1 s 2 (s 1 + s 2 ) -s 2 1 s 2 2 =0: (74)

We make Eq. ( 74) explicit by introducing the notation Á deÿned by

Á = s 1 s 2 = ( -c 2 )= ; (75) so that c 2 = -Á 2 (76)
and hence, using Eq. ( 70),

s 2 1 + s 2 2 = Á 2 + (2ÿ -)= ; s 1 + s 2 = [Á 2 + 2Á + (2ÿ -)= ] 1=2 : (77)
Corresponding to Eq. ( 71) we have 0 6 Á 6 = = 2 ; (78) but if 2ÿ ¡ the left-hand side limit is replaced by Á L 6 Á, where Á L ¿ 0 is the positive solution of

Á 2 L + 2Á L + (2ÿ -)= = 0; (79)
as discussed in [START_REF] Ogden | On interfacial waves in pre-stressed layered incompressible elastic solids[END_REF] and [START_REF] Chadwick | Interfacial waves in pre-strained isotropic elastic media[END_REF]. Note that a in Eq. ( 73) is independent of c, while b; u; v, when expressed in terms of Á, may be written

b = b -k Á 2 ; u = u + rÁ 2 ; v = v + Á 2 ; ( 80
) where b = k( -U )= ; u = (k 3 U ÄÄ + kU )= -4 r; r = -1 Ik 3 = + k ; v = (2ÿ -)= -a (81)
and is deÿned as

= -1 0 = : (82)
Except for k and the parameters deÿned above are dimensionless. Using Eqs. ( 75) and ( 77), and the notation deÿned in Eqs. ( 80) -( 82) we may re-express Eq. ( 74) as an equation for Á in the form

(k Á 3 + rÁ 2 -bÁ + u)[Á 2 + 2Á + (2ÿ -)= ] 1=2 + k rÁ 4 + Á 3 + (k u -r b + 1)Á 2 + ( v -a)Á -a 2 -bu = 0:
(83) This is the secular equation which determines Á, and hence the wave speed through Eq. ( 76), in terms of the various material parameters of the bulk and ÿlm material, the deformation and the wave number k. Note that at this point no specialization of Ŵ ( ) or U ( ; Ä) has been adopted. When there is no surface coating we have = b = r = u = 0 and a = -1, in which case Eq. ( 83) reduces to the cubic

Á 3 + Á 2 + (2ÿ -+ 2 )Á= -1 = 0; (84)
which is a special case (corresponding to 2 = 0) of an equation given by [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF].

Eq. ( 83) simpliÿes considerably for materials with 2ÿ = + , which, for the present (plane strain) situation, forces the strain energy to be of the neo-Hookean form, so that

Ŵ ( ) = 1 2 ( 2 + -2 -2); ( 85
)
where is the shear modulus in the natural conÿguration. Eq. ( 83) then becomes a quartic for Á, namely

k (r + 1)Á 4 + (k + r + 1)Á 3 + [k u -(r + 1)( b -1)]Á 2 + (u -b + 1 -2a)Á -u( b -1) -a 2 = 0; ( 86
)
in contrast to cubic equation ( 84).

For an uncoated half-plane (with 2 = 0) Á satisÿes cubic equation ( 84) and the material constants enter in the combination (2ÿ + 2 -)= , which also features in Eq. ( 83) through the term v -a. For the coated half-space, by contrast, Eq. ( 83) is not in general a polynomial for Á, but for materials with 2ÿ = + , in particular, Eq. ( 86) is a quartic for Á. While for the uncoated case waves are non-dispersive, the dependence of Eqs. ( 83) and ( 86) on the wave number k ensures that waves on a coated half-plane are dispersive.

In order to illustrate the e ect that the coating has on the propagation of surface waves we focus on Eq. ( 86), which captures the main features involved. Speciÿcally, in Section 7, we solve Eq. ( 86) numerically for Á for given values of the constants and hence obtain the wave speed in the form (for a neo-Hookean material)

≡ c 2 = = 2 --2 Á 2 ;
(87) which, recalling Eq. ( 76), deÿnes the notation .

Material parameters and non-dimensionalization

Residual stress and moment

Industrial thin-ÿlm deposition processes invariably leave residual stresses and moments in the ÿlm. It is therefore appropriate to incorporate both residual stress and moment in the constitutive description of the ÿlm material. They can be accounted for by using the quadratic energy function

U = F 0 ( -1) + M 0 Ä + 1 2 ( -1) 2 + CÄ( -1) + 1 2 Ä 2 ; ( 88
)
where ; ; F 0 ; M 0 and C are constants, and hence

F = F 0 + ( -1) + CÄ; M = M 0 + C( -1) + Ä; (89)
with U = ; U Ä = C; U ÄÄ = . The energy Eq. ( 88) is a generalization of the simple model used in (I) that did not include the terms in F 0 ; M 0 and C.

For the considered problem Ä = 0 and F 0 ; M 0 respectively are the residual stress and couple in the undeformed conÿguration. Since Eq. ( 89) are constants, the undeformed conÿguration is equilibrated with S = 0 in the bulk material. Note that M 0 does not feature in the incremental equations, but that the moment M in the deformed conÿguration is detected in these equations through C. The term in C is therefore included to re ect moment e ects in the incremental equations. In accordance with Eq. ( 17), we must have

¿ 0; ¿ 0; ¿ C 2 : (90) L.B.
Freund has shown us a simple argument, based on a ÿlm of ÿnite thickness h, which may be used to motivate expression Eq. ( 88) for the strain-energy function. Let the stretch and curvature of the ÿlm at its centre-line be and Ä, respectively. Let and Ä be the corresponding quantities at the interface between the ÿlm and substrate. In general, these will not be equal to and Ä. For example, if the ÿlm undergoes pure bending, then, with b = -h=2,

-1 = -1 + Äb; Ä = Ä;
(91) provided that |Äb|1.

