Propagation of surface waves in anisotropic solids: theoretical calculation and experiment

Propagation of surface waves in anisotropic solids: theoretical calculation and experiment T.-T. W u and J.-F. Chai Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan Received 4 August 1992; revised 24 May 1993

This paper utilizes the Stroh-Barnett's integral formalism of surface waves in anisotropic solids to calculate the variation of surface wave speeds of a unidirectional fibre-reinforced composite in an efficient way. The decaying pattern of the displacements and tractions along the depth were also calculated using the determined surface wave speeds. To study the propagations of surface waves in the fibre-reinforced composite plate experimentally, point-source/point-receiver (PS/PR) experiments were conducted. A pulsed laser was utilized to generate a point disturbance in the composite plate and the transient waves were recorded by a PZT transducer with an acting diameter of about 1.5 mm. The slowness curves of the surface wave energy velocity were determined from the experimental data and the results were in accordance with those calculated theoretically.

Keywords: surface waves; ray surfaces; fibre-reinforced composites The early studies of the propagations of elastic surface waves in crystals can be found in a review article by Farnell 1. Basically, most of the studies in the anisotropic surface wave problems adopted the formulation utilized by Synge 2, and hence iterations between the secular equations of the equations of motion and the complex surface boundary determinants are required to determine the surface wave velocity. A formal mathematical proof for the existence of surface waves in all directions of an anisotropic material has been given by Barnett and Lothe 3. Their proof was based upon the works of Stroh 4 in the development of theory of dislocations. In the Stroh formalism, the material properties and the elastic symmetries are contained in the so-called fundamental elasticity tensor 5 and the solution of a particular problem can be expressed in terms of the eigenvalues and eigenvectors of a six-dimensional tensor of rank two 6. Barnett, Lothe and their coworkers 7'8 extended the Stroh formalism to the integral formalism and showed that the surface wave velocity in an anisotropic material can be determined by the vanishing of the determinant of a real symmetric 2 x 2 matrix without solving the eigenvalue problem in the Stroh formalism. In an anisotropic solid, the energy velocity of a surface wave is, in general, different from the phase velocity in both magnitude and direction. The direction of energy velocity is fluctuated along the depth of an anisotropic solid, but remains parallel to the surface 9. However, with the integration of the surface wave energy from surface to infinity, it can be shown that the direction of energy propagation is perpendicular to the slowness surface of the phase velocity 9. In this paper, we utilized the integral formalism proposed by Barnett et al. 7"8 to study the propagation of surface waves in a unidirectional fibre-reinforced composite. The phase velocities, as well as the field 0041 - 624X/94/010021-09 © 1994 Butterworth-Heinemann Ltd

distributions of surface waves, were calculated first. The ray surface of the fibre-reinforced composite, which was different only in scale from the actual wavefront propagating in all directions from a point source, was then given. For comparisons, the propagation of surface waves on a steel plate was also studied. Finally, the slowness curves of the energy velocity of both the composite plate and the steel plate were measured using the point-source/point-receiver (PS/PR) technique x°. A pulsed laser in the ablation range was employed as a point source and transient wave signals were received through the utilization of a PZT transducer with a small acting area.

S u r f a c e w a v e s in a n i s o t r o p i c solids Conventional solutions

Consider an anisotropic half space with inner normal n o (Figure 1) and an inhomogeneous plane wave propagating

on the surface along m °, the displacements ul can be expressed as u i = A i exp {ik(m°xj + p n ° x i - vt)}

(1)

where A i are the amplitudes of the displacement vector ul, k is the wave-number, v is the phase velocity, and p is a complex constant governing the decay of the amplitude along depth. For an anisotropic linear elastic material with no body force, the displacements satisfy the equations of motion ~2ut C'ijkl = Cijkl

(2)

- 0 --

pv2m°m°6jk

(3)

where C'qu is the dynamic elastic constant tensor components of the material with Cuu the anisotropic

