NASA-CR-4001 19860018194 

NASA Contractor Report 4001 

Wave Propagation in Anisotropic 

Medium Due to an Oscillatory 

Point Source With Application 

to Unidirectional Composites llBRAR'{ CGPl 

James H. Williams, Jr., Elizabeth R. C. Marques, 

and Samson S. Lee 

GRANT NAG3-328 

JULY 1986 

tjl.NGlEY RESEARCH cctHER 

LIBRARY, ~!?_SA 

HAMPTON VIRGINIA 

-! 

WOT TO a& TAKEN nOM fHJS 10011 

NI\S/\ 111111111111111111111111111111111111111111111 

NF01994 
NASA Contractor Report 4001 

Wave Propagation in Anisotropic 

Medium Due to an Oscillatory 

Point Source With Application 

to Unidirectional Composites 

James H. Williams, Jr., Elizabeth R. C. Marques, 

and Samson S. Lee 

Massachusetts Institute 0/ Technology 

Cambridge, Massachusetts 

Prepared for 

Lewis Research Center 

under Grant NAG3-328 

NI\SI\ 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1986 
INTRODUCTION 

The expanding uses of composite materials, especially of the 

filamentary type, require the generation of a variety of data on 

material behavior so that adequate design can be achieved. Many 

test techniques have been developed for such purposes. Among these 

are the nondestructive evaluation (NDE) techniques of ultrasonics 

and acoustic emission. 

The application and interpretation of results obtained from various 

NDE techniques are strongly related to the stress wave propagation 

characteristics of the material. The theoretical prediction of the 

stress wave fields under the action of well defined excitation can 

give information regarding the proper use of the NDE techniques, and 

may provide improved precision of a material's diagnosis by 

establishing more quantitative standards. 

Via an asymptotic approximation, the stress wave field generated 

by a point source in an infinite anisotropic medium can be described 

in analogy with the problem of magneto hydrodynamic waves discussed 

by Lighthill [1]. For the present work, the particular case of 

transversely isotropic media is presented by following the same 

procedure used by Buchwald [2], where here the solution is extended 

such that the displacements are obtained and expressed in cartesian 

form. 

Here, the solution is given for the case of a glass fiber 

-1-
reinforced epoxy unidirectional composite. The slowness and wave 

surfaces are determined as well as the displacements for points 

throughout the medium. Polar diagrams for displacement amplitudes 

are constructed illustrating the patterns of the displacement field. 

Other aspects of the problem of stress waves in homogeneous 

anisotropic media can be found in the studies of Synge [3], Carrier 

[4] and Musgrave [5]. Synge analyzed the behavior of stress waves 

in a single layer of anisotropic material subjected to uniform stress 

at one of its surfaces and extended the study to the limiting case 

where the layer is transformed into a half space. Carrier proposed 

a solution for waves in an infinite medium using potential functions 

and integral techniques. Musgrave established the conditions to be 

satisfied for the existence of the general equation of motion 

when a plane wave solution is adopted and described the geometry of 

the spreading disturbances by means of the wave surfaces and 

slowness surfaces. 

The work on layered media was carefully explored but was not 

pursued for this research. The primary reason for this is that 

general anisotropy is a more convenient approach for representing 

a composite because the effective modulus theory can be used to 

model the material as homogeneous [6]. It is assumed that the 

material behaves as homogeneous since the layer thicknesses (fiber 

diameter for filamentary composites) are small as compared with the 

wavelengths. If this requirement is satisfied, the effective 

-2-
modulus theory appears to describe the behavior of stress waves in 

laminated composites better than the laminated media theory [7,8]. 

-3-
EQUATION OF" MOTION AND ITS SOLUTION , 

A schematic of the system under investigation is shown in Fig. 1. 

An infinite transversely isotropic medium is subjected to a point 

force source which generates stress waves that propagate throughout 

the medium. A cartesian coordinate system x, y and z is adopted such 

that the x-y plane is the isotropic plane in the medium. In a 

unidirectional composite, z would be the fiber direction. The point 

source is located at the origin and oscillates in the x direction as 

shown. The displacements at any point P in the medium due to a 

steady-state sinusoidal point force excitation are analyzed. 

Force-Dynamic Equations 

The force-dynamic equations for an infinite medium can be derived 

from the equilibrium conditions of a differential volumetric element 

and are given as [9] 

(1) 

where the implied summation is over the repeated subscripts, p is the 

mass density, the indices i and j have values x, y and z in a 

rectangular cartesian coordinate system, ujis the displacement in the 

j direction, Tij is the stress tensor and F

j is the body force (force 

per unit mass) in the j direction. The indices following the comma 

-4-
refer to partial derivatives such that Uj,tt represents the second 

derivative of the jth component of displacement with respect to t, 

the time variable. 

For the point sinusoidal force shown in Fig. 1, the body 

forces can be written in exponential form as 

F = F 6(x)6(y)6(z) x 0 

-iwt 

e and F = F = 0 Y z 

(2) 

where i = (-1), w is the radian frequency of the harmonic excitation, 

6( ) is the Dirac delta function and F is the force amplitude. The 

o 

actual oscillatory force is taken as the real part of the expression 

for F in eqn. (2). x 

Constitutive Relations 

The constitutive relations based on Hooke's law are [10] 

= C.. e: ~Jpq pq (3) 

where Ci . are the elastic coefficients that constitute the stiffness Jpq 

matrix and e: is the strain tensor that can expressed in terms of 

pq 

displacements using the engineering definitions for strains as 

e:pq = !(u + u ) 2 p.q q.P (4) 

where the suffixes each represent the x, y and z directions. The 

-5-
elastic coefficients satisfy the symmetry conditions such that [11] 

cae = cae ijpq jipq ijqp pqij (5) 

The four indices notation for the elastic coefficients can be 

modified to the two indices compact form [9]. In this case, the 

stress tensor is written as a vector 

(6) 

where the equivalence in the cartesian system is 'I a, , '2 a, , XX yy 

'3 = 'zz' '4 a 'xz' '5 = 'yz' '6 = 'xy. The same correspondence 

is applicable for the strains. 

For a transversely isotropic medium as shown in Fig. 1, where 

the x - y plane is the isotropic plane, z is the symmetry axis 

of the medium. It can be shown that for transversely isotropic 

materials, there are five independent elastic coefficients [11]. 

The nonzero elastic coefficients according to the definition of 

the stress components in eqn. (6) are 

(7) 

Constraints 

The constraints are introduced by the application of the 

"radiation condition". Since the medium is infinite, the constraint 

-6-
is that no waves originate at infinity or are reflected at infinity; 

that is, the only source of disturbance is due to the point force 

located at the origin of the coordinate system. From this 

assumption, the amplitudes of the displacements must decrease with 

distance from the source. In the limit, as the distance approaches 

infinity, the displacement amplitudes must vanish. 

