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, S11,S22,S33,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 (Bij,Qi) 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+∂s31∂z+ρΔx+ρ(ωzY-ωyZ)-ρeX-exx∂S11∂x-eyy∂S12∂y-ezz∂S31∂z-eyz(∂S31∂y+∂S12∂z)-ezx(∂S11∂z+∂S31∂x)-exy(∂S12∂x+∂S11∂y)+(S11-S22)∂Wz∂y-2S12∂Wz∂x+S12∂Wx∂z+(S33-S11)∂Wy∂z+2S12∂Wy∂x-S31∂Wx∂y+S23(∂Wy∂y-∂Wz∂z)=ρ∂2u1∂t2.
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. ΔXi are the components of incremental body force, which are assumed to satisfy the equationΔXi=ujXi,j=0,e=exx+eyy+ezz.
The Sij are the components of prestress, which are assumed to satisfy the equilibrium equation.Sij,j+ρXi=0,
and are related to the initial strain ϵij by Hooke’s lawSij=λϵiiδij+2μϵij,
The incremental stresses sij are supposed to be linearly related to the incremental strains eij through the incremental elastic coefficients Bij and Qis11=B11exx+B12eyy+B13ezz,s22=B21exx+B22eyy+B23ezz,s33=B31exx+B32eyy+B33ezz,s23=2Q1eyz,s31=2Q2ezx,s23=2Q3exy.
We assume that(u1,u2,u3)=(u,v,w),(X1,X2,X3)=(X,Y,Z),exx=∂u∂x,eyy=∂v∂y,ezz=∂w∂z,exy=12(∂u∂y+∂v∂x),eyz=12(∂v∂z+∂w∂y),ezx=12(∂w∂x+∂u∂z),Wx=12(∂w∂y-∂v∂z),Wy=12(∂u∂z-∂w∂x),Wz=12(∂v∂x-∂u∂y).
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∂x∂z+(B12+Q3+P12)∂2v∂x∂y+12S31∂2v∂y∂z-12S23∂2v∂x∂z+(B13+Q2-P32)∂2w∂x∂z+12S12∂2w∂y∂z-12S23∂2w∂x∂y-(∂S11∂x+ρX)∂u∂x-12(∂S12∂x+∂S11∂y)∂u∂y-12(∂S31∂x+∂S11∂z+ρZ)∂u∂z-12(∂S12∂x+∂S11∂y-ρY)∂v∂x+(∂S12∂y+ρX)∂v∂y-12(∂S31∂y+∂S12∂z)∂v∂z-12(∂S31∂x+∂S11∂z-ρZ)∂w∂x-12(∂S31∂y+∂S12∂z)∂w∂y-(∂S31∂z+ρX)∂w∂z=ρ∂2u∂t2.
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.

3. Propagation of Waves

For elastic waves propagating in a direction specified by direction cosines (l,m,n) along the axes, we takeu=Uiexp(iP),v=Viexp(iP),w=Wiexp(iP).(Ui,Vi,Wi) are amplitude factors along the axes and P is phase factor:P=k{ct-(lx+my+nz)},
where c is phase velocity and k is wave number.

Putting (3.1) and (3.2) in (2.7), we get(Ω1+Ω1′+ρc2)Ui+(I1+I1′)Vi+(K1+K1′)Wi=0,(K2+K2′)Ui+(Ω2+Ω2′+ρc2)Vi+(I2+I2′)Wi=0,(I3+I3′)Ui+(K3+K3′)Vi+(Ω3+Ω3′+ρc2)Wi=0,
whereΩ1=-{B11l2+(Q3-P12)m2+(Q2+P32)n2+S12lm+S23mn+S31nl},I1=S12l2+(S122)n2-(B12+Q3+P12)lm-(S312)mn+(S232)ln,K1=S31l2+(S312)m2-(B13+Q2-P32)ln-(S122)mn+(S232)ln,Ω1′=ik{(∂S11∂x+ρX)l+12(∂S12∂x+∂S11∂y+ρY)m+12(∂S31∂x+∂S11∂z+ρZ)n},I1′=ik{12(∂S12∂x+∂S11∂y-ρY)l-(∂S12∂y+ρX)m+12(∂S31∂y+∂S12∂z)n},K1′=ik{12(∂S31∂x+∂S11∂z-ρZ)l+12(∂S31∂y+∂S12∂z)m+12(∂S31∂z+ρX)n}.
The explicit expressions for (Ω2,Ω3),(ℑ2,ℑ3), (𝒦2,𝒦3),(Ω2′,Ω3′),(ℑ2′,ℑ3′), and (𝒦2′,𝒦3′) 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 Ui,Vi and Wi 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-HG)+(ABD+EFG+HIJ-AFJ-EID-HGB)=0,
whereA=Ω1+Ω1′,E=I1+I1′,H=K1+K1′,B=Ω2+Ω2′,F=I2+I2′,I=K2+K2′,D=Ω3+Ω3′,G=I3+I3′,J=K3+K3′.

