epoxy composite laminate

Composites Science and Technology 46 (1993) 345-351

THE T H E R M A L R E S P O N S E TO D E F O R M A T I O N TO F R A C T U R E OF A C A R B O N / E P O X Y COMPOSITE LAMINATE

a

A . D . M e l v i n , "'b A . C. Lucia" & G. P. S o l o m o s " Commission of the European Communities, Institute for Systems Engineering and Informatics, J.R.C. lspra, 21020 lspra (Varese), Italy b Department of Physics, Loughborough University of Technology, Leics., UK, LE11 3TU (Received 9 September 1991; revised version received 10 March 1992; accepted 24 March 1992)

Abstract The thermoelastic response to deformation of carbonfibre~epoxy-resin composite laminates has been considered theoretically, and compared with experimental results. It is found that the surface temperature is strongly dependent on the near-surface lay-up. The total thermal response to tensile testing to failure is also reported. The results indicate that a simple, but accurate, estimation of the yield strength in composites is possible by measuring the thermoelastic limit strength, and that microstructural 'events' are detectable by measuring the surface temperature with sufficiently high resolution.

It has been found in isotropic materials that analysis of the thermal response to deformation can provide information on a body's thermomechanical properties. 2 In particular, the yield point is clearly recognised from the response of the surface temperature, and in creep tested metals, an estimation of the level of creep damage is possible. 3 The extension of the technique to composite materials requires a more in-depth analysis of heat conduction because of the laminated structure. A model is developed of the thermoelastic response to deformation of a multidirectional laminate, a specific lay-up is then chosen and the predicted surface temperature is compared to experimental results. The experimental data contain more information than simply the thermoelastic response. Before brittle fracture there exists a brief period of thermoplasticity visible by a net temperature increase of the specimen surface. The point of inversion clearly reveals the yield strength of the body, a parameter that is very difficult to estimate from only stress/strain data. This definition of the yield strength is based on the thermoelastoplastic limit and is, therefore, founded on sound thermodynamic principles, rather than on arbitrary convention.

Keywords: thermal response analysis, composite laminates, thermoelastic-plastic limit

INTRODUCTION The thermal response to deformation depends critically on whether the strain is elastic or non-elastic. For a purely elastic tensile strain in an adiabatic configuration the thermoelastic effect results in a cooling of isotropic specimens with positive thermal expansion coefficients, first described in the early 19th century, and explained thermodynamically by Thomson (Lord Kelvin) in 1853.1 However, if inelastic deformation is also present, then heating due to the dissipation of the applied mechanical energy will normally overcome the thermoelastic contribution and result in a net specimen temperature increase. It is well known that mechanical deformation causes temperature changes, but they go unnoticed if the loading is slow, resulting in an isothermal process. At the other extreme, under very fast loading conditions, considerable viscous heating can occur, possibly leading to localised melting.

ANISOTROPIC THERMOELASTIC THEORY Multidirectional fibre reinforced composites are usually manufactured by stacking laminae of unidirectional sheets of fibres pre-impregnated with resin. The structure is therefore heterogeneous and so consideration should be made of the stress discontinuities between the fibres and matrix for the purposes of calculating the overall temperature field. It would be expected that the fibres and matrix would experience different temperature fluctuations due to their dissimilar thermoelastic constants. By modelling a typical carbon fibre as an infinite cylinder and

Composites Science and Technology 0266-3538/93/$06.00 (~ 1993 Elsevier Science Publishers Ltd. 345