Let F and M be the axial force and bending moment of the ÿlm at its centre-line. Then, if F and M are the corresponding axial force and moment acting at the ÿlm-substrate interface, we have

F = F; M = M + Fb: (92)
Suppose the constitutive equations for the ÿlm are F = A( -1) and M = B Ä, where A and B are constants. Then, F( ; Ä) d + M ( ; Ä) dÄ = dU , where

U ( ; Ä) = 1 2 A( -1) 2 + Ab( -1)Ä + 1 2 (B + b 2 A)Ä 2 (93)
is the strain-energy function for the interface. This is a special case of Eq. ( 88) with

= A; = B + b 2 A; C = Ab: (94)
It is noteworthy that Eq. ( 90) then yields the restrictions

A ¿ 0; B¿ 0 (95)
which are independent of h. We then have C = -Ah=2, which is negative.

Returning to the present model, in (I) it was also noted that the ratio ( = ) 1=2 deÿnes a local length scale for the problem, and this was used as a basis for a non-dimensionalization scheme. We now make this explicit by writing this length scale as H , which is then deÿned by

H = ( = ) 1=2 : (96)
We use this to deÿne dimensionless quantities

k = k H ; ˜ = = H = = H 3 : (97)
The length scale H is deÿned through the elastic moduli and is not explicitly a physical dimension which requires measurement. The moduli themselves should be determined from wave propagation experiments based on the theory. The use of H , however, enables us to make contact with classical engineering plate theory. Comparison of the appropriate specialization of Eq. ( 88) with the corresponding expression in (plane strain) engineering plate theory, as described, for example, in [START_REF] Timoshenko | Theory of Elasticity, 2nd Edition[END_REF], would suggest the identiÿcations

= EH=(1 -2 ); = EH 3 =12(1 -2 ); (98)
where E and are, respectively, Young's modulus and Poisson's ratio for a plate and H is the plate thickness. A comparison of Eq. ( 96) with the same ratio obtained from Eq. ( 98) then gives H = 2 √ 3 H . It is also useful for deÿning other physical quantities associated with the coating. For example, we may use the standard connection E = 2 c (1 + ), with c analogous to the shear modulus of the coating material. It then follows that the dimensionless material constant ˜ , deÿned in Eq. ( 97), may be given in the form

˜ = 4 √ 3 1 - c : (99)
Thus, the correspondence between our model and engineering plate theory makes explicit the interpretation, discussed in (I), of ˜ as a measure of the shear sti ness of the ÿlm material relative to that of the bulk material.

In addition to Eq. ( 97) two further material constants appear in Eq. ( 86) when specialized in respect of Eq. ( 88), namely , deÿned by Eq. ( 82), and I , deÿned by

I = -1 Ik 3 = : (100)
Let 0 = c H , where c (again using the plate theory correspondence) is interpreted as the mass (per unit reference area) of the coating. Then

k = 2 √ 3 -1 k c = ≡ m -1 k; (101)
where we have introduced the notation

m = 2 √ 3 c = : (102)
Now, since I is deÿned per unit reference length, in plate theory it may be taken to have the form

I = 0 h 2 =12 = c h 2 H=12; (103)
obtained from that for a rod of length h about an axis through its centre and perpendicular to its length. Again using the plate theory correspondence it is convenient to use this form of I for the model considered here. Moreover, we assume the ÿlm material to be incompressible (so that = 1=2). Then, h = -1 H and hence, using Eqs. ( 96), (97) 1 , (100) and ( 103),

I = m k3 -3 : (104)
This is consistent with our assumption that I is constant since the underlying deformation is homogeneous (and hence is constant) and any associated increment in I does not contribute to the linearized equations (it features only in second-order terms).

It also avoids the need to treat I as an additional independent parameter.

Thus, the coe cients in Eq. ( 86) may now be simpliÿed according to

a = -1; b = (m -˜ ) k 3 ; r = (1 + -2 k2 )m k -1 ; u = ( k2 + -1) ˜ k 2 -4 r: (105)
Note that I is negligible if and only if k2 1, which corresponds to the low-frequency (or long wavelength) limit. As mentioned in Section 4, it is sometimes assumed in the classical theory that I can be neglected, so that rotatory inertia is regarded as unimportant compared with the linear inertia and bending sti ness terms (for example, [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF]. Clearly, this is not the case in general since bending sti ness (proportional to ˜ ) and rotatory inertia (proportional to m) each contribute a term of order k3 to u. Thus, we retain the rotatory inertia term here and note that the relative importance of the terms depends on the ratio ˜ =m.

When, however, k is small then it is straightforward to obtain a solution of Eq. ( 86) for Á expanded in powers of k. The zero-order solution, say Á 0 , satisÿes the equation

Á 3 0 + Á 2 0 + 3Á 0 -1 = 0; (106)
which is the secular equation for an uncoated half-plane of neo-Hookean material [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF]. This is the Rayleigh wave limit, and hence one branch of the solutions of Eq. ( 86) must pass through the Rayleigh wave speed value at k = 0. Eq. ( 106) has a unique positive solution, given approximately by Á 0 = 0:2956, and the associated wave speed is given by setting Á = Á 0 in Eq. ( 87). The ÿrst-order solution for Á depends on ; m and ˜ but it is a lengthy expression and it is not instructive to give it explicitly here.

