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The propagation of ultrasonic waves in CFRP laminates
The propagation of ultrasonic waves in CFRP laminates W. N. REYNOLDS and S. J. WILKINSON
A complete survey is presented of studies of ultrasonic wave propagation in flat carbon fibre reinforced plastic (CFRP) laminates. After a summary of earlier work on uniaxial specimens, results are given for two and three-ply laminates and sandwich structures, with the current extent of interpretation. Velocities measured perpendicular to fibre directions show an interesting dependence on porosity of the matrix and the measuring technique used, which could lead to wider exploitation.
Introduction
Early experiments
The outstanding technical advantage of carbon fibres lies in the high ratio of elastic modulus E to density p which they exhibit. Since this ratio is directly measured by the propagation velocity V of acoustic waves, it has been evident from the outset that precise measurements of the velocity of ultrasonic pulses would be of great interest in the non-destructive inspection and quality control of carbon fibre reinforced materials. This is quite apart from the use of ultrasonics in the search for defects, which in these heterogeneous anisotropic media is itself a subject of special interest.
The results of the early work on CFRP were given by Curtis. s In an experiment designed to elucidate the transition from rod to bulk waves, he measured the group velocity of a short pulse in the frequency range 0.5-1.25 MHz as a function of d/k, the diameter/wavelength ratio. As is well known, for isotropic materials there is a sharp increase from the rod to the bulk value as d/~ approaches unity. For uniaxial CFRP there was no effect in this region, although there was some evidence of a much smaller effect of a similar kind at a value of d / ~ ~ 0.2.
Nevertheless, experimental investigations so far described in the literature are incomplete and in some cases even misleading. Thus direct correlations between V2p and E for laminates as measured in various materials and in various directions completely ignore the theoretical basis provided notably by Musgrave 1 for the study of wave propagation in anisotropic crystals where the wavelengths are much smaller than the specimen dimensions. The importance of this approach has been emphasized by Markham, 2 Smith 3 and Dean and Turner, 4 who have used the technique of measuring the time of flight of a short pulse between two transducers in a water bath, the specimen being mounted between them at various orientations.
The reasons for this behaviour have been discussed elsewhere& For isotropic materials the jump in velocity corresponds to a change in the condition p V z = E for rod waves to
E(1 - ' )
pV 2 =
(1 +")(1
--
2V)
for bulk waves. If the Poisson's ratio v = 0.3, this corresponds to an increase of about 35%. For a uniaxial anisotropic material; with the '3' axis the axis of symmetry, the corresponding transition is from p V 2 = E 1/$33 to P V2 = 6"33- From the general relations between the compliances and stiffnesses, it can easily be shown that =
In this paper it is proposed to summarize previous AERE work on ultrasonic propagation in uniaxial materials, including the current interpretation of the effects observed, and then to present some new results obtained on various flat panels of cross-ply laminated construction. The experimental technique most commonly used was the direct timing of a pulse passing between a separate transmitter and receiver across a flat disc, but both reflection goniometry and the National Physical Laboratory (NPL) immersion bath technique have also been employed. The authors are at the Nondestructive Testing Centre, Atomic Energy Research Establishment, Harwell, Didcot, Berkshire, UK Paper received 10 December 1973.
ULTRASONICS. MAY 1974
1
2C23 -
Saa
6"33
Cll + C12
and Cl3
Cll + C12
Sl3
- v the (axial) Poisson's ratio
$33
109
These are particular instances of the general relationships Ac As s i / - A c q ' cq - Asq where Ac, As are the determinant arrays of the stiffnesses and compliances and Acij, Asij are the corresponding minors. Experimental values of ~, reported in the literature are widely scattered, between 0.2 and 0.6, but it is clear that 1
g =
$33
__~ C33 - C13
apparent moduli which lay between the Musgrave and Young's modulus values. This was rather surprising, although similar results have in fact been reported for other fibrereinforced materials. It was felt, however, that the use of non-axially symmetric specimens introduced possibilities of internal refractions, reflections and mode changes and that other techniques and specimen shapes should be investigated. In this connexion the development of a practical motorized reflection goniometer 7 was of great interest. In principle this device permits the measurement of the velocities of the L, Tl and T2 waves on the selected plane of reflection. In practice it has not been found possible to cover the complete range of orientations but a number of important results have been obtained.
