- Email: [email protected]
A non-adiabatic thermoelastic theory for composite laminates
J. Phys. Chem. Solids Vol. 52, No. 3, pp. 483494, Printed in Great Britain.
1991
OOZZ-3697191 S3.00 + 0.00 Q 1991 Pergamon Press plc
A NON-ADIABATIC THERMOELASTIC THEORY FOR COMPOSITE LAMINATES A. K. WONG~ Materials and Structures Department,
Royal Aerospace Establishment,
(Received 10 April 1990; accepted 19 September
Famborough,
Hants, U.K.
1990)
Abstract-The heat transferring characteristics of fibre-reinforced composites under cyclic loading are analysed and a non-adiabatic thermoelastic theory for a general laminated composite plate is formulated. The theory successfully explains the observed dependency of measured thermoelastic response on ply lay-up configuration and loading frequency, and provides an insight to the mechanics of interlaminar heat transfer. It is shown that non-adiabatic conditions can persist even for relatively high loading frequencies and consequently, interpretation of SPATE data can be extremely difficult. However, the analysis revealed that this new found complexity can, in principle, be exploited to yield individual strain components, thereby making SPATE potentially a more powerful device for analysing laminated composites than it is for isotropic
materials.
Keywords: Thermoelasticity,
SPATE, composites, thermal diffusion.
1. INTRODUCTION The coupling between elastic deformation and temperature change of a body under load, known as the
thermoelastic effect, was first analysed by Lord Kelvin [l]. However, it took over a century before its potential as a means for stress analysis was realized by Belgen [2] and some further 10 years before a practical device, SPATE (stress pattern analysis by measurement of thermal emission), became readily available. Though relatively young, SPATE’s ability to obtain full-field non-contact measurements has quickly established itself as a useful stress analysis tool and its application has become increasingly widespread, particularly for metallic structures, e.g. [3-S]. Unfortunately, the use of SPATE on composite structures has so far been limited to simple comparative studies (e.g. [6]) or to qualitative NDT-type applications (e.g. [7]). The reason is perhaps due to the lack of an adequate theory which can describe the thermoelastic process of a general laminated composite component. For instance, all existing theories are based on an adiabatic approach first derived by Biot [8] which all lead to the misconception that since SPATE is a surface measurement device, the measured data reflect only the stresses on the surface ply (e.g. [9]). However, experimental results of Dunn et al. [lo] showed that subsurface ply orientations can markedly influence SPATE’s frequency response. This is clear evidence that, depending on the ply
t Permanent address: Aircraft Structures Division, Aeronautical Research Laboratory, P.O. Box 4331, Melbourne, Victoria 3001, Australia. Crown Copyright. KS
52,3--B
lay-up configuration, true adiabatic conditions may not be achieved over the usual operational frequency range (5-30 Hz). Such interaction between plies has undoubtedly been a major cause of confusion over the use of SPATE on composites. In this paper, the problem of interlaminar heat transfer is modelled and examined in detail, and the question of how deep SPATE “sees” on a laminated composite component is tackled.
2. HEAT TRANSFER BETWEEN MATRIX AND FIBRES Thermoelastic stress analysis on isotropic materials works on the principle that when a sinusoidal load is applied at a sufficiently high frequency, the corresponding stress-induced temperature fluctuation has insufficient time to diffuse within the material. The attainment of such an adiabatic condition allows
Kelvin’s Law to be applied, leading to a simple relationship between the amplitude of temperature cycle to the amplitude of the bulk-stress cycle. By studying the stress pattern of a metallic plate containing a circular hole for which a well-known analytical solution exists, it was found that a frequency of greater than approx. 10 Hz is sufficient (see SPATE user manual [l 11). Of course, this lower bound frequency should not be taken as universally applicable as the diffusion rate would depend, amongst other parameters, on the temperature (and therefore stress) gradient. In the case of fibre-reinforced composites, large stress gradients in the form of stress discontinuities exist between fibres and the matrix material. This raises a rather interesting question: In a uni483
A. K.
484
WONG
directional CFRP, do the fibres and matric experience different temperature fluctuations as a result of cyclic loading? To resolve this question, consider a single fibre which is subjected to a sudden change in its surface temperature. This may be idealized as an infinitely long circular cylinder of radius r = R, and which is initially at thermal equilibrium such that T(r, t) = 0. At t = 0, the boundary condition T(R, 0) = T, is imposed. The conduction process in this case may be described by the one-dimensional diffusion equation in cylindrical coordinates: 1dT
------y at
a2T
1aT
ar2
r ar
= 0,
(I) I I -Z/l
where y = k/PC, in which k is the conduction coefficient, p is the density and C is the specific heat. The solution to eqn (1) with the above boundary conditions is well documented (e.g. [ 12]), and may be expressed as a series of Bessel functions: @=
s =f u,J,(~ir/R)exp(-~LTFo), (2)
0
”
i=l
where a = 2/~,5, (pi), Jo and JI are respectively the zeroth and first order Bessel functions of the first kind, pi are the roots of Jo and Fo = yt/R2 is known as the Fourier number. Figure 1 shows the development of the temperature profile across the radius of the cylinder. It is seen that the diffusion process is essentially completed at Fo = 0.8. This means that for a typical carbon fibre used in composite laminates in which y =9x 10-‘m2sand R = 3.5 x 10-6m, a uniform temperature would be attained in 1 x 10m5s. As this time interval is much less than the quarter period of even a 30 Hz loading cycle (8.3 x 10m3s), it may be concluded that whilst there are stress discontinuities
Fig. 1.Temperature profiles on an infinite cylinder subjected to a step change in ambient condition.
I
Fig. 2. Temperature profiles of an infinite plate subjected to a step change in ambient condition.
between fibres and matrix, the two materials would experience essentially the same stress-induced temperature fluctuations over the normal loading frequency range considered. In other words, a typical unidirectional CFRP laminate may be treated as a homogeneous material from a thermoelastic stress analysis viewpoint. 3. HEAT TRANSFER BETWEEN PLIES
3.1. A jirs t assessment In general, composite laminates consist of many plies of different fibre orientations. Consequently, a situation analogous to the fibre/matrix system arises as stress discontinuities could exist at the iaminar interfaces. To assess the effect of heat transfer plies, a similar analysis is applied. Consider an infinite flat plate initially at a uniform temperature being subjected to a sudden change in temperature at both its surfaces. Again, the solution to this problem is well known (e.g. [12]), and is best illustrated in the nondimensional form shown in Fig. 2. The Fourier number in this case is given by
in which 21 is the lamina thickness. From Fig. 2, it is seen that diffusion is essentially completed when Fo = 2, and for a typical CFRP lamina in which y = 8 x lo-’ m2 S-I and 21= 1.25 x 10m4m, this corresponds to a time of approx. 0.01 s. As this is of the same order as the quarter period of say a 30 Hz cycle (8.33 x lo-’ s), we conclude that homogenization of temperatures between different layers would not occur completely. On the other hand, adiabatic conditions would not be achieved either as this would require Fo < 0.005, or t 6 2 x IO-‘s. This means that adiabatic conditions will be achieved only for extremely high loading frequencies.
Composite laminates
3.2. A numerical model The analytical model in the previous section can provide only an approximate assessment of the heat transfer process between plies as it assumes a fixed temperature boundary condition. In reality, the boundary temperature would be time dependent due both to the applied time-varying load and to the fact that the adjoining laminae cannot be considered as infinite sources or sinks. For a more accurate representation of the problem, the interfacial temperatures must form part of the solution rather than be applied as boundary conditions. Consider a single lamina from a multi-ply composite plate subjected to a uniform uniaxial oscillating stress. Since each ply can be treated as a homogeneous orthotropic plate, it may be shown from principles of mechanics and thermodynamics [9] that, for quasi-static conditions and small temperature changes (dT 6 T,)
’ aQI
I-= i=l
axi
-pCef-
To
x (a,,k,, + a22d22 + X09),
(4)
where Qi is the heat flux passing through the point under consideration, p is the density, C, is the specific heat under constant strain, To is the absolute temperature of the material, a,, , az2are the coefficients of linear thermal expansion and uII, uz2 are the stress components in the respective inplane orthotropic directions. .%?“8Y is a higher-order term which gives rise to the “mean-stress effect” which was first discussed for isotropic materials by Wong et al. [13, 141, and may be neglected when the applied static stress is small. In all previous derivations it has been assumed that adiabatic conditions are achieved so that Qi vanishes and a closed form solution to eqn (4) is obtained. However, the preliminary analysis in Section 3.1. showed that heat transfer could be significant in the case of composite laminates and thus should not be neglected. Assuming that heat transfer during thermoelastic cycling is dominated by diffusion in the throughthickness direction (this would be true if the in-plane stress gradients were much less than the stress gradients between plies), the application of Fourier’s Law of heat conduction gives
485
eqn (4) yields the classical one-dimensional equation
de --
at
where y = k/pC, diffusivity and S=
is
2 g=s,
the
coefficient
of
-~(a,,d,,+a,,d,,+~~~) e
thermal
(7)
represents an internal heat source. Since the diffusion coefficient in the through-thickness direction is independent of ply orientation, eqns (6) and (7), though derived for a single lamina may, in the case where all plies are of the same material, be applied for the whole laminate. We note, however, that since stresses can vary from ply to ply, the heat source term [eqn (7)] will in general be a discontinuous function through the laminate thickness. With this complexity, the solution is best sought by the numerical method outlined in the following. Discretizing both t and z, a Crank-Nicolson finite difference respresentation of eqn (7) may be written (e.g. [IS]) as L9_, + A& + I!?:, , = B,,
(8)
where
and Bi=
-2s,$.!-e;9;+2-A..? yAt e:-1-e;;:
( > 1
(9)
in which superscript t and subscript i indicate the location on the t-z grid, whilst At and AZ are the corresponding time and spatial increments. To complete the system, initial and boundary conditions are required. At t = 0, it can be assumed that the body is at equilibrium with the surroundings so that f3(i,0) = 0. For t 2 0, two types of boundary conditions are applicable. Since composite laminates are usually made symmetric about their mid-planes, modelling is necessary only up to the plane of symmetry. By virtue of symmetry, and adiabatic boundary condition is applicable on this surface. That is,
de -= az where I is the coordinate in the through-thickness direction and k is the corresponding heat conduction coefficient. Defining the non-dimensional temperature 0 as 6 = (T - T,)/T, and substituting eqn (5) into
diffusion
0,
(10)
or in finite difference form, assuming i = N represents the boundary at the plane of symmetry,
e N+I =b-,.
