CATERTALIA Pergamon Acta mater.48(20004841-4849 www.elsevier.com/locate/actamat ANISOTROPIC DAMAGE EVOLUTION IN A%/90 LAMINATED CERAMIC-MATRIX COMPOSITE Y.-M. LIUT, T E. MITCHELL and H N G. WADLEY+2 Center for Materials Science, Mail Stop K- 765, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and Department of Materials Science and Engineering, School of Engineering and Applied cience, University of Virginia, Charlottesville, VA 22903, USA Received 22 February 2000: accepted 2 August 2000) Abstract-Anisotropic damage evolution in a 0/90 laminated Nicalon"SiC fiber-reinforced calcium alumi- nosilicate(CAS)glass-ceramic composite during uniaxial tensile deformation has been investigated using a variety of non-invasive characterization techniques. The elastic constant reduction in the three principal direc- tions was measured from in situ laser-generated ultrasonic velocity measurements in various sound prop gation directions. They indicate that, in addition to a large drop in elastic stiffness in the loading directio he constants characterizing the nominal elastic stiffness transverse to the loading direction were al degraded Surface replicas taken intermittently during loading revealed that transverse softening of the elastic stiffness was associated with fiber/matrix interface damage mainly in the 0 plies, while the large softenin of the elastic stiffness in the loading direction was the result of multiple matrix cracking in both the 0o 90 plies. While the ultrasonic data allowed a detailed characterization of the anisotropic damage evolution in this laminate, acoustic emission measurements and surface replica data identified the crack initiation stress the 90 plies and correlated it to macroscopically observable deviations of the stress-strain curve fr lies t behavior. These matrix cracks were found to have initiated preferentially in the weak 9 90/0 ply boundaries. 2000 Acta Metallurgica Inc. Published by elsevier Science Ltd. All Keywords: Acoustic, Non-destructive testing, Multilayers; Ceramics; Composites 1 INTRODUCTION loaded with the tensile axis aligned parallel to the 0o Laminated composites sometimes have significant ply direction, it is generally believed that transverse advantages over unidirectional composites for multi- matrix cracks first appear in the weaker 90 plies[3]) axially loaded structural applications because the These small cracks are thought to extend laterally to orientation,thickness and stacking sequence of the span the entire 90 layer and only then penetrate into fiber-reinforced laminae can be varied to satisfy the neighboring 5]. Finite element analyses needed stiffness and strength requirements in differ- have indicated that the degradation of the elastic stiff- ent loading directions [1, 2). However, ensuring the ness constants of cross-ply laminates is related to the acking,fiber/matrix interface debonding and inter- ure damage evolution in situ and to confirm this view- oly delamination are more complicated than in the point. Conventionally, damage evolution is charac- case of a unidirectional composite loaded in one terized by an analysis of the uniaxial stress-strain direction. A detailed understanding of the anisotropic behavior [3, 7, 8]. In this approach, the effective aspects of damage evolution is therefore important for Young's modulus in the loading direction at various both designing improved laminated composites and deformation stages is determined from utilizing them more reliably in structural applications. loading/unloading curves and compared with metallo When a brittle-matrix 0/90 cross-ply laminate is graphic observations of the crack density. The recog nition of anisotropic damage and the assessment of its mechanical significance have been limited, in part, 四cmg Intel Corporation, Assembly Tech- by the absence of effective experimental method CH5-263,5000 West Chandler Boul- ologies for its measurement respondence should be addressed. Fax: Recent ultrasonic studies on an sic/sic bi-direc 1359-6454100/520.00@ 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved PI:S1359-6454(00)00286-X
Acta mater. 48 (2000) 4841–4849 www.elsevier.com/locate/actamat ANISOTROPIC DAMAGE EVOLUTION IN A 0°/90° LAMINATED CERAMIC-MATRIX COMPOSITE Y. -M. LIU†1 , T. E. MITCHELL1 and H. N. G. WADLEY†2 1 Center for Materials Science, Mail Stop K-765, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and 2 Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22903, USA ( Received 22 February 2000; accepted 2 August 2000 ) Abstract—Anisotropic damage evolution in a 0°/90° laminated Nicalon SiC fiber-reinforced calcium aluminosilicate (CAS) glass–ceramic composite during uniaxial tensile deformation has been investigated using a variety of non-invasive characterization techniques. The elastic constant reduction in the three principal directions was measured from in situ laser-generated ultrasonic velocity measurements in various sound propagation directions. They indicate that, in addition to a large drop in elastic stiffness in the loading direction, the constants characterizing the nominal elastic stiffness transverse to the loading direction were also degraded. Surface replicas taken intermittently during loading revealed that transverse softening of the elastic stiffness was associated with fiber/matrix interface damage mainly in the 0° plies, while the large softening of the elastic stiffness in the loading direction was the result of multiple matrix cracking in both the 0° and 90° plies. While the ultrasonic data allowed a detailed characterization of the anisotropic damage evolution in this laminate, acoustic emission measurements and surface replica data identified the crack initiation stress in the 90° plies and correlated it to macroscopically observable deviations of the stress–strain curve from linear elastic behavior. These matrix cracks were found to have initiated preferentially in the weak 90° plies near the 90°/0° ply boundaries. 