July 2005 Ceramic Composites with Three-Dimensional Architectures within the layers separating the polyhedra, which will increase VIL. Conclusions he tensile stresses at the surface, which will promote edge crack ng. Likewise, at a given mullite content, increasing the lay Ceramic ickness will increase the tensile stresses within both the poly thin compressive layers were fabricated by consolidating sphe hedra themselves, and within the compressive layer. Thus, for ical alumina agglomerates coated with thin layers of different large mullite contents, and/or for thicker layers, edge cracking mixtures of mullite and alumina. a fracture mechanics model was developed to predict the strengths, assuming failure due to a will be the flaw that initiates failure, and the crack will try to particular flaw type, as a function of the different variables extend within the compressive layer on a path that separates the stress or the laver thickness increases. more of the crack will ness of the compressive layer material, and the architectural di- extend within the compressive layers to reveal the underlying mensions of the composite. This model suggested that if fracture Studies to determine how to avoid edge cracks by covering rger than that exhibited by a laminate composite of the same erials with similar architectural dimensions. Finite element the surface with a compressive layer, e. g, with the same material modeling of the stresses within the composites showed that ten- used to form the compressive regions between the polyhedra, are currently in progress; these studies already show that the tensile sile stresses exist within the compressive layers in a direction stresses at the surface can be reduced to zero if the thickness of perpendicular to the polyhedra interface; the magnitude of these he surface coating is equal to one-half the thickness of the tensile stresses is less than the tensile stress within the polyhedra compressive layer separating the polyhedra. Other studies are The strengths of the materials, machined into bar specimens eeded to learn how to prevent the voids formed due to incom- were dominated by two different flaws that existed within the plete deformation of the polyhedra during processing. These oppressive layers. One flaw population comprised large voids studies would also aid in improving ceramics formed with located within the compressive layer at the junction of three or agglomerated, spray-dried powders four polyhedra that did not fully deform during consolidation The second flaw population comprised edge cracks that exist on the surface of the compressive layers. The edge cracks were clearly visible when either the ive stress or the thickness (3) Prospects for Realizing Threshold Strength Behavior of the compressive layer was large. Until these flaw populations Although the intent of the current study was to achieve a thresh can be controlled, the possible ability of this architecture to ex- old strength in a 3-D col e using compress hibit threshold strength behavior analogous to that observed for uld stop and hinder cracks that extend from within the larger laminate systems" will remain unknown. In addition, a single lyhedral regions, it was discovered that flaws that exist with valued threshold strength would require a periodic array of pe the composite layers dominate the fracture behavior. These yhedra; a non-periodic array would only exhibit a distribution flaws also highlight a limitation of the proposed strengthening of"threshold strengths concept descibed in the early sections of the paper: the failure causing faws are not those assumed in the derivation of the Acknowledgments threshold strength. In both the fracture mechanics model for the hreshold strength of the 3-D architecture derived above and The authors would like to thank Randall Hay, ronal hat derived for the laminate architecture, a particular type of Scientific for their insightful discussions regarding the ment of residual failure-causing flaw must be assumed. The assumed faw is fur her assumed to be located in a location. such that it interacts ith the compressive layers of the composite architectures upe propagation due to an applied stress. If such flaws are not present and/or not located in the appropriate location,no threshold strength behavior is observed. Nevertheless. the cur- der Biaxial, Residual Compressive Stress. "J.A. Ceram. Soc., 78, 2353-9(1995). rent derivation illustrates another way in which threshold ninar Ceramics That Exhibit a Threshold Strength, "J. Am. Ceram Soc., 84. strength behavior may be realized for a particular flaw type in a manner analogous to that previously demonstrated for the SM. P Rao. A.J. Sanchez-Herencia. G.E. Beltz. R.M. McMeeking, and F.F laminate architecture 2-4 Lange, "Laminar Ceramics That Exhibit a Threshold Strength. Science. 