COMPOSITES SCIENCE AND TECHNOLOGY ELSEⅤIER Composites Science and Technology 62(2002)967-976 www.elsevier.com/locate/compscitech Thermal expansion of unidirectional and cross-ply fibrous monoliths M.Y. Hea, D. Singh, J.C. mcnulty, F w. Zok, s Department, University of California, Santa Barbara, CA 93106-5050, US.A lergy Technology Division, Argonne National Laboratory, Argonne, IL 60439 USA Received 24 April 2001; received in revised form 2 January 2002: accepted 2 January 2002 Abstract An investigation of the thermal expansion behavior of ceramic fibrous monoliths(FMs) is presented. The emphasis is on the devel- pment and validation of material models applicable to Si3 N4/BN FMs in both unidirectional and cross-ply architectures. Approximate analytical models are developed for the coefficient of thermal expansion(CtE) based on the analysis of representative unit cells of the Si3 N4 fibers and the surrounding BN interphase. The pertinent cell shapes are identified from quantitative measurements on real Sin BN FMs Corresponding finite element analyses are performed on the same unit cells for the purpose of validating the analytic models. Good agreement is obtained between the model predictions and experimental measurements of CtE. A rudimentary modification to the analytical model to account for texturing and anisotropy of the bn appears to yield adequate results. 2002 Elsevier Science Ltd. All rights reserved Keywords: A. Ceramic matrix composites: B. Thermal properties; C. Elastic properties; C. Finite element analysis(FEA): Fibrous monoliths 1. Introduction (comparable to that of conventional SiC/SiC compo- sites with the same architecture [5]). Because of the pre- Fibrous monoliths(FMs)comprising Si3 N4 fibers and sence of the weak BN interphase in the transverse plies a bn interphase exhibit good potential for high tem- the strength is dictated by the axial plies: the inference perature thermostructural applications. The attractive being that the fiber strength, or >20o/9o 500 MPa. The characteristics include low costs [1], reasonably high latter value is consistent with that of some monolithic strength and high fracture toughness. The low cost is Si3N4 materials although there may be some room fo derived largely from the use of straightforward powder- improvement through enhanced process control. Similar processing routes, including: (i)co-extrusion of the con- strength levels are obtained in the +45 orientation stituent phases into green monofilaments, (i) the forma-(O45-250-300 MPa [4]. The initiation fracture resis- tion of unitapes by filament winding, (iii) lay-up of the tance of these FMs is Al0 MPa,m: about 3 times the unitapes to produce the requisite(multidirectional) fiber fracture toughness of neat Si3N4 [4]. This difference is architectures, and (iv) densification of the laminates attributable to the deflection of cracks that are in the through conventional hot-pressing, HIPing or pressure- Si3 N4 into the BN interphase and the associated mitigat less sintering(see, for example, the review by Kovar et al. ing effects on the crack tip stress intensity. Subsequent to [2]and the references therein). Variations on this process cracking of the fibers, the fracture resistance increases that involve solid freeform fabrication have also shown further with increasing crack length, a consequence of promise for producing three-dimensional fiber archi- pullout of the broken fibers past the surrounding BN, and lectures and component shapes and for further reducing at steady-state, reaches values of Ks 20-25 MPa,m[4] the costs, especially those associated with the hand lay-up With the expectation that this class of FMs will find methods [3]. In cross-ply architectures, the strength, aoo, use in high temperature thermostructural applications, a in the 0%/90 loading orientation is about 250 MPa [4] further understanding of their thermo-elastic properties will be required, both for component design and for Corresponding author. Tel: +1-805-893-8699 materials development. The present study addresses one aspect of this property group, notably the coefficient of E-mail address: zok(@ engineering. ucsd. edu(F w. Zok) thermal expansion (CTE), over a temperature range 0266-3538/02/S. see front matter C 2002 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(02)00033-7
Thermal expansion of unidirectional and cross-ply fibrous monoliths M.Y. Hea , D. Singhb, J.C. McNultya , F.W. Zoka,* a Materials Department, University of California, Santa Barbara, CA 93106-5050, USA bEnergy Technology Division, Argonne National Laboratory, Argonne, IL 60439-4838, USA Received 24 April 2001; received in revised form 2 January 2002; accepted 2 January 2002 Abstract An investigation of the thermal expansion behavior of ceramic fibrous monoliths (FMs) is presented. The emphasis is on the development and validation of material models applicable to Si3N4/BN FMs in both unidirectional and cross-ply architectures. Approximate analytical models are developed for the coefficient of thermal expansion (CTE) based on the analysis of representative unit cells of the Si3N4 fibers and the surrounding BN interphase. The pertinent cell shapes are identified from quantitative measurements on real Si3N4/ BN FMs. Corresponding finite element analyses are performed on the same unit cells for the purpose of validating the analytical models. Good agreement is obtained between the model predictions and experimental measurements of CTE. A rudimentary modification to the analytical model to account for texturing and anisotropy of the BN appears to yield adequate results. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic matrix composites; B. Thermal properties; C. Elastic properties; C. Finite element analysis (FEA); Fibrous monoliths 1. Introduction Fibrous monoliths (FMs) comprising Si3N4 fibers and a BN interphase exhibit good potential for high temperature thermostructural applications. The attractive characteristics include low costs [1], reasonably high strength and high fracture toughness. The low cost is derived largely from the use of straightforward powderprocessing routes, including: (i) co-extrusion of the constituent phases into green monofilaments, (ii) the formation of unitapes by filament winding, (iii) lay-up of the unitapes to produce the requisite (multidirectional) fiber architectures, and (iv) densification of the laminates through conventional hot-pressing, HIPing or pressureless sintering (see, for example, the review by Kovar et al. [2] and the references therein). Variations on this process that involve solid freeform fabrication have also shown promise for producing three-dimensional fiber architectures and component shapes and for further reducing the costs, especially those associated with the hand lay-up methods [3]. In cross-ply architectures, the strength, 0/90, in the 0/90 loading orientation is about 250 MPa [4] (comparable to that of conventional SiC/SiC composites with the same architecture [5]). Because of the presence of the weak BN interphase in the transverse plies, the strength is dictated by the axial plies: the inference being that the fiber strength, f 520/90500 MPa. The latter value is consistent with that of some monolithic Si3N4 materials although there may be some room for improvement through enhanced process control. Similar strength levels are obtained in the 45 orientation (45250–300 MPa [4]). The initiation fracture resistance of these FMs is 10 MPapm: about 3 times the fracture toughness of neat Si3N4 [4]. This difference is attributable to the deflection of cracks that are in the Si3N4 into the BN interphase and the associated mitigating effects on the crack tip stress intensity. Subsequent to cracking of the fibers, the fracture resistance increases further with increasing crack length, a consequence of pullout of the broken fibers past the surrounding BN, and, at steady-state, reaches values of Kss20–25 MPapm [4]. With the expectation that this class of FMs will find use in high temperature thermostructural applications, a further understanding of their thermo-elastic properties will be required, both for component design and for materials development. The present study addresses one aspect of this property group, notably the coefficient of thermal expansion (CTE), over a temperature range 0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00033-7 Composites Science and Technology 62 (2002) 967–976 www.elsevier.com/locate/compscitech * Corresponding author. Tel.: +1-805-893-8699; fax: +1-805-893- 8486. E-mail address: zok@engineering.ucsd.edu (F.W. Zok).