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Acoustic; Non-destructive testing; Multilayers; Ceramics; Composites 1. INTRODUCTION Laminated composites sometimes have significant advantages over unidirectional composites for multiaxially loaded structural applications because the orientation, thickness and stacking sequence of the fiber-reinforced laminae can be varied to satisfy the needed stiffness and strength requirements in different loading directions [1, 2]. However, ensuring the durability of brittle-matrix laminates can be problematic because damage modes such as transverse ply cracking, fiber/matrix interface debonding and interply delamination are more complicated than in the case of a unidirectional composite loaded in one direction. A detailed understanding of the anisotropic aspects of damage evolution is therefore important for both designing improved laminated composites and utilizing them more reliably in structural applications. When a brittle-matrix 0°/90° cross-ply laminate is * Present address: Intel Corporation, Assembly Technology Development, CH5-263, 5000 West Chandler Boulevard, Chandler, AZ 85226, USA. † To whom all correspondence should be addressed. Fax: 1804 924 3032. 1359-6454/00/$20.00 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S13 59-6454(00)00286-X loaded with the tensile axis aligned parallel to the 0° ply direction, it is generally believed that transverse matrix cracks first appear in the weaker 90° plies [3]. These small cracks are thought to extend laterally to span the entire 90° layer and only then penetrate into neighboring 0° plies [3–5]. Finite element analyses have indicated that the degradation of the elastic stiffness constants of cross-ply laminates is related to the transverse matrix crack density [1, 5, 6], but relatively little experimental work has been conducted to measure damage evolution in situ and to confirm this viewpoint. Conventionally, damage evolution is characterized by an analysis of the uniaxial stress–strain behavior [3, 7, 8]. In this approach, the effective Young’s modulus in the loading direction at various deformation stages is determined from loading/unloading curves and compared with metallographic observations of the crack density. The recognition of anisotropic damage and the assessment of its mechanical significance have been limited, in part, by the absence of effective experimental methodologies for its measurement. Recent ultrasonic studies on an SiC/SiC bi-directional composite immersed in a water bath (for ultra-
4842 LIU et al.: LAMINATED COMPOSITE sonic coupling) indicated that ultrasonic techniques mal residual stresses. Note that each ply is under a have the potential to characterize anisotropic damage biaxial residual stress state [9]. A laser-ultrasonic (LU) approach has sub- Tensile testing was performed with a screw-driven sequently been developed and used to monitor the Instron 4200 machine at a crosshead speed of 0.03 anisotropic damage evolution in a unidirectional mm/min. The axial strain was measured with an Nicalon" SiC fiber-reinforced calcium aluminosilic- extensometer(I in. gage length). The specimen was ate( CAS/SiC)composite in situ during uniaxial ten- loaded to a pre-set stress level, held at that level for sile loading in its fiber direction [10]. This laser-ultra- laser-ultrasonic measurements, and the stress then sonic technique revealed that extensive transverse reduced to 10 MPa. During the loading and unloading softening of the elastic moduli accompanied degra- process, acoustic emission events were also recorded dation of the axial elastic modulus. this was believed Details of the lu measurement and acoustic emission to be caused by fiber-matrix debonding near the inter- recording techniques are identical to those reported section of fibers with transverse matrix cracks. Here, previously [101 we apply this LU method to investigate anisotropic damage accumulation in a0°90°cros- ply CAs/s composite loaded in uniaxial tension 3, STRESS-STRAIN BEHAVIOR AND VISUAL In this study, anisotropic damage in 0%/90%cross- OBSERVATIONS OF DAMAGE EVOLUTION ply samples was characterized using ultrasonic velo- 3.L. Stress-strain behavior cities measured in situ within two principal planes The anisotropic damage behavior is represented by a A typical loading/unl stress-strain curve for deterioration of the elastic stiffness constants Cu. C2 a CAS/SiC cross-ply d the corresponding and C33(determined from the ultrasonic velocities) acoustic emission(A Its are shown in Fig. 2 in the three principal directions. In addition, acoustic The deviation of the stress-strain curve from the lin emission sensing, intermittent surface replica charac- ear regime occurred at around 50 MPa, and correlated finesse ns and loading/unloading hysteresis curve with an acceleration of AE at about the same stress ere acquired Correlations between the elastic level. A detailed correlation between AE events, the deterioration and tropic damage crack initiation stress and the residual stress will be accumulation are discussed and compared with exist- discussed in Section 5 ing views of damage progression in unidirectional 3.2. Matrix cracking evolution CAS/SIC composites. Figures 3-5 are optical micrographs of surface rep- licas exhibiting matrix cracks at different stages of 2. EXPERIMENTAL APPROACH loading. At low stress levels(<75 MPa) as shown in Fig 3(a) and(b). some(marked) matrix cracks in the Laminated 0%/90 CAS/SIC composite material was 90 ply extended across the entire layer, however a provided by Corning, Inc.(Corning, NY). The majority of the cracks were rather short and confined material was composed of 16 plies with a ply thick- within the ply boundary by fiber bridgment. The met- ness of about 170 um. The material properties have allograph clearly indicated that, in contrast to th been presented elsewhere [8, 11]. The dimensions of viewpoint adopted by many modeling studies, matrix the tensile specimens were -150 mmX10 mmx cracks, once initiated, do not always span an entire .7 mm. The sample ends were bonded to low-modu- ply. Instead, the initiation of new matrix cracks and lus fiberglass tabs for gripping. The edges of the the growth of pre-existing cracks(to span an entire specimens were polished before testing so that acetate ply) progressed simultaneously. For example, Fig replicas could be taken at various stages of loading 4(aHc)reveals a series of surface cracks at three dif- for crack detection and monitoring of its progression. ferent stress levels. A newly initiated crack in the 90o The coordinate system used to characterize both ply at the cross-ply boundary at 93 MPa [marker I the fiber architecture and the test configuration is in Fig. 4(a)l had partially propagated across the ply illustrated in Fig. 1, where, for clarity, only four plies after the stress was increased to 106 MPa [Fig. 4(b) near the middle plane are shown. Direction I was the At a higher stress of 120 MPa, this crack had spread oading direction while 3 was the laminate thickness across the