4844 LIU et al.: LAMINATED COMPOSITE Fig. 3. Optical micrographs of replicas showing surface cracking in the early stages of deformation at: (a)60 (b)75 MPa. 8(b)) and failure(Fig. 7)is most likely a result of the state Cu+C2 and, for such an orthotropic material, partial closing of crack opens rather than crack "heal- the ultrasound velocities of the three different wave modes in the 1-3 plane can be related to the elastic stiffness constants [151 For the quasi-longitudinal mode(QL), the velocity Anisotropic reductions in the elastic stiffness tensor a t vb2-4c components of damaged ceramic-matrix composites 2 (CMCs) result from the collective effects of microcracks. When the wavelength of sound propa gating in a body is much larger than the size of the and for the quasi-shear mode(QT) microcracks it contains the wave velocities and the elastic stiffness constants, Ci are related through the Christoffel equation [15]. While the symmetry of the as-received material with even numbers of 0 and 90o plies can be regarded as tetragonal(Cu= C2), it is reduced to orthotropic after damage in the form of transverse matrix cracks is induced. In the damaged where4844 LIU et al.: LAMINATED COMPOSITE Fig. 3. Optical micrographs of replicas showing surface cracking in the early stages of deformation at: (a) 60 MPa; (b) 75 MPa. 8(b)] and failure (Fig. 7) is most likely a result of the partial closing of crack opens rather than crack “heal￾ing”. 4. LASER-ULTRASONIC CHARACTERIZATION 4.1. Ultrasonic velocity and elastic stiffness Anisotropic reductions in the elastic stiffness tensor components of damaged ceramic-matrix composites (CMCs) result from the collective effects of microcracks. When the wavelength of sound propa￾gating in a body is much larger than the size of the microcracks it contains, the wave velocities and the elastic stiffness constants, Cij, are related through the Christoffel equation [15]. While the symmetry of the as-received material with even numbers of 0° and 90° plies can be regarded as tetragonal (C11 5 C22), it is reduced to orthotropic after damage in the form of transverse matrix cracks is induced. In the damaged state C11ÞC22 and, for such an orthotropic material, the ultrasound velocities of the three different wave modes in the 1–3 plane can be related to the elastic stiffness constants [15]. For the quasi-longitudinal mode (QL), the velocity is given by VQL 5 ! a 1 √b2 24c 2r , (1) and for the quasi-shear mode (QT) VQT 5 ! a2√b2 24c 2r , (2) where