(11)
486
A. K.
On the free surface (i = 1), allowance must be made for convective heat losses to the surroundings. This may be expressed as
Q, = he,
(12)
in which Q, is the heat leaving a unit area at the surface and h is the convective coefficient. Equating this to the heat arriving from beneath the surface, we have he,
=kE az +,’
Expressing eqn (13) in its finite difference form and upon rearrangement, we have 2h AZ
e0 = 0, - -0,
k
Writing eqn (8) for all i, i = 1 . N, and eliminating the terms ON+, and B, using eqns (11) and (14) a tridiagonal set of N equations in 0, is formed, and can be solved efficiently by well-known algorithms. 4. EXPERIMENTAL
To validate the model presented, the surface temperature response of a cyclically-loaded CFRP specimen was examined. Since the work of Dunn et al. [lo], it has been learnt that the application of the usual high-emissivity paint on composite specimens can, depending on its thickness, introduce substantial phase shifts to the SPATE signal. Furthermore, the result presented in their paper was the measured SPATE response, not the temperature response. As will be shown later, the system electronics can introduce amplitude attenuation and phase shifts, particularly at low frequencies, and must be corrected in order to infer the actual temperature response; because there is some uncertainty in the thickness of the paint layer used in their tests, and the characteristics of the equipment employed was unknown, new tests were conducted on a similar but unpainted specimen, taking into account the characteristics of SPATE’s system response. The specimen was made from Ciba-Geigy XAS914C prepreg and the lay-up was identical to that in Aluminium end [lo], namely [(O’, +45”, -4Y)J5. tabs were bonded to the composite using a room temperature setting adhesive and the finished specimen has a rectangular gauge section of approx. 15Ox23x3mm. Sinusoidal loading was achieved on a servohydraulic testing machine running under load control, and the load-cell signal was used as the reference input to SPATE’s correlator. Prior to testing, an area scan was initially performed on the gauge section of the specimen to ensure a uniform emission and the
WONG
absence of any bending. When proper alignment was achieved, testing was performed with SPATE focused on a fixed point. With a selected load amplitude and frequency, the amplitude and phase of the SPATE signal were recorded; because of the relatively low signals (<4 mV r.m.s.), a long integration time was required for adequate noise rejection. This was achieved by setting the correlator time constant to a maximum of 10 s (see SPATE manual [1 1]), and a data acquisition system was used to sample the amplitude and phase output channels of the correlator as rapidly as possible (between 1 and 2 s intervals) for a duration of 4 min. Two load amplitudes (4 and 6 kN) covering a frequency range of 0.5-30 Hz were tested. For the 20 Hz tests, the sampling duration was extended to 8 min in an attempt to provide a more accurate datum as, in adopting the convention in Dunn et al. [lo], all amplitude data were to be normalized by the 20 Hz signal. A thermocouple was used to monitor the reference temperature, ensuring that no significant viscoelastic heating occurred. 5. EFFECTS OF CONVECTIVE HEAT TRANSFER AND FILTERING The experimental results in Dunn et al. [lo] showed that the SPATE frequency response for a (+45”, -45”) lay-up behaves essentially the same as that for aluminium. That is, a response which is independent of frequency beyond approx. 5 Hz. This is easily explained by the current model. Because all plies have identical fibre and transverse stresses in this case, the source term S would, like an isotropic material, be constant through the thickness of the plate. The reason for the signal drop-off at frequencies less than about 5 Hz is mainly due to the characteristics of the SPATE electronics which were designed to eliminate any dc. component. Another possible cause for such a signal drop-off at low frequencies might be the presence of convective heat transfer between specimen and the surroundings. To investigate the significance of this effect, the present model was used to analyse a unidirectional composite plate with various convective boundary conditions. Because amplitude attenuation is most apparent at low frequencies, a heat transfer coefficient based on natural convection is most appropriate. Hence, taking a 200 mm long vertical plate at O.Ol”C above ambient to be typical, a value of h = 1 W me2 “C-r was estimated (see [16]). In a series of runs, it was found that using such a value for h, and indeed for h up to 100 W me2 ‘C-l, insignificant heat losses occurred for frequencies above 1 Hz. This suggests that no part of the SPATE signal drop-off at low frequencies is attributable to heat losses to the surroundings. The result also means that an adiabatic surface boundary condition may be used on all subsequent modelling. To account for the effects of the system electronics, a series of experiments were carried out using an
Composite laminates aluminium specimen. Since we know that, with the absence of any significant heat loss, the thermoelastic response of aluminium should be independent of frequency and should lag the applied load by 180”, the resulting SPATE response for such a material may therefore be used as a source for transforming SPATE data to temperature data. Figure 3 shows the amplitude correction curve used in the present study which was derived from the smoothed aluminium results. Because the phase offset curve (Fig. 4) did not exhibit an asymptotic behaviour, further investigation using the “edge effect” was undertaken. As SPATE detects changes in infrared flux, it is sensitive not only to changes in temperature, but also to changes in emissivity. This is easily seen from the expression for the infrared irradiation flux @, where @ = BeT4,
(15)
in which B is the Stefan-Boltzmann constant, e is the emissivity, and T the absolute temperature. The fractional change in @ is therefore given by d@ de _=_+4dT. @ e
T
(16)
Normally, the specimen under investigation would have a uniform emissivity so that SPATE detects only changes in temperature. However, when SPATE is focused onto an edge which is moving at the reference frequency, the contribution from de/e can become significant. Given that SPATE can detect a O.OOl”C temperature change at room temperature (T = 300 K), it is easy to see from eqn (16) that SPATE would be sensitive to an emissivity change of merely 0.0013%. This is precisely why SPATE can produce spurious results at free edges. In the current study, this effect was exploited in a series of tests to
487
Fig. 4. Phase shift of SPATE signal. (0) Aluminium response (shifted by 180”), (0) edge effect (displacement amplitude = 1.5 mm), (*) edge effect (load amplitude = 4 kN), (-) smoothed data which may be treated as the difference between the phase obtained by SPATE and the phase of the temperature response.
verify that the phase offsets observed in the aluminium tests were indeed not due to some real temperature phase lags. By attaching a two-tone strip (a reflective metallic foil whose lower half has been painted matt-black) on the testing machine’s actuator, and with SPATE focused onto the reflective/black edge, a sizable SPATE signal was obtained once the actuator was set in motion. In the first case considered, no specimen was mounted between the grips and the testing machine was run under displacement control at a fixed amplitude of 1.5 mm. Another test was run with the actual composite specimen mounted and using similar loading conditions (i.e. mean load of 8 kN, and load amplitude of 4 kN). In both cases, the change in the average emissivity of the field of view was the sole contributor to the SPATE signal as the strip was not loaded and thus could not be subjected to any thermoelastic effect. Setting dT = 0 in eqn (16) and noting that de must be in phase with the actuator motion, we deduce that the infrared irradiation flux should also be in phase with the actuator motion. However, as can be seen in Fig. 4, the phase offset for base cases was extremely similar to that found for the aluminium test, suggesting that such offset was an artifact of the equipment system response. A smooth curve was thus fitted through the data in Fig. 4, and was used for correcting the SPATE data to infer the actual temperature response.
6. MATERIAL PROPERTIES Before the numerical simulation can proceed, various material properties are required. Unfortunately, material properties for composites are not Fig. 3. Amplitude attenuation of SPATE signal. (0) Stress amplitude = 53 MPa, (0) stress amplitude = 34 MPa, (*) stress amplitude = 17 MPa, (-) smoothed data which may be treated as the ratio between SPATE amplitude and temperature amplitude.
as well documented as for metals and obtaining a consistent set of data was difficult as different sources report slightly different values. Nevertheless, best estimates of the properties for the XAS-914C
488
A. K. WONG
material used were obtained from British Aerospace [17-191, viz. E,, = 130,000 MPa, E22 =
9000 MPa,
G,, =
5800 MPa,
vu =
0.3,
surfaces for two different frequencies (10 and 15 Hz) and two different load amplitudes (3 and 5 kN). The resulting scans typically showed three distinct uniform regions with the signal from the middle strip being approx. 6.7 times greater than those on the sides. The scan data of each strip were averaged and, from the adiabatic thermoelastic equation [8], it may be shown that CQ,/CQ~ is given simply by ~II Ezz SP, -=_.az2
p = 1630 kg mm3,
(17)
in which E,, , Ez2, G,, are the elastic constants in the fibre, transverse and in-plane shear directions, v,~ is the Poisson’s ratio in the (12) direction and p is the density. The remaining parameters which are required for solving the thermoelastic/diffusion problem are the thermal properties. However, reliable values for these properties were not readily available. Values of k=0.87Wm-‘“C-’ and C,=990Jkg-‘“C-’ were reported for similar composites by Freeman and Kuebeler [20] whilst k = 1.5 W m-’ “C-’ and C, = 880 J kg-’ “C-’ were reported in Caurtaulds’ data sheet [21]. Because of such discrepancies, it was decided that simulation should be carried out for both sets of data. Another material parameter which lacks a reliable source is the coefficient of thermal expansion in the fibre direction (a,]). This is probably due to the fact that cl], for CFRPs in general is extremely small. Whilst the transverse expansion coefficient (ccZ2)is comparable in magnitude to that of aluminium and an agreed value of az2 = 28 x 10m6“C-’ has been found in many sources, reported values of a,, ranged between -0.7 x 10e6 [21] and 0.22 x lo-‘“C-’ [17]. In view of such diversity in data and suspecting that this small value could well be sensitive to batch-to-batch variations and local production procedures, experiments were carried out to measure this quantity. 6.1. Measurement qf a,,jazz As will be seen later, it is the ratio u,,/az2 rather than their respective absolute values which will be important in the present model. Measurement of this value was achieved using SPATE and a specially constructed specimen. The specimen comprised three separate strips of unidirectional XAS-914C composites bonded side-by-side between common aluminium end plates. The plates ensured that all three strips would undergo the sum displacements. The two outer strips had the fibres aligned with the loading direction (0”) whilst the middle strip fibres were perpendicular (90”). All three strips had identical gauge dimensions, namely, 160 x 23 x 2 mm. The surface of the composite were lightly grit blasted to remove most of the surface resin layer and an area SPATE scan was made on the unpainted
4,
(18)
SP, ’
whre Sp, and Sp, are the averaged SPATE signals for the 0” and 90” composites, respectively. The results on the range of tests considered consistently yielded the ratio
tl
=
2 = 0.010.