346

A. D. Melvin, A. C. Lucia, G. P. Solomos

imposing a step temperature change on the outer surface, Wong 4 showed that the fibre would attain a uniform temperature in only 10#s. It can thus be concluded that, although large stress discontinuities may exist between fibres and matrix, for thermal purposes the laminate may be considered as macroscopicaUy homogeneous. The situation regarding heat transfer between the laminae is not as simply analysed. In the following analysis, we calculate the temperature field in a laminate due to uniaxial sinusoidal loading. The laminate can be treated as a homogeneous, orthotropic plate from the macroscopic point of view, but with anisotropic elastic properties. The derivation of the linear theory of thermoelasticity 5 is, therefore, still valid for each lamina:

elastic stress analysis on composites, have completely disregarded the effects of heat conduction on the specimen temperature field. This has been justified by executing the mechanical tests at 'high' frequencies and declaring the thermodynamic system as quasiadiabatic. However, Wong 4 recently showed that this can lead to significant errors in estimates of the specimen temperature up to testing frequencies as high as 30 Hz. In this work the thermal diffusion term in eqn (1) is, therefore, retained. To avoid any confusion over the coordinate axis directions it is preferable to change notation such that x3 = z. If there are no applied stresses in the z(x3) direction, then the differential equation for the temperature field for each lamina will be O2T

0 k,.. pco ¢ - Ox--:

= -ro%o.O~.o

(1)

where p is the density, co is the specific heat at constant stress, T is the absolute temperature, x~, are the coordinate axes, k~,~ is the thermal conductivity tensor, te,,o is the thermal expansion tensor, o~,o is the stress tensor, and the dot notation denotes differentiation with respect to time. Equation (1) indicates that temperature changes due to thermoelasticity are as a result of the rate of change of stress, and not the absolute stress magnitude. In a more exact theory, 6 taking into consideration the temperature dependence of the elastic properties of the material, a correction term is derived which does depend on the stress magnitude. The influence of this term may be considerable in some materials at high values of static stress (most notably a~-titanium7), but in this work it may be neglected. Local stress concentrations in the body, such as at the interface between two laminae, result in differences in the rate of change of temperature locally, as can be seen from eqn (1). The presence of interfaces in the laminate results in thermal diffusion being dominated by thermal gradients in the through-the-thickness direction when the component of the thermal conductivity tensor in the fibre direction is not much greater than the component in the through-the-thickness (x3) direction. Specifically, assuming that kl! )P k33 and k22 = k33 , then 02T k33 ~

ox3

02T >> kit ~ ,

ox~

02T k22

Ox 2

(2)

It is, therefore, justified to include in the analysis only the x3 component of the thermal diffusion term in eqn (1). Also, since k33 is independent of ply orientation, because of transverse isotropy, the term will be valid over the whole laminate. Most previous workers, when performing thermo-

pcoT - k

8Z 2 =

-T(tellOll + t1"22022 Jr" 1~'12012) (3)

Note that o~12 0 for orthotropic materials, (x, y, z) is the coordinate system of the laminate, and (1, 2, 3) are the lamina axes. Also, since the only component of the thermal conductivity tensor of interest is k33 it is represented by the scalar k. Transformation between the two coordinate systems is achieved by using classical laminate theory. A model is now developed of a symmetric, multidirectional composite specimen undergoing fully reversed sinusoidal elastic deformation, Ox = Ao sin tot. The laminate is composed of 2/ laminae, each of thickness a (total thickness 2h = 21a), but only one-half of the laminate needs to be modelled because of the symmetry. For each lamina i there will be an equation of the form eqn (3), thus =

a2T,(z, t) Xi

8z 2

Z~ + kiig'(t) = Ti(z, t)

i = 1, 2 , . . . ,

l

(4)

where X~and k i are the thermal diffusivity (X = k/pco) and thermal conductivity (in the z direction), respectively, and g i ( t ) = - T ( o q l O l l + a~22022)i is the relevant heat source term for lamina i. The set of equations will now be solved for T(z, t) for the case of temperature changes small in comparison with the absolute temperature. Initially, constant temperature is assumed throughout the specimen. The constant can be taken as zero, for convenience, by solving the problem for the temperature change 0 rather than for the absolute temperature T (0 = T - To). The initial condition of the problem is then O(z, 0 ) = 0 for O~_z<_h. The centre line of the specimen is taken to be z = 0. This boundary must be adiabatic, since it is the central plane of a symmetric body. The boundary at z = h is one of the outside surfaces of the specimen, and is subject to linear heat transfer according to Newton's law of cooling, with heat transfer coefficient H, given

