Dispersion of Love Waves in a Composite Layer Resting on Monoclinic Half-Space

Abstract

Dispersion of Love waves is studied in a fibre-reinforced layer resting on monoclinic half-space. The wave velocity equation has been obtained for a fiber-reinforced layer resting on monoclinic half space. Shear wave velocity ratio curve for Love waves has been shown graphically for fibre reinforced material layer resting on various monoclinic half-spaces. In a similar way, shear wave velocity ratio curve for Love waves has been plotted for an isotropic layer resting on various monoclinic half-spaces. From these curves, it has been observed that the curves are of similar type for a fibre reinforced layer resting on monoclinic half-spaces, and the shear wave velocity ratio ranges from 1.14 to 7.19, whereas for the case isotropic layer, this range varies from 1.0 to 2.19.

1. Introduction

Fiber-reinforced composite materials have become very attractive in many engineering applications recently due to their superiority over the structural materials in applications requiring high strength and stiffness in light-weight material. Consequently, the characterization of their mechanical behavior is an utmost requirement. The monoclinic system is the largest symmetry system with almost a third of all minerals belonging to one of its classes. This system contains two nonequal axes (𝑎 and 𝑏) that are perpendicular to each other and a third (𝑐), that is, inclined with respect to the axes. The 𝑎 and 𝑐 axis lie in a plane. The 𝑎-𝑐 plane can be, but is not always, a mirror plane with left side of 𝑏-axis a reflection of the right side. Fledspar which is an example of monoclinic material is the name of a group of rock-forming minerals which make up as much as 60% of earth’s crust. Feldspars crystallize from magma in both intrusive and extrusive igneous rocks, and they also can occur as compact minerals, as veins, and are also present in many types of metamorphic rock. Rock formed entirely of plagioclase feldspar is known as an orthosite. Feldspars are also found in many types of sedimentary rock. The wave propagation in reinforced medium was studied by Chattopadhyay and Choudhury [1] and in crystalline monoclinic plate was studied by Chattopadhyay and Bandyopadhyay [2]. Propagation of elastic waves in laminated composite plates was studied by Datta et al. [3]. Chattopadhyay et al. [4] studied the propagation, reflection, and transmission of shear waves in monoclinic media and obtained a dispersion equation for a monoclinic layer overlying monoclinic half-space. Besides these, a large number of papers on elastic wave propagation have been published in different journals. Without going into details of research works in this field, we mention papers by Kim [5] and Nayfeh [6]. In this paper, we have computed the ranges of shear wave velocity ratio and the corresponding wave numbers for Love waves at a layer of fiber-reinforced material resting on monoclinic half-space and compared them with shear wave velocity ratio of Love waves at isotropic layer resting on monoclinic half-space. Using these values the dispersion curves have also been obtained.

2. Formulation of the Problem

The constitutive equations for fibre-reinforced linearly elastic medium whose referred direction is that of 𝐚 are (Spencer [7])𝜏_𝑖𝑗=𝜆𝑒_𝑘𝑘𝛿_𝑖𝑗+2𝜇_𝑇𝑒_𝑖𝑗+𝛼𝑎_𝑘𝑎_𝑚𝑒_𝑘𝑚𝛿_𝑖𝑗+𝑒_𝑘𝑘𝑎_𝑖𝑎_𝑗+2𝜇_𝐿−𝜇_𝑇𝑎_𝑖𝑎_𝑘𝑒_𝑘𝑗+𝑎_𝑗𝑎_𝑘𝑒_𝑘𝑖+𝛽𝑎_𝑘𝑎_𝑚𝑒_𝑘𝑚𝑎_𝑖𝑎_𝑗, (2.1) where 𝜏_𝑖𝑗, are components of stress, 𝑒_𝑖𝑗, are components of infinitesimal strain, and 𝑎_𝑖𝑗 are components of 𝑎, all referred to Cartesian coordinates. The vector a may be a function of position. The coefficients 𝜆,𝜇_𝐿,𝜇_𝑇,𝛼, and 𝛽 are elastic constants with dimension of stress. If 𝐚 is so chosen that its components are (0,0,1). The stress components (2.1) become𝜏₁₁=𝜆+2𝜇_𝑇𝑒₁₁+𝜆𝑒₂₂+(𝜆+𝛼)𝑒₃₃,𝜏₂₂=𝜆+2𝜇_𝑇𝑒₂₂+𝜆𝑒₁₁+(𝜆+𝛼)𝑒₃₃,𝜏₃₃=𝜆+2𝛽−2𝜇_𝑇+2𝛼+4𝜇_𝐿𝑒₃₃+(𝜆+𝛼)𝑒₂₂+(𝜆+𝛼)𝑒₁₁,𝜏₁₂=2𝜇_𝑇𝑒₁₂,𝜏₁₃=2𝜇_𝐿𝑒₁₃,𝜏₂₃=2𝜇_𝐿𝑒₃₂, (2.2) where 2𝑒_𝑖𝑗=(𝑢_𝑖,𝑗+𝑢_𝑗,𝑖) and 𝑢_𝑖(𝑖=1,2,3) are the displacement components.

