INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 23, 1567-1 578 (1986)

WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

EDUARDO KAUSEL*

Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, M A 021 39, U.S.A.

SUMMARY The determination of the natural modes of wave propagation in an anisotropic layered medium requires the solution of a transcendental eigenvalue problem that is usually approached numerically with the aid of search techniques. Such computations require great effort. The method presented in this paper provides an alternate solution to this problem in terms of a quadratic eigenvalue problem involving tridiagonal matrices, for which the eigenvalues can be found with great speed and accuracy. The technique is then illustrated by means of an example involving a cross-anisotropic Gibson solid.

INTRODUCTION

Solutions to the wave equation for plane waves in unbounded, homogeneous, anisotropic media have been known for many years, and can be found in many good textbooks dealing with the subjects of theory of elasticity and/or wave propagation. More general wave types occurring in bounded and/or layered anisotropic media are, on the other hand, vastly more complicated to investigate because the determination of the natural modes of wave propagation in such media requires the solution of analytically intractable transcendental equations. The method presented in this paper avoids the difficulties inherent in search procedures by. _,arting to a discretization of the displacement field in the direction of layering. The technique constitutes a generalization of a procedure originally proposed by Waas and Lysmer in 1972, and which was extended by the author and by others in a number of related papers; the reader is referred to these works for further details.’-13

WAVE EQUATION IN AN ANISOTROPIC MEDIUM

Consider a layered, anisotropic soil medium (Figure 1). The material properties are homogeneous within each layer, although they may change from layer to layer. Subdividing each layer into thin sublayers we obtain a soil profile consisting of n distinct layers.

Any particular sublayer in the stratum is characterized by a thickness h, a mass density p and a (symmetric) constitutive matrix D = I d i j ) (i, j = 1, . . . ,6). Cutting out this individual layer (Figure 2), we preserve equilibrium by applying appropriate tractions at the upper ( j = 1) and lower ( j = 2) interfaces:

* Associate Professor of Civil Engineering.

0029-5981/86/111567-12$06.00 0 1986 by John Wiley & Sons, Ltd.

Received 7 M a y 1985 Revised 20 January 1986

1568

layer H

sublayer

E. KAUSEL

X

Figure 1. Discretization of anisotropic stratum

A

I hi jth sublayer + j + l

Figure 2. Isolated sublayer

where

The components of the vector Sj are the stresses at the two bounding interfaces. Within the layer, the motion is governed by the wave equation (a superscript T stands for the transposed matrix or vector):

w = pti-LTa =Lo (3)

WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA I569

where

and

Also

a az -

U = D E

E=LU

with strain tensor (listed as a vector)

and displacement vector

Hence

Next, we write the operator matrix L as

a a a L==LX--+L --+L,- ax y a y a2

The matrices L,,L,,L, follow trivially by inspection of equation (4). This definition leads to the expansion of the product LTDL as

in which the matrices D, are defined by

Dij = $(LTDLj + LfDL,), i, j = X,Y, z (1 3)

These matrices are given in Appendix 1. The interface stresses S j can also be written as

S j = LTdj = LTDLUI

where oj is the stress vector for the jtk interface.

DISCRETE LAYER METHOD

In a continuum solution, the wave equation is satisfied identically t h r o ~ g ~ o u t the medium, and the applied tractions balance the interface stresses exactly. If, on the other hand, we approximate the displacement field within the layer by a linear expansion

U = % U , + ( l - % ) U , (1 5)

1570 E. KAUSEL

(with ( = z/h), then the wave equation (3) and the boundary conditions in equation (1) will not be satisfied exactly; instead, introduction of the expansion (15) into (3) will lead to unbalanced internal body forces W # 0. We seek then a finite element solution in which the tractions applied at the boundaries of the layers balance the internal forces in a weak sense. For this purpose, the principle of virtual work is applied:

GVTT = 6VTS + 6UTW dz (16)

6VT = (SUT su;> (17)

s: with

that is, the virtual work performed by the externai tractions is equal to the virtual work performed by the interface stresses and the unbalanced internal body forces.