286. In order to convincingly demonstrate the concept for the 3-D G. Pontin, M. P. Rao. A.J. Sanchez-Herencia and F. F. Lange. Laminar architecture considered here, the processing defects discovered in the flexural testing of the composites, namely voids due to ncomplete consolidation and edge cracks, need to be eliminat- .J. Am. Ceram. Soc-85, 3041-8(2002 ed. Controlled inter-polyhedra voids of varying size could then Ceram. Soc. in pre duce a Threshold Strength-. Processing J. An. K.K. Chawla, Ceramic Matrix Composites. Chapman& Hall, London, 1993. these controlled flaws would elucidate t composites contra of used to produce the agglomerates that become the the composite. Mechanical testing of the fectiveness of the Ravichandran,""Elastic Properties of Two-Phase Composites, J.Am. proposed strengthening mechanism. Furthermore, it should be noted that a single-valued threshold strength would require a ada, P C. Paris, and G. R. Irwin, The Stress Analysis of Cracks Handbook. riodic array of polyhedra with uniform compressive layer Del Research Corp, St. Louis, 1978. thickness; a non-periodic array with varying compressive layer thickness, similar to the one produced here, would only exhibit a id, Spherical Agglomerates During Drying. J. Am. Ceram. Soc. distribution of * threshold strengths This is another limitation M. R. Snyder and F. F. Lange, ""Prismatic Composites with a Threshold of the current approach; periodic architectures with uniform compressive layer thicknesses are most easily produced in a ered Ceramic Composites by Edge Coating ' Int. J. Solids Struct, 42, 581-90 laminate architecturewithin the layers separating the polyhedra, which will increase the tensile stresses at the surface, which will promote edge crack￾ing. Likewise, at a given mullite content, increasing the layer thickness will increase the tensile stresses within both the poly￾hedra themselves, and within the compressive layer. Thus, for large mullite contents, and/or for thicker layers, edge cracking will be the flaw that initiates failure, and the crack will try to extend within the compressive layer on a path that separates the polyhedra. Thus, as shown in Fig. 4, as either the compressive stress or the layer thickness increases, more of the crack will extend within the compressive layers to reveal the underlying polyhedra. Studies to determine how to avoid edge cracks by covering the surface with a compressive layer, e.g., with the same material used to form the compressive regions between the polyhedra, are currently in progress; these studies already show that the tensile stresses at the surface can be reduced to zero if the thickness of the surface coating is equal to one-half the thickness of the compressive layer separating the polyhedra.12 Other studies are needed to learn how to prevent the voids formed due to incom￾plete deformation of the polyhedra during processing. These studies would also aid in improving ceramics formed with agglomerated, spray-dried powders. (3) Prospects for Realizing Threshold Strength Behavior Although the intent of the current study was to achieve a thresh￾old strength in a 3-D composite using compressive layers that could stop and hinder cracks that extend from within the larger polyhedral regions, it was discovered that flaws that exist with the composite layers dominate the fracture behavior. These flaws also highlight a limitation of the proposed strengthening concept descibed in the early sections of the paper: the failure causing flaws are not those assumed in the derivation of the threshold strength. In both the fracture mechanics model for the threshold strength of the 3-D architecture derived above and that derived for the laminate architecture, a particular type of failure-causing flaw must be assumed. The assumed flaw is fur￾ther assumed to be located in a location, such that it interacts with the compressive layers of the composite architectures upon propagation due to an applied stress. If such flaws are not present and/or not located in the appropriate location, no threshold strength behavior is observed. Nevertheless, the cur￾rent derivation illustrates another way in which threshold strength behavior may be realized for a particular flaw type in a manner analogous to that previously demonstrated for the laminate architecture.2–4 In order to convincingly demonstrate the concept for the 3-D architecture considered here, the processing defects discovered in the flexural testing of the composites, namely voids due to incomplete consolidation