Rayleigh waves in an incompressible fibre-reinforced elastic solid with impedance boundary conditions

1 Introduction

Fibre-reinforced composite concrete structures are significant due to their low weight and high strength. A reinforced composite has the characteristic property where its components act together as a single anisotropic unit till they remain in the elastic condition. During an earthquake, the artificial structures on the surface of the earth are excited which gives rise to violent vibrations in some cases. The material structures which resist the oscillatory vibration are of much interest to engineers and architects. The idea of introducing a continuous self-reinforcement at every point of an elastic solid is given by Belfield et al. [1].

The propagation of time-harmonic elastic waves in a fibre-reinforced composite were studied by Bose and Mal [2]. Scott and Hayes [3] discussed the small vibrations of a fibre-reinforced composite. Scott [4, 5] studied the waves in a fibre-reinforced elastic material. Sengupta and Nath [6] considered the surface waves in fibre-reinforced anisotropic elastic media. The reflection of qP and qSV waves at the free surface of a fibre-reinforced anisotropic elastic half-space was studied by Singh and Singh [7]. Singh [8] obtained the reflection coefficients from the free surface of an incompressible transversely isotropic fibre-reinforced elastic half-space for the case when the outer slowness section is a re-entrant.

Surface waves play an important role in the study of earthquakes, geophysics and geodynamics. Rayleigh waves cause destruction to the structure due to their slower attenuation of the energy than that of the body waves. Surface waves in elastic solids were first studied by Lord Rayleigh [9] for an isotropic elastic solid. The extension of surface wave analysis and other wave propagation problems to anisotropic elastic materials has been the subject of many studies; see, for example ([10–24]).

Impedance boundary conditions mean a linear combination of the unknown functions and their derivatives is prescribed on the boundary. It is common to use impedance boundary conditions in various fields of physics like acoustics and electromagnetism. Rayleigh waves with impedance boundary conditions are significant in many fields of science and technology. However, very little literature on Rayleigh waves with impedance boundary conditions is available. For example, Malischewsky [25] studied Rayleigh waves with Tiersten’s impedance boundary conditions and obtained a secular equation. Godoy et al. [26] studied the existence and uniqueness of Rayleigh waves with impedance boundary conditions. Recently, Vinh and Hue [27] investigated the propagation of Rayleigh waves in an orthotropic and monoclinic half-space with impedance boundary conditions. In this paper, the propagation of Rayleigh wave in an incompressible transversely isotropic fibre-reinforced elastic medium is considered and an explicit secular equation is obtained under impedance boundary conditions. The iteration method is applied to compute the non-dimensional wave speed of a Rayleigh wave. The non-dimensional speed is plotted against a non-dimensional material parameter to show the effect of impedance and transverse isotropy.

2 Equations of motion

We consider an incompressible transversely isotropic fibre-reinforced elastic medium. The constitutive equation explaining the stress-strain response to small deformation of such a material is given by

where ε, σ and I denote the infinitesimal strain and stress tensors and the 3×3 identity tensors, respectively, and e is a unit vector defining the axis of transversely isotropy. In equation (1), p is the hydrostatic pressure required to maintain the incompressibility constraint

In equation (1), μ_L and μ_T are longitudinal and transverse shear moduli and μ_E is a weighted shear modulus, given by

where E_L and E_T are longitudinal and transverse Young’s moduli.

Consider a Cartesian coordinate system Ox₁x₂x₃, such that Ox₁ is parallel to the direction of transversely isotropy, the constitutive relation (1) is written in component form as

Let us consider a transversely isotropic fibre-reinforced elastic material occupying the half-space x₂<0, with boundary x₂=0. We consider a plane motion in the (x₁, x₂) plane with displacement components (u₁, u₂, u₃) such that

where t is time.

For an incompressible material, we have

from which we deduce the existence of a scalar function, denoted ψ(x₁, x₂, t), such that

Using equation (7) and the strain-displacement relation 2ε_ij =u_i,_j +u_j,_i , the stress components are written in terms of scalar function ψ and pressure function p as

where p=p(x₁, x₂, t) is the hydrostatic pressure associated with the incompressibility constraint, and c₁²=4μ_E -μ_T , c₂²=μ_T , c₃²=μ_L . The requirements of a physically reasonable response ensure that c₁², c₂² and c₃² are all positive.

In absence of body forces, the equations of motion are

which upon using (8)–(10) give

Elimination of p by cross differentiation leads to the following equation of motion

3 Rayleigh waves

We consider the Rayleigh surface waves propagating along the direction x₁ and we write ψ in the form

where y=k x₂.