Thermal response to deformation o f a carbon~epoxy composite laminate in W / ( m zK). At each of the internal boundaries, perfect thermal contact and continuous heat flux are assumed. These boundary conditions can be expressed

Substitution of eqn (12) into eqn (6) yields

O(z, t ) =

dr Z =O

as

at

z =0

Oi(z,t)=Oi+l(z,t)

at

z = a , 2a . . . . .

at

z=a, 2a,...,(l-1)a

at

z=h

a0, az

OOi+l

ki--=--ki+l--

az

ktOOt+HOt=O az

a~ =0

a,j(z, t l z',

gj(z', t

dz'

(6)

I. The Green's function is given by

~, ~ 1 kj - -

,,:l

as

)----~in(Z)~)jn(Z ) N.,Z/

= --TOW AO(Kll~ll

= ~ k__/fz'+' Ip~.(z')dz'

1

Z

xcos T][y

(9)

The constants Aj, and Bj,,, and the eigenvalues ft, are derived from the boundary conditions. In this problem, simplification is possible because each of the thermal diffusivities and conductivities of the laminae are equal. The constants A j,, and Bj,, are found to be 0 and 1, respectively, and the eigenvalues are given by the following transcendental equation:

(10)

where r / = h f l / V ~ and H* = H h / k . For small H*, only the first few roots of eqn (10) need to be found numerically, since for r/>>H* the equation can be approximated to tan r / = 0, with the solution r/, = n:r. The normalisation function eqn (8) is then

kh ( N,=~

1+

sin 2r/~] 2r/. /

(11)

Consequently it is found that

c,/(~,t[ ~', l-) = %(z, t I z', 1

l-) ~,

,~,

/fl,,z\

/fl,,z'\

= -X ~ - e -t~"( - ~cos/----~| .--1=-N, \ ~/X) c o s k ~ - x ]

(:-:+,) s,n(,v) +: sin fl~e-~' 1

(12)

(15)

where ~ = --T0(Kll~ll + Ka2~22)/AOco

and the infinite set of eigenfunctions is given by

r/tan r/= H*

I-1

x [fl~ cos cot + co sin cot (8)

(14)

O(z, t) = h ~= =, rl,(fl4, + coZ)N,

(7)

j=l ~j Jz'=z,

flnz]+ B/. COS(~) ~,j.(z) =Ajnsin(Vx/

+ K 2 2 0 ' 2 2 ) j c o s col"

where the constants K,, and Kz2 are derived from laminate theory by transformation of the stress tensor for lamina j to the coordinate system of the laminate. On performing the integrations in eqn (13) the result is

where the normalisation function is defined by

k ~

\Vx:

The heat source terms in eqn (13), gj(l-), are sinusoidally varying functions that depend on the fibre orientation of the laminae with respect to the coordinate axes of the laminate. They can be written

(l-1)a

(5)

Gij(z, t I z', l-

7:) = z.., e -a"(

N,

~n z

gj(l-) = - To(CrllO,, + azzOz2)j

..'z'=zj

for i, j = 1, 2 . . . . .

1 e_,~(,_~) N,

\Vx/Jz'=z,

The solution to eqn (4), subject to eqn (5), is a special case of the general problem solved by Ozisik. 8 It is stated here in terms of the composite medium Green's function Gij(z, t [ z', lr):

Oi(z , t) =

n=l

×cos(-~nZ f zj+l

a0a --=0 az

347

(16)

