NDT.net • May 2006 • Vol. 11 No.5

Love Surface Waves for Materials Evaluation

Edouard G. Nesvijski
Civil Engineering Department
University of Minnesota
Minneapolis, MN 55455
E-mail: nesvi002@umn.edu

Introduction

Original Love surface wave could be considered as wave propagating through a superficial solid layer covering a half-space solid body [1]. Historical overview of these types of surface waves starts with the first papers published by Love [2,3]. Then this problem was rediscovered and deeply analyzed in several papers [4-7]. Interest to Love surface waves has been supported by problems of geophysics and nondestructive testing of layered materials [8-12]. Results of a number of theoretical and applied researches of Love waves propagation were published during early nineties of the last century [13-19]. Some new approaches to practical applications of Love waves were made [20-23] and contemporary computational features for analysis of Love waves were published during the latest decade [24,25]. An important potential field of application of these types of waves is evaluation of laminated materials and coatings, including layered-silicates and thin film nanocomposites. Surface acoustic Love waves may become a powerful instrument for materials evaluation in laboratory as well as in situ keeping in mind high level of existing low frequency and ultrasonic measurement means for nondestructive testing (NDT). However, these types of surface waves have not yet become an everyday tool for practical NDT applications. One of the main reasons explaining this delay is a gap between the pure theory and practical needs laying in interpretation of measuring results. This chapter is devoted to the main features of Love surface wave propagation and their consideration for several important practical cases using computational approaches and graphical visualization of wave velocities and attenuation. Implementation of the computational approach given in this paper fills the gap between the theory and practical needs using supplemental MATLAB codes, that allow to apply Love wave technique to practical tasks of materials evaluation.

Models and Formulations

For easy understanding of the problem it has been assumed that the half-space solid body and the layer covering it are both homogeneous and isotropic, but each of them has different elastic properties. Thickness of the layer h is and its elastic properties could be described by moduli of elasticity such as shear module µL , and density rL . The half-space body could be also described by the similar parameters as shear module µ and density r correspondently. A computational model in Cartesian coordinates X and Z for analysis of Love wave propagation in a complex media is shown in the Figure 1.

Fig 1: A computational model for analysis of Love wave propagation in a complex media in Cartesian coordinates X and Z: 1- layer with a thickness h, shear module µL and density rL ; 2 - half-space solid body with shear µ module and density r.

Boundary conditions for this model are considered as continuous motions in the layer and the half-space body, which are determined by continuity of displacements and stresses along the verge of the half-space body and bonded surface of the layer where and absence of stress on the free surface of the layer, where z = -h .

Wave motion in the model presented in the Figure 1 is considered for horizontal polarization, where layer and plane body generate special types of surface waves with characteristics depending on material properties and layer thickness.

For boundary conditions for this model displacements UY ≠ 0 , UX = 0 , UZ = 0 , and derivation for and plane waves propagating in the direction X with displacement UY which perpendicular to this direction, wave equation will be displayed as a partial differential equation [1]

(1)

Where U UY · y0 and y0 - directional vector for wave propagation in horizontal plane.

The simple example of these types of waves with horizontal polarization in plane could be expressed as an expression:

(2)

Where A - arbitrary amplitude, kt -wave number of Love waves, w - angular frequency and t - time domain for waves satisfies boundary conditions described above.

This wave is unstable, and any small change in boundary conditions could transfer it to a different mode of surface wave. That is why this wave could exist only for limited cases of surface wave propagating through a superficial layer of a certain thickness over half-space body and materials elastic properties of the layer and the half-space body. The presented explanation allows to approached Love wave problem solution by putting limitation to permanent layer thickness h = const.

(3)

The displacement (2) should satisfy boundary conditions for half-space body with in wave equation (1) as combination of two plane surface waves in synchronized propagation along borders of layer and half-space body where A and B - arbitrary amplitudes, x,z - coordinates, and wave parameters and could be described by expressions for two branches of wave equation solution

(4)

Where kt1 and kt2 - wave numbers for shear waves in layer and body for the case of bulk waves propagation in two unbounded spaces, and k - wave number for Love wave. It is necessary to note that wave number k connected with wave velocity V and angular frequency w for each unbounded space under consideration as

(5)

where Vt1,Vt2 - shear wave velocities in materials of those unbounded spaces. Boundary conditions for this model are continues displacements and stresses for and unstressed condition for z = 0 . Substituting (3) by these conditions receive three linear homogeneous equations regarding arbitrary amplitudes AB and C

(6)

Assuming that determinant of linear system is equal to zero receive dispersion equation for Love wave propagation with wave velocities formulations. Using equation (6) it possible to present dispersion equation as

(7)

Where are - Vt1,Vt2 shear wave velocities in unbounded spaces of layer and body, V - velocity of Love wave.