Uniaxial discs For a 50% type I uniaxial material, 6"33 ~ 200 GNm "2 and C13 "~ 7 GNm "2, so t h a t E differs from C33 by only a few per cent, and the velocity of bulk waves differs from that of rod waves by less than 2%. Furthermore, results to be discussed below suggest that the value of X effective in the criterion for the transition is that of shear waves travelling in a direction transverse to the fibres, which is only about one fifth of the longitudinal value. In terms of the longitudinal wavelength, the transition should therefore be very small and occur at d/X "~ 0.2, which is quite consistent with the data published, s The next problem was to attempt to verify the theory of Musgrave for the propagation of waves in anisotropic media for the case of CFRP. The equations for the velocity of propagation of the L, T2 and TI waves at an angle ¢ to the fibre-reinforcement direction are
P~L' T2 = (?44 + ~-
-
4m2n 2 (ah - d2)] w }
PV~l = n2C44 + where:
m
=
m2a + n2h +- [(m2a + n2h2) 2
CII - C12 ) m 2
sin
n = cos ~b a = e l l - C44 d = C 13 "t"C44 h = (?33 - C44 The T2 wave is polarized on the plane of rotation and the TI wave at right angles. If the stiffnesses are converted to compliances, the rotated elastic moduli E ' may be found from 1
E ' - $33n4 + (2S13 + $44) n2m2 + SI t m 4 with corresponding expressions for the rotated shear moduli. Curtis s cut rectangular rod specimens at different angles to the fibre direction in uniaxial material and obtained
110
The foregoing developments led to the idea that a more decisive test could be obtained on a disc-shaped specimen cut from flat uniaxial sheet. Velocities at any angle to the fibre axis could then be measured by direct contact across the appropriate diameter and the possible effects of internal reflections eliminated by comparison with measurements on shorter non-diametral chords with the pulse travelling in either direction. The same specimen could be used without modification in a goniometer, and also in an NPL immersion bath so that ideally there would be at least three independent measurements of each velocity at each chosen angle. The results obtained have been described elsewhere by the present authors s and in a report by Elliott 9 on the goniomerry. They can be summarized by reference to Fig.1. In general, the shear wave velocities measured were entirely consistent with the Musgrave model, although the different methods did not cover the entire theoretical range. For the Tz curve, polarized in the plane of the disc, results were obtained from about 10 to 70 degrees to the fibre axis by the immersion bath, from about 30 to 80 degrees by direct contact to the disc periphery, and from 0 to 80 degrees by the goniometer. No method gave results in the 80 to 90 degree range where the velocity is decreasing rapidly. Results at 0 to 90 degrees could be obtained by the use of shear wave probes in direct contact. For the T1 wave, polarized normal to the disc, data at 0 to 90 degrees was obtained by direct contact and points in the range 0 to 30 degrees by the goniometer. Some points in the range 60 to 70 degrees were also obtained in the immersion bath. This range can be extended by tilting the specimen about two axes simultaneously, but as the results yield no new information this method was not pursued. The most interesting results were obtained from measurements of the L wave. For non-axial directions this is a quasi-compressive wave for which the particle displacement and wave velocity vectors are not parallel. Its behaviour in accordance with theory was most clearly demonstrated by the goniometer, with measurements extending from 90 degrees down to about 40 degrees, or to about 70 degrees by the immersion-transmission method. The results from the direct contact method were, however, completely at variance with the theory for non-axial directions. As has been explained elsewhere 8 on the basis of these and other experiments, it is inferred that the L wave is attenuated by multiple reflections at the carbon-resin interfaces, and that this attenuation is very strong for angles above the critical
ULTRASONICS
. MAY
1974
4-0
about 0.24 mm to the thickness of the finished specimen after curing and machining to size with the removal of potentially resin-rich surfaces. Two types of specimen were made - one in which each layer of the laminate was laid at 60 or 90 degrees to the previous layer and one in which all the layers in a given direction were laid together. These were called respectively the interleaved and sandwich specimens. The important difference is that for the former the layer thickness is much less, and for the latter much greater than some critical wavelength. For propagation on the laminate plane, the two types of specimen gave entirely different results.
Theory • Direct c o n t a c t o Immersion ~onk • Shear wave probes Goniometer
T',n
~> 3-0 O
o *" 2 . 0 u
o
Measurements
o n t h e p l a n e o f t h e disc
>
I0
I IO
0
a
I 20
I 30
I 40
I 50
I 60
I 70
I 80
90
A n g l e o f p r o p a g a t i o n t o fibre axis [ degrees] 12 w • o A
II .--, ion
IO
Theory Direct c o n t a c t Immersiontonk Goniomcter
9 8
The velocity of compressional waves was measured as a function of direction on the plane of each disc. The results are given in Fig.3. For the uniaxial case the velocity falls from just under 12 km s"1 parallel to the fibres to just over 2.5 km s"1 at right angles. For the sandwich specimens, as might be expected, the velocity parallel to any fibre direction is about the same - the small differences between specimens being attributable to differences in volume fraction of fibres. For the 0•90 degree sandwich the minimum value at 45 degrees is just under 8 km s"1 and for the 0/60/120 degree sandwich at 30 degrees it is just under 10 km s-1 . For the interleaved specimens the results are lower, falling from 8 to 6 km s"1 for the 0/90 degree material and remaining almost independent of direction at just over 7 km s"1 for the 0[60/120 degree material.
.13
o
6S
-
•
•
3
21 O
b
•
I IO
I 20
I 30
I 40
I 50
•
I 60
°
Oo
•
•
I 70
q
1 80
90
Angle o f p r o p o g G t i o n t o f i b r e axis [ d e g r e e s ]
a -- Velocities of propagation of shear waves as a function Fig.1 of angle of propagation to fibre axis in uniaxial material. The T 1 and T 2 waves are polarized respectively perpendicular and on the plane of rotation; b -- velocity of compressional wave as a function of angle to fibre axis as measured by different techniques
In the light of the work on uniaxial material, the behaviour of the sandwich specimens is not difficult to understand. For the 0[90 degree case the shortest path (in time) across a given diameter consists of two parts, each parallel to one set of fibres, and therefore involving scattering through a right angle at a point on the surface. The ratio of maximum to minimum transit time is therefore 1.4:1 and this explanation fits the upper curve of Fig.3b exactly. In the 0/60/120 degree sandwich, the transit time across diameters between fibre directions is even shorter as favourable fibre directions can be found to cover most of the path, leaving only a short section to be covered by shear waves after conversion. For the interleaved laminates, the velocity parallel to a fibre direction or along a line of symmetry such as the
F4
for ideally total reflection. Apparant propagation in these directions is in fact an indirect effect resulting from shear wave transfer in the lateral direction of compressive wave energy travelling in the fibre direction. This is illustrated in Fig.2. The use of CFRP as a converter of compressive waves to shear or vice-versa is discussed in the appendix.
,O
F2
FI New
B
Fibre oxis
-~
experiments
In the light of the principles elucidated by the study of uniaxial materials, it is now possible to discuss experiments by different techniques on two and three-ply laminates and to interpret the results obtained. Discs of these materials were made from commercial prepreg sheet, the curing conditions being such that each layer of prepreg contributed
ULTRASONICS.
