Propagation of Rayleigh waves in an incompressible rotating orthotropic elastic solid half-space with impedance boundary conditions

1 Introduction

Rayleigh [1] investigated elastic surface waves for compressible isotropic elastic solids. These surface waves have been studied extensively and have a wide range of applications in seismology, acoustics, geophysics and materials science. The explicit secular equations of Rayleigh waves are important in solving the direct (forward) problems, in evaluating the dependence of the wave velocity on material parameters and in determining material parameters from measured values of wave velocity.

Godoy et al. [2] mentioned that it is common to use impedance boundary conditions in many fields of physics. In impedance boundary conditions, a linear combination of a unknown function and their derivatives is prescribed on the boundary. See, for example, Vinh and Hue [3, 4] and the references therein. In study of Rayleigh waves propagation in a half-space coated by a thin layer, the researchers (Vinh et al. [5] and references therein) often replace the effect of the thin layer on the half-space by the effective boundary conditions on the surface of the half-space. These conditions result into the impedance boundary conditions on the surface. The Rayleigh wave is then considered as a surface wave which propagates in a half-space without coating whose surface is not traction-free but is subjected the impedance boundary conditions. However, investigations on Rayleigh waves with impedance boundary conditions are limited. For example, Malischewsky [6] considered the propagation of Rayleigh waves with Tiersten [7]’s impedance boundary conditions and obtained a secular equation. Godoy et al. [2] investigated the existence and uniqueness of Rayleigh waves with impedance boundary conditions in an isotropic elastic half-space. Vinh and Hue [3, 4] studied the Rayleigh waves with impedance boundary conditions in incompressible and compressible anisotropic half-spaces. Vinh and Xuan [8] derived a formula for the velocity of Rayleigh waves with impedance boundary condition and discussed the existence and uniqueness. Singh [9] studied the Rayleigh waves in an incompressible fibre-reinforced elastic solid with impedance boundary conditions. Singh [10] also studied the Rayleigh wave in a thermoelastic solid half-space with impedance boundary conditions.

Vinh and Hue [3] obtained a non-dimensional secular equation for Rayleigh wave in an incompressible orthotropic elastic half-space with impedance boundary conditions. However, they did not solve the secular equation numerically to obtain the wave speed of Rayleigh wave. In this paper, the problem of Vinh and Hue [3] is revisited with rotation as an additional parameter. The propagation of Rayleigh waves with impedance boundary conditions in a rotating orthotropic incompressible elastic half-space is studied. The secular equation is obtained by employing the traditional techniques as in Ogden and Vinh [11]. The non-dimensional secular equation for Rayleigh wave is solved numerically in MATLAB in presence as well as in absence of rotation.

2 Equations of motion

Consider an incompressible orthotropic elastic half-space occupying the region x₂≥0. In this problem, the displacement components are taken as

where t is the time. Following Nair and Sotiropoulo [12, 13], the strain–stress relations are given by

and the strain energy of the material is positive semi-definite if and only if

where c_ij are the material constants, σ_ij are the stress components, p=p(x₁, x₂, t) is the hydrostatic pressure associated with the incompressibility constraints, commas indicate differentiation with respect to spatial variables x_k. For an incompressible material, we have

From equation (4), there exists a scalar function ψ(x₁, x₂, t) such that

Following Schoenberg and Censor [14], the equations of motion for rotating half-space in absence of body forces are

where Ω is component of rotation vector ω in x₃ direction, ρ is the mass density, a superposed dot denotes differentiation with respect to t. Using equations (2) and (5) in equation (6), an equation in ψ is obtained after elimination p,

3 Rayleigh waves

We consider the propagation of Rayleigh wave, travelling with velocity c(>0) and wave number k(>0) in the x₁-direction and decaying in the x₂-direction, that is

Following Godoy et al. [2] and Malischewsky [6], the surface x₂=0 is assumed to be subjected to following impedance boundary conditions

where ω=kc is the wave circular frequency, Z₁, Z₂ are real numbers known as impedance parameters with dimension of stress/velocity. With the use of equations (2), (5) and the first equation of (6), the impedance boundary conditions (9) are written as

From equations (5) and (8), we obtain following radiation condition

Following Ogden and Vinh [11], the scalar function ψ(x₁, x₂, t) is obtained as

where y=kx₂. On substituting equation (12) in equation (7), we obtain

where a prime indicates differentiation with respect to y, δ=(c11+c22−2c12)/c66, x=c2/c22, c22=c66/ρ, Ωo=Ω/kc.