Nonlinear equation (7) allows analyzing values of Love wave velocities V for different elastic properties of materials in layer and half-space body presented by shear modules µ,µL , densities r,rL and layer thickness h for different angular frequencies w.

Computational analysis of Love waves propagation

Generally the nonlinear equation (7) contains complex roots, and for some combinations of materials elastic parameters and thickness are generating real or imaginary roots. Moreover, Love waves may content different modes depending on frequency w of propagating waves and materials elastic properties. It is very important to present Love equation as a function regarding wave velocity and layer thickness for visualization of dispositions of Love equation roots as cross-sections of "zero plane" and the function below:

Computational images of this equation are presented in the Figures 2-5 for different angular frequencies w of propagating waves. Using common engineering notions of linear frequencies f = w/2p, the Figure 2 demonstrates the function for f = 100 kHz:

Fig 2: Love equation presented as a function calculated for shear modules µL/µ=1.4, shear velocity relation VL/V=1.15 and frequency f=100 kHz.

The Figure 3 presents Love equation as a function for frequency f=500 kHz:

Fig 3: Love equation presented as a function calculated for shear modules µL/µ=1.4, shear velocity relation VL/V=1.15 and frequency f=500 kHz.

The Figure 4 presents Love equation as the function for frequency f=1000 kHz:

Fig 4: Love equation presented as a function calculated for shear modules µL/=1.4, shear velocity relation VL/V=1.15 and frequency f=1000 kHz.

Visible cross-sections of the calculated functions with the "zero plane" demonstrate dispositions of the roots of Love equation.

Solutions of equation (7) could be obtained by computational approach using the Gauss-Newton algorithm. The least-squares optimization method was implemented for calculation of complex roots of Love equation (7) using MATLAB codes.

Calculated roots of Love equation are wave numbers. Love waves velocities could be calculated using those wave numbers and values of frequency. Typical graphs of calculated Love wave velocities are presented in the Figure 5, where relative parameter of ratio between wave length l = 2pV/w and layer thickness h:

Fig 5: Calculated Love wave velocities V versus relative ratio l/h for frequency f=100 kHz: a - real roots; b - imaginary roots.

Increase of frequency changes conditions of Love waves propagation and affects calculation results. Calculated data of Love waves velocities demonstrate these changes for frequency f=500 kHz in the Figure 6.

Fig 6: Calculated Love waves velocities V versus ratio l/h for frequency f=500 kHz: a - real roots; b - imaginary roots.

Considerable increase of frequency (over >1000 kHz) is displaying special behavior of complex roots. There is a "dead zone" for Love wave propagations, which is shown in the Figure 7.

Fig 7: Calculated Love waves velocities versus V ratio l/h for frequency f>1000 kHz: a - real roots; b - imaginary roots.

It is possible to observe different zones of roots:

a) a zone where imaginary roots are equal to zero and real roots exist;
b) a zone where imaginary roots have non-zero value and real roots exist;
c) a zone where complex solution does not exist (a "dead zone").

A physical explanation of the meaning of complex roots and complex Love waves velocities could be presented as complex data, where the real roots present velocity of pure Love waves and imaginary roots describe transformation of these waves to other types of waves. In the case (a) real roots present velocities of pure Love waves (imaginary roots are equal to zero) propagating in media. In the case (b), when imaginary roots have nonzero values, reflection of waves or their transformation to other types of surface and bulk waves takes place. Some combinations of material shear modules µ,µL , densities rrL and frequency w of propagating waves are generating conditions for the "dead zone" in the case (c).

Freelance amplitudes of Love waves could be extracted from (3), (4) and (5) as displacements in the following forms:

(8)

where UY(1) - displacement in layer along border; UY(2) - displacement in the half-space along border.

Calculation of these displacements (8) is shown in the Figure 8.

Fig 8: Displacements in Love waves for layer UY(1) (a) and half-space UY(2) (b) components for verity of frequencies ( f=100-1000 kHz).

The Figure 8 demonstrates that horizontal component for Love waves propagation. does not demonstrate any dependency of waves attenuation on frequency of propagating waves.

