Motion of the particles due to viscoelastic surface waves of an assigned frequency

Abstract

The motion of the material particles due to viscoelastic properties of the half-space is considered when the propagating surface wave has an assigned frequency. The detailed examination shows that for Rayleigh waves in classical elasticity as well as for the viscoelastic Rayleigh-type surface waves the sense of the particle path along the ellipse is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. When there is a viscoelastic surface wave that does not satisfy the adopted four criteria for behaviour at infinity (the case of viscoelastic Rayleigh-type surface wave) but only two of them, this additional wave is direct on the surface of the half-space and does not change or may change many times the sense of the particle path along the ellipse with the distance from the stress-free surface of the half-space. In contrast to the elastic case, the ellipse axes in the viscoelastic case are not parallel and orthogonal to the surface of the half-space, respectively. Their orientation depends on the magnitude of the viscous part of the Lamé moduli and tends to a constant at great depth. The numerical computations in the paper refer to some typical values of the complex Lamé moduli and to some real materials.

Keywords

Viscoelasticity surface waves frequency

1. Introduction

The characteristic features of Rayleigh waves [1] do not remain valid when viscous properties are considered. Then we try to find waves with a structure and mechanical characteristics which are very similar to these of the Rayleigh waves [2, 3]. We call them Rayleigh-type surface waves. These waves are dispersive and it is possible to consider two different cases of wave propagation: waves of an assigned frequency or waves of an assigned wavelength. They may be interpreted as a superposition of two inhomogeneous plane waves. For each of them the plane of constant phase and the plane of constant amplitude are not orthogonal. The superposed waves have different directions of propagation which are not parallel to the surface, different phase velocities and attenuation coefficient values. The planes of constant phase of the superposed waves intersect the surface of the half-space at one and the same straight line. This line moves parallel to itself with a speed usually called the speed of the surface wave.

In order to preserve these most characteristic features of Rayleigh waves known from classical elasticity, the following criteria for behaviour at infinity are adopted (it is assumed that the semi-infinite body occupies the region z ≥ 0, where Oxyz is a rectangular Cartesian coordinate system and the wave propagates along the x-axis):

the amplitudes of the superposed waves do not increase for the points of intersection of the propagating planes of constant phase with any fixed line parallel to the x-axis (surface waves of an assigned frequency) or with the time at a fixed point of the half-space (surface waves of an assigned wavelength);

the amplitudes of the superposed waves decay exponentially with the increasing of the distance from the surface of the half-space on any fixed line parallel to the z-axis;

the amplitudes of the superposed waves do not increase for any fixed point on a plane of constant phase in the direction of its propagation;

the amplitudes of the superposed waves tend to zero with the increasing of the distance from the surface of the half-space for the points of the lines in which the planes of constant phase at any fixed time intersect the coordinate plane Oxz.

If only the first two criteria for behaviour at infinity are adopted, another surface wave with quite different character from the classical Rayleigh wave could appear and we could not consider it as a Rayleigh-type surface wave. This is the reason these two additional criteria for behaviour at infinity to be accepted. It is perhaps worth recalling also that in classical elasticity the criteria (iii) and (iv) coincide with the criteria (i) and (ii), respectively, and do not add additional restrictions. We could summarize the obtained results in [2, 3] as follows:

- a unique viscoelastic surface wave (Rayleigh-type surface waves) of an assigned frequency or an assigned wavelength always exists for viscoelastic materials, when the four criteria for behaviour at infinity are satisfied;

- for some viscoelastic materials there exists a second surface wave of an assigned frequency that satisfied only first two criteria for behaviour at infinity; this additional wave is quite different from the classical Rayleigh wave and cannot be considered as a Rayleigh-type surface wave.