entire ply [Fig. 4(c). Figure 4(c)also direction. Continuous SiC fibers were aligned along shows an example of a matrix crack(marker 2)that oth the 1-and 2-directions. Since the load was was initiated at the high stress level applied in the 1-direction, the plies with fibers in this Figure 5 presents a higher magnification of a sur- rection were the 0 plies. Double 90 layers were face replica that shows cracks deflecting around th present at the center of the lay-up, and an even num- fibers in the 90 layer leaving a debonded interface ber of 0%and 90 plies existed in the sample Conse- The matrix cracks in the 90 plies exhibited both juently, the 0/90 cross-ply used for the tests is a straight and curved paths, depending on the local symmetric laminate, and there should be no coupling arrangement of fibers within the ply. The curved between stretching and bending [2]. Also shown sche- cracks( Fig 4, marker 1) appeared to have extended matically in Fig. I are the directions of laminate ther- more slowly than the straight cracks [see cracks in
4842 LIU et al.: LAMINATED COMPOSITE sonic coupling) indicated that ultrasonic techniques have the potential to characterize anisotropic damage [9]. A laser-ultrasonic (LU) approach has subsequently been developed and used to monitor the anisotropic damage evolution in a unidirectional Nicalon SiC fiber-reinforced calcium aluminosilicate (CAS/SiC) composite in situ during uniaxial tensile loading in its fiber direction [10]. This laser-ultrasonic technique revealed that extensive transverse softening of the elastic moduli accompanied degradation of the axial elastic modulus. This was believed to be caused by fiber–matrix debonding near the intersection of fibers with transverse matrix cracks. Here, we apply this LU method to investigate anisotropic damage accumulation in a 0°/90° cross-ply CAS/SiC composite loaded in uniaxial tension. In this study, anisotropic damage in 0°/90° crossply samples was characterized using ultrasonic velocities measured in situ within two principal planes. The anisotropic damage behavior is represented by a deterioration of the elastic stiffness constants C11, C22 and C33 (determined from the ultrasonic velocities) in the three principal directions. In addition, acoustic emission sensing, intermittent surface replica characterizations and loading/unloading hysteresis curve data were acquired. Correlations between the elastic stiffness deterioration and anisotropic damage accumulation are discussed and compared with existing views of damage progression in unidirectional CAS/SiC composites. 2. EXPERIMENTAL APPROACH Laminated 0°/90°CAS/SiC composite material was provided by Corning, Inc. (Corning, NY). The material was composed of 16 plies with a ply thickness of about 170 µm. The material properties have been presented elsewhere [8, 11]. The dimensions of the tensile specimens were |150 mm310 mm3 2.7 mm. The sample ends were bonded to low-modulus fiberglass tabs for gripping. The edges of the specimens were polished before testing so that acetate replicas could be taken at various stages of loading for crack detection and monitoring of its progression. The coordinate system used to characterize both the fiber architecture and the test configuration is illustrated in Fig. 1, where, for clarity, only four plies near the middle plane are shown. Direction 1 was the loading direction while 3 was the laminate thickness direction. Continuous SiC fibers were aligned along both the 1- and 2-directions. Since the load was applied in the 1-direction, the plies with fibers in this direction were the 0° plies. Double 90° layers were present at the center of the lay-up, and an even number of 0° and 90° plies existed in the sample. Consequently, the 0°/90° cross-ply used for the tests is a symmetric laminate, and there should be no coupling between stretching and bending [2]. Also shown schematically in Fig. 1 are the directions of laminate thermal residual stresses. Note that each ply is under a biaxial residual stress state. Tensile testing was performed with a screw-driven Instron 4200 machine at a crosshead speed of 0.03 mm/min. The axial strain was measured with an extensometer (1 in. gage length). The specimen was loaded to a pre-set stress level, held at that level for laser-ultrasonic measurements, and the stress then reduced to 10 MPa. During the loading and unloading process, acoustic emission events were also recorded. Details of the LU measurement and acoustic emission recording techniques are identical to those reported previously [10]. 3. STRESS–STRAIN BEHAVIOR AND VISUAL OBSERVATIONS OF DAMAGE EVOLUTION 3.1. Stress–strain behavior A typical loading/unloading stress–strain curve for a CAS/SiC cross-ply sample and the corresponding acoustic emission (AE) events are shown in Fig. 2. The deviation of the stress–strain curve from the linear regime occurred at around 50 MPa, and correlated with an acceleration of AE at about the same stress level. A detailed correlation between AE events, the crack initiation stress and the residual stress will be discussed in Section 5. 3.2. Matrix cracking evolution Figures 3–5 are optical micrographs of surface replicas exhibiting matrix cracks at different stages of loading. At low stress levels (,75 MPa) as shown in Fig. 3(a) and (b), some (marked) matrix cracks in the 90° ply extended across the entire layer; however a majority of the cracks were rather short and confined within the ply boundary by fiber bridgment. The metallography clearly indicated that, in contrast to the viewpoint adopted by many modeling studies, matrix cracks, once initiated, do not always span an entire ply. Instead, the initiation of new matrix cracks and the growth of pre-existing cracks (to span an entire ply) progressed simultaneously. For example, Fig. 4(a)–(c) reveals a series of surface cracks at three different stress levels. A newly initiated crack in the 90° ply at the cross-ply boundary at 93 MPa [marker 1 in Fig. 4(a)] had partially propagated across the ply after the stress was increased to 106 MPa [Fig. 4(b)]. At a higher stress of 120 MPa, this crack had spread across the entire ply [Fig. 4(c)]. Figure 4(c) also shows an example of a matrix crack (marker 2) that was initiated at the high stress level. Figure 5 presents a higher magnification of a surface replica that shows cracks deflecting around the fibers in the 90° layer leaving a debonded interface. The matrix cracks in the 90° plies exhibited both straight and curved paths, depending on the local arrangement of fibers within the ply. The curved cracks (Fig. 4, marker 1) appeared to have extended more slowly than the straight cracks [see cracks in
