Arch Appl Mech (2006) 76: 65–74 DOI 10.1007/s00419-006-0005-0
O R I G I NA L
A. Chattopadhyay
Wave reflection in triclinic crystalline medium
Received: 2 April 2005 / Accepted: 8 November 2005 / Published online: 8 February 2006 © Springer-Verlag 2006
Abstract In this paper, the reflection of a plane wave at a traction free boundary of a half -space composed of triclinic crystalline material is considered. It is shown that an incident plane wave generates three plane waves, namely quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH) waves governed by the propagation condition involving the acoustic tensor. A simple procedure is presented for the calculation of all the three phase velocities of these waves. It is demonstrated that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation. A procedure is established for the calculation of the amplitude vector in terms of the phase velocity, the propagation vector, and the stiffness coefficients of the medium. Closed form solutions are obtained for the reflection coefficients of qP, qSV and qSH waves. Using the parameters of Vosges sandstone exhibiting triclinic symmetry, the graphical representations of the reflection coefficients due to an incident qP wave are given. It is observed that, in triclinic medium, the reflection coefficients are significantly different from those in an isotropic medium. Keywords Reflection · Incident wave · Triclinic medium · Quasi-P · Quasi-SV · Quasi-SH waves 1 Introduction The theoretical study of reflection and refraction of elastic waves is of considerable interest in the field of seismology, in particular, in seismic prospecting since the Earth’s interior is modeled by layers of different material properties. The elastic properties of a crystalline material depend on its internal structure. Wave propagation in crystalline media plays an important role in geophysics, ultrasonics and signal processing. Seismic waves of high energy can produce structural change of the rocks through which they propagate. Seismic wave velocities are usually greater in crystalline rocks than in sedimentary rocks. These theoretical studies help to obtain knowledge about rock structure and elastic properties, and information about minerals and fluids inside the Earth. The effect of the material anisotropy on the reflection coefficients is of great practical importance. The effect of earthquakes on structures is one such an area where structural engineers have to take a high factor of safety due to non-availability of an exact information about a crystalline bedrock. Propagation of body and surface waves in anisotropic media is fundamentally different from that in isotropic media. In seismology, the anisotropy manifests most straightforwardly by variation of the phase speed of seismic waves with their direction of propagation. An elastic material displaying velocity anisotropy must have its effective elastic constants arranged in some form of crystalline symmetry. Crampin [6] pointed out that the behavior of both body and surface waves in anisotropic structures differs from that in isotropic structures, and variation of velocity with propagation A. Chattopadhyay Department of Applied Mathematics Indian School of Mines, Dhanbad- 826004, Jharkhand, India E-mail:
[email protected] Fax: +91-326-2210028