Ultrasonics 1994 Vol 32 No 1

21

P r o p a g a t i o n o f surface waves in anisotropic sofids." T. - T. Wu a n d J.-F. Chai

saggital

Figure 1

determinant (Equation (6)). The calculated p~, Aio are then substituted into Equation (10) to check whether the boundary conditions are satisfied. If the complex determinant does not vanish, another phase velocity has to be chosen, until both the real and complex determinants are satisfied, and then this velocity is the surface wave velocity vR. Knowing the Rayleigh wave velocity VR, one can calculate the decaying numbers of p~ and amplitudes Ai~ by Equation (5), and then calculate the linear combination coefficients Co by Equation (9), and finally, the displacement field can be determined by Equation (7).

plane

Stroh-Barnett's

C o o r d i n a t e s of t h e a n i s o t r o p i c haft s p a c e

surface

elastic constants, p is the mass density, and fig is the Kronecker delta. On defining a 3 × 3 matrix 4)

(ab)j k = aiC'ijktb t

Equation (2) can be arranged in a form as {p2(n°n°) + p[(m°n °) + (n°m°)] + (m°m°)}A = 0

integral formalism

for

waves

In this subsection, the Stroh-Barnett's integral formalism for surface waves in anisotropic solids is introduced. To form a six-dimensional eigenvalue problem which incorporates both the equations of motion and the tractions on the plane with inner normal n °, Stroh defined a vector L L = - {(n°m °) + p(n°n°)}A

(11)

or, rewritten in the form (5)

pA

( n ° n ° ) - X(n°m°)A - ( n ° n ° ) - tL

(12)

In order to have a non-trivial solution for A~, A2, A3, the determinant of the coefficients of Equation (5) must vanish, i.e.

The equation of motion in the form of Equation (51 can then be expressed as

Det{p2(n°n °) + p[(m°n °) + (n°m°)] + (re°m°)} = 0

pL = { ( m ° m °) - ( m ° n ° ) ( n ° n °) l ( n ° m ° ) } A

(6)

For a given v, Equation (6) is a six-order polynomial with real coefficients in p and the six roots p~, ~ = 1 - 6, occur in three complex conjugate pairs. For surface waves propagating in the half space (n°x~ > 0) only the roots p~ with positive imaginary parts are selected to construct a surface wave. Therefore, the surface wave displacements can be expressed as a linear combination of the fields with the chosen p~ as (7)

where Ai~ are the complex eigenvectors associated with the corresponding eigenvalues p=, and Co are constants to be determined from the surface boundary conditions. For a traction free boundary condition at

x3=0

(8)

Substituting Equation (7) into Equation (8) gives 3

3 ~

+

(n°m°)]ikAkaC ~ = 0

1

the so-called six-dimensional eigenvalue problem forms

(15)

where N is a 6 × 6 matrix

= 0

(10)

Ultrasonics 1994 Vol 32 No 1

(n°n°) -1

(n°n°) - l(n°m°) N = -- L (m °n°Xn°n°)- l(n°m°) - ( m ° m °)

(mOnOXnOnO)-1A

(16) The six eigenvalues p~ of N occur in three complex conjugate pairs while the eigenvectors ~ satisfy the closure relations 3 6

A,~Aio = 0; 1

It is noted that the components R~= are dependent on the anisotropic elastic constants as well as the eigenvalues (Po) and their corresponding eigenvectors (A~o). The determinant in Equation (10) is, in general, complex since the eigenvectors and the corresponding eigenvalues are complex. To determine the fields for surface waves, one first has to choose a phase velocity v, and then calculate the eigenvalues p~ and eigenvectors A~ which satisfy the real

22

,14,

(9)

The determinant of the coefficients of C~ must vanish for a non-trivial solution, i.e. Det {Rio }

On combining Equations (12) and (13) and introducing a six-dimensional vector

6

Ri~C~ = ~ [po(n°n °) o=1

(13)

N~ = pC

3

Ui = ~ C~eAice exp {ik(m°xj + p=n°xj - vt)}

cTij~jo = 0

- ( m ° n ° ) ( n ° n °) - 1L

~ LioL~o = 0; ~--1

6

~ Ai=L~o = 6ii z~l

(17) Consider the saggital plane formed by m ° and n o

(Figure 1) and rotate m ° and n o through an angle ~, to a new basis m and n in this plane, i.e. m = m ° c o s ~ + n o sin ~9 n = -re°sinS