MAY
1974
Fig.2 Propagation of compressive waves in uniaxial CFRP. The effective wavefront of the pseudo-L wave is given by F 1 F 2 F3 F4. Points such as P on this f r o n t can be reached by any path between OAP and OBP, where the transit time is composed of a compressive wave path parallel t o the fibres and a shear wave path at 80 degrees. A transducer such as T 2 eventually picks up a compressive wave, whereas 7"3 situated on the long side receives a shear wave
111
12
12, O(
o
'o
o oo O
I0
O
o
Itn
8
o
0
O
0
OO
Sandwich o q ooo
O
000000
O
0
'
Interleaved
~04~d~=~-C~d~
uuuuO
uOUO
u O'O~n
o ~oO
Theoretical
4--
O OO
o A 9.O.O.
6
u 0
0
O
4
>-
0
O
O
000000
8
5
6
>
=m
0
5
--
IC
O
OOOOOoooOooOOOOO
2
2-
C
I 20
, a
I 40
I 60
I 80
I IOO
Ar~l¢ of propagation
I 120
I 140
C
I 160
180
[degrees]
C
12 o I0
_ tao
Sandwich o
- 0
0
[ 20
I 40
I 60
I 80
1 I00
Angle o f p r o p a g a t i o n
[ 120
I 140
[ t60
180
[degrees]
Fig.3 Velocities of propagation of compressional waves as a function of angle to fibre axis, measured by direct contact to a disc: a -- uniaxial material; b -- 0 / 9 0 degree sandwich and interleaved specimens; c -- 0 / 6 0 / 1 2 0 degree sandwich and interleaved specimens
8 E >.
6
Interleaved
u O
from which V = 7.4 km s"1 , again in good agreement with experiment.
Theoretical
4 2
M e a s u r e m e n t s on the f l a t sides o f t h e disc I 20
b
I 40
I 60
I 80
I I00
I 120
I 140
160
180
Angle of propagation [ degrees ]
45 degree line in the 0/90 degree material is governed by the stiffness of the material in the corresponding direction which is always greater than the Young's modulus. In terms of the uniaxial values 1
1
C0, 90° = s- Cll + s- C33 2 2 C45 o = Cll cos4 45° + C33 sin4 45° + (4C44 + 2C13) sin2 45 ° cos2 45 ° 1
--
4
(Cll + C33 +4C44 + 2C13)
This leads to theoretical estimates, based on the known values for uniaxial material, 8 of F0,9o° = 8.1 km s"1 and F4so = 6.2 km s"1 , in good agreement with the lower curve of Fig.3b. For values at smaller angles to the fibre axes it would be necessary to extend the Musgrave theory of wave propagation to cases of tetragonal symmetry. As this has not yet been done, values of sufficient accuracy can be obtained from the mean values of stiffness constants for the two layers in the required direction. For the interleaved 0/60/120 degree material, the stiffness on the plane is independent of direction and in terms of uniaxial values is given by 3 C =-
112
8
1 (Cll
-I-C33) -I-
(2C44 +C13)
Goniometer measurements by reflection from the flat sides of the discs give information about the velocities of propagation of waves travelling in the plane of the disc. However, the mechanics of the reflection process are such that this information applies only to that region of the material within a wavelength of the surface, the wavelength in question referring in this case to compressional waves travelling normally to the surface. It has been verified experimentally that when this wavelength is less than the thickness of a single layer of prepreg in the composite (0.24 mm) the reflections are essentially the same as from uniaxial materials. If the wavelength used is larger than this no clearly identifiable reflections are obtained. Transmission measurements
In many ways the measurements of compressional wave velocity perpendicular to the planes of the discs have proved to be the most interesting, for reasons which were not appreciated when the work was begun. It has, however, been pointed out by Dean and Turner 4 that the velocity measured in this direction is affected by the attenuation and dispersion arising from distributed porosity. The results for our specimen are summarized in Table 1 which gives the compressive wave velocity obtained by various techniques, as well as the velocities in unreinforced resins and in the fibres for the transverse direction calculated from the data of Dean and Turner. The average porosity of the specimens was found by measurement of the densities and of the proportion of fibre remaining after chemical removal of the resin in a destructive test on an off-cut sample from each specimen. This is expressed as a volume fraction of the resin constituent of each sample. The important features shown in the table are: 1. For the specimen Ul of low porosity there is good agreement between all the velocity measurements at different points, in different directions and by different methods.
ULTRASONICS.
MAY
1974
Table 1.
