Propagation of Elastic Waves in Prestressed Media

Abstract

3D solutions of the dynamical equations in the presence of external forces are derived for a homogeneous, prestressed medium. 2D plane waves solutions are obtained from general solutions and show that there exist two types of plane waves, namely, quasi-P waves and quasi-SV waves. Expressions for slowness surfaces and apparent velocities for these waves are derived analytically as well as numerically and represented graphically.

1. Introduction

In fact, the Earth is prestressed medium, due to many physical causes, that is, gravity variation, slow process of creep and variation of temperature, and so forth. Therefore, these problems are of much interest to seismologists due to its application in mineral prospecting and prediction of earthquakes.

For studying the propagation of elastic waves in prestressed solids of infinite extent, Sidhu and Singh [1, 2], Norris [3], and Day et al. [4] used a much simpler form of equation of motion of Biot [5]. The medium considered by earlier investigators is a homogenous prestressed with incremental elastic coefficients possessing orthotropic anisotropy due to normal components of initial stresses.

In the present paper, 3D problem of propagation of 𝑃, SV and SH waves for a homogeneous prestressed medium is discussed. The model of prestressed medium used here is much more general than that used by earlier investigators.

The 2D plane waves solutions are obtained from general solutions for different conditions of initial stresses with or without external forces. Graphs of slowness surfaces are derived.

