critical angle measurement of elastic constants in composite material

Critical angle measurement of elastic constants in composite material

S.I. Rokhlin and W. Wang Department of Welding Engineering, The Ohio State University, 190 West 19th Avenue, Columbus, Ohio 43210

(Received 21 January 1989; accepted for publication 10 July 1989)

Application of critical angle measurement for determination of elastic constants of composite materials is described. For general orientation of the composite material relative to the plane of incidence, three critical angles exist, for quasilongitudinal and fast and slow quasitransverse waves. It is shown that, for any two-dimensional composite material, the critical angles are directly related to the phase velocities of bulk waves in the sample plane. A simple and novel type of goniometer is developed for these measurements which utilize a cylindrical reflector for measurement of the double reflection coefficient. The values of the elastic constants are

reconstructed from the measured velocity surfaces by using nonlinear least-squares optimization fitting. Analytical relations between ultrasonic velocity and elastic constants valid for arbitrary anisotropy are implemented in the fitting optimization.

PACS numbers: 43.35.Cg

INTRODUCTION

Determination of the ultrasonic velocity of a solid by critical angle measurements on a liquid-immersed sample was first suggested by Mayer I in the 1960s. This method has the advantages that (1) there is no need for the sample to have two parallel surfaces as there is in the pulse echo method and (2) access to only one surface is required. It had been shown by Mayer that a sharp strong maximum corresponds to the first critical angle for not-very-dense materials; therefore, velocity measurements may be done with sufficient precision if the temperature of the liquid is stabilized. A detailed description of the critical angle method was also given later by Fountain, 2 who, in addition, observed in his experiments a dip corresponding to the Rayleigh critical angle.

While generalization of the critical angle method to anisotropic materials seems natural, it cannot be done in the general case as has been discussed by Henneke and Jones. 3 The critical angle condition for anisotropic material is that the propagation of the energy flow of the elastic wave in the solid be parallel to the surface (90 ø refraction angle for energy flow), which does not necessarily mean 90 ø refraction for the phase velocity. 3 This means that phase velocity cannot be measured using the critical angle in the general case in anisotropic materials. Possibly for this reason, further work for anisotropic materials has been done by measuring the Ray- leigh-wave critical angle. 4-6 As long as the Rayleigh wave exists on the free surface without attenuation, both the energy flow and the phase velocity for this wave lie in the plane of the sample surface (this condition will not be satisfied for pseudo-Rayleigh waves).

In this paper, the applicability of the critical angle method will be demonstrated for composite materials. The method of bulk-wave critical angles has the advantage of simple and unique calculation of the elastic constants of the material from the experimental data. It is shown that, for two- dimensional composite materials, all three critical angles may be observed and at the critical angle the phase velocity

always lies in the surface plane and, therefore, it may be calculated from Snell's law. It is shown that all three in-plane velocity surfaces can be measured and the in-plane elastic constants of the composite material can be reconstructed from experimental data. It has recently been demonstrated 7 that, using this technique in combination with the double- through-transmission method, the full matrix of elastic constants for an orthotropic material can be reconstructed. For discussion of other methods for elastic constant measure-

ments of composite materials, the reader is referred to Refs. 8-14.

I. THEORY

In an orthotropic material, in general, three elastic waves may propagate with different velocities: one quasilongitudinal and two quasitransverse waves. If the plane of incidence from liquid to sample is oriented arbitrarily to the planes of symmetry [ as, for example, in Fig. 1 (a) ], then all three elastic waves will be excited in the sample since all of them in general will have a normal component of displacement on the surface of the sample and will be coupled to the liquid. If the incident plane coincides with a plane of symmetry [for example, with plane 1-2 or 1-3 in Fig. 1 (a) ], only two elastic waves will be excited in the solid since one of the

transverse waves in this plane will be a purse SH wave with displacement parallel to the surface of the sample and will not be coupled to the liquid. Therefore, one may expect two critical angles for the plane of symmetry and three critical angles for an arbitrary plane of incidence.

