4846 LIU et al.: LAMINATED COMPOSITE (a)137 MPa the off-diagonal elastic constants have relatively large uncertainties compared with diagonal elastic con stants. and so we concentrate on the use of c. c and C33 to study anisotropic damage in the cross- ply laminates. 4.2. Elastic constant data in the undamaged state The elastic properties of undamaged0°90° CAS/SiC have been evaluated precisely using res- onant ultrasound spectroscopy(RUS)[11]. To com- pare the results from RUS with LU measurements d the wave propagation velocity in the 1-3 plane was calculated based on elastic constants measured by (b)150MPa→10MPa RUS technique. These are plotted along with the LU measured wave velocities in Fig. 9. In Fig. 9, the square symbols represent the LU measured longitudi nal wave velocities and triangles represent the shear velocities. We note that the RUS predicted and LU measured velocities are in good agreement. In the undamaged states, both the RUS and LU techniques indicate that the Pt and Qt shear wave velocities the 1-3 plane are nearly identical. Elastic constants for the as-received material base 20四 n are listed in Table 137 MPa:,(b) unloaded from 150 MPa of the same surface ing loading tion of elastic stiffness constants dur- Figure 10(a)and(b) shows the elastic stiffness con- stants in the three primary directions determined from the LU velocities as a function of applied stress using a compilation of data from two tests. The Youngs c=(Cu sin20+ Css cos e)Css sin-e(5) modulus along the loading direction(E1) was determ- C33 cos20-(C1+Css)sin20 cos 0 ined from the stress-strain curve by partial unloading (Ao =20 MPa) and is also plotted for comparison For the pure shear mode(Pt) we have 66 sin-0+ Ca4 cos 6 where 0 is the angle between the wave propagation direction and the 3-axis. The expressions for the a sound velocities in the 2-3 plane are similar; one only E needs to replace the subscript I by 2, and 5 by 4 It is worth noting that the above equations are only alid for homogeneous materials. However, since the ultrasound wavelength is relatively large (1000 um g ogeneities involved(fibers with diameter 15 um and s3 in CAS/SiC)compared with the scale of the inhom- plies with thickness -170 um), these CAS/SiC ply composites can be regarded as homoge neous media to a good approximation and the above equa- tions can therefore be applied to convert measured elocity data to elastic stiffness constants [16] Wave velocity(mm/us)-3 direction Anisotropic damage along the three principal direc- Fig. 9. A comp velocities in the 1-3 plane tions is characterized by CIl, C2 and C33. In the pro- RUS measurements and the oly laminate deduced from cess of deducing Cll, C2 and C33, a nonlinear curve trasonic method. Square fitting method, similar to that described previously symbols represent the shear mode. Experimental data are base [10], has been used. An4846 LIU et al.: LAMINATED COMPOSITE Fig. 8. Optical micrographs of surface replicas: (a) at stress of 137 MPa; (b) unloaded from 150 MPa of the same surface region, where the crack indicated by an arrow is partially closed after unloading. c 5 (C11 sin2 q 1 C55 cos2 q)(C55 sin2 q (5) 1 C33 cos2 q)2(C13 1 C55) 2 sin2 q cos2 q. For the pure shear mode (PT) we have VPT 5 ! C66 sin2 q 1 C44 cos2 q r , (6) where q is the angle between the wave propagation direction and the 3-axis. The expressions for the sound velocities in the 2–3 plane are similar; one only needs to replace the subscript 1 by 2, and 5 by 4. It is worth noting that the above equations are only valid for homogeneous materials. However, since the ultrasound wavelength is relatively large (>1000 µm in CAS/SiC) compared with the scale of the inhom￾ogeneities involved (fibers with diameter 15 µm and plies with thickness |170 µm), these CAS/SiC cross￾ply composites can be regarded as homogeneous media to a good approximation and the above equa￾tions can therefore be applied to convert measured velocity data to elastic stiffness constants [16]. Anisotropic damage along the three principal direc￾tions is characterized by C11, C22 and C33. In the pro￾cess of deducing C11, C22 and C33, a nonlinear curve fitting method, similar to that described previously [10], has been used. An error analysis indicates that the off-diagonal elastic constants have relatively large uncertainties compared with diagonal elastic con￾stants, and so we concentrate on the use of C11, C22 and C33 to study anisotropic damage in the cross￾ply laminates. 4.2. Elastic constant data in the undamaged state The elastic properties of undamaged 0°/90° CAS/SiC have been evaluated precisely using res￾onant ultrasound spectroscopy (RUS) [11]. To com￾pare the results from RUS with LU measurements, the wave propagation velocity in the 1–3 plane was calculated based on elastic constants measured by the RUS technique. These are plotted along with the LU measured wave velocities in Fig. 9. In Fig. 9, the square symbols represent the LU measured longitudi￾nal wave velocities and triangles represent the shear velocities. We note that the RUS predicted and LU measured velocities are in good agreement. In the undamaged states, both the RUS and LU techniques indicate that the PT and QT shear wave velocities in the 1–3 plane are nearly identical. Elastic constants for the as-received material based on both methods are listed in Table 1. 4.3. Degradation of elastic stiffness constants dur￾ing loading Figure 10(a) and (b) shows the elastic stiffness con￾stants in the three primary directions determined from the LU velocities as a function of applied stress using a compilation of data from two tests. The Young’s modulus along the loading direction (E1) was determ￾ined from the stress–strain curve by partial unloading (Ds 5 20 MPa) and is also plotted for comparison. Fig. 9. A comparison of ultrasonic velocities in the 1–3 plane of the as-received CAS/SiC cross-ply laminate deduced from RUS measurements and the laser-ultrasonic method. Square symbols represent the longitudinal wave mode and triangle symbols represent the shear mode. Experimental data are based on two specimens