Abstract

This work investigates the waves propagation in an isotropic saturated porous-thermoelastic medium. The incidence of a seismic wave is considered at the free surface of a thermoelastic semi-infinite space. The surface of semi-infinite space is taken as isothermal. Reflection ratios of waves for the incidence of longitudinal waves are derived for impedance boundary conditions. The graph depicts the effect of porosity on reflection ratios versus incident angle and impedance parameter. Graph plotting is done with the help of the MATLAB programming language. Results are presented for a particular model in this paper.

Keywords: Porous, thermoelastic, reflection, impedance, isothermal

9.1 Introduction

Thermoelasticity is the study of the change in shape and dimension of a solid object when the temperature of the object varies. With the theory of elasticity, Madan et al. [1] and Kumari and Madan [2] explore the propagation of waves. By extending the classical coupled theory of thermoelasticity by introducing relaxation time, Lord and Shulman [3] and Green and Lindsay [4] presented the generalized theory of thermoelasticity. A number of researchers [5–11] have discussed the phenomena of waves reflecting on a free surface. Rani and Madan [12] investigated the effect of initial stress and imperfect interface on waves.

In porous media, the thermo-poroelasticity theory describes the relationship between temperature and stress components. A thermoelastic saturated porous material with solid-fluid pores is a thermally conducting porous solid and these materials are generally found in reservoir rocks of the earth’s crust. The phenomenon of co-existence of thermoelasticity and porosity plays a crucial role in the non-destructive evaluation of composite materials and structures. The propagation of thermoelastic waves in fluidsaturated soil was studied by Haibing [13]. Kumar et al. [14] investigated the Rayleigh waves propagation in thermoelastic semi-infinite space with double porosity under isothermal and insulated boundary conditions in thermoelastic semi-infinite space with double porosity. Wilson et al. [15] discuss the propagation of acoustic waves. In a thermo-poroelastic medium, Wang et al. [16] explored the reflection phenomenon of inhomogeneous elastic waves. The reflection phenomenon of seismic waves at insulated and free surfaces of thermo-poroelastic material was explored by Zorammuana and Singh [17].

Impedance boundary-conditions are defined as a linear combination of unknown parameters and corresponding derivatives that are predictable on the surface. Electromagnetism, acoustics, seismology, science, and technology all benefit from these conditions. The propagation of waves for impedance boundary-conditions was studied by Singh [18], Vinh and Hue [19], Vinh and Xuan [20], and others.

To conclude the wave propagation effects, a quantitative analysis of reflection at the surface of a thermo-poroelastic medium is required. This work examines the waves propagation at the surface of a saturated porous-thermoelastic isothermal medium with an impedance boundary condition. The influence of solid porosity and the Biot parameter for the coupling of fluid and solid phases on the reflection ratios versus impedance parameter and incident angle of a longitudinal wave is studied.

9.2 Basic Equations

The stress-tensors in a non-viscous thermal-porous saturated solid are given by [21]:

(9.1)

where

τ_ij: components of stress,

P_f: pressure of fluid,

β: Biot parameter,

δ_ij: Kronecker delta

The stress-strain relation is given by [22, 23]

(9.2)

(9.3)

where

λ, μ: lame’s constants,

e_ij: components of strain,

ϴ: temperature,

γ_s: coefficient of thermal stress corresponding to solid,

γ_f: coefficient of thermal stress corresponding to fluid,

N: elastic parameter for bulk-coupling of fluid,

v_i = f (U_i − u_i),

f: solid porosity,

U_i: component of displacement corresponding to fluid,

u_i: component of displacement corresponding to solid,

Also, the strain components are

The equation of motion in an isotropic thermal-porous solid saturated with a non-viscous fluid are given by [24]:

(9.4)

(9.5)

(9.6)

where

ρ: porous-aggregate density,

ρ_f: pore-fluid density,

q: coupling parameter between porous-aggregate solid and pore-fluid,

γ = γ_s + βγ_f

9.3 Problem Formulation

Using Equations (9.1)-(9.3) in Equation (9.4), we have:

(9.7)

Using Equations (9.1)-(9.3) in Equation (9.5), we have:

(9.8)

Using Equations (9.1)-(9.3) in Equation (9.6), we have:

(9.9)

Displacement components in terms of potentials are

(9.10)

(9.11)

Using Equations (9.10) and (9.11) in Equations (9.7)-(9.9), we have:

(9.12)

(9.13)

(9.14)

(9.15)

(9.16)

where is the Laplace operator.

The coupling structure of potentials p_f, p_s with thermal wave can be easily observed from Equations (9.12), (9.14), and (9.16), while the coupling structure of potentials q_f and q_s can be easily observed from Equations (9.13) and (9.15).

The velocities of waves are [17]

(9.17)

(9.18)

(9.19)

(9.20)

where,

(9.21)

9.4 Reflection at the Free Surface

Consider an isotropic porous-thermoelastic semi-infinite space, z > 0 (Figure 9.1). If a longitudinal wave is incident with an angle e at the surface, then it will give four reflected waves. Let e₁, e₂, e₃ be angles which reflect coupled longitudinal waves, making them normal, and e_s be the angle which reflected transverse waves makes normal in medium M.