Surface waves of an assigned frequency in viscoelastic materials have been examined by many authors. Currie et al. [4] and Currie and O’Leary [5] showed that in contrast with elastic materials the waves may be either direct or retrograde at the surface. They also found that two surface waves exist for some values of the complex Lamé moduli when the two criteria for behaviour at infinity adopted in their papers are satisfied. On the other hand Romeo [6, 7] found “that the secular equation for Rayleigh waves in viscoelastic half-space always admits only one complex root”. He explains this result with correspondence principal between elastic and viscoelastic problems and supposes that when the imaginary parts of the complex Lamé moduli tend to zero the solution of the secular equation “tends to a non admissible root of the elastic problem, which is spurious solution introduced by squaring the secular equation”. In our earlier paper [2] we brought attention to this problem and confirmed that for some materials there are two surface waves of an assigned frequency which correspond to two different complex roots of the secular equation if only the criteria (i) and (ii) for behaviour at infinity, adopted by Currie et al. [4] and Currie and O’Leary [5], are satisfied. The numerical results for different combinations of complex Lamé moduli and for some real materials show that the second surface wave could exist only for some special values of material parameters, but these results do not allow us to find mathematical conditions for coexistence of both waves.

The motion of the particles due to viscoelastic properties of the half-space is considered in this paper when the propagating surface wave has an assigned frequency. It is important to mention here that the sense in which the particle path (ellipse in our case) is described is a matter of definition. The sense depends on the side from which we look at the ellipse. Nevertheless it is useful to investigate how the sense of particle paths changes in the different cases when viscous properties are taken into account. The motion of the particles due to viscoelastic properties of the half-space is considered in [8] when the propagating surface wave has an assigned wavelength. It is proved that in the elastic case as well as in the viscoelastic case at different values of the viscous part of the Lamé moduli the sense of the particle path is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. The only difference is that, in contrast to the elastic case, the ellipse axes in the viscoelastic case are not parallel and orthogonal respectively to the stress-free surface. More results connecting the elastic case only could be found in the paper by Malischewsky et al. [9] and the references therein.

In this paper the main attention is brought to compare the particle path behaviour between the Rayleigh-type surface waves and the additional wave in the case of viscoelastic waves of an assigned frequency. Our motivation is connected with the description of an additional mechanical reason which shows again that the second surface wave is quite different from the Rayleigh-type surface wave. The numerical computations refer to some typical values of the Lamé moduli and to some real materials. The detailed examination shows that for Rayleigh waves in classical elasticity as well as for the viscoelastic Rayleigh-type surface waves the sense of the particle path along the ellipse is retrograde on the surface of the half-space and changes into direct only once at depth between $\frac{1}{7}$ and $\frac{1}{4}$ of the wavelength. When an additional surface wave exists, this wave is direct on the surface of the half-space and does not change or may change many times the sense of the particle path along the ellipse with the distance from the stress-free surface of the half-space. This wave decays much slower than the Rayleigh-type wave.

2. Solution for surface waves of an assigned frequency

Let the semi-infinite body with mass density ρ occupy the region z ≥ 0. The displacement components are denoted by u, v and w. The motion is governed by the linearized equations of isotropic viscoelasticity, where external body forces are disregarded. It is a straightforward task to construct a surface-wave solution when it is assumed that, throughout the motion, the boundary of the body z = 0 is stress-free. This solution does not depend on y and could be interpreted as a superposition of two dispersive inhomogeneous plane waves [2] where

\begin{matrix} u = e^{- q_{x} x} R e (B e^{i γ}), v = 0, w = e^{- q_{x} x} R e (C e^{i γ}), \end{matrix}

(1)

with the following notation introduced:

\begin{array}{l} B = (G_{1} + A_{2} b_{2} G_{2}) A, C = i (b_{1} G_{1} + A_{2} G_{2}) A, γ = ω (\frac{x}{v_{x}} - t), \\ b_{1}^{2} = 1 - v_{0}^{2} s^{2}, b_{2}^{2} = 1 - s^{2}, s = \frac{ω}{p v_{S}}, v_{0} = \frac{v_{S}}{v_{T}}, \\ v_{S} = {v^{'}}_{S} v_{S}^{*}, {v^{'}}_{S} = \sqrt{\frac{μ^{+}}{ρ}}, v_{S}^{*} = \sqrt{1 + i \frac{μ^{-}}{μ^{+}}}, v_{T} = \sqrt{\frac{λ + 2 μ}{ρ}}, \\ A_{2} = \frac{2 b_{1}}{s^{2} - 2}, Λ = \frac{2 π}{ω} v_{x}, G_{n} = e^{- 2 π \frac{v_{x}}{{v^{'}}_{S}} (\frac{q_{z}^{(n)}}{q_{0}^{'}} - i \frac{v_{S}^{'}}{v_{z}^{(n)}}) \frac{z}{Λ}}, \\ q_{x} = {q^{'}}_{0} I m [{(s v_{S}^{*})}^{- 1}], q_{z}^{(n)} = {q^{'}}_{0} R e [b_{n} {(s v_{S}^{*})}^{- 1}], {q^{'}}_{0} = \frac{ω}{{v^{'}}_{S}}, \\ v_{x} = \frac{v_{S}^{'}}{R e [{(s v_{S}^{*})}^{- 1}]}, v_{z}^{(n)} = - \frac{v_{S}^{'}}{I m [b_{n} {(s v_{S}^{*})}^{- 1}]}, (n = 1, 2) . \end{array}