In respect of Eq. ( 88) Eq. ( 86) takes the form

m k( 3 + 2 m k + m k3 )Á 4 + ( 3 + 2 2 m k + m k3 )Á 3 + [ 3 + ( 2 -6 + k2 )m k + ˜ k 6 + m ˜ k2 3 ( 2 k2 + k2 + 2 2 -) -2m 2 k2 3 ( 2 + k2 ) + 4 k2 m F0 ]Á 2 + 4 [ ˜ k 2 ( k2 + 2 -1) -m k (2 2 + k2 ) + 3 + 2 k F0 -2 3 k2 C]Á -[ ˜ ( k2 + -1) -m( 2 + k2 ) + F0 ][(m -˜ ) k 3 -1] k 5 -4 ( 3 k2 C -1) 2 = 0; ( 107
)
where the dimensionless constants F0 and C are deÿned by

F0 = F 0 = H ; C = C= H 2 : (108)
In Section 7 we shall present numerical solutions of Eq. ( 107) for using connection Eq. ( 87) with k as the independent variable and for speciÿc choices of the parameters , m, ˜ , C and F0 . Note that it follows from Eqs. ( 97), ( 90) and ( 108) that

| C| 6 ˜ : (109)
We recall that for the example considered earlier in this section C is negative, but in general this need not be the case.

When Eq. ( 107) is specialized by setting = 1; F0 = 0; C = 0 it may be compared with the corresponding equation obtained by [START_REF] Tiersten | Elastic surface waves guided by thin ÿlms[END_REF], although the latter is for a compressible half-space. Tiersten's equation ( 58) does not include the cubic and quartic terms in k associated with rotatory inertia, but it does include other cubic and quartic powers of k that are associated with bending sti ness. This omission has a signiÿcant e ect on the results even at relatively small values of k.

Specialization to a membrane ÿlm

When specialized to the membrane case we have =0, C = 0 and I = 0, but 0 and are taken to be ÿnite and the previous non-dimensionalization based on H , which now vanishes, is no longer appropriate. This limit corresponds to c H and c H tending to non-zero ÿnite values as H → 0. In terms of the wave number k, as deÿned by Eq. ( 82), and the notations deÿned by = = ; F 0 = F 0 = ;

(110)

Eq. ( 107) simpliÿes to

k (1 + k )Á 4 + (1 + 2k )Á 3 + [1 + (1 -4 )k + 3 k + 2 (2 -1)k 2 -2 4 k 2 2 + 2 k 2 F 0 ]Á 2 + [ 2 (2 -1)k -2 4 k + 3 + 2 k F 0 ]Á -2 k[( -1) -2 + F 0 ][ 3 k( -) -1] -1 = 0: (111)
In respect of Eq. ( 111) we seek solutions for with k (instead of k) as the independent variable and for chosen values of the reduced number of parameters , with and F 0 instead of ˜ and F0 . The parameters ; ; F 0 may be made dimensionless by multiplication by k but this makes interpretation of the asymptotic results for large k unclear. Hence we leave them in dimensional form. Note that, in accordance with Eq. ( 18), we must have

F 0 + ( -1) ¿ 0: (112)
In the undeformed conÿguration with = 1 Eq. ( 111) simpliÿes further to

k (k + 1)Á 4 + (2k + 1)Á 3 + [k 2 F 0 + (k + 1)k -2k 2 2 + 1]Á 2 + (k F 0 + k -2k + 3)Á + k 2 F 0 -(k -1)k F 0 -k 2 + k 2 2 -k -1 = 0: (113)
It can be shown that this is equivalent to the incompressible limit of a result obtained by [START_REF] Murdoch | The propagation of surface waves in bodies with material boundaries[END_REF] for compressible materials for this special case by identifying F 0 ; and Á, respectively, with the notations ; and S used by Murdoch.

Asymptotic results

As already noted in Section 5, in the limit k → 0 there is a unique solution of Eq. ( 107) corresponding to the Rayleigh wave speed. Away from this limit the character of the Rayleigh wave branch and of the other solutions is strongly in uenced by the behaviour of Eq. ( 107) at large k. We therefore now study the limiting case k → ∞, which requires that the coe cient of the highest power of k in Eq. ( 107) must vanish. This yields a quadratic in which has solutions given by 2m = ( 2 + 1) 2 ˜ ± 2 [( -1) 2 ˜ 2 + 4 2 C2 ] 1=2 :

(114)

If, with reference to Eqs. ( 78) and ( 87), these two solutions both lie in the interval [0; 2 ] then it follows that two branches of the solution to Eq. ( 107) have large wave number (or high-frequency) asymptotes.

The larger solution lies in the required interval if and only if

2 ˜ 6 m; ˜ 6 m; 2 C2 6 ( 2 ˜ -m)( ˜ -m): (115)
If Eq. ( 115) holds then the smaller solution is also in the required interval if

| C| 6 ˜ ; (116)
which is just Eq. ( 109). If Eq. ( 115) holds but Eq. ( 116) does not then there is only one asymptote. If Eq. ( 115) does not hold then for a single asymptote we must have, in addition to Eq. ( 116), either

( 2 + 1) ˜ 6 2m;

or

( 2 + 1) ˜ ¿ 2m; 2 C2 ¿ ( 2 ˜ -m)( ˜ -m): (117)
If Eq. ( 115) fails and either of Eq. ( 116) or Eq. ( 117) fails then there are no asymptotes. Note that inequalities Eq. ( 115)-( 117) are independent of the residual stress F0 and, further, depend only on the ratios ˜ =m and C=m.

For the special case in which = 1 ( C = 0), for example, Eq. ( 114) reduces to

m = ˜ ± C (118)
which, by Eq. ( 109), are both non-negative. There are two asymptotes if

˜ + | C| 6 m; (119) one if ˜ -| C| 6 m ¡ ˜ + | C|; (120)
and none if

m ¡ ˜ -| C|: (121)
For the situation of a ÿnite thickness layer on a half-space large wave number (high-frequency) asymptotes were discussed by [START_REF] Ogden | On interfacial waves in pre-stressed layered incompressible elastic solids[END_REF]. One asymptote in that case corresponds to the speed of a surface wave in the layer material and the other to an interfacial wave speed between two half-spaces consisting of the two materials. This interpretation is not applicable in the present circumstances.