A specificmodel The solution eqn (15) can be applied to any material and symmetric lay-up provided that all laminae have the same thermal properties. As an example, a quasi-isotropic, carbon/epoxy laminate was chosen with a fibre volume fraction of 0-42 + 0-02 and lay-up [(+45/-45/0/90)2]s. The stiffness matrix was calculated by using the following elastic constants (from manufacturer's specifications and4): Ell = 130GPa, E22 = 9 GPa, Gl2 = 5"8 G P a and v12 0.3. (v21 was calculated from the relationship: vztE22=vlzElt). The matrix was then inverted to give --

e, = 1-911 x 10 -ll A o s i n cot ev = - 5 . 5 5 6 × 10 -12 A o s i n co

(17)

exy=0 The constants Kll and K22 , calculated from eqn (17), are shown in Table 1. The density of the specimens was known to be 1470 kg/m 3, but no information was available on the thermal properties. The values used by Wong 4 on a similar composite were, therefore, utilised, i.e. c~=990J/(kgK), k=0-87W/(mK), oq~=0-28× 10 -6 K - 1 , a'22 = 28.0 x 10 -6 K -l. Equation (15) was approximated after assessing the number of terms

348

A. D. Melvin, A. C. Lucia, G. P. Solomos

Table 1. [~minm stress factors calculated for carbon/epoxy specimen with lay-up [(+45/45/0/90),1,

Ktt

Ply direction

K~2

0°

2.485

0.002

19° +45 °

-0.675 0.920

0.158 0-095

required to obtain a convergent solution. It was found that convergence of the series was closely related to the value of the angular frequency 09. For 10 Hz cycling, 20 terms were sufficient, whereas at 100 Hz, 100 terms were required to guarantee convergence. At low frequencies, at room temperature, a heat transfer coefficient dependent on natural convection is usually quoted. 9. Whitaker ~° estimated a value of H, based on natural convection, of 1 W/(m2K) for a 200mm long vertical plate at 0.01°C above room temperature. This value, although not specifically derived from a carbon/epoxy composite, is not greatly material dependent and was utilised initially for the modelling. Various other values up to 100 W/(m 2 K) were also applied (see below). It should be pointed out that this estimate of the coefficient for surface heat transfer is only valid for tests taking place at room temperature. Higher ambient temperatures would involve a greater contribution from radiation. Figure 1 shows the predicted temperature change through the thickness of the specimen for stress controlled cycling at various frequencies, with a stress amplitude of 300 MPa. In each case the curve was plotted at the instant in time in one cycle when the surface temperature (z = 1 mm) was at its minimum. Each of the tick marks on the abscissa of the graph corresponds to a boundary between laminae. It can be seen that the troughs and peaks developing at higher frequency are due to the 90° and 0° laminae, --5

Hz

--10

Hz

respectively. The conduction of heat has the effect of smoothing out temperature variations through the thickness of the specimen. In Fig. 2, through-thethickness temperature changes are again shown for the composite specimen, but at higher frequencies. At 1 kHz the lay-up of the specimen is clearly defined and adiabatic test conditions have been almost attained. However, thermoelastic stress analysis at this frequency is impossible with present-day equipment. The results show, unequivocally, that adiabatic testing conditions are not attained even at high test frequencies. It can be expected, therefore, that the measured surface temperature will be due to a combination of the thermal response from several of the plies nearest to the surface. This is in contrast to the generally held view that when measuring temperature during sinusoidal loading of composites only the surface ply is of importance, although in agreement with the conclusions of Wong, 4 using a finite element technique. These considerations need to be taken into account when performing stress pattern analyses, using the SPATE instrument for example. Failure to do so could result in significant errors in estimates of the surface stress field. The effect of increased surface convective heat transfer was also analysed. The specimen was modelled with convective coefficients H of 1.0, 10 and 100W/(m 2 K). It was found that for test frequencies above 1 Hz there was no variation in the temperature patterns observed for the range of H used. For frequencies less than 1 Hz the magnitude of the surface temperature was reduced as H was increased, and the phase lag between the load cycle and surface temperature response decreased from the initial value of 180°. Figure 3 shows the situation using the three values of H for the low test frequency of 0.01 Hz. For monotonic uniaxial loading, the derivation of the equation for the temperature field is slightly simpler. For applied stress proportional to time,

.... 30 Hz

100 Hz

O-

E ~-100-

//

~

/

.

~

0 -50-

o -I00.c o -150.

."