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),(Ω2′,Ω3′),(ℑ2′,ℑ3′), and (𝒦2′,𝒦3′), we getΩ11=-(Q2+P32),Ω21=-(Q1-P22),Ω31=-B33,I11=S122,I21=0,I31=S31,K11=0,K21=S122,K31=S23,Ω11′=i2k(∂S31∂x+∂S11∂z+ρz),Ω21′=i2k(∂S23∂y+∂S22∂z+ρz),Ω31′=ik(∂S33∂x+ρz),I11′=i2k(∂S31∂y+∂S12∂z),I21′=-ik(∂S23∂z+ρY),I31′=ik(∂S33∂x+∂S31∂z-ρX),K11′=ik(∂S31∂z+ρX),K21′=i2k(∂S23∂x+∂S12∂z),K31′=i2k(∂S33∂y+∂S23∂z+ρY).
Equation (3.10) takes the formρ3c6+ρ2c4(A1+B1+D1)+ρc2(A1B1+B1D1+D1A1-F1J1-E1I1-H1G1)+(A1B1D1+E1F1G1+H1I1J1-A1F1J1-E1I1D1-H1G1B1)=0,
whereA1=Ω11+Ω11′,B1=Ω21+Ω21′,D1=Ω31+Ω31′,E1=I11+I11′,F1=I21′,G1=I31+I31′,H1=K11′,I1=K21+K21′,J1=K31+K31′.

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+ρ2c4(Ω11+Ω21+Ω31)+ρc2(Ω11Ω21+Ω21Ω31+Ω31Ω11-I11K21)+(Ω11Ω21Ω31-I11K21Ω31)=0.

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 form(Ω11+ρc2)(Ω21+ρc2)(Ω31+ρc2)=0;
on simplification, we get three values of c2:c2=Q2+P3/2ρ,c2=Q1-P2/2ρ,c2=B33ρ.

4.2. Two Dimensional Prestressed Medium4.2.1. 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+Ω111′+ρc2)Ui+(I111+I111′)Vi=0,(K211+K211′)Ui+(Ω211+Ω211′+ρc2)Vi=0,
whereΩ111=-{B11l2+(Q3-P12)m2+S12ml},I111={S12l2-(B12+Q3+P12)ml},Ω211=-{(Q3+P12)l2+B22m2+S12ml},K211={S12m2-(B12+Q3-P12)nl},Ω111′=ik{(∂S11∂x+ρX)l+12(∂S12∂x+∂S11∂y+ρY)m},Ω211′=ik{12(∂S22∂x+∂S12∂y+ρX)l+(∂S22∂y+ρY)m},I111′=ik{12(∂S12∂x+∂S11∂y-ρY)l-(∂S12∂y+ρX)m},K211′=ik{(∂S12∂x+ρY)l+12(∂S22∂x+∂S12∂y+ρX)m}.
Equation (4.8) has nontrivial solution whenρ2c4+(Ω111+Ω211+Ω111′+Ω211′)ρc2-(I111+I111′)(K211+K211′)=0.
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 get(Ω111+ρc2)Ui+I111Vi=0,K211Ui+(Ω211+ρc2)Vi=0.
If S12=S23=S31=0, then (4.11) takes the form(Ω1111+ρc2)Ui+I1111Vi=0,K2111Ui+(Ω2111+ρc2)Vi=0,
whereΩ1111=-{B11l2+(Q3-P12)m2},I1111=-(B12+Q3+P12)ml,Ω2111=-{(Q3+P12)l2+B22m2},K2111=-(B21+Q3-P12)lm.
The set of homogeneous (4.12) in Ui,Vi has a nontrivial solution when |(Ω1111+ρc2)I1111K2111(Ω2111+ρc2)|=0.
This quadratic equation in ρc2 may be solved to obtain 2ρc2={(B11+Q3+P12)l2+(B21+Q3-P12)m2}±{(B11-Q3-P12)l2+(B22-Q3+P12)m2}2+4(B21+Q3-P12)(B12+Q3+P12)l2m2.
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+P12)sin2θ+(B21+Q3-P12)cos2θ}±{(B11-Q3-P12)sin2θ+(B22-Q3+P12)cos2θ}2+A,
where 𝔸=4(B12+Q3+P1/2)(B12+Q3+P1/2)sin2θcos2θ, whereB21-P1=B12.
Let CP(θ) and CSV(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+P1,B12=λ+P1,B22=λ+2μ,Q3=μ.
Using (5.1) in (4.17), we get a nondimensional form of velocity equations asĈP(θ)=12[(δ+3+p)+{(δ+1+p)2+δp(δ+1+p)sin2θ}1/2],ĈSV(θ)=12[(δ+3+p)-{(δ+1+p)2+δp(δ+1+p)sin2θ}1/2],
whereδ=λμ,p=P12μ,β2=μρ.
The apparent velocities for quasi-P waves and quasi-SV waves can be obtained from (5.2) asCPa(θ)=ĈP(θ)sinθ,CSVa(θ)=ĈSV(θ)sinθ.
The numerical values of the dimensionless slowness (1/ĈP(θ),1/ĈSV(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.

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.

SidhuR. S.SinghS. J.Comments on “Reflection of elastic waves under initial stress at a free surface: P and SV motionSidhuR. S.SinghS. J.Reflection of P and SV waves at the free surface of a prestressed elastic half-spaceNorriesA. N.Propagation of plane waves in a prestressed elastic mediumDayS. RayN.DattaA.Propagation of P and S waves and reflection of P in a medium under normal initial stressesBiotM. A.