Further information about the structure of the branches (other than the Rayleigh branch) is determined by considering where they cut the line Á = 0 ( = 2 ). The value of the wave speed at such a point is that of a shear wave speed in the half-space material. By setting Á = 0 in Eq. ( 107) we obtain a quartic for k, namely

[( 2 ˜ -m)( ˜ -m) -2 C2 ] 4 k4 + ( 2 ˜ -m) k3 + 2 [ 3 ( ˜ -m){ F0 + ˜ ( -1) -m } + 2 C] k2 + 2 [ F0 + ˜ ( -1) -m ] k -1 = 0: (122)
Note that, in contrast to the asymptotic results, this does depend on F0 .

For completeness we now note the specialization of Eq. ( 107) associated with bifurcation. This is obtained by setting = 0 (Á = 2 ) and again this yields a quartic in k, in this case

6 ( ˜ 2 - C2 ) k4 + 3 ( 2 + 1) ˜ k3 + 3 [ 2 ( -1) ˜ 2 + 2 ˜ F0 -2( 2 -1) C] k2 + 2 ( 2 + 1)[( 3 + -1) ˜ + F0 ] k + 6 + 4 + 3 2 -1 = 0: (123)
As noted in Section 4, when F0 and C are set to zero Eq. ( 123) agrees with Eq. (6.57) in (I), although the latter has to be expanded in terms of k to make this explicit.

From Eq. ( 123) we may deduce immediately that necessary conditions for the underlying conÿguration to be stable, i.e. for the left-hand side of Eq. ( 123) to be positive for all k ¿ 0, are The ÿrst of these is the condition for stability of an uncoated half-plane [START_REF] Dowaikh | On surface waves and deformations in a pre-stressed incompressible elastic solid[END_REF], while the second (in its non-strict form) appeared in Eq. ( 116) in connection with the enumeration of asymptotes. General necessary and su cient conditions may be written down by considering the detailed properties of the left-hand side of Eq. ( 123) but they are very complicated and it is not instructive to include them here.

As indicated above, the asymptotic behaviour has a strong in uence on that at smaller values of k. If situations in which bifurcation can occur are discounted, that is if dispersion curves are not allowed to cut the axis = 0, then analysis of the properties of Eq. ( 122) yields the following results: (a) if there are zero or two high-frequency asymptotes then there are either one or three values of k where Á = 0; (b) if there is just one asymptote then there are zero or two (or possibly four) points where Á = 0. In principle, the latter possibility in (b) may arise since the permissible ranges of parameter values admit coe cients in Eq. ( 122) with alternating signs. However, for the parameter values for which numerical results have been obtained this possibility has not been encountered. It may be possible to rule it out by more detailed consideration of the properties of the quartic in Eq. ( 122) but thus far attempts to do this have not proved successful. A selection of the other cases is illustrated in the numerical results shown in Section 7.

Results for a membrane corresponding to those discussed above may be deduced on noting that the two asymptotic values of are

= = ; = [ F 0 + ( -1) ]= ; (125)
the latter depending on the residual stress. The membrane equation corresponding to Eq. ( 122) is a quadratic in k, namely

5 ( -)[( -1) -2 + F 0 ]k 2 -2 [( -1) -2 + F 0 ]k + 1 = 0; (126)
and that corresponding to Eq. ( 123) is also a quadratic, in this case

5 [ F 0 + ( -1)]k 2 + 2 ( 2 + 1)[ F 0 + ( 3 + -1)]k + 6 + 4 + 3 2 -1 = 0; (127)
which is independent of . It follows from Eq. ( 127) that necessary and su cient conditions for stability of the underlying conÿguration with a membrane coating are (124) 1 together with the strict form of Eq. ( 112), namely

F 0 + ( -1) ¿ 0: (128)
On consideration of Eq. ( 126), it is easy to show that for a membrane the following two possibilities arise: (a) if zero or two high-frequency asymptotes exist then there is exactly one value of k where Á = 0; (b) if there is only one asymptote then there are zero or two points with Á = 0.

Numerical results and discussion

Results for = 1

Fig. 1 shows the e ect of a positive residual moment in the absence of residual stress, with the values of the parameters given by = 1; ˜ = 1 and, respectively in (a) -(f), m = 0:5; 0:9; 1; 1:5; 3; 5. The long-, medium-and short-dashed curves are for C = 0:5; 0:8 and 1.2, respectively. Note that for C = 1:2 inequality (124) 2 required for stability is violated. As expected from the discussion in Section 6, it can be seen that the presence of non-zero C a ects the number and disposition of the asymptotes. Results for negative values of C, which are not shown here, are very similar to those for positive C although there are some di erences of detail, particularly for small values of m. Note, however, that for the example constructed in Section 5.1 C is negative. If C = 0 the e ect of varying F0 is very similar to the e ect of applying a stretch , which is discussed below, so we do not show such results separately.

Results for = 1

In order to illustrate the e ect of extensional and compressive stretches in the substrate, we now show results, in Figs. 2 and 3 for = 1:4 and 0.7, respectively. In each case the same values of ˜ and m as in Fig. 1 are used. In Fig. 2 the upper limit for is 1.96 (compared with 1 in Fig. 1) so that generally a value of ¿ 1 has the e ect of increasing the wave speed. This is similar to the e ect of a tensile residual stress, but results for this are not shown separately. Fig. 2 shows results for positive values of C, with F0 = 0. The general character of the results is similar to that in Fig. 1. One value of | C| (i.e. 1.6) has been chosen so that the necessary condition for stability (124) 2 is violated. Again, results are not shown for negative values of C since they are very similar to those shown in Fig. 2. In Fig. 3 the upper limit for is 0.49 and generally the wave speed is reduced by the compressive stretch. With F0 =0, Fig. 3 shows results for positive values of C. The values of C used are 0:3; 0:6 and 1, corresponding to long-, medium-and short-dashed curves, respectively. The di erences between the results for positive and negative values of C are somewhat more marked than for larger stretches, but again we do not show these results here. Calculations not illustrated here also show, as expected, that stability is lost for a smaller value of the compressive residual stress than is the case for = 1 or 1.4. Calculations have also been carried out for combined non-zero values of F0 and C, but these do not reveal features that are essentially di erent from those shown here, so these results are not described separately. Corresponding results for the membrane limit are not included here so as to save space.