The definition of critical angle for an anisotropic material is different from that of an isotropic, due to the fact that the direction of the wave vector and the energy flow do not coincide in the anisotropic case. This question was first addressed by Henneke •5 (see, also, the discussion in Rokhlin et al.•6), who defines critical angle as that at which energy flow (group velocity) in the refracted wave is parallel to the interface separating the two media. At this angle, the wave vector

1876 J. Acoust. Soc. Am. 86 (5), November 1989 0001-4966/89/111876-07500.80 ¸ 1989 Acoustical Society of America 1876

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 141.217.58.222 On: Thu, 27 Nov 2014 04:59:04

(a)

(b)

FIG. 1. Schematic illustration of the deviation of the refracted wave vector

K from group velocity (ray vector): (a) in general case and (b) for incident angle ?'i equal to critical angle. Here, ½ is out-of-incident plane deviation angle of the group velocity vector and angle/3 corresponds to the in-plane deviation. Also, ?'r is the refraction angle and a is the angle between incident plane and fiber direction.

of the refracted wave, in general, will deviate from the interface. Therefore, in an anisotropic material at the critical angle, both the refraction angle and the wave velocity are un- known and cannot be calculated from Snell's law. This

question was addressed theoretically by Henneke and Jones, 3 where they demonstrated this phenomenon for the example of quartz and showed that the value of a critical angle may not be used in general for phase velocity measurements in anisotropic materials. This limitation does not hold, as we will see in a moment, for two-dimensional composite materials.

Let us consider a unidirectional composite material with fibers in the direction of axis 3 [Fig. 1 (a) ]. Plane 1-3 will be parallel to the sample surface and plane 1-2 will be orthogonal to the surface and to the fiber direction. Let us select a plane of incidence having arbitrary angle a with the fiber direction. Let the incident angle in liquid be ?'i and select one of the refracted waves with wave vector K lying in the incidence plane at angle of refraction ?'r' The ray vector, which coincides with the energy flow and group velocity directions, will deviate from the wave vector direction. This deviation may be represented by an out-of-incidence-plane deviation angle q• and in-incidence-plane deviation angle/3. As was discussed previously, the critical angle condition ?, = •r is satisfied when the group velocity vector ray is parallel to the interface plane 1-3. For an arbitrary direction of propagation in a two-dimensional composite, the angle of

deviation between phase and group velocity vectors has both in-plane and out-of-plane components.

For two-dimensional composite materials with fibers lying in plane (1-3), at the critical angle both the group velocity (ray) vector and the wave vector will lie in the same plane, 1-3 [ see Fig. 1 (b) ]. This is clear from symmetry con- siderations since the in-plane elastic properties of a composite plate are invariant under change of direction of the normal to that plane.

This means that, at the critical angle, the wave vector (phase velocity vector) will be parallel to the plate surface and will lie in the plane of incidence. The group velocity vector [Fig. 1 (b) ] will also be parallel to the plate surface but may not lie in the plane of incidence and will deviate from this plane by an angle ½, remaining in the plate plane. This is illustrated for a graphite-epoxy composite material by calculations of the deviation angle (Fig. 2 ) versus angle of refraction. The incidence plane is twisted at an angle a = 45 ø to the fiber direction. It can be seen that the in-plane deviation angle (solid curve) equals zero at a refraction angle of 90 deg. The practical conclusion from this is that Snell's law may be used at the critical angle for the calculation of the in- plane phase velocity (in-plane means here in the plane of the composite plate).

This conclusion will hold for three-dimensional (3-D) composite materials if the 1-3 plane is a plane of symmetry and may not be satisfied if one of the fiber directions is not parallel or not orthogonal to the sample surface. In this last case, the deviation angle between the phase and group velocity may have an out-of-plane component and the wave vector will deviate from the interface at the critical angle.

The energy reflection coefficients have been calculated using the general algorithm previously described•6 for treat- ing the general problem of reflection and refraction on an interface between two anisotropic media, which has also been applied to composite materials. 17 The results of calculations of the energy reflection coefficient from a water-composite interface are shown in Fig. 3. The data are shown for a unidirectional composite material at different angles of rotation a of the plane of incidence relative to the fiber direction. Figure 3 (a) shows the reflection coefficient along the fiber direction (a = 0) versus incident angle. The very sharp maximum at approximately 10 ø corresponds to the critical

5O

4O

'- 30 o

"- 20

• •o • 0 o -IC)

• -20

-40

-50

-90

---Out of Plane , \ / •

--InPIone/9 \,, / '•__ I

-60 -30 0 30 60 90

Refraction Angle

FIG. 2. Angles of deviation of group velocity from wave normal as function of refraction angle for quasilongitudinal wave. Incidence plane deviation from fiber direction a equals 45 ø.