Ultrasonic wave velocities measured perpendicular to fibres in flat discs Velocities [km S"1 ]
Width
Thickness Resin porosity
Specimen
Type
[%]
Goniometer
Transmission
Goniometer
Transmission
U
Uniaxial
< 1
2.62
2.59 -+ 0.03
2.64 +- 0.05
2.62
5
Uniaxial
24 -+ 4
2.71 -+0.11
1.94-+0.02
2.81 -+0.22
2.31
6
Uniaxial
25.7 -+ 0.2
2 . 3 6 -+ 0.20
1.93 -+ 0.05
2.80 -+ 0.20
2.18
7
Uniaxial
19 -+ 3
2.24 -+ 0.02
1.94 +- 0.02
2.52 -+ 0.17
2.25
1
0/90 ° sandwich
14 -+ 5
--
2.56 -+ 0.04
2.74 -+ 0.06
--
2
0/90 ° laminate
10
--
2.38 + 0.04
2.80 + 0.10
-
3
0 / 6 0 / 1 2 0 ° laminate
23
-
1.99 +- 0.03
2.77 + 0.18
-
Velocities in pure resin: 2.94 km s"1 ; type 1 fibre: 2.47 km s"1
2. All the other specimens show a considerable amount of porosity in the resin matrix. In these specimens the transmission velocity measured by direct contact is lower than that in the fibres and therefore outside the range between that and the resin value. The thickness value is lower than that in the width direction for all the uniaxial porous specimens. 3. Goniometer values, although showing a greater spread than the transmission values, are consistently higher. In the width direction they are close to the value for the nonporous specimens. Discussion
Study of the results presented in Table 1 suggests that several important factors are involved in producing the low velocities in porous specimens. In the first place, of course, the difference between the thickness and width values obtained by direct contact suggests that there is less discontinuity in the structure in the latter case than in the former. This would be explained by the type of porosity produced, for example, by the trapping of air or gas between layers of prepreg in the initial lay-up. The second factor is the straightforward decrease in elastic stiffness of the composite due to the effective decrease in the elastic stiffness of the porous resin matrix. Theoretical estimates for this may be made on the basis of the theories of Hashin and Shtrikman lo for isotropic spherical inclusions in an isotropic matrix, and then for the transverse stiffness constant of the composite in terms of the weakened matrix using the theory of Hashin. 11 Appropriate allowance for the decreased density is then made, and the velocity of the transverse L wave calculated. The value obtained is in fact an upper limit and lower values would be obtained for special non-uniform distributions of the porosity, but the calculations are still of considerable interest, as illustrated in Fig.4. This shows that for a non-porous composite, the velocity should be between 2.60 and 2.67 km s"z , and that the value at 30% resin porosity falls to about 2.25 km s"1 . It is notable that over most of the range the dependence on fibre/resin ratio is slight. Velocities obtained from the width measurements are in better agreement with
U L T R A S O N I C S . M A Y 1974
the calculations than those obtained from thickness measurements. There is, however, considerable uncertainty in the porosity estimation by current techniques and, as mentioned above, non-uniform distributions will produce values lower than those calculated. There is also a further important point. The significant differences shown in Table 1 between the velocity measurements in the same direction by the transit time and goniometer techniques for porous specimens lends support to the process of preferential attenuation of the higher frequency components suggested by Dean and Turner. 4 The critical angle method is much less influenced by this effect and therefore yields a higher velocity in each case. A more detailed comparison of the two methods over a range of frequencies would be of great interest in the more precise characterization of the structure.
~40°/o'~x Fibre/resin 2 6 ~ , ~ . ~ 5 0 O/o"r~ by volume I ~60Cyo ~ o 2.51 ~ ~
>. 2-3 u o
Experimental velocities o Thickness A Width
A
60%
2.2
.
>
~ 2. I
40% Theoretical maxima
2-0 I-9
~soo/o
o o
o
I
I
O-IO
0.20
o
I
0-30
Resin void fraction by volume Fig.4 Comparison of compressive wave velocities perpendicular to fibres, as measured by direct contact, with theoretical upper limits for material with different matrix porosity. For uniaxial specimens the experimental values for width and thickness directions differ
113
iI CFRP
Resin specimen
:::::::::::::::::::/::::::::.:::
iii!iiiiiiiiiiiiiiiiiiiiiiiiii
9
i!!ii!!!iiiiiiiiiiiiiiii 10 Fig.5 Use of compressional transducers X and Y to measure the shear wave velocity in a resin specimen by the employment of t w o uniaxial CFRP strips as converters
Conclusions
It is concluded that the propagation of ultrasonic waves in the plane of a CFRP laminate is now adequately understood in that theoretical estimates can be made and checked by various techniques. Standard quantitative tests for production lines can therefore be devised to check the distribution and orientation of fibres in a resin matrix. Propagation normal to the laminate plane has raised questions of wider interest, and further studies of scattering, attenuatior and dispersion by distributed voidage would be of interest. In the meantime, the sensitivity of pulse velocity measurements to porosity has been demonstrated, and practical field equipment (to be described elsewhere) has been developed to scan laminates in terms of this velocity and to record the results as an indication of mean voidage distribution. This approach has obvious theoretical advantages over the conventional use of attenuation techniques, and can now be evaluated directly as a non-destructive test of shear strength. References
Musgrave, M. J. P. On the propagation of elastic waves in aeleotropic media: II media of hexagonal symmetry, Proceedings of the Royal Society 226 (1954) 356
114
11 12
Markham, M.F. Measurement of the elastic constants of fibre composites by ultrasonics, Composites 1 (1970) 145 Smith, R. E. Ultrasonic elastic constants of carbon fibres and their composites, Journal o f Applied Physics 43 (1972) 2555 Dean, G. D., Turner, P. The elastic properties of carbon fibres and their composites, Composites 4 (1973) 174 Curtis, G. Harwell equipment for materials evaluation by ultrasonics, Ultrasonics for Industry, IPC Science and Technology Press (1969) Reynolds, W. N. Problems of nondestructive testing in carbon fibres and their composites, International Conference on Carbon Fibres, London (1971) The Plastics Institute Curtis, G., Richards, R., Berry, J. A. The Harwell ultrasonic goniometer, unpublished information Wilkinson, S. J., Reynolds, W. N. The propagation of ultrasonic waves in carbon fibre reinforced plastics, Journal o f Physics D: Applied Physics 7 (1974) 50 EIliott, J. G. An investigation of ultrasonic goniometry methods applied to carbon fibre composite materials, AERE report NDT 64 (1973) Hashin, Z., Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials, Journal o f the Mechanics and Physics of Solids 11 (1963) 127 Hashin, Z. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry, Journal o f the Mechanics and Physics o f Solids 13 (1965) 119 Smith, A., Wilkinson, S. J., Reynolds, W. N. The elastic constants of some epoxy resins, to be published in Journal o f Materials Science (1974)
Appendix Use of uniaxial CFRP strip as an ultrasonic wave converter
The mixed type of compressive-shear wave propagation illustrated in Fig.2 shows that a strip of uniaxial material can be used to convert compressive waves to shear or viceversa, as shown in Fig.5. The compressional transducers can be used to measure the shear modulus of the specimen because a shear wave is emitted in a direction normal to the converter at the point of contact. A single converter strip could be used for example to monitor shear wave acoustic emission signals with a compressional transducer as receiver. In practice it is often found that the converter-specimen interface causes some reconversion so that there is an interfering compressional wave masking the shear wave. This difficulty can be overcome by using a shear transducer, such as a y-cut quartz crystal, as receiver. This system is still much more sensitive than using two shear transducers, and has been used elsewhere 12 to measure the ultrasonic shear moduli of resin samples over 35 m m in length.