LIU et al.: LAMINATED COMPOSITE 4843 9o°ply 0°ply addd Fig. 1. Coordinates and layer residual stress states for the 0/90 cross-ply laminate. Loading direction is along the I-direction became more prevalent than in the 90 ply because of the overlapping of matrix cracks from the 90% plies on either side of the 0o ply. The penetration of pre existing cracks into neighboring 0 plies was accompanied by the initiation of new matrix cracks in the 90 ply. This again contrasts with a previous report that matrix cracks in the 90 ply were saturated a when the stress had reached 80 MPa [3]. It has also been noted that the ultimate tensile strength of five composite samples tested here ran 130 to 160 MPa, while in previous studies the failure stress for this composite was reported from 120 MPa Strain (%) [12] to 220 MPa 13, 13]. These differences may be attributed to sample variations [14] Fig. 2. Loading/unloading stress-strain curve of CAS/SiC Metallographic observations of surface replicas cross-ply laminate and the recorded acoustic emission events. taken after unloading and failure are consistent with the notion that matrix cracks initiated in the 90 ply the top 90 ply in Fig 3(b)]. Although previous stud- near the ply boundary, then gradually extended into es have also indicated that the matrix cracks initiated the 90 ply. As shown in Fig. 7(taken after sampl first in the 90 plies [3, 12], the observations of crack failure), matrix cracks in the central double 90 layer growth presented here are not in total agreement with are arrested near ply boundaries. This is believed to all of the prior investigations. For example, Beyerle be due to uneven crack openings in the 90 ply: crack et al. [3] suggested that matrix cracks in the 90 ply openings near the ply boundary(initiation site)were ahvays spanned the entire ply once initiated, whereas generally larger than those at the ply center for a Mall and Kim [12] believed that matrix cracks given stress level. Additionally, fibers in the 0 plies occurred in a random manner and that no single crack exert frictional forces upon crack closure during w across an entire ply unloading. which could also contribute to a lower The average crack density was determined as a extent of crack closure in the 90o ear the ply function of stress from the surface replicas and is boundary. This is further illustrated by comparing presented in Fig. 6. Crack initiation stress was taken Fig. 8(a), taken at a stress of 137 MPa, with Fig. 8(b) as 50 MPa, which was estimated from the stress- taken after unloading to 10 MPa from 150 MPa. It strain and AE measurement. At stress levels can be seen that crack openings near the central part below -95 MPa the crack density in the weaker 90 of the 90 ply crack(shown by arrow)are signifi ply was higher than in the 0 ply. As the applied cantly reduced after unloading. Since there was no stress increased, the 90 ply cracks then began to pen- obvious increase in the reloading modulus compared etrate into the 0 ply. By the time the stress had with the unloading modulus, what has been seen in reached above 95 MPa, matrix cracks in the 0 ply the central region of 90 ply after unloading [Fig
LIU et al.: LAMINATED COMPOSITE 4843 Fig. 1. Coordinates and layer residual stress states for the 0°/90° cross-ply laminate. Loading direction is along the 1-direction. Fig. 2. Loading/unloading stress–strain curve of CAS/SiC cross-ply laminate and the recorded acoustic emission events. the top 90° ply in Fig. 3(b)]. Although previous studies have also indicated that the matrix cracks initiated first in the 90° plies [3, 12], the observations of crack growth presented here are not in total agreement with all of the prior investigations. For example, Beyerle et al. [3] suggested that matrix cracks in the 90° ply always spanned the entire ply once initiated, whereas Mall and Kim [12] believed that matrix cracks occurred in a random manner and that no single crack grew across an entire ply. The average crack density was determined as a function of stress from the surface replicas and is presented in Fig. 6. Crack initiation stress was taken as 50 MPa, which was estimated from the stress– strain curve and AE measurement. At stress levels below |95 MPa the crack density in the weaker 90° ply was higher than in the 0° ply. As the applied stress increased, the 90° ply cracks then began to penetrate into the 0° ply. By the time the stress had reached above 95 MPa, matrix cracks in the 0° ply became more prevalent than in the 90° ply because of the overlapping of matrix cracks from the 90° plies on either side of the 0° ply. The penetration of preexisting cracks into neighboring 0° plies was accompanied by the initiation of new matrix cracks in the 90° ply. This again contrasts with a previous report that matrix cracks in the 90° ply were saturated when the stress had reached 80 MPa [3]. It has also been noted that the ultimate tensile strength of five 0°/90° composite samples tested here ranged from 130 to 160 MPa, while in previous studies the failure stress for this composite was reported from 120 MPa [12] to 220 MPa [3, 13]. These differences may be attributed to sample variations [14]. Metallographic observations of surface replicas taken after unloading and failure are consistent with the notion that matrix cracks initiated in the 90° ply near the ply boundary, then gradually extended into the 90° ply. As shown in Fig. 7 (taken after sample failure), matrix cracks in the central double 90° layer are arrested near ply boundaries. This is believed to be due to uneven crack openings in the 90° ply: crack openings near the ply boundary (initiation site) were generally larger than those at the ply center for a given stress level. Additionally, fibers in the 0° plies exert frictional forces upon crack closure during unloading, which could also contribute to a lower extent of crack closure in the 90° ply near the ply boundary. This is further illustrated by comparing Fig. 8(a), taken at a stress of 137 MPa, with Fig. 8(b) taken after unloading to 10 MPa from 150 MPa. It can be seen that crack openings near the central part of the 90° ply crack (shown by arrow) are signifi- cantly reduced after unloading. Since there was no obvious increase in the reloading modulus compared with the unloading modulus, what has been seen in the central region of 90° ply after unloading [Fig