2 -200-

-200E

kHz

50-

-50E

-150-

--1

100-

& -250

-250-

F- - 3 0 0 . -300 i i J 0.000 0.125 0.250 0.375

i 0.500

i i 0.625 0.750

i 0,875

1,000

Distonce t h r o u g h s p e c i m e n thickness [ m m ]

1. Temperature variations through the specimen thickness for fully reversed sinusoidal loading at various test frequencies, with stress amplitude 300MPa. The graph represents the instant in one cycle when the surface temperature is at its minimum. Fig,

-.350 0.000

0.1t25

I I 0.250 0.575

I I ~ 0.500 0.625 0.750

I 0.875

1.000

Distonce t h r o u g h specimen thickness [ r a m ]

Fig. 2. Temperature variations through the specimen thickness for fully reversed sinusoidal loading at higher test frequencies, with stress amplitude 300MPa. The graph represents the instant in one cycle when the surface temperature is at its minimum.

Thermal response to deformation of a carbon~epoxy composite laminate --1

--10

100

1

./

,oo I

///

o[

\'

/

-50

I

k--

-150

"\

I 1 O0

I 120

~ 140

I 160

I 180

200

Time [ s ] (1 cycle)

Fig. 3. Temperature change versus time at the surface of the specimen for a test frequency of 0.01Hz, and various values for the convective heat transfer coefficient H (units: W/(m: K)).

ox=St, the heat source terms, eqn (14), for the laminae become g / ( r ) = - T,,(~,, a , , + ~r::a:~)j

(18)

= - T o S ( K t l O q , + K::c~:e)j

and the integration of eqn (13) gives the result:

O(z, t) = h

1

c°sky )

2

(g/-gj+Osm

x

T

+glsinr/,

j=l

x [1 - e -t~'t]

] (19)

The graph shown in Fig. 4 compares the thermal response, calculated from eqn (19), at various stress rates, to the response expected if the test were adiabatic, i.e. the thermal response of the surface ply alone. The convective heat transfer coefficient used in this case was H = 10 W / ( m 2 K). Even though the test performed at the stress rate of 50 MPa/s resulted in a linear thermal response plot, its gradient was lower 1

--5

. . . . 10

. . . . 50

than the adiabatic case, indicating the influence of the subsurface laminae on the surface temperature. There are both advantages and disadvantages to being able to sense the temperature of the subsurface laminae. Although the p h e n o m e n o n indicates that the thermoelastic stress analysis technique may be invalid, the implication is that defect detection, using thermal response measurements, may be assisted by the effect. Subsurface defects (such as delaminations), which become heat sources under stress, should be detected on the surface and thus characterised and perhaps localised if enough sensors are bonded to the specimen. The analogy with acoustic emission monitoring is apparent, and it would be expected that the two techniques would be used concurrently. Equation (19) and graphs such as Fig. 4 serve to select suitable stress rates for tensile tests. The stress rate should be chosen so that the temperature/stress curve is linear when the specimen is in the elastic strain region, so that yielding may be detected by divergence from linearity, in the same way as for isotropic materials. 2 RESULTS A N D DISCUSSION

[~/,z\

:l ~.fl.N.

349

--adiabatic

-5o~

An experimental investigation was made into the thermal response from carbon/epoxy laminates undergoing tensile tests. The surface temperature was measured by 0-5 mm diameter thermistor beads; the experimental arrangement is described in Ref. 11. The specimens, supplied by Agusta Helicopters S.p.A., had the dimensions shown in Fig. 5; the end tabs were made of glass fibre reinforced plastic. The lay-up and mechanical properties were the same as those used for the theoretical study above. They were tested at various strain rates with the surface temperature being monitored locally on one side at the midpoint. Figure 6 shows the stress/strain graph produced from a uniaxial monotonic tensile test to failure at a strain rate of 2 . 3 × 1 0 - 4 S - l . The dotted line represents a linear elastic response; it can be seen that departure from linearity occurred at a stress of approximately 375 MPa. The accompanying temperature/stress graph is shown in Fig. 7. In this

-100-

-150-

,

T

-200k--

-25C 0

I O0

200

300

400

500

Stress [ M P o ] 4

Fig. 4. Temperature change plotted against stress at the specimen surface for various monotonic loading rates (in MPa/s); H = 10 W/(m 2 K). The solid line corresponds to the adiabatic response of the surface ply.