In this problem, we consider a fiber-reinforced of anisotropic layer −ℎ≤𝑥₂≤0 resting on the monoclinic half-space (𝑥₂≥0) in the 𝑥₃−𝑥₂ plane. The 𝑥₃-axis is chosen parallel to the layer in the direction of propagation of the disturbance. The strain-displacement relations for a monoclinic crystal medium are𝑆₁=𝜕𝑢^′₁𝜕𝑥₁,𝑆₂=𝜕𝑢^′₂𝜕𝑥₂,𝑆₃=𝜕𝑢^′₃𝜕𝑥₃,𝑆₄=𝜕𝑢^′₂𝜕𝑥₃+𝜕𝑢^′₃𝜕𝑥₂,𝑆₅=𝜕𝑢^′₁𝜕𝑥₃+𝜕𝑢^′₃𝜕𝑥₁,𝑆₆=𝜕𝑢^′₂𝜕𝑥₁+𝜕𝑢^′₁𝜕𝑥₂, (2.3) where 𝑢^′₁, 𝑢^′₂,and 𝑢^′₃ are displacement components in the directions 𝑥₁,𝑥₂, and 𝑥₃, respectively, and 𝑆_𝑖(𝑖=1,2,…6) are the strain components. The stress-strain relations for a rotating 𝑥₂-cut plate of quartz which exhibits monoclinic symmetry with 𝑥₁ being the diagonal axis are𝑇₁=𝐶₁₁𝑆₁+𝐶₁₂𝑆₂+𝐶₁₃𝑆₃+𝐶₁₄𝑆₄,𝑇₂=𝐶₂₁𝑆₁+𝐶₂₂𝑆₂+𝐶₂₃𝑆₃+𝐶₂₄𝑆₄,𝑇₃=𝐶₃₁𝑆₁+𝐶₃₂𝑆₂+𝐶₃₃𝑆₃+𝐶₃₄𝑆₄,𝑇₄=𝐶₄₁𝑆₁+𝐶₄₂𝑆₂+𝐶₄₃𝑆₃+𝐶₄₄𝑆₄,𝑇₅=𝐶₅₅𝑆₅+𝐶₅₆𝑆₆,𝑇₆=𝐶₆₅𝑆₅+𝐶₆₆𝑆₆, (2.4) where 𝑇_𝑖(𝑖=1,2,3) are the normal stresses 𝑇_𝑖(𝑖=4,5,6) are the shearing stresses, 𝐶_𝑖𝑗=𝐶_𝑗𝑖 (𝑖,𝑗=1,2,3,4,5,6) are the elastic constants. In the study of seismic waves, when s-pulses are polarized so that all particles of the substance move horizontally during its passage, the wave motion is called SH-wave. Our problem is to investigate the propagation of such waves in the media which consist of two separate media in which upper medium is a layer of thickness ℎ, and the lower one is monoclinic half-space.

3. Solution of the Problem

For wave propagating in the 𝑥₃-direction and causing displacements in the 𝑥₁-direction only, we assume that 𝑢₂=𝑢₃=0 and 𝑢₁=𝑢₁(𝑥₃,𝑥₂,𝑡)=𝑢₁(𝑥₂,𝑥₃,𝑡). For the shear wave propagating in the 𝑥₃-𝑥₂ plane, 𝜕/𝜕𝑥₁=0. So the equation of motion for shear wave takes the form𝜕𝜕𝑥₂𝜏₁₂+𝜕𝜕𝑥₃𝜏₁₃𝜕=𝜌²𝑢₁𝜕𝑡². (3.1) Putting the values of 𝜏₁₂ and 𝜏₁₃, the above equation takes the form given below𝜇_𝑇𝜕²𝑢₁𝜕𝑥²₂+𝜇_𝐿𝜕²𝑢₁𝜕𝑥²₃𝜕=𝜌²𝑢₁𝜕𝑡². (3.2)

