The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional ‘quasi-static’ elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.
The theory of elastic surface waves for coated bodies is highly important for many modern applications, for example, thin-film technology. As examples of ongoing interest to the problem, we mention recent publications (Fu 2007; Steigmann & Ogden 2007; Qian et al. 2009 and references therein).
The potential of the exact analysis in the framework of three-dimensional elasticity is rather limited due to a sophisticated dispersion relation (e.g. Achenbach & Keshava 1967). This motivates the development of approximate formulations including modelling the effect of the coating by introducing certain effective boundary conditions imposed on the substrate surface (Auld 1990 and references therein). In particular, such conditions were derived in Tiersten (1969) within the isotropic context using physical assumptions based on the classical theory of plate extensions. Later on, it was claimed by Bövik (1996) that the results of Tiersten (1969) are not asymptotically consistent in the case of a thin coating. A standard asymptotic procedure presented in §3 of the paper justifies the consistency of the consideration in Tiersten (1969). Moreover, as shown in §5, the dispersion relation associated with the aforementioned boundary conditions in Tiersten (1969) coincides with the two-term long-wave expansion of the exact dispersion relation studied in Shuvalov & Every (2008).
To our best knowledge, very little, if any, explicit asymptotic results have been obtained for the transient dynamic response of a coated half-space subject to surface loading. A way forward is to extract from the wave equations in linear elasticity an approximate model oriented for the surface wave phenomena only. A two-dimensional version of the model was established in Kaplunov et al. (2006) by perturbing the original plane strain equations about the homogeneous Rayleigh wave solution expressed in terms of harmonic functions (Friedlander 1948; Chadwick 1976). The resulting relations consist of ‘quasi-static’ elliptic equations over the interior for the wave Lame potentials satisfying at the surface the ‘dynamic’ boundary conditions involving an inhomogeneous hyperbolic equation. These clearly indicate a dual elliptic–hyperbolic nature of the Rayleigh wave. A high accuracy of the predictions of the model in Kaplunov et al. (2006) was demonstrated in Kaplunov et al. (in press) by comparison with the exact solution of the transient moving load problem. We also remark that the philosophy underlying the approach in Kaplunov et al. (2006, in press) seems to be analogous to that in Achenbach (1998) and Parker & Kiselev (2009) dealing with specialized formulations for the free surface waves of general profile.
In §4, we extend the results of Kaplunov et al. (2006) in two directions. First, we incorporate the effect of a coating via the effective boundary conditions leading to a weak long-wave dispersion of the surface wave. In contrast to an uncoated half-space, the hyperbolic equation for the surface now contains a singular perturbation in the form of an integro-differential or a pseudo-differential operator (see discussion in §5). Second, we proceed to a three-dimensional problem starting from the Radon transform that has been traditionally used for the three-dimensional to two-dimensional reduction in linear elasticity (e.g. Willis 1971; Wang & Achenbach 1996; Georgiadis & Lykotrafitis 2001).
In §6, the proposed model is applied to plane problems for surface-impulsive and moving loads. As might be expected, the behaviour of the phase velocity in the vicinity of its maximum or minimum corresponding to the long-wave Rayleigh limit determines the advancing or receding type of the surface wave front induced by a point instantaneous impulse. This behaviour also affects the resonant response in case of a moving load.
2. Statement of the problem
We consider a three-dimensional half-space covered by an elastic coating of constant thickness h. The Cartesian axes Ox1 and Ox2 are lying on the upper surface of the coating, with the interface between the coating and the half-space given by x3=h (figure 1).
The equations of motion in three-dimensional elasticity are written as 2.1 where un are the components of displacement vector, σin,σ3n are the components of the Cauchy stress tensor and ρ is the volume density of solid media. We suppose, throughout the paper, unless otherwise stated, that i≠j=1,2 and n=1,2,3. The constitutive relations for a linear isotropic elastic solid are given by 2.2 where λ and μ are elastic moduli. The associated bulk wave speeds may then be expressed as 2.3 We start from the relations (2.1) and (2.2) to describe the substrate (x3≥h). In the case of the coating (0≤x3≤h), we supply with suffix ‘0’ the material parameters in the above equations, operating with ρ0, λ0, μ0 as well as c10 and c20.