It is clear that condition Vt2<Vt2 generates real roots of equation (7) and Love wave numbers exist for limited values between wave numbers of shear waves if layer kt1 and half-space kt2 have relation:

kt1 < k < kt2 (9)

Nonlinear equation (7) has several solutions, which present different modes of Love waves, which depend on types of materials in layer and half-space as well as on thickness of layers h . Traditionally only the first mode of the solution is assumed as Love waves. But in practice, when wideband pulse signals is used, propagation of a number of modes is observed.

Interpretation of Love waves behavior in layered materials

It is possible to see from the graphs and formulae (4) and (8) that the displacements are constant through the layer thickness, but they depreciate along the border between the layer and the half-space. Different conditions could be considered:
  • If ratio between the layer thickness h and wave length l is growing, the phase velocity V of Love wave is approaching shear wave velocity Vt1 in the layer;
  • If the layer is very thin h/l <<1, Love wave velocity V is approaching shear wave velocity Vt2 in the half-space;
  • If the layer is very thick h/l >>1, Love wave velocity V in approaching Rayleigh wave velocity VR ;
  • If condition of wave conductivity between the layer and the half-space is changing to high impedance value, Love wave is transferring to Lamb wave in the layer;
  • If wave velocity in the layer is higher than in the half-space, effect of velocity acceleration of Love wave propagation is observed;
  • If velocity in the layer is less than in the half-space, a "brake" effect in the Love wave propagation is observed.

Some practical cases of Love wave applications

"Hard" layer on "soft" half-space:

The case of the "Hard" layer on the "soft" half-space represents a condition, when velocity in the layer material is higher than velocity in the half-space. Values of shear bulk waves velocities in the layer and half-space material are used for determination of elastic properties of these materials. An example of Love wave velocities calculation from equation (7) for shear modules relation µ/µL=1.55 and shear wave velocity in layer Vt1=1200 m/s, shear wave velocity in half-space Vt2=1000 m/s, layer thickness h=0.015 m, and frequency f=100 kHz is given in the Figure 9 below:

Fig 9: Love wave velocities calculation form equation (7) for shear modules relation µ/µL=1.55 and shear wave velocity in layer Vt1=1200 m/s, shear wave velocity in half-space Vt2=1000 m/s, layer thickness h=0.015 m, and frequency f=100 kHz. : a - real roots; b - imaginary roots.

"Soft" layer on the "hard" half-space:

The case of the "soft" layer on the "hard" half-space represents a condition, when velocity in the layer material is lower than velocities in the half-space. An example of Love wave velocities calculation form equation (7) for shear modules relation µ/µL=0.8 and shear wave velocity in layer Vt1=1000 m/s, shear wave velocity in half-space Vt2=1200, layer thickness h=0.015 m, and frequency f=500 kHz is given in the Figure 10.

Fig 10: Love wave velocities calculation form equation (7) for shear modules relation µ/µL=0.8 and shear wave velocity in layer Vt1=1000 m/s, shear wave velocity in half-space Vt2=1200, layer thickness h=0.015 m, and frequency f=500 kHz: a - real roots; b - imaginary roots.

Multi-mode solution for Love waves in a thin layer:

The case of a very thin layer on the half-space represents a condition, when multi-mode solution for Love wave propagation exists. It is possible to demonstrate that for some combinations of parameters Love waves may transform to other types of waves or may be presented by a multi-mode propagation. A solution for a thin layer is shown in the Figure 11 for shear modules relation µ/µL=1.16 and shear wave velocity in layer Vt1=1000 m/s, shear wave velocity in half-space Vt2=900, layer thickness h=0.001 m, and frequency f=700 kHz.

Fig 11: Love wave velocities calculation form equation (7) for shear modules relation µ/µL=1.16 and shear wave velocity in layer Vt1=1000 m/s, shear wave velocity in half-space Vt2=1200, layer thickness h=0.001 m, and frequency f=700 kHz: a - real roots; b - imaginary roots.

Conclusions

Love waves applications for materials testing need special attention because of complicity of problem and high dependency of these waves on a combination of parameters of materials in the layer and the half-space as well as a relation between the layer thickness and frequency of waves utilized for material evaluation. For each case of evaluation it is possible to apply computational approach for data analysis, optimization of measuring procedure and modeling of result. Therefore, practical applications of Love surface waves require preliminary analysis and adjustment of materials evaluation procedure. MATLAB codes for Love waves propagation analysis allow to facilitate evaluation procedure and interpretation of phenomena of these waves for different practical cases.

References

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