Equation (15) can be written as

so that

GVTT = 6VTS + GVT{ [ jl pNTN d z ] ~ - [ f NTLTDLN dz]V) 0

We seek now harmonic solutions of the form u = U e i ( m t - & - rtyt

in which U = O(z). Hence, the displacements at the interfaces ( j 4 1, j + 1 + 2) are

and similar expressions for S, T in terms of S, T. Hence

Also, the term a2V/az2 vanishes, because of the linear expansion L..osen. ibstituting equations (14), (21), (22) into (19), integrating over the thickness of the layer, and requiring the result to be valid for arbitrary virtual displacements 6V leads, after lengthy but simple algebra, to an expression of the form

T = (Ak' + iBk + G - w2M)P (23) in which the wave numbers r, q have been written as

and

5 = kcos6 q=ksin6

A = A, cos2 8 + A,, sin 6 cos 6 + A,, sin2 6 B = B, cos6 + B, sin0

WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA 1571

The matrices A l l , A12, AZ2, B,, B,, G and M are presented in Appendix 11. Equation (23) can be written in compact form as

T = K,V (26) where K, is the stiffness matrix of the mth layer under consideration. This matrix is algebraic in the wave numbers (i.e. g and q do not appear as arguments in transcendental functions).

In order to analyse all layers in the soil profile (continuity of displacements and stresses requires o, <, q to be the same for all layers), we must overlap the stiffness matrices of adjacent layers, obtaining the dynamic equilibrium equation

in which P = K U , K=(K,)

PT = (PT, P;, * .., P;f)

UT = (UT, u;, , . . , U;f) (29) The elements of Pj represent the net tractions applied at the interfaces, that is the result of overlapping the contributions of Tj for neighbouring layers.

An alternative representation of equation (23) can be obtained by rearranging columns and rows in the matrices by degrees of freedom rather than by interface, i.e. all x-d.0.f.s are grouped first, followed by all the y and then all the z dofs. The resulting expression is of the form {:I= {r :: ~~~ k Z + ~-~~ - BZ, - :: BTz $]ik

(sym.)

+ iGxx sym.1 zz: i!!]-cozr My M~~~~~ (30)

where all the submatrices are tridiagonal (see Appendix 111). In the particular case where the (symmetric) constitutive matrix D has the form

D =

'dll d12 d13 *

d21 d 2 2 d23 . d31 d32 d33 *

. d44 d45

ld61 d62 d63 . (which is somewhat more general than the orthotropic case), then equation (30) can be expressed in the fully symmetric form

{![ iP, p, = {r: A,, A,, :]k2+[: K z B T Z : " ] k

(32)

(M, = M, = M,; for an orthotropic material G,, = 0). (This equation is obtained from equation

1572 E. KAUSEL

(30) by multiplying the third row by i = ,/( - 1) and factoring out i from the third column.) Equation (32) in turn leads to the representation

B = (Ak’ + e)O in which

(33)

(344

G,, - 02M, (34b) G,, - w2Mx GXY BXZ

G,, - w2MZ E = { G,x

and

(344

NATURAL MODES O F WAVE PROPAGATION

The determination of the natural modes of wave propagation requires setting the external traction vector P in equation (27) equal to zero, i.e. KU = 0. For a given azimuth of propagation 6, this leads to a quadratic eigenvalue problem in the wave number k, with corresponding propagation mode 4 = U. Since the global stiffness matrix has a banded structure, it is not difficult to determine all the eigenvalues and eigenvectors at a particular frequency o. Waas12 discusses the procedure at length for the special case of isotropic soils.

If the alternative form of the relationship between interface loads and displacements given by equation (33) is used instead of equation (27), then the associated eigenvalue problem (Ak’ + e)o =I 0 is linear in k2 although the matrices A, are not symmetric in such case. Hence, the system of equations (33) (with P = 0) may admit complex eigenvalues k 2 , and it will have distinct left and right eigenvectors satisfying orthogonality conditions which are similar to those described in Reference 5.