M.Y. He et al. Composites Science and Technology 62(2002)967-976 relevant to the targeted service conditions(ambient to (described later). The thermoplastic binder was removed 1200C). The study includes CtE measurements on by heating slowly to 700C in a nitrogen atmosphere unidirectional and cross-ply FMs along the principal The plates were then hot-pressed at 1750C for 2 h to a material directions and neat specimens of both BN and final density of 98% of the theoretical value. This also Si3N4, all processed in the same manner. Additionally, leads to the transformation of the Si, n4 from o to B[7 composite models are developed to describe the CtE of FM plates in both unidirectional (0%)and cross-ply(0%/ the FMs in terms of the topology and the thermoelastic 90)architectures were produced, each with dimensions properties of the constituent phases. Measurements and of 155 mmx155 mmx 3.2 mm. For comparison, plates models of elastic moduli are also included, as needed for of neat bn and Si3 N4 were also made using the same understanding the thermal expansion behavior of the starting powders, thermoplastic binder and sintering FMs. Comparisons between the experimental measure- additives, and following the same extrusion, lay-up and ments and model predictions are made and used to draw hot-pressing procedures insights regarding the utility of the models and to identify Quantitative metallography was employed to deter- deficiencies associated with texturing of the constituents mine the volume fractions of Si3 N4 and bn in the Fms The paper is organized in the following way. Section 2 as well as to characterize the cross-sectional shape of the describes the materials and their processing history. fibers. In both the 0o and 00/90 materials, the volume Section 3 presents the experimental techniques and ractions of bn and si3N4 were f≈20%andl-f≈80% measurements. The models are detailed in Section 4. respectively. However, the fiber shapes in these two Comparisons between the model predictions and the cases were different, as described below experimental measurements are made in Section 5 In the unidirectional FM, the fibers exhibited a shape that resembles a flattened hexagonal prism(Fig. 1). This 2. Materials All materials were manufactured by Advanced Ceram ics Research. The powders for the constituent phases were prepared separately and mixed with a thermoplastic bin der. The fiber powder consisted of equiaxed particles of Si3N4, a0. 5 um in diameter, and small amounts of yttria nd alumina powders to aid in sintering. The Si3N4 was primarily a-phase, with small amounts(5%) of B-phase [7. The final composition was Si3 N4-6 wt% Y2O3-2 wt AlO3. The Bn powder was composed of well-crystallized HCP platelets, 10 um in diameter and 0.2 um thick. After mixing, the Si3 N4 compound was compression molded into a 20 mm diameter rod The Bn compound was compression molded into a matching pair of split cylindrical shells, I mm thick and 20 mm in diameter. The bn shells were snugly fitted around the Si3 N4 rod to make a cylindrical composite feedrod comprising a central core of the Si3 N4 and a concentric shell of bn the feedrod was then extruded through a heated die to create continuous 220 um diameter green filaments, comprising a Si3 N4 core and a thin uniform bn coating Shear-induced re-orientation of the bn platelets results in a highly textured structure with the basal planes of the bn aligned parallel to the surface of the Si3 N4 [7] Green unitapes were produced by winding the fila ments around a mandrel and fixing them into place with a spray adhesive. The unitapes were then cut and stacked into the desired architecture(either 0 or 0/90o) through a hand lay-up procedure. The stacked assembly 100m was then warm-pressed at 100-150oC at a pressure of 2 Fig. 1. SEM micrographs of cross-sections through the (a) unidire MPa. During the warm pressing, the filaments were tional Si,N/BN FM, and (b)cross-ply FM. The overlaid outlines deformed to a shape dictated by the fiber lay-up the idealized fiber shapes
relevant to the targeted service conditions (ambient to 1200 C). The study includes CTE measurements on unidirectional and cross-ply FMs along the principal material directions and neat specimens of both BN and Si3N4, all processed in the same manner. Additionally, composite models are developed to describe the CTE of the FMs in terms of the topology and the thermoelastic properties of the constituent phases. Measurements and models of elastic moduli are also included, as needed for understanding the thermal expansion behavior of the FMs. Comparisons between the experimental measurements and model predictions are made and used to draw insights regarding the utility of the models and to identify deficiencies associated with texturing of the constituents. The paper is organized in the following way. Section 2 describes the materials and their processing history. Section 3 presents the experimental techniques and measurements. The models are detailed in Section 4. Comparisons between the model predictions and the experimental measurements are made in Section 5. 2. Materials All materials were manufactured by Advanced Ceramics Research. The powders for the constituent phases were prepared separately and mixed with a thermoplastic binder. The fiber powder consisted of equiaxed particles of Si3N4, 0.5 mm in diameter, and small amounts of yttria and alumina powders to aid in sintering. The Si3N4 was primarily a-phase, with small amounts (5%) of b-phase [7]. The final composition was Si3N4–6 wt.% Y2O3–2 wt.% Al2O3. The BN powder was composed of well-crystallized HCP platelets, 10 mm in diameter and 0.2 mm thick. After mixing, the Si3N4 compound was compression molded into a 20 mm diameter rod. The BN compound was compression molded into a matching pair of split cylindrical shells, 1 mm thick and 20 mm in diameter. The BN shells were snugly fitted around the Si3N4 rod to make a cylindrical composite feedrod comprising a central core of the Si3N4 and a concentric shell of BN. The feedrod was then extruded through a heated die to create continuous 220 mm diameter green filaments, comprising a Si3N4 core and a thin uniform BN coating. Shear-induced re-orientation of the BN platelets results in a highly textured structure with the basal planes of the BN aligned parallel to the surface of the Si3N4 [7]. Green unitapes were produced by winding the filaments around a mandrel and fixing them into place with a spray adhesive. The unitapes were then cut and stacked into the desired architecture (either 0 or 0/90) through a hand lay-up procedure. The stacked assembly was then warm-pressed at 100–150 Cat a pressure of 2 MPa. During the warm pressing, the filaments were deformed to a shape dictated by the fiber lay-up (described later). The thermoplastic binder was removed by heating slowly to 700 Cin a nitrogen atmosphere. The plates were then hot-pressed at 1750 Cfor 2 h to a final density of 98% of the theoretical value. This also leads to the transformation of the Si3N4 from a to b [7]. FM plates in both unidirectional (0) and cross-ply (0/ 90) architectures were produced, each with dimensions of 155 mm155 mm3.2 mm. For comparison, plates of neat BN and Si3N4 were also made using the same starting powders, thermoplastic binder and sintering additives, and following the same extrusion, lay-up and hot-pressing procedures. Quantitative metallography was employed to determine the volume fractions of Si3N4 and BN in the FMs as well as to characterize the cross-sectional shape of the fibers. In both the 0 and 0/90 materials, the volume fractions of BN and Si3N4 were f 20% and 1f 80%, respectively. However, the fiber shapes in these two cases were different, as described below. In the unidirectional FM, the fibers exhibited a shape that resembles a flattened hexagonal prism (Fig. 1). This Fig. 1. SEM micrographs of cross-sections through the (a) unidirectional Si3N4/BN FM, and (b) cross-ply FM. The overlaid outlines are the idealized fiber shapes. 968 M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976