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 where
4844 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 “healing”. 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 propagating 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
LIU et al.: LAMINATED COMPOSITE 4845 200m Fig. 5. Optical micrograph showing that the matrix crack propagation(stress 106 MPa)in the 90 layer deflects around fibers, fiber ends in 90 plies are visible due to debonding 2—0o900 90o ply Fg.6. Average matrix crack densities in0°90° plies,,0°and 90 plies as a function of stresses 200 same area at different loading stages: (a)93 MPa, arrow I indi- cates short matrix cracks initiated at 90 plies near ply bound- ries; (b)106 MPa, showing the crack growth within a 90 ply (arrow 1);(c)120 MPa, crack I spans the 90 ply, arrow 2 shows a newly initiated matrix crack a=(Cl+ Css)sin20+(C33+ Css)cos0 b= C55 Cu sin28+ C33 cos20(4)
LIU et al.: LAMINATED COMPOSITE 4845 Fig. 4. Optical micrographs of surface replicas taken in the same area at different loading stages: (a) 93 MPa, arrow 1 indicates short matrix cracks initiated at 90° plies near ply boundaries; (b) 106 MPa, showing the crack growth within a 90° ply (arrow 1); (c) 120 MPa, crack 1 spans the 90° ply, arrow 2 shows a newly initiated matrix crack. a 5 (C11 1 C55) sin2 q 1 (C33 1 C55) cos2 q, (3) b 5 C55 1 C11 sin2 q 1 C33 cos2 q (4) and Fig. 5. Optical micrograph showing that the matrix crack propagation (stress 106 MPa) in the 90° layer deflects around fibers; fiber ends in 90° plies are visible due to debonding. Fig. 6. Average matrix crack densities in 0°/90° plies, 0° and 90° plies as a function of stresses. Fig. 7. Surface cracks after failure, where cracks in the 90° ply are only visible near the ply boundary
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. An
4846 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 inhomogeneities involved (fibers with diameter 15 µm and plies with thickness |170 µm), these CAS/SiC crossply composites can be regarded as homogeneous media to a good approximation and the above equations can therefore be applied to convert measured velocity data to elastic stiffness constants [16]. Anisotropic damage along the three principal directions is characterized by C11, C22 and C33. In the process 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 constants, and so we concentrate on the use of C11, C22 and C33 to study anisotropic damage in the crossply laminates. 4.2. Elastic constant data in the undamaged state The elastic properties of undamaged 0°/90° CAS/SiC have been evaluated precisely using resonant ultrasound spectroscopy (RUS) [11]. To compare 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 longitudinal 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 during loading Figure 10(a) and (b) shows the elastic stiffness constants 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 determined 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
LIU et al.: LAMINATED COMPOSITE 4847 Table 1. Elastic stiffness constants of 0/90 cross-ply CAS/SiC composites determined by RUS [11] and laser-generated ultrasound(units: GPa) C(±%) C3(±2%) C(土%) C1(%0) C4(土%) C.(% 14745(0.11)145.19(0.28 5223(0.39) 499 461200.01) 46.35(0.02) Ultrasound method 150.0(3.3) 542(14.6) 42.5(10.1) 45.3(4.9 90° ply o0°pam have only half the number of 0 plies unidirectional laminates. the stiffness transverse direction for the o°/90°lar there- fore less than that observed in unidirectional 120- CAS/SIC [10, 18 Figure 8 indicates that partial crack closure appears to have occurred in the 90 ply upon unloading. This is consistent with the observation of a slight increase The deduced values of Cul and Cax durig to 10 MPa in ultrasonic velocity during unl 33 during loading and unloading are shown in Fig. 11. Error bars(similar to 0 30 60 90 120 150 180 those given in Fig. 10) are not included for clarity Since partial crack closure occurs mainly in the load direction. an Cu is seen during 0Soo0N unloading while C33 appears to have remained con- stant. It should be pointed out that the results in Fig Il are from two tests. and some of the scatter can be ributed to sample 5. RESIDUAL STRESS AND MATRIX CRACK INITIATION The initiation stress for matrix cracking is an Stress(MPa important issue for fiber-reinforced CMCs, because it denotes not only the onset of damage but also the los laser-ultrasonic wave velocity measurements. Young's modu. of protection provided by the matrix against environ- lus E, measured from unloading stress-strain curve is also mental corrosion and/or oxidation of the fibers [191 shown.(b) Normalized curves showing the relative trends of Prediction of the matrix crack initiation stress(omd) stiffness reduction requires knowledge of the(statistical flaw population controlled)strength of the ceramic matrix and an Normalized data are also included [Fig. 10(b) to understanding of the residual stress state. There are show the relative reductions. Similar to unidirectional ponents to the residual stress in la CAS/SiC, Cu in the loading direction exhibited the nates: that of the individual lamina, and that within largest stiffiness reduction as a result of many trans- verse matrix cracks. Its relative trend with stress [ Fig 10(b)] is about identical to the change of E, measured by mechanical testing. Cll and C22 have approxi- mately the same initial value, but lose their degener- a acy as damage develops. Normalized data indicate that C2? and C33 have a similar reduction trend [Fig. 120 o(b)]. The softening of C2 and C33 implies that 9 120 acks with opening displacement in the 2-and 3 direction exist and are most likely a consequence of 80 fiber/matrix interface debonding C,(unloading Although debonding occurs both in the 0 and 90 g plies, softening in the transverse plane C.(unloading be a consequence of interfacial debonding in the 0 plies. This is because the matrix cracks are nearly par- allel to the transverse plane, and micromechanical calculations show that the elastic constants in the Stress(MPa) directions parallel to the crack plane are insensitive Fig. 11. Comparison of elastic stiffiness constants Cu and C to the crack density [17]. Because 0/90 laminates during loading and unloading at different stress levels