+ n° cos ~0

(18)

Then, the 6 x 6 matrix N becomes a function of ~b and it can be shown that 0N($)

-

(1 + N ( ¢ ) ) 2

(19)

Propagation of surface waves in anisotropic sofids. T.- T. Wu and J.-F. Chai

and

(29), the real matrix B must satisfy

OP=(~) = --(1 + p2)

(20)

where ~= are invariant with the rotation angle ~. On the new plane basis, the eigenvalue problem becomes N($)~, = P~(0)~

(21)

and from Equation (20), the eigenvalues p, satisfy the following identity

f]

~ p~(~) d~, = _ 2hi

(22)

where, + depends on whether p= has a positive or negative imaginary part. On integrating Equation (21) and utilizing Equation (22), the dependence of eigenvalues p= can be removed and the following form results 1 ~ = __+i~

(23)

where N(~b) d~k

IB11B22 -- B12B211 = 0

(31)

With the above derivations, it follows that finding the surface wave velocity of an anisotropic solid can be replaced simply by finding the velocity which causes the determinant of a 2 x 2 real matrix to vanish. Once the surface wave velocity along a particular propagation direction m ° is known, calculations of the distributions of the displacements and tractions along the depth of an anisotropic half space are straightforward, i.e.

ct=l 3

Ji = - i k ~ C=Li=exp {ik(m°xj + p=n°xj - v~t)} where Ca can be determined by Equation (29).

sT

Energy velocity waves

1 (nn)- X(nm) d~k -2--n fO2~

lfo" lfo"

Q = - 2-n

(nm)-' d~b

B = - 2~

[(mn)(nn)- l ( m ) - (mm)] dO

(26)

The limiting velocity vL is defined as the smallest velocity for which (nn)- ~ becomes singular for some orientation of n in the saggital plane (with angle 0L); thus, in the subsonic range (0 ~< v ~< VL) (an)- ~ and therefore N and 1N always exist• From the definition of (nn), one can show that vL = C1(0L) sec(~/L), where cl(~bL) is the smallest bulk wave velocity propagating along the ~kL direction. From Equations (9) and (11), the traction free boundary conditions (Equation (8)) can be expressed as 3

L~=C~ = 0

(27)

(33)

where W is the energy density and P is the time average of the Poynting vector per unit area 1 .* *" Pj = - ~(alju i + oijui)

(34)

The energy density W is a sum of the average kinetic (Eki,) and strain (Estr) energy densities Eki n +

Est r

= ~u~ui + ~(oljuij + %u~.) (28)

Equation (27) can be written as 3 0

of surface

ve = = W

•

3

Bij = 2i ~ Li~Lj= ~t=l

and ray surface

For an anisotropic solid, the phase velocity vector and the energy velocity vector ve do not, in general 11, have the same direction. The ray surface is the locus of ve centred at some point against its direction. Thus, for harmonic waves, the time-averaged energy flux in all directions can be represented by the radius vector of the ray surface. The ray surface is therefore different only in scale from the actual wave front propagating in all directions from a point source. The energy velocity of an elastic wave is defined as

W=

By the definition of matrix B

Z CctAi~Bik=

(32)

~=1

and

cc=l

simultaneously, where P~ = ~3= 1C=Ai=is the polarization vector, and Re, Im denote real, imaginary parts, respectively. From Equation (30), it is obvious that the rank of the real matrix B equals one when the traction free boundary condition is satisfied. Thus, the determinants of all the 2 x 2 submatrices of B are zero, for example

(24)

with the components of l'q defined as

S=

(30)

3 ui = ~ C~Ai~ exp {ik(m°xj + p~n°xj - vRt)}

2n s= -lfo

N=

Bik Re {Pi} = ~ } Bik Im {el}

(29)

From the closure relations (Equation (17)), it can be 3 lLi=Lj= is pure imaginary, and therefore, shown that ~== the matrix B is real and symmetric. Thus, from Equation

.*

1

-*

*

•

(35)

The (e) in Equation (33) denotes the integration with depth from 0 to infinity, e.g. P = S~ P(xa)dx3" Consider the propagation of a surface wave along the m ° direction, the displacements and the constitutive relation have the forms

ui = { ~_l C~Ai~eikP~"~XJ}exp {ik(m°xj -- vRt)} (36)

O'ij ~" CijklUk,l

From Equations (33)-(36), the components of the

Ultrasonics 1 9 9 4 Vol 32 N o 1

23

Propagation of surface waves in anisotropic sofids: T.-T. Wu and J.-F. Chai

velocity, and is perpendicular to the slowness surface of the phase velocity 9.