2. Basic Equations

Consider a homogenous prestressed solid. The state of prestresses is, therefore, defined by six components, that is, 𝑆_11,𝑆_22,𝑆_33,𝑆₁₂=𝑆₂₁,𝑆₃₁=𝑆₁₃, and 𝑆₂₃=𝑆₃₂. Let all stress components be functions of (𝑥,𝑦,𝑧). The state of initial stress introduces anisotropy so that even for an initially isotropic solid defined by two Lame’s constants 𝜆,𝜇, the number of the incremental elastic coefficients (𝐵_𝑖𝑗,𝑄_𝑖) will be always larger than 2. Let (𝑋,𝑌,𝑍) be components of body forces along coordinate axes, respectively, where 𝑋, 𝑌, 𝑍 are all constant. The general form of dynamical equation for prestressed solids in the presence of external forces is given by Biot [5, page 52]:𝜕𝑠₁₁+𝜕𝑥𝜕𝑠₁₂+𝜕𝑦𝜕𝑠₃₁𝜔𝜕𝑧+𝜌Δ𝑥+𝜌_𝑧𝑌−𝜔_𝑦𝑍−𝜌𝑒𝑋−𝑒_𝑥𝑥𝜕𝑆₁₁𝜕𝑥−𝑒_𝑦𝑦𝜕𝑆₁₂𝜕𝑦−𝑒_𝑧𝑧𝜕𝑆₃₁𝜕𝑧−𝑒_𝑦𝑧𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂𝜕𝑧−𝑒_𝑧𝑥𝜕𝑆₁₁+𝜕𝑧𝜕𝑆₃₁𝜕𝑥−𝑒_𝑥𝑦𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁+𝑆𝜕𝑦¹¹−𝑆₂₂𝜕𝑊_𝑧𝜕𝑦−2𝑆₁₂𝜕𝑊_𝑧𝜕𝑥+𝑆₁₂𝜕𝑊_𝑥+𝑆𝜕𝑧³³−𝑆₁₁𝜕𝑊_𝑦𝜕𝑧+2𝑆₁₂𝜕𝑊_𝑦𝜕𝑥−𝑆₃₁𝜕𝑊_𝑥𝜕𝑦+𝑆₂₃𝜕𝑊_𝑦−𝜕𝑦𝜕𝑊_𝑧𝜕𝜕𝑧=𝜌²𝑢₁𝜕𝑡². (2.1) The two other equations are obtained by cyclic permutation of 𝑥, 𝑦, 𝑧, 1, 2, 3 and 𝑋, 𝑌, 𝑍. 𝜌 is the density and (𝑢₁,𝑢₂,𝑢₃) the displacement components along the axes. Δ𝑋_𝑖 are the components of incremental body force, which are assumed to satisfy the equationΔ𝑋_𝑖=𝑢_𝑗𝑋_𝑖,𝑗=0,𝑒=𝑒_𝑥𝑥+𝑒_𝑦𝑦+𝑒_𝑧𝑧. (2.2) The 𝑆_𝑖𝑗 are the components of prestress, which are assumed to satisfy the equilibrium equation.