LIU et al.: LAMINATED COMPOSITE 4847 Table 1. Elastic stiffness constants of 0°/90° cross-ply CAS/SiC composites determined by RUS [11] and laser-generated ultrasound (units: GPa) C11 (±%) C33 (±%) C13 (±%) C12 (±%) C44 (±%) C66 (±%) RUS 147.45 (0.11) 145.19 (0.28) 52.23 (0.39) 49.96 (0.88) 46.12 (0.01) 46.35 (0.02) Ultrasound method 150.0 (3.3) 144.2 (2.4) 54.2 (14.6) – 42.5 (10.1) 45.3 (4.9) Fig. 10. (a) Elastic constants C11, C22 and C33 determined from laser-ultrasonic wave velocity measurements. Young’s modulus E1 measured from unloading stress–strain curve is also shown. (b) Normalized curves showing the relative trends of stiffness reduction. Normalized data are also included [Fig. 10(b)] to show the relative reductions. Similar to unidirectional CAS/SiC, C11 in the loading direction exhibited the largest stiffness reduction as a result of many transverse matrix cracks. Its relative trend with stress [Fig. 10(b)] is about identical to the change of E1 measured by mechanical testing. C11 and C22 have approximately the same initial value, but lose their degeneracy as damage develops. Normalized data indicate that C22 and C33 have a similar reduction trend [Fig. 10(b)]. The softening of C22 and C33 implies that cracks with opening displacement in the 2- and 3- direction exist and are most likely a consequence of fiber/matrix interface debonding. Although debonding occurs both in the 0° and 90° plies, softening in the transverse plane is presumed to be a consequence of interfacial debonding in the 0° plies. This is because the matrix cracks are nearly parallel to the transverse plane, and micromechanical calculations show that the elastic constants in the directions parallel to the crack plane are insensitive to the crack density [17]. Because 0°/90° laminates have only half the number of 0° plies compared with unidirectional laminates, the stiffness reduction in the transverse direction for the 0°/90° laminates is therefore less than that observed in unidirectional CAS/SiC [10, 18]. Figure 8 indicates that partial crack closure appears to have occurred in the 90° ply upon unloading. This is consistent with the observation of a slight increase in ultrasonic velocity during unloading to 10 MPa. The deduced values of C11 and C33 during loading and unloading are shown in Fig. 11. Error bars (similar to those given in Fig. 10) are not included for clarity. Since partial crack closure occurs mainly in the loading direction, an increase in C11 is seen during unloading while C33 appears to have remained constant. It should be pointed out that the results in Fig. 11 are from two tests, and some of the scatter can be attributed to sample variations. 5. RESIDUAL STRESS AND MATRIX CRACK INITIATION The initiation stress for matrix cracking is an important issue for fiber-reinforced CMCs, because it denotes not only the onset of damage but also the loss of protection provided by the matrix against environmental corrosion and/or oxidation of the fibers [19]. Prediction of the matrix crack initiation stress (smc) requires knowledge of the (statistical flaw population controlled) strength of the ceramic matrix and an understanding of the residual stress state. There are two main components to the residual stress in laminates: that of the individual lamina, and that within Fig. 11. Comparison of elastic stiffness constants C11 and C33 during loading and unloading at different stress levels
4848 LIU et al.: LAMINATED COMPOSITE an isolated ply caused by the thermal mismatch stress on reaches 140 MPa for unidirectional between the fiber and the matrix [4]. CAS/SiC 18, 101, which corresponds to om The first part (ply residual stress)arises because of 187.4 MPa from equation( 8). When the maximum the anisotropy in the coefficients of thermal expansion stress criterion is applied, based on the matrix crack (CTE). In CAS/SiC, the Cte transverse to the fiber initiation stress of the unidirectional composite, the direction(ar =4.5X10-6K-)is larger than in the predicted applied stress for matrix crack initiation in fiber direction(a,=4.3X10-6K- )[20]. Clearly, if the 0/900 cross-ply laminate is 56 MPa [equation aL=ar, there will be no thermal residual stress (7). This is in good agreement with the lowest stress resulting from differently oriented plies. Using lami- level where surface cracks were visually detected nation theory [2] for a symmetric laminate(the case [ Fig. 3(a). Mechanical testing shows that the stress- in our study), the residual stress from this source is strain curve starts to deviate from linear behavior at uniform within each layer, but changes discontinu- 50 MPa, and the corresponding AE recording also Following lamination theory and assuming that about the same stress level(Fig. 2). Other work lEB ously across the ply bor indicates that the onset of matrix cracking occurs each layer is homogeneous [2], the lamination also shown that the matrix cracking stress o ranges residual stress in each ply is calculated to be biaxial from 40 to 60 MPa [3]. Some of this discrepancy can with a magnitude of about 13 MPa. The 90 plies are be attributed to sample differences. Additionally, it under tension in the loading (i.e, global 1-)direction has also been noted that in unidirectional CMCs, while the 0 plies are in compression in the l-direc- matrix cracking initiation depends on the local fiber tion, Fig. 1. Because the residual stresses in the 0 distribution; matrix cracking can be initiated in the and 90 plies are of identical magnitude but opposite matrix-rich region at a lower stress level than where in sign, the net residual force on the laminate is zero. the fibers are more densely and uniformly distributed To this stress state we a Id the effects due to the [23 Also, free edge effects(although considered to CTE mismatch between the fiber and the matrix. be small in this case)on surface crack initiation in a Since the matrix has a higher Cte than the fiber, the 0/90 laminate [24] were not investigated in above matrix is under axial and circumferential(hoop)ten- simplified analysis. Regardless, the overall agreement sion, and the fiber/matrix interface(radial direction) between theory and experimental observation is good is under compression. Using an elastic composite cyl- Finally, we note that the residual stress analysis inder model [21, 22], the calculated residual matrix above showed that the maximum matrix tensile stress axial stress Oa=81 MPa and the radial interfacial occurred in the loading direction, within the 90 pl stress OR=-57 MPa, assuming 0.35 fiber volume The observation that matrix cracks initiated preferen- fraction. These values are in reasonable agreement tially near the boundaries of 90 plys must