PARTICULAR CASE: CROSS-ANISOTROPIC MATERIAL

The elements of the constitutive matrix for a cross-anisotropic (transversely anisotropic, or ortho- isotropic) material are

Lame constant d66 = G shear modulus

in the isotropic plane

= 1, + 2G, constrained in the transverse direction

4 3

d,, = d 2 3 = 2, d,, = d,, = G,

Lame constant shear modulus

All other elements dij are zero (except, of course, for the symmetric ones such as d,,, etc.). Introducing the above relations into the matrices in Appendix 111, it can be shown that they can

WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA 1573

be expressed in the form

A,, = A, cos2 8 + A, sin2 8 A,, = A, sin2 0 + A, cos2 8

A,, = (A, - A,) sin 8 cos 8 B,, = BCOS 8 By, = B sin 8 G,, = G,, = G

A,, = A,

G,, = G , M, = M, = M, = M

(35)

The characteristic equation for the natural modes of wave propagation is then

A, cos2 6 + A, sin2 8 (A, - A,) sin 6 cos 8 (A, - A,) sin 8 cos 8 A, sin2 0 + A, cos2 8

A, ]k2

q5 = O (36) BsinB i k i G - 02M -BTsin8 1

It can further be shown that the solution 14, = ( 4 ) of this equation is of the form

(37) 0

where the modal matrices CDx = {q5,}, @, = (q5,>, CD, = (4,) are obtained by solving the following two uncoupled eigenvalue problems (which correspond to solutions of equation (36) for 8 = 0 and 8 = x/2):

G, - 02M ik+

[A,k2 + (G - 0’M)]q5, = 0 (38b)

These two equations can be interpreted as formulations for generalized Rayleigh and Love wave propagation problems. In this case, the wave numbers k (eigenvalues) do not depend on 8, as could have been anticipated because of the existence of the plane of isotropy. These (algebraic) characteristic equations are of the same form as those in Reference 5 (equations (15) there), so that the same orthogonality conditions for the modes and the same solution techniques apply here as well.

EXAMPLE

Consider a stratum of total depth If = 1 made of a cross-anisotropic Gibson material (i.e. a solid in which the elastic moduli increase linearly with depth). The material properties are as follows:

Top layer: Shear modulus, G, = 1.5 G (G = 1)

1574 E. KAUSEL

Im (kH) Figure 3. Spectral lines for the wave numbers of a cross-anisotropic Gibson solid over a rigid base

Position’s ratio, v, = v = 113 -+ A = 2G; Mass density, p = 1 Shear wave velocity, C, = J(G/p) = S

A, = 2G, = 3G

Other layers: as above, but with shear moduli increasing linearly in increments AG = G/10, AG, = GJSO.

Assembling the tridiagonal matrices indicated in Appendix 111, and solving the eigenvaiue problem at various frequencies by inverse iteration with shift by Rayleigh quotient, one obtains the wave number spectrum shown in Figure 3. (Note that although in this example G = p = H = C, =I 1 were used in the computation, the plot in Figure 3 is valid for arbitrary values of these parameters.)

CONCLUSION

Finding the natural modes of wave propagation in a layered anisotropic stratum normally requires the solution of intractable transcendental eigenvalue equations, which are characterized by a countable infinity of eigenvalues (wave numbers) and propagation modes. In practice, such

1575 WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

solutions are found numerically using search procedures. However, these numerical schemes are often ill-behaved, or it may be dificult to find the complex roots sequentially without skipping some of them. Moreover, many solution methods require approximate starting values that may be impossible to provide in the case of clearly layered anisotropic media, and particu~ar~y when the higher modes are of interest. The semidiscrete procedure proposed herein circumvents all of these difiicdties by casting the wave propagation problem in terms of an algebraic eigenvalue equation having a finite number of modes, and involving narrowly banded matrices. As a result, the eigenvalues can be found efficiently using standard methods in matrix algebra.

APPENDIX I

APPENDIX I1

h 6

A,, =-

1576 E. KAUSEL

Ax, = <

APPENDIX 111

' h h l \

Ld:, 3 -7;d:l

h 'd : , (!$ +$d:,) 7 4 1 h2 2

($(ifl +$d:,)

6 ' cos2 0

h2 -d11 6 2

\ etc. for all layers

WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA 1577

I etc. for a11 layers J

+

etc. for all layers)

sin2 %

with ht and dij being the thickness and the material properties, respectively, of the lth layer. The angle 19 is the propagation azimuth. The matrices are tridiagonal.