(9.22)

(9.23)

(9.24)

(9.25)

(9.26)

where

(9.27)

Using Snell’s Law, angles and corresponding wave vectors are associated by:

(9.28)

and in terms of velocities, are given by:

(9.29)

9.4.1 Boundary Conditions

At surface z = 0, impedance boundary conditions are given by [18]:

1. σ₃₃ = 0, which gives:
   (9.30)
2. σ₃₁ + ωZu₁ = 0 gives:
   (9.31)
3. gives:
   (9.32)
4. for thermally-insulated surface, h → ∞ for isothermal surface) gives
   (9.33)

where

(9.34)

The matrix equation from these equations can be written as

(9.35)

where

(9.36)

Here, X_i, i = 1, 2, 3, 4 represents reflection ratios in isotropic thermoelastic porous medium.

9.4.2 Energy Ratios

The average energy transmission per unit surface area per unit time is given by [25]:

(9.37)

Using Equations (9.22)-(9.28) and (9.37),

(9.38)

where P_refL1, P_refL2, P_refL3, and P_refT are the average energy of reflected coupled longitudinal waves at e₁, e₂, e₃, and transverse wave at e₄, respectively, and

(9.39)

9.5 Numerical Results and Discussion

For the numerical computation of amplitude ratios, we employed the following parameters [24]:

The problem of isothermal surfaces has been studied here. To plot the graphs and execute numerical calculations, the MATLAB programming language is employed. Figure 9.2 explored the impact of porous-parameter f on the reflection ratio of a coupled longitudinal wave at an angle e₁ to the incident angle e of the longitudinal wave. It is concluded that as the value of f increases, the reflection ratio of a longitudinal wave decreases for e < 70° and then for e > 70°, attaining the same value for f = 0.03 (circle-line), f = 0.04 (star-line), and f = 0.05 (triangle-line).

Variation of the reflection ratios of coupled longitudinal waves at an angle e₂, e₃ against the incident angle for β = 0.4, Z = 10, f = 0.03 (circle-line), f = 0.04 (star-line), f = 0.05 (triangle-line) is shown in Figures 9.3 and 9.4, respectively. It is concluded that the reflection ratio of a coupled-longitudinal wave against the incident angle attains minimum value at e = 71°, then e > 71° attains the same value for distinct values of f. Qualitatively, the same behavior of the reflection ratios of a longitudinal wave at an angle e₂, e₃ against the incident angle is observed.

Figure 9.5 explores the variation in amplitude ratio of transverse wave against incident angle e of the longitudinal wave. The reflection ratio of a coupled longitudinal wave attains a maximum value at e = 8° and then decreases gradually. The reflection ratio of a transverse wave behaves qualitatively the same as a coupled longitudinal wave at an angle e₂, e₃.

Figures 9.6 and 9.7 exhibit variation in the amplitude ratio of a coupled longitudinal wave at angle e₁, e₂ against impedance parameter Z for f = 0.01, β = 0.1 (represented by circle-line), β = 0.2 (represented by triangle-line), and β = 0.3 (represented by star-line). The magnitude of amplitude ratios of coupled longitudinal wave at angle e₁, e₂ increases as the value of β increases and attains a maximum value at Z = 50. The amplitude ratios of a coupled longitudinal wave at an angle e₃ and transverse wave increases against the impedance parameter Z and attains a maximum value at Z = 50 in Figures 9.8 and 9.9, respectively. It is noticed that the quantitative value of the reflection ratio decreases as the value of β increases.

Effect of Biot parameter on energy ratios of transverse and longitudinal waves against incident angle of longitudinal wave is shown in Figures 9.10-9.13. It is noticed that the energy ratios of longitudinal and transverse waves achieve a fixed value for incident angle e > 70°. As the value of the Biot parameter grows, the energy ratio of longitudinal waves reflected at an angle e₁ decreases, while the energy ratio of longitudinal and transverse waves reflected at angles e₁, e₂ and e₃, respectively, increases against the incident angle of longitudinal waves.

9.6 Conclusion

The effects of porosity and impedance on waves propagating in isotropicporous thermoelastic semi-infinite space is investigated in this work. The following key points are drawn from graphs in this paper:

• As solid porosity increases, the amplitude ratio of the longitudinal wave reflected at an angle e₁, drops against the incident angle of longitudinal waves, reaching a minimum value at e = 65° and subsequently, a fixed value for e > 70°
• The reflection ratios of longitudinal waves (at angle e₂, e₃) and transverse wave falls against incident angle of longitudinal wave, as solid porosity increases, reaching a minimum for e > 70°
• Amplitude ratios of reflected waves in isotropic porous-thermoelastic medium increases with the impedance parameter
• The reflection ratio of the coupled longitudinal waves at angle e₁, e₂, increases against the impedance parameter with increment of Biot parameter
• The reflection ratio of the longitudinal wave at e₃ and transverse wave decreases against the impedance parameter with increment of Biot parameter
• The Biot parameter has less impact on the reflection ratio of the transverse waves
• The energy ratio of the reflected longitudinal waves at an angle e₁ drops against the incident angle of longitudinal waves, achieving a minimum value at e = 65° and subsequently a fixed value for e > 70°
• The sum of all energy ratios approaches unity

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