For wave changing harmonically 𝑢₁(𝑥₃,𝑥₂,𝑡)=𝑢(𝑥₂)exp[𝑖(𝑘𝑥₃−𝜔𝑡)], where 𝑘= wave number, 𝜔=(𝑘𝑐) is the angular frequency, and 𝑐 is the speed of simple harmonic waves of wavelength (2𝜋/𝑘). Substituting this value of 𝑢₁ in the equation of motion, it transforms into𝜕²𝑢𝜕𝑥²₂+𝜌𝜔²𝜇_𝑇−𝜇_𝐿𝜇_𝑇𝑘²𝑢𝑥₂=0. (3.3) The solution of this equation is𝑢𝑥₂=𝐴cos𝜒𝑥₂+𝐵sin𝜒𝑥₂where𝜒=𝜌𝜔²𝜇_𝑇−𝜇_𝐿𝜇_𝑇𝑘². (3.4) Therefore,𝑢₁𝑥₃,𝑥₂,𝑡=𝐴cos𝜒𝑥₂+𝐵sin𝜒𝑥₂exp𝑖𝑘𝑥₃−𝜔𝑡. (3.5) Equation of motion for the lower half-space has been obtained by 𝜕𝑇₆𝜕𝑥₂+𝜕𝑇₅𝜕𝑥₃=𝜌^′𝜕²𝑢^′₁𝜕𝑡² (3.6) using the values of 𝑇₅ and 𝑇₆. Finally, the equation has been obtained as𝐶₆₆𝜕²𝑢^′₁𝜕𝑥²₂+2𝐶₅₆𝜕²𝑢^′₁𝜕𝑥₂𝜕𝑥₃+𝐶₅₅𝜕²𝑢^′₁𝜕𝑥²₃=𝜌^′𝜕²𝑢^′₁𝜕𝑡², (3.7) where 𝑢^′₁ is the displacement in the 𝑥₁ direction in the lower half-space. We assume that 𝑢^′₁𝑥₂,𝑥₃,𝑡=𝑢^′𝑥₂exp𝑖𝑘𝑥₃−𝜔𝑡, (3.8) is the solution of the above differential equation. So, by substituting this to the above equation, we obtain the solution as𝑢^′₁=𝐴₁𝑒^{((−𝛼′/2)−(1/2)(𝛼′2−4𝛽′))𝑥2}exp𝑖𝑘𝑥₃,𝛼−𝜔𝑡_′𝐶=2𝑖𝑘⁵⁶𝐶₆₆,𝛽^′=𝜌^′𝜔²−𝐶₅₅𝑘²𝐶₆₆. (3.9)

3.1. Boundary Conditions

The boundary conditions in the plane 𝑥₂=−ℎ at the top of the system𝜏₁₂=0. (3.10) At the interface, that is, at 𝑥₂=0,𝜏₁₂𝑥₃,𝑥₂=0,𝑡=𝑇₆𝑥₃,𝑥₂=0,𝑡, (3.11)𝑢₁𝑥₃,𝑥₂=0,𝑡=𝑢^′₁𝑥₃,𝑥₂=0,𝑡. (3.12) From (3.10),𝐵tan𝜒ℎ=−𝐴. (3.13) From (3.11), we obtain𝜇_𝑇𝐶𝐵𝜒=−⁶⁶2𝐴₁𝛼^′2−4𝛽^′. (3.14) From (3.12), we obtain𝐴=𝐴₁. (3.15) Therefore, 𝜇_𝑇𝐶𝐵𝜒=−⁶⁶2𝐴𝛼^′2−4𝛽^′. (3.16) So, 𝐵𝐴𝐶=−⁶⁶2𝜒𝜇_𝑇𝛼^′2−4𝛽^′, (3.17) where𝛼^′2−4𝛽^′=2𝑘𝐶₅₅𝐶₆₆−𝐶₅₆𝐶₆₆²−𝑐²𝐶₆₆/𝜌^′. (3.18) Finally,𝐵tan𝜒ℎ=−𝐴=𝐶₆₆𝜇_𝑇𝑘𝐶₅₅/𝐶₆₆−𝐶₅₆/𝐶₆₆²−𝑐²/𝐶₆₆/𝜌^′𝜒 (3.19) this equation (3.19) gives the velocity of love wave in an elastic fiber-reinforced layer of finite thickness ℎ resting on the monoclinic half-space which may be naturally a stable rock layer of large thickness. So, it is a possible case that a road of fiber-reinforced composite material has been constructed on a rock surface. To make it seismically stable, the knowledge of probable velocity of Love wave will be a necessary precondition. In this context, this study will be helpful. The real root of this equation can be found for the values of 𝑐 given below𝜇_𝑇𝜌𝜇_𝐿𝜇_𝑇<𝑐<𝐶₆₆𝜌^′𝐶₅₅𝐶₆₆−𝐶₅₆𝐶₆₆². (3.20) If we transform the upper and lower layer as isotropic medium with different rigidity and density, the wave equation (3.19) takes the form shown below, by taking 𝐶₅₆=0,𝐶₆₆=𝐶₅₅=𝜇₀, and 𝜇_𝑇=𝜇_𝐿=𝜇,tan𝜌𝑐²𝜇=𝜇−1𝑘ℎ⁰𝜇1−𝑐²/𝜇₀/𝜌^′𝜌𝑐²/𝜇−1, (3.21) where rigidity and density for upper and lower media are 𝜇,𝜌 and 𝜇₀,𝜌^′, respectively. This equation is well-known Love wave equation for classical case. By taking, 𝐶₆₆=𝜇_𝑇,𝐶₅₅=𝜇_𝐿,𝐶₅₆=0, the wave equation (3.19) transforms to the case of wave equation for a fiber-reinforced layer resting on fiber-reinforced half-space derived by Sengupta and Nath [8].