𝑆_{𝑖𝑗,𝑗}+𝜌𝑋_𝑖=0, (2.3) and are related to the initial strain 𝜖_𝑖𝑗 by Hooke’s law𝑆_𝑖𝑗=𝜆𝜖_𝑖𝑖𝛿_𝑖𝑗+2𝜇𝜖_𝑖𝑗, (2.4) The incremental stresses 𝑠_𝑖𝑗 are supposed to be linearly related to the incremental strains 𝑒_𝑖𝑗 through the incremental elastic coefficients 𝐵_𝑖𝑗 and 𝑄_𝑖𝑠₁₁=𝐵₁₁𝑒_𝑥𝑥+𝐵₁₂𝑒_𝑦𝑦+𝐵₁₃𝑒_𝑧𝑧,𝑠₂₂=𝐵₂₁𝑒_𝑥𝑥+𝐵₂₂𝑒_𝑦𝑦+𝐵₂₃𝑒_𝑧𝑧,𝑠₃₃=𝐵₃₁𝑒_𝑥𝑥+𝐵₃₂𝑒_𝑦𝑦+𝐵₃₃𝑒_𝑧𝑧,𝑠₂₃=2𝑄₁𝑒_𝑦𝑧,𝑠₃₁=2𝑄₂𝑒_𝑧𝑥,𝑠₂₃=2𝑄₃𝑒_𝑥𝑦. (2.5) We assume that𝑢₁,𝑢₂,𝑢₃=(𝑢,𝑣,𝑤),𝑋₁,𝑋₂,𝑋₃=(),𝑒𝑋,𝑌,𝑍_𝑥𝑥=𝜕𝑢𝜕𝑥,𝑒_𝑦𝑦=𝜕𝑣𝜕𝑦,𝑒_𝑧𝑧=𝜕𝑤,𝑒𝜕𝑧_𝑥𝑦=12𝜕𝑢+𝜕𝑦𝜕𝑣𝜕𝑥,𝑒_𝑦𝑧=12𝜕𝑣+𝜕𝑧𝜕𝑤𝜕𝑦,𝑒_𝑧𝑥=12𝜕𝑤+𝜕𝑥𝜕𝑢,𝑊𝜕𝑧_𝑥=12𝜕𝑤−𝜕𝑦𝜕𝑣𝜕𝑧,𝑊_𝑦=12𝜕𝑢−𝜕𝑧𝜕𝑤𝜕𝑥,𝑊_𝑧=12𝜕𝑣−𝜕𝑥𝜕𝑢.𝜕𝑦 (2.6) Substituting (2.2), (2.5), and (2.6) in (2.1), we have 𝐵₁₁𝜕²𝑢𝜕𝑥²+𝑄₃−𝑃₁2𝜕²𝑢𝜕𝑦²+𝑄₂+𝑃₃2𝜕²𝑢𝜕𝑧²−𝑆₁₂𝜕²𝑣𝜕𝑥²−12𝑆₁₂𝜕²𝑣𝜕𝑧²−𝑆₃₁𝜕²𝑤𝜕𝑥²−12𝑆₃₁𝜕²𝑤𝜕𝑦²+𝑆₁₂𝜕²𝑢𝜕𝑥𝜕𝑦+𝑆₂₃𝜕²𝑢𝜕𝑦𝜕𝑧+𝑆₃₁𝜕²𝑢+𝐵𝜕𝑥𝜕𝑧¹²+𝑄₃+𝑃₁2𝜕²𝑣+1𝜕𝑥𝜕𝑦2𝑆₃₁𝜕²𝑣−1𝜕𝑦𝜕𝑧2𝑆₂₃𝜕²𝑣+𝐵𝜕𝑥𝜕𝑧¹³+𝑄₂−𝑃₃2𝜕²𝑤+1𝜕𝑥𝜕𝑧2𝑆₁₂𝜕²𝑤−1𝜕𝑦𝜕𝑧2𝑆₂₃𝜕²𝑤−𝜕𝑥𝜕𝑦𝜕𝑆₁₁𝜕𝑥+𝜌𝑋𝜕𝑢−1𝜕𝑥2𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁𝜕𝑦𝜕𝑢−1𝜕𝑦2𝜕𝑆₃₁+𝜕𝑥𝜕𝑆₁₁𝜕𝑧+𝜌𝑍𝜕𝑢−1𝜕𝑧2𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁𝜕𝑦−𝜌𝑌𝜕𝑣+𝜕𝑥𝜕𝑆₁₂𝜕𝑦+𝜌𝑋𝜕𝑣−1𝜕𝑦2𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂𝜕𝑧𝜕𝑣−1𝜕𝑧2𝜕𝑆₃₁+𝜕𝑥𝜕𝑆₁₁𝜕𝑧−𝜌𝑍𝜕𝑤−1𝜕𝑥2𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂𝜕𝑧𝜕𝑤−𝜕𝑦𝜕𝑆₃₁𝜕𝑧+𝜌𝑋𝜕𝑤𝜕𝜕𝑧=𝜌²𝑢𝜕𝑡². (2.7) The two other equations are obtained from (2.7) by cycle permutation of 𝑥, 𝑦, 𝑧, 1, 2, 3, 𝑋, 𝑌, 𝑍 and 𝑢, 𝑣, 𝑤.