Concluding remarks

The model for a thin ÿlm discussed in this paper is an exact (one-dimensional) rod model and the resulting rod equations are exact boundary conditions for the substrate. Unlike classical engineering rod or plate theory, it is not derived by descent from the three-dimensional continuum theory and should be considered on its own merits. Indeed, the ÿlms being modelled are too thin to support the continuum hypothesis and our model is therefore designed as an alternative basis for the discussion of thin ÿlms. The surface energy U has been taken to depend on just the stretch and the curvature Ä, but may, more generally, be allowed to depend on higher gradients in order to re ect more structure in the ÿlm.

Classical plate theory is based on various approximations regarding the threedimensional displacement ÿeld, and in the version of the theory derived from continuum theory it is known that the conclusions are reliable for k1 [START_REF] Mindlin | In uence of rotatory inertia and shear on exural motions of isotropic, elastic plates[END_REF]. The nature of the approximations discounts the validity of the theory for larger k. By contrast, since the model considered here is exact, there is no a priori physical reason why its validity should be restricted to small k. Even if this were not the case, from the mathematical point of view the results for large k would be valuable not least because they dictate the general character of the results for smaller k.

Fig. 1 .

 1 Fig.1. Plot of = c 2 = (vertical scale) against k (horizontal scale) for = 1; ˜ = 1; F0 = 0 and the following values of m: (a) 0.5, (b) 0.9, (c) 1, (d) 1.5, (e) 3, (f) 5. In each ÿgure the continuous (long-dashed, dashed, short-dashed) curves are for C = 0 (0:5; 0:8; 1:2).

Fig. 2 .

 2 Fig.2. Plot of = c 2 = (vertical scale) against k (horizontal scale) for =1:4; ˜ =1; F0 =0 and the following values of m: (a) 0.5, (b) 0.9, (c) 1, (d) 1.5, (e) 3, (f) 5. In each ÿgure the continuous (long-dashed, dashed, short-dashed) curves are for C = 0 (0:5; 1; 1:6).

Fig. 3 .

 3 Fig.3. Plot of = c 2 = (vertical scale) against k (horizontal scale) for =0:7; ˜ =1; F0 =0 and the following values of m: (a) 0.5, (b) 0.9, (c) 1, (d) 1.5, (e) 3, (f) 5. In each ÿgure the continuous (long-dashed, dashed, short-dashed) curves are for C = 0 (0:3; 0:6; 1).

+ 4 + 3 2 -1 ¿ 0; | C| ¡ ˜ : (124)

Acknowledgements

This work was supported by a NATO Collaborative Research Grant CRG950152. R.W. Ogden is grateful to the Department of Mechanical Engineering, University of California at Berkeley, for his appointment as Springer Visiting Professor in October 1997.


Didier Galmiche

Guy Perrier

On Proof Normalization in Linear Logic


Didier Galmiche

Guy Perrier

Foundations of Proof Search Strategies Design in Linear Logic


Sibele Fausto
email: sifausto@usp.br

Pascal Aventurier
email: pascal.aventurier@paca.inra.fr

Scientific literature on Twitter as a subject research: Findings based on Bibliometric Analysis

Keywords: Twitter, scientific literature, bibliometric analysis

Since its launch in 2006, Internet platform Twitter has rapidly expanded. As a phenomenon of the digital era, Twitter generates a new type of research data that has received a good deal of attention in the academic literature. It has turned into a popular subject research that has been widely investigated in the academic world in different fields ranging from the Social Sciences to Health Sciences, addressing various questions, methods approaches, and covering multiple data sets. This study provides some findings of a bibliometric study which was conducted to describe the scientific literature available on Twitter with descriptive, quantitative information and also in a qualitative approach, in addition to the previous studies and designed as a contribution to a broader picture of how the evolution of the current scientific literature about Twitter is related to bibliographic data sets. Results show a variety of findings that can provide a better comprehension of this social media platform which evolved from a data source for the research to, nowadays, being a research subject itself.

Introduction

Since its launch in 2006, founded by Jack Dorsey and associates in San Francisco, California, USA, Internet platform Twitter (https://twitter.com) has rapidly expanded, "conceived as part of a long line of squawk media, dispatch, short messaging, as well as citizen communications services" [START_REF] Rogers | Debanalising Twitter: The Transformation of an Object of Study[END_REF], with Dorsey trying to define Twitter as a new medium in itself, a public instant messaging system, meaning also a public information utility [START_REF] Sarno | Twitter creator Jack Dorsey illuminates the site's founding document. Part I[END_REF].

Correspondence concerning this article should be addressed to Sibele Fausto, Technical Department of the Integrated Library System, University of São Paulo, Rua da Biblioteca, s/n, Complexo Brasiliana, São Paulo, SP, CEP 05508-050 (Brazil). Email: sifausto@usp.br.

As a phenomenon of the digital era, Twitter generates a new type of research data related to this role as public information utility, emerging as an important source of the new digital paradigm, and has received a good deal of attention in the academic literature. It has turned into a popular research subject that has been widely investigated in the academic world in different fields, ranging from the Social Sciences to the Health Sciences, addressing various questions, method approaches, and covering multiple data sets, with early studies focusing on characterizing tweet types, determining how many of them are of value, and evaluating Twitter as, more or less, of interesting content [START_REF] Rogers | Debanalising Twitter: The Transformation of an Object of Study[END_REF]).