Equations of Motion 

The equations of motion can be represented in terms of the 

displacements by substituting eqns. (4) and (5) into eqn. (3) and 

then substituting the result into eqn. (1) to give [10] 

pUJ',tt = Ci ' u i + P F, (8) Jpq p,q J 

where the symmetry condition given in eqn. (5) is utilized. The 

forcing function is as given in eqn. (2). For a transversely 

isotropic medi~, the elastic coefficients obey eqn. (7). 

Solution of Equations of Motion 

The equations of motion are given in eqn. (8) and must be 

solved in accordance with the imposed constraints and the applied 

forces described earlier. 

Egn. (8) can be rewritten using u, v and w for the 

displacements in the x, y and z directions, respectively, instead 

of u ,u and u ; and using x, y and z for the subscripts denoting x y z 

-7-
derivatives. This gives three equations in u, v and w. Then 

these equations can be differentiated with respect to the space 

and time variables to. give terms in u, ,u v v t tx ' tty' , t tx ' , tty 

and W'ttz which can be combined by addition and subtraction to 

give [2] 

2 

!I. + as A F (9) lI.'tt = a4V'1 + 'zz x,y 

r'tt = a3A,zz + as 

A2 r + a2r,zz (10) 1 

2 

+ as A,zz + al 'i12

A'tt = a A (11) 3

'i1 l r + F 1 X,x 

where the new variables are 

A = v, - U'y x 

r = w'z (12) 

A = u'x + v, y 

A represents the z component of rotation, r is the strain in the 

z direction and (r + A) is the dilatation, all in accordance with 

standard definitions in elasticity. Physically, the fact that 

a separate equation for the z rotation can be written, namely eqn. 

(9), indicates that the propagation of these rotational waves is 

independent of the other possible modes. The same is not true 

for the displacement functions r and A, since eqns. (10) and (11) 

cannot be decoupled, The a's are constants given by 

-8-
p 

(13) 

and ~i is the laplacian operator in two dimensions such that 

(14) 

If eqn. (2) is substituted into eqns. (9), (10) and (11), the 

asymptotic solution can be obtained by using Fourier integrals and 

the method of stationary phase [12]. Basically, eqns. (9), (10) 

and (11) are written in terms of Fourier integrals. In doing this, 

plane wave solutions are assumed with associated wave number vectors 

that are represented by their components a along the x direction, 

S along the y direction and y along the z direction. Then, the 

values of the Fourier integrals are approximated by their residues 

such that the stationary points are determined by the singularities 

of the integrands. In this approximation, terms involving the 

factor 1/r2 (where r is the radial distance from the origin to P the 

point of interest) are neglected. Thus, the expressions obtained 

are adequate for the calculation of the displacements of points 

in the thus defined far field. This leads to the following results [2] 

i 6 F 

A ~ ________ ~o~~= exp [i(ax + Sy + yz - wt)] 

2~rl~GI (I KGI)1/2 

(15) 

-9-
tJ. ~ _1_ 

21Tr 

N An i Fo 2 2 2 2 I --=---=-""':'1""7/'::"2 an [aS(an+8 )+a2Yn- w )exp[i(a x+8 y+y z-wt») 

n:al Iwl (I~I) n n n n 

'V 1 N 

r '" 21Tr I 

n=l 

2 -A i Fay a

non n 3 ..-.,;~-;;;.....--=:....~/~..;;.. exp[i(a x+8 y+y z-wt») Iwl (1~1)1 2 n n n 

(16) 

(17) 

where the symbol I I denote "the magnitude of". Now a, 8 and yare 

the wave number components defining the singularities of the integrands 

in the integration process for which the residues are calculated. 

Eqn. (15) represents the residue corresponding to a single singularity 

point. For the eqns. (16) and (17) the wave number components are 

followed by the index n indicating that it is possible to have more 

than one singularity point and in this case the expressions for tJ. and 

r represent the sum of the residues corresponding to each of the 

singularity points. N is the total number of singularity points over 

which the summation must be performed. Also 

(18) 

(19) 

2 22 2 22 22 2222 H '" easY +al(a +e )-w )[as(a +8 )+a2y -w ]-a3

y (a +8 ) 

(20) 

-10-
(21) 

2 C, -2G, G, + G2 

G'SS[G'a C G'aa] 

KG = yy a y 'eLr 'r (22) 

(G2 + G2 )2 'a 'y 

2 H, -2H, H, H +H2 H ] H'SS[H'a yr a r 'eLr 'r 'eLa ~= (23) 

(H2 + H2 )2 'a 'y 

(24) 

The values KG and ~ are commonly known as gaussian curvatures [1], 

and the combination of IVHI and (1~1)1/2after simplification [2] 

is usually expressed by 

A 

n 

H, + H, 

~ 2 2 ] 1/2 

(25) 

A is a phase coefficient that is determined by the geometric properties n 

of the H = 0 surface as follows: 

a) A = + 1 if K > 0 and VH is in the +r direction; n H 

b) A = - 1 if K > 0 and VH is in the -r direction; 

n H 

-11-
c) An = + 1 if KH< 0 and H = 0 is convex in the VH direction; 

and 

d) A = - 1 if K < 0 and H = 0 is concave in the VH direction. 

n H 

The values for A can be determined only by numerical calculations. 

n 

As mentioned before, eqns. (15), (16) and (17) are residues or 

sums of residues resulting from the integration process. In wave 

number space, G = 0 and H = 0 are surfaces containing all possible 

values of wave numbers that are able to generate such residues, 

considering the elastic properties of the material. Each point 

(a, a, y) on the surfaces can then be associated with a particular 

wave train in a particular direction which effectively contributes 

to the displacement functions. It is important to observe that the 

H = 0 surface is in reality a quartic of two sheets [2]; so H = 0 

defines two surfaces. 

According to the method of stationary phase, the displacements 

at a point P in the medium are calculated by summing the contributions 

of plane waves passing through such a point [1]. These waves and the 

corresponding wave numbers can be found by locating the point 

(a, a, y) or points (a , a , y ) on the surfaces G = 0 and H = 0, n n n 

respectively, where the normal is parallel to the direction OP (in 

Fig. 1), which corresponds to finding the singularity points that 

generate the residues for the specific point P. 