ULTRASONICS.
MAY
1974
A complete survey is presented of studies of ultrasonic wave propagation in flat carbon fibre reinforced plastic (CFRP) laminates. After a summary of earlier work on uniaxial specimens, results are given for two and three-ply laminates and sandwich structures, with the current extent of interpretation. Velocities measured perpendicular to fibre directions show an interesting dependence on porosity of the matrix and the measuring technique used, which could lead to wider exploitation.
Introduction
Early experiments
The outstanding technical advantage of carbon fibres lies in the high ratio of elastic modulus E to density p which they exhibit. Since this ratio is directly measured by the propagation velocity V of acoustic waves, it has been evident from the outset that precise measurements of the velocity of ultrasonic pulses would be of great interest in the non-destructive inspection and quality control of carbon fibre reinforced materials. This is quite apart from the use of ultrasonics in the search for defects, which in these heterogeneous anisotropic media is itself a subject of special interest.
The results of the early work on CFRP were given by Curtis. s In an experiment designed to elucidate the transition from rod to bulk waves, he measured the group velocity of a short pulse in the frequency range 0.5-1.25 MHz as a function of d/k, the diameter/wavelength ratio. As is well known, for isotropic materials there is a sharp increase from the rod to the bulk value as d/~ approaches unity. For uniaxial CFRP there was no effect in this region, although there was some evidence of a much smaller effect of a similar kind at a value of d / ~ ~ 0.2.
Nevertheless, experimental investigations so far described in the literature are incomplete and in some cases even misleading. Thus direct correlations between V2p and E for laminates as measured in various materials and in various directions completely ignore the theoretical basis provided notably by Musgrave 1 for the study of wave propagation in anisotropic crystals where the wavelengths are much smaller than the specimen dimensions. The importance of this approach has been emphasized by Markham, 2 Smith 3 and Dean and Turner, 4 who have used the technique of measuring the time of flight of a short pulse between two transducers in a water bath, the specimen being mounted between them at various orientations.
The reasons for this behaviour have been discussed elsewhere& For isotropic materials the jump in velocity corresponds to a change in the condition p V z = E for rod waves to
E(1 - ' )
pV 2 =
(1 +")(1
--
2V)
for bulk waves. If the Poisson's ratio v = 0.3, this corresponds to an increase of about 35%. For a uniaxial anisotropic material; with the '3' axis the axis of symmetry, the corresponding transition is from p V 2 = E 1/$33 to P V2 = 6"33- From the general relations between the compliances and stiffnesses, it can easily be shown that =
In this paper it is proposed to summarize previous AERE work on ultrasonic propagation in uniaxial materials, including the current interpretation of the effects observed, and then to present some new results obtained on various flat panels of cross-ply laminated construction. The experimental technique most commonly used was the direct timing of a pulse passing between a separate transmitter and receiver across a flat disc, but both reflection goniometry and the National Physical Laboratory (NPL) immersion bath technique have also been employed. The authors are at the Nondestructive Testing Centre, Atomic Energy Research Establishment, Harwell, Didcot, Berkshire, UK Paper received 10 December 1973.
ULTRASONICS. MAY 1974
1
2C23 -
Saa
6"33
Cll + C12
and Cl3
Cll + C12
Sl3
- v the (axial) Poisson's ratio
$33
109
These are particular instances of the general relationships Ac As s i / - A c q ' cq - Asq where Ac, As are the determinant arrays of the stiffnesses and compliances and Acij, Asij are the corresponding minors. Experimental values of ~, reported in the literature are widely scattered, between 0.2 and 0.6, but it is clear that 1
g =
$33
__~ C33 - C13
apparent moduli which lay between the Musgrave and Young's modulus values. This was rather surprising, although similar results have in fact been reported for other fibrereinforced materials. It was felt, however, that the use of non-axially symmetric specimens introduced possibilities of internal refractions, reflections and mode changes and that other techniques and specimen shapes should be investigated. In this connexion the development of a practical motorized reflection goniometer 7 was of great interest. In principle this device permits the measurement of the velocities of the L, Tl and T2 waves on the selected plane of reflection. In practice it has not been found possible to cover the complete range of orientations but a number of important results have been obtained.