(19)
__
This result certainly confirms that the fibre-direction coefficient of thermal expansion is much smaller than its counterpart for the transverse direction. It is, however, interesting to note that this value was determined to be positive rather than the negative values frequently reported for CFRP composites. 7. RESULTS
AND DISCUSSION
With all necessary parameters, we can proceed to solve the nonadiabatic thermoelastic problem for the (O“, +45”, -45”) plate undergoing cyclic loading. It may be shown from laminate theory that the plate under consideration has a Poisson’s ratio v,,, of 0.683 (where x is the 0” direction). Assuming an applied uniaxial sinusoidal stress of the form 0 = Aa sin wt in the 0” (x) direction (where Aa is an arbitrary constant), and it may be shown that the laminar stresses are given by u ,,
= 2.256 Au sin wt,
022= -0.0607 Au sin at
for the 0” plies, and u ,,
= 0.370 Au sin cot, az2=
0.0326 A(Tsin ot (21)
for the f45” plies. Due to a lack of any reliable data for the temperature dependent parameters which give rise to the “mean-stress effect” discussed by Wong et al. [ 131and Potter and Greaves [22], the term ,%?“09 in eqn (7) will be neglected. In any case, the mean stress in the experiments will be kept low so as to avoid this term becoming significant. Hence, the substitution of eqns (20) and (21) into eqn (7) gives S = 0.038law cos wt
(22)
Composite laminates
489
for the 0” plies, and S = -0.0363aw
cos cot
(23)
for the f45” plies. The constant a in both cases is given by a = a2rAa/(pC,). In the present tests, a was taken as an arbitrary constant as its effect as a scaling factor was eliminated by normalization of the results by some chosen reference data (e.g. result at 20 Hz). Equations (22) and (23) were used in modelling the thermoelastic process of the laminate under consideration. The half-depth thickness consisting of 12 plies was discretized by 481 nodes, and a marching timestep of 0.02n/w s (period/lOO) was used for all frequencies studies. Computation was performed on a Vax 1l/780 using double precision arithmetic and subsequent mesh refinement analyses showed that the selected parameters were adequate. The full temperature field was solved at every time increment and the surface temperature was monitored as this represented the values which would be detected by any surface temperature sensor. It was found that the surface temperature quickly settled to a sinusoidal waveform and computation was terminated once steady-state was obtained. For most cases considered, the steady condition was achieved within 10 cycles and the resulting amplitude and phase of the surface temperature waveform were recorded. Figure 5 shows the through-thickness temperature profile at the moment for which the surface temperature is a maximum within a cycle. It is clearly seen that even though the source term is a discontinuous step function from ply to ply, the temperature profiles are smoothly continuous as a result of diffusion. At low frequencies, the time-scale is such that significant conduction can take place and therefore effectively flatten the temperature profiles. As the frequency is increased, the time-scale is contracted and less diffusion is allowed, yielded much more severe temperature gradients. In any case, it is seen that even at 30 Hz, the temperature profile is still far removed
Fig. 5. Temperature profiles at the instant of maximum surface temperature. ( ---) 5 Hz, (--) 10 Hz), (-) 30 Hz (abscissa scale: z = 0 represents the free surface and z = 1corresponds to the plane of symmetry. Ordinate scale: arbitrary).
Fig. 6. Temperature profiles at the instant of maximum surface temperature. (-) 100 Hz, (---) 1 MHz (same coordinate scales as for Fig. 5).
from the step function representing an adiabatic process, and the omission of the diffusion term in the thermoelastic equation cannot possibly suffice for such cases. For the purpose of illustrating the convergence to the adiabatic state, computation was carried out for frequencies up to 1 MHz (see Fig. 6). It is interesting to note from Fig. 6 that the maximum surface temperature was in fact not attained in the fully adiabatic state as might be expected. Slightly higher temperatures were achieved at moderate frequencies (of the order of 100 Hz) in which the dynamics of the system allowed temperature overshoots to occur near the locations of source discontinuity (ply interfaces). This phenomenon is somewhat analogous to the response of many other second order dynamic systems which exhibit a resonance behaviour. Figures 7 and 8 show both the computed and measured specimen surface temperature response when subjected to cyclic loading. Maintaining the convection of Dunn et al. [lo], and to avoid the need for various scaling factors, the amplitude data have been normalized by the corresponding result at 20 Hz. The experimental results have also been corrected for amplitude attenuation and phase shift introduced by the system response as discussed in Section 5. The measured results for the two different load amplitudes (4 and 6 kN) agree well, although the amplitude data appear to diverge from one another beyond about 20 Hz. This was perhapd due to a possibly higher noise level for the 6 kN case as, above this frequency, a great degree of load-frame vibration and a noticeably distortion loading waveform were experienced. Despite this, and the fact that a large number of material parameters were involved in the computation, the agreement between numerical and measured results is relatively good. It may be seen that the computed results for the two different published sets of thermal properties form bounds which encompass most parts of the experimental results.
490
A. K. WON0
One interesting feature which emerged clearly from the computed results showed the existence of a minimum in the temperature amplitudes between 0.5 and 2 Hz. This minimum represents the transition from the domination of the surface temperature by the angle ply the~oelastic efI’ect to that of the 0” surface ply. Since the effective heat source of each angle ply is roughly equivalent but opposite in phase to that of the 0” ply [see eqns (22) and (23)], and that there are twice as many angle plies as there are 0” plies, the resulting surface temperature when complete diffusion is permitted (as frequency tends to zero) would be in phase with the angle ply source. This may be seen in Fig. 8 as the phase between the surface temperature and load tends to 180” as the frequency decreases towards 0 Hz. Conversely, as frequency increases the process becomes increasingly adiabatic such that the phase of the surface temperature should in principle approach 0”.
8. NUMERICAL
STUDY
Having validated the numerical model, we can now use it to gain a better understanding of the thermal interaction between plies. It has been shown tht subsurface layers do affect the surface temperature characteristics. How deep SPATE can “see” when applied to composites is therefore the next obvious question, This is best answered by considering the problem in a decomposed form. Because the problem involved (neglecting the mean stress effect) is linear, the principle of superposition may be applied. This means that the overall probfem may be treated as a linear combination of a set of simpler but fundamental problems. Because of the step nature of the effective heat source, and again assuming an applied sinusoidal remote load, eqn (7) may be expressed as
x
0
cos U)f
(24)
where a = -au(pC,), n is the number of plies, and Hi is a unit-pulse function of z in which H, = 1 over the jth ply and vanishes elsewhere. For the jth ply, Ai is a parameter which is a function of the laminar stresses, namely A, = @da,, + Au&
(25)
Since only the source term in the diffusion equation is affected by stresses (and hence ply orientation), a basis set of solutions which is independent of ply lay-up configuration for a given material and number of plies (n) may be obtained by solving Fig. 7. Amplitude plot of 0,. (--) Computed (y = 1.04 x 10e6 mz s-l); (--f computed Q = 5.80 x IOw7m2 SS’); (0) measured (load amplitude = 6 kN); (0) measured (load amplitude = 4 kN). All results have been normalized such that the responses at 20 Hz are unity.
az;_ at
8’7,
coswt
y-$=Hp
for j = 1 to n. The
for a specific lay-up compiled as 8
=a
(24)
actual temperature field B configuration may then be
i
47,.
(27)
f=i
Since only the surface temperature is of interest in our case, we can apply eqn (27) specifically at the surface giving (28)
Fig. 8. Phase plot of B,. (- -) Computed (‘/ = 1.04 x computed (p = 5.80 x IO- ’ rn’ S-‘; 10m6m’ s-‘): (-) (0) measured (load amplitude = 6 kN); (0) measured (load amplitude = 4 kN).
in which 8, and TV, are, respectively, the values of 0 and 7, at the free surface (i = 1). To illustrate this, the set of basic solutions T+ was obtained for the 24 ply XAS914C material Computation was carried out using the averaged published values for thermal conductivity and specific heat, viz. 1.185 W m-l “C’ and 900 J kg-’ “C-l, respectively.
Composite laminates
Frequency
491
composite plate. For such plates, it is easy to see the relative contribution of each ply from these results. For example, it may be seen that at 5 Hz, the contribution from even the fourth layer can be considered significant whilst at 30 Hz, only the first two layers are important. Of course, how deep SPATE sees would much depend on the ply lay-up as the parameters Aj have the effect of scaling each of the curves shown in Fig. 9 by different amounts. Take the (O’, 45”, -45”) plate studied earlier for instance, and consider the relative contributions from the first three layers. From eqns (22) and (23), we have
(Hz1
Fig. 9. Amplitude plot of T,,. Ordinate scale is such that rs, tends to 1 as frequency tends to infinity.
Fig. 10. Phase
plot of T,,
Figures 9 and 10 show the computed surface temperature response for each of the individually excited plies. As expected, the amplitude of the surface temperature due to a unit source in the surface layer 7s, increases with frequency whilst the contributions from subsurface layers show a general decrease with increasing frequency. This set of solutions has the property that if a vectorial sum of the surface response is taken over all plies, it yields a constant which is independent of frequency. This is because such a sum is equivalent to the solution to the problem where the source term is uniform throughout the thickness of the plate as would be the case for a unidirectional Table
I. Relative contributions
Relative
Total response
0, :
Phase
.i I
magnitude 0.8269
(deg) 25.6
2 3
0.3790 0.1738
160.9 116.2
0.6939
58.3
(29)
Combining this with the computed results 7s,, we find that the relative contributions from the first three layers are as shown in Table 1. It is seen that even at 30 Hz, the contribution from the second layer is still considerable, and the notation that SPATE is a “surface” stress analysis device is simply untrue in this case. The fact that SPATE can see different amounts of each layer depending on the ply orientation poses a major problem for practical applications. For the simple rectangular plate studied, a fair prediction of the measured results was achieved mainly because the nature of the stresses with respect to the ply orientations was known. In practice however, the stress field for each lamina generally varies in the plane of the laminate, and since the coefficients A, are functions of stresses, interpretation of SPATE results could become extremely difficult. Furthermore, because the contribution of each layer to the surface temperature is also frequency dependent, a different SPATE scan pattern might emerge depending on loading frequency. Whilst this phenomenon makes the straightforward application of SPATE to composites difficult, it does open one new possibility. It will be revealed in the following section that this effect can be exploited, at least in principle, to obtain individual strain components in laminated plates.
to the surface temperature from each lamina (all amplitudes 20 Hz surface temperature response) 10
45
A, A, A, = A = -0.953. 1
Frequency 20 Relative
have been normalized
(Hz) 30 Phase
Relative
magnitude
(de&
magnitude
1.0464 0.3307 0.11099
19.2 136.0 72.8
1.1350 0.2800 0.0727
41.3
1.1670
I .oOOo
by the
Phase
(de& 15.0 117.6 40.2 30.7
A. K. WONG
492 9. STRESS/STRAIN
DECOMPOSITION-A POSSIBILITY
NEW
SPATE’s inability to deliver a tensorial quantity (which stresses and strains are) has long been recognized as its major disadvantage. This has led to many investigations of decomposition techniques (e.g. [23], [24]). However, all such techniques require the solution of the equilibrium equation with almost all of the normally required boundary conditions. This means that their application to components or arbitrary geometry is not readily achievable. For laminated CFRP plates, the fact that different loading frequencies might yield different scan patterns can be exploited for the purpose of obtaining the individual strain components. Providing in-plane diffusion is small compared with that of the through-thickness direction, the following technique can be independently applied point-bypoint, thus removing any necessity for boundary conditions. Combining eqn (24) and (29, the effective source term may be expressed as
seen that the final surface temperature obtained as
may be
0, = b i [(@ COS*4j + sin* $j) Ac, j-l
+
(j3
sin* 4j + cos* 4j) AcYY
+ 2(8 - 1)sin 4) cos 4j Ac,]r,.