M.Y. He et al. Composites Science and Technology 62(2002)967-976 shape is a manifestation of the nearly-hexagonal pack- ing of the green fibers in the stacked assembly coupled with the large axial compressive strains that are produced during warm pressing. A schematic of the idealized fiber cross-section is shown in Fig. 2. The cross-section is defined by a 1 that has been re-scaled in the y-direction(defined in Fig. 2) by a factor, k300 fibers. The predicted adjacent laminae give rise to a flattening of the fibers values of these parameters are into the observed rectangular shape. The average width, a and thickness b of the resultant fibers were determined +√1+3/k2 (1) from quantitative metallography measurements on >100 fibers, yielding an average aspect ratio, ab=3.4. Again, for comparison with the real fibers, the idealized shape has been superimposed on the micrograph in Fig. I(b) A=6V3ke2 The preceding characterizations of fiber shape were used to select appropriate unit cell shapes for the models The relevant non-dimensional ratio of these parameters described in Section 4. +√1+3/k 3. Experiments 3. Mec The distribution in the measured ratio P/A is plotted in Fig 3. Its average value is P/VA=4.44. From Eq The coefficients of thermal expansion of the FMs and (3), the scaling factor that yields the same ratio of P/va the neat constituents were measured using a dual-rod is k=0.35. To further demonstrate the correlation. a ifferential dilatometer(Model Dilatronic, Theta Indus- flattened hexagon for which k=0.35 has been super- tries, Port Washington, NY). The technique involves imposed on the micrograph of the real fibers in Fig. 1(a). measurement of the difference of expansion between the The fibers in the cross-ply materials were essentially test sample and a standard material for which the rectangular in cross-section [Fig. 1(b)]. In this case, no expansion is known (dense polycrystalline Al2O3), opportunity exists for nesting of fibers in adjacent laminae thereby eliminating the expansion of the system. The displacements were measured by LvDTs and the tem perature profiles were measured by thermocouples; both were monitored and recorded using a computer. Peri- odic calibration checks of the dilatometer were made using a platinum standard sample Measurements of thermal expansion were made in either two or three orthogonal directions, depending on material symmetry. The neat BN and Si3N4 specimens are transversely isotropic, the plane perpendicular to the hot pressing direction is thermally isotropic, with a CtE that differs(in general) from the through-thickness value Similar symmetry exists in the 0/90 FM because of the gon that resembles the cross-section of the Si,N4 fibers in the balanced cross-ply lay-up. By contrast, the 0 FM exhi- directional FM bits orthotropic symmetry: the three principal directions
shape is a manifestation of the nearly-hexagonal packing of the green fibers in the stacked assembly coupled with the large axial compressive strains that are produced during warm pressing. A schematic of the idealized fiber cross-section is shown in Fig. 2. The cross-section is defined by a regular hexagon that has been re-scaled in the y-direction (defined in Fig. 2) by a factor, k300 fibers. The predicted values of these parameters are: P ¼ 4k‘ 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 3=k2 h i p ð1Þ and A ¼ 6 ffiffiffi 3 p k‘2 ð2Þ The relevant non-dimensional ratio of these parameters is P ffiffiffiffi Ap ¼ a ffiffiffiffiffiffiffiffiffi k 6 ffiffiffi 3 p s 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 3=k2 h i p ð3Þ The distribution in the measured ratio P/ pA is plotted in Fig. 3. Its average value is P/ pA=4.44. From Eq. (3), the scaling factor that yields the same ratio of P/ pA is k=0.35. To further demonstrate the correlation, a flattened hexagon for which k=0.35 has been superimposed on the micrograph of the real fibers in Fig. 1(a). The fibers in the cross-ply materials were essentially rectangular in cross-section [Fig. 1(b)]. In this case, no opportunity exists for nesting of fibers in adjacent laminae into a hexagonal arrangement (as in the unidirectional FM). Upon warm pressing, the mutual constraints of adjacent laminae give rise to a flattening of the fibers into the observed rectangular shape. The average width, a, and thickness, b, of the resultant fibers were determined from quantitative metallography measurements on >100 fibers, yielding an average aspect ratio, a/b=3.4. Again, for comparison with the real fibers, the idealized shape has been superimposed on the micrograph in Fig. 1(b). The preceding characterizations of fiber shape were used to select appropriate unit cell shapes for the models described in Section 4. 3. Experiments 3.1. Measurement techniques The coefficients of thermal expansion of the FMs and the neat constituents were measured using a dual-rod differential dilatometer (Model Dilatronic, Theta Industries, Port Washington, NY). The technique involves measurement of the difference of expansion between the test sample and a standard material for which the expansion is known (dense polycrystalline Al2O3), thereby eliminating the expansion of the system. The displacements were measured by LVDTs and the temperature profiles were measured by thermocouples; both were monitored and recorded using a computer. Periodic calibration checks of the dilatometer were made using a platinum standard sample. Measurements of thermal expansion were made in either two or three orthogonal directions, depending on material symmetry. The neat BN and Si3N4 specimens are transversely isotropic; the plane perpendicular to the hot pressing direction is thermally isotropic, with a CTE that differs (in general) from the through-thickness value. Similar symmetry exists in the 0/90 FM because of the balanced cross-ply lay-up. By contrast, the 0 FM exhibits orthotropic symmetry; the three principal directions Fig. 2. Re-scaling of a regular hexagon to produce a flattened hexagon that resembles the cross-section of the Si3N4 fibers in the unidirectional FM. Fig. 3. Characterization of fiber shape in the unidirectional FM. M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976 969
M.Y. He et al. Composites Science and Technology 62(2002)967-976 are:(i) parallel to the fiber direction(denoted"in-plane are presented in the form of a secant CtE relative to longitudinal), (i) transverse to the fibers and aligned ambient temperature, defined by: with the plane of hot pressing( denoted"in-plane trans verse), and (ii) parallel to the hot pressing direction a=[E()-e(Ta)l/(T-Ta) (4) (denoted"through-thickness") The in-plane measurements were made using rectan- where E(n) is the thermal strain at temperature, T, and gular bars, a3 mmx 4 mm in cross-section and 30 mm E(Ta) is the thermal strain at ambient temperature long(the long axis coinciding with the principal direc- Ta=20C tion of interest). The through-thickness measurements The longitudinal (in-plane)elastic moduli of the FMs were made using of stacked specimens, each a3 and the neat constituent specimens were determined in mm thick and each polished carefully to ensure flat and four-point flexure. The specimen dimensions were 3 parallel faces. The use of several such specimens in a mmx5 mmx 50 mm, and the inner and outer loading ack rather than a single thin specimen increases the spans were 15 and 40 mm, respectively. Strain gauges total displacement associated with thermal expansion were bonded to the tensile and compressive surfaces and hence increases the precision in the measurements. within the constant moment region. The stresses were To verify the stacking technique, some measurements calculated using standard beam theory. The moduli were made on stacked specimens cut from the in-plane were determined through linear regression of the initial orientation and the results were then compared with the (linear) part of the stress-strain curves. Three tests were ones obtained from the long(contiguous) rectangular performed for each material and architecture bars. The two sets of measurements were found to be essentially equal to one another. The measurements 3. 2. Experimental result were made in a N2 atmosphere during both heating and cooling over the temperature range 20-1200C. The The cte measurements on the constituents and the heating and cooling rates were 2.5C/min. The results FMs are summarized in Fig. 4. In the neat Si3N4, the *7 Si,N2 9859280页日9OE In-Plane 8=80西E In-Plane 20040060080010001200140 00400600800100012001400 Temperature(C) Temperature(°c 0°/90°FM plane longitudinal oEx8tE95页580E9 In-Plane 600800100012001400 200400 0800100012001400 Temperature(c) Fig 4. Summary of CTE measurements on the neat constituents and the FMs