Here𝑃₁=𝑆₁₁−𝑆₂₂,𝑃₂=𝑆₂₂−𝑆₃₃,𝑃₃=𝑆₃₃−𝑆₁₁. (2.8)

3. Propagation of Waves

For elastic waves propagating in a direction specified by direction cosines (𝑙,𝑚,𝑛) along the axes, we take𝑢=𝑈_𝑖exp(𝑖𝑃),𝑣=𝑉_𝑖exp(𝑖𝑃),𝑤=𝑊_𝑖exp(𝑖𝑃). (3.1)(𝑈_𝑖,𝑉_𝑖,𝑊_𝑖) are amplitude factors along the axes and 𝑃 is phase factor:𝑃=𝑘𝑐𝑡−𝑙𝑥+𝑚𝑦+𝑛𝑧, (3.2) where 𝑐 is phase velocity and 𝑘 is wave number.

Putting (3.1) and (3.2) in (2.7), we getΩ₁+Ω^′₁+𝜌𝑐²𝑈_𝑖+ℑ₁+ℑ^′₁𝑉_𝑖+𝒦₁+𝒦^′₁𝑊_𝑖𝒦=0,₂+𝒦^′₂𝑈_𝑖+Ω₂+Ω^′₂+𝜌𝑐²𝑉_𝑖+ℑ₂+ℑ^′₂𝑊_𝑖ℑ=0,₃+ℑ^′₃𝑈_𝑖+𝒦₃+𝒦^′₃𝑉_𝑖+Ω₃+Ω^′₃+𝜌𝑐²𝑊_𝑖=0, (3.3) whereΩ₁=−𝐵₁₁𝑙²+𝑄₃−𝑃₁2𝑚²+𝑄₂+𝑃₃2𝑛²+𝑆₁₂𝑙𝑚+𝑆₂₃𝑚𝑛+𝑆₃₁ℑ𝑛𝑙,₁=𝑆₁₂𝑙²+𝑆₁₂2𝑛²−𝐵₁₂+𝑄₃+𝑃₁2𝑆𝑙𝑚−³¹2𝑆𝑚𝑛+²³2𝒦𝑙𝑛,₁=𝑆₃₁𝑙²+𝑆₃₁2𝑚²−𝐵₁₃+𝑄₂−𝑃₃2𝑆𝑙𝑛−¹²2𝑆𝑚𝑛+²³2Ω𝑙𝑛,_′1=𝑖𝑘𝜕𝑆₁₁1𝜕𝑥+𝜌𝑋𝑙+2𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁1𝜕𝑦+𝜌𝑌𝑚+2𝜕𝑆₃₁+𝜕𝑥𝜕𝑆₁₁𝑛ℑ𝜕𝑧+𝜌𝑍,_′1=𝑖𝑘12𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁𝜕𝑦−𝜌𝑌𝑙−𝜕𝑆₁₂1𝜕𝑦+𝜌𝑋𝑚+2𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂𝑛𝒦𝜕𝑧_′1,=𝑖𝑘12𝜕𝑆₃₁+𝜕𝑥𝜕𝑆₁₁1𝜕𝑧−𝜌𝑍𝑙+2𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂1𝜕𝑧𝑚+2𝜕𝑆₃₁𝑛𝜕𝑧+𝜌𝑋. (3.4)(3.5)(3.6)(3.7)(3.8)(3.9) The explicit expressions for (Ω₂,Ω₃),(ℑ₂,ℑ₃),  (𝒦₂,𝒦₃),(Ω^′₂,Ω^′₃),(ℑ^′₂,ℑ^′₃), and (𝒦^′₂,𝒦^′₃) are obtained from (3.4)–(3.9), by cyclic permutation of 𝑥, 𝑦, 𝑧, 1, 2, 3, 𝑙, 𝑚, 𝑛 and 𝑋, 𝑌, 𝑍, respectively.