(33)
Since the same strains are experienced by all plies, these terms may be factored, yielding es = cXXC (/I Cos*f$j+ sin* tij)r,, j-l
+ Cyyi
(fi sin* $j + cos* 4j)r,
j=l
+ 2(/I - l)Q 1 sin 4jcos 4jrX,,
(34)
j=l
where the quantities & = b A+,,,
S = -F
$, (a Aol, + Au,,),H,w cos wt. PI
(35)
(30)
Since there is discontinuity in stresses from layer to layer, it is convenient to express eqn (30) in terms of strains, viz.
S = b 5 (p AE,, + Ac,,),H,w cos cut i= I
may be considered as scaled strain amplitude components. In practice, the effects of the constant b, and indeed any other scaling factors which may be introduced by measurement instrumentations, can be eliminated by normalization with the data at a point where the strain is known (e.g. a far-field point where strain gauge data are available). Because 0, and rs, may be treated as vectors (i.e. possessing both magnitude and phase), we can resolve eqn (34) along any chosen direction giving
= b i [(fi cos2 4, + sin* 4,) AcII ,=I + (fi sin* 4, + cos2 +j) AcY,.- 2(/I - 1) where x sin 4, cos 4, At,,,]H,o
cos cot,
(31)
where u = i (B cos* 4, + sin* +j[fs, (cos($,~, - I(lrer)7 ,=I >
V = i: (P sin* 4j
and
+
COS*
~j)lzs,~cOs(~~., -
+dh
,=I
w
(32)
in which E = E,, /E2*. The subscripts x and y refer to the global reference axes, Ac,,~,AE,). and AE,, are the normal and shear strain amplitudes in .the respective directions and &j is the orientation of the fibre direction of the jth ply relative to the global axes. From eqns (24)-(28), it is
=
2(/? - 1) i
,= I
sin & cos 4j COS($,~,- tircr), (37)
and in which (8,) and /7+ ( are the magnitudes, and 1(10, I(lrgtheir corresponding phases. I(lrrris an arbitrarily chosen reference phase. Now suppose that measurement of the surface temperature at the same applied load was taken at three different frequencies (keeping in mind that the highest frequency should not exceed that which may
Composite laminates
cause significant viscoelastic or dynamic effects). The repeated application of eqn (36) gives
I[1
04)
w(w,)
utwz)
w(wz)
t&)
r+%)
L cYY7 (38) CXP
where w,, c2 and o, represent the three selected frequencies. Equation (38) is a set of linear equations in three unknowns and, provided they are linearly independent, a unique solution in &., .& and Z_ may be obtained. Because we do not have the analytical form of 7+, the conditions for eqn (38) to have a unique solution are not readily derived. However, it may be easily shown that there are two special cases in which no unique solution exists. As mentioned earlier, t,, has the property that its sum over all plies is a frequency independent constant. That is, i
7+ =
const
(39)
/=I
for all frequencies. From this, it is easy to show that the determinant of the matrix in eqn (38) vanishes for the case where B = 1,
(40)
or alternately, tan* 4, = tan2 I#J*=
. = tan* 4,.
493
Figs 9 and 10) into eqn (38), we get (setting ljlref arbitarily to 00)
II
0.3646 0.7513 = [ 1.0034
-0.5405 -0.4931 -0.4623
1
E,, CYY.
-0.9015 -0.9490
0.1331 0.1317
-0.9799
0.0935 I[ E,,
(42) Equations
(42) may be solved algebraically,
L_.= 4.8410,
c,,Y= -3.3097,
cXY = 0.0190,
or for the purpose of eliminating
%’ = -0.684 L
and
yielding (43)
any scaling effect,
2 = 0.004.
(44)
Recalling that the laminate has a Poisson’s ratio vXY = 0.683 and that from symmetry considerations no shear strain was applied, the inverse solution algorithm indeed deciphered the actual loading conditions to a good degree of accuracy. Of course, the success of this procedure would depend very much on the availability of accurate data for the many material parameters involved. The application of the above algorithm using the actual measured surface temperature data (averaged between the two tests) yielded
g= -0.646 LXX
and
3=0.107, %r
(46)
(41)
The first [eqn (40)] is applicable for an isotropic material, and the second [eqn (41)] is true for laminates in which the absolute angles of all plies are equal. Since the assignment of the global reference is arbitrary, the latter condition implies that laminates with two or less ply orientations would not be amenable to solution by the above method. This result is not entirely surprising given materials, uniour experience with isotropic directional and (+45”, -45”) composites. What is surprising is that for other practical ply lay-up configurations which consist of at least three fibre orientations, a non-vanishing determinant invariably results, indicating that a unique solution is possible. To illustrate this principle, the decomposition algorithm is applied to the foregoing model assuming, for the purpose of this exercise, the simulated surface temperature response is without error. That is, selecting the three frequencies to be 10, 20 and 30 Hz say, we assume that the corresponding measured values of 0, are as given in Table 1. Substituting these, together with the necessary material parameters and the solution of 7,y, (as shown in
which, whilst partly acceptable, could be improved with more accurate material parameters. Nonetheless, this unique potential for decomposing strain components of laminated plates from thermoelastic data is clearly demonstrated, and could perhaps make SPATE a much more powerful tool for analysing composites than it is for isotropic materials.
10. CONCLUSION
A first order analysis of the heat diffusion characteristics of CFRP from a micro-level has been presented. For the normal frequency range suitable for thermoelastic stress analysis (5-30 Hz), it is concluded that whilst fibres and matrix may be treated as a homogeneous material from SPATE’s point of view, the thermoelastic process of different layers may, in general, neither be treated as homogeneous nor adiabatic. A non-adiabatic model for the thermoelastic effect for laminated composite plates has been developed, and its validity has been demonstrated by the good agreement between predicted and measured surface temperature/load response. A superpositional analysis provided an insight to the
A. K. WONG
494
mechanics of interlaminar heat transfer and the extent to which each layer can influence the surface temperature depending on frequency and ply orientation. It is concluded that the process of interlaminar heat transfer makes interpretation of SPATE results difficult unless an extremely high loading frequency is used. On the other hand, there now exists a possibility, at least in principle, for exploiting the frequency-dependent nature of the surface temperature for determining individual strain components from thermoelastic data. Acknowledgements-This work was undertaken during the author’s visit to RAE, and the encouragement and support from members of the MS6E Section was greatly appreciated. The author also wishes to express thanks to British Aerospace, Warton, for providing the material data for the composite used.
REFERENCES I. Thomas W. (Lord Kelvin), Trans. R. Sot. Edinburgh 20, 261 (1853).
2. Belgen M. H., ISA Trans. 6, 49 (1967). 3. Harwood N. and Cummings W. M., Strain 2, 7 (1986). 4. Stanley P. and Chan W. K., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 131, 102 (1987). 5. Wong A. K., Dunn S. A. and Sparrow J. G., Nature 332, 613 (1988). 6. Owens R. H., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 731, 74 (1987). 7. Cox B. N. and Pettit D. E., Proc. SEM Spring Co& Sot. Exp. Mech.. Housron, Texas, p. 545 (1987).
8. Biot M. A., .I. appl. Phys. 21, 240 (1956). 9. Potter R. T., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 731, 110 (1987). 10. Dunn S. A., Lombard0 D. and Sparrow J. G., Proc. SPIE Conf, Stress and Vibration: Recent Developments in Industrial Measurement and Analysis, London. p. 129
(1989). 11. SPATE 8000 MANUAL. Ometron Ltd, p. 129 (1989). 12. Luikov A. V., Analytical Heat Diffusion Theory (Edited by J. P. Hartnett). Academic Press, New York (1968). 13. Wong A. K., Jones R. and Sparrow J. G., J. Phys. Chem. Solids 48, 749 (1987). 14. Wong A. K., Sparrow J. G. and Dunn S. A., J. Phys. Chem. Solids 49, 395 (1988). 15. Ames. W. F., Numerical Methods for Partial Differential equations, 2nd edn. Academic Press, New York (1977). 16. Whitaker S., Fundamental Principles of Heat Transfer, p. 262. Robert E. Krieger Publishing, Malabar, Florida (1985). 17. British Aerospace, Design Data Sheet, Vol. 3, Section 2, Chap. 2, pp. 2-3, Amendment No. 2 (1985). 18. British Aerospace, Report No. SOR(P) 157 (1986). 19. British Aerospace, Report No. BAe-WSO-RP-RESSOR-000177, Addendum 1 (1989). 20. Freeman W. T. and Keubeler G. C., Composite Materials: Testing and Design (3rd Conf.) ASTM STP 546, ASTM, p. 435 (1974). 21. Caurtaulds Ltd, Dafa sheet G14, Sheet 2 (1974). 22. Potter R. T. and Greaves L. J., Proc. SPIE Conf, Optomechanical Systems Engineering, San Diego, California 817, 134 (1987). 23. Ryall T. G. and Wong A. K., Mech. Mater. 7, 205 (1988).
24. Huang Y. M., Determination of individual stresses from thermoelastically measured trace of stress tensor. Ph.D. thesis, University of Wisconsin, Madison (1989).