With the help of equation (12), the impedance boundary conditions (10) are expressed in terms of ϕ as

where δj=Zj/ρc66(j=1, 2) are dimensionless real impedance parameters. From Equations (11) and (12) it follows that

Following Ogden and Vinh [11], the general solution ϕ(y) of equation (13) which satisfies the radiation condition (15) is as

where the constants A and B are to be determined and s12 and s22 are the roots of following equation

with positive real parts. From Equation (17), the sum and product of roots s12 and s22 are

In present medium, the Rayleigh wave exist if s12⋅s22>0, i.e.

The solutions (16) will satisfy the boundary conditions (14) under following conditions

For a non trivial solution, the determinant of coefficients of the system (20) must vanish. After the removal of the factor (s₂−s₁), we obtain

Using equation (18), the equation (21) becomes

Equation (22) is a dimensionless secular equation of Rayleigh waves propagating in an incompressible rotating orthotropic elastic half-space whose surface is subjected to the impedance boundary conditions. In absence of rotataion (Ω=0), this equation reduces to equation (23) in Vinh and Hue [3].

For δ₁=δ₂=0, the equation (22) reduces to following dimensionless secular equation of Rayleigh waves propagating along a traction-free surface of incompressible rotating orthotropic elastic half-space

which in absence of rotation coincides with those obtained by Ogden and Vinh [11].

For incompressible transversely isotropic rotating elastic case (with the isotropic axis being x₃-axis), c₁₁=c₂₂, c₁₁−c₁₂=2c₆₆, and δ=4. Then equation (22) reduces to

For incompressible rotating isotropic materials, the secular equation is also of the form (24) in which x=ρc²/μ, μ is the shear modulus.

4 Numerical results

The secular equation for Rayleigh surface wave is solved numerically with the help of programming in MATLAB software. The non-dimensional speed of the Rayleigh wave in an incompressible rotating orthotropic medium is computed and plotted in Figures 1–14 against the non-dimensional constant, rotation parameter and impedance parameters.

Figure 1:

Variations of non-dimensional phase speed (x) against non-dimensional constant δ for impedance parameters δ₁=−5; δ₂=−5.

Figure 2:

Variations of non-dimensional phase speed (x) against non-dimensional constant δ for impedance parameters δ₁=0; δ₂=0.

Figure 3:

Variations of non-dimensional phase speed (x) against non-dimensional constant δ for impedance parameters δ₁=10; δ₂=10.

Figure 4:

Variations of non-dimensional phase speed (x) against impedance parameter δ₁ when δ₂=0 and Ω₀=0.

Figure 5:

Variations of non-dimensional phase speed (x) against impedance parameter δ₁ when δ₂=0 and Ω₀=5.

Figure 6:

Variations of non-dimensional phase speed (x) against impedance parameter δ₁ when δ₂=0 and Ω₀=10.

Figure 7:

Variations of non-dimensional phase speed (x) against impedance parameter δ₁ when δ₂=0 and Ω₀=15.

Figure 8:

Variations of non-dimensional phase speed (x) against impedance parameter δ₂ when δ₁=0 and Ω₀=0.

Figure 9:

Variations of non-dimensional phase speed (x) against impedance parameter δ₂ when δ₁=0 and Ω₀=5.

Figure 10:

Variations of non-dimensional phase speed (x) against impedance parameter δ₂ when δ₁=0 and Ω₀=10.

Figure 11:

Variations of non-dimensional phase speed (x) against impedance parameter δ₂ when δ₁=0 and Ω₀=15.