Numerical implementation

0 Deviations between the phase velocity and energy

Figure 2 velocity

read p~nd Cok,

/ m°(O),n°(O) L

t C~jta = Ciiki - PV2 6ikrn~m~

equation (3)

T Integral formalism (Gaussian integration) _____J[__.__ .

{

.

.

.

.

.

.

.

.

.

.

.

.

.

¢

ii '

:o~2.

(ram), (ran), (nn), (nn)-' is defined as equation (4)

', i

(mm)

i

." . . . . f B : 2~ 1 f ° , ' { (ram) - (mn)(nn) '(rim)} - - - 1 ~d

~v

1 F=O t is the surface wave velocity vRJ

From the previous discussions, the surface wave velocities in anisotropic solids as well as the displacement and traction distributions along the depth can be calculated in an efficient way. That is, the interations between the vanishing of a 3 x 3 complex determinant are avoided. Furthermore, the energy velocity of a surface wave and the corresponding ray surface can also be determined. In the following, the numerical implementation is described. An inhomogeneous plane wave propagating along m ° = (cos 0, sin 0, 0) in an anisotropic half-space with inner normal n o was considered in the present study. Figure 3 shows the block diagram that describes how to calculate the surface wave speed. The notation utilized in this block diagram is the same as in the previous section. The Gaussian integration method was employed in the integration of qJ from 0 2m Once the surface wave velocity along the propagation direction m is known, calculations of the distributions of the displacements and tractions are straightforward. The first step is to construct the six-dimensional matrix N using the determined surface-wave velocity. The eigenvalues and the corresponding eigenvectors of the matrix can then be determined from an eigenvalue solver (IMSL). The mechanical displacements and the tractions obtained from Equation (32) are referenced to the laboratory coordinates x~x2x3; however, the mechanical displacements are usually expressed in the components of the space coordinates X~, X 2, X 3 (Figure 4) where the X ~-axis is the wave propagation direction and the X3-axis is the inner normal of the half space. In the sequence, the mechanical displacements are expressed in terms of the space cordinates X 1, X 2, X 3 through a coordinate transformation. In this paper, we utilized the aforementioned formalism to study the propagations of surface waves in a unidirectional fibre-reinforced composite. The elastic constants utilized in the calculations were ~2

Figure 3 Relations between the laboratory coordinates and the space cordinates C6x 6

surface wave energy velocity can be expressed as *

UcJ ~ UR

*

0

* )

lm{~,flC~Cl~Ak~AiflCijkt(mt + p~n°)/(p~ -- pfl),, 2 * * * Im{pvR~,~43C~CflAi~Aifl/(p~ -- pfl)} (37)

In arriving at the above equation, the equality of the kinetic and the strain energy densities has been used. From the components of energy velocity, the deviation angle ¢ (4~ = 0 e - 0p) between the energy velocity (0e) and the phase velocity (0p) directions can be determined. Furthermore, it is easy to show from Equation (37) that the phase velocity is just the projection of the energy velocity on the propagation direction, i.e. vc.m° = VR, (Figure 2). Besides, it has been shown that the surface wave energy velocity (Equation (33)) is equal to the group

24

Ultrasonics 1 9 9 4 Vol 32 No 1

"142.4

7.5

7.5

0

0

0

7,5

14.6

7.3

0

0

0

7.5

7.3

14.6

0

0

0

0

0

0

4.0

0

0

0

0

0

0

6.4

0

0

0

0

0

0

6.4

GPa

(38)

and the density was p = 1569.4kgm-3. The fibre direction of the composite plate was pointing along the xl-axis and the plate normal was along the x3-axis. Shown in Figure 5 are the variations of the limiting velocities and surface wave velocities on the surface of the composite plate, with the surface normal along the x3-axis. It is noted that the differences in the surface wave speeds along different propagational directions can be as large as 500 m s- 1 The distributions of displacement and traction along the depth (for example, 0 = 40 °) can be calculated from Equation (32) and are shown in Figures 6 and 7. One can see from Figure 7 that the traction free boundary