1991
OOZZ-3697191 S3.00 + 0.00 Q 1991 Pergamon Press plc
A NON-ADIABATIC THERMOELASTIC THEORY FOR COMPOSITE LAMINATES A. K. WONG~ Materials and Structures Department,
Royal Aerospace Establishment,
(Received 10 April 1990; accepted 19 September
Famborough,
Hants, U.K.
1990)
Abstract-The heat transferring characteristics of fibre-reinforced composites under cyclic loading are analysed and a non-adiabatic thermoelastic theory for a general laminated composite plate is formulated. The theory successfully explains the observed dependency of measured thermoelastic response on ply lay-up configuration and loading frequency, and provides an insight to the mechanics of interlaminar heat transfer. It is shown that non-adiabatic conditions can persist even for relatively high loading frequencies and consequently, interpretation of SPATE data can be extremely difficult. However, the analysis revealed that this new found complexity can, in principle, be exploited to yield individual strain components, thereby making SPATE potentially a more powerful device for analysing laminated composites than it is for isotropic
materials.
Keywords: Thermoelasticity,
SPATE, composites, thermal diffusion.
1. INTRODUCTION The coupling between elastic deformation and temperature change of a body under load, known as the
thermoelastic effect, was first analysed by Lord Kelvin [l]. However, it took over a century before its potential as a means for stress analysis was realized by Belgen [2] and some further 10 years before a practical device, SPATE (stress pattern analysis by measurement of thermal emission), became readily available. Though relatively young, SPATE’s ability to obtain full-field non-contact measurements has quickly established itself as a useful stress analysis tool and its application has become increasingly widespread, particularly for metallic structures, e.g. [3-S]. Unfortunately, the use of SPATE on composite structures has so far been limited to simple comparative studies (e.g. [6]) or to qualitative NDT-type applications (e.g. [7]). The reason is perhaps due to the lack of an adequate theory which can describe the thermoelastic process of a general laminated composite component. For instance, all existing theories are based on an adiabatic approach first derived by Biot [8] which all lead to the misconception that since SPATE is a surface measurement device, the measured data reflect only the stresses on the surface ply (e.g. [9]). However, experimental results of Dunn et al. [lo] showed that subsurface ply orientations can markedly influence SPATE’s frequency response. This is clear evidence that, depending on the ply
t Permanent address: Aircraft Structures Division, Aeronautical Research Laboratory, P.O. Box 4331, Melbourne, Victoria 3001, Australia. Crown Copyright. KS
52,3--B
lay-up configuration, true adiabatic conditions may not be achieved over the usual operational frequency range (5-30 Hz). Such interaction between plies has undoubtedly been a major cause of confusion over the use of SPATE on composites. In this paper, the problem of interlaminar heat transfer is modelled and examined in detail, and the question of how deep SPATE “sees” on a laminated composite component is tackled.
2. HEAT TRANSFER BETWEEN MATRIX AND FIBRES Thermoelastic stress analysis on isotropic materials works on the principle that when a sinusoidal load is applied at a sufficiently high frequency, the corresponding stress-induced temperature fluctuation has insufficient time to diffuse within the material. The attainment of such an adiabatic condition allows
Kelvin’s Law to be applied, leading to a simple relationship between the amplitude of temperature cycle to the amplitude of the bulk-stress cycle. By studying the stress pattern of a metallic plate containing a circular hole for which a well-known analytical solution exists, it was found that a frequency of greater than approx. 10 Hz is sufficient (see SPATE user manual [l 11). Of course, this lower bound frequency should not be taken as universally applicable as the diffusion rate would depend, amongst other parameters, on the temperature (and therefore stress) gradient. In the case of fibre-reinforced composites, large stress gradients in the form of stress discontinuities exist between fibres and the matrix material. This raises a rather interesting question: In a uni483
A. K.
484
WONG
directional CFRP, do the fibres and matric experience different temperature fluctuations as a result of cyclic loading? To resolve this question, consider a single fibre which is subjected to a sudden change in its surface temperature. This may be idealized as an infinitely long circular cylinder of radius r = R, and which is initially at thermal equilibrium such that T(r, t) = 0. At t = 0, the boundary condition T(R, 0) = T, is imposed. The conduction process in this case may be described by the one-dimensional diffusion equation in cylindrical coordinates: 1dT
------y at
a2T
1aT
ar2
r ar
= 0,
(I) I I -Z/l
where y = k/PC, in which k is the conduction coefficient, p is the density and C is the specific heat. The solution to eqn (1) with the above boundary conditions is well documented (e.g. [ 12]), and may be expressed as a series of Bessel functions: @=
s =f u,J,(~ir/R)exp(-~LTFo), (2)
0
”
i=l
where a = 2/~,5, (pi), Jo and JI are respectively the zeroth and first order Bessel functions of the first kind, pi are the roots of Jo and Fo = yt/R2 is known as the Fourier number. Figure 1 shows the development of the temperature profile across the radius of the cylinder. It is seen that the diffusion process is essentially completed at Fo = 0.8. This means that for a typical carbon fibre used in composite laminates in which y =9x 10-‘m2sand R = 3.5 x 10-6m, a uniform temperature would be attained in 1 x 10m5s. As this time interval is much less than the quarter period of even a 30 Hz loading cycle (8.3 x 10m3s), it may be concluded that whilst there are stress discontinuities
Fig. 1.Temperature profiles on an infinite cylinder subjected to a step change in ambient condition.
I
Fig. 2. Temperature profiles of an infinite plate subjected to a step change in ambient condition.
between fibres and matrix, the two materials would experience essentially the same stress-induced temperature fluctuations over the normal loading frequency range considered. In other words, a typical unidirectional CFRP laminate may be treated as a homogeneous material from a thermoelastic stress analysis viewpoint. 3. HEAT TRANSFER BETWEEN PLIES
3.1. A jirs t assessment In general, composite laminates consist of many plies of different fibre orientations. Consequently, a situation analogous to the fibre/matrix system arises as stress discontinuities could exist at the iaminar interfaces. To assess the effect of heat transfer plies, a similar analysis is applied. Consider an infinite flat plate initially at a uniform temperature being subjected to a sudden change in temperature at both its surfaces. Again, the solution to this problem is well known (e.g. [12]), and is best illustrated in the nondimensional form shown in Fig. 2. The Fourier number in this case is given by
in which 21 is the lamina thickness. From Fig. 2, it is seen that diffusion is essentially completed when Fo = 2, and for a typical CFRP lamina in which y = 8 x lo-’ m2 S-I and 21= 1.25 x 10m4m, this corresponds to a time of approx. 0.01 s. As this is of the same order as the quarter period of say a 30 Hz cycle (8.33 x lo-’ s), we conclude that homogenization of temperatures between different layers would not occur completely. On the other hand, adiabatic conditions would not be achieved either as this would require Fo < 0.005, or t 6 2 x IO-‘s. This means that adiabatic conditions will be achieved only for extremely high loading frequencies.
Composite laminates
3.2. A numerical model The analytical model in the previous section can provide only an approximate assessment of the heat transfer process between plies as it assumes a fixed temperature boundary condition. In reality, the boundary temperature would be time dependent due both to the applied time-varying load and to the fact that the adjoining laminae cannot be considered as infinite sources or sinks. For a more accurate representation of the problem, the interfacial temperatures must form part of the solution rather than be applied as boundary conditions. Consider a single lamina from a multi-ply composite plate subjected to a uniform uniaxial oscillating stress. Since each ply can be treated as a homogeneous orthotropic plate, it may be shown from principles of mechanics and thermodynamics [9] that, for quasi-static conditions and small temperature changes (dT 6 T,)
’ aQI
I-= i=l
axi
-pCef-
To
x (a,,k,, + a22d22 + X09),
(4)
where Qi is the heat flux passing through the point under consideration, p is the density, C, is the specific heat under constant strain, To is the absolute temperature of the material, a,, , az2are the coefficients of linear thermal expansion and uII, uz2 are the stress components in the respective inplane orthotropic directions. .%?“8Y is a higher-order term which gives rise to the “mean-stress effect” which was first discussed for isotropic materials by Wong et al. [13, 141, and may be neglected when the applied static stress is small. In all previous derivations it has been assumed that adiabatic conditions are achieved so that Qi vanishes and a closed form solution to eqn (4) is obtained. However, the preliminary analysis in Section 3.1. showed that heat transfer could be significant in the case of composite laminates and thus should not be neglected. Assuming that heat transfer during thermoelastic cycling is dominated by diffusion in the throughthickness direction (this would be true if the in-plane stress gradients were much less than the stress gradients between plies), the application of Fourier’s Law of heat conduction gives
485
eqn (4) yields the classical one-dimensional equation
de --
at
where y = k/pC, diffusivity and S=
is
2 g=s,
the
coefficient
of
-~(a,,d,,+a,,d,,+~~~) e
thermal
(7)
represents an internal heat source. Since the diffusion coefficient in the through-thickness direction is independent of ply orientation, eqns (6) and (7), though derived for a single lamina may, in the case where all plies are of the same material, be applied for the whole laminate. We note, however, that since stresses can vary from ply to ply, the heat source term [eqn (7)] will in general be a discontinuous function through the laminate thickness. With this complexity, the solution is best sought by the numerical method outlined in the following. Discretizing both t and z, a Crank-Nicolson finite difference respresentation of eqn (7) may be written (e.g. [IS]) as L9_, + A& + I!?:, , = B,,
(8)
where
and Bi=
-2s,$.!-e;9;+2-A..? yAt e:-1-e;;:
( > 1
(9)
in which superscript t and subscript i indicate the location on the t-z grid, whilst At and AZ are the corresponding time and spatial increments. To complete the system, initial and boundary conditions are required. At t = 0, it can be assumed that the body is at equilibrium with the surroundings so that f3(i,0) = 0. For t 2 0, two types of boundary conditions are applicable. Since composite laminates are usually made symmetric about their mid-planes, modelling is necessary only up to the plane of symmetry. By virtue of symmetry, and adiabatic boundary condition is applicable on this surface. That is,
de -= az where I is the coordinate in the through-thickness direction and k is the corresponding heat conduction coefficient. Defining the non-dimensional temperature 0 as 6 = (T - T,)/T, and substituting eqn (5) into
diffusion
0,
(10)
or in finite difference form, assuming i = N represents the boundary at the plane of symmetry,
e N+I =b-,.