Figure 12:

Variations of non-dimensional phase speed (x) against rotation parameter Ω₀ when impedance parameters δ₁=−5 and δ₂=−5.

Figure 13:

Variations of non-dimensional phase speed (x) against rotation parameter Ω₀ when impedance parameters δ₁=0 and δ₂=0.

Figure 14:

Variations of non-dimensional phase speed (x) against rotation parameter Ω₀ when impedance parameters δ₁=10 and δ₂=10.

The variations of non-dimensional phase speed x=c/c66/ρ against non-dimensional constant δ=(c₁₁+c₂₂−2c₁₂)/c₆₆ are shown graphically in Figures 1–3 for three different sets of impedance parameters (δ₁=−5, δ₂=−5);(δ₁=0, δ₂=0) and (δ₁=10, δ₂=10) when rotation rate Ω₀=0, 5, 10 and 15. In Figures 1–3, the variations of non-dimensional speed against non-dimensional constant δ are similar for three different sets of impedance parameters. The minimum non-dimensional phase speed is nearly zero at δ=0. It increases sharply for the range 0<δ<2 and then increases slowly its maximum value at δ=10. For Ω₀=0, it attains maximum values 0.6187, 0.9940 and 0.3145 for three different sets of impedance parameters (δ₁=−5, δ₂=−5);(δ₁=0, δ₂=0) and (δ₁=10, δ₂=10), respectively. These variations are shown by solid lines in Figures 1–3. The variations corresponding to rotation parameter Ω₀=5, 10 and 15 in these figures are shown by dashed lines. With the increase in value of rotation parameter, the non-dimensional phase speed decreases at each value of δ. Significant effects of impedance parameters δ₁;δ₂ and rotation parameter Ω₀ are observed on non-dimensional phase speed of Rayleigh wave at each value of non-dimensional constant δ. In particular, the effects of impedance parameters are observed on non-dimensional speed of Rayleigh wave, when we compare the results in Figures 1 and 3 with those Figure 2.

The variations of non-dimensional phase speed x against impedance parameter δ₁ are shown graphically in Figures 4–7 when δ₂=0. The solid lines in these figures correspond to the variations of non-dimensional speed for δ=2 and for different values of Ω₀, whereas dashed lines in these figures correspond to δ=4, 6 and 8. From Figures 4–7, the effects of rotation parameter Ω₀ and non-dimensional constant δ are observed at each value of impedance parameter δ₁. Similarly, the effects of rotation parameter Ω₀ and non-dimensional constant δ are observed at each value of impedance parameter δ₂ as shown in Figures 8–11.

The variations of non-dimensional phase speed x against rotation parameter Ω₀ are shown graphically in Figures 12–14 for three different sets of impedance parameters (δ₁=−5, δ₂=−5);(δ₁=0, δ₂=0) and (δ₁=10, δ₂=10) when non-dimensional constant δ₀=2, 4, 6 and 8. The solid lines in these figures correspond to the variations of non-dimensional speed for δ=2 for different combinations of impedance parameters, whereas dashed lines in these figures correspond to δ=4, 6 and 8. Significant effects of non-dimensional constant is observed at each value of rotation parameter. In particular, the effects of impedance parameters are observed on non-dimensional speed of Rayleigh wave at each value of Ω₀, when we compare the results in Figures 12 and 14 with those in Figure 13.

5 Conclusion

A problem on propagation of Rayleigh wave in an incompressible orthotropic rotating elastic half-space is considered. A non-dimensional secular equation for Rayleigh wave is obtained by using traditional techniques. The non-dimensional secular equation for Rayleigh wave is solved numerically with use of programming in MATLAB software. The numerical values of non-dimensional wave speed of Rayleigh are plotted against non-dimensional material constant, impedance parameters and rotation parameter. In absence of rotation, the present problem reduces to as considered by Vinh and Hue [3]. These numerical results also include the results in absence of rotation, which validate the theoretical results obtained by Vinh and Hue [3] for orthotropic case.