Setting the determinant of the coefficients of 𝑈_𝑖,𝑉_𝑖 and 𝑊_𝑖 of (3.3) equal to zero, on simplification, we get the cubic equation in 𝜌𝑐²:𝜌³𝑐⁶+𝜌²𝑐⁴(𝐴+𝐵+𝐷)+𝜌𝑐²()+()𝐴𝐵+𝐵𝐷+𝐷𝐴−𝐹𝐽−𝐸𝐼−𝐻𝐺𝐴𝐵𝐷+𝐸𝐹𝐺+𝐻𝐼𝐽−𝐴𝐹𝐽−𝐸𝐼𝐷−𝐻𝐺𝐵=0, (3.10) where𝐴=Ω₁+Ω^′₁,𝐸=ℑ₁+ℑ^′₁,𝐻=𝒦₁+𝒦^′₁,𝐵=Ω₂+Ω^′₂,𝐹=ℑ₂+ℑ^′₂,𝐼=𝒦₂+𝒦^′₂,𝐷=Ω₃+Ω^′₃,𝐺=ℑ₃+ℑ^′₃,𝐽=𝒦₃+𝒦^′₃. (3.11)

4. Special Cases

Two special cases may be dealt with immediately.

4.1. Three Dimensional Prestressed Medium

4.1.1. Propagation of Waves along the Unique Axis

Putting 𝑛=1, 𝑙=𝑚=0 in (3.3)–(3.8) and in the expressions of (Ω₂,Ω₃), (ℑ₂,ℑ₃),  (𝒦₂,𝒦₃),(Ω^′₂,Ω^′₃),(ℑ^′₂,ℑ^′₃), and (𝒦^′₂,𝒦^′₃), we getΩ₁₁=−𝑄₂+𝑃₃2,Ω₂₁=−𝑄₁−𝑃₂2,Ω₃₁=−𝐵₃₃,ℑ₁₁=𝑆₁₂2,ℑ₂₁=0,ℑ₃₁=𝑆₃₁,𝒦₁₁=0,𝒦₂₁=𝑆₁₂2,𝒦₃₁=𝑆₂₃,Ω^′₁₁=𝑖2𝑘𝜕𝑆₃₁+𝜕𝑥𝜕𝑆₁₁𝜕𝑧+𝜌𝑧,Ω^′₂₁=𝑖2𝑘𝜕𝑆₂₃+𝜕𝑦𝜕𝑆₂₂𝜕𝑧+𝜌𝑧,Ω^′₃₁=𝑖𝑘𝜕𝑆₃₃,ℑ𝜕𝑥+𝜌𝑧_′11=𝑖2𝑘𝜕𝑆₃₁+𝜕𝑦𝜕𝑆₁₂𝜕𝑧,ℑ^′₂₁𝑖=−𝑘𝜕𝑆₂₃𝜕𝑧+𝜌𝑌,ℑ^′₃₁=𝑖𝑘𝜕𝑆₃₃+𝜕𝑥𝜕𝑆₃₁,𝒦𝜕𝑧−𝜌𝑋_′11=𝑖𝑘𝜕𝑆₃₁𝜕𝑧+𝜌𝑋,𝒦^′₂₁=𝑖2𝑘𝜕𝑆₂₃+𝜕𝑥𝜕𝑆₁₂𝜕𝑧,𝒦^′₃₁=𝑖2𝑘𝜕𝑆₃₃+𝜕𝑦𝜕𝑆₂₃.𝜕𝑧+𝜌𝑌 (4.1)(4.2) Equation (3.10) takes the form𝜌³𝑐⁶+𝜌²𝑐⁴𝐴₁+𝐵₁+𝐷₁+𝜌𝑐²𝐴₁𝐵₁+𝐵₁𝐷₁+𝐷₁𝐴₁−𝐹₁𝐽₁−𝐸₁𝐼₁−𝐻₁𝐺₁+𝐴₁𝐵₁𝐷₁+𝐸₁𝐹₁𝐺₁+𝐻₁𝐼₁𝐽₁−𝐴₁𝐹₁𝐽₁−𝐸₁𝐼₁𝐷₁−𝐻₁𝐺₁𝐵₁=0, (4.3) where𝐴₁=Ω₁₁+Ω^′₁₁,𝐵₁=Ω₂₁+Ω^′₂₁,𝐷₁=Ω₃₁+Ω^′₃₁,𝐸₁=ℑ₁₁+ℑ^′₁₁,𝐹₁=ℑ^′₂₁,𝐺₁=ℑ₃₁+ℑ^′₃₁,𝐻₁=𝒦^′₁₁,𝐼₁=𝒦₂₁+𝒦^′₂₁,𝐽₁=𝒦₃₁+𝒦^′₃₁. (4.4)

4.1.2. In the Absence of Body Forces

When 𝑋=𝑌=𝑍=0 and 𝑆₁₁,𝑆₂₂, 𝑆₃₃, 𝑆₁₂ and so forth are all constant, then (4.3) becomes, with the help of (4.2) and (4.4),𝜌³𝑐⁶+𝜌²𝑐⁴Ω₁₁+Ω₂₁+Ω₃₁+𝜌𝑐²Ω₁₁Ω₂₁+Ω₂₁Ω₃₁+Ω₃₁Ω₁₁−ℑ₁₁𝒦₂₁+Ω₁₁Ω₂₁Ω₃₁−ℑ₁₁𝒦₂₁Ω₃₁=0. (4.5)

4.1.3. Prestressed Is Defined by Normal Components

There are only normal components of prestress present in the medium, that is putting 𝑆₁₂=𝑆₃₁=𝑆₂₃=0 in (4.1), (4.5) takes the formΩ₁₁+𝜌𝑐²Ω₂₁+𝜌𝑐²Ω₃₁+𝜌𝑐²=0; (4.6) on simplification, we get three values of 𝑐²:𝑐²=𝑄₂+𝑃₃/2𝜌,𝑐²=𝑄₁−𝑃₂/2𝜌,𝑐²=𝐵₃₃𝜌. (4.7)