(2)

Here ω is the angular frequency, taken to be real and positive and Λ is the projection of the wavelength of the superposed waves on the x-axis. The constant A is an arbitrary real or complex number and t is the time. The secular equation which determines s and hence through (2)₆ the complex wave number p can be stated as

{(2 - s^{2})}^{2} = 4 \sqrt{1 - v_{0}^{2} s^{2}} \sqrt{1 - s^{2}} .

(3)

The complex Lamé moduli λ = λ⁺ + iλ⁻ and μ = μ⁺ + iμ⁻ satisfy the following inequalities

μ^{+} > 0, 3 λ^{+} + 2 μ^{+} > 0, μ^{-} \leq 0, 3 λ^{-} + 2 μ^{-} \leq 0 .

(4)

In the case of surface waves of an assigned frequency [5, 10] from

\begin{matrix} \frac{μ^{-}}{μ^{+}} = - ω β, \frac{λ^{-}}{μ^{+}} = ω [\frac{2}{3} β - (\frac{λ^{+}}{μ^{+}} + \frac{2}{3}) η] \end{matrix}

for the Kelvin–Voigt material and

\begin{matrix} \frac{μ^{-}}{μ^{+}} = - \frac{1}{ω β}, \frac{λ^{-}}{μ^{+}} = \frac{1}{ω} [\frac{2}{3} \frac{1}{β} - (\frac{λ^{+}}{μ^{+}} + \frac{2}{3}) \frac{1}{η}] \end{matrix}

for the Maxwell material we could determine for each of these viscoelastic models the dimensionless Lamé moduli $\frac{μ^{-}}{μ^{+}}$ and $\frac{λ^{-}}{μ^{+}}$ if we know the dimensionless Lamé modulus $\frac{λ^{+}}{μ^{+}}$ and the constant relaxation times β and η and conversely we could determine the relaxation times β and η if we know $\frac{λ^{+}}{μ^{+}}$ , $\frac{μ^{-}}{μ^{+}}$ and $\frac{λ^{-}}{μ^{+}}$ . We assume for definiteness hereafter that v_x is positive.

3. Particle paths

At a given point with coordinates (x, y, z), the elliptical orbit is found from displacements (1) after eliminating exponent eⁱ^γ, connecting with time. Then

C \bar{C} u^{2} - (B \bar{C} + \bar{B} C) u w + B \bar{B} w^{2} = - \frac{e^{- 2 q_{x} x}}{4} {(B \bar{C} - \bar{B} C)}^{2} .

(5)

The bar over a function denotes the complex conjugate value of this function. The angle φ made by one of the axes of the ellipse (5) with the x-axis is given by

t g 2 φ = \frac{B \bar{C} + \bar{B} C}{B \bar{B} - C \bar{C}} .

(6)

The ellipse is described in a direct sense if P > 0 and retrograde when P < 0 where

P = i (B \bar{C} - \bar{B} C) .

(7)

Since P is a continuous function with respect to z, the particle path could change from retrograde to direct or vice versa with the distance from the stress-free surface of the half-space. The ellipse degenerates into a straight line if P = 0, as it follows from (5). It means that at the depth before the direction of particle paths to be changed one of the axis of the ellipse tends to zero. At the depth where the direction of particles changes one of the axes of the ellipse is zero, i.e. the particle paths are straight lines coinciding with the other axis. After that, with increasing the depth the first axis begins to increase, but now the particle paths change to opposite. For elastic materials it is known that Rayleigh waves are always retrograde at the top surface, but that they change from retrograde to direct at a certain level below the surface. The axes of the ellipse in this case are parallel to the coordinate axes since φ = 0. When the ellipse degenerates to a straight line this line is vertical. Then $B \bar{C} = \bar{B} C = 0$ .