-12-
Note that normals to the G = 0 or H = 0 surfaces at more than 

one point may lie in the same direction (parallel to OP). In such 

a direction waves of different settings (direction of wave number 

vector) and spacings (magnitude of wave number vector) can be superposed and for this reason the summation signs appear in eqns. (16) 

and (17). It must be noted that this remark does not apply if the 

points are simply symmetric such as (a, B, y) and (-a, -B, -y) which 

correspond to waves of the same setting and spacing. For the solution 

satisfying the radiation condition, only one out of each such pair 

of points is selected. Geometrically, the surface G - 0 for a 

transversely isotropic medium is an ellipsoid and as a consequence 

there is only one point on the surface where the normal is parallel 

to OPe 

With the use of eqns. (18) through (25), the expressions given 

in eqns. (15), (16) and (17) for the displacement functions can be 

rewritten as 

exp [i(ax + By + yz - wt)] 

(26) 

exp [i{a x + B .. J + Y z - wt)] n nn 

(27) 

-13-
~ 1 N 2 

r = --- I A i A Fay exp[i(a x + S y + y z - wt)] 2~r n=l n non n n n n 

(2S) 

where 

(29) 

The displacements u, v and w can be determined by assuming forms 

compatible with eqns. (26), (27) and (2S) such that eqns. (12) are 

satisfied. Since the displacement functions are of exponential 

form, identical exponential forms are chosen to represent the 

displacements as 

where kJ through kS are constants to be determined. If the 

derivatives of eqns. (30), (31) and (32) are calculated and terms 

involvi~g the factor l/r2 are neglected, the substitution of the 

resulting expressions for the derivatives into eqns. (12) gives the 

values of these constants as 

-14-
-a 

a 

. k == k2 

n = 1 i(a2 + ( 2) i(a2 

+ a

2

) n n 

(33) 

a a 

k5 

n = k4 = Ha2 + ( 2) 2 i(a a 2) n 

+ 

n 

(34) 

k7 

1 :::z-- i y n 

The constants k

3 , k6 and kg are determined using the radiation 

condition presented earlier. Since there are no sources in the medium 

except at the origin, the displacements at infinity must vanish. 

This is possible only if the constants k3 , k6 and kg are zero; so 

tV 

U = 

k3 == k == k = 0 6 g (36) 

Therefore the final expressions for the displacements are 

1 N 

+ 21Tr L 

n=1 

2 

a. 

n 

exp[i(ax + ay + yz - wt)] 

2 2 2 2 A A F [as(a +a )+a2

y -w ] n non n n 

exp[i(a x + a y + y·z - wt)] n n n 

(37) 

-15-
'V 

'It = -

N 

+ 1 \' 27Tr L n=l 

exp[i(ax + Sy + Y - wt)] z 

exp [i(a x + S Y + y z - wt)] n n n 

(38) 

'V 1 N 

w = - - \' A AF a y a exp [i (a x + S Y + y z - wt)] 27Tr L n non n 3 n n n n=l 

(39) 

Observe that a different procedure could have been followed 

to obtain the expressions for the displacements. Recall. that the 

expressions in eqns. (12) relate the displacements u, v and w with 

the displacement functions A, rand 6. Then, from the expressions 

in eqns. (26), (27) and (28), the system represented by eqn. (12) 

could have been solved by direct· integration. This procedure 

is mathematically more complex than the method of assuming 

exponential forms for u, v and wand determining constants as was 

done here. 

-16-
PHASE VELOCITY AND WAVE SURFACE 

In order to understand the propagation behavior of waves in 

anisotropic media, some concepts concerning the plane wave propagation 

characteristics are reviewed below. For this, consider the general 

equation of motion for an anisotropic medium when no body forces are 

present. From eqn. (8) withF. ,. 0 

J 

C. u - pu = 0 ~jpq p,qi j,tt 

Assume a plane wave front of the form [5] 

(40) 

(41) 

where ~ represents a characteristic with unit normal n and scalar ~ 

phase velocity 1£1 along the normal direction. Observe that the 

characteristics represent the wave front in time and space. By the 

theory of characteristics, the displacements are functions of ¢ such 

that the spatial dependence of u is as n • x • ~ 

For the nontrivial solution (that is, for the propagation of 

the front) the "characteristic condition" must be applied. This 

condition can be written as [5] 

(42) 

or substituting the derivatives according eqn. (41) into eqn. (42) 

gives 

-17-
(43) 

where ni and n are the projections (direction cosines) of n in the q -r . 

directions i and q, respectively, and 0iP is the Kronecker delta • 

. The solution of eqn. (43) is an eigenvalue problem for the phase 

velocitylsJfor any specified direction n. In general, there are 

-r 

three velocities (that is, eigenvalues), distinct in magnitude for each 

n • 

-r 

Now suppose that the actual point source introduces the excitation 

into the medium. If composition of plane wave fronts is used to 

represent the actual disturbance front, according to Huygen's principle 

[5], the front can be traced out by the envelope of plane wave fronts 

as if they were generated at the origin in all possible directions, 

at time t = O. So the normals n of the component plane fronts must 

-r 

assume all possible directions in order to sweep the entire space 

around the source. For each direction n a new determinant of the 

-r 

form shown in eqn. (42) is generated and three new eigenvelocities can 

be determined. Polar diagrams can be constructed for each of the 

three velocities as functions of the corresponding normal direction. 

The surfaces expressed in polar representation (diagrams) are called 

the velocity surf~ces. Moreover, the envelope of plane wave fronts 

at unit'time is called the wave surface [3]. There is a correspondence 

-18-
between the velocity surface and the wave surface such that if three 

velocity surfaces exist, the same number of wave surfaces can be 

generated. The existence of three surfaces indicates that there are 

three possible modes of propagation, each mode with its particular 

phase velocities. If just one mode were to exist, the physical meaning 

of the wave .surface could simply be stated as the boundary between 

the disturbed and undisturbed regions in the medium at unit time after 

the source is set into action. 

Consider now the eigenvectors corresponding to the eigenvelocities. 

The symmetry of the forms in eqn. (42) indicates that the eigenvectors 

(that is, displacement modes) corresponding to each of the three 

eigenvalues are mutually orthogonal but in general none of them is 

parallel to the wave front normal direction n [2,5]. If one of the 

-r 

eigenvectors is coincident with the wave normal direction~, the 

corresponding displacement mode is purely longitudinal and the above -

mentioned orthogonality ensures that the remaining displacement modes 

are purely transverse. This condition is fulfilled for all wave 

propagation directions in isotropic media where each of the wave 

surfaces can be associated with a pure propagation mode, namely, 

one longitudinal mode and two transverse modes with perpendicular 

polarization planes. 

-For the transversely isotropic medium as opposed to the isotropic 

medium, the eigenvectors are not coincident with the wave normals 

in all-directions, so plane wave fronts travel along directions 

oblique to their normal n [5]. This means that the modes of 

-r 

-19-
propagati9n are combinations of longitudinal and transverse modes. 

Exceptions occur for certain directions. These directions are: 

(1) the principal direction perpendicular to the isotropic plane and 

(2) any direction contained in the isotropic plane. The values of 

the velocities for these directions, according to the convention 

adopted for the coordinate axes, are given in Table 1. 

-20-
SLOWNESS SURFACE 

The condition given in eqn. (43) for the existence of the solution 

of the equation of motion given in eqn. (40) can also be represented ifi 

terms of slownesses. The slowness s is generally defined as a vector 

whose magnitude is the inverse of the velocity magnitude I~I, and whose 

direction is the same direction of ~, or 

(44) 

For a system with three different velocities there are three such 

slowness vectors. 