are: (i) parallel to the fiber direction (denoted ‘‘in-plane longitudinal’’), (ii) transverse to the fibers and aligned with the plane of hot pressing (denoted ‘‘in-plane transverse’’), and (iii) parallel to the hot pressing direction (denoted ‘‘through-thickness’’). The in-plane measurements were made using rectangular bars, 3 mm4 mm in cross-section and 30 mm long (the long axis coinciding with the principal direction of interest). The through-thickness measurements were made using a series of stacked specimens, each 3 mm thick and each polished carefully to ensure flat and parallel faces. The use of several such specimens in a stack rather than a single thin specimen increases the total displacement associated with thermal expansion and hence increases the precision in the measurements. To verify the stacking technique, some measurements were made on stacked specimens cut from the in-plane orientation and the results were then compared with the ones obtained from the long (contiguous) rectangular bars. The two sets of measurements were found to be essentially equal to one another. The measurements were made in a N2 atmosphere during both heating and cooling over the temperature range 20–1200 C. The heating and cooling rates were 2.5 C/min. The results are presented in the form of a secant CTE relative to ambient temperature, defined by: ¼ ½ "ð Þ T "ð Þ Ta =ðÞ ð TTa 4Þ where "(T) is the thermal strain at temperature, T, and "(Ta) is the thermal strain at ambient temperature, Ta=20 C. The longitudinal (in-plane) elastic moduli of the FMs and the neat constituent specimens were determined in four-point flexure. The specimen dimensions were 3.2 mm5 mm50 mm, and the inner and outer loading spans were 15 and 40 mm, respectively. Strain gauges were bonded to the tensile and compressive surfaces within the constant moment region. The stresses were calculated using standard beam theory. The moduli were determined through linear regression of the initial (linear) part of the stress–strain curves. Three tests were performed for each material and architecture. 3.2. Experimental results The CTE measurements on the constituents and the FMs are summarized in Fig. 4. In the neat Si3N4, the Fig. 4. Summary of CTE measurements on the neat constituents and the FMs. 970 M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976
M.Y. He et al. Composites Science and Technology 62(2002)967-976 in-plane CtE was almost independent of temperature. thermoelastic properties of the constituents, th Its average value(2.0x10-6K-)is comparable to the approach yields essentially exact results. The second values reported previously for a wide range of Si3 N4 approach involves partitioning the unit cell for a uni ceramics [6]. The average through-thickness value was directional lamina(comprising a rectangular Si3 N4 fiber slightly higher: 2x10-6 K-. This anisotropy is con- and a uniform BN layer ) into a convenient set of sub-cell sistent with slight texturing of the hexagonal B-grains, and approximating the behavior of each of the sub-cells with the basal plane oriented preferentially in the plane through standard upper or lower bound estimates for of hot pressing. It occurs through alignment of the pre- the CTE. In this way, approximate analytical solutions existing B-Si3 N4 particles during pressing and preferred are obtained for the principal CTEs of the laminae within growth of the new p grains on the pre-existing B particles. the cross-ply FM. Despite the differences in the cell gec [7]. The neat BN exhibited a similarly low in-plane Cte metry in the unidirectional and cross-ply materials(flat (1.8x10-6K-), but a much higher through-thickness tened hexagon vs rectangle) it is demonstrated that the CTE (13.5x10-6K-) Here, the anisotropy is attrib- results obtained using the sub-cell approach for the uted to shear-induced alignment of the hexagonal bl angular cell(relevant to the laminae in the cross-ply platelets during pressing. Similar effects have been FM)are also applicable to the hexagonal fiber in the observed within the bn phase of Si3 N4/BN FMs after unidirectional FM. The results for the individual(uni- extrusion, using both TEM and X-ray diffraction [7-9]. directional) laminae are used subsequently to model the The FMs also exhibited some anisotropy in Cte, behavior of the cross-ply laminate using classical lami- somewhat greater in magnitude than that of the neat nation theory. Comparisons are made between the ana Si3N4 but substantially less than that of the BN. This lytical solutions and the exact numerical results for correlation can be rationalized qualitatively on the basis select cases, mainly for the purpose of validating the that the major constituent of the FM is Si3 N4 and thus analytical solutions. The model predictions are com- the behavior of the FMs should mimic that of the neat pared with the experimental results in Section 5. Fur Si3N4. The models and analysis presented below validate thermore, a rudimentary approach to the incorporation his assertion in a quantitative manner. of the anisotropy of the constituents is described and The relevant elastic properties are summarized in shown to produce adequate results for the through Table 1. These results are used in subsequent modeling, thickness CtE especially for establishing connections between the Ctes of the constituents and those of the 0 and 0/90 FMs. A 4. 2. Finite element analyses particularly notable feature is the large difference in the elastic moduli of the constituents. the modulus of the Finite element analyses(FEA)of thermal expansion Si3N4 being more than an order of magnitude greater were performed on unit cells of both the unidirectional than that of the BN. A consequence is that the bn and the cross-ply FMs using the aBaQUs code. F produces only minimal constraint on the thermal strain the unidirectional FM, the fiber cross-section was taken in the Si3 N4, especially that parallel to the fiber axis to be a flattened hexagon, characterized by k=0.35 Because of the geometry and packing arrangement of the fibers, the calculations were two-dimensional. The finite 4. Modeling element mesh for the hexagonal cell comprised 168 10- node biquadratic quadrilateral generalized plane strain 4.1. Approach elements(Fig. 5). For the cross-ply FM, three-dimen- sional calculations were performed using the unit cell Two complementary modeling approaches are devel- model shown in Fig. 6. In this case, the fibers were taken oped to describe the Cte of the FMs. The first is based to be rectangular in cross-section with an aspect ratio, on finite element analyses(FEA)of two unit cell models, a b=3. 4, and arranged in a two-dimensional orthogonal each being representative of either the unidirectional or pattern, commensurate with the real fiber arrangement in the cross-ply FM. When combined with the appropriate the cross-ply FM [Fig. I(b)]. The mesh comprised 2016 20-node quadratic brick elements and 9725 nodes. For Table I comparison with the hexagonal cell model for the uni- Summary of the measured elastic moduli directional FM, finite element calculations were also per Material formed for a two-dimensional arrangement of the E(GPa) rectangular fibers, again with an aspect ratio of 3. 4. The 21±2 latter results demonstrate the rather weak sensitivity of 42±4 the material behavior to the fiber cross-section(flattened 16±2 hexagon vs. rectangle). For the CtE calculations, the 0190°FM l8l±6 surfaces of the cell boundaries were constrained to Values for FMs correspond to longitudinal orientation. remain planar with zero shear traction and zero average