(11)
486
A. K.
On the free surface (i = 1), allowance must be made for convective heat losses to the surroundings. This may be expressed as
Q, = he,
(12)
in which Q, is the heat leaving a unit area at the surface and h is the convective coefficient. Equating this to the heat arriving from beneath the surface, we have he,
=kE az +,’
Expressing eqn (13) in its finite difference form and upon rearrangement, we have 2h AZ
e0 = 0, - -0,
k
Writing eqn (8) for all i, i = 1 . N, and eliminating the terms ON+, and B, using eqns (11) and (14) a tridiagonal set of N equations in 0, is formed, and can be solved efficiently by well-known algorithms. 4. EXPERIMENTAL
To validate the model presented, the surface temperature response of a cyclically-loaded CFRP specimen was examined. Since the work of Dunn et al. [lo], it has been learnt that the application of the usual high-emissivity paint on composite specimens can, depending on its thickness, introduce substantial phase shifts to the SPATE signal. Furthermore, the result presented in their paper was the measured SPATE response, not the temperature response. As will be shown later, the system electronics can introduce amplitude attenuation and phase shifts, particularly at low frequencies, and must be corrected in order to infer the actual temperature response; because there is some uncertainty in the thickness of the paint layer used in their tests, and the characteristics of the equipment employed was unknown, new tests were conducted on a similar but unpainted specimen, taking into account the characteristics of SPATE’s system response. The specimen was made from Ciba-Geigy XAS914C prepreg and the lay-up was identical to that in Aluminium end [lo], namely [(O’, +45”, -4Y)J5. tabs were bonded to the composite using a room temperature setting adhesive and the finished specimen has a rectangular gauge section of approx. 15Ox23x3mm. Sinusoidal loading was achieved on a servohydraulic testing machine running under load control, and the load-cell signal was used as the reference input to SPATE’s correlator. Prior to testing, an area scan was initially performed on the gauge section of the specimen to ensure a uniform emission and the
WONG
absence of any bending. When proper alignment was achieved, testing was performed with SPATE focused on a fixed point. With a selected load amplitude and frequency, the amplitude and phase of the SPATE signal were recorded; because of the relatively low signals (<4 mV r.m.s.), a long integration time was required for adequate noise rejection. This was achieved by setting the correlator time constant to a maximum of 10 s (see SPATE manual [1 1]), and a data acquisition system was used to sample the amplitude and phase output channels of the correlator as rapidly as possible (between 1 and 2 s intervals) for a duration of 4 min. Two load amplitudes (4 and 6 kN) covering a frequency range of 0.5-30 Hz were tested. For the 20 Hz tests, the sampling duration was extended to 8 min in an attempt to provide a more accurate datum as, in adopting the convention in Dunn et al. [lo], all amplitude data were to be normalized by the 20 Hz signal. A thermocouple was used to monitor the reference temperature, ensuring that no significant viscoelastic heating occurred. 5. EFFECTS OF CONVECTIVE HEAT TRANSFER AND FILTERING The experimental results in Dunn et al. [lo] showed that the SPATE frequency response for a (+45”, -45”) lay-up behaves essentially the same as that for aluminium. That is, a response which is independent of frequency beyond approx. 5 Hz. This is easily explained by the current model. Because all plies have identical fibre and transverse stresses in this case, the source term S would, like an isotropic material, be constant through the thickness of the plate. The reason for the signal drop-off at frequencies less than about 5 Hz is mainly due to the characteristics of the SPATE electronics which were designed to eliminate any dc. component. Another possible cause for such a signal drop-off at low frequencies might be the presence of convective heat transfer between specimen and the surroundings. To investigate the significance of this effect, the present model was used to analyse a unidirectional composite plate with various convective boundary conditions. Because amplitude attenuation is most apparent at low frequencies, a heat transfer coefficient based on natural convection is most appropriate. Hence, taking a 200 mm long vertical plate at O.Ol”C above ambient to be typical, a value of h = 1 W me2 “C-r was estimated (see [16]). In a series of runs, it was found that using such a value for h, and indeed for h up to 100 W me2 ‘C-l, insignificant heat losses occurred for frequencies above 1 Hz. This suggests that no part of the SPATE signal drop-off at low frequencies is attributable to heat losses to the surroundings. The result also means that an adiabatic surface boundary condition may be used on all subsequent modelling. To account for the effects of the system electronics, a series of experiments were carried out using an
Composite laminates aluminium specimen. Since we know that, with the absence of any significant heat loss, the thermoelastic response of aluminium should be independent of frequency and should lag the applied load by 180”, the resulting SPATE response for such a material may therefore be used as a source for transforming SPATE data to temperature data. Figure 3 shows the amplitude correction curve used in the present study which was derived from the smoothed aluminium results. Because the phase offset curve (Fig. 4) did not exhibit an asymptotic behaviour, further investigation using the “edge effect” was undertaken. As SPATE detects changes in infrared flux, it is sensitive not only to changes in temperature, but also to changes in emissivity. This is easily seen from the expression for the infrared irradiation flux @, where @ = BeT4,
(15)
in which B is the Stefan-Boltzmann constant, e is the emissivity, and T the absolute temperature. The fractional change in @ is therefore given by d@ de _=_+4dT. @ e
T
(16)
Normally, the specimen under investigation would have a uniform emissivity so that SPATE detects only changes in temperature. However, when SPATE is focused onto an edge which is moving at the reference frequency, the contribution from de/e can become significant. Given that SPATE can detect a O.OOl”C temperature change at room temperature (T = 300 K), it is easy to see from eqn (16) that SPATE would be sensitive to an emissivity change of merely 0.0013%. This is precisely why SPATE can produce spurious results at free edges. In the current study, this effect was exploited in a series of tests to
487
Fig. 4. Phase shift of SPATE signal. (0) Aluminium response (shifted by 180”), (0) edge effect (displacement amplitude = 1.5 mm), (*) edge effect (load amplitude = 4 kN), (-) smoothed data which may be treated as the difference between the phase obtained by SPATE and the phase of the temperature response.
verify that the phase offsets observed in the aluminium tests were indeed not due to some real temperature phase lags. By attaching a two-tone strip (a reflective metallic foil whose lower half has been painted matt-black) on the testing machine’s actuator, and with SPATE focused onto the reflective/black edge, a sizable SPATE signal was obtained once the actuator was set in motion. In the first case considered, no specimen was mounted between the grips and the testing machine was run under displacement control at a fixed amplitude of 1.5 mm. Another test was run with the actual composite specimen mounted and using similar loading conditions (i.e. mean load of 8 kN, and load amplitude of 4 kN). In both cases, the change in the average emissivity of the field of view was the sole contributor to the SPATE signal as the strip was not loaded and thus could not be subjected to any thermoelastic effect. Setting dT = 0 in eqn (16) and noting that de must be in phase with the actuator motion, we deduce that the infrared irradiation flux should also be in phase with the actuator motion. However, as can be seen in Fig. 4, the phase offset for base cases was extremely similar to that found for the aluminium test, suggesting that such offset was an artifact of the equipment system response. A smooth curve was thus fitted through the data in Fig. 4, and was used for correcting the SPATE data to infer the actual temperature response.
6. MATERIAL PROPERTIES Before the numerical simulation can proceed, various material properties are required. Unfortunately, material properties for composites are not Fig. 3. Amplitude attenuation of SPATE signal. (0) Stress amplitude = 53 MPa, (0) stress amplitude = 34 MPa, (*) stress amplitude = 17 MPa, (-) smoothed data which may be treated as the ratio between SPATE amplitude and temperature amplitude.
as well documented as for metals and obtaining a consistent set of data was difficult as different sources report slightly different values. Nevertheless, best estimates of the properties for the XAS-914C
488
A. K. WONG
material used were obtained from British Aerospace [17-191, viz. E,, = 130,000 MPa, E22 =
9000 MPa,
G,, =
5800 MPa,
vu =
0.3,
surfaces for two different frequencies (10 and 15 Hz) and two different load amplitudes (3 and 5 kN). The resulting scans typically showed three distinct uniform regions with the signal from the middle strip being approx. 6.7 times greater than those on the sides. The scan data of each strip were averaged and, from the adiabatic thermoelastic equation [8], it may be shown that CQ,/CQ~ is given simply by ~II Ezz SP, -=_.az2
p = 1630 kg mm3,
(17)
in which E,, , Ez2, G,, are the elastic constants in the fibre, transverse and in-plane shear directions, v,~ is the Poisson’s ratio in the (12) direction and p is the density. The remaining parameters which are required for solving the thermoelastic/diffusion problem are the thermal properties. However, reliable values for these properties were not readily available. Values of k=0.87Wm-‘“C-’ and C,=990Jkg-‘“C-’ were reported for similar composites by Freeman and Kuebeler [20] whilst k = 1.5 W m-’ “C-’ and C, = 880 J kg-’ “C-’ were reported in Caurtaulds’ data sheet [21]. Because of such discrepancies, it was decided that simulation should be carried out for both sets of data. Another material parameter which lacks a reliable source is the coefficient of thermal expansion in the fibre direction (a,]). This is probably due to the fact that cl], for CFRPs in general is extremely small. Whilst the transverse expansion coefficient (ccZ2)is comparable in magnitude to that of aluminium and an agreed value of az2 = 28 x 10m6“C-’ has been found in many sources, reported values of a,, ranged between -0.7 x 10e6 [21] and 0.22 x lo-‘“C-’ [17]. In view of such diversity in data and suspecting that this small value could well be sensitive to batch-to-batch variations and local production procedures, experiments were carried out to measure this quantity. 6.1. Measurement qf a,,jazz As will be seen later, it is the ratio u,,/az2 rather than their respective absolute values which will be important in the present model. Measurement of this value was achieved using SPATE and a specially constructed specimen. The specimen comprised three separate strips of unidirectional XAS-914C composites bonded side-by-side between common aluminium end plates. The plates ensured that all three strips would undergo the sum displacements. The two outer strips had the fibres aligned with the loading direction (0”) whilst the middle strip fibres were perpendicular (90”). All three strips had identical gauge dimensions, namely, 160 x 23 x 2 mm. The surface of the composite were lightly grit blasted to remove most of the surface resin layer and an area SPATE scan was made on the unpainted
4,
(18)
SP, ’
whre Sp, and Sp, are the averaged SPATE signals for the 0” and 90” composites, respectively. The results on the range of tests considered consistently yielded the ratio
tl
=
2 = 0.010.
