Joumal of the American Ceramic Sociery-Kovar et al. Vol. 80. No. 10 Cell Boundary ■圜 ■國圜塵圖 Fig. 7. Schematic of the "brick model" used to calculate the Youngs modulus of fibrous monoliths Table 1. Measured and predicted values of elastic EBN, ESN, and VBN with a model that combines elastic elements Properties for Uniaxially Aligned Fibrous Monoliths in series and in parallel, 9 and yields the following equation: Er(GPa) Ez(GPa) G12(GPa) Measured 276 PredictionVoigt or Reuss 268 E2,3EBN+P*EsNEBN(I-I*) Prediction-brick model 268 124 likely to be different from other BN materials. The presence of This predicts a value for the se modulus, E,, of 124 etermined value of 127 the section on microstructure)also can influence the modulus GPa for uniaxially aligned tested in the off-axis direction an approximate number of 20 GPa is used To calculate the Youngs moduli, E, and E2, for uniaxially D) Poisson's Ratio and Shear Modulus for Uniaxially aligned fibrous monoliths from the constituent values for SiaN Aligned Architecture: Predicting Poissons ratio for a com- lex architecture is not as straightforward as it is for the nd BN, the cross-sectional structure of the materials is mod- Youngs modulus. However, because there is only a weak de eled using a"brick model, in which the cells are represented pendence of architecture on Poissons ratio, we use a simple by square brick and the cell boundaries are treated as mortar rule-of-mixtures approach. Assuming a condition of uniform surrounding the brick. This brick model is shown schematically in Fig. 7. The measured values of the elastic constants are listed strain exists in Table I, with predicted values from the sections that follo (3) (B) Direction 1-Longitudinal Modulus of Uniaxial Archi tecture: The well-known Voigt rule-of-mixtures is used as a nd v,i can be found from first approximation for El, the Youngs modulus in the longi- tudinal direction E= EBN BN ESN(-VBNd (1) This is a plausible model for the longitudinal Young's modulus solids. 22Mogeneral result from the elasticity for orthotropic which is for the uniaxially aligned architecture, because the SiaN4 cells fiber-reinforced composites, but these all require accurate and BN cell boundaries reasonably approximate the assump- knowledge of the shear modulus of the constituents. The shear tions made in the Voigt model (equal strain in elastic elements modulus of polycrystalline BN is unknown. Because an inde- connected in parallel). Table I lists the predicted value from the dent measurement of the shear modulus for bn could not Voigt model, which shows agreement within% compared to be obtained the shear modulus of fibrous monolithic ceramics not actually a tensile modulus, because the impulse-excitation suring G, on bars aligned at 0o and then at 90.Because hnique excites the specimen in a flexural model. The lor the two measured values should be itudinal flexural modulus of the uniaxial architecture is mod eled by calculating the effective section modulus using the average was (E) Elastic Modulus as a Function of Ply Angle for Uni brick model shown in Fig. 7.28 This leads to a straightforward but quite lengthy expression in terms of EBN, EsN, and v 29 axially Aligned Architecture: Classical laminate theory can This model converges to the Voight rule of mixtures for fibrous aligned architecture as a function of ply angle using the elastic monoliths with six or layers. Because our specimens are typically 25 layers, the Voight model is used properties calculated in the previous section(El, E2, G12, and (C) Direction 2-Transverse Modulus of Uniaxial Archi v12). In terms of the angular orientation of the ply angle, the tecture: The well-known Reuss model (uniform stress to elas- Youngs modulus is given by tic elements connected in series)serves as a lower bound for m2n2 the transverse modulus Ex, but usually badly underestimates the Youngs modulus of composites. 2 Other models for pre Ee dicting E2 for fiber-reinforced composites typically require ad- where itional experimental data and empirical factors. We have been able to accurately predict the transverse modulus using only
likely to be different from other BN materials. The presence of secondary phases and microcracks (discussed in more detail in the section on microstructure) also can influence the modulus of BN. Given the uncertainty in the Young’s modulus of BN, an approximate number of 20 GPa is used. To calculate the Young’s moduli, E1 and E2, for uniaxially aligned fibrous monoliths from the constituent values for Si3N4 and BN, the cross-sectional structure of the materials is modeled using a ‘‘brick model,’’ in which the cells are represented by square brick and the cell boundaries are treated as mortar surrounding the brick. This brick model is shown schematically in Fig. 7. The measured values of the elastic constants are listed in Table I, with predicted values from the sections that follow. (B) Direction 1—Longitudinal Modulus of Uniaxial Architecture: The well-known Voigt rule-of-mixtures is used as a first approximation for E1, the Young’s modulus in the longitudinal direction: E1 4 EBNVBN + ESN(1 − VBN) (1) This is a plausible model for the longitudinal Young’s modulus for the uniaxially aligned architecture, because the Si3N4 cells and BN cell