in-plane CTE was almost independent of temperature. Its average value (2.0106 K1 ) is comparable to the values reported previously for a wide range of Si3N4 ceramics [6]. The average through-thickness value was slightly higher: 2106 K1 . This anisotropy is consistent with slight texturing of the hexagonal b-grains, with the basal plane oriented preferentially in the plane of hot pressing. It occurs through alignment of the preexisting b-Si3N4 particles during pressing and preferred growth of the new b grains on the pre-existing b particles. [7]. The neat BN exhibited a similarly low in-plane CTE (1.8106 K1 ), but a much higher through-thickness CTE (13.5106 K1 ). Here, the anisotropy is attributed to shear-induced alignment of the hexagonal BN platelets during pressing. Similar effects have been observed within the BN phase of Si3N4/BN FMs after extrusion, using both TEM and X-ray diffraction [7–9]. The FMs also exhibited some anisotropy in CTE, somewhat greater in magnitude than that of the neat Si3N4 but substantially less than that of the BN. This correlation can be rationalized qualitatively on the basis that the major constituent of the FM is Si3N4 and thus the behavior of the FMs should mimic that of the neat Si3N4. The models and analysis presented below validate this assertion in a quantitative manner. The relevant elastic properties are summarized in Table 1. These results are used in subsequent modeling, especially for establishing connections between the CTEs of the constituents and those of the 0 and 0/90 FMs. A particularly notable feature is the large difference in the elastic moduli of the constituents: the modulus of the Si3N4 being more than an order of magnitude greater than that of the BN. A consequence is that the BN produces only minimal constraint on the thermal strain in the Si3N4, especially that parallel to the fiber axis. 4. Modeling 4.1. Approach Two complementary modeling approaches are developed to describe the CTE of the FMs. The first is based on finite element analyses (FEA) of two unit cell models, each being representative of either the unidirectional or the cross-ply FM. When combined with the appropriate thermoelastic properties of the constituents, this approach yields essentially exact results. The second approach involves partitioning the unit cell for a unidirectional lamina (comprising a rectangular Si3N4 fiber and a uniform BN layer) into a convenient set of sub-cells and approximating the behavior of each of the sub-cells through standard upper or lower bound estimates for the CTE. In this way, approximate analytical solutions are obtained for the principal CTEs of the laminae within the cross-ply FM. Despite the differences in the cell geometry in the unidirectional and cross-ply materials (flattened hexagon vs. rectangle), it is demonstrated that the results obtained using the sub-cell approach for the rectangular cell (relevant to the laminae in the cross-ply FM) are also applicable to the hexagonal fiber in the unidirectional FM. The results for the individual (unidirectional) laminae are used subsequently to model the behavior of the cross-ply laminate using classical lamination theory. Comparisons are made between the analytical solutions and the exact numerical results for select cases, mainly for the purpose of validating the analytical solutions. The model predictions are compared with the experimental results in Section 5. Furthermore, a rudimentary approach to the incorporation of the anisotropy of the constituents is described and shown to produce adequate results for the throughthickness CTE. 4.2. Finite element analyses Finite element analyses (FEA) of thermal expansion were performed on unit cells of both the unidirectional and the cross-ply FMs using the ABAQUS code. For the unidirectional FM, the fiber cross-section was taken to be a flattened hexagon, characterized by k=0.35. Because of the geometry and packing arrangement of the fibers, the calculations were two-dimensional. The finite element mesh for the hexagonal cell comprised 168 10- node biquadratic quadrilateral generalized plane strain elements (Fig. 5). For the cross-ply FM, three-dimensional calculations were performed using the unit cell model shown in Fig. 6. In this case, the fibers were taken to be rectangular in cross-section with an aspect ratio, a/b=3.4, and arranged in a two-dimensional orthogonal pattern, commensurate with the real fiber arrangement in the cross-ply FM [Fig. 1(b)]. The mesh comprised 2016 20-node quadratic brick elements and 9725 nodes. For comparison with the hexagonal cell model for the unidirectional FM, finite element calculations were also performed for a two-dimensional arrangement of the rectangular fibers, again with an aspect ratio of 3.4. The latter results demonstrate the rather weak sensitivity of the material behavior to the fiber cross-section (flattened hexagon vs. rectangle). For the CTE calculations, the surfaces of the cell boundaries were constrained to remain planar with zero shear traction and zero average Table 1 Summary of the measured elastic moduli Material E (GPa) BN 212 Si3N4 2424 0 FM 2162 0/90 FM 1816 Values for FMs correspond to longitudinal orientation. M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976 971
h.Y. He et al. Composites Science and Technology 62(2002 )967-976 assumed to be elastic. In most cases, the youngs mod- uli of the constituents were taken to be those measured experimentally: Er=242 GPa(for the Si3 N4 fibers)and E;=20.7 GPa(for the BN interphase). The Poissons for both phases was taken to or com- Unit Cell parison between the FEA and the analytical models some FEA calculations were performed for a range of modulus values, characterized by the ratio B=Ei/Er The results from the FEA and comparisons with the Si、 N Fiber analytical models are presented in the next section 4.3. Analytical model The longitudinal thermoelastic properties of the uni directional FM are obtained using a standard Voigt averaging procedure, assuming the Poisson's ratios of he two constituent phases to be equal to one another The longitudinal modulus, Er, and thermal expansion E=f(1-+B and Fig. 5.(a) Schematic of the rray of uni nal fibers and f+(1-0) the associated unit cell; (b) finit h of the unit cell where a; and af are the thermal expansion coefficients of normal traction. FEA of the elastic moduli of the uni- the interphase and fiber, respectively. For the purpose directional FMs were also performed, as needed to of comparing the analytical solutions for the CTE with ply FMs. For the latter calculations, similar boundary convenient to re-express the result in Q (6) 9.,it is nderstand the thermal expansion behavior of the cross- those from the finite element calculations, it is conditions were used with the exception that the surface normal to the loading direction had an average stress a= Ciz ai+ Cr:ar equal to the applied stress In all cases, the relative amounts of Si3 N4 and bn where Ciz and Cf represent the respective contributions were selected to be consistent with the respective volume from the interphase and fiber to the thermal expansion fractions, /=0.8 and l-f=0. 2. Both phases were coefficient in the z-direction; they are Fig. 6. Finite element mesh of the interphase(left)and the fibers(right)in the idealized cell geometry for the cross-ply FM