1877 J. Acoust. Soc. Am., Vol. 86, No. 5, November 1989 S.I. Rokhlin and W. Wang: Critical angle measurement 1877


Woter / Composite 0 ø

o O.T

o •_. o.B

• o.õ

• o.• 0.3

o.I

o •o zo ao •o •o •o 7o

Incident Wave Vector Angle

I (a)

1.0 Water/Composite 45 ø

(b)

0.9 --

ß •'•-• 0.8 -- • 0.? -- ._o 0.6 -

• 0.5 --

•- 0.4

05 -

0.2 -.

Ol

0 •0 20 30 40 50 •0 zo so 90

[nciden$ Wave Vector An•le

Woter/ Composite 90 ø I 0ø9--

o.8-

0.?-- 0.6--

0.õ--

0.4--

0.5

0.2

0.0 o •o 2o 5o 4o •o 6o 7o •o

FIG. 3. Energy reflection coefficient from water-unidirectional composite interface for different angles between plane of incidence and fiber direction: (a) a = 0 ø, (b) a = 45 ø, and (c) a = 90 ø. In (c), the first maximum corresponds to the critical angle for the quasilongitudinal wave, the second to fast quasitransverse, and the reflection coefficient reaches unity for the slow quasitransverse of the third wave.

angle for a longitudinal wave propagating along the fiber direction. The reflection coefficient becomes unity at the second critical angle, which corresponds to the transverse SV wave propagating along the fiber direction and polarized perpendicular to the interface. A transverse SH wave with polarization parallel to the surface is not coupled to the liquid and is not excited in the solid.

The energy reflection coefficient for the sample position with incident plane at 45 ø to the fiber direction is shown in Fig. 3 (b). At oblique incidence, three elastic waves are excited in the solid in this case. The first very sharp maximum corresponds to the critical angle for the quasilongitudinal wave, the second for the fast quasitransverse wave, and the reflection coefficient becomes unity at the third critical angle, which corresponds to the slow quasitransverse wave. At different angles a between the fiber direction (which is an axis of symmetry) and the plane of incidence, the behavior of the reflection coefficient versus angle of incidence resembles

that shown for a = 45 ø, except that the positions of critical angles will be shifted according to angular dependence of the ultrasonic velocities in the plate plane.

The case of a -- 90 ø, when the incident wave is perpendicular to the fibers, is shown in Fig. 3 (c). There, the maximum corresponds to the critical angle for a longitudinal wave, which is propagated perpendicular to the fibers. After the first critical angle, only a slow transverse wave having $V character is excited. It has displacements in the plane perpendicular to the fiber direction and has velocity below that in water from some angle of refraction. Therefore, a critical angle does not exist for this wave. The reflection coefficient reaches unity at grazing incidence. The fast transverse wave has SH character and is not coupled to the liquid. At a = 90 ø, it propagates perpendicular to the fiber direction and is polarized in the fiber direction. It has velocity equal to the transverse SV wave, which propagates along the fiber direction and is polarized perpendicular to the fibers and the

1878 J. Acoust. Sac. Am., Vol. 86, No. 5, November 1989 S.I. Rokhlin and W. Wang: Critical angle measurement 1878


plate surface [to this wave corresponds the second critical angle in Fig. 3 (a) ].

II. NOVEL ULTRASONIC GONIOMETER AND EXPERIMENTAL APPROACH

The experimental apparatus that implements a novel type of ultrasonic geniemeter is shown in Fig. 4. It is distinguished by its ability to make simultaneous measurements of both doubly transmitted (through the sample) and doubly reflected (from the sample) ultrasonic signals with the use of only one ultrasonic transducer. A plane sample is mounted in the center of an aluminum cylinder. The sample is rotated around a cylinder axis that lies in the frontal plane of the sample.

Incident and reflected ultrasonic beams propagate along cylinder radii. After being reflected from the cylinder wall, the ultrasonic signal returns to the sample, reflects a second time from its surface, and returns to the receiving transducer. The ultrasonic signal transmitted through the sample is reflected from the plane reflector and returned to the transducer after the second through transmission. The amplitude values of the received signal after analog peak detection are digitized, averaged, and collected by computer as a function of the angle of rotation.