be either with previously reported values(axial 89 MPa and a consequence of a locally higher matrix fraction at radial -65 MPa)[8]. The hoop stress, oBe, in the the ply interface or variations in stress at the 0%/900 matrix calculated from the composite cylinder model ply boundary [25]. Further studies are needed to is tensile with a maximum at the fiber/matrix bound- improve the incorporation of matrix crack initiation ary of -120 MPa. For the laminate system investi- in models of laminate mechanics gated here, the contribution from the laminae residual tress is much less than the residual stress resulting from CtE mismatch between the fiber and the matrix 6. CONCLUSIONS When an external stress o, is applied, the stress The elastic stiffness degradation of a 0%/90cross- is distributed among the layers. Using lamination ply laminate has been evaluated by a combination of theory, the stress distributions on 0 and 90 layers a laser-ultrasonic characterization, acoustic emission are approximately 1.0350, and 0.965o11-Thus, the and surface replicas. The LU technique allowed a maximum total stress along the loading direction, detailed characterization of the anisotropic damage including contributions from applied load, lamination through ultrasonic velocity measurements in various residual stress and hoop residual stress, on the 90 propagation directions during loading and unloading ply prior to matrix crack initiation is f the composite. Damage along the three principal directions was characterized by the elastic constants Gm=0.965G1+dB+o (7) CIl, C2 and C33. The largest stiffness reduction was observed in Cll, the loading-direction modulus. This has been linked to transverse matrix cracking initiat For unidirectional CAS/SiC, the total stress carried in 90 plies with an initiation stress of about 50 by the matrix is MPa. Smaller reductions for C2? and C33 in the trans- verse plane are attributed to interface damage in th +oR=0.760+oR.(8)0 plies that initiated at about 75 MPa, Matrix cracks died stress levels than in unidirec- tional CAS/SiC composites. This observation is Surface matrix cracks are detected when the applied rationalized by residual stress analy
4848 LIU et al.: LAMINATED COMPOSITE an isolated ply caused by the thermal mismatch between the fiber and the matrix [4]. The first part (ply residual stress) arises because of the anisotropy in the coefficients of thermal expansion (CTE). In CAS/SiC, the CTE transverse to the fiber direction (aT 5 4.531026 K21 ) is larger than in the fiber direction (aL 5 4.331026 K21 ) [20]. Clearly, if aL 5 aT, there will be no thermal residual stress resulting from differently oriented plies. Using lamination theory [2] for a symmetric laminate (the case in our study), the residual stress from this source is uniform within each layer, but changes discontinuously across the ply boundary. Following lamination theory and assuming that each layer is homogeneous [2], the lamination residual stress in each ply is calculated to be biaxial with a magnitude of about 13 MPa. The 90° plies are under tension in the loading (i.e., global 1-) direction while the 0° plies are in compression in the 1-direction, Fig. 1. Because the residual stresses in the 0° and 90° plies are of identical magnitude but opposite in sign, the net residual force on the laminate is zero. To this stress state we add the effects due to the CTE mismatch between the fiber and the matrix. Since the matrix has a higher CTE than the fiber, the matrix is under axial and circumferential (hoop) tension, and the fiber/matrix interface (radial direction) is under compression. Using an elastic composite cylinder model [21, 22], the calculated residual matrix axial stress sR aa 5 81 MPa and the radial interfacial stress sR rr 5 257 MPa, assuming 0.35 fiber volume fraction. These values are in reasonable agreement with previously reported values (axial 89 MPa and radial 265 MPa) [8]. The hoop stress, sR θθ, in the matrix calculated from the composite cylinder model is tensile with a maximum at the fiber/matrix boundary of |120 MPa. For the laminate system investigated here, the contribution from the laminae residual stress is much less than the residual stress resulting from CTE mismatch between the fiber and the matrix. When an external stress s11 is applied, the stress is distributed among the layers. Using lamination theory, the stress distributions on 0° and 90° layers are approximately 1.035s11 and 0.965s11. Thus, the maximum total stress along the loading direction, including contributions from applied load, lamination residual stress and hoop residual stress, on the 90° ply prior to matrix crack initiation is sm 5 0.965s11 1 sR θθ 1 sR 90. (7) For unidirectional CAS/SiC, the total stress carried by the matrix is sm 5 Em Ec s11 1 sR aa 5 0.76s11 1 sR aa. (8) Surface matrix cracks are detected when the applied stress s11 reaches 140 MPa for unidirectional CAS/SiC [8, 10], which corresponds to sm 5 187.4 MPa from equation (8). When the maximum stress criterion is applied, based on the matrix crack initiation stress of the unidirectional composite, the predicted applied stress for matrix crack initiation in the 0°/90° cross-ply laminate is 56 MPa [equation (7)]. This is in good agreement with the lowest stress level where surface cracks were visually detected [Fig. 3(a)]. Mechanical testing shows that the stress– strain curve starts to deviate from linear behavior at |50 MPa, and the corresponding AE recording also indicates that the onset of matrix cracking occurs at about the same stress level (Fig. 2). Other work has also shown that the matrix cracking stress s11 ranges from 40 to 60 MPa [3]. Some of this discrepancy can be attributed to sample differences. Additionally, it has also been noted that in unidirectional CMCs, matrix cracking initiation depends on the local fiber distribution; matrix cracking can be initiated in the matrix-rich region at a lower stress level than where the fibers are more densely and uniformly distributed [23]. Also, free edge effects (although considered to be small in this case) on surface crack initiation in a 0°/90° laminate [24] were not investigated in above simplified analysis. Regardless, the overall agreement between theory and experimental observation is good. Finally, we note that the residual stress analysis above showed that the maximum matrix tensile stress occurred in the loading direction, within the 90° ply. The observation that matrix cracks initiated preferentially near the boundaries of 90° plys must be either a consequence of a locally higher matrix fraction at the ply interface or variations in stress at the 0°/90° ply boundary [25]. Further studies are needed to improve the incorporation of matrix crack initiation in models of laminate mechanics. 