4. Numerical results

In this section we investigate corresponding results for viscoelastic materials in the case of surface waves of an assigned frequency. The numerical computations refer to 64 different values of the ratios λ⁺/μ⁺, λ⁻/μ⁺, μ⁻/μ⁺ and to some real materials, considered in [4, 5]. Since the results are similar we restrict ourself to six typical viscoelastic cases and corresponding elastic one, given in Table 1.

Table 1.

Values of the used material ratios.

Materials	λ⁺/μ⁺	λ⁻/μ⁺	μ−/μ⁺
1	1	−0.06	−0.04
2	0.5	−0.05	−0.15
3	0.2	−0.2	−0.24
4	0.2	−2.4	−2
5 (Granite)	0.776	−0.063	−0.035
6 (Shale)	5.43	−0.024	−0.103
7 (Elastic)	1	0	0

The numerical results show that one (materials 2, 6 and 7) or two (materials 1, 3, 4 and 5) solutions exist, satisfying Equation (3) when the boundary of the body z = 0 is stress-free, but one and only one of them satisfies the adopted four criteria for behaviour at infinity. This wave could be considered as a Rayleigh-type surface wave. Its characteristics are given for each of the materials in the first row of the table. In the next row the same characteristics are shown for the second solution (if it exists) when this additional wave satisfies only first two criteria for behaviour at infinity. From the numerical results in Table 2 it follows that in the elastic case as well as in the viscoelastic case at different values of the viscous part of the Lamé moduli the sense of the particle paths for Rayleigh-type surface wave is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. For the additional wave the particle paths are direct on the surface of the half-space and do not change or may change many times the sense of the particle path along the ellipse with the distance from the surface of the half-space. These changes for materials 1, 3 and 5 after the first change (see Table 2) are shown in Table 3. The first particle path change of the additional wave is closer to the surface of the half-space than this change for the Rayleigh-type surface wave. The last change of the particle path is shown at the end of Table 3 after dots. Material 4 is not mentioned in Table 3 since its particle path does not change direction. In contrast to the elastic case the ellipse axes in the viscoelastic case are not parallel and orthogonal respectively to the stress-free surface since φ≠0. Their orientation depends on the magnitude of the viscous part of the Lamé moduli.

Table 2.

Changes in the particle path with the distance from the surface of the half-space.

Material		z/Λ = 0	No	direction		z/Λ = 1		z/Λ = 10		z/Λ = 100
	Direction	φ	z/Λ	φ	Direction	φ	Direction	φ	Direction	φ
1	R	0.0023	0.192	−0.006	D	−0.0026	D	−0.0011	D	−0.0011
	D	−0.375	0.009	−0.374	D	−0.287	D	−0.2899	D	−0.2232
2	R	−0.005	0.212	−0.027	D	−0.0081	D	0.0025	D	0.0025
3	R	0.043	0.228	−0.027	D	−0.0274	D	−0.0215	D	−0.0215
	D	−0.109	0.088	−0.11	R	−0.0693	D	−0.0085	D	−0.0085
4	R	0.0436	0.162	−0.165	D	−0.0474	D	−0.0218	D	−0.0218
	D	−0.349	No	change	D	−0.1383	D	−0.1376	D	−0.1376
5	R	0.0052	0.2	−0.004	D	−0.0034	D	−0.0026	D	−0.0026
	D	−0.295	0.011	−0.295	R	0.2371	D	−0.1492	D	−0.1229
6	R	−0.006	0.154	−0.017	D	0.0004	D	0.0028	D	0.0028
7	R	0	0.193	0	D	0	D	0	D	0

Here R and D denote retrograde and direct particle path respectively.

Table 3.

Additional changes in the particle path including last one.