When necessary, ultrasonic rf signals are digitized and analyzed in the frequency domain by a LeCroy 9400 125- MHz digital oscilloscope. Both signal and spectrum are fed to the computer through an IEEE interface.

For sample rotation by angle Yi around the cylinder axis, a computer-controlled dc motor is used. Angle resolution and repeatability of 0.01 ø are achieved in our setup. In addition, the sample may be rotated in its plane by an angle a, so different orientations of the incident plane relative to fiber direction may be selected. The water tank is temperature stabilized within 0.1 øC.

Pulser/

Receiver

I Oscilloscope •'• t

Peak

Detector Computer

DC Motor

Controller

Back Reflector

Cylinder Reflector

FIG. 4. Schematic of the experimental apparatus. During measurement, the sample is rotated by angle y, around a cylindrical axis lying in the frontal plane of the sample.

The measurements have been performed for the 1-3 plane of the sample, as schematically shown in Fig. 5. The incident plane has been selected for different angles a with the fiber direction. At the selected angle a, the sample was rotated around the axis 0'-0', which is perpendicular to the incidence plane Q. The amplitude of the doubly reflected signal was recorded as a function of the incident angle The double reflection coefficient corresponds to the single energy reflection coefficient, that is represented in Fig. 3.

Ultrasonic phase velocity in the plane of the sample may be calculated at each critical angle

V• = Vo/sin •'i, ( 1 )

where Vo is the velocity in water. The measurements have been done at the temperature 29.8 øC.

III. EXPERIMENTAL RESULTS AND RECONSTRUCTION OF THE ELASTIC CONSTANTS

As an example, several recordings of the amplitude of the doubly reflected signal versus incident angle Yi are shown in Fig. 6. The data were taken with an incident angle step of 0.05 ø . The doubly reflected signal disappears for incident angles •,• below 4 ø and above 75 ø due to the openings in the cylinder wall made for the ultrasonic transducer and back reflector (see Fig. 4). The first recording is taken in the plane coinciding with the fiber direction (the angle a equals zero). The experiments were repeated stepping the sample rotation by 5 ø from parallel to the fiber direction to perpendicular to the fiber direction.

As can be seen from the data, the first maximum is not pronounced. Its position may be located with the help of the sharp minimum that follows it (see also Fig. 3). Polishing of the sample surface, which is originally rough, practically does not change the results. The maximum is weak because it is narrow (Fig. 3 ), so its appearance depends critically upon the ultrasonic attenuation and the effect of finite beam

width. These effects have been discussed theoretically in Ref. 18 for isotropic material. To test the view that the minimum is weak because it is narrow, we repeat the measurement for the same experimental conditions for perspex. Perspex has

FIG. 5. Schematic diagram showing orientation of the coordinate system, incidence plane Q and axis 0'-0' of the sample rotation during measurement of the double-reflection coefficient as function of incident angle y,. When the scan on the angle 7, is finished, the angle between fiber direction and the plane of incidence a is changed, and the scan over angle y, is repeated.

1879 J. Acoust. Sec. Am., Vol. 86, No. 5, November 1989 S.I. Rokhlin and W. Wang: Critical angle measurement 1879


1.0

0.8

0.6

ß 0.4

.i

0.2

a=10 ø

3rd

(a)

1st •

I I I I 1

2O 4O

Incident Angle (deg)

1.0

0.8

0.6

0.4

0.2

a=35 ø

2nd

t

/

0 20 40

1st

3r

(c) i i I t 1

0 60 80 60 80

Incident Angle (deg)

a=20ø / ,.• I.I _[ - a=55 ø

-- , 0.8

3r / {

3r

o.4 o.4

• • l•t 2rid t tl < O• < 0.2 l s•

o ....... 't o , , , i i i i

0 20 40 60 80 0 20 • 60 80

Inoi•ent An•le (•e•) Inoi•ent An•le

FIG. 6. Examples of recordings of the amplitude of the doubly reflected signal versus incident angle ?,for different angles of deviation a of the incidence plane from the fiber direction: (a) a = 10 ø, (b) a = 20 ø, (c) a = 35 ø, and (d) a = 55 ø.

acoustic characteristics similar to that of the composite matrix, and theoretical calculations show a very strong, wide maximum at the first critical angle. The experimental maximum at the first critical angle in this case was very strong, and the results confirm the measurements on perspex done by Mayer. I In spite of the weakness of the maximum at the first critical angle, it can be located easily, as mentioned above, from the experimental data with the help of the sharp minimum by which it is followed.