Propagation of surface waves in anisotropic sofids." T.- T. Wu and J.-F. Chai

X2:2"~,

X I

F i g u r e 4 Block diagram for the calculation of surface speeds using Stroh-Barnett's integral formalism

conditions are satisfied. We note again that the displacements and tractions are referenced to the (X~, X2, X3) coordinate system on which the X~-axis was pointed along the propagating direction and the X3-axis along the inner surface normal of the half space. Shown in Figure 8 is a plot of the deviation angles (¢) of the energy and phase velocities versus the phase velocity directions, while Figure 9 shows the slowness curves of the phase velocity as well as the energy velocity of the surface waves. The ray surface can be obtained as the inverse of the slowness curve of the energy velocity. It is seen that both the largest energy and phase velocities of the surface waves in the unidirectional composite are along the fibre direction. We note that the slowness curve of the phase velocities on the present surface is convex, hence there is no focussing effect on the ray surface.

UD composite material 2100

Traction Xl-a,xis is the fiber direction

2000

4.5

zlz2 plane

4

1900 "~

xlz2 plane

3.5

0 = 40°fromxl-axis vn=1781.13m/s

o

1800

3

8 1700 "U >

"~

1600

2.5

~

2

= o

1.5

'~

1

/

'\\

/

1500

0.5

1400

50

100

150 0

10

20

30

40

Phase angle (O) from xl-axis (deg.) F i g u r e 5 Variations of the surface wave and limiting velocities along different propagation directions

Depth (wXs × lO-Sm/s) Figure 7 Surface wave tractions along the depth of a unidirectional fibre-reinforced composite with propagating direction along 0 = 40 °

Displacement

UD composite material

1.2--

20 15

0.8 /'3

E

x]x2 plane

m

8 = 40° from xl-axis

tq

VR=1781.13 m/s

o

10

"o-

5 0

~0

0"6f 0.4

t~ o

-10

0.2

-15 O0

10

20

30

40

"200

- 1-~

200

300

Depth (wXs x lO-Sm/s) F i g u r e 6 Surface wave displacements along the depth of a unidirectional fibre-reinforced composite with propagating direction along 0 = 40 °

Phase angle 8 from xl-axis Figure 8 Variations of the deviation angle ~b as a function of propagation direction

Ultrasonics 1994 Vol 32 No 1

25

Propagation of surface waves in anisotropic sofids. T.-T. Wu and J.-F. Chai xlO 4

SAW slowness curve z2 -axis / // // // ///

4 =

\

w //

2~ zraxis m

x,

-4!

,q

"?'<),vn-1

Figure 9 Slowness curves of the phase and energy velocities of surface waves in a unidirectional fibre-reinforced composite

20i I ,source point

To stack the transient elastic wave signals obtained in the PS/PR experiments, an accurate control of the triggering of these point sources is required, to have a reference time for all the receiving transient elastic wave signals. For this reason, a pulsed laser (Excimer Laser LPX 200) with XeF gas (wavelength 351 nm) was utilized as the scanning point sources. The laser beam, with a rectagular shape, from the pulsed laser was focussed through a convex lens into approximately a 1 m m line segment to simulate a point source. However, most of the energy is concentrated around the central part of the 1 m m line segment. In the experiments, the direction of the line segment was perpendicular to the line of the scanning source points to minimize the finite aperture effects. The duration of the laser pulse utilized is 20 ns and the energy carried is about 100 mJ. A thin layer of glycerin was coated on the loading surface of the specimen. In this case, the coated glycerin is ablated from the surface to cause a reactive force on the surface 13. The generated elastic wave signals from the laser sources were recorded utilizing a PZT-type point sensor (Valpey-Fisher) with an acting diameter of about 1.5 ram. Shown in Figure 11 is the block diagram of the present experimental set-up. A trigger signal synchronized with the laser source was utilized to trigger the 10 bits digitizer (390AD, Tektronix). The recorded signals were sent to a personal computer via GPIB. Instead of scanning the laser source, the specimen was translated via a precision translation stage.