Material	Direction	z/Λ	φ	Amplitude	Direction	z/Λ	φ	Amplitude
1	R-D	0.295	0.335	0.404	D-R	0.599	−0.368	1.108
3	R-D	0.491	0.093	0.862	D-R	0.983	−0.072	0.991
5	R-D	0.341	0.284	0.569	D-R	0.69	−0.29	1.161

Material	Direction	z/Λ	φ	Amplitude	Direction	z/Λ	φ	Amplitude

1	R-D	0.885	0.265	0.431	D-R	1.189	−0.362	0.537
3	R-D	1.359	0.05	0.836	D-R	1.886	−0.046	0.886
5	R-D	1.019	0.26	0.415	D-R	1.368	−0.286	1.113

Material	Direction	z/Λ	φ	Amplitude	…Direction	z/Λ	φ	Amplitude

1	R-D	1.474	0.206	0.419	…R-D	26.7	−0.222	0.266
3	R-D	2.217	0.025	0.794	…R-D	3.06	0.008	0.747
5	R-D	1.697	0.238	0.55	…R-D	51.1	−0.115	0.043

With Ampl. we denote maximal displacement amplitude along x and z axes, i.e. max(|ReB|, |ImB|, |ReC |, |ImC|) where A = 1.

The magnitudes of these amplitudes are shown in Table 4. For the Rayleigh-type surface wave they are very much smaller at a distance 10 times the wavelength from the surface than at the surface of the half-space. The additional wave decays many times slower than the Rayleigh-type wave. At a distance 10 times the wavelength from the surface, the amplitude of the additional wave is very much larger than the amplitude of the Rayleigh-type wave.

Table 4.

Changes in amplitude values with the distance from the surface of the half-space.

Material	Amplitude	No	direction	Amplitude	Amplitude	Amplitude
	z/Λ = 0	z/Λ	Amplitude	z/Λ = 1	z/Λ = 10	z/Λ = 100
1	0.62048	0.19243	0.60833	0.11975	10⁻¹⁰	10⁻¹⁰⁸
	1.57733	0.0087	1.57577	0.83091	0.72902	0.00951
2	0.57	0.21192	0.51302	0.08988	10⁻¹²	10⁻¹¹⁶
3	0.53227	0.22766	0.44616	0.06629	10⁻¹²	10⁻¹²²
	1.11707	0.08833	1.12209	0.9873	0.33706	0.0002
4	0.62537	0.16212	0.65663	0.13666	10⁻¹¹	10⁻¹⁰⁴
	1.47959	No	change	0.68319	0.0211	10⁻¹⁷
5	0.60022	0.19983	0.57088	0.108	10⁻¹¹	10⁻¹¹¹
	1.40982	0.01088	1.40899	0.47221	0.56022	0.00363
6	0.75724	0.15378	0.85776	0.21471	10⁻⁹	10⁻⁸⁹
7	0.62	0.1925	0.61	0.12	10⁻¹¹	10⁻¹⁰⁸

5. Conclusions

The motion of the particles due to viscoelastic properties of the half-space is considered in this paper when the propagating surface wave has an assigned frequency. The numerical computations refer to some typical values of the Lamé moduli and to some real materials. The detailed examination shows the following:

- In the elastic case, as well as in the viscoelastic case at different values of the viscous part of the Lamé moduli and Rayleigh-type surface waves, the sense of the particle paths is retrograde on the surface of the half-space and changes into direct only once at depth about one seventh to one quarter of the wavelength. The amplitude connected with the ellipse axes is very much smaller at a distance from the surface 10 times the wavelength than at the surface.

- When an additional surface wave exists, this wave is direct on the surface of the half-space and does not change or may change many times the sense of the particle path along the ellipse with the distance from the stress-free surface of the half-space. The additional wave decays much slower than the Rayleigh-type wave. At a distance 10 times the wavelength from the surface the amplitude of this wave is very much larger than the amplitude of the Rayleigh-type wave.

- In contrast to the elastic case, the ellipse axes in the viscoelastic case are not parallel and orthogonal to the stress-free surface, respectively. Their orientation depends on the magnitude of the viscous part of the Lamé moduli and leads to a constant value when the distance from the stress-free surface of the half-space tends to infinity.

Footnotes

Declaration of conflicting interest

None declared.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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