4.2. Two Dimensional Prestressed  Medium

4.2.1. Plane Waves Solution

Here, we consider the behaviour of plane waves in 𝑥𝑦-plane perpendicular to the 𝑧-axis; putting 𝑛=0 in (3.4)–(3.9) and in the expressions of (Ω₂,Ω₃) and so forth, we get the set of equations from (3.3):Ω₁₁₁+Ω^′₁₁₁+𝜌𝑐²𝑈_𝑖+ℑ₁₁₁+ℑ^′₁₁₁𝑉_𝑖𝒦=0,₂₁₁+𝒦^′₂₁₁𝑈_𝑖+Ω₂₁₁+Ω^′₂₁₁+𝜌𝑐²𝑉_𝑖=0, (4.8) whereΩ₁₁₁=−𝐵₁₁𝑙²+𝑄₃−𝑃₁2𝑚²+𝑆₁₂,ℑ𝑚𝑙₁₁₁=𝑆₁₂𝑙²−𝐵₁₂+𝑄₃+𝑃₁2,Ω𝑚𝑙₂₁₁=−𝑄₃+𝑃₁2𝑙²+𝐵₂₂𝑚²+𝑆₁₂,𝒦𝑚𝑙₂₁₁=𝑆₁₂𝑚²−𝐵₁₂+𝑄₃−𝑃₁2,Ω𝑛𝑙_′111=𝑖𝑘𝜕𝑆₁₁1𝜕𝑥+𝜌𝑋𝑙+2𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁𝑚,Ω𝜕𝑦+𝜌𝑌_′211=𝑖𝑘12𝜕𝑆₂₂+𝜕𝑥𝜕𝑆₁₂𝜕𝑦+𝜌𝑋𝑙+𝜕𝑆₂₂𝑚,ℑ𝜕𝑦+𝜌𝑌_′111=𝑖𝑘12𝜕𝑆₁₂+𝜕𝑥𝜕𝑆₁₁𝜕𝑦−𝜌𝑌𝑙−𝜕𝑆₁₂𝑚,𝒦𝜕𝑦+𝜌𝑋_′211=𝑖𝑘𝜕𝑆₁₂1𝜕𝑥+𝜌𝑌𝑙+2𝜕𝑆₂₂+𝜕𝑥𝜕𝑆₁₂𝑚.𝜕𝑦+𝜌𝑋 (4.9) Equation (4.8) has nontrivial solution when𝜌²𝑐⁴+Ω₁₁₁+Ω₂₁₁+Ω^′₁₁₁+Ω^′₂₁₁𝜌𝑐²−ℑ₁₁₁+ℑ^′₁₁₁𝒦₂₁₁+𝒦^′₂₁₁=0. (4.10) It is quadratic equation in 𝜌𝑐², and it has two values of 𝑐² corresponding to quasi-SV waves and quasi-𝑃 waves.

In the absence of body forces and when 𝑆₁₁, 𝑆₂₂, and so forth are constants, then from (4.8) and (4.9), we getΩ₁₁₁+𝜌𝑐²𝑈_𝑖+ℑ₁₁₁𝑉_𝑖𝒦=0,₂₁₁𝑈_𝑖+Ω₂₁₁+𝜌𝑐²𝑉_𝑖=0. (4.11) If 𝑆₁₂=𝑆₂₃=𝑆₃₁=0, then (4.11) takes the formΩ₁₁₁₁+𝜌𝑐²𝑈_𝑖+ℑ₁₁₁₁𝑉_𝑖𝒦=0,₂₁₁₁𝑈_𝑖+Ω₂₁₁₁+𝜌𝑐²𝑉_𝑖=0, (4.12) whereΩ₁₁₁₁=−𝐵₁₁𝑙²+𝑄₃−𝑃₁2𝑚²,ℑ₁₁₁₁=−𝐵₁₂+𝑄₃+𝑃₁2Ω𝑚𝑙,₂₁₁₁=−𝑄₃+𝑃₁2𝑙²+𝐵₂₂𝑚²,𝒦₂₁₁₁=−𝐵₂₁+𝑄₃−𝑃₁2𝑙𝑚. (4.13) The set of homogeneous (4.12) in 𝑈_𝑖,𝑉_𝑖 has a nontrivial solution when Ω₁₁₁₁+𝜌𝑐²ℑ₁₁₁₁𝒦₂₁₁₁Ω₂₁₁₁+𝜌𝑐²=0. (4.14) This quadratic equation in 𝜌𝑐² may be solved to obtain 2𝜌𝑐²=𝐵₁₁+𝑄₃+𝑃₁2𝑙²+𝐵₂₁+𝑄₃−𝑃₁2𝑚²±𝐵₁₁−𝑄₃−𝑃₁2𝑙²+𝐵₂₂−𝑄₃+𝑃₁2𝑚²²+4𝐵₂₁+𝑄₃−𝑃₁2𝐵₁₂+𝑄₃+𝑃₁2𝑙²𝑚². (4.15) Thus, in general, in this two-dimensional model of the prestressed medium, there exist two types of plane waves, namely, quasi-𝑃 waves and quasi-SV waves whose phase velocities correspond to upper and lower signs of (4.15).