normal traction. FEA of the elastic moduli of the unidirectional FMs were also performed, as needed to understand the thermal expansion behavior of the crossply FMs. For the latter calculations, similar boundary conditions were used with the exception that the surface normal to the loading direction had an average stress equal to the applied stress. In all cases, the relative amounts of Si3N4 and BN were selected to be consistent with the respective volume fractions, f=0.8 and 1f=0.2. Both phases were assumed to be elastic. In most cases, the Young’s moduli of the constituents were taken to be those measured experimentally: Ef=242 GPa (for the Si3N4 fibers) and Ei=20.7 GPa (for the BN interphase). The Poisson’s ratio for both phases was taken to be =0.2. For comparison between the FEA and the analytical models, some FEA calculations were performed for a range of modulus values, characterized by the ratio Ei/Ef. The results from the FEA and comparisons with the analytical models are presented in the next section. 4.3. Analytical model The longitudinal thermoelastic properties of the unidirectional FM are obtained using a standard Voigt averaging procedure, assuming the Poisson’s ratios of the two constituent phases to be equal to one another. The longitudinal modulus, Ez, and thermal expansion coefficient, z, are given by: Ez Ef ¼ f ð Þþ 1 ð5Þ and z ¼ ið1 f Þ þ f f f þ ð1 f Þ ð6Þ where i and f are the thermal expansion coefficients of the interphase and fiber, respectively. For the purpose of comparing the analytical solutions for the CTE with those from the finite element calculations, it is convenient to re-express the result in Eq. (6) as: z ¼ Ciz i þ Cfz f ð7Þ where Ciz and Cfz represent the respective contributions from the interphase and fiber to the thermal expansion coefficient in the z-direction; they are: Fig. 5. (a) Schematic of the periodic array of unidirectional fibers and the associated unit cell; (b) finite element mesh of the unit cell. Fig. 6. Finite element mesh of the interphase (left) and the fibers (right) in the idealized cell geometry for the cross-ply FM. 972 M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976
M.Y. He et al./ Composites Science and Technology 62(2002)967-976 (1-f)B This composite sub-cell is then combined with the f+(1-f)β f+(1-)B remaining interphase sub-cell I, and the Youngs mod- li of the entire cell obtained from the Voigt and Reuss Comparisons between the elastic modulus in Eq. (5) averages(as before), yielding the results and that obtained from the finite calculations are shown in Fig. 7. Similarly, comparisons of the coefficients Ci Ex-B(+D(+BD t(1+B1)+B(1+1) (lla) and Cfz in the Cte relationships [Eq ( 8)] are shown in Fig &(a). In all cases, the differences are extremely small In deriving analytical expressions for the transverse elastic moduli and thermal expansion coefficients, the rectangular unit cell (applicable to the cross-ply FM)is first subdivided into three sub-cells: one representing the fiber and two representing the surrounding interphase material. Two such subdivision schemes are used. In the first [Fig 9(b). the interphase is partitioned into the two 6 parts labeled I, and 12. A composite sub-cell is created by combining the fiber, labeled F, with the interphase o FEM: Rectangular cell EM: Hexagonal cell sub-cell 12. The Youngs moduli of this composite sub-cell in the x-and y-directions are evaluated using the Voigt and Reuss averages, respectively, yielding + Br Er 1+t E/E Ey B(1+r) Er B+I where i and t are normalized cell dimensions [defined in Eqn.(15) Fig 9(a)] and are related to the fiber volume fraction o FEM: Rectangular cell through FEM: Hexagonal cell f (1+1)(2+1) E 08E- Eqn.(14) ……Eqn.(17 Eqn.(12) ▲FEM: Hexagonal ce‖l FEM 02 E,/Er (lines) and those obtained from finite element calculations(symbols) Fig. 8. Coefficients dictating the thermal expansion coefficients for the for a unidirectional fm unidirectional FM in(a)z-direction, (b)y-direction and (c)x-direction
Ciz ¼ ð1 f Þ f þ ð1 f Þ and Cfz ¼ f f þ ð1 f Þ ð8Þ Comparisons between the elastic modulus in Eq. (5) and that obtained from the finite calculations are shown in Fig. 7. Similarly, comparisons of the coefficients Ciz and Cfz in the CTE relationships [Eq. (8)] are shown in Fig. 8(a). In all cases, the differences are extremely small (<<1%). In deriving analytical expressions for the transverse elastic moduli and thermal expansion coefficients, the rectangular unit cell (applicable to the cross-ply FM) is first subdivided into three sub-cells: one representing the fiber and two representing the surrounding interphase material. Two such subdivision schemes are used. In the first [Fig. 9(b)], the interphase is partitioned into the two parts labeled I1 and I2. A composite sub-cell is created by combining the fiber, labeled F, with the interphase sub-cell I2. The Young’s moduli of this composite sub-cell in the x- and y-directions are evaluated using the Voigt and Reuss averages, respectively, yielding: EB x Ef ¼ 1 þ t 1 þ t ð9aÞ and EB y Ef ¼ ð Þ 1 þ t þ t ð9bÞ where l and t are normalized cell dimensions [defined in Fig. 9(a)] and are related to the fiber volume fraction through: f ¼ l ð Þ 1 þ t ð Þ l þ t ð10Þ This composite sub-cell is then combined with the remaining interphase sub-cell I1 and the Young’s moduli of the entire cell obtained from the Voigt and Reuss averages (as before), yielding the results: Ex Ef ¼ ð Þ l þ t ð Þ 1 þ t t ð Þþ 1 þ t l ð Þ 1 þ t ð11aÞ and Fig. 7. Comparisons of the analytical solutions for elastic moduli (lines) and those obtained from finite element calculations (symbols) for a unidirectional FM. Fig. 8. Coefficients dictating the thermal expansion coefficients for the unidirectional FM in (a) z-direction, (b) y-direction and (c) x-direction. M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976 973
M.Y. He et al./ Composites Science and Technology 62(2002)967-976 cell shape(at least over the range of shapes applicable to the two FMs studied here). A similar approach (involving subdivision of the unit F cell) is used to obtain the CTEs. Following the scheme shown in Fig. 9(b), the coefficients in the x-and y- directions for the composite sub-cell comprising F and 12 are. (13a) 2vt(1-B) +t(1+)(1+t) vt(1-P) (13b) +(1+′(1+1)(1+ Then, upon combining this composite sub-cell with Fig 9. Schematic showing the subdivision of the unit cell into rectan- the sub-cell I,, the coefficients for the entire unit cell ular slabs become E_βt(β+D)+B(1+D (+1)(B+1) (11b) Bt =(+1)(1+B)1+Bt Because of the way in which this averaging is (+1)(1+B1)1+B formed, it is expected that the results in Eqs.(6)and(7) represent lower bounds to the actual values Fig 9(c)shows an alternate way to subdivide the unit B+0++2u(1-B cell. In this case. the fiber. f is first combined with the (1+tB) interphase sub-cell labeled I3 and the properties of the t(B+D)+(1+t) (15) composite sub-cell obtained in the manner described 2vt(1-B) above. This composite sub-cell is then combined with (1+tB) e remaining interphase sub-cell, 14. Following thi B+1)+(1+t) procedure, the Youngs moduli of the entire composite cell are found to be where Ex Bt(t+Bi)+B(+t (12a) i(1-B) Er(t+BA)(1+1) (1+1)+(1+t6)(+1) E1 B(1+1)(Bt+) Similarly, for the partitioning scheme shown in Er t(Bt+2)+B(+o (12b) Fig 9(c), the coefficients are Again Eqs. (12a)and (12b)are expected to yield lower l(t+B)+(+1) bound estimates 2vir(1-B) Comparisons between the predictions of Eqs. (11)and (+B)+t+-B (t+B) ( (t+B) (12)and the Fea results are shown in Fig. 7. For the (17) range of properties studied here, the correlation between the two sets of results is good, with the difference being r(Bt+i)+Br consistently<5%. Furthermore +元 lie slightly below the exact results for the rectangular cell, consistent with the expectation that the former 2vi2t(1-B) should yield lower bound estimates. The FEA result (1+1)(t+)(1+)(Bt+A)(B+t)t+t+ for the hexagonal cell with k=0.35 and Youngs moduli of the Si3 N4/ BN FMs are also included in Fig. 7. The 5%, demonstrating the insensitivity of the CTE to the a By analogy to Eq (7), it is convenient to re-write the differences in moduli between the two cell shapes is TEs in the fo