According to the form of the argument of the wave front function 

in eqn. (41) and the definition in eqn. (44), the component of ~ in 

the direction i, si' can be written as 

(45) 

Substituting eqn. (45) into eqn. (43) gives 

(46) 

The solution for the slownesses is the same as the solution for 

the velocities. As before, if each direction in space is considered, 

three surfaces can be traced out which are the reciprocals of the 

velocity surfaces [3]. The resulting surfaces are called slowness 

surfaces. Each radius vector from the origin to the surface on the 

slowness surface has its corresponding inverse on the velocity surface. 

-21-
The slowness surface is then an alternative way of geometrically 

representing the propagation characteristics of materials. The solution 

for slowness is often pref~dto the solution for velocities since 

the former involves simpler algebraic transformations. This argument 

is supported by the fact that the determinant from eqn. (43) leads to 

an expression of twelfth order in the velocities as opposed to the 

determinant from eqn •. (46) which is a sixth order equation in si[5]. 

Recall, the case under study is that of a transversely isotropic 

medium subjected to an oscillatory point source. The motion is 

represented by eqns. (9), (10) and (11) in terms of the variables A, 

r, and~. The slowness surfaces can be identified when the differential 

equations (9), (10) and (11) are written in terms of their Fourier 

transforms [1], and the inverse transforms written for each variable A, 

r and~. In the inverse transform expressions, the expressions in 

~, 8 and y defining the singularities of the integrand are automatically 

the slowness surfaces. 

For the variable A, the associated slowness surface is the G = 0 

surface where G is given in eqn. (18). This surface will be called 

SH. For the variables r and ~, the remaining slowness surfaces are 

represented by H - 0 where H is given in eqn. (20). However, there 

are two surfaces that satisfy H = 0, resulting in two slowness 

surfaces [2,5]. These two surfaces will be called SV and P. , 

The physical meaning of the slowness surfaces SH, SV and P can 

be better pursued if the isotropic medium case is considered. For 

isotropic materials 

-22-
(47) 

Further it can be shown that all three slowness surfaces are spherical. 

SV corresponds to purely rotational waves (purely transverse waves 

if plane waves are considered); SH corresponds to purely rotational 

waves (purely transver6e waves if plane waves are considered); and P 

corresponds to purely dilatational waves (purely longitudinal waves 

if plane waves are considered). Moreover, the surfaces SH and SV 

are identical, meaning that shear waves with any polarization have 

exactly the same behavior in isotropic media. 

Formally, the identification of the surfaces SV and SH can be 

accomplished by the corresponding intersections with coordinate axes 

as follows: SH is defined as the surface that contains the shear 

mode for plane waves in the x direction or the z direction with 

particle displacements in the y direction. SV is the surface that 

contains the shear mode for plane waves in the x and z directions 

with particle displacement in the z and x directions, respectively. 

Finally, the slowness surface P is defined as the surface that 

contains the longitudinal modes in the x and z directions. The 

correspondence between wave number components (a, a, y) and 

geometric coordinates (x,y,z) is maintained here. 

For transversely isotropic media the rotational and dilational 

propagation modes corresponding to the slowness surfaces are not 

pure, since in general the displacement vectors are not aligned 

-23-
with the normals of the plane front segments that constitute the 

wave front. For wave propagation in the principal directions of the 

medium though, the waves can be purely rotational or purely 

dilatational. Specifically, the plane wave propagation along a 

principal direction, the rotational mode corresponds to a shear mode 

and the dilational mode corresponds to a longitudinal mode, just 

as for the isotropic medium. The identification of the surfaces SH, 

SV and P is done as explained above, by the intersections with the 

principal directions. .An itlustration of the slowness surfaces for 

a transversely isotropic medium is shown in Fig. 2. Observe that 

the values of the slownessess at the intersections of the surfaces 

with the coordinate axes are the inverses of the phase velocities 

presented in the previous section. The character of the corresponding 

plane waves along the principal directions is also sketched, showing 

the propagation and particle displacement directions. 

-24-
GROUP VELOCITY 

The group velocity Q (or velocity of energy propagation) for 

the general anisotropic medium can be written according to Rayleigh's 

energy argument [13] extended to three dimensional propagation as 

[1] 

'VG 

U =-

- -G,w (48) 

where G is given by eqn. (18) and Q has the direction of 'VG or, 

equivalently, the same direction as the normal to the slowness surface. 

For points on the slowness surface G = 0, eqn. (48) can also 

be written as 

w'VG 

!!. = k.'VG (49) 

where k is the wave number vector (a, a, y). Observe that the 

correspondence between the wave number vector and the slowness vector 

is [1,3] 

k = ws (50) 

such that k and s have the same orientation. The phase velocity in 

the k direction can be represented vectorially by 

wk 

(51) 

-25-
If eqns. (49) and (51) are compared, it can be seen that 

u • k - c • k (52) 

Thus, the resultant of U in the c direction is £, which is 

equivalent to saying that the velocity of energy propagation in 

the direction normal to the wave front is equal to the phase 

velocity. 

Eqns. (48) through (52) are also valid for the two other 

slowness surfaces. The same results are obtained if G is replaced 

by H in the eqns. (48) through (52), provided that k is taken 

accordingly. Thus, there are three phase velocities and three 

group velocities for each direction in the medium. 

Although eqn. (52) is satisfied, eqns. (49) and (51) indicate 

that U and c are different. There is a component of energy velocity 

in the direction parallel to the wave front, meaning that energy 

transmission is not in the same direction as wave motion. This is 

true for media where the phase speed varies with direction [1]. 

Thus, plane waves oblique to the principal directions can propagate 

along their own normals only if energy is also supplied in a direction 

parallel to the wave fronts. 

-26-
APPLICATION EXAMPLE 

Material Description 

As an application example, a unidirectional composite material 

was chosen. The material was 3M Scotchply type 1002 fiberglass epoxy 

consisting of unidirectional E-glass in a 1650 C curing epoxy 

matrix. The composite prepreg tape was cured in a heated press at 

6 2 0.69 10 N/m. The resulting resin content was 36 percent by 

weight [18]. The properties of such composites depend not only on 

the volume fraction of the components but also on the fabrication 

method. Since knowledge of the elastic properties of the material 

was necessary for the numerical calculations, samples were 

fabricated for experiments. Ultrasonic methods [14,15,16,17] were 

used for the experimental measurements, relating wave speeds and 

elastic constants c ij • The wave speeds were determined in the 

through - transmission configuration using tone bursts to generate 

either longitudinal or shear waves. The tests were performed at 

frequencies from 0.5 to 2.25 MHz in increments of 0.25 MHz. 