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direction is only one of the anomalies which may occur. In an anisotropic material, three body waves can propagate in any direction, having different velocities and different polarizations. Moreover, in an anisotropic medium, the qP, qSV, and qSH are coupled. This coupling introduces polarization anomalies that can be applied to investigate anisotropy within the Earth. The problem of reflection and refraction of elastic waves was discussed by many authors. Knott [12] discussed the problem of reflection and refraction of elastic waves with seismological applications. He was first to derive an analytical expression for the reflection of ultrasound wave-packets. The square roots of the energy ratio of the reflected and transmitted waves, as functions of incident angles, were evaluated numerically by Gutenberg [8] for various solid–solid interfaces. Crampin and Taylor [7] studied surface wave propagation in anisotropic media by a generalized technique permitting any combination of plane layers possessing any anisotropic symmetry using a numerical code based on the Thompson–Haskell matrix formulation. They presented some numerical examples of surface wave propagation in anisotropic media to identify geophysical structures. Crampin [5] calculated the particle motion of surface waves propagating in particular symmetry directions in anisotropic media and demonstrated that propagation in some directions showed particle motion anomalies diagnostic of the symmetry. A full account of reflection and refraction of elastic waves was given in the monograph by Achenbach [1]. Three papers by Keith and Crampin [9–11] presented synthetic seismograms for (plane wave) body wave propagation in multi-layered azimuthally anisotropic media. This was the first time that shear wave splitting was demonstrated in synthetic seismograms. Tolstoy [19] and Norris [15] studied elastic waves in pre-stressed solids. Pal and Chattopadhyay [18] studied reflection phenomena of plane waves at a free boundary in a pre-stressed elastic half-space. Auld [2] discussed in his monograph (volume I) acoustic wave propagation in both isotropic and anisotropic media. In the volume II of this monograph, he discussed in detail reflection and refraction problems. Recent advances in the study of the dynamic behavior of layered materials in general, and laminated fibrous composites in particular, are presented in the book by Nayfeh [14]. These two monographs explore the fundamental question of how mechanical waves propagate in layered anisotropic media. Three-dimensional reflection and transmission coefficients for plane waves in isotropic elastic media (applicable to analyzing waves in nonparallel-sided waveguides) were derived by Borejko [3], where an extensive survey of literature pertinent to the problem of reflection and transmission of plane waves was also given. Ogden and Sotirropoulos [16,17] investigated the problems of reflection in both incompressible and compressible elasticity. Chattopadhyay and Rogerson [4] studied reflection of plane waves at an incrementally traction-free boundary of a half-space composed of linearly incompressible elastic material. In this paper, we study reflection of plane waves at the traction-free boundary of a triclinic crystalline medium. Analytical formulas are derived for the amplitude vector (the polarization vector) in terms of the phase velocity, the propagation vector (the propagation direction), and the stiffness coefficients of the medium. The principal results of the paper are the analytical formulas for the three phase speeds and the analytical formulas for the reflection coefficients at the traction-free boundary of a triclinic elastic half-space. The reflection coefficients due to an incident qP wave are computed for Vosges sandstone. It is observed that, in triclinic medium, the amplitude ratios for reflected waves are remarkably different from those in an isotropic medium.