The prescribed boundary conditions at the surface x3=0 of the coating are 2.4 where P=P(x1,x2,t) is a normal force. We also assume continuity of all displacements and stresses σn3 at the interface x3=h.
The aim of the consideration below is to establish a long-wave model for the surface wave propagating in the system ‘coating-substrate’. The derivation contains two stages. First, we reduce the effect of the coating on the substrate by effective boundary conditions at the interface. Then, we perturb the equations for the substrate subject to these boundary conditions about the homogeneous surface wave solution.
3. Effective boundary conditions
In this section, we take into separate consideration the problem for an elastic coating, setting the following boundary conditions at the interface x3=h: 3.1 where vn=vn(x1,x2,t) are prescribed displacements and n=1,2,3 as above.
We apply to the boundary value problem for the coating (2.1), (2.2), (2.4) and (3.1) the standard scheme of direct asymptotic integration (e.g. Goldenveizer et al. 1993 and references therein). Let us specify a small parameter 3.2 corresponding to the long-wave assumption, where L is a typical wave length, and scale the original variables as 3.3 We also define the dimensionless quantities 3.4 where V is the maximal displacement amplitude. Here and below in this section all quantities with the asterisk are of the same asymptotic order.
The equations of motion and the constitutive relations may now be rewritten as 3.5 where κ0=c10/c20.
The boundary conditions become 3.6 Next we expand the displacements and stresses in asymptotic series in terms of the small parameter ε: 3.7
Substituting these expansions into the equations (3.5), and boundary conditions (3.6), we obtain at the leading order 3.8 Equations (3.8)2 and (3.8)4 together with the related boundary conditions (3.8)5 and (3.8)6 may be integrated to yield 3.9 and 3.10 The last formula represents the sought for expression for the normal stress σ33. Evaluation of the stresses σi3 requires further analysis.
At the next order, equation (3.5)5 becomes 3.11 It is also clear that 3.12 Then, from equations (3.11) and (3.12), we have 3.13 Similarly, we derive from the formula (3.5)6 taken at order ε, and the boundary conditions (3.12) 3.14 Now, using the formulae (3.5)4, (3.9) and (3.14), we obtain 3.15 Finally, starting from equation (3.8)1 and using equations (3.8)3 and (3.15) with the boundary conditions (3.8)7, we are in a position to determine the leading order terms in the asymptotic expansions for the stresses σ*i3. They are 3.16 In the original dimension form, the stresses at the interface x3=h may be presented as 3.17 where sn3=σn3(x1,x2,h,t). The assumed continuity of the displacements and stresses at x3=h now reads as σn3=sn3 and un=vn, where σn3 and un denote the stresses and displacements in the substrate, respectively. As a result, the effective boundary conditions on the surface of the substrate can be written as 3.18 These boundary conditions in case of P=0 coincide with those obtained in Tiersten (1969) using an ad hoc physical approach based on modelling of the coating in the framework of the classical theory of plate extension. A more recent analysis in Bövik (1996), with the methodology later extended to anisotropy in, for example, Niklasson et al. (2000), claims that boundary conditions (3.18) neglect a few terms of O(h).
Let us consider expressions (35) and (36) from Bövik (1996) in the notation of the present paper. 3.19 The underlined terms in the r.h.s. of the above formulae do not appear in the effective boundary conditions (3.18). It can easily be verified by introducing in equations (3.19) the scaled variables (3.3) and dimensionless displacements and stresses (3.4) that all the extra terms are of the next asymptotic order in ε, that is, O(h2) in the notation of Bövik (1996).
4. Derivation of the asymptotic model
First, we express the equations of motion in terms of displacements. On substituting equation (2.2) into equation (2.1), we obtain 4.1 where u=(u1,u2,u3) is the displacement vector, and a conventional notation is adopted for three-dimensional differential operators.