The test specimens were cut in the form of rectangular prisms 

with uniform square cross sections from a 25.4 mm thick composite 

plate. The cross-sectional area was square with 12.7 mm sides. 

The lengths of the specimens ranged from 2.54 mm to 25.4 mm. The 

test specimen axes were oriented at various angles with respect 

to the fiber direction. The elastic constants were determined 

-27-
from the measured longitudinal and shear wave speeds. The 

experimental values for the elastic constants are given in Table 2. 

The convention adopted for the transversely isotropic medium 

in Fig. 1 was maintained here, so the z axis corresponded to the 

fiber direction. 3 The density was 1850 kg/m according to the 

manufacturer's data [18]. 

Slowness Surfaces 

The slowness-surfaces for the fiberglass epoxy material were 

calculated from G = 0 and H = 0, where G and H are given in eqns. 

(18) and (20) respectively. The forms of these equations indicate 

that both the surfaces are symmetric about y for a rectangular 

coordinate system ~, S, and y. Thus, in order to determine the 

surfaces, only their intecsections with the ~-y plane need to be 

calculated. Then, the entire slowness surfaces can be generated 

by revolving the intersection curves around the y axis. Additionally, 

the intersection curves of the slowness surfaces G = 0 and H = 0 

with the ~-y plane are symmetric with respect to ~ axis, which can 

be checked by setting S = 0 in eqns. (18) and (20). By the 

combination of all symmetries, only 1/4 of the intersections of 

the surfaces on the ~-y plane need to be calculated. The positive 

quadrant was chosen for the calculation. 

The ranges of the variables are limited by the values shown 

in Fig. 2 which can be obtained by setting S = 0 and alternately, 

one at a time, ~ = 0 or y = 0 in eqns. (18) and (20). For the positive 

quadrant the limits are: 

-28-
For slowness surface SH: a < ~< 1/(a4

)1/2 a :t.. 1/2 

< ~ l/(aS) ; -w- w 

For slowness surface SV: a a. 1/(as

)1/2 0 <:t.. < l(as

)1/2; < -< and -w- -w 

For slowness surface P: o < ~ < 11 (a ) 1/2 

-w- 1 o ~ ~ ~ 1/(a2

)1/2. 

The surfaces are generated by the successive combination of the 

pairs (a.,y) satisfying eqn. (18) for SH and eqn. (20) for SV and P. 

Fig. 3 shows the slowness surfaces SH, SV and P for the fiberglass epoxy composite. In Fig. 3, points A and B are the so-called 

parabolic points on the slowness surface. At these points, the 

gaussian curvature based on eqn. (23) is zero. Geometrically, these 

are inflection points on the slowness surface. For the points of 

zero gaussian curvature, the amplitude coefficient A (which corresponds n 

to the inverse of a decay factor along the direction of the normal 

to the surface) goes to infinity which would give infinite 

displacement. The physical implications of the existence of such 

points can be better understood by analyzing the wave surfaces. 

From the values obtained for slownesses, the velocity surfaces 

were calculated using eqn. (44) and are shown in Fig. 4. The 

velocity surfaces, namely V(SH) , V(SV) and V(P) correspond, respectively, 

to the "inverses" of the slowness surfaces SH, SV and P. 

Wave Surfaces 

The wave surfaces can be directly obtained from the slowness 

surfaces. It can be shown [1] that the coordinates x, y and z 

of the wave surfaces can be calculated from -G, IG, ; -G, /G, ; 

a. w B w 

and -G, /G, , respectively, for the wave surface corresponding to 

y w 

-29-
the slowness surface SH, and by -H, /H, ;-H'a/H, ; and -H, /H, a w w y w 

for the wave surfaces corresponding to the slowness surfaces SV 

and P. Fig. 5 shows the wave surfaces as calculated from the 

slowness surfaces SH, SV and P. The wave surface W (SH), which 

corresponds to SH, has the smallest phase velocities. The wave 

surface W(P), which corresponds to P, has the largest phase 

velocities. 

As shown in Fig. 5, the wave surface corresponding to SV has 

two finite cuspidal edges [1,5]. The points at the tips of the 

cuspida1 edges correspond to the inflection points A and B shown 

in Fig. 3. The directions defined by the lines from the origin 

to the tips of the cuspida1 edges (Fig. 5) represent the 

directions for which the solution is not valid. These directions 

can be found to be at 41.970 0 and 62.545 0 with r~spect to the z 

axis. Since the wave fronts are geometrically similar in time, 

the cuspidal edges will be always located in the same direction 

with respect to the coordinate axes. Observe that for directions 

between the angles defined above, radii from the origin to the 

W(SV) wave surface may assume three distinct valu'es as can be seen 

in Fig. 5, indicating that there are three contributions for the 

displacements of points in this region of space. The contributions 

arise because it is possible to find three points on the 

corresponding slowness surface where the normals have the same 

direction, according to the method of stationary phase. At a 

-30-
point P in a direction in space between the angles 41.970° and 62.545, 

three plane wave fronts can be seen traveling along the direction 

OP from the origin (where the excitation is located). Thus, it is 

possible to have plane wave fronts of different orientations and 

different phase velocities passing through the same point in the 

medium at different times. In reality these plane wave fronts 

are segments that constitute the actual wave front. 

-31-
Displacement Calculations 

The displacements in a transversely isotropic medium subjected 

to a point force excitation as shown in Fig. 1 were also calculated 

for the fiberglass epoxy material. 

A computer program was written that calculates the displacements 

at a given point P(x,y,z) in the material. The following steps were 

performed: 

(1) Search for the point or points on each of the slowness 

surfaces in which the outward normals are parallel to the direction 

OP. This is done by comparing the direction defined by OP with the 

direction of the gradients to the slowness surfaces. The program 

retains the closest values to the direction given (one value for 

slowness surface SH; up to three values for slowness surface SV and 

up to two values for slowness surface P, in accordance with the 

possible number of points having the same normal direction). The 

wave numbers are determined from the coordinates of the selected 

points on the slowness surfaces via eqns. (18) and (20). 

(2) Check the signs of the gaussian curvature and the 

direction of the gradient vector of the slowness surfaces, thus 

establishing the resulting phase coefficient A • 

n 

(3) Substitute into eqns. (36) through (38) the values found 

for a, e and y and A above and calculate the contributions of each 

n 

slowness surface to the displacements u, v and w. 

-32-
Points in space were chosen and the corresponding displacements 

calculated. For simplicity the points were placed on the coordinate 

planes. The set of points chosen were: 

(1) On the plane, points along the 2 

+ 

2 :::0 4 2 

x-z arc x z m , and 

(2) On the x-y plane, points along the 2 

+ 

2 :::0 4 2 

arc x y m. 

The frequency was set at 1 MHz and a load (body force) of 

unit amplitude (F = 1 N/kg) was applied. From eqns. (15), (16) and 

o 

(17) it can be seen that if any of the gaussian curvatures is zero, 

the correspond~ng displacement function assumes an infinite value. 