2 Formulation of the problem Consider a homogeneous triclinic medium having 21 elastic constants. For a plane wave propagating in the x2 x3 -plane, the components of the displacement vector u are u i = u i (x2 , x3 , t),
∂ = 0, ∂ x1
i = 1, 2, 3,
(1)
where the symbols t, x2 , and x3 denote the time, and the respective orthogonal Cartesian coordinate axes. The stress–strain relations are τ11 = C11 e11 + C12 e22 + C13 e33 + 2(C14 e23 + C15 e13 + C16 e12 ), τ22 = C12 e11 + C22 e22 + C23 e33 + 2(C24 e23 + C25 e13 + C26 e12 ), τ33 = C13 e11 + C23 e22 + C33 e33 + 2(C34 e23 + C35 e13 + C36 e12 ),
(2a)
Wave reflection in triclinic crystalline medium
67
τ23 = C14 e11 + C24 e22 + C34 e33 + 2(C44 e23 + C45 e13 + C46 e12 ), τ13 = C15 e11 + C25 e22 + C35 e33 + 2(C45 e23 + C55 e13 + C56 e12 ), τ12 = C16 e11 + C26 e22 + C36 e33 + 2(C46 e23 + C56 e13 + C66 e12 ), where Ci j = C ji , 2ei j = (u i, j + u j,i ), τi j are the components of the stress tensor τ , Ci j are the stiffness coefficients, and ei j are the components of the infinitesimal strain tensor e. The equations of motion in the absence of body force are τi j, j = ρ u¨ i , i = 1, 2, 3,
(2b)
where ρ and u¨ i denote the mass density and the components of the acceleration, respectively. The following equations of motion are obtained after recalling Eqs. (1) and (2) ∂ 2u1 ∂ 2u1 ∂ 2u1 ∂ 2u2 ∂ 2u2 ∂ 2u2 C55 2 + 2C56 + C66 2 + C45 2 + (C46 + C25 ) + C26 2 ∂ x2 ∂ x3 ∂ x2 ∂ x3 ∂ x3 ∂ x2 ∂ x3 ∂ x2 ∂ 2u3 ∂ 2u3 ∂ 2u1 ∂ 2u3 + C35 2 + (C36 + C45 ) + C46 2 = ρ 2 , ∂ x2 ∂ x3 ∂t ∂ x3 ∂ x2 ∂ 2u1 ∂ 2u1 ∂ 2u1 ∂ 2u2 ∂ 2u2 ∂ 2u2 + C26 2 + C44 2 + 2C24 + C22 2 C45 2 + (C25 + C46 ) ∂ x2 ∂ x3 ∂ x2 ∂ x3 ∂ x3 ∂ x2 ∂ x3 ∂ x2 ∂ 2u2 ∂ 2u3 ∂ 2u3 ∂ 2u3 + C24 2 = ρ 2 , + C34 2 + (C23 + C44 ) ∂ x2 ∂ x3 ∂t ∂ x3 ∂ x2 ∂ 2u1 ∂ 2u1 ∂ 2u1 + C46 2 C35 2 + (C45 + C36 ) ∂ x2 ∂ x3 ∂ x3 ∂ x2 ∂ 2u2 ∂ 2u2 ∂ 2u2 + C24 2 + C34 2 + (C23 + C44 ) ∂ x2 ∂ x3 ∂ x3 ∂ x2 ∂ 2u3 ∂ 2u3 ∂ 2u3 ∂ 2u3 + C44 2 = ρ 2 . + C33 2 + 2C34 ∂ x2 ∂ x3 ∂t ∂ x3 ∂ x2
(n)
(n)
(n)
(3)
(n)
Let p = p2 e2 + p3 e3 denote the unit propagation vector with components 0, p2 , and p3 , where e2 and e3 are the unit vectors along the x2 and x3 coordinate axes, respectively. Let cn denote the phase velocity and kn denote the wavenumber of a plane wave propagating in the x2 x3 -plane. Consider a plane wave solution of Eqs. (3) of the form
(n) (n) u1 d1 (n) (n) u 2 = An d2 exp(iηn ), (n) (n) u3 d3
(4)
where (n) (n) (n) d = d1 e1 + d2 e2 + d3 e3 is the unit amplitude vector, An is the wave amplitude and (n)
(n)
ηn = kn (x2 p2 + x3 p3 − cn t)
(5)
is the phase function of the plane wave. Inserting the representation (4) into Eqs. (3), we obtain the Christoffel equation of the form (n) d1 (S − c¯2 ) T P (n) R T (Q − c¯2 ) d2 = 0, 2 (n) P R (W − c¯ ) d3
(6)
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where C ρcn2 i j = i j , , C C44 C44 56 p2 p3 + C 66 p22 , 55 p32 + 2C S=C 46 + C 25 ) p2 p3 + C 26 p22 , 45 p32 + (C T =C 36 + C 45 ) p2 p3 + C 46 p22 , 35 p32 + (C P=C
c¯2 =
(7)
24 p2 p3 + C 22 p22 , 44 p32 + 2C Q=C 23 + C 44 ) p2 p3 + C 24 p22 , 34 p32 + (C R=C 34 p2 p3 + C 44 p22 . 33 p32 + 2C W =C From the Eqs. (6), follows the characteristic equation, derived by setting the characteristic determinant of the Christoffel equation equal to zero, c¯6 + a1 c¯4 + a2 c¯2 + a3 = 0,
(8)
a1 = −(S + Q + W ), a2 = Q S + W S + QW − R 2 − T 2 − P 2 , a3 = −(S QW − S R 2 − W T 2 + 2P T R − P 2 Q).