Let us apply the Radon transforms to equation (4.1) (bib14 for more details). The result is 4.2 where 4.3 and with the angle α varying over the interval 0≤α<2π. We also introduce the transformed displacements in the Cartesian frame (χ,ζ), (figure 2). They are 4.4
Now we set assuming that the anti-plane motion is not induced by the studied normal force. Then equations (4.2) rewritten in terms of the displacements (4.4) take the form of the plane problem of elasticity, that is, 4.5 The associated boundary conditions at the surface x3=h follow from the transformation of the original boundary conditions (3.18). They can be written as 4.6 where and P(α) denote the transformed stresses and normal force, respectively.
In case of h=0, the boundary value problems (4.5) and (4.6) are identical to those considered in Kaplunov et al. (2006), dealing with an explicit asymptotic model for the plane Rayleigh waves induced by the transient loads applied to the surface of an elastic half-space. Below we extend the asymptotic methodology in Kaplunov et al. (2006) to a coated half-space.
We begin with the scaling 4.10 where ε=h/L≪1, L is a typical wavelength and cR denotes a surface wave speed, which is not supposed yet to be the classical Rayleigh one. The parameter ε here states a small deviation of the phase velocities of interest from the wave speed cR.
The two-term asymptotic solution of equation (4.11) is given by Kaplunov et al. (2006) 4.12 where P0 is the maximal amplitude of the normal force P, ϕ(0)=ϕ(0)(ξ,k1γ,τ) and ψ(0)=ψ(0)(ξ,k2γ,τ) are arbitrary plane harmonic functions in the first two arguments and 4.13 where ϕ(1,0)=ϕ(1,0)(ξ,k1γ,τ) and ψ(1,0)=ψ(1,0)(ξ,k2γ,τ) are again plane harmonic functions and 4.14 with bar denoting a harmonic conjugate.
On substituting the two-term asymptotic behaviours (4.12) into the boundary conditions (4.9), we have at the leading order 4.15 Next we exploit the Cauchy–Riemann identities for a plane harmonic function f(x,ky) 4.16 having from equations (4.15) 4.17 The compatibility of equation (4.17) leads to the classical Rayleigh surface wave equation presented as 4.18 Thus, the wave speed cR introduced in equations (4.10) is precisely the conventional Rayleigh wave speed. The boundary conditions (4.17) also result in the relation 4.19 between the transformed potentials at the surface.
At the next order, the boundary conditions (4.9) become 4.20 where After a fairly straightforward algebra making use of the Cauchy–Riemann identities and the formula (4.19), we obtain from equations (4.20) the following: 4.21 Finally, we eliminate from these equations and perform algebraic manipulations involving the Rayleigh equation (4.18). The result is 4.22 where and 4.23 Now we suppose that in equations (4.12) 4.24 Then, equation (4.22) may be re-cast in terms of the dimensional variables (χ,x3,t) as 4.25 with .
Similarly, at the leading order, the transformed potentials over the interior satisfy the elliptic equations obtained from equations (4.11), that is, 4.26 and the potential ψ(α) at the surface is determined from the relation (4.19) in the form 4.27 In what follows it is convenient to operate with a pair of the potentials and along with the potential ϕ. By inverting Radon transform in (4.25)–(4.27), we arrive at the long-wave model given by the elliptic equations over the interior (x3≥h) 4.28 and 4.29 with the boundary conditions at x3=h 4.30 and 4.31 where Δ2 denotes two-dimensional Laplacian in the coordinates (x1,x2).
5. Discussion of the model
The asymptotic model derived in the previous section is oriented to the surface wave only, ignoring the effect of the bulk waves. The contribution of the surface wave is dominant for a number of practically important problems including, in particular, the resonance phenomena caused by moving loads (e.g. Kaplunov et al. in press). The model includes the three-dimensional elliptic equations (4.28) and (4.29) governing the spatial decay over the interior along with the conditions (4.30) and (4.31) imposed on the surface. The second of them relates the potentials ψ1 and ψ2 with the potential ϕ, whereas the first one is the only relation in the model involving the dynamic factor; in doing so, it operates with the component of the original load P inducing the surface wave. In the absence of a coating (h=0), this reduces to an inhomogeneous two-dimensional wave equation generalizing the elliptic–hyperbolic formulation in Kaplunov et al. (2006) for the plane strain problem.