It is known physically that no point in space may undergo an infinite 

displacement since the amplitude of excitation F is finite. It can 

o 

be shown [2] mathematically that the discontinuities can be eliminated 

and the decay of the displacement amplitudes is found to vary as 

-5/6 

r In order to generate the amplitude coefficients, the points of 

discontinuity on the slowness surface SV are "avoided", which is 

equivalent to excluding the points A and B from the calculation set. 

Figs. 6 through 9 show the polar diagrams of the displacement 

amplitudes for the points 2 m away from the origin on the x-y plane. 

Figs. 10 through 13 show the polar diagrams of the displacements 

amplitudes for the points 2 m away from the origin on the x-z plane. 

Observe that by reasons of loading symmetry, v displacements for 

points on the x-z plane are zero as well as w displacements for points 

on x··y plane. The curves are plotted separately according to the 

slowness surface that generates the contribution. 

For points on the x-y plane, the contributions of the slowness 

surface SV are zero which implies that there are no shear waves 

-33-
propagating along x direction with particle displacement in the z 

direction. Also, there are no shear waves propagating along the 

y direction with particle displacement in the z direction. 

The contributions from the slowness surface P, Figs. 6 and 

8, show that (1) there are pure longitudinal waves traveling along 

the x direction and this is the direction of maximum amplitude for 

u displacements (as expected since x is coincident with the force 

line); (2) no longitudinal waves travel in the y direction (zero 

amplitude component of v displacement for points in the y direction), 

and no shear waves with y polarization travel in the x direction 

(v displacements are zero in x direction). 

The contributions from the slowness surface SH, Figs. 7 and 9, 

show that (1) there are pure shear waves in the y direction (with 

x polarization direction) but since v displacements along the y 

direction are zero, these are standing waves (y direction vibrates 

as a string); (2) no longitudinal waves travel in the y direction 

(zero amplitude component of v displacements for points in the 

y direction), and no shear waves with y polarization travel in the x 

direction (v displacements are zero in x direction). 

For points on the x-z plane, the contributions of slowness 

surface SH are zero which implies that there are no shear waves 

propagating along the x direction with particle displacement in 

the y direction. Also there are no waves propagating along the z 

direction with particle motion in the y direction. 

-34-
The contributions from the slowness surface SV, Figs. 10 and 

11, show that (1) there are pure shear waves in the z direction (with 

x polarization direction) but since the w displacements along the 

z direction are zero these are standing waves (z direction vibrates as 

a string); (2) there are no shear waves traveling along the 

x direction (zero amplitude component of w displacement in the x 

direction). 

The contributions from the slowness surface P, Figs. 11 and 13, 

show that (1) there are pure longitudinal waves propagating along 

the x direction and the u displacement amplitude along the x direction 

is larger than the corresponding u displacement amplitude along the 

z direction; (2) there are pure shear waves in the z direction (with 

x polarization direction) but since the w displacements along the 

z direction are zero these are standing waves. 

By observing the polar diagrams of displacements (Figs. (6) 

through (13», it can be seen that there are characteristic directions 

defining maxima for the displacement amplitudes. For comparison 

purposes these directions are listed in Table 3. It can be observed 

that there are two maxima for the contributions of slowness surface 

SV corresponding to the inflection points A and B in Fig. 3. For 

the contributions of the slowness surface P, the maxima occur at 

two other distinct positions. For the points on the plane x-y, 

the maxima occur along the line of force (x axis itself) for u 

displacements except for the small contribution of SH (see Fig. 7), 

-35-
and at 45° for v displacements as expected, since x-y is isotropic. 

Slowness furfaces not I-is ted in Table 3 have zero contributions to 

the corresponding displacements. 

Recalling the relationship between strain energy and displacement 

amplitude (energy is proportional to the square of amplitude), it 

is clear that for points in regions of larger amplitudes the energy 

will assume the higher values. The difference between anisotropic 

and isotropic media is that the energy density for anisotropic media 

is not a uniform function for equal solid angles having vertexes at 

the origin, constructed around different directions in space. 

Lighthill [2] referred to this phenomenon saying that the energy is 

"confined" to some cone in space, or in other words, most of the 

energy travels along preferential directions. The preferential 

directions for the unidirectional fiberglass epoxy composite are 

shown in Table 3. It is understood that the maxima are the center 

directions of the regions around which the energy is confined. When 

the numerical process is carried out the preferential directions are 

automatically determined. 

-36-
CONCLUSIONS 

The far-field displacement pattern in an infinite transversely 

isotropic medium subjected to an oscillatory point force was determined 

and evaluated for a specific fiberglass epoxy composite. The solution 

describes the stress wave field in terms of the geometric aspects 

of the disturbance spreading by means of slowness and wave surfaces. 

The solution for displacements allows the prediction of the amplitude 

distribution as shown by the construction of polar diagrams. 

"It was seen that the energy from a point source travels along 

radial directions even if the wave surfaces assume complicated shapes, 

meaning that the phase and group velocities are different. 

The existence of preferential directions is an important aspect 

to be considered when experimental tests are to be designed. 

Knowledge of the displacement field allows a better choice to be 

made for the positioning of components of the measuring system. 

There appear to be no restrictions regarding the applicability 

of this method to other types of anisotropy. If appropriate symmetry 

relations can be applied as in the case of eqn. (7), the same 

procedurescan be followed for the displacement field solution. This 

possibility is attractive for use in the description of the stress 

wave field of filamentary composite materials with fiber arrangements 

other than unidirectional. 

Another possible application is the study of acoustic emission 

(AE) phenomena in composites or other anisotropic materials. It is 

known that acoustic emission is a transient disturbance of relatively 

small dimensions occurring inside the material, and normally 

-37-
containing a range of frequency components. If the method is applied 

for each component of frequency, superposition can be used for the 

determination of the field resulting from the AE source. Knowledge 

of the displacement field might allow inferences regarding the 

source such as strength and orientation, and consequently perhaps 

the degree of damage within the material. 

-38-
REFERENCES 

(1] M.J. Lighthill, "Studies on Magneto-Hydrodynamic Waves and 

other Anisotropic Wave Motions", Philosophical Transactions of the 

Royal Society, Series A, Vol. 252, 1960, pp. 397-430. 

[2] V.T. Buchwald, "Elastic Waves in Anisotropic Media", Proceedings 

of the Royal Society, Series A, Vol. 253, 1959, pp. 563-580. 

(3] J.L. Synge, "Elastic Waves in Anisotropic Media", Journal of 

Mathematics and Physics, Vol. 25, 1956, pp. 323-334. 

(4] G.F. Carrier, "The Propagation of Waves in Orthotropic Media", 

Quarterly of Applied Mathematics, Vol. 4, 1946, pp. 160-165. 