(19)
__
This result certainly confirms that the fibre-direction coefficient of thermal expansion is much smaller than its counterpart for the transverse direction. It is, however, interesting to note that this value was determined to be positive rather than the negative values frequently reported for CFRP composites. 7. RESULTS
AND DISCUSSION
With all necessary parameters, we can proceed to solve the nonadiabatic thermoelastic problem for the (O“, +45”, -45”) plate undergoing cyclic loading. It may be shown from laminate theory that the plate under consideration has a Poisson’s ratio v,,, of 0.683 (where x is the 0” direction). Assuming an applied uniaxial sinusoidal stress of the form 0 = Aa sin wt in the 0” (x) direction (where Aa is an arbitrary constant), and it may be shown that the laminar stresses are given by u ,,
= 2.256 Au sin wt,
022= -0.0607 Au sin at
for the 0” plies, and u ,,
= 0.370 Au sin cot, az2=
0.0326 A(Tsin ot (21)
for the f45” plies. Due to a lack of any reliable data for the temperature dependent parameters which give rise to the “mean-stress effect” discussed by Wong et al. [ 131and Potter and Greaves [22], the term ,%?“09 in eqn (7) will be neglected. In any case, the mean stress in the experiments will be kept low so as to avoid this term becoming significant. Hence, the substitution of eqns (20) and (21) into eqn (7) gives S = 0.038law cos wt
(22)
Composite laminates
489
for the 0” plies, and S = -0.0363aw
cos cot
(23)
for the f45” plies. The constant a in both cases is given by a = a2rAa/(pC,). In the present tests, a was taken as an arbitrary constant as its effect as a scaling factor was eliminated by normalization of the results by some chosen reference data (e.g. result at 20 Hz). Equations (22) and (23) were used in modelling the thermoelastic process of the laminate under consideration. The half-depth thickness consisting of 12 plies was discretized by 481 nodes, and a marching timestep of 0.02n/w s (period/lOO) was used for all frequencies studies. Computation was performed on a Vax 1l/780 using double precision arithmetic and subsequent mesh refinement analyses showed that the selected parameters were adequate. The full temperature field was solved at every time increment and the surface temperature was monitored as this represented the values which would be detected by any surface temperature sensor. It was found that the surface temperature quickly settled to a sinusoidal waveform and computation was terminated once steady-state was obtained. For most cases considered, the steady condition was achieved within 10 cycles and the resulting amplitude and phase of the surface temperature waveform were recorded. Figure 5 shows the through-thickness temperature profile at the moment for which the surface temperature is a maximum within a cycle. It is clearly seen that even though the source term is a discontinuous step function from ply to ply, the temperature profiles are smoothly continuous as a result of diffusion. At low frequencies, the time-scale is such that significant conduction can take place and therefore effectively flatten the temperature profiles. As the frequency is increased, the time-scale is contracted and less diffusion is allowed, yielded much more severe temperature gradients. In any case, it is seen that even at 30 Hz, the temperature profile is still far removed
Fig. 5. Temperature profiles at the instant of maximum surface temperature. ( ---) 5 Hz, (--) 10 Hz), (-) 30 Hz (abscissa scale: z = 0 represents the free surface and z = 1corresponds to the plane of symmetry. Ordinate scale: arbitrary).
Fig. 6. Temperature profiles at the instant of maximum surface temperature. (-) 100 Hz, (---) 1 MHz (same coordinate scales as for Fig. 5).
from the step function representing an adiabatic process, and the omission of the diffusion term in the thermoelastic equation cannot possibly suffice for such cases. For the purpose of illustrating the convergence to the adiabatic state, computation was carried out for frequencies up to 1 MHz (see Fig. 6). It is interesting to note from Fig. 6 that the maximum surface temperature was in fact not attained in the fully adiabatic state as might be expected. Slightly higher temperatures were achieved at moderate frequencies (of the order of 100 Hz) in which the dynamics of the system allowed temperature overshoots to occur near the locations of source discontinuity (ply interfaces). This phenomenon is somewhat analogous to the response of many other second order dynamic systems which exhibit a resonance behaviour. Figures 7 and 8 show both the computed and measured specimen surface temperature response when subjected to cyclic loading. Maintaining the convection of Dunn et al. [lo], and to avoid the need for various scaling factors, the amplitude data have been normalized by the corresponding result at 20 Hz. The experimental results have also been corrected for amplitude attenuation and phase shift introduced by the system response as discussed in Section 5. The measured results for the two different load amplitudes (4 and 6 kN) agree well, although the amplitude data appear to diverge from one another beyond about 20 Hz. This was perhapd due to a possibly higher noise level for the 6 kN case as, above this frequency, a great degree of load-frame vibration and a noticeably distortion loading waveform were experienced. Despite this, and the fact that a large number of material parameters were involved in the computation, the agreement between numerical and measured results is relatively good. It may be seen that the computed results for the two different published sets of thermal properties form bounds which encompass most parts of the experimental results.
490
A. K. WON0
One interesting feature which emerged clearly from the computed results showed the existence of a minimum in the temperature amplitudes between 0.5 and 2 Hz. This minimum represents the transition from the domination of the surface temperature by the angle ply the~oelastic efI’ect to that of the 0” surface ply. Since the effective heat source of each angle ply is roughly equivalent but opposite in phase to that of the 0” ply [see eqns (22) and (23)], and that there are twice as many angle plies as there are 0” plies, the resulting surface temperature when complete diffusion is permitted (as frequency tends to zero) would be in phase with the angle ply source. This may be seen in Fig. 8 as the phase between the surface temperature and load tends to 180” as the frequency decreases towards 0 Hz. Conversely, as frequency increases the process becomes increasingly adiabatic such that the phase of the surface temperature should in principle approach 0”.
8. NUMERICAL
STUDY
Having validated the numerical model, we can now use it to gain a better understanding of the thermal interaction between plies. It has been shown tht subsurface layers do affect the surface temperature characteristics. How deep SPATE can “see” when applied to composites is therefore the next obvious question, This is best answered by considering the problem in a decomposed form. Because the problem involved (neglecting the mean stress effect) is linear, the principle of superposition may be applied. This means that the overall probfem may be treated as a linear combination of a set of simpler but fundamental problems. Because of the step nature of the effective heat source, and again assuming an applied sinusoidal remote load, eqn (7) may be expressed as
x
0
cos U)f
(24)
where a = -au(pC,), n is the number of plies, and Hi is a unit-pulse function of z in which H, = 1 over the jth ply and vanishes elsewhere. For the jth ply, Ai is a parameter which is a function of the laminar stresses, namely A, = @da,, + Au&
(25)
Since only the source term in the diffusion equation is affected by stresses (and hence ply orientation), a basis set of solutions which is independent of ply lay-up configuration for a given material and number of plies (n) may be obtained by solving Fig. 7. Amplitude plot of 0,. (--) Computed (y = 1.04 x 10e6 mz s-l); (--f computed Q = 5.80 x IOw7m2 SS’); (0) measured (load amplitude = 6 kN); (0) measured (load amplitude = 4 kN). All results have been normalized such that the responses at 20 Hz are unity.
az;_ at
8’7,
coswt
y-$=Hp
for j = 1 to n. The
for a specific lay-up compiled as 8
=a
(24)
actual temperature field B configuration may then be
i
47,.
(27)
f=i
Since only the surface temperature is of interest in our case, we can apply eqn (27) specifically at the surface giving (28)
Fig. 8. Phase plot of B,. (- -) Computed (‘/ = 1.04 x computed (p = 5.80 x IO- ’ rn’ S-‘; 10m6m’ s-‘): (-) (0) measured (load amplitude = 6 kN); (0) measured (load amplitude = 4 kN).
in which 8, and TV, are, respectively, the values of 0 and 7, at the free surface (i = 1). To illustrate this, the set of basic solutions T+ was obtained for the 24 ply XAS914C material Computation was carried out using the averaged published values for thermal conductivity and specific heat, viz. 1.185 W m-l “C’ and 900 J kg-’ “C-l, respectively.
Composite laminates
Frequency
491
composite plate. For such plates, it is easy to see the relative contribution of each ply from these results. For example, it may be seen that at 5 Hz, the contribution from even the fourth layer can be considered significant whilst at 30 Hz, only the first two layers are important. Of course, how deep SPATE sees would much depend on the ply lay-up as the parameters Aj have the effect of scaling each of the curves shown in Fig. 9 by different amounts. Take the (O’, 45”, -45”) plate studied earlier for instance, and consider the relative contributions from the first three layers. From eqns (22) and (23), we have
(Hz1
Fig. 9. Amplitude plot of T,,. Ordinate scale is such that rs, tends to 1 as frequency tends to infinity.
Fig. 10. Phase
plot of T,,
Figures 9 and 10 show the computed surface temperature response for each of the individually excited plies. As expected, the amplitude of the surface temperature due to a unit source in the surface layer 7s, increases with frequency whilst the contributions from subsurface layers show a general decrease with increasing frequency. This set of solutions has the property that if a vectorial sum of the surface response is taken over all plies, it yields a constant which is independent of frequency. This is because such a sum is equivalent to the solution to the problem where the source term is uniform throughout the thickness of the plate as would be the case for a unidirectional Table
I. Relative contributions
Relative
Total response
0, :
Phase
.i I
magnitude 0.8269
(deg) 25.6
2 3
0.3790 0.1738
160.9 116.2
0.6939
58.3
(29)
Combining this with the computed results 7s,, we find that the relative contributions from the first three layers are as shown in Table 1. It is seen that even at 30 Hz, the contribution from the second layer is still considerable, and the notation that SPATE is a “surface” stress analysis device is simply untrue in this case. The fact that SPATE can see different amounts of each layer depending on the ply orientation poses a major problem for practical applications. For the simple rectangular plate studied, a fair prediction of the measured results was achieved mainly because the nature of the stresses with respect to the ply orientations was known. In practice however, the stress field for each lamina generally varies in the plane of the laminate, and since the coefficients A, are functions of stresses, interpretation of SPATE results could become extremely difficult. Furthermore, because the contribution of each layer to the surface temperature is also frequency dependent, a different SPATE scan pattern might emerge depending on loading frequency. Whilst this phenomenon makes the straightforward application of SPATE to composites difficult, it does open one new possibility. It will be revealed in the following section that this effect can be exploited, at least in principle, to obtain individual strain components in laminated plates.
to the surface temperature from each lamina (all amplitudes 20 Hz surface temperature response) 10
45
A, A, A, = A = -0.953. 1
Frequency 20 Relative
have been normalized
(Hz) 30 Phase
Relative
magnitude
(de&
magnitude
1.0464 0.3307 0.11099
19.2 136.0 72.8
1.1350 0.2800 0.0727
41.3
1.1670
I .oOOo
by the
Phase
(de& 15.0 117.6 40.2 30.7
A. K. WONG
492 9. STRESS/STRAIN
DECOMPOSITION-A POSSIBILITY
NEW
SPATE’s inability to deliver a tensorial quantity (which stresses and strains are) has long been recognized as its major disadvantage. This has led to many investigations of decomposition techniques (e.g. [23], [24]). However, all such techniques require the solution of the equilibrium equation with almost all of the normally required boundary conditions. This means that their application to components or arbitrary geometry is not readily achievable. For laminated CFRP plates, the fact that different loading frequencies might yield different scan patterns can be exploited for the purpose of obtaining the individual strain components. Providing in-plane diffusion is small compared with that of the through-thickness direction, the following technique can be independently applied point-bypoint, thus removing any necessity for boundary conditions. Combining eqn (24) and (29, the effective source term may be expressed as
seen that the final surface temperature obtained as
may be
0, = b i [(@ COS*4j + sin* $j) Ac, j-l
+
(j3
sin* 4j + cos* 4j) AcYY
+ 2(8 - 1)sin 4) cos 4j Ac,]r,.