Using equation (15) into equation (14), we obtain

Following Godoy et al. [26] and Vinh and Hue [27], the impedance boundary conditions at the surface x₂=0 are

where ω=kc is circular frequency of wave. Z₁, Z₂ are impedance real valued parameters. With the help of equations (9) and (10), the above boundary conditions are written in terms of scalar function and pressure function as

Differentiating (19) with respect to x₁ and eleminating the terms in p with the help of (12), we have

In addition to conditions (18) and (20), we need also the following condition on ψ

In terms of ϕ, the conditions (18), (20) and (21) become

The general solution ϕ(y) of equation (16) that satisfies radiation condition (24) is

where A, B are constants and s₁, s₂ are solutions of following quadratic equation in s²

If the roots s₁² and s₂² of the quadratic equation (26) are real, then they must be positive to ensure that s₁ and s₂ can have a positive real part. If they are complex then they are conjugate. In either case, the product s₁²s₂² must be positive and hence a real (surface) wave speed satisfies the inequalities

Using solutions (25) into boundary conditions (22) and (23), we have

For non-trivial solution, we require the determinant of coefficients of the system of equations (28) and (29) to vanish, which after removal of factor (s₂-s₁), yields the secular equation in implicit form as

In order to make secular equation explicit, we introduce the notation η=1-ρc2c32. In view of (27), we may take η to be positive and its satisfies 0<η<1. Then s₁²s₂²=η² and we also have s₁²+s₂²=η²+ξ-3, where ξ=c12+c22c32, and thus (s₁±s₂)²=η²±2η+ξ-3. Since s₁² and s₂² are complex conjugates, therefore (s₁+s₂)²(s₁-s₂)²=(s₁²-s₂²)²<0 so that (η²+2η+ξ-3)(η²–2η+ξ-3)<0 and hence, since η>0, η²+2η+ξ-3>0, η²-2η+ξ-3<0. Thus s₁+s₂ is real and without loss of generality we may take s1+s2=+η2+2η+ξ-3, since s₁ and s₂ must have positive real parts. It follows that 4s₁s₂=(s₁+s₂)²-(s₁-s₂)²=4η>0.

Then, the secular equation (30) is expressed in explicit form

where

For ξ=4(μ_E =μ_L ), the above equation corresponds to isotropic case. In absence of impedance parameters, i.e. for Z₁=0 and Z₂=0, the secular equation (31) reduces to

The equation (31) has atleast one solution in the interval (0, 1), if Z2∗(Z1∗+ξ-3)<1. Keeping in view of this inequality, the values of ξ, Z₁^* and Z₂^* are chosen for numerical computation of non-dimensional speed of a Rayleigh wave.

4 Numerical results

Using iteration method, the equation (31) is solved for the non-dimensional wave speed x=ρc2c32 for given values of ratio ξ of material constant and impedance parameters Z₁^* and Z₂^*. The non-dimensional wave speed x=ρc2c32 is plotted against ξ in Figure 1 for Z₁^*=0, Z₂^*=0 (dotted curve), for Z₁^*=-0.02, Z₂^*=-0.03 (solid curve),and for Z₁^*=0.02, Z₂^*=0.03 (solid curve with centre symbols). The non-dimensional wave speed increase sharply with the increase in value of ξ. However, this sharpness decreases slowly with the increase in ξ. The comparison of these curves in Figure 1 shows the effect of impedance on non-dimensional wave speed. The non-dimensional wave speed is also computed and plotted against Z₁^* and Z₂^* in Figures 2 and 3, respectively for ξ=5. From Figures 2 and 3, effect of impedance parameters Z₁^* and Z₂^* on non-dimensional wave speed is observed significantly.

Figure 1:

Variation of non-dimensional speed of a Rayleigh wave against the non-dimensional material constant ξ, when Z1∗=0.02, Z2∗=0.03 (solid curve with centre symbols), Z1∗=-0.02, Z2∗=-0.03 (solid curve), and Z1∗=0, Z2∗=0 (dotted curve).

Figure 2:

Variation of non-dimensional speed of a Rayleigh wave against an impedance parameter Z1∗, when ξ=0 and for Z2∗=0.5 (solid curve with centre symbols), Z2∗=-0.5 (solid curve), and Z2∗=0 (dotted curve).

Figure 3:

Variation of non-dimensional speed of a Rayleigh wave against an impedance parameter Z2∗, when ξ=5 and for Z1∗=0.5 (solid curve with centre symbols), Z1∗=-0.5 (solid curve), and Z1∗=0 (dotted curve).