(9)
where
From Eqs. (6), we obtain (n)
d3
d1(n)
=
T 2 − (Q − c¯2 )(S − c¯2 ) , P(Q − c¯2 ) − RT
(n)
d2
(n) d1
=
R(S − c¯2 ) − P T , P(Q − c¯2 ) − RT
(10)
and (n)
d3
(n)
d2
=
T 2 − (Q − c¯2 )(S − c¯2 ) . R(S − c¯2 ) − P T
Equations (10) can be used to calculate the amplitude vector (the polarization vector) in terms of the phase velocity, the propagation vector (the propagation direction), and the stiffness coefficients of the medium. Solving Eq. (8), we obtain the phase velocities of the quasi-P (qP), quasi-SV (qSV), and quasi-SH (qSH) waves ρcL2 = −2r cos
2 = 2r cos ρcSV
2 = 2r cos ρcSH
ϕ
π 3 π 3
3
−
a1 , 3
+
ϕ a1 − , 3 3
−
ϕ a1 − , 3 3
(11)
Wave reflection in triclinic crystalline medium
69
where 2q =
2a13 a1 a2 − + a3 , 27 3 r = − | p|,
3p =
ϕ = cos−1
3a2 − a12 , 3
q
. (12) r3 Equations (11) state the principal results of the present paper since they provide explicit analytical expressions for the three phase velocities in a triclinic medium. In an isotropic medium, C11 = C22 = C33 = λ + 2µ, C12 = C13 = C23 = λ, C44 = C55 = C66 = µ,
(13)
and all other elastic constants are zero, where λ and µ are Lamé constants. Substituting Eqs. (13) in Eqs. (9) and (12), we obtain −(λ + µ)2 −(λ + µ) −(λ + µ)3 (14) ,r = ,q = , a1 = −(λ + 4µ). 9 3 27 Finally, substituting Eqs. (14) into Eqs. (11), we obtain the compressional velocity cL and the repeated roots cSV and cSH for shear velocity as p=
c2L =
λ + 2µ 2 µ , c SV = c2S H = . ρ ρ
(15)
Equations (11) provide three real roots of the characteristic equation (8). The largest root provides the phase velocity of the qP wave; the second largest provides the phase velocity of the qSV wave, and the lowest root provides the phase velocity of qSH wave. The phase velocities of the two quasi-transverse waves (qSV and qSH) are not identical in a triclinic medium. This result is tested with a set of data as mentioned in Sect. 4. If any geophysical evidence exists that the qSH wave velocity is greater than qSV wave velocity, the graphs of the reflected qSV wave and reflected qSH wave should be interchanged. The present method of calculation of the velocities of all the three quasi-waves is general and it can thus be applied to identify the phase velocities for different types of anisotropy. 3 Solution of the problem Consider reflection of plane qP, qSV, and qSH waves at the boundary plane x2 = 0 of a half-space, defined by −∞ < x1 < ∞ , −∞ < x3 < ∞ , and x2 0, composed of a triclinic material. Since the displacement components u i are independent of the coordinate x1 , the plane of incidence is then given by x1 =0, and the incident and reflected plane waves can be at most functions of the coordinates x2 and x3 . The plane qP, qSV, or qSH wave incident at the boundary x2 =0 generates three plane reflected qP, qSV, and qSH waves since all three displacement components ui are coupled [Eqs. (3)]. We introduce the index n taking the values 0,1,2,3 which identify the incident, reflected qP, reflected qSV, and reflected qSH wave, respectively. In the plane x2 = 0, the displacements and the stresses of the incident and reflected waves are represented by (n)
(n)
u j = An d j exp(iηn ), j = 1, 2, 3. (n)
τ12 = P1n ikn An exp(iηn ), (n)
τ22 = Q n ikn An exp(iηn ), (n) τ23
= Rn ikn An exp(iηn ),
(16)
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where (n) (n)
(n) (n)
(n) (n)
(n) (n)
P1n = C26 p2 d2 + C36 p3 d3 + C46 {d2 p3 + d3 p2 } (n) (n)
(n) (n)
+C56 d1 p3 + C66 d1 p2 , Qn = Rn =
(17)
(n) (n) (n) (n) (n) (n) C22 p2 d2 + C23 p3 d3 + C24 {d2 p3 (n) (n) (n) (n) +C25 d1 p3 + C26 d1 p2 , (n) (n) (n) (n) (n) (n) C24 p2 d2 + C34 p3 d3 + C44 {d2 p3 +C45 d1(n) p3(n) + C46 d1(n) p2(n) , (n) kn (x3 p3 − cn t),
(n) (n) + d3 p2 }
(18) (n) (n) + d3 p2 }
ηn = and n = 0, 1, 2, 3.
(19)
For the incident plane wave, (0)
(0)
p2 = − cos θ0 , p3 = sin θ0 , c0 = c I . For the reflected qP wave, p2(1) = cos θ1 , p3(1) = sin θ1 , c1 = c L1 . For the reflected qSV wave,
(20)
p2(2) = cos θ2 , p3(2) = sin θ2 , c2 = cT , For the reflected qSH wave, p2(3) = cos θ3 ,
p3(3) = sin θ3 , c3 = cT 1 ,
where c I , c L1 , cT , and cT 1 are the phase velocities of the incident plane wave, the reflected qP wave, reflected qSV wave, and reflected qSH wave, respectively. At the free boundary plane x2 = 0, the boundary conditions are (0)
(1)
(2)
(3)
τ12 + τ12 + τ12 + τ12 = 0,
(21)
(0) (1) (2) (3) τ22 + τ22 + τ22 + τ22 = 0, (0) (1) (2) (3) τ23 + τ23 + τ23 + τ23 = 0.