The second maximum that corresponds to the critical angle for the fast transverse wave is very clear and sharp, and its position may be easily measured.

Experimental data for velocities of the quasilongitudinal and two quasitransverse waves are summarized in Fig. 7 as functions of the deviation angle from the fiber direction. They represent the dependence of the phase velocity on the direction of wave propagation in the 1-3 plane of the composite material.

At a close to 90 ø, the fast transverse wave has $H char-

acter and, therefore, cannot be excited from the liquid. Data for a slow transverse wave at angles above a = 55 ø cannot be obtained since it has velocity below that of the liquid, and the critical angle for this wave does not exist.

The value of the elastic constants may be calculated from phase velocity measurements in several specific directions of the anisotropic material. While this approach is fea- sible, it is somewhat sensitive to errors of measurement when reconstruction of the full matrix of elastic constants is de-

sired. We used a nonlinear least-squares technique for best data fitting at different angles. The algorithm is valid for materials of arbitrary anisotropy and for arbitrary directions.

To do optimization, it is more efficient to have an analytical function that relates the phase velocity and the elastic constants. Since the Christoffel equation may be written for arbitrary anisotropy as a third-order polynomial in the square of the velocity, it is useful to use Cardan's solution of the cubic equation, which may be written in the form



8

I o IO •0 •0 •0 •0 60 70 80 90

C• (deg)

FIG. 7. The phase velocity of ultrasonic waves as function of the angle between the direction of wave propagation and the fiber direction. Wave propagation is in the 1-3 plane, which is parallel to the sample surface. Experi- mental velocity values are points extracted from critical angle measurements. Solid lines are calculated values based on elastic constants

reconstructed from experimental data.

p V• -- Zk -- a/3,

Z• = 2x/p/3 cos[ (•b + 2k•r)/3],

where

k = 1,2,3,

arc cos( -- q/[2(p/3)3/2]•,

•-b, q=c--•+2 P=3 3 ' -- (Gii -'l- G22 -3- G33),

_ (a:: + +

-- GiiG22- GiiG33 -- G22G33),

-- (Gi1G22G33 + 2Gi2Gi3G23 - GiiG•3

-- G22G •3 -- G33G •2 ),

O,,. =

and Cijlm are the elastic constants in tensorial form. Also, Vk is phase velocity and p is density. The values in the arc cos argument should be calculated with double precision.

The program minimizes the sum of all squares of the deviations between the experimental and calculated velocities considering the elastic constants as variables in a multidimensional space. The method has the same advantages as the conventional least-squares method. For example, small random deviations of experimental points have minimal effect on the reconstruction results. Even if measurement at

several angles deviates significantly from the actual values, there will be only a moderate effect on the precision of the reconstruction.

The best fit to the experimental velocity data obtained from critical angle measurements is shown in Fig. 7 by a solid line. Elastic constants reconstructed for this fit are given in Table I. It is seen that comparison of the calculated and the experimental data is satisfactory.

TABLE I. Reconstructed constants in GPa and density in g/cm 3.

CI I'-' C22 •3 Ci2 Ci3 '-' C23 C44 '-' C55 C66 p

17.0 162 8.2 11.8 8.0 4.4 1.61

IV. SUMMARY

During angle beam immersion evaluation of a two-dimensional composite material, three critical angles may be observed, corresponding to quasilongitudinal and fast and slow quasitransverse waves. It is shown that, for two-dimensional composite materials at the critical angle, both phase and group velocities are oriented parallel to the sample surface and, therefore, Snell's law can be used for phase velocity extraction from critical angle measurements. By measuring phase velocity from the critical angles at different angles of deviation between the incident plane and the fiber direction, the cross section of three phase velocity surfaces in the plane of the composite plate is determined experimentally.

By using a nonlinear least-squares optimization technique for the best fitting of experimental data the in-plane elastic constants of a composite are reconstructed. The program minimized the squares of the deviations between the calculated and experimental velocities, considering the elastic constants as variables in a multidimensional space.