An analysis of the coverage of Twitter's first three years, performed by [START_REF] Arceneaux | Seems stupid until you try it: Press coverage of Twitter, 2006-9[END_REF], pointed out the prevalence of "food tweets" and the more general "mindless stream", emanating from Twitter in its early years, leading [START_REF] Rogers | Debanalising Twitter: The Transformation of an Object of Study[END_REF] to ask what value lies in breakfast and lunch tweets, and responding that "geo-located food tweets may be of interest to those studying the geography of taste and other questions of cultural preference" (p. xiii-xiv).

Thus, together with other social web platforms, such as Facebook, blogs, Wikipedia, YouTube, and others, Twitter becomes a new source of information for many researchers in different disciplines, providing a variety of data to extract [START_REF] Metaxas | Sifting the sand on the river bank: Social media as a source for research data[END_REF]. Despite being behind services like Facebook in numbers (with 1.3 billion active users) and WhatsApp (500 million), this platform currently has about 284 million active users worldwide, with 500 million tweets being sent out every day in more than 30 different languages (Twitter, 2015). In 2014 the eMarketer Consultancy predicted that Twitter should reach 300 million monthly active users by 2016, and pass the 400 million mark by the end of 2018.

Since its launch Twitter has incorporated growing facilities for sharing content, with options to add links, images and videos, and efficient mechanisms to aggregate specific topics by hashtags: labels assigned to a term, preceded by the hash symbol (#), making the term an active link that allows recovery of all tweets with the hashtag. Hashtags not only gather and record the tweets on the same subject but also make it possible to recover its track record, through monitoring or mining. Moreover, Twitter's dynamics, where users develop a virtual community made up of a public identified by common interests -followers, who share and replicate content freely with each other in an intense interaction through actions, such as the "re-tweet" (RT), generating affiliations around themes, topics and identities [START_REF] Zappavigna | Ambient affiliation: A linguistic perspective on Twitter[END_REF]. All these characteristics make Twitter a major data source to track the performance of personal and institutional microblogs, communities and the most diverse subjects through systems that monitor the interactions of real-time messages generated on the platform, targeting to various analyses, hence the growth of scientific literature on Twitter.

Bruns and Weller (2014) pointed out that scientific publications, with "Twitter" explicitly in the title, are as follows: 1,400 on Scopus R database, 470 on Web of Science R (WoS) database and another 10,000 available via Google Scholar. In turn, Zimmer and Proferes (2014) conducted a content analysis study with a set of 382 academic publications that used Twitter from 2006 to 2012. Also, Weller et al (2014) published a comprehensive study on Twitter, addressing aspects of this social media related to its concepts and methods, even as its perspectives and practices in use in society, as those from popular culture, brand and crisis communication, politics and activism, journalism, and in academia.

This study provides some more insights in this matter through a bibliometric study which was conducted to describe the scientific literature available on Twitter, with quantitative information and also in a qualitative approach, in addition to the previous studies and designed as a contribution to a broader picture of how the evolution of the current scientific literature about Twitter is related to bibliographic data sets.

Methods

The methodological approach chosen for this study is Bibliometric Analysis, an original area of study generally covering books and publications and whose principle, according to Tague-Sutcliffe (1992), is the analysis of scientific or technical activity through quantitative studies of publications. Bibliometric Analysis provides insights on which stage a given research area lies, and it is a methodology also used in several areas, with the objective of characterizing the published research output and the forecasting of trends, yielding results that are also useful to support decision-making [START_REF] Rostaing | La bibliométrie et ses techniques[END_REF]. Through Bibliometric Analysis it is possible to study some quantitative parameters of the publications, periodicals, authors, keywords, users and citations [START_REF] Pao | Concepts of information retrieval[END_REF]. But nowadays Bibliometric Analysis integrates the broader area of metric studies of the information, with different scopes and finalities in the discovery of information characteristics in general, not just about science, comprising and combining different methodologies and tools to provide a more comprehensive view of its objects of study (Cronin & Sugimoto, 2014). This Bibliometric Analysis was carried out for the indexed literature in the period from 2006 (when Twitter was released) until 2014, and only included articles with the word "Twitter" in the title, abstract and/or keywords. Analysis was conducted on the results of the Scopus R database (from Reed-Elsevier), which has a wider coverage than WoS R (from Thomson Reuters). The 2,338 documents retrieved were analyzed on two levels. The first was a basic one related to quantitative indicators, according to the average growth in the number of publications per year, identifying main research areas that have published about Twitter, as well as journal titles, most prolific authors, institutions and countries. This was followed by a second level with a deeper analysis, considering citations and the 'hot topics' of the literature, through the combination of techniques and tools of lexical analysis and content about the corpora, with the software Sphinx Lexica R , that allows the researcher to differentiate in a semi-automatic form the main terms, identifying those that are 'hot topics' in the publications about Twitter, and CorText Manager3 , that was used to discover the relationships between the different topics and countries involved, providing maps of heterogeneous networks as thematic and collaborative maps, sorted by term frequency per countries and by a chi-squared (χ 2 ) distribution of the "hot topics".

Results

Quantitative indicators

The academic literature with Twitter as research subject has seen an average growth rate of 52.23% per year, from one sole article published in 2006 to the 720 published in 2014. Figure 1 shows that this growth was ongoing and has accelerated after 2009, reaching the peak in 2014. This result points out that in fact Twitter is becoming a consolidated object of research. As for the research areas which have published about Twitter, Figure 2 shows the dominance of certain disciplines with more than 100 publications, such as the Social Sciences, with 1,014 articles (33% of the total sample); and Computer Science, with 826 articles (27%), followed by Medicine, with 327 articles (11%). But we can observe that other disciplines also appear with relevant participation, such as Business Management and Accounting, and Engineering, which had 284 and 273 papers, respectively, both reaching 9% of the total sample; followed by Arts and Humanities, with 215 (7%) and Mathematics, with 105 (4%).