B

-':=

G

--~"

Fig. 5. Carbon/epoxy composite laminate speomen design, lay-up: [(+45/-45/0/90)2],. Dimensions (in mm): L = 271, B=60, C = 7 , D = 9 , G = 150, T = 2-16, W=25.

A. D. Melvin, A. C. Lucia, G. P. Solomos

350 500 400 -

~300200100-

O,

0.0

t

I

0.2

0.4

I

0.6 Stroin

I

0.8

~. 10

12 1.

1.4

[~]

1 ~ . 6. Stress/strain response to a monotonic tensile test on

a carbon/epoxy specimen with linear elastic response shown by a dotted line. The stress drops marked A and B are discussed in the text. case the dashed line represents the predicted thermoelastic temperature response given by eqn (19). Agreement between experimental and theoretical data is good until a stress of about 270 MPa is reached, when the thermal response of the specimen begins to invert prior to brittle fracture. The observed inversion in the thermal response to the imposed deformation signals the upper limit of the thermoelastic region. At higher applied stresses, heat dissipation is promoted by plasticity in the body, leading to a net temperature increase, overcoming the temperature drop due to the thermoelastic effect. In isotropic bodies, under adiabatic test conditions, a sharp inversion point is observed. This value of the applied stress (strictly, the stress at which the thermoelastic gradient ( a T / a o ) tends to zero) is known as the thermoelastic-plastic limit Oo." For the case of the composite laminate under study, adiabatic conditions were not attained and consequently the inversion of the thermoelastic gradient was less abrupt due to heat diffusion over an area

- -

Experimentel

Theoreticot

0 v

-25-

& & -50g®

/ //B

f/

-75-100-

E

-125-150 0

1O0

200 .300 Stress [MPa]

400

500

Fig, 7. Temperature change versus stress response to a monotonic tensile test on a carbon/epoxy specimen, with theoretical elastic response shown by the dashed line. The stress drops marked A and B are discussed in the text.

surrounding the site of the temperature sensor. It is, therefore, not possible to determine o0 in the same manner as for isotropic bodies. However, a representative 'elastic limit' can still be defined as the first deviation of the temperature/stress curve from linear thermoelastic cooling. At stresses beyond this point, thermodynamically irreversible heat producing events took place, implying entropy production and therefore 'permanent' structural changes. It has been found 11 that, if tests are performed on metallic specimens up to loads between the thermoelastic limit and the conventionally defined 'yield point', then signs indicative of damage can be detected from the temperature/stress curve, although not always from only the stress/strain data. The form of damage depends on the material, but in this case it is believed to be mainly due to localised interlaminar debonding on a microscopic scale. For the specimens reported here, the thermoelastic limit was at a stress approximately 100MPa lower than the estimate of the elastic limit given by analysing only the stress/strain graph. It is possible that there was also non-linear elasticity present during the test. However, this would be revealed by the deviation from linearity of the conventional stress/strain graph (Fig. 6) and could, therefore, only have occurred at a much higher stress than the thermoelastic limit. The thermoelastic limit may consequently provide a more accurate estimation of the true elastic limit than the value deduced from analysis of the stress/strain data, although it should be remembered that it is dependent on the near-surface lay-up of the specimen. The extent to which subsurface laminae influence the observed thermoelastic limit depends on the strain rate, as shown in Fig. 4, and is, therefore, to some extent controllable. Continuous fibre composites are often treated as elastic to failure in engineering applications, a fact justified by the global stress/strain response (Fig, 6). The results described above provide an indication that non-linear behaviour may begin at a strain of about 0-5% in the carbon/epoxy laminate tested. Further work is being carried out into the physical significance of this find. It is hypothesised that the thermoelastic limit may correspond to the fatigue limit, i.e. the stress below which 'infinite' fatigue life is expected. It should also be recognised that oo is a local estimate, because of the point-contact temperature sensors, whereas the stress and strain (measured by extensometer) are averages over the specimen gauge length. This has the implication that the temperature sensors may be distributed over the specimen surface to obtain high spatial resolution of yield information, a distinct advantage over strain gauges, the alternative method for revealing local yield information, which are much larger in size. The difficulties involved in applying strain gauges to composite materials are