(5] M.J.P. Musgrave, Crystal Acoustics; Introduction to the Study 

of Elastic Waves and Vibrations in Crystals, Holden Day, San 

Francisco, 1970. 

(6] J.D. Achenbach, A Theory of Elasticity with Microstructure for 

Directionally Reinforced Composites, OISM-Courses and Lectures, 

No. 167 (International Center for Mechanical Sciences), SpringerVerlag, N.Y., 1965. 

(7] ,J .E. White and F .A. Angona, "Elastic Waves in Laminated Media", 

The Journal of the Acoustical Society of America,' Vol. 17, No.2, 

1955, pp. 310-317. 

(8] B.W. Rosen, "Stiffness of Fibre Composite Materials", Composites, 

Vol. 4, No.1, 1973, pp. 16-25. 

(9] A.E.H. Love, A Treatice on the Mathematical Theory of Elasticity, 

Dover, 4th Edition, 1944. 

(10] S.G. Lekhnistskii, Theory of Elasticity of an Anisotropic Elastic 

Body, Holden Day, San Francisco, 1963, (translation). 

(11] I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw Hill, 

NY, 1956. 

[12] K.F. Graff, Wave Motion in Elastic Solids, Ohio State University 

Pre~s, 1975. 

[13] L. Rayleigh, Theory of Sound, Vol. 2, Dover, 1945. 

[14] G.D. Dean and F.J. Lockett, "Determination of the Mechanical 

Properties of Fiber Composites by Ultrasonic Techniques", Analysis 

of the Test Methods for High Modulus Fibers and Composites, 

ASTM STP52l, American Society for Testing Materials, 1973, pp. 

326-345. 

-39-
[151 G.D. Dean, "Characterization of Fibre Composites using Ultrasonics", 

Composites-Standards, Testing and Design, National Physics Laboratory 

Conference Proceedings, Middlesex, England, 1974, pp. 126-130. 

[161 J.H. Williams, Jr., H.N. Hashemi and S.S. Lee, "Ultrasonic 

Attenuation and Velocity in AS/3501-6 Graphite/Epoxy Fiber 

Composite", Journal of Nondestructive Evaluation, Vol. 1, No.2, 

1980, pp. 137-148. 

[171 M.F. Markham, "Measure~ents of the Elastic Constants of Fibre 

Composites by Ultrasonics", Composites, Vol. 1, No.3, 1970, 

pp. 145-149. 

[181 !'Scotchply Reinforced Plastics Technical Data - Type 1002", 

3M Company, Minnesota, May 1, 1969. 

-40-
TABLE 1 Velocities for Plane Waves along Principal 

Directions of Transversely Isotropic Media. 

(x-y is isotropic plane) 

Direction of Direction of 

Propagation x 

x (a1

)1/2 

y (a4

)1/2 

z (a

5

)1/2 

The a. are defined in eqn. (13). ~ 

-41-

Displacement Vector 

y z 

(a4)1/2 (a5)1/2 

(a

1

)1/2 (a5

)1/2 

(a

5

)1/2 (a2

)1/2 
TABLE 2 Elastic Constants for Unidirectional Fiberglass 

Epoxy Composite (Scotchply 1002). 

Elastic Computed Value from 

Constant Wave Speed Measurements 

C

ll 10.580 109 N/m2 

C

12 4.098 109 N/m 2 

C

13 4.679 109 N/m2 

C

33 40.741 109 N/m2 

C

44 4.422 109 N/m2 

-42-
TABLE 3 Points of Maximum Displacement Amplitude for Lines 

on x-z Plane and x-y Plane 

Slowness Maximum 

Line Equation Displm Surfaces Location 

u sv 41 0; 62.5° (*) 

P 90° (*) 

2 

+ z 

2 

x = 4 

W. SV 42°; 62.6° (*) 

P .15 ° (*) 

u sa 0° (**) 

x2 + y2 

P 90° (**) 

= 4 

v SH 45° (**) 

t--

P 45° (**) 

(*) Angles with respect to z axis (in degrees) 

(**) Angles with respect to y axis (in degrees) 

(***) These amplitudes correspond to a unit body force. 

-43-

Order of 

Magnitude 

of Amplitudes 

10-8 

10-8 

10-9 

10-9 

10-8 

10-9 

10-9 

10-9 

(***) 

I 
Fig. 1 

----~ /1 o~------~:...----::..-z 

I 

x 

I 

I 

I 

I /' 1/ ------~ 

I 

, 

p (X,Y. z) 

Schematic illustrating sinusoidal point load 

exciting an infinite transversely isotropic 

medium, where xy is isotropic plane in cartesian 

coordinate system defined by (x,y,z). 

-44-
z z L 

w 

z 

t 11./02 

a- wave number vector component in x direction 

y-wave number vector component in z direction 

- Propagation direction of associated plane wave (along 

principal directions) 

--- Particle displacement of associated plane wOIe (along 

· principal d irecti ons) 

Fig. 2 Plane representation of slowness surfaces for transversely 

isotropic medium where z axis corresponds to the larger 

elastic constant value and x-y is the isotropic plane. 

-45-

1/-1ci4 

a 
8 x 10-4 

-

e 

" en 7 -:3 

~ 

c 6 

0 .-

+-

CJ 

Q) 

.. 5 . -

~ 

N 

c: 4 

fit 

fit 

., 

c 3 

:s 

0 

fit 

.. 2 

0 

+-

c: 

G) 

c 

0 

0. 

E 

0 0 (,) 

0 2 3 4 5 6 7 

Component of slowness in x direction a/w (s/m) 

Slowness surfaces SR, SV and P for unidirectional 

fiberglass composite for positive x-z quadrant. 

-46-

8 x 10-4 
5xI03,-----------------------------------------~ 

-en 

....... 

E 

-N 4 

(J 

c 

o 

-

-o 

-c: 

w 

c: 

o 

Co 

E 

o 

u 

2 

o ~--------~~~--_r--~----~------~--------~ 

o 2 3 4 5 x 103 

Component of phase velocity in x direction Cx em/s) 

Fig. 4 Velocity surfaces V(SH), V(SV) and V(P) for 

unidirectional fiberglass composite for 

positive x-z quadrant. 

-47-
5x 103 -------------------------------------------

c 

o 

-CJ 

~ 4 

:a 

N 

c: 

--

0_ c e 3 ~ N ~ 

CD-a 

:> c: 

c 0 3 u ~Q,) 

~ f/J 2 

-c:Jc: 

~-CD 

:> 

o 

~ 

-

CD 

(J 

c 

c 

-(J) .-o o 

o 2 3 4 

Distance traveled by wavefront in x directton 

in I second x (m) 

Fig. 5 I,ave surfaces t~(SH), W(SV} and W(P) for unidirectional 

fiberglass composite for positive x-z quadrant. 