Related to journal titles with more than 10 publications in this matter, results were as follows: PLoS One The most prolific authors were H. W. Park with 13 articles and A. Bruns, with 12 ones, but others authors also figured with a good research output in this matter, such as J. J. Jung and T. Highfield, both with 9 articles; P. R. Spence and S. H. Burton, with 8 articles; and A. O. Larsson, I. Himelboim, J. K. Harris and M. Thelwall with 7 articles each [Table 1]. As for institutions with more than 15 contributions, results show Yeungnam University with the most number of articles published (30 contributions); followed by Queensland University of Technology The following countries had more than 50 articles: the United States (USA) with 912 publications (39% of the total sample); the United Kingdom (UK) with 216 (9.23%); Spain, 148 (6.33%); Australia, 119 (5.1%); South Korea, 116 (4.9%); China, 99 (4.23%); Canada, 98 (4.2%); Germany, 65 (2.8%); Japan, 63 (2.7%); and The Netherlands, 51 (2.2%) [Figure 5]. Ahead, in this analysis, we will show those same countries according to the topics most published by them, forming clusters by equivalent subject.

Citations and textual analysis

Most cited authors and journals. In addition to the basic quantitative indicators shown before, we also examined the data sample to verify the most cited papers in order to get insights about the impact of that scientific literature with Twitter as subject research. Table 2 shows articles with more than 100 citations, summarized by authors, article titles, publication year, journal titles, number of citations received and the topics.

The most cited authors were Kaplan and Haenlein (2010), with an article published in the "Business Horizons" journal, which reached 1,296 citations. As the journal title points out, the main topic of this paper is Business. Next, the authors cited about 400 times were Jansen et al ( 2009 2].

Only four journals with most-cited articles are also in the top journals list with more than 10 articles published about Twitter, as follows: "PLoS One" has published 46 articles on Twitter and also figured among the most-cited with two articles, from Chew and Eysenbach So there isn't a clear relationship between the largest number of published articles on Twitter and a greater number of citations received, as only four (18, 18%) from a total of 22 journals with more than 10 publications on Twitter received more than 100 citations.

"Hot Topics" in the literature about Twitter. Continuing our analysis, we proceeded in the mining of the most frequent topics into literature about Twitter, through applying the software Sphinx Lexica R in a semi-automatic lexical corpus analysis, after removing the term "Twitter" from it, identifying specific terms with more than 100 occurrences by year to observe their evolution in frequency during the analyzed period of 2006-2014. All terms were joined together in groups of equivalent ones.

Results show several most frequent topics, with "social media" leading in a total of 3,197 occurrences, with an ongoing growth in its frequency over the period, followed by the topic "social networks," with the total of 1,981 occurrences, also with an ongoing growth in its frequency over the same period.

"Hot topics" also very often include the topic of "learning" (human learning), with 360 occurrences; "health" (330); "political activity" (324), among others represented in Table 3.

Representative keywords were evolving rapidly. For example, "Second Life" was representative in the early years. In turn "big data" and "data mining" appear over the whole time period. It was observed that in general most of the topics had a continued growth in frequency since the year in which they first appeared in the literature on Twitter, and this evolution of their frequency can predict the trends of these topics. This topic analysis also allowed the mapping of these topics by clusters, according to related lexical similarities, e.g. a topic cluster related to "health" groups terms such as "public health," "patient" and "health care," (represented in clear blue in Figure 6), as well as a topic cluster related to terms such as "political activity," "social movements," "public relations" and "mass communication" (represented in clear yellow in Figure 6, that also shows the other topic clusters that emerged in this analysis).

This "hot topic" analysis also allowed clustering topics related to their frequency by countries, as shown in Figure 6, and also pointing out the relations between countries by thematic collaborations, next in Figure 7.

In Figure 6 there are countries that are more present in each cluster, addressing related topics sorted by decreased term frequency, such as USA, China, UK, Spain, Australia, Canada, South Korea, Japan, etc. But in Figure 7 it is possible to observe the thematic collaborations between these countries and others that were checked through the institutional affiliations (by country) in the recovered articles, with a chi-squared (χ 2 ) distribution of the "hot topics". It was observed that there are only 340 papers with international collaborations, 13% of the whole corpus, which is a very low percentage compared to other domains in international research. These clusters, for example, show such themes as "collective action", "big data", "text and web mining", "marketing", "political activity" and "user generated content" and were common topics in the collaborations between Brazil, France, India, Italy, UK, and others. In turn, topics such as "crisis communication", "corporate communication", "traditional education", "political activity" and "sentiment classification", were most common in collaborations between Belgium, Switzerland, The Netherlands, and Turkey. This analysis is useful to realize which countries have collaborations in the research of specific topics about Twitter.

Discussion and concluding remarks

These findings indicate that there was a steady growth in the academic literature which includes Twitter as research subject in the period from its founding, in 2006, to 2014. The subject areas with most contributions were the Social Sciences, followed by Computer Science and Medicine. There is a long tail of journal titles that have published articles on this subject, with 22 titles in more than 10 publications, all of them in English, apart from a single title in Spanish. Two authors stand out in terms of the number of articles, and various institutions have made contributions, most of them being universities from North America (USA and Canada) and Europe (UK, Spain, Germany and The Netherlands), although Asia (China, Japan and South Korea) and Oceania (Australia) also stood out.

With a second-level analysis in a deeper manner, it was possible also consider citations and the "hot topics" of the literature about Twitter, in order to verify their impact, and to identify which topics were more discussed in this literature, as well as to find out with a mapping tool how these topics were distributed by subject groups and countries, including collaborations between these countries. It was observed that there is a group of papers which received more than 100 citations, but there was not a clear relation between the largest number of published articles on Twitter and a greater number of citations received.

There are several "hot topics" that were addressed in the literature on Twitter, with "social media" and "social networks" leading with 3,197 and 1,981 occurrences, respectively, and the majority of these topics had a continued growth in their frequency in the literature during the analyzed period. Twitter analysis is strongly related to social issues and enhanced problems linked to youth populations, as health (obesity), public health, international disasters, drug abuse, but also to the learning and education process. Three other important fields of analysis are social behavior (opinion), political activity and branding/media strategy. It was interesting to see how these "hot topics" were organized under common subjects and how they were distributed between different countries, including collaborations between these countries in the research of certain topics about Twitter. There are a few international collaborations (13% of the corpus) and it is mostly delimited to traditional collaborator countries (between USA, UK, and those from Asia -China and South Korea).