Uniaxial discs For a 50% type I uniaxial material, 6"33 ~ 200 GNm "2 and C13 "~ 7 GNm "2, so t h a t E differs from C33 by only a few per cent, and the velocity of bulk waves differs from that of rod waves by less than 2%. Furthermore, results to be discussed below suggest that the value of X effective in the criterion for the transition is that of shear waves travelling in a direction transverse to the fibres, which is only about one fifth of the longitudinal value. In terms of the longitudinal wavelength, the transition should therefore be very small and occur at d/X "~ 0.2, which is quite consistent with the data published, s The next problem was to attempt to verify the theory of Musgrave for the propagation of waves in anisotropic media for the case of CFRP. The equations for the velocity of propagation of the L, T2 and TI waves at an angle ¢ to the fibre-reinforcement direction are
P~L' T2 = (?44 + ~-
-
4m2n 2 (ah - d2)] w }
PV~l = n2C44 + where:
m
=
m2a + n2h +- [(m2a + n2h2) 2
CII - C12 ) m 2
sin
n = cos ~b a = e l l - C44 d = C 13 "t"C44 h = (?33 - C44 The T2 wave is polarized on the plane of rotation and the TI wave at right angles. If the stiffnesses are converted to compliances, the rotated elastic moduli E ' may be found from 1
E ' - $33n4 + (2S13 + $44) n2m2 + SI t m 4 with corresponding expressions for the rotated shear moduli. Curtis s cut rectangular rod specimens at different angles to the fibre direction in uniaxial material and obtained
110
The foregoing developments led to the idea that a more decisive test could be obtained on a disc-shaped specimen cut from flat uniaxial sheet. Velocities at any angle to the fibre axis could then be measured by direct contact across the appropriate diameter and the possible effects of internal reflections eliminated by comparison with measurements on shorter non-diametral chords with the pulse travelling in either direction. The same specimen could be used without modification in a goniometer, and also in an NPL immersion bath so that ideally there would be at least three independent measurements of each velocity at each chosen angle. The results obtained have been described elsewhere by the present authors s and in a report by Elliott 9 on the goniomerry. They can be summarized by reference to Fig.1. In general, the shear wave velocities measured were entirely consistent with the Musgrave model, although the different methods did not cover the entire theoretical range. For the Tz curve, polarized in the plane of the disc, results were obtained from about 10 to 70 degrees to the fibre axis by the immersion bath, from about 30 to 80 degrees by direct contact to the disc periphery, and from 0 to 80 degrees by the goniometer. No method gave results in the 80 to 90 degree range where the velocity is decreasing rapidly. Results at 0 to 90 degrees could be obtained by the use of shear wave probes in direct contact. For the T1 wave, polarized normal to the disc, data at 0 to 90 degrees was obtained by direct contact and points in the range 0 to 30 degrees by the goniometer. Some points in the range 60 to 70 degrees were also obtained in the immersion bath. This range can be extended by tilting the specimen about two axes simultaneously, but as the results yield no new information this method was not pursued. The most interesting results were obtained from measurements of the L wave. For non-axial directions this is a quasi-compressive wave for which the particle displacement and wave velocity vectors are not parallel. Its behaviour in accordance with theory was most clearly demonstrated by the goniometer, with measurements extending from 90 degrees down to about 40 degrees, or to about 70 degrees by the immersion-transmission method. The results from the direct contact method were, however, completely at variance with the theory for non-axial directions. As has been explained elsewhere 8 on the basis of these and other experiments, it is inferred that the L wave is attenuated by multiple reflections at the carbon-resin interfaces, and that this attenuation is very strong for angles above the critical
ULTRASONICS
. MAY
1974
4-0
about 0.24 mm to the thickness of the finished specimen after curing and machining to size with the removal of potentially resin-rich surfaces. Two types of specimen were made - one in which each layer of the laminate was laid at 60 or 90 degrees to the previous layer and one in which all the layers in a given direction were laid together. These were called respectively the interleaved and sandwich specimens. The important difference is that for the former the layer thickness is much less, and for the latter much greater than some critical wavelength. For propagation on the laminate plane, the two types of specimen gave entirely different results.
Theory • Direct c o n t a c t o Immersion ~onk • Shear wave probes Goniometer
T',n
~> 3-0 O
o *" 2 . 0 u
o
Measurements
o n t h e p l a n e o f t h e disc
>
I0
I IO
0
a
I 20
I 30
I 40
I 50
I 60
I 70
I 80
90
A n g l e o f p r o p a g a t i o n t o fibre axis [ degrees] 12 w • o A
II .--, ion
IO
Theory Direct c o n t a c t Immersiontonk Goniomcter
9 8
The velocity of compressional waves was measured as a function of direction on the plane of each disc. The results are given in Fig.3. For the uniaxial case the velocity falls from just under 12 km s"1 parallel to the fibres to just over 2.5 km s"1 at right angles. For the sandwich specimens, as might be expected, the velocity parallel to any fibre direction is about the same - the small differences between specimens being attributable to differences in volume fraction of fibres. For the 0•90 degree sandwich the minimum value at 45 degrees is just under 8 km s"1 and for the 0/60/120 degree sandwich at 30 degrees it is just under 10 km s-1 . For the interleaved specimens the results are lower, falling from 8 to 6 km s"1 for the 0/90 degree material and remaining almost independent of direction at just over 7 km s"1 for the 0[60/120 degree material.
.13
o
6S
-
•
•
3
21 O
b
•
I IO
I 20
I 30
I 40
I 50
•
I 60
°
Oo
•
•
I 70
q
1 80
90
Angle o f p r o p o g G t i o n t o f i b r e axis [ d e g r e e s ]
a -- Velocities of propagation of shear waves as a function Fig.1 of angle of propagation to fibre axis in uniaxial material. The T 1 and T 2 waves are polarized respectively perpendicular and on the plane of rotation; b -- velocity of compressional wave as a function of angle to fibre axis as measured by different techniques
In the light of the work on uniaxial material, the behaviour of the sandwich specimens is not difficult to understand. For the 0[90 degree case the shortest path (in time) across a given diameter consists of two parts, each parallel to one set of fibres, and therefore involving scattering through a right angle at a point on the surface. The ratio of maximum to minimum transit time is therefore 1.4:1 and this explanation fits the upper curve of Fig.3b exactly. In the 0/60/120 degree sandwich, the transit time across diameters between fibre directions is even shorter as favourable fibre directions can be found to cover most of the path, leaving only a short section to be covered by shear waves after conversion. For the interleaved laminates, the velocity parallel to a fibre direction or along a line of symmetry such as the
F4
for ideally total reflection. Apparant propagation in these directions is in fact an indirect effect resulting from shear wave transfer in the lateral direction of compressive wave energy travelling in the fibre direction. This is illustrated in Fig.2. The use of CFRP as a converter of compressive waves to shear or vice-versa is discussed in the appendix.
,O
F2
FI New
B
Fibre oxis
-~
experiments
In the light of the principles elucidated by the study of uniaxial materials, it is now possible to discuss experiments by different techniques on two and three-ply laminates and to interpret the results obtained. Discs of these materials were made from commercial prepreg sheet, the curing conditions being such that each layer of prepreg contributed
ULTRASONICS.
