410 V, Bianchi et al material elastic properties. This technique is described aI fiber transverse CTE in a previous papct The composites were treated in an argon gas flow uH fiber major Poissons coefficient, vertical and tubular electric furnace which could be matrix Poissons coefficient heated to 1550C, with the temperature being controlled E fiber longitudinal Youngs modulus near to the sample by a Pt/Pt-Rh 10% thermocouple matrix Youngs modulus Linear heating and cooling rates of 5. minwere Vm- Vr matrix and fiber volume fractions used. The furnace pressure can be controlled from 10-2 to 1000 hPa. The sample temperature and the transdu Measurements of af, af and am with a vertical dilat cer signals were recorded simultaneously by a computer. ometer (Setaram TMA 92), in a neutral atmosphere. Because of errors in the material density(20.5%), the with linear heating and cooling rates of 3 up to from the phase displacement owing to coupling with the For this, the experimental curves of relative mipid or sample length(20-1%), the propagation time resulting 1000 C, should therefore allow calculation of aa wave guide(sl.5%)and the changes in density and versus temperature were approximated by a third-order CtE which were taken into account, the uncertainty in the Youngs modulus value is *4%. Several parameters can affect the accuracy of the impulse propagation. such △L =A+Br+C72+D73 as the ratio between the sizes of the heterogeneities and ple dimensions. Techniques based on a temporal signal At a time t,(AL/Lo) and(AL/Lo) can be differ- analysis can then become imprecise. The implementa- entiated to obtain af(T)and af(n), and thus ai(n)and tion of a owever allowed an af(T). a(T)and a' (n) can then be integrated to obtain improvement in the measurement of propagation delays (△L/Lo)and(△L/ Lo)i backed with the following by intercorrelation techniques. boundary conditions. 2.2 Coefficients of thermal expansion (△LLo)(T=20°C)=0 If the fiber/matrix bonding is strong and the poisson ratios of both fibers and matrix are equal, Schapery shows that the rule of mixtures can be applied to the (△L/Lo{(=1000eas case of the cte of a unidirectional. continuous-fibe (△L/Lo):(7=1000° C)Td composite a=“m2mm+E 2. 3 Determination of residual thermal stresses Em Vm+EVr Because of the high ratio of fiber length to diameter in continuous-fiber-reinforced materials, these composites When the Poisson s ratios are different, which is the can be represented by two semi- infinite concentric usual case, the author shows that eqn(1)is still a good cylinders with radii a and b( Fig. 1). 1.2 From this approximation. representation of a composite, Hsueh and Becher have For isotropic fibers, the transverse CTE of the com- developed a model -itself a simplified version of the posite is model previously proposed by Mikata and Toya4 allowing the determination of the thermal stresses ;, oI af=(1+Vm )am Im +(+vidaVr-(vm Vm+v vr) d om in composites and their influence on the expansion (2) For Kevlar- fiber composites, Rojstaczer et al. have considered the anisotropic thermoelasticity of the fiber (1+Vm)Vm +a Vr+vir Vr-(vm Vm +vl vr)ai The following nomenclature has been used(the matrix is taken to be isotropic composite longitudinal cte. omposite transverse CtE, fiber longitudinal CTE Fig. I, Schematic illustration of the composite cylinder model410 I/. Biunchi et al material elastic properties. This technique is described in a previous paper.3 The composites were treated in an argon gas flow in a vertical and tubular electric furnace which could be heated to 155O”C, with the temperature being controlled near to the sample by a Pt/Pt-Rh 10% thermocouple. Linear heating and cooling rates of 5”Cmin’ were used. The furnace pressure can be controlled from lop2 to 1000 hPa. The sample temperature and the transdu￾cer signals were recorded simultaneously by a computer. Because of errors in the material density (~0.5%) the sample length (Z O.l%), the propagation time resulting from the phase displacement owing to coupling with the wave guide (Z 1.5%) and the changes in density and CTE which were taken into account, the uncertainty in the Young’s modulus value is * 4%. Several parameters can affect the accuracy of the impulse propagation, such as the ratio between the sizes of the heterogeneities and the ultrasonic wavelength, and insufficiently large sam￾ple dimensions. Techniques based on a temporal signal analysis can then become imprecise. The implementa￾tion of a frequency analysis, however, allowed an improvement in the measurement of propagation delays by intercorrelation techniques.x 2.2 Coefficients of thermal expansion If the fiber/matrix bonding is strong and the Poisson’s ratios of both fibers and matrix are equal, Schapery” shows that the rule of mixtures can be applied to the case of the CTE of a unidirectional, continuous-fiber composite: When the Poisson’s ratios are different, which is the usual case, the author shows that eqn (1) is still a good approximation. For isotropic fibers, the transverse CTE of the com￾posite is: For Kevlar-fiber composites, Rojstaczer et ~11.‘~’ have considered the anisotropic thermoelasticity of the fiber: The following nomenclature has been used (the matrix is taken to be isotropic): CY y composite longitudinal CTE, o; composite transverse CTE, or fiber longitudinal CTE, I a( fiber transverse CTE, om matrix CTE. ‘& fiber major Poisson’s coefficient, 2 matrix Poisson’s coefficient, EL fiber longitudinal Young’s modulus, matrix Young’s modulus, V,. V’ matrix and fiber volume fractions. Measurements of a:, cry and CX, with a vertical dilat￾ometer (Setaram TMA 92) in a neutral atmosphere, with linear heating and cooling rates of 3°C min ’ up to lOOO”C, should therefore allow calculation of of and of. For this, the experimental curves of relative expansion versus temperature were approximated by a third-order polynomial: AL -=A+BT+CT2$DT’ &I (4) At a time t, (AL/Lo): and (AL/Lo): can be differ￾entiated to obtain c+ 7’) and $( 7’), and thus of( 7) and o:(7). c1f(7’) and of(7’) can then be integrated to obtain (AL/Lo); and (AL/Lo); backed with the following boundary conditions: (Af&)f;,( T = 20°C) = 0 (5) (AW&(T= lOOO”C),,,,,,,,, = WILdftV = lOOO”%,,,,,,,, (6) 2.3 Determination of residual thermal stresses Because of the high ratio of fiber length to diameter in continuous-fiber-reinforced materials, these composites can be represented by two semi-infinite concentric cylinders with radii a and h (Fig. l).“,‘2 From this representation of a composite, Hsueh and Becher” have developed a model&itself a simplified version of the model previously proposed by Mikata and Toya14-- allowing the determination of the thermal stresses ci, af and cm in composites and their influence on the expansion 7 Fig. I. Schematic illustration of the composite cylinder model