boundaries reasonably approximate the assumptions made in the Voigt model (equal strain in elastic elements connected in parallel). Table I lists the predicted value from the Voigt model, which shows agreement within ∼3% compared to the experimental values. However, the experimental value is not actually a tensile modulus, because the impulse-excitation technique excites the specimen in a flexural model. The longitudinal flexural modulus of the uniaxial architecture is modeled by calculating the effective section modulus using the brick model shown in Fig. 7.28 This leads to a straightforward but quite lengthy expression in terms of EBN, ESN, and VBN. 29 This model converges to the Voight rule of mixtures for fibrous monoliths with six or more layers. Because our specimens are typically 25 layers, the Voight model is used. (C) Direction 2—Transverse Modulus of Uniaxial Architecture: The well-known Reuss model (uniform stress to elastic elements connected in series) serves as a lower bound for the transverse modulus E2, but usually badly underestimates the Young’s modulus of composites.22 Other models for predicting E2 for fiber-reinforced composites typically require additional experimental data and empirical factors. We have been able to accurately predict the transverse modulus using only EBN, ESN, and VBN with a model that combines elastic elements in series and in parallel,29 and yields the following equation: E2,3 = EBNV* + ~1 − V*!EBNESN V*ESN + EBN ~1 − V*! (2a) where V* = 1 − =1 − VBN (2b) This predicts a value for the transverse modulus, E2, of 124 GPa, compared to the experimentally determined value of 127 GPa for uniaxially aligned specimens tested in the off-axis direction. (D) Poisson’s Ratio and Shear Modulus for Uniaxially Aligned Architecture: Predicting Poisson’s ratio for a complex architecture is not as straightforward as it is for the Young’s modulus. However, because there is only a weak dependence of architecture on Poisson’s ratio, we use a simple rule-of-mixtures approach. Assuming a condition of uniform strain exists, n12 4 VSNn1,2,SN + VBNn1,2,BN (3) and n21 can be found from n12 E1 = n21 E2 (4) which is a general result from the elasticity for orthotropic solids.22 Models exist that accurately predict shear modulus for fiber-reinforced composites, but these all require accurate knowledge of the shear modulus of the constituents. The shear modulus of polycrystalline BN is unknown. Because an independent measurement of the shear modulus for BN could not be obtained, the shear modulus of fibrous monolithic ceramics instead was measured directly. This was accomplished by measuring G12 on bars aligned at 0° and then at 90°. Because, theoretically, G12 4 G21, the two measured values should be equal, and, thus, the average was used. (E) Elastic Modulus as a Function of Ply Angle for Uniaxially Aligned Architecture: Classical laminate theory can be used to predict the Young’s modulus of the uniaxially aligned architecture as a function of ply angle using the elastic properties calculated in the previous section (E1, E2, G12, and n12). In terms of the angular orientation of the ply angle, the Young’s modulus is given by22 1 Eu = m2 E1 ~m2 − n2 n12! + n2 E2 ~n2 − m2 n21! + m2 n2 G12 (5a) where m 4 cos u (5b) Fig. 7. Schematic of the ‘‘brick model’’ used to calculate the Young’s modulus of fibrous monoliths. Table I. Measured and Predicted Values of Elastic Properties for Uniaxially Aligned Fibrous Monoliths E1 (GPa) E2 (GPa) G12 (GPa) Measured 276 127 78 Prediction—Voigt or Reuss 268 88 Prediction—brick model 268 124 2476 Journal of the American Ceramic Society—Kovar et al. Vol. 80, No. 10
October 1997 Fibrous monolithic ceramics 2477 sin e Table l. Measured and Predicted values of young Modulus for Several Multilayer Fibrous Monoliths This expression is plotted in Fig. 8 with the experimentally measured values, showing that there is very good agreement ±6o][o45/ between experiment and prediction Measured modulus(GPa)198±2205±7202±3 (F Young's Modulus for Multiaxial Architectures: The 198 198 Young's modulus for multiaxial architectures is calculated terms of the engineering properties E1, E2, G12, and v12 using baut icat t ound ihestaquarid te xrs o lengtate th present Tare by an individual cell is related to the geometry of the specimen Il shows the measured and predicted Young's moduli for three 90]. The experimental values and the predictions from laminate earie d can be calculated using laminate theory as described architectures with simple stacking: [0/90), [0/460), and [0/45/ ary, and theory agree within 2.5% Unlike most monolithic ceramics or layered A(G) Summary of Elastic Properties: Because of the tex- strophic failure of the entire layer of cells on the tensile surface of the specimen. In other words, local failure of a single cell properties along the principal axes differ from the values pre- does not always cause global failure. If the stress that was have been presented that allow the elastic properties to be carried by the fractured cell can be transferred to neighboring accurately predicted. Principal moduli were determined from the elastic properties of the constituent materials and, with possible for the layer to remain intact after the failure of indi- vidual cells. Presumably, this