MAY
1974
Fig.2 Propagation of compressive waves in uniaxial CFRP. The effective wavefront of the pseudo-L wave is given by F 1 F 2 F3 F4. Points such as P on this f r o n t can be reached by any path between OAP and OBP, where the transit time is composed of a compressive wave path parallel t o the fibres and a shear wave path at 80 degrees. A transducer such as T 2 eventually picks up a compressive wave, whereas 7"3 situated on the long side receives a shear wave
111
12
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o
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8
o
0
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OO
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'
Interleaved
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u O'O~n
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Theoretical
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6
u 0
0
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4
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0
O
O
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8
5
6
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=m
0
5
--
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O
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2
2-
C
I 20
, a
I 40
I 60
I 80
I IOO
Ar~l¢ of propagation
I 120
I 140
C
I 160
180
[degrees]
C
12 o I0
_ tao
Sandwich o
- 0
0
[ 20
I 40
I 60
I 80
1 I00
Angle o f p r o p a g a t i o n
[ 120
I 140
[ t60
180
[degrees]
Fig.3 Velocities of propagation of compressional waves as a function of angle to fibre axis, measured by direct contact to a disc: a -- uniaxial material; b -- 0 / 9 0 degree sandwich and interleaved specimens; c -- 0 / 6 0 / 1 2 0 degree sandwich and interleaved specimens
8 E >.
6
Interleaved
u O
from which V = 7.4 km s"1 , again in good agreement with experiment.
Theoretical
4 2
M e a s u r e m e n t s on the f l a t sides o f t h e disc I 20
b
I 40
I 60
I 80
I I00
I 120
I 140
160
180
Angle of propagation [ degrees ]
45 degree line in the 0/90 degree material is governed by the stiffness of the material in the corresponding direction which is always greater than the Young's modulus. In terms of the uniaxial values 1
1
C0, 90° = s- Cll + s- C33 2 2 C45 o = Cll cos4 45° + C33 sin4 45° + (4C44 + 2C13) sin2 45 ° cos2 45 ° 1
--
4
(Cll + C33 +4C44 + 2C13)
This leads to theoretical estimates, based on the known values for uniaxial material, 8 of F0,9o° = 8.1 km s"1 and F4so = 6.2 km s"1 , in good agreement with the lower curve of Fig.3b. For values at smaller angles to the fibre axes it would be necessary to extend the Musgrave theory of wave propagation to cases of tetragonal symmetry. As this has not yet been done, values of sufficient accuracy can be obtained from the mean values of stiffness constants for the two layers in the required direction. For the interleaved 0/60/120 degree material, the stiffness on the plane is independent of direction and in terms of uniaxial values is given by 3 C =-
112
8
1 (Cll
-I-C33) -I-
(2C44 +C13)
Goniometer measurements by reflection from the flat sides of the discs give information about the velocities of propagation of waves travelling in the plane of the disc. However, the mechanics of the reflection process are such that this information applies only to that region of the material within a wavelength of the surface, the wavelength in question referring in this case to compressional waves travelling normally to the surface. It has been verified experimentally that when this wavelength is less than the thickness of a single layer of prepreg in the composite (0.24 mm) the reflections are essentially the same as from uniaxial materials. If the wavelength used is larger than this no clearly identifiable reflections are obtained. Transmission measurements
In many ways the measurements of compressional wave velocity perpendicular to the planes of the discs have proved to be the most interesting, for reasons which were not appreciated when the work was begun. It has, however, been pointed out by Dean and Turner 4 that the velocity measured in this direction is affected by the attenuation and dispersion arising from distributed porosity. The results for our specimen are summarized in Table 1 which gives the compressive wave velocity obtained by various techniques, as well as the velocities in unreinforced resins and in the fibres for the transverse direction calculated from the data of Dean and Turner. The average porosity of the specimens was found by measurement of the densities and of the proportion of fibre remaining after chemical removal of the resin in a destructive test on an off-cut sample from each specimen. This is expressed as a volume fraction of the resin constituent of each sample. The important features shown in the table are: 1. For the specimen Ul of low porosity there is good agreement between all the velocity measurements at different points, in different directions and by different methods.
ULTRASONICS.
MAY
1974
Table 1.
Ultrasonic wave velocities measured perpendicular to fibres in flat discs Velocities [km S"1 ]
Width
Thickness Resin porosity
Specimen
Type
[%]
Goniometer
Transmission
Goniometer
Transmission
U
Uniaxial
< 1
2.62
2.59 -+ 0.03
2.64 +- 0.05
2.62
5
Uniaxial
24 -+ 4
2.71 -+0.11
1.94-+0.02
2.81 -+0.22
2.31
6
Uniaxial
25.7 -+ 0.2
2 . 3 6 -+ 0.20
1.93 -+ 0.05
2.80 -+ 0.20
2.18
7
Uniaxial
19 -+ 3
2.24 -+ 0.02
1.94 +- 0.02
2.52 -+ 0.17
2.25
1
0/90 ° sandwich
14 -+ 5
--
2.56 -+ 0.04
2.74 -+ 0.06
--
2
0/90 ° laminate
10
--
2.38 + 0.04
2.80 + 0.10
-
3
0 / 6 0 / 1 2 0 ° laminate
23
-
1.99 +- 0.03
2.77 + 0.18
-
Velocities in pure resin: 2.94 km s"1 ; type 1 fibre: 2.47 km s"1
2. All the other specimens show a considerable amount of porosity in the resin matrix. In these specimens the transmission velocity measured by direct contact is lower than that in the fibres and therefore outside the range between that and the resin value. The thickness value is lower than that in the width direction for all the uniaxial porous specimens. 3. Goniometer values, although showing a greater spread than the transmission values, are consistently higher. In the width direction they are close to the value for the nonporous specimens. Discussion
Study of the results presented in Table 1 suggests that several important factors are involved in producing the low velocities in porous specimens. In the first place, of course, the difference between the thickness and width values obtained by direct contact suggests that there is less discontinuity in the structure in the latter case than in the former. This would be explained by the type of porosity produced, for example, by the trapping of air or gas between layers of prepreg in the initial lay-up. The second factor is the straightforward decrease in elastic stiffness of the composite due to the effective decrease in the elastic stiffness of the porous resin matrix. Theoretical estimates for this may be made on the basis of the theories of Hashin and Shtrikman lo for isotropic spherical inclusions in an isotropic matrix, and then for the transverse stiffness constant of the composite in terms of the weakened matrix using the theory of Hashin. 11 Appropriate allowance for the decreased density is then made, and the velocity of the transverse L wave calculated. The value obtained is in fact an upper limit and lower values would be obtained for special non-uniform distributions of the porosity, but the calculations are still of considerable interest, as illustrated in Fig.4. This shows that for a non-porous composite, the velocity should be between 2.60 and 2.67 km s"z , and that the value at 30% resin porosity falls to about 2.25 km s"1 . It is notable that over most of the range the dependence on fibre/resin ratio is slight. Velocities obtained from the width measurements are in better agreement with
U L T R A S O N I C S . M A Y 1974
the calculations than those obtained from thickness measurements. There is, however, considerable uncertainty in the porosity estimation by current techniques and, as mentioned above, non-uniform distributions will produce values lower than those calculated. There is also a further important point. The significant differences shown in Table 1 between the velocity measurements in the same direction by the transit time and goniometer techniques for porous specimens lends support to the process of preferential attenuation of the higher frequency components suggested by Dean and Turner. 4 The critical angle method is much less influenced by this effect and therefore yields a higher velocity in each case. A more detailed comparison of the two methods over a range of frequencies would be of great interest in the more precise characterization of the structure.