Ey Ef ¼ t ð Þþ þ t l ð Þ 1 þ t ð Þ l þ t ð Þ þ t ð11bÞ Because of the way in which this averaging is performed, it is expected that the results in Eqs. (6) and (7) represent lower bounds to the actual values. Fig. 9(c) shows an alternate way to subdivide the unit cell. In this case, the fiber, F, is first combined with the interphase sub-cell labeled I3 and the properties of the composite sub-cell obtained in the manner described above. This composite sub-cell is then combined with the remaining interphase sub-cell, I4. Following this procedure, the Young’s moduli of the entire composite cell are found to be: Ex Ef ¼ t tð Þþ þ l ð Þ l þ t ð Þ t þ l ð Þ 1 þ t ð12aÞ Ey Ef ¼ ð Þ 1 þ t ð Þ t þ l t ð Þþ t þ l ð Þ l þ t ð12bÞ Again Eqs. (12a) and (12b) are expected to yield lower bound estimates. Comparisons between the predictions of Eqs. (11) and (12) and the FEA results are shown in Fig. 7. For the range of properties studied here, the correlation between the two sets of results is good, with the difference being consistently <5%. Furthermore, the analytical results lie slightly below the exact results for the rectangular cell, consistent with the expectation that the former should yield lower bound estimates. The FEA results for the hexagonal cell with k=0.35 and Young’s moduli of the Si3N4/BN FMs are also included in Fig. 7. The differences in moduli between the two cell shapes is <5%, demonstrating the insensitivity of the CTE to the cell shape (at least over the range of shapes applicable to the two FMs studied here). A similar approach (involving subdivision of the unit cell) is used to obtain the CTEs. Following the scheme shown in Fig. 9(b), the coefficients in the x- and ydirections for the composite sub-cell comprising F and I2 are: B x ¼ t 1 þ t i þ 1 1 þ t f ð13aÞ B y ¼ t 1 þ t þ 2tð Þ 1 ð Þ 1 þ t ð Þ 1 þ t i þ 1 1 þ t 2tð Þ 1 ð Þ 1 þ t ð Þ 1 þ t f ð13bÞ Then, upon combining this composite sub-cell with the sub-cell I1, the coefficients for the entire unit cell become: x ¼ t l ¼ t þ l t ð Þ l þ t ð Þ 1 þ t þ 2 B 1 þ t i þ l ð Þ l þ t ð Þ 1 þ t 2 B 1 þ t f ð14Þ y ¼ tð Þþ þ t lt þ 2tð Þ 1 l ð Þ 1 þ t tð Þþ þ t ð Þ 1 þ t l i þ l 2tð Þ 1 l ð Þ 1 þ t tð Þþ þ t ð Þ 1 þ t l f ð15Þ where B ¼ ltð Þ 1 ½ ð Þ 1 þ t l þ tð Þ 1 þ t ð Þ l þ t ð16Þ Similarly, for the partitioning scheme shown in Fig. 9(c), the coefficients are: x ¼ 1 t tð Þþ þ l ð Þ l þ t t tð Þþ þ l t þ 2ltð Þ 1 ð Þ t þ l i þ ðl 2ltð Þ 1 ð Þ t þ l Þf ð17Þ y ¼ tð Þþ t þ l t ð Þ 1 þ t ð Þ t þ l þ 2l2 tð Þ 1 ð Þ 1 þ t ð Þ t þ l ½ ð Þ l þ t t þ t þ l i þ l ð Þ 1 þ t ð Þ t þ l 2l2 tð Þ 1 ð Þ 1 þ t ð Þ t þ l ½ ð Þ l þ t t þ t þ l f ð18Þ By analogy to Eq. (7), it is convenient to re-write the CTEs in the form: Fig. 9. Schematic showing the subdivision of the unit cell into rectangular slabs. 974 M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976
h.Y. He et al. Composites Science and Technology 62(2002 )967-976 ax= Cira+ Cfar ( 19) given by Eqs. (15)and(18). Similarly good agreement is obtained between this model and the three-dimensional ay=Ciya+ Cryar (20) FEA calculations, as shown in Fig10 where Civ and Cfr represent the respective contributions CTE 5. Comparisons between theory and experiment -direction, and Cix and Cix are the corresponding values for the cte in the x-direction These are The analytical models for CTE(Section 4) have been obtained readily from Eqs. (14),(15),(17)and(18) used to predict the Cte of both0°and09190°FMs Comparisons between the analytical models and the fea results for the transverse ctes of the unidirec tional lamina within the cross-ply FM are shown in Fig 8(b)and(c). Excellent agreement is obtained in all 70FM cases, providing confidence in the analytical approach Because of this correlation, all subsequent comparisons 地如Mmmm: with experimental measurements are based on the ana lytical models xox ceding results for the unidirectional lamina combined (a) with classical lamination theory. For this purpose, the in-plane strains in each of the laminae are partitioned 0200400600 into thermal and mechanical terms(the latter arising Temperature,T(°c) from the anisotropy of the CTE), the strains are set equal to one another to maintain compatibility between adjoining laminae, and the mechanical strains are pre- o°/90°F scribed in such a way that internal stresses are self- equilibrating. The final result for the in-plane Cte is 1+ν)(ax-a:) Analytical model (Eqn. 21) 1+E=/Ex+2v Comparisons between this result and the 3D FEA Experimental calculations are presented in Fig. 10, for values of elastic constants applicable to the Si3 N4/BN FMs and for range of values of the CTE mismatch, ai af. The two are in very good agreement. 0200400600800100012001400 The predicted through-thickness CtE of the cross-ply Temperature, T(C) material is identical to that of the unidirectional material Analytical models Through-thicknes 1.5 su5O Hybrid model o°9o°sN/BNFM 0.5 f=08.E/E,=0.087 0200400600800100012001400 CTE ratio, o/ Fig. I1. Comparisons between the measured and predicted CTEs Fig 10. CtE predictions for cross-ply FN (a, b) in-plane and (c)through-thickness
x ¼ Cixi þ Cfxf ð19Þ y ¼ Ciyi þ Cfyf ð20Þ where Ciy and Cfy represent the respective contributions from the interphase and fiber to the CTE in the y-direction, and Cix and Cfx are the corresponding values for the CTE in the x-direction. These are obtained readily from Eqs. (14), (15), (17) and (18). Comparisons between the analytical models and the FEA results for the transverse CTEs of the unidirectional lamina within the cross-ply FM are shown in Fig. 8(b) and (c). Excellent agreement is obtained in all cases, providing confidence in the analytical approach. Because of this correlation, all subsequent comparisons with experimental measurements are based on the analytical models. For the cross-ply FM, material symmetry dictates that the CTE will be isotropic in the plane of hot pressing. The in-plane CTE can be derived readily using the preceding results for the unidirectional lamina combined with classical lamination theory. For this purpose, the in-plane strains in each of the laminae are partitioned into thermal and mechanical terms (the latter arising from the anisotropy of the CTE), the strains are set equal to one another to maintain compatibility between adjoining laminae, and the mechanical strains are prescribed in such a way that internal stresses are selfequilibrating. The final result for the in-plane CTE is c z ¼ c x ¼ z þ ð Þ 1 þ ð Þ x z 1 þ Ez=Ex þ 2 ð21Þ Comparisons between this result and the 3D FEA calculations are presented in Fig. 10, for values of elastic constants applicable to the Si3N4/BN FMs and for a range of values of the CTE mismatch, i/f. The two are in very good agreement. The predicted through-thickness CTE of the cross-ply material is identical to that of the unidirectional material, given by Eqs. (15) and (18). Similarly good agreement is obtained between this model and the three-dimensional FEA calculations, as shown in Fig. 10. 5. Comparisons between theory and experiment The analytical models for CTE (Section 4) have been used to predict the CTE of both 0 and 0/90 FMs Fig. 10. CTE predictions for cross-ply FM. Fig. 11. Comparisons between the measured and predicted CTEs: (a, b) in-plane and (c) through-thickness. M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976 975