A,, A,,, Axy, A,,, A,, have the same structure as A,,, with the following substitution (super- script I implied):

d46

Notice that for material obeying equation (3 1) A,, = A,, = 0.

- ( 4 3 t- 4 5 )

-(di3 -dks) t(d!3 - d ; s ) -fd:, +d;s ) etc. for all layers

- (di6 f d i 5 1

- ~ d : 6 - d : ~ ~ + ~ ~ ~ ~ - ~ : 5 ~

B,., By,,, B,,, Bxyr By, have the same tridiagonal structure as B,,, with the following substitutions (superscript I implied):

1578 E. KAUSEL

Notice that B,,, By,, B,, are antisymmetric (tridiagonal) matrices, and that B,, = - B:,, B,, = - B:,, B,, = - B:=. Also, for a material obeying equation (31), we have B,, = By, = B,, = B,, = 0.

G,,= - I [ * etc. for ail layers J

The matrices G,,, G,,, G,,, G,,, G,, have the same structure as G,,, with the following substitutions (superscript I implied):

~ G,, G,, G,, G,, G,,

Again, for a material of the type given by equation (31), G,, = G,, = 0. Finally,

iP2h2 etc. for all layers

with p l = mass density of the Ith layer. M, and M, are identical to M,.

REFERENCES

1. S. Hull and E. Kausel, ‘Dynamic loads in layered halfspaces’, Proceedings of the ASCE EMD Specialty Conference,

2. E. Kausel, ‘Forced vibrations of circular foundations on layered media’, Research Report R74-11, Soils Publication No.

3. E. Kausel, ‘An explicit solution for the Green’s functions for dynamic loads in layered media,’ Research Report R81-13,

4. E. Kausel and J. M. Roesset, ‘Stiffness matrices for layered soils’, Bull. Seism. SOC. Am., 71, 1743-1761 (1981). 5. E. Kausel and R. Peek, ‘Dynamic loads in the interior of a layered stratum: an explicit solution’, Bull. Seism. SOC. Am., 72,

6. E. Kausel and R. Peek, ‘Boundary integral method for stratified soils’, Research Report R82-50, Order No. 746, Dept. of

7. J. Lysmer and G. Waas, ‘Shear waves in plane infinite structures’, J . Eng. Mech. Dio.. ASCE, 18, 85-105 (1972). 8. J. W. Schlue, ‘Finite element matrices for seismic surface waves in three-dimensional structures’, Bull. Seism. Sac. Am.,

9. H. Tajimi, ‘A contribution to theoretical prediction of dynamic stiffness of surface foundations’, Proc. 7th World Con$

10. J. L. Tassoulas, ‘Elements for the numerical analysis of wave motion in layered media’, Research Report R81-2, Order

11. J. L. Tassoulas and E. Kausel, ‘Elements for the numerical analysis of wave motion in layered strata’, Int. j . nurner.

12. G. Waas, ‘Linear two-dimensional analysis of soil dynamics problems in semi-infinite layer media’. Ph.D. Thesis,

13. G. Waas, ‘Dynamish belastete Fundamente auf geschichtetem Baugrund‘, VDI Berichte N o . 381, 185-189 (1980).

University of Wyoming, Laramie, Wyoming, August 1984.

336, Dept. of Civil Engineering, M.I.T., Cambridge, Mass, 1974.

Publication No. 699, Dept. of Civil Engineering, M.I.T., Cambridge, Mass 1981.

(5), 1459-1981 (1982). (See also the Errata, BSSA 79, (4), p. 1508.)

Civil Engineering, M.I.T., Cambridge, Mass., 1982.

69, 1425-1437 (1979).

on Earthquake Eng., Istanbul, Turkey, 1980, Vol. 5, pp. 105-112.

No. 689, Dept. of Civil Engineering, M.I.T., Cambridge, Mass., 1981.

methods eng., 19, 100-1032 (1983).

University of California, Berkeley, Calif., 1972.