The presence of a coating leads to a coupling between the elliptic equation (4.28) and the singularly perturbed hyperbolic equation (4.30) due to the normal derivative with respect to x3 entering the latter. To a certain extent, the coupled problem (4.28) and (4.30) may be interpreted as a membrane resting on an incompressible fluid.
Although the scaling (4.10) underlying the model assumes that the effect of coating has to be taken into consideration at the leading order approximation, it is obvious that the equation on the surface is valid for a weak coupling as well. In this case, the dynamic response of a coated half-space can be calculated using a relatively simple regular perturbation scheme.
It is well known that the equation (4.30) can be reduced to the form of an integro-differential or pseudo-differential equation on the surface. To this end, we have to express the normal derivative ∂ϕ/∂x3 in terms of the potential at the surface ϕ(x1,x2,0,t). In particular, the solution of equation (4.28) can be rewritten in a symbolic form as 5.1 where is understood as a pseudo-differential operator. Then, we have 5.2 and equation (4.30) becomes 5.3 In case of a plane strain problem (say if ∂ϕ/∂x2=0), the last equation reduces to a one-dimensional pseudo-differential equation. It is 5.4 Similar to Kovalev et al. (2002), equation (5.3) can also be presented as an integro-differential equation. Using the identities 5.5 where H(f)(x) denotes the Hilbert transform, given by 5.6 we have 5.7 The dispersion relation associated with the homogeneous equations (5.4) and (5.7) is 5.8 where k is the wave number normalized by h−1 and vph is the phase speed normalized by cR.
The above expression coincides with the long-wave asymptotic expansion of the ‘exact’ dispersion relation analysed by Shuvalov & Every (2008). In the special case in which μ0=μ and λ0=λ, it reduces to the dispersion relation displayed in Kovalev et al. (2002) with 5.9 These comparisons provide one more illustration of the asymptotic consistency of the effective boundary conditions in Tiersten (1969) demonstrated in §3.
It is also worth mentioning that, as it has been shown in Shuvalov & Every (2008), the parameter b in the proposed model may take both positive and negative values corresponding to the local minimum and maximum of the phase velocity equal to the Rayleigh wave speed, (figure 3 and also formula (5.8)). As might be expected, the sign of b is crucial for the surface dynamic behaviour see examples in §6.
6. Illustrative examples
In this section, we evaluate the dynamic response of a coated half-space within the framework of the long-wave model for two particular types of surface loading, namely for an instantaneous point impulse and also for a distributed moving load. For the sake of simplicity, we deal with plane strain problems.
(a) Impulse loading
Let us set P=P0δ(x1)δ(t) in the r.h.s. of equation (5.3) and specify dimensionless variables by 6.1 where L is a chosen linear scale. Then, equation (5.4) becomes 6.2 where θ=−2ϕ/AcRP0 and the parameter hL=h|b|/L is assumed to be small, that is, hL≪1.
Below we develop the scheme in Emri et al. (2001) of the method of matched asymptotic expansions. We begin with the boundary layer at the characteristic of the degenerated equation (6.2). The inner co-ordinate is 6.3 indicating that the width of the boundary layer is proportional to h1/2.
On substituting the scaling (6.3) into the homogeneous equation (6.2), we obtain at leading order 6.4 The solution of the latter can be presented in the form of a Fourier integral as 6.5 where ζ0=τ−r and arbitrary function C(ω) has to follow from matching with the appropriate outer expansion.