-48-
6 x 10-9 

-e 

-

c 5 

0 

-u 

OJ 

... 

. -

~ 4 

>-

c 

-c 

OJ 3 

c 

0 

~ 

e 

0 

u 2 

OJ 

~ 

~ 

-

-

~ 

e 

« 

o 

o 1 2 3 4 5 

Ampl itude component in x direct ion (m) 

Fig. 6 Polar diagram for u displacement amplitudes for 

points along the line x2 + y2 = 4m2 , due to 

slowness surface P only. 

-49-
-8 2.0x10 

-

E 

-

c: 

0 

-(,J 

Q,) 

~ 

"C 

~ 

c: 

-c: 

Q,) 

c: 

o 

1.6 

1.2 

E 0.8' 

o 

(,J 

Q,) 

"C 

::l 

- 0.4 

a. 

E 

« 

o 

Fig. 7 

0.4 0.8 1.2 1.6 

Amplitude component in x direction (m) 

Polar diagram for u displac~ment arnpli~udes 

for points along the line x + y2 = 4m , due 

to slowness surface SH only. 

-50-

-8 2.0 It 10 
2.8xIO~~--------------------------------------~ 

-

e 

-

c: 

o 

-u 

2.4 

~ 2.0 

"0 

:>. 

c: 

1.2 1.6 2.0 2.4 

Amplitude component in x direction (m) 

Fig. 8 Polar diagram for v displacement amplitudes for 

points along the line x2 + y2 = 4m2 , due to 

slowness surface P only. 

-51-
-9 

IOxlO ~--------------------------------------~ 

-

E 

-

g 8 

-u 

Q) 

~ 

"0 

~ 

c: 

-c: 

Q) 

c: 

0 

c. 

E 

0 

u 

Q) 

"0 

::J 

-

a. 

E 

<r 

6 

4 

2 

0 

o 246 8 

Amplitude component in x direction (m) 

Fig. 9 Polar diagram for v displacement amplitudes for 

points along the line x2 + y2 = 4m2 , due to 

slowness surface SH only. 

-52-
2.8xIO~;------------------------------------------' 

-e 

- 2.4 

c 

o 

-

-

Co) 

~ 2.0 

~ 

N 

C 

-c 

CD 

c 

1.6 

&. 1.2 

e 

o 

Co) 

CD 0.8 ~ 

~ 

-

~ 0.4 

<t 

o 

/ 

o 0.4 0.8 1.2 1.6 2.0 2.4 

Ampl itude component in x direction (m) 

-8 2.8x10 

Fig. 10 Polar diagram for u displacement amplitudes for 

points along the line x2 + z2 = 4m2 , due to slowness 

surface SV only. 

-53-
6xIO-9~--------------------______________________ ~ 

-

E 

-

c: 

0 

-0 

Q) 

'-

~ 

N 

c: 

-c: 

Q) 

c: 

0 

~ 

E 

0 

0 

Q) 

~ 

~ 

-

c. 

E 

<l: 

5 

4 

3 

2 

o 

o 2 3 4 5 

Amplitude component in x direction (m) 

'Fig. 11 Polar diagram for u displacement amplitudes for 

points along the line x2 + z2 = 4m2 , due to 

slowness surface P only. 

-54-

-9 6xlO 
6 x 10-9 

-e 

- 5 

C 

0 

-u 

Q) 

~ 4 

" N 

c 

- 3 

C 

Q) 

c 

0 

aE 

0 2 

u 

Q) 

" :J 

.---

aS 

ex 

0 

/ 

/ 

/ 

/' 

/ , 

,. / 

/ 

.,,/ 

/'" 

0 I 2 3 4 5 6 x 10-9 

Amplitude component in x direction (m) 

Fig. 12 Polar diagram for w displacement amplitudes for 

points along the line x2 + z2 = 4m2 , due to slowness 

surface SV only. 

-55-
-9 

3.0x10 

-

E 

-

c: 

0 

-(,J 

Q) 

~ 

~ 

N 

c::: 

-c::: 

Q) 

c:: 

0 

Co 

E 

0 

u 

Q) 

~ 

::l 

-

Co 

E 

<X: 

2.4 

1.8 

1.2 

0.6 

0 

-9 

0 0.6 I. 2 1.8 2.4 3.0 x 10 

Amplitude component in x direction (m) 

Fig. 13 Polar diagram for w displacement amplitudes for 

points along the line x2 + z2 = 4m2 , due to slowness 

surface P only. 

-56-
1. Report No. 2. Government Accession No. 

NASA CR-4001 

4. Title and Subtitle 

Wave Propagation in Anisotropic Medium Due to an 

Oscillatory Point Source With Application to 

Unidirectional Composites 

7. Author(s) 

James H. Williams, Jr., Elizabeth R. C. Marques, 

and Samson S. Lee 

9. Performing Organization Name and Address 

Massachusetts Institute of Technology 

Department of Mechanical Engineering Cambridge, Massachusetts 02139 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

15. Supplementary Notes 

3. Recipient's Catalog No. 

5. Report Date 

July 1986 

6. Performing Organization Code 

8. Performing Organization Report No. 

None 

10. Work Unit No. 

11. Contract or Grant No. 

NAG3-328 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

506-43-11 (E-3093) 

Final report. Project Manager, Alex Vary, Structures Division, NASA Lewis 

Research Center, Cleveland, Ohio 44135. 

16. Abstract 

The far-field displacements in an infinite transversely isotropic elastic medium 

subjected to an oscillatory concentrated force are derived. The concepts of 

velocity surface, slowness surface and wave surface are used to describe the 

geometry of the wave propagation process. It is shown that the decay of the wave 

amplitudes depends not only on the distance from the source (as in isotropic 

media) but also depends on the direction of the point of interest from the 

source. As an example, the displacement field is computed for a laboratoryfabricated unidirectional fiberglass epoxy composite. The solution for the 

displacements is expressed as an amplitude distribution and is presented in polar 

diagrams. This analysis has potential usefulness in the acoustic emission (AE) 

and ultrasonic nondestructive evaluation of composite materials. For example, 

the transient localized disturbances which are generally associated with AE 

sources can be modeled via this analysis. In which case, knowledge of the displacement field which arrives at a receiving transducer allows inferences regarding the strength and orientation of the source, and consequently perhaps the 

degree of damage within the composite. 

17. Key Words (Suggested by Author(s)) 

Elastic waves; Ultrasonics; Acoustic 

emission; Wave propagation analysis; 

Nondestructive testing; Fiber 

reinforced composites 

18. Distribution Statement 

Unclassified - unlimited 

STAR Category 38 

19. Security Class if. (of this report) 

Unclassified 

20. Security Classlf. (of this page) 

Unclassified 

21. No. of pages 

58 

"For sale by the National Technical Information Service, Springfield, Virginia 22161 

22. Price· 

A04 

NASA-Langley, 1986 
End of Document