6. CONCLUSIONS The elastic stiffness degradation of a 0°/90° crossply laminate has been evaluated by a combination of a laser-ultrasonic characterization, acoustic emission and surface replicas. The LU technique allowed a detailed characterization of the anisotropic damage through ultrasonic velocity measurements in various propagation directions during loading and unloading of the composite. Damage along the three principal directions was characterized by the elastic constants C11, C22 and C33. The largest stiffness reduction was observed in C11, the loading-direction modulus. This has been linked to transverse matrix cracking initiating in 90° plies with an initiation stress of about 50 MPa. Smaller reductions for C22 and C33 in the transverse plane are attributed to interface damage in the 0° plies that initiated at about 75 MPa. Matrix cracks in this cross-ply composite are found to initiate at much lower applied stress levels than in unidirectional CAS/SiC composites. This observation is rationalized by residual stress analysis
LIU et al.: LAMINATED COMPOSITE 4849 Acknowledgements-This work was co-supported by the 10. LIu, Y. M, Mitchell, T. E. and wadley, H.N. G, Acta Defense Advanced Research Projects Agency (DARPA mater.,1997,45(10,3981 through a Research Initiative coordinated by the Uni M, He, Y, Chu, F, Mitchell, T. E and Wadl Califomia at Santa Barbara under ONR Contract No. No0014-. H. N. G.,J. Am. Ceram. Soc., 1997, 80(1), 14 92-J-1808(Drs S. Fishman and W. Coblinz, Program 12. Mall,S and Kim,R. Y, Composites. 1992, 23(4), 215 Managers)and the US Department of Er Ofice of Basic 13. Cady, C, Heredia, F. E and Evans, A. G.,J. Am. Ceram Inc)for supplying the material, and Drs D. T. Queheillalt and 14. Larsen, D. C, Coming Inc, personal communication C.T.Herakovich, both of the University of Virginia, for help- 15. Auld,B A Acoustic Fields and Waves in Solids. Krieger 16. Christensen, R. M, Mechanics of Composite Materials 17. Huang, Y, Hu, K. X and Chandra, A, Int J. Solids REFERENCES Struct,1993,30,1907. 18. Liu, Y. M, Mitchell, T. E. and Wadley, H. N. G 1. Aboudi, J, Lee, S. W. and Herakovich, C. T.,J. Appl. Interfacial Engineering for Optimied Properties. ed C. ech,1988,55,389 er. Res. Symp 2. Herakovich, C. T, Mechanics of Fibrous Composites. John roc.. Vol. 458 Materials Research Society. Pittsburgh. Wiley Sons Inc, New York, 1998 PA,1997,p.161 3. Beyerle, D S, Spearing, S M. and Evans, A G,J.Am. 19. Marshall, D. B, Cox, B. N and Evans, A. G, Acta metall. Ceran.Soc,192,75(12),3321 1985,33(11)2013 4. Cox, B N. and Marshall, D. B.,J. Am. Ceram. Soc., 1996, 20. Chyung, K, Corning Inc, personal communication. 79(5,1181 21. Poristsky, H, Physics. 1934, 5, 406 5. Xia, Z C, Car, R. R and Hutchinson, J. W, Acta. metall. 22. Sypeck, D, Damage evolution in titanium matrix com- marer.,1993,41(8) sites.Ph. D. Dissertation, University of Virginia, Char 6. Herakovich, C. T, Aboudi, J, Lee, S. W. and Strauss, E ttesville. 1996 A. Mech. Mater 1988.7.91 23. Dutton, R. E, Pagano, N J and Kim, R.Y.,J.Am. Ceram. 7. Drissi-Habti, M, Scripta metall. mater., 1995, 33, 967. Soc,1996,794),865 8. Beyerle, D. S, Spearing, S. M, Zok, F. W. and Evans, A. 24. Clarke, D. R, University of California at Santa Barbara, G.,J.Am. Ceram. Soc,1992,75(10,2719 rsonal communication 9. Baste, S, Guerjouma, R E and Audoin, B, Mech. Mater, 25. Herakovich, C. T, University of Virginia, personal com- 1992.14.15
LIU et al.: LAMINATED COMPOSITE 4849 Acknowledgements—This work was co-supported by the Defense Advanced Research Projects Agency (DARPA) through a Research Initiative coordinated by the University of California at Santa Barbara under ONR Contract No. N00014- 92-J-1808 (Drs S. Fishman and W. Coblinz, Program Managers) and the US Department of Energy, Office of Basic Energy Sciences. We are grateful to Mr D. C. Larsen (Corning Inc.) for supplying the material, and Drs D. T. Queheillalt and C. T. Herakovich, both of the University of Virginia, for helpful discussion of this work. REFERENCES 1. Aboudi, J., Lee, S. W. and Herakovich, C. T., J. Appl. Mech., 1988, 55, 389. 2. Herakovich, C. T., Mechanics of Fibrous Composites. John Wiley & Sons Inc., New York, 1998. 3. Beyerle, D. S., Spearing, S. M. and Evans, A. G., J. Am. Ceram. Soc., 1992, 75(12), 3321. 4. Cox, B. N. and Marshall, D. B., J. Am. Ceram. Soc., 1996, 79(5), 1181. 5. Xia, Z. C., Carr, R. R. and Hutchinson, J. W., Acta. metall. mater., 1993, 41(8), 2365. 6. Herakovich, C. T., Aboudi, J., Lee, S. W. and Strauss, E. A., Mech. Mater., 1988, 7, 91. 7. Drissi-Habti, M., Scripta metall. mater., 1995, 33, 967. 8. Beyerle, D. S., Spearing, S. M., Zok, F. W. and Evans, A. G., J. Am. Ceram. Soc., 1992, 75(10), 2719. 9. Baste, S., Guerjouma, R. E. and Audoin, B., Mech. Mater., 1992, 14, 15. 10. Liu, Y. M., Mitchell, T. E. and Wadley, H. N. G., Acta. mater., 1997, 45(10), 3981. 11. Liu, Y. M., He, Y., Chu, F., Mitchell, T. E. and Wadley, H. N. G., J. Am. Ceram. Soc., 1997, 80(1), 142. 12. Mall, S. and Kim, R. Y., Composites. 1992, 23(4), 215. 13. Cady, C., Heredia, F. E. and Evans, A. G., J. Am. Ceram. Soc., 1995, 78(8), 2065. 14. Larsen, D. C., Corning Inc., personal communication. 15. Auld, B. A., Acoustic Fields and Waves in Solids. Krieger Publishing Company, Malabar, FL, 1990. 16. Christensen, R. M., Mechanics of Composite Materials. Wiley, New York, 1979. 17. Huang, Y., Hu, K. X. and Chandra, A., Int. J. Solids Struct., 1993, 30, 1907. 18. Liu, Y. M., Mitchell, T. E. and Wadley, H. N. G., in Interfacial Engineering for Optimized Properties. ed. C. L. Briant, C. B. Carter and E. L. Hall, Mater. Res. Symp. Proc., Vol. 458 Materials Research Society, Pittsburgh, PA, 1997, p. 161. 19. Marshall, D. B., Cox, B. N. and Evans, A. G., Acta metall., 1985, 33(11), 2013. 20. Chyung, K., Corning Inc., personal communication. 21. Poristsky, H., Physics. 1934, 5, 406. 22. Sypeck, D., Damage evolution in titanium matrix composites. Ph.D. Dissertation, University of Virginia, Charlottesville, 1996. 23. Dutton, R. E., Pagano, N. J. and Kim, R. Y., J. Am. Ceram. Soc., 1996, 79(4), 865. 24. Clarke, D. R., University of California at Santa Barbara, personal communication. 25. Herakovich, C. T., University of Virginia, personal communication