The measurements have been done in a novel type of ultrasonic goniometer, which is distinguished by the possi- bility of making simultaneous measurements of doubly reflected and doubly transmitted ultrasonic signals using only one transducer. Using the experimental apparatus, a computer-controlled recording of the reflected signal may be taken as a function of incident angle with resolution of 0.01ø.

ACKNOWLEDGMENTS

This work was, in part, sponsored by the Center for Ad- vanced NDE, operated by the Ames Laboratory, USDOE, for the Air Force Wright Patterson Aeronautical Laborato- ries under Contract No. SC-88-148B. The assistance of L.

Wang with calculations is also appreciated.

•W. G. Mayer, "Determination of ultrasonic velocities by measurement of angles of total reflection," J. Acoust. Soc. Am. 32, 1213-1215 (1960).

2L. S. Fountain, "Experimental evaluation of the total-reflection method of determining ultrasonic velocity," J. Acoust. Soc. Am. 42, 242-247 (1966).

3E. G. Henneke II and G. L. Jones, "Critical angle for reflection at a liquid- solid interface in single crystals," J. Acoust. Soc. Am. 59, 204-205 (1976).

4H. Enfan, K. A. Ingerbrigtsen, and A. Tonning, "Elastic surface wave in a-quartz: Observation of leaky surface waves," Phys. Lett. 10, 311-313 (1967).

5F. R. Rollins, Jr., T. C. Lim, and G. W. Farnell, "Ultrasonic reflectivity and surface wave phenomenon on surfaces of copper single crystals," Appl. Phys. Lett. 12, 236-238 (1968).

60. I. Diachok, R. J. Hallermier, and W. G. Mayer, "Measurement of ultrasonic surface wave velocity and absorptivity on single-crystal copper," Appl. Phys. Lett. 17, 288-289 (1970).

7S. I. Rokhlin and W. Wang, "Ultrasonic evaluation of in-plane and out- plane elastic properties of composite materials," in Review of Progress in Quantitative Nondestructive Evaluation, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1989), Vol. 8B, pp. 1489-1496.



aM. F. Markham, "Measurements of the elastic constants of fiber composites by ultrasonics," Composites 1, 145 (1970).

9R. E. Smith, "Ultrasonic elastic constants of carbon fibers and their composites," J. Appl. Phys. 43, 2555-2562 (1972).

•oj. H. Gieske and R. E. Allred, "Elastic constants of B-A 1 composites by ultrasonic velocity measurements," Exp. Mech. 14, 158-165 (1974).

•R. D. Kriz and W. W. Stinchomb, "Elastic moduli of transversely isotropic graphite fibers and their composites," Exp. Mech. 19, 41-49 (1979).

•2L. H. Pearson and W. J. Murri, "Measurement of ultrasonic wavespeeds in off-axis directions of composite materials," in Review of Progress in Quantitative Nondestructive Evaluation, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1987), Vol. 6B, pp. 1093-1101.

13B. Hosten, M. Deschamps, and B. R. Tittman, "Inhomogeneous wave generation and propagation in lossy anisotropic solids. Application to the characterization ofviscoelastic composite materials," J. Acoust. Soc. Am. 82, 1763-1770 (1987).

•4R. A. Kline and Z. T. Chen, "Ultrasonic technique for global anisotropic property measurement in composite materials," Mater. Eval. 46, 986-992 (1988).

•SE. G. Henneke II, "Reflection-refraction of a stress wave at a plane boundary between anisotropic media," J. Acoust. Soc. Am. 51, 210-217 (1972).

•6S. I. Rokhlin, T. K. Bolland, and L. Adler, "Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media," J. Acoust. Soc. Am. 79, 906-918 (1986).

•7S. I. Rokhlin, T. K. Bolland, and L. Adler, "Effects on reflection and refraction of ultrasonic waves on the angle beam inspection of anisotropic composite materials," in Review of Progress in Quantitative Nondestruc- tive Evaluation, edited by D. O. Thompson and D. E. Chimenti (Plenum, New York, 1987), Vol. 6B, pp. 1103-1110.

laT. D. K. Ngoc and W. G. Mayer, "Ultrasonic nonspecular reflectivity near longitudinal critical angle," J. Appl. Phys. 50, 7948-7951 (1979).



critical angle measurement of elastic constants in composite material

Documents