(33)
Since the same strains are experienced by all plies, these terms may be factored, yielding es = cXXC (/I Cos*f$j+ sin* tij)r,, j-l
+ Cyyi
(fi sin* $j + cos* 4j)r,
j=l
+ 2(/I - l)Q 1 sin 4jcos 4jrX,,
(34)
j=l
where the quantities & = b A+,,,
S = -F
$, (a Aol, + Au,,),H,w cos wt. PI
(35)
(30)
Since there is discontinuity in stresses from layer to layer, it is convenient to express eqn (30) in terms of strains, viz.
S = b 5 (p AE,, + Ac,,),H,w cos cut i= I
may be considered as scaled strain amplitude components. In practice, the effects of the constant b, and indeed any other scaling factors which may be introduced by measurement instrumentations, can be eliminated by normalization with the data at a point where the strain is known (e.g. a far-field point where strain gauge data are available). Because 0, and rs, may be treated as vectors (i.e. possessing both magnitude and phase), we can resolve eqn (34) along any chosen direction giving
= b i [(fi cos2 4, + sin* 4,) AcII ,=I + (fi sin* 4, + cos2 +j) AcY,.- 2(/I - 1) where x sin 4, cos 4, At,,,]H,o
cos cot,
(31)
where u = i (B cos* 4, + sin* +j[fs, (cos($,~, - I(lrer)7 ,=I >
V = i: (P sin* 4j
and
+
COS*
~j)lzs,~cOs(~~., -
+dh
,=I
w
(32)
in which E = E,, /E2*. The subscripts x and y refer to the global reference axes, Ac,,~,AE,). and AE,, are the normal and shear strain amplitudes in .the respective directions and &j is the orientation of the fibre direction of the jth ply relative to the global axes. From eqns (24)-(28), it is
=
2(/? - 1) i
,= I
sin & cos 4j COS($,~,- tircr), (37)
and in which (8,) and /7+ ( are the magnitudes, and 1(10, I(lrgtheir corresponding phases. I(lrrris an arbitrarily chosen reference phase. Now suppose that measurement of the surface temperature at the same applied load was taken at three different frequencies (keeping in mind that the highest frequency should not exceed that which may
Composite laminates
cause significant viscoelastic or dynamic effects). The repeated application of eqn (36) gives
I[1
04)
w(w,)
utwz)
w(wz)
t&)
r+%)
L cYY7 (38) CXP
where w,, c2 and o, represent the three selected frequencies. Equation (38) is a set of linear equations in three unknowns and, provided they are linearly independent, a unique solution in &., .& and Z_ may be obtained. Because we do not have the analytical form of 7+, the conditions for eqn (38) to have a unique solution are not readily derived. However, it may be easily shown that there are two special cases in which no unique solution exists. As mentioned earlier, t,, has the property that its sum over all plies is a frequency independent constant. That is, i
7+ =
const
(39)
/=I
for all frequencies. From this, it is easy to show that the determinant of the matrix in eqn (38) vanishes for the case where B = 1,
(40)
or alternately, tan* 4, = tan2 I#J*=
. = tan* 4,.
493
Figs 9 and 10) into eqn (38), we get (setting ljlref arbitarily to 00)
II
0.3646 0.7513 = [ 1.0034
-0.5405 -0.4931 -0.4623
1
E,, CYY.
-0.9015 -0.9490
0.1331 0.1317
-0.9799
0.0935 I[ E,,
(42) Equations
(42) may be solved algebraically,
L_.= 4.8410,
c,,Y= -3.3097,
cXY = 0.0190,
or for the purpose of eliminating
%’ = -0.684 L
and
yielding (43)
any scaling effect,
2 = 0.004.
(44)
Recalling that the laminate has a Poisson’s ratio vXY = 0.683 and that from symmetry considerations no shear strain was applied, the inverse solution algorithm indeed deciphered the actual loading conditions to a good degree of accuracy. Of course, the success of this procedure would depend very much on the availability of accurate data for the many material parameters involved. The application of the above algorithm using the actual measured surface temperature data (averaged between the two tests) yielded
g= -0.646 LXX
and
3=0.107, %r
(46)
(41)
The first [eqn (40)] is applicable for an isotropic material, and the second [eqn (41)] is true for laminates in which the absolute angles of all plies are equal. Since the assignment of the global reference is arbitrary, the latter condition implies that laminates with two or less ply orientations would not be amenable to solution by the above method. This result is not entirely surprising given materials, uniour experience with isotropic directional and (+45”, -45”) composites. What is surprising is that for other practical ply lay-up configurations which consist of at least three fibre orientations, a non-vanishing determinant invariably results, indicating that a unique solution is possible. To illustrate this principle, the decomposition algorithm is applied to the foregoing model assuming, for the purpose of this exercise, the simulated surface temperature response is without error. That is, selecting the three frequencies to be 10, 20 and 30 Hz say, we assume that the corresponding measured values of 0, are as given in Table 1. Substituting these, together with the necessary material parameters and the solution of 7,y, (as shown in
which, whilst partly acceptable, could be improved with more accurate material parameters. Nonetheless, this unique potential for decomposing strain components of laminated plates from thermoelastic data is clearly demonstrated, and could perhaps make SPATE a much more powerful tool for analysing composites than it is for isotropic materials.
10. CONCLUSION
A first order analysis of the heat diffusion characteristics of CFRP from a micro-level has been presented. For the normal frequency range suitable for thermoelastic stress analysis (5-30 Hz), it is concluded that whilst fibres and matrix may be treated as a homogeneous material from SPATE’s point of view, the thermoelastic process of different layers may, in general, neither be treated as homogeneous nor adiabatic. A non-adiabatic model for the thermoelastic effect for laminated composite plates has been developed, and its validity has been demonstrated by the good agreement between predicted and measured surface temperature/load response. A superpositional analysis provided an insight to the
A. K. WONG
494
mechanics of interlaminar heat transfer and the extent to which each layer can influence the surface temperature depending on frequency and ply orientation. It is concluded that the process of interlaminar heat transfer makes interpretation of SPATE results difficult unless an extremely high loading frequency is used. On the other hand, there now exists a possibility, at least in principle, for exploiting the frequency-dependent nature of the surface temperature for determining individual strain components from thermoelastic data. Acknowledgements-This work was undertaken during the author’s visit to RAE, and the encouragement and support from members of the MS6E Section was greatly appreciated. The author also wishes to express thanks to British Aerospace, Warton, for providing the material data for the composite used.
REFERENCES I. Thomas W. (Lord Kelvin), Trans. R. Sot. Edinburgh 20, 261 (1853).
2. Belgen M. H., ISA Trans. 6, 49 (1967). 3. Harwood N. and Cummings W. M., Strain 2, 7 (1986). 4. Stanley P. and Chan W. K., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 131, 102 (1987). 5. Wong A. K., Dunn S. A. and Sparrow J. G., Nature 332, 613 (1988). 6. Owens R. H., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 731, 74 (1987). 7. Cox B. N. and Pettit D. E., Proc. SEM Spring Co& Sot. Exp. Mech.. Housron, Texas, p. 545 (1987).
8. Biot M. A., .I. appl. Phys. 21, 240 (1956). 9. Potter R. T., Proc. SPIE Conf, Stress Analysis by Thermoelastic Techniques, London 731, 110 (1987). 10. Dunn S. A., Lombard0 D. and Sparrow J. G., Proc. SPIE Conf, Stress and Vibration: Recent Developments in Industrial Measurement and Analysis, London. p. 129
(1989). 11. SPATE 8000 MANUAL. Ometron Ltd, p. 129 (1989). 12. Luikov A. V., Analytical Heat Diffusion Theory (Edited by J. P. Hartnett). Academic Press, New York (1968). 13. Wong A. K., Jones R. and Sparrow J. G., J. Phys. Chem. Solids 48, 749 (1987). 14. Wong A. K., Sparrow J. G. and Dunn S. A., J. Phys. Chem. Solids 49, 395 (1988). 15. Ames. W. F., Numerical Methods for Partial Differential equations, 2nd edn. Academic Press, New York (1977). 16. Whitaker S., Fundamental Principles of Heat Transfer, p. 262. Robert E. Krieger Publishing, Malabar, Florida (1985). 17. British Aerospace, Design Data Sheet, Vol. 3, Section 2, Chap. 2, pp. 2-3, Amendment No. 2 (1985). 18. British Aerospace, Report No. SOR(P) 157 (1986). 19. British Aerospace, Report No. BAe-WSO-RP-RESSOR-000177, Addendum 1 (1989). 20. Freeman W. T. and Keubeler G. C., Composite Materials: Testing and Design (3rd Conf.) ASTM STP 546, ASTM, p. 435 (1974). 21. Caurtaulds Ltd, Dafa sheet G14, Sheet 2 (1974). 22. Potter R. T. and Greaves L. J., Proc. SPIE Conf, Optomechanical Systems Engineering, San Diego, California 817, 134 (1987). 23. Ryall T. G. and Wong A. K., Mech. Mater. 7, 205 (1988).
24. Huang Y. M., Determination of individual stresses from thermoelastically measured trace of stress tensor. Ph.D. thesis, University of Wisconsin, Madison (1989).