4. Numerical Results and Discussion

The range of shear wave velocity ratio (𝑐/𝜇_𝑇/𝜌) of Love wave has been obtained from (3.20), and within this range, the corresponding values of 𝑘ℎ are obtained from (3.19) by considering different material constants of fiber-reinforced layer and monoclinic half-space. These values have been plotted to obtain the shear wave velocity ratio curve. We have obtained a set of shear wave velocity ratio curve for fiber-reinforced layer resting on half-spaces of different materials (monoclinic and isotropic). Similarly, a set of shear wave velocity curves for isotropic layer resting on monoclinic and isotropic half-spaces have also been obtained. These are specified in details in Cases 2 and 1, respectively, below.

Case 1. Here we have considered three sets of arrangements; in all these, top layer is isotropic-I, and lower half-spaces are (i) isotropic-II with different material constants from isotropic-I, (ii) lower half-space is monoclinic, and (iii) lower half-space is monoclinic II, respectively. The material constants are =6×10⁹N/m², 𝜌=2300kg/m³ for isotropic-I and 𝜇=25×10⁹N/m², 𝜌=2700kg/m³ for isotropic-II. The material constants for monoclinic are as 𝐶₅₅=0.94×10¹¹N/m², 𝐶₅₆=−0.11×10¹¹N/m², 𝐶₆₆=0.93×10¹¹N/m², and 𝜌=7800kg/m³ and for monoclinic-II are 𝐶₅₅=0.60×10¹¹N/m², 𝐶₅₆=0.09×10¹¹N/m², 𝐶₆₆=0.75×10¹¹N/m², 𝜌=4700kg/m³given by Tiersten [9]. Using these values, shear wave velocity ratio curves have been plotted for corresponding values of 𝑘ℎ obtained from (3.19). The range of values for shear wave velocity ratio and 𝑘ℎ are given in Table 1, and curves are given in Figure 1.

Table 1 

Ranges of shear wave velocity ratio and ranges of 𝑘ℎ for isotropic layer.

Figure 1 

Shear wave velocity ratio (𝑐/𝜇/𝜌) curve for Love waves at isotropic layer resting on monoclinic half-Space.

Case 2. In this case, we have plotted three dispersion curves as in Case 1. Here the upper layer is taken as fibre-reinforced layer, and the lower half-spaces are taken as in Case 1 of three separate arrangements, and the material constants are the same as earlier. The material constants for fibre-reinforced layer are𝜇_𝐿=2.45×10⁹N/m², 𝜇_𝑇=1.89×10⁹N/m², and𝜌=7800kg/m³. Shear wave velocity ratio curve for Love waves at fibre-reinforced layer has been plotted for all these three cases and shown in Figure 2, and the ranges of shear wave velocity ratio and wave number are shown in Table 2.
From the results it has been observed that shear wave velocity ratio for Love waves at fibre-reinforced layer is very much higher than at isotropic layer.

Table 2 

Ranges of shear wave velocity ratio and 𝑘ℎ for fiber-reinforced layer.

Figure 2 

Shear wave velocity ratio curve (𝑐/𝜇𝑇/𝜌) for Love waves at fibre-reinforced layer resting on monoclinic half-Space.

5. Conclusions

From the curves plotted and results tabulated, it has been clearly observed that the shear wave velocity ratio for a fiber-reinforced layer resting on any layer whether it is isotropic or monoclinic is always much higher than on the isotropic layer resting on similar half-spaces. For the case of fiber-reinforced layer, shear wave velocity ratio ranges from 1.14 to 6.15, 1.14 to 7.16, and 1.14 to 7.19, for isotropic half-spaces, monoclinic and monoclinic-II half-spaces, respectively. In the contrary for the case of an isotropic layer, the shear wave velocity ratio ranges from 1.0 to 1.85, 1.0 to 2.19, and 1.0 to 2.19 for the case isotropic, monoclinic, and monoclinic-II half-spaces, respectively.

Acknowledgments

S. Saha is very much thankful to Director CRRI, for his permission to publish this paper. S. Saha is also very much thankful to the reviewer for valuable comments to modify this paper.