~40°/o'~x Fibre/resin 2 6 ~ , ~ . ~ 5 0 O/o"r~ by volume I ~60Cyo ~ o 2.51 ~ ~
>. 2-3 u o
Experimental velocities o Thickness A Width
A
60%
2.2
.
>
~ 2. I
40% Theoretical maxima
2-0 I-9
~soo/o
o o
o
I
I
O-IO
0.20
o
I
0-30
Resin void fraction by volume Fig.4 Comparison of compressive wave velocities perpendicular to fibres, as measured by direct contact, with theoretical upper limits for material with different matrix porosity. For uniaxial specimens the experimental values for width and thickness directions differ
113
iI CFRP
Resin specimen
:::::::::::::::::::/::::::::.:::
iii!iiiiiiiiiiiiiiiiiiiiiiiiii
9
i!!ii!!!iiiiiiiiiiiiiiii 10 Fig.5 Use of compressional transducers X and Y to measure the shear wave velocity in a resin specimen by the employment of t w o uniaxial CFRP strips as converters
Conclusions
It is concluded that the propagation of ultrasonic waves in the plane of a CFRP laminate is now adequately understood in that theoretical estimates can be made and checked by various techniques. Standard quantitative tests for production lines can therefore be devised to check the distribution and orientation of fibres in a resin matrix. Propagation normal to the laminate plane has raised questions of wider interest, and further studies of scattering, attenuatior and dispersion by distributed voidage would be of interest. In the meantime, the sensitivity of pulse velocity measurements to porosity has been demonstrated, and practical field equipment (to be described elsewhere) has been developed to scan laminates in terms of this velocity and to record the results as an indication of mean voidage distribution. This approach has obvious theoretical advantages over the conventional use of attenuation techniques, and can now be evaluated directly as a non-destructive test of shear strength. References
Musgrave, M. J. P. On the propagation of elastic waves in aeleotropic media: II media of hexagonal symmetry, Proceedings of the Royal Society 226 (1954) 356
114
11 12
Markham, M.F. Measurement of the elastic constants of fibre composites by ultrasonics, Composites 1 (1970) 145 Smith, R. E. Ultrasonic elastic constants of carbon fibres and their composites, Journal o f Applied Physics 43 (1972) 2555 Dean, G. D., Turner, P. The elastic properties of carbon fibres and their composites, Composites 4 (1973) 174 Curtis, G. Harwell equipment for materials evaluation by ultrasonics, Ultrasonics for Industry, IPC Science and Technology Press (1969) Reynolds, W. N. Problems of nondestructive testing in carbon fibres and their composites, International Conference on Carbon Fibres, London (1971) The Plastics Institute Curtis, G., Richards, R., Berry, J. A. The Harwell ultrasonic goniometer, unpublished information Wilkinson, S. J., Reynolds, W. N. The propagation of ultrasonic waves in carbon fibre reinforced plastics, Journal o f Physics D: Applied Physics 7 (1974) 50 EIliott, J. G. An investigation of ultrasonic goniometry methods applied to carbon fibre composite materials, AERE report NDT 64 (1973) Hashin, Z., Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials, Journal o f the Mechanics and Physics of Solids 11 (1963) 127 Hashin, Z. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry, Journal o f the Mechanics and Physics o f Solids 13 (1965) 119 Smith, A., Wilkinson, S. J., Reynolds, W. N. The elastic constants of some epoxy resins, to be published in Journal o f Materials Science (1974)
Appendix Use of uniaxial CFRP strip as an ultrasonic wave converter
The mixed type of compressive-shear wave propagation illustrated in Fig.2 shows that a strip of uniaxial material can be used to convert compressive waves to shear or viceversa, as shown in Fig.5. The compressional transducers can be used to measure the shear modulus of the specimen because a shear wave is emitted in a direction normal to the converter at the point of contact. A single converter strip could be used for example to monitor shear wave acoustic emission signals with a compressional transducer as receiver. In practice it is often found that the converter-specimen interface causes some reconversion so that there is an interfering compressional wave masking the shear wave. This difficulty can be overcome by using a shear transducer, such as a y-cut quartz crystal, as receiver. This system is still much more sensitive than using two shear transducers, and has been used elsewhere 12 to measure the ultrasonic shear moduli of resin samples over 35 m m in length.
ULTRASONICS.
MAY
1974