h.Y. He et al. Composites Science and Technology 62(2002 )967-976 using the appropriate constituent properties and the the cte of the BN, since the bn is the""continuous results have been compared with the experimental mea- phase"in this orientation and thus plays a more sig- surements on the FMs(Section 3). The key results are nificant role in the Cte of the FM. A rudimentary plotted in Fig. ll. For the in-plane properties, the rele- modification to the model to account for texturing and vant constituent properties (CTE and E)were those anisotropy of the Bn appears to yield adequate results, measured on the neat Si3N4 and BN in the in-plane although further work is needed to ascertain the extent orientation. Very good agreement is obtained between of texturing and its effect on various thermo-mechanical the calculated and measured values. For the through properties of FMs. hickness properties, calculations are presented for two limiting cases: (i) using the in-plane properties of both constituents [lower line in Fig. 11(c)]; and (ii)using the Acknowledgements through-thickness properties of both constituents [upper line in Fig. 11(c)). These two sets of results appear to Funding for this work was provided by the US bound the experimental data. The results from a hybrid Department of Energy(Prime Contract No. W-31-10 model that accounts for the anisotropy of the con- Eng-98)to the Argonne National Laboratory and by stituents in a rudimentary way is also shown in Fig. 11(c) subcontract from the Argonne National Laboratory to (middle line). The latter model is based on the unit cell the University of California, Santa Barbara( Contract ub-division scheme shown in Fig 9(b). In this case, the No. 981752401). The authors gratefully acknowledge CtE values for the fiber and the interphase sub-cell I2 Advanced Ceramics Research for the provision of are taken to be those measured in the through-thickness materials used in this study, Dr. S. Dorris for helpful orientation whereas the Cte of the interphase sub-cell discussion on the thermal expansion measurements, and II is taken to be that measured in-plane(assuming that Drs K. Goretta and w. Coblenz for support and helpful the c-axis of the bn crystals is aligned circumferentially comments around the Si3N4 fiber). This approach yields a pre- dicted Cte that agrees better with the experimental data than does either of the two preceding limiting References pecially in comparison with the°/90°FM. n A, Advanced Ceramics Research, private 2Kovar D, King BH, Trice RW, Halloran Jw. Fibrous monolithic Analytical models for the Cte of FMs have been olzin Bj, et al. Development of advanced fibrous monoliths developed, taking into account the fiber shape and the final report for projects of 1998-2000. ANL 01/04, Argonn fiber architecture. The models have been validated National Laboratory, June 2001 (4 McNulty JC, Begley MR, Zok Fw. In-plane fracture resistance through the use of finite element calculations. Further of a cross-ply fibrous monolith. J Am Ceram Soc 2001: 84: 367-75 validation of the models has been accomplished through [5] Evans AG,Zok he physics and mechanics of fibre-rein- comparisons between the model predictions and the forces brittle matrix composites. J Mater Sci 2001: 84: 3857-96. experimental measurements. The correlations between Ceramic source, voL 6. Westerville(OH): Am. Ceram. Soc.: 1990 theory and experiment appear to be good. Some uncer .355 [7 Lienard SY, Kovar D, Moon R, Bowman KJ. Halloran JE. tainty remains with regard to the degree of texturing Texture development in Si3 N4/BN fibrous monolithic ceramics. J (especially of the BN) and its effect on the properties of Mater Sci2000;35:3365-71 he FMs. This texturing does not appear to have a large 8 Edgar JH. Crystal structure, mechanical properties, and thermal influence on the in-plane Cte of either the 0 or 0/900 es of BN. In: Edgar JH, editor. Properties of group l FMs, provided that the in-plane CTEs of the con p 7-21. London: INSPEC: 1994 tituents are used in the models. In contrast the Halloran Jw. Investigation of mechanical properties of hot-pressed boron nitride/oxide com- through-thickness cte is sensitive to the selection of posites. J Am Ceram Soc 1999: 82: 2563-5
using the appropriate constituent properties and the results have been compared with the experimental measurements on the FMs (Section 3). The key results are plotted in Fig. 11. For the in-plane properties, the relevant constituent properties (CTE and E) were those measured on the neat Si3N4 and BN in the in-plane orientation. Very good agreement is obtained between the calculated and measured values. For the throughthickness properties, calculations are presented for two limiting cases: (i) using the in-plane properties of both constituents [lower line in Fig. 11(c)]; and (ii) using the through-thickness properties of both constituents [upper line in Fig. 11(c)]. These two sets of results appear to bound the experimental data. The results from a hybrid model that accounts for the anisotropy of the constituents in a rudimentary way is also shown in Fig. 11(c) (middle line). The latter model is based on the unit cell sub-division scheme shown in Fig. 9(b). In this case, the CTE values for the fiber and the interphase sub-cell I2 are taken to be those measured in the through-thickness orientation whereas the CTE of the interphase sub-cell I1 is taken to be that measured in-plane (assuming that the c-axis of the BN crystals is aligned circumferentially around the Si3N4 fiber). This approach yields a predicted CTE that agrees better with the experimental data than does either of the two preceding limiting cases, especially in comparison with the 0/90 FM. 6. Summary Analytical models for the CTE of FMs have been developed, taking into account the fiber shape and the fiber architecture. The models have been validated through the use of finite element calculations. Further validation of the models has been accomplished through comparisons between the model predictions and the experimental measurements. The correlations between theory and experiment appear to be good. Some uncertainty remains with regard to the degree of texturing (especially of the BN) and its effect on the properties of the FMs. This texturing does not appear to have a large influence on the in-plane CTE of either the 0 or 0/90 FMs, provided that the in-plane CTEs of the constituents are used in the models. In contrast, the through-thickness CTE is sensitive to the selection of the CTE of the BN, since the BN is the ‘‘continuous phase’’ in this orientation and thus plays a more significant role in the CTE of the FM. A rudimentary modification to the model to account for texturing and anisotropy of the BN appears to yield adequate results, although further work is needed to ascertain the extent of texturing and its effect on various thermo-mechanical properties of FMs. Acknowledgements Funding for this work was provided by the US Department of Energy (Prime Contract No. W-31-109- Eng-98) to the Argonne National Laboratory and by a subcontract from the Argonne National Laboratory to the University of California, Santa Barbara (Contract No. 981752401). The authors gratefully acknowledge Advanced Ceramics Research for the provision of the materials used in this study, Dr. S. Dorris for helpful discussion on the thermal expansion measurements, and Drs. K. Goretta and W. Coblenz for support and helpful comments. References [1] Mulligan A, Advanced Ceramics Research, private communication. [2] Kovar D, King BH, Trice RW, Halloran JW. Fibrous monolithic ceramics. J Am Ceram Soc 1997;80(10):2471–87. [3] Goretta KC, Singh D, Cruse TA, Ellingson WA, Picciolo JJ, Polzin Bj, et al. Development of advanced fibrous monoliths: final report for projects of 1998–2000. ANL 01/04, Argonne National Laboratory, June 2001. [4] McNulty JC, Begley MR, Zok FW. In-plane fracture resistance of a cross-ply fibrous monolith. J Am Ceram Soc 2001;84:367–75. [5] Evans AG, Zok FW. The physics and mechanics of fibre-reinforces brittle matrix composites. J Mater Sci 2001;84:3857–96. [6] Ceramic source, vol. 6. Westerville (OH): Am. Ceram. Soc.; 1990 p. 355. [7] Lienard SY, Kovar D, Moon RJ, Bowman KJ, Halloran JE. Texture development in Si3N4/BN fibrous monolithic ceramics. J Mater Sci 2000;35:3365–71. [8] Edgar JH. Crystal structure, mechanical properties, and thermal properties of BN. In: Edgar JH, editor. Properties of group III nitrides. p. 7–21. London: INSPEC; 1994. [9] Trice RW, Halloran JW. Investigation of the physical and mechanical properties of hot-pressed boron nitride/oxide composites. J Am Ceram Soc 1999;82:2563–5. 976 M.Y. He et al. / Composites Science and Technology 62 (2002) 967–976