behavior is favored when there to accurately predict modulus as a function of ply angle within are many cells in the specimen and the variability in the models can be extended to predict the elastic moduli for fibrous If loading is continued beyond the point where the fracture onolithic ceramics with multiaxial architectures of an individual cell occurs, eventually enough cells fracture so that the remaining cells on the tensile surface can no longer bear the applied stress, and an entire layer of cells fractures. In IV. Failure Mechanisms in Fibrous monoliths Two modes of failure have been observed during the flexural simple laminates, where fracture is controlled by weak-link atistics. Because the failure of fibrous monoliths is controlled of the beam because of tensile stress, or failure can initiate near by damage accumulation, the strength should be less sensitive the midsection of the beam because of shear stress. Either can to preexisting flaws than either monolithic ceramics or simple ditions. and on the fracture resistance of the cell and the cell laminates boundary. Tensile failure is favored when the cell boundaries load-bearing capacity of the bar is reduced, because the effec- are tough in comparison to that of the cells. Shear failure is tive cross section of the bar is smaller. In the case of uniaxially favored when the cell boundaries are weak compared to the aligned specimens with cells aligned at zero degrees(on-axis), cells. In a flexural test, the span-to-depth ratio of the bar de termines the relative magnitude of the normal stress to the the maximum applied load is typically achieved at the poi have a strength of -450 MPa. If the test is conducted in dis- tested as a short, thick beam. We consider each failure mecha- placement control, it is possible for the specimen to continue to nism separately bear a substantial load to large deflections even after the peak () Tensile Failure by Cell fracture load is achieved. An example of a typical stress-deflection for For an individual cell, the failure criterion is simply that a specimen in which failure initiated on the tensile surface is ailure occurs when the normal, tensile stress carried by that fa shown in Fig. 9(a). Each stress drop is associated with the cell exceeds the strength of the cell. Because the cells are made fracture of one or several layers of cells. The progressive nature from SiNa, the strength of an individual cell depends on the of the fracture process is shown in Fig. 9(b), the side surface of flaw size and fracture resistance of the cell. The stress carried this specimen after testing. The area under the stress-deflection curve is related to the energy dissipated by the sample during his noncatastrophic fracture. Typically, uniaxially aligned ecimens tested on-axis have a work-of-fracture of.5 Measure Predicted ( Tensile Failure by Cell-Boundary fracture In architectures where cells are misaligned with respect to the axis of the applied load, it is possible for the cells to remain intact, but for the specimen to fail when the surrounding cell boundary fractures. An SEM micrograph of the fracture surface of a BN cell boundary is shown in Fig 3, which shows that fracture in the interphase occurs by separation of the platelike grains between the weak, basal planes of the BN. It is likely that the preexisting Mrozowski microcracking in the Bn in terphases weakens the BN interphase by introducing large pre- existing defects that can propagate to failure Figure 10(a) shows examples of stress-deflection curves for specimens tested with cells oriented at 90 and at 30 with ersus orient hs Measu he Bn interphase on the tensile surface. Thus, the strength is
n 4 sin u (5c) This expression is plotted in Fig. 8 with the experimentally measured values, showing that there is very good agreement between experiment and prediction. (F) Young’s Modulus for Multiaxial Architectures: The Young’s modulus for multiaxial architectures is calculated in terms of the engineering properties E1, E2, G12, and n12 using laminate theory. The equations are too lengthy to present here, but can be found in standard texts on laminate theory.22 Table II shows the measured and predicted Young’s moduli for three architectures with simple stacking: [0/90], [0/±60], and [0/±45/ 90]. The experimental values and the predictions from laminate theory agree within 2.5%. (G) Summary of Elastic Properties: Because of the texture associated with fibrous monolithic ceramics, the elastic properties along the principal axes differ from the values predicted using rule-of-mixture models. However, simple models have been presented that allow the elastic properties to be accurately predicted. Principal moduli were determined from the elastic properties of the constituent materials and, with experimentally measured shear modulus data, have been used to accurately predict modulus as a function of ply angle within the plane of hot pressing. It also has been shown that these models can be extended to predict the elastic moduli for fibrous monolithic ceramics with multiaxial architectures. IV. Failure Mechanisms in Fibrous Monoliths Two modes of failure have been observed during the flexural testing of fibrous monoliths. Failure can initiate on the surface of the beam