Thermal response to deformation o f a carbon~epoxy composite laminate comparable to the particular requirements for recording very small temperature changes during mechanical testing. T o achieve a measurement resolution of 1 m K the test must take place in still air (by enclosing the specimen in a chamber), and a 1/~V resolution voltmeter must be made available. The experimental procedure is described in detail in Ref. 11. Figure 7 also reveals that two small stress drops, observable in Fig. 6 just prior to fracture (labelled A and B), were heat producing events. These events were repeated with other tests and are thought to have arisen from failure of the +45 ° laminae. By means of thermal response analysis it would be possible, given enough sensors, to localise these events and thus deduce where the failure was initiated in the specimen. CONCLUSION Analytical solution of the thermoelastic equations for a composite medium demonstrates that quasiadiabatic conditions are not possible when deforming laminate specimens. The implication is that the measured surface temperature is due to contributions from several subsurface laminae as well as from the surface ply. Analysis of the thermal response to deformation allows for a novel definition of 'yielding' based on the thermoelastic limit, the measurement of which is highly localised due to the point-contact temperature sensors. The thermoelastic limit is clearly identified from a temperature/stress graph, even in the case of specimens showing little or no ductility, in which a practical yield limit is difficult to estimate. In the carbon/epoxy specimens tested, the thermoelastic limit was approximately 25% lower than the elastic limit determined from stress/strain data. Analysis of the thermal response to deformation of composite

351

laminates has the potential to localise heat producing defects and reveal more about the mechanical behaviour of such bodies than can conventional stress/strain data.

REFERENCES 1. Thomson, W., On the dynamical theory of heat. Trans. Royal Soc. Edinburgh, 20 (1853) 261-83. 2. Melvin, A. D., Lucia, A. C., Solomos, G. P., Volta, G. & Emmony, D. C., Thermal emission measurements from creep damaged specimens of AISI 316L and Alloy 800H. Proc. 9th Int. Conf. on Experimental Mechanics. Technical University of Denmark, Lyngby, Denmark, 1990 pp. 765-73. 3. Melvin, A. D., Lucia, A. C., Solomos, G. P., Volta, G. & Emmony, D. C., A note on the use of the thermal response to deformation as a damage assessment tool. J. Mater. Sci. Lett., 9 (1990) 752-3. 4. Wong, A. K. A non-adiabatic thermoelastic theory and the use of SPATE on composite laminates. Proc. 9th Int. Conf. on Experimental Mechanics. Technical University of Denmark, Lyngby, Denmark, 1990 pp. 793-802. 5. Beghi, M. G., Bottani, C. E. & Caglioti, G., Irreversible thermodynamics of metals under stress. Res Mechanica, 19 (1986) 365-79. 6. Wong, A. K., Jones, R. & Sparrow, J. G., Thermoelastic constant or thermoelastic parameter? J. Phys. Chem. Solids, 48 (1987) 749-53. 7. Wong, A, K., Sparrow, J. G. & Dunn, S. A., On the revised theory of the thermoelastic effect. J. Phys. Chem. Solids, 49 (1988) 395-400. 8. Ozisik, M. N., Heat Conduction, Wiley-Interscience, New York, 1980. 9. Gr6ber, E., Erk, G. & Grigull, F., Wiirmeiibertragung, Springer-Verlag, Berlin, 1955. 10. Whitaker, S., Fundamental Principles of Heat Transfer, R. E. Krieger, Malabar, India, 1985. 11. Melvin, A. D., The application of thermal emission analysis to damage assessment and material characterisation, PhD Thesis, Loughborough University of Technology, UK, 1991.