Using the boundary conditions and Eqs. (16), we obtain, (0)
(1)
P10 A0 k0 exp{ik0 (x3 p3 − c I t)} + P11 A1 k1 exp{ik1 (x3 p3 − c L1 t)} (2)
(3)
+P12 A2 k2 exp{ik2 (x3 p3 − cT t)} + P13 A3 k3 exp{ik3 (x3 p3 − cT 1 t)} = 0, (0) Q 0 A0 k0 exp{ik0 (x3 p3
− c I t)} +
(22)
(1) Q 1 A1 k1 exp{ik1 (x3 p3
− c L1 t)} (2) +Q 2 A2 k2 exp{ik2 (x3 p3 − cT t)} + Q 3 A3 k3 exp{ik3 (x3 p3(3) − cT 1 t)} = 0, (0) (1) R0 A0 k0 exp{ik0 (x3 p3 − c I t)} + R1 A1 k1 exp{ik1 (x3 p3 − c L1 t)} (2) (3) +R2 A2 k2 exp{ik2 (x3 p3 − cT t)} + R3 A3 k3 exp{ik3 (x3 p3 − cT 1 t)} = 0.
These equations are valid for all values of x3 and t, so that we have k0 (x3 sin θ0 − c I t) = k1 (x3 sin θ1 − c L1 t) = k2 (x3 sin θ2 − cT t) = k3 (x3 sin θ3 − cT 1 t),
(23)
which gives k0 c I = k1 c L1 = k2 cT = k3 cT 1 = k,
(24)
Wave reflection in triclinic crystalline medium
71
and k0 sin θ0 = k1 sin θ1 = k2 sin θ2 = k3 sin θ3 = ω,
(25)
where k and ω are apparent wave number and circular frequency, respectively. The amplitude ratios of qP, qSV, and qSH waves are denoted by A1 /A0 , A2 /A0 and A3 /A0 , respectively. Solving Eqs. (22) for these ratios, the explicit analytical expressions for the reflection coefficients of qP, qSV, and qSH waves are obtained D1 A1 = , A0 D0
A2 D2 = , A0 D0
A3 D3 = , A0 D0
(26)
where a1 a2 a3 D0 = b1 b2 b3 , c1 c2 c3 a1 − 1 a3 D2 = b1 − 1 b3 , c1 − 1 c3
−1 a2 a3 D1 = −1 b2 b3 , −1 c2 c3 a1 a2 − 1 D3 = b1 b2 − 1 c1 c2 − 1
,
and ai =
1i ki P , P10 k0
1i = P1i , P C44
bi =
i ki Q , Q 0 k0
i = Q i , Q C44
ci =
i ki R , R0 k 0
i = Ri , R C44
i = 1, 2, 3
i = 0, 1, 2, 3.
(27)
4 Numerical examples and conclusions Numerical calculations were performed using data of Vosges sandstone exhibiting triclinic anisotropy [13]. The elastic constants of Vosges sandstone are: C11 C12 C23 C34 C44 C55
= 16.248 GPa, C22 = 11.88 GPa, C33 = 12.216 GPa, = 1.488 GPa, C13 = 2.4 GPa, C14 = −1.152 GPa, C15 = 0.0 GPa, C16 = −0.561 GPa, = 1.032 GPa, C24 = 0.912 GPa, C25 = 1.608 GPa, C26 = 1.248 GPa, = 0.672 GPa, C35 = 0.216 GPa, C36 = 0.216 GPa, = 5.64 GPa, C45 = 2.16 GPa, C46 = 0.0 GPa, = 5.88 GPa, C66 = 6.912 GPa, C56 = 0.0 GPa, ρ = 2.40 g/cm3 .