4.2.2. Propagation of Plane Waves in Orthotropic Medium

Consider a homogenous prestressed elastic solid. The material is either isotropic in finite strain or anisotropic with orthotropic symmetry. The principal directions of initial stress are chosen to coincide with the directions of elastic symmetry and the coordinate axes. Let the state of uniform initial stresses have principal stresses 𝑆₁₁,𝑆₂₂, and 𝑆₃₃. We further assume that 𝑆₂₂=𝑆₃₃ and 𝑆₁₁ and 𝑆₂₂ are constant. The principal stress 𝑆₃₃ does not enter explicitly into the equations of motion. Its influence is, however, included indirectly in the values of the incremental elastic coefficients. We put 𝑙=sin𝜃 and 𝑚=cos𝜃; (4.15) can be written as2𝜌𝑐²(𝜃)=𝐵₁₁+𝑄₃+𝑃₁2sin²𝜃+𝐵₂₁+𝑄₃−𝑃₁2cos²𝜃±𝐵₁₁−𝑄₃−𝑃₁2sin²𝜃+𝐵₂₂−𝑄₃+𝑃₁2cos²𝜃²+𝔸, (4.16) where 𝔸=4(𝐵₁₂+𝑄₃+𝑃₁/2)(𝐵₁₂+𝑄₃+𝑃₁/2)sin²𝜃cos²𝜃, where𝐵₂₁−𝑃₁=𝐵₁₂. (4.17) Let 𝐶_𝑃(𝜃) and 𝐶_SV(𝜃) be the values of 𝑐 associated with upper and lower signs in (4.16), corresponding to the velocities for quasi-𝑃 waves and quasi-SV waves, respectively. Hence these expressions coincide with the expressions obtained by Sidhu and Singh [1] for velocities of quasi-𝑃 waves and quasi-SV waves.

5. Numerical Calculation and Discussions

Here we consider the model for an initially stressed medium:𝐵₁₁=𝜆+2𝜇+𝑃_1,𝐵₁₂=𝜆+𝑃₁,𝐵₂₂=𝜆+2𝜇,𝑄₃=𝜇. (5.1) Using (5.1) in (4.17), we get a nondimensional form of velocity equations as𝐶_𝑃(𝜃)=12𝛿+3+𝑝+𝛿+1+𝑝²+𝛿𝑝𝛿+1+𝑝sin²𝜃^1/2,𝐶_SV(𝜃)=12𝛿+3+𝑝−𝛿+1+𝑝²+𝛿𝑝𝛿+1+𝑝sin²𝜃^1/2, (5.2) where𝜆𝛿=𝜇𝑃,𝑝=¹2𝜇,𝛽²=𝜇𝜌. (5.3) The apparent velocities for quasi-𝑃 waves and quasi-SV waves can be obtained from (5.2) as𝐶_𝑃𝑎(𝜃)=𝐶_𝑃(𝜃),𝐶sin𝜃_SV𝑎(𝜃)=𝐶_SV(𝜃).sin𝜃 (5.4) The numerical values of the dimensionless slowness (1/𝐶_𝑃(𝜃),1/𝐶_SV(𝜃)) have been calculated from (5.2) assuming that 𝛿=1 for different values of 𝑝 vary from −0.8to0.8 and for different values of 𝜃 vary from 0^∘to90^∘. Figures 1 and 2 show the variation of the dimensionless slowness for the quasi-𝑃 and quasi-SV waves with the angle of incidence −0.8, −0.6, 0.2, 0.4, 0.6, and 0.8.

Figure 1 

Figure 2 

6. Conclusion

The study shows that the phase velocities of quasi-𝑃 waves and quasi-SV waves are highly affected by the initial stresses present in the medium and also the direction of propagation.

Acknowledgment

The authors are thankful to Dr. George Jaiani, Tbilisi State University, Georgia for his valuable comments and encouragement to prepare the paper.