because of tensile stress, or failure can initiate near the midsection of the beam because of shear stress. Either can occur depending on the specimen geometry and loading conditions, and on the fracture resistance of the cell and the cell boundary. Tensile failure is favored when the cell boundaries are tough in comparison to that of the cells. Shear failure is favored when the cell boundaries are weak compared to the cells. In a flexural test, the span-to-depth ratio of the bar determines the relative magnitude of the normal stress to the shear stress; therefore, the same material might fail because of tensile stress if tested as a long, slim beam, but fail in shear if tested as a short, thick beam. We consider each failure mechanism separately. (1) Tensile Failure by Cell Fracture For an individual cell, the failure criterion is simply that failure occurs when the normal, tensile stress carried by that cell exceeds the strength of the cell. Because the cells are made from Si3N4, the strength of an individual cell depends on the flaw size and fracture resistance of the cell. The stress carried by an individual cell is related to the geometry of the specimen and the constituent elastic properties of the cell and cell boundary, and can be calculated using laminate theory as described earlier. Unlike most monolithic ceramics or layered ceramics, the failure of an individual cell does not necessarily cause catastrophic failure of the entire layer of cells on the tensile surface of the specimen. In other words, local failure of a single cell does not always cause global failure. If the stress that was carried by the fractured cell can be transferred to neighboring cells that are strong enough to bear the increased stress, it is possible for the layer to remain intact after the failure of individual cells. Presumably, this behavior is favored when there are many cells in the specimen and the variability in the strength of the cells is high.30 If loading is continued beyond the point where the fracture of an individual cell occurs, eventually enough cells fracture so that the remaining cells on the tensile surface can no longer bear the applied stress, and an entire layer of cells fractures. In this case, failure of the layer involves the accumulation of failure of a number of cells. Contrast this with monoliths or simple laminates, where fracture is controlled by weak-link statistics. Because the failure of fibrous monoliths is controlled by damage accumulation, the strength should be less sensitive to preexisting flaws than either monolithic ceramics or simple laminates. Once a layer of cells fractures during flexural loading, the load-bearing capacity of the bar is reduced, because the effective cross section of the bar is smaller. In the case of uniaxially aligned specimens with cells aligned at zero degrees (on-axis), the maximum applied load is typically achieved at the point just prior to failure in the layer of cells closest to the tensile surface. Uniaxially aligned specimens tested on-axis typically have a strength of ∼450 MPa. If the test is conducted in displacement control, it is possible for the specimen to continue to bear a substantial load to large deflections even after the peak load is achieved. An example of a typical stress–deflection for a specimen in which failure initiated on the tensile surface is shown in Fig. 9(a). Each stress drop is associated with the fracture of one or several layers of cells. The progressive nature of the fracture process is shown in Fig. 9(b), the side surface of this specimen after testing. The area under the stress–deflection curve is related to the energy dissipated by the sample during this noncatastrophic fracture. Typically, uniaxially aligned specimens tested on-axis have a work-of-fracture of ∼7.5 kJ/m2 . (2) Tensile Failure by Cell-Boundary Fracture In architectures where cells are misaligned with respect to the axis of the applied load, it is possible for the cells to remain intact, but for the specimen to fail when the surrounding cell boundary fractures. An SEM micrograph of the fracture surface of a BN cell boundary is shown in Fig. 3, which shows that fracture in the interphase occurs by separation of the platelike grains between the weak, basal planes of the BN. It is likely that the preexisting Mrozowski microcracking13 in the BN interphases weakens the BN interphase by introducing large preexisting defects that can propagate to failure. Figure 10(a) shows examples of stress–deflection curves for specimens tested with cells oriented at 90° and at 30° with respect to the applied load. The pattern of cracking is shown in Figs. 10(b) and (c), where the side surfaces of the specimens are shown after testing. Failure is catastrophic and initiates in the BN interphase on the tensile surface. Thus, the strength is Fig. 8. Young’s modulus versus orientation for uniaxially aligned fibrous monoliths. Measured values are indicated by points. Line is the predicted behavior using the brick model and laminate theory. Table II. Measured and Predicted Values of Young’s Modulus for Several Multilayer Fibrous Monoliths [0/90] [0/±60] [0/±45/90] Measured modulus (GPa) 198 ± 2 205 ± 7 202 ± 3 Predicted modulus (GPa) 201 198 198 October 1997 Fibrous Monolithic Ceramics 2477