2 i~interval 0.5 mm

receiver

Figure I 0

ot

fiber direction -

Relative positions of the point sources and the receiver

Experimental w o r k s In the present work, an experimental study is made of propagations of surface waves in a steel specimen and in a unidirectional carbon-fibre composite (T300). The dimensions of the steel specimen are 120mm length, 60 m m width and 25 m m thickness, while those of the composite plate are 300 m m for both length and width, and 11 m m thickness. The Young's modulus, Poisson's ratio and the density of the steel specimen are 210.89 GPa, 0.294 and 7835.0kgm -3, respectively. Since the ray surface is different only in scale from the actual wavefront propagating in all directions from a point source, the PS/PR technique was utilized in this paper to measure the surface wave ray surfaces of a uni-directional fibre-reinforced composite and a steel specimen. Instead of using one point source and an array of receivers, one point receiver and a series of point sources were utilized in the present study. Shown in Figure 10 are the arrangements of the source and receiver positions. The source points were lying along a line perpendicular to the fibre direction with a uniform spacing of 0.5 mm. 21 point sources starting from point "o' were utilized in the study. The normal distance between the line of point sources and the receiver was 9.0 mm. Therefore, the angles between the fibre direction and the surface rays ranged from 0 to 48". In the case of the steel specimen, the spacing between the point sources was 1.0 mm, while the normal distance between the receiver and the source line was 19.0 mm.

26

Ultrasonics 1994 Vol 32 No 1

Results and discussions Shown in Figures 12 and 13 are the grey-scale plots of the elastic wave signals received from the aforementioned receiver array (seismogram) due to a laser point source on the steel specimen and the unidirectional composite, respectively. The horizontal axis is the elapsed time and the vertical axis is the positions of the sources. As compared with the amplitudes of the surface waves, the amplitudes of the P and SV waves are small, and hence the P and SV wavefronts cannot be seen in the grey-scale plots. The wavefronts behind the surface waves are the reflections between the surfaces of the plate specimens. Since the distances between the point sources and receivers are approximately (or less than) one half of the round trip distances in the plate specimens, most of the first reflected wavefronts (P P wave) are behind the Rayleigh wavefronts. The differences in the arrival times

oscilloscope

LPX 200

J

convex len i

'

iJ

i

Laser pulse controller

~'

{

syncd

-q

!) ! ),~ ~:?' receiver specimen

t

~)" :~'translation stage

Figure 11

digitizer

~GPIB i[J PC ' :: _

',! i

Experimental set-up of the PS/PR experiment

Propagation of surface waves in anisotropic sofids. T.- T. Wu and J.-F. Chai

= o

o

' 0

5

i0

15

Time (#sec) Figure 12 Grey-scale plot of the signals recorded in the PS/PR experiments of the steel specimen

I

0

5

i0

15

Time (,usee) Figure 13

Grey-scale plot of the signals recorded in the PS/PR experiments of the unidirectional fibre-reinforced composite specimen

of the surface waves in the steel specimen are due to the different travelling distances between the sources and the receiver (Figure 10), while the differences in those of the composite specimen are due to both the different travelling distances and the anisotropy of the composite plate. In order to compare the measured surface-wave speeds with those calculated in the previous section, the arrival times of different rays from the sources to the receivers were adjusted to let every ray have the same travelling distance. Since the arrival times of the wavefronts cannot be easily identified, the times which correspond to the positive peaks of the bipolar surface waves were utilized. Shown in Figure 14 are the comparisons of the calculated and the measured arrival times of the surface waves in

the steel specimen. The vertical axis represents the angle between the surface-wave ray and the xl-axis. In the figure, the deviations of the measured arrival times of the surface waves were about 0.2 #s. Similar to Figure 14, Figure 15 is the results for the unidirectional composite plate. The results show that the measured surface wave anisotropy was in accordance with the theoretical predictions; however, we noted that there are inaccuracies for ray angles beyond approximately 30 ° in the composite specimen. This may be due to the finite width (1 mm) of the laser source utilized in this study, as previously described. To observe the isotropic and the anisotropic slowness curves of the energy velocity, the grey-scale plots in Figures 12 and 13 were transformed to eliminate the effect