The elastic constants for the isotropic material are: µ = 6.0 GPa, λ = 8.0 GPa, ρ = 2.8 g/cm3 . Curve-1 and curve-2 were obtained considering the anisotropic (Vosges sandstone) and isotropic data. The amplitude ratios for the reflected qP, qSV, and qSH waves were computed and curves were plotted for different values of the angle of incidence ranging from 0◦ to 90◦ (Figs.1,2,3). The ordinates in Figs. 1, 2, and 3 were taken as the reflection coefficients of qP,qSV, and qSH waves and the abscissa in all these figures have been taken as incident angle (in degrees). Figure- 1 shows the variation of the reflection coefficient Rpp = A1 /A0 (the incident qP wave is reflected as the qP wave). In curve-1, around 8◦ –12◦ , the values of the reflected coefficient Rpp increase sharply and
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2.00
curve-1
Rpp--->
1.00
curve-2
0.00
-1.00
-2.00 0.00
20.00
40.00
60.00
80.00
100.00
Incident angle (in degree)--->
Fig. 1 Amplitude ratios of reflected qP waves
2.00
curve-2
Rpsv--->
1.00
0.00
curve-1
-1.00
-2.00 0.00
20.00
40.00
60.00
80.00
100.00
Incident angle (in degree)--->
Fig. 2 Amplitude ratios of reflected qSV waves
then show a regular trend till 90◦ . It is clear that the magnitude of the reflected qP wave in an anisotropic medium (curve-1) is much higher compared to the isotropic case (curve-2). In this figure, the maximum value of curve-1 is 0.97 at 59◦ , and that for curve-2 is 0.15 at 69◦ . Figure- 2 shows the variation of the reflection coefficient R psv = A2 /A0 (the incident qP wave is reflected as qSV wave). In curve-1, around 8◦ –12◦ , the reflected coefficient Rpsv decreases sharply and then shows a regular trend till 90◦ . It is clear that the magnitude of the reflected qSV wave in an anisotropic medium (curve-1) is much lower compared to the isotropic case (curve-2). In this Figure, the maximum value of curve-1 is 0.24 at 69.5◦ , and that for curve-2 is 1.06 at 43.5◦ . Figure-3 shows the variation of the reflection coefficient Rpsh = A3 /A0 (the incident qP wave is reflected as qSH wave). The maximum value of curve-3 is 1.05 at 24◦ .
Wave reflection in triclinic crystalline medium
73
1.20
Rpsv--->
0.80
0.40
0.00
curve-3
-0.40 0.00
20.00
40.00
60.00
80.00
100.00
Incident angle (in degree)--->
Fig. 3 Amplitude ratios of reflected qSH waves
It can be concluded that, in a triclinic medium, the phase velocity is significantly different from that in an isotropic medium. The amplitude ratios of reflected waves are affected significantly and, therefore, the results can possibly be utilized in the interpretation and analysis of geophysical data. The present findings can be helpful in forecasting geophysical parameters at greater depths through signal processing and seismic data analysis. The present study can possibly be useful to geophysicist and metallurgists for analysis of rock and material structures through non-destructive testing . Acknowledgements The work was completed while I was visiting the Technical University of Vienna, Department of Civil Engineering as visiting Professor. The author is grateful to Professor Franz Ziegler for providing fellowship and all the facilities for conducting research. I am also grateful to Docent Piotr Borejko for various discussions about the research work and also for all kind of help during my stay at Vienna, Austria. This award is very gratefully acknowledged. The author is also grateful to the reviewers for their constructive suggestions for the improvement of this paper.
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14. Nayfeh, A,H.: Wave propagation in layered anisotropic media with applications to Composites. Elsevier, Amsterdam (1995) 15. Norris, A.N.: Propagation of plane waves in a pre-stressed elastic media. J Acoust Soc Am 74, 1642–1643 (1983) 16. Ogden, R.W. Sotirropoulos, D.: The effect of pre-stress on the propagation and reflection of plane waves in incompressible elastic solids. IMA J Appl Math 59, 95–121 (1997) 17. Ogden, R.W., Sotirropoulos, D.: Reflection of plane waves from the boundary of a pre-stressed compressible elastic halfspace. IMA J Appl Math 61, 61–90 (1998) 18. Pal, A.K., Chattopadhyay, A.: The reflection phenomena of plane waves at a free boundary in a pre-stressed elastic half-space. J Acoust Soc Am 76(3), 924–925 (1984) 19. Tolstoy, I.: On elastic waves in pre-stressed solids. J Geophy Res 87, 6823–6827 (1982)