All these findings add to and group together with previous studies about Twitter and aim at a contribution to a better comprehension of this social media platform which evolved from a data source for the research to, nowadays, being a research subject itself.

Future directions of the investigation of this role of Twitter as an object of study through Bibliometric Analysis are a continuous task that point to other approaches which can provide more detailed insights, such as how the research network is drawn from the cited references.

Fig. 1 :

 1 Fig. 1: Growth of the scientific literature about Twitter (2006-2014).

  (n = 46); Computers in Human Behavior (n = 37); Journal of Medical Internet Research (n = 37); First Monday (n = 36); Information Communication and Society (n = 35); Public Relations Review (n = 34); Estudios Sobre El Mensaje Periodistico (n = 19); Social Science Computer Review (n = 18); E-content (n = 17); Cutting Edge Technologies in Higher Education (n = 16); New Media and Society (n = 15); Expert Systems with Applications (n = 14); ACM Transactions on Intelligent Systems and Technology, Journal of Communication, Government Information Quarterly, and PC World San Francisco CA, 13 articles each; Journal of the American Society for Information Science and Technology, and

Fig. 2 :

 2 Fig. 2: Number of articles by disciplines.

Fig. 3 :

 3 Fig. 3: Top journal titles with articles published about Twitter (2006-2014).

  (n = 22); Pennsylvania State University, the University of Oxford and the University of Toronto (n = 18 each); University of Texas at Austin (n = 17); University of Maryland and Indiana University (n = 16 each); Seoul National University, University College, (London), Carnegie Mellon University, University of New York State at Buffalo, with 15 articles each [Figure 4].

Fig. 4 :

 4 Fig. 4: Top institutions with more than 15 articles published about Twitter (2006-2014).

Fig. 5 :

 5 Fig. 5: Top countries with more than 50 articles published about Twitter (2006-2014).

  ), with an article cited 438 times, in the topic of Information in the "Journal of the American Society for Information Science and Technology;" and Bollen, Mao and Zeng (2011), cited 430 times, with a paper on the topic of Computation, in the " Journal of Computational Science". The third group of most-cited authors, which reached about 200 citations, were Kietzmann et al (2011) with an article on the topic of Business, again in the "Business Horizons" journal, with 283 citations; Marwick and Boyd (2011) on the topic of Information in "New Media and Society," with 243 citations; and, finally, Yardi, Romero and Schoenebeck (2009), with an article in "First Monday" on the General topic that reached 239 citations. The fourth and last group of most-cited authors spanned 116 to 176 citations and includes 14 journals, with topics like Science (3 occurrences), Health (5 occurrences), Information (2 occurrences), Computation (2 occurrences) and Business (1 occurrence) [Table

Fig. 6 :

 6 Fig.6: "Hot topics" by countries sorted by decreased term frequency.

Fig. 7 :

 7 Fig.7: "Hot topics" by country collaborations sorted by χ 2 distribution.

Table 1 :

 1 Most prolific authors who published about Twitter(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)

	Author	n. Articles
	Park, H.W.	13
	Bruns, A.	12
	Jung, J.J.	9
	Highfield, T.	9
	Spence, P.R.	8
	Burton, S.H.	8
	Larsson, A.O.	7
	Himelboim, I.	7
	Harris, J.K.	7
	Thelwall, M.	7

Table 2 :

 2 Most prolific authors who published about Twitter (2006-2014) (2010) which received 176 citations, and from Signorini et al (2011), with 161 citations. The "First Monday" has published 36 articles on Twitter and received 239 citations in the article by Yardi, Romero and Schoenebeck (2009). The "Information Communication and Society" has published 35 articles and received 167 citations from Boyd and Crawford (2012). Finally, the "New Media and Society" journal published 15 articles on Twitter and received 243 citations to the Marwick and Boyd's article (2011).

Table 3 :

 3 "Hot topics" and their evolution in literature about Twitter(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)

	Topics	2006 2007 2008 2009 2010 2011 2012 2013 2014 TOTAL
	social_media	0	0	1	23	10	370	494	984 1211	3197
	social_networks	0	4	19	50	128	221	373	592	593	1981
	learning	0	0	0	1	10	38	61	93	157	360
	health	0	0	0	4	11	30	35	122	127	330
	political_activity	0	0	0	0	3	38	44	114	125	324
	mobile	0	2	2	14	28	32	67	68	69	282
	real_time	0	0	6	1	11	28	38	60	53	197
	content_analysis	0	0	0	0	10	20	30	58	73	191
	sentiment_analysis	0	0	0	0	0	12	25	56	89	182
	marketing	0	0	0	5	8	22	34	61	45	175
	twitter_users	0	0	1	2	2	11	29	48	69	162
	case_study	0	0	0	5	5	11	30	47	57	155
	opinion	0	0	0	0	0	13	23	45	66	147
	public_health	0	0	0	4	5	18	21	40	50	138
	facebook	0	0	0	3	9	24	17	25	48	126
	news	0	0	1	1	4	22	24	32	41	125
	qualitative_analysis_data	0	0	0	0	3	18	19	24	54	118
	web_sites	0	0	1	4	19	19	23	25	23	114
	big_data	0	0	0	0	0	0	27	30	55	112
	large_scale	0	0	0	0	6	7	23	26	43	105
	higher_education	0	0	0	1	10	18	16	36	22	103
	user_activities	0	0	0	0	1	2	19	28	49	102
	second_life	0	1	0	1	5	10	0	2	0	19

CorText Manager: urlhttp://www.cortext.net/.

Author