U l t r a s o n i c s 1 9 9 4 Vol 32 N o 1

27

Propagation of surface waves in anisotropic solids. T.-T. Wu and J.-F. Chat ,+

4{)!
3O

o o

2O Steel

10

--: o

c~

<

{/

block

cal. value : exp.

d~tta

0 <,

- 10

-20 -31) o

-40 K5

9

10

10.5

the propagations of surface waves in a unidirectional fibre-reinforced composite. The phase velocities, displacement and traction distributions, as well as the energy velocities of the anisotropic surface waves in the unidirectional fibre-reinforced composite have been calculated in an efficient way. The calculation of the surface wave speed in an anisotropic solid, based oil Stroh Barnett's integral formalism is simply the calculation of the speed which caused the determinant of a 2 >: 2 real matrix to vanish. A laser point s o u r c e has been utilized in the PS/PR experiments to study experimentally' the anisotropic wave propagations in the composite specimen. The results have shown that tile measured and theoretically calculated slowness curves of the surfacc wave energy velocity were in good agreement. With the experimental implementation and the fast algorithm for the forward calculation of surface wave speeds, the

Travel t i m e (,asec) Figure 14 Comparisons of the calculated and measured arrival times of the surface waves in the steel specimen

i

o

40 31) 20

UD c o m p o s i t e

10

--1 *rJ

i) <

C~II. V~IIuc

(~ : exp. data

-l(},

+2tl' "o

-30 -40 6.5

+

715 ''

S

~.5

Travel t i m e (/*see)

a

F i g u r e 15 Comparisons of the calculated and measured arrival times of the surface waves in the unidirectional fibre-reinforced composite specimen

of unequal travelling distances of the surface waves. The time axis of each trace was multiplied by a factor such that the distances between the sources and the receiver (Figure 10) remain constant. Shown in Figure 16a is the polar plot of the measured signals in the steel specimen and one can see that the arrivals of the surface waves are at almost the same time. Figure 16b shows the comparisons between the calculated and measured slowness curves of the energy velocities of surface waves. Figures 17a and 17b are the results for the unidirectional fibre-reinforced composite. It is seen that the differences in the arrivals of the surface waves are solely due to the anisotropy of the composite specimen. Conclusions In this paper, we have utilized the Stroh-Barnett's integral formalism of surface waves in anisotropic solids to study

28

Ultrasonics 1994 Vol 32 No 1

S i m u l a t e d s l o w n e s s of e n e r g y velocity _ _

10!

/

4 !:

L

~'o

:

/

,

--:

cal. value

~, j

i

ii b

-io

i ] I

i 1

i

Figure16 (a) Polar plot of the measured surface wave signals in the steel specimen. The travel distances of each ray have been corrected. (b) Comparisons between the calculated and measured slowness curves of surface wave energy velocity in the steel specimen

Propagation of surface waves in anisotropic sofids." T.- 7-. Wu and J.-F. Chai anisotropic elastic constants can be recovered from the surface wave group velocity measurements based on an i n v e r s i o n a l g o r i t h m 1..

Acknowledgements Part of this investigation was supported by the National Science Council of the Republic of China through the grant NSC82-0401-E002-152. The authors thank Professor C.-C. C h u f o r l e t t i n g us u s e his p u l s e l a s e r s y s t e m .

References 1

a Simulated slowness of energy velocity 10 8

6

/

/

\

\

/ : cal. value

4

o : exp. data

.~

o

-4 -6 -8

b

/

\ \

\

/

1

/

-lO

Figure 17 (a) Polar plot of the measured surface wave signals in the unidirectional fibre-reinforced composite specimen. The travel distances of each ray have been corrected. (b) Comparisons between the calculated and measured slowness curves of surface wave energy velocity in the unidirectional fibre-reinforced composite specimen

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