In the outer region, equation (6.2) reduces to the one-dimensional wave equation 6.6 from which we have θout=H(ζ0), or, identically, 6.7
Matching of the expansions (6.5) and (6.7) yields for the former (see also Cole 1968 for more detail) 6.8 resulting in the uniform asymptotic behaviour 6.9 where (Prudnikov et al. 1986) 6.10 and C(x) and S(x) are Fresnel integrals The obtained solution, due to the restrictions of the surface wave model, does not take into account the slowly varying component of the overall wave field corresponding to the bulk waves. The latter becomes negligible in the case of the derivatives of the function ϕ in equation ((6.9)). For an applied force P, equation (6.9) provides the analytical expression of the wave profile for given cR, bh and A (cf. equations (5.4) and (5.7)). On the other hand, if for an applied P one measures the wave profile and compares it with equation (6.9), cR, bh and A can then be determined. Thus, the analytical results presented here may be used for the non-destructive testing of the coating thickness h and the Lamé constants λ and μ of the substrate for the known elastic moduli of the coating.
Numerical results are displayed in figure 4 for τ=1 and hL=0.01. In case b>0 corresponding to the local maximum of the phase velocity at the Rayleigh wave speed (figure 3a, we observe a receding front. Case b<0 (figure 3b) is related to the minimum of the phase velocity. As a result, we arrive at an advancing front.
We also remark that the receding and advancing fronts of plate extension were studied in Dai & Cai (1999, 2000) and Kaplunov et al. (2000) for pre-stressed structures. Although the solution of Kaplunov et al. (2000) is expressed through the Airy functions, the graphs in the cited paper look pretty similar to those given in figure 4.
(b) Steady-state moving load problem
Now, we consider the effect of a distributed load steadily moving along the surface with the constant speed c. Taking the load in the form where the parameter l determines the load distribution, we obtain from equation (5.4) 6.11 with where σ corresponds to the normal surface stress, s is the dimensionless moving co-ordinate and g and hl are the key problem parameters characterizing the thickness of the coating and the proximity of the speed of the load to the Rayleigh wave speed, respectively.
Next, using the Fourier transform, we have 6.12 In this paper, we restrict ourselves to the case (cR−c)b<0, in which the denominator in equation (6.12) has no poles. Then, equation (6.12) becomes 6.13 where and Ei is the integral exponent given by
Now, we investigate two limiting cases. In the limit |g|≫|hl| associated with an uncoated half-space, we have the estimate (for details, see Abramowitz & Stegun 1982) 6.14 coinciding with the well-known near-resonant regime of the moving load at c→cR (e.g. Cole & Huth 1958).
In the other limit |g|≪|hl|, we arrive at 6.15 where γ≈0.577 is the Euler’s constant, (figure 5b). The above formula demonstrates that the presence of a coating does not remove the resonance at c=cR. The reason is that, despite the dispersion of the surface wave due to the influence of the coating, the extremal value of the phase speed is still given by the Rayleigh wave speed. This resonant mechanism is also characteristic of the famous example of a moving load on a beam supported by a Winkler foundation (Timoshenko 1927; also see Timoshenko 1953). For the latter, the critical speed of the load coincides with the minimal phase speed of flexural waves.
7. Concluding remarks
In this paper, we have not achieved a complete separation of the wave problem on the surface in a differential form as it has been done in Kaplunov et al. (2006) for an uncoated elastic half-space. At the same time, the scalar boundary value problem interpreted in §4, as a sort of analogue of a vibrating membrane supported by an incompressible fluid, is still a major simplification in comparison with the original three-dimensional vector hyperbolic system for a coated elastic half-space. In addition, the pseudo-differential equation (5.4) treated as a singular perturbed hyperbolic equation may be analysed using the method of matched asymptotic expansions (see the first example in §6). On the application side, the explicit analytical results obtained for the illustrative examples may be used for non-destructive testing of the coating thickness and the elastic moduli of the substrate.
The derived long-wave model may easily be extended to linear surface waves in case of anisotropic and pre-stressed coatings. The generalizations to nonlinear coatings are less obvious. Nevertheless, there is a possibility of hybrid numeric–analytic approaches combining numerical treatment of nonlinear surface behaviour with analytic formulae for the interior. Moreover, in case of a weak nonlinear coupling, there is a room for asymptotic considerations.
J.K. and D.A.P. thank the City University of Hong Kong for its support and hospitality. The authors also gratefully acknowledge E. Nolde and A. Pichugin for fruitful discussions.
- Received March 2, 2010.
- Accepted April 19, 2010.
- © 2010 The Royal Society