and edge cracks, need to be eliminat￾ed. Controlled inter-polyhedra voids of varying size could then be introduced by incorporating graphite flakes into the slurries used to produce the agglomerates that become the polyhedra of the composite. Mechanical testing of the composites containing these controlled flaws would elucidate the effectiveness of the proposed strengthening mechanism. Furthermore, it should be noted that a single-valued threshold strength would require a periodic array of polyhedra with uniform compressive layer thickness; a non-periodic array with varying compressive layer thickness, similar to the one produced here, would only exhibit a distribution of ‘‘threshold strengths’’. This is another limitation of the current approach; periodic architectures with uniform compressive layer thicknesses are most easily produced in a laminate architecture.2–4 VII. Conclusions Ceramic composites consisting of large polyhedra separated by thin compressive layers were fabricated by consolidating spher￾ical alumina agglomerates coated with thin layers of different mixtures of mullite and alumina. A fracture mechanics model was developed to predict the strengths, assuming failure due to a particular flaw type, as a function of the different variables, namely, the compressive stress in the layers, the fracture tough￾ness of the compressive layer material, and the architectural di￾mensions of the composite. This model suggested that if fracture initated within the polyhedra, the threshold strength would be larger than that exhibited by a laminate composite of the same materials with similar architectural dimensions. Finite element modeling of the stresses within the composites showed that ten￾sile stresses exist within the compressive layers in a direction perpendicular to the polyhedra interface; the magnitude of these tensile stresses is less than the tensile stress within the polyhedra. The strengths of the materials, machined into bar specimens, were dominated by two different flaws that existed within the compressive layers. One flaw population comprised large voids located within the compressive layer at the junction of three or four polyhedra that did not fully deform during consolidation. The second flaw population comprised edge cracks that exist on the surface of the compressive layers. The edge cracks were clearly visible when either the compressive stress or the thickness of the compressive layer was large. Until these flaw populations can be controlled, the possible ability of this architecture to ex￾hibit threshold strength behavior analogous to that observed for laminate systems2–4 will remain unknown. In addition, a single￾valued threshold strength would require a periodic array of pol￾yhedra; a non-periodic array would only exhibit a distribution of ‘‘threshold strengths’’. Acknowledgments The authors would like to thank Randall Hay, Ronald Kerans, and Triplicane Parthasarathy of Air Force Research Laboratory and David Marshall of Rockwell Scientific for their insightful discussions regarding the development of residual stresses in the composites. References 1 S. Ho, C. Hillman, F. F. Lange, and Z. Suo, ‘‘Surface Cracking in Layers Un￾der Biaxial, Residual Compressive Stress,’’ J. Am. Ceram. Soc., 78, 2353–9 (1995). 2 M. P. Rao, J. Rodel, and F. F. Lange, ‘‘Residual Stress Induced R-Curves in Laminar Ceramics That Exhibit a Threshold Strength,’’ J. Am. Ceram. Soc., 84, 2722–4 (2001). 3 M. P. Rao, A. J. Sanchez-Herencia, G. E. Beltz, R. M. McMeeking, and F. F. Lange, ‘‘Laminar Ceramics That Exhibit a Threshold Strength,’’ Science, 286, 102–5 (1999). 4 M. G. Pontin, M. P. Rao, A. J. Sanchez-Herencia, and F. F. Lange, ‘‘Laminar Ceramics Utilizing the Zirconia Tetragonal-to-Monoclinic Phase Transformation to Obtain a Threshold Strength,’’ J. Am. Ceram. Soc., 85, 3041–8 (2002). 5 G. E. Fair and F. F. Lange, ‘‘Ceramic Composites with Three-Dimensional Architectures Designed to Produce a Threshold Strength—I. Processing J. Am. Ceram. Soc., in press. 6 K. K. Chawla, Ceramic Matrix Composites. Chapman & Hall, London, 1993. 7 B. R. Marple and D. J. Green, ‘‘Mullite/Alumina Particulate Composites by Infiltration Processing: IV, Residual Stress Profiles,’’ J. Am. Ceram. Soc., 75, 44–51 (1992). 8 K. Ravichandran, ‘‘Elastic Properties of Two-Phase Composites,’’ J. Am. Ceram. Soc., 77, 1178–84 (1994). 9 H. Tada, P. C. Paris, and G. R. Irwin, The Stress Analysis of Cracks Handbook. Del Research Corp., St. Louis, 1978. 10G. E. Fair and F. F. Lange, ‘‘Effect of Interparticle Potential on Forming Solid, Spherical Agglomerates During Drying,’’ J. Am. Ceram. Soc., 87, 4–9 (2004). 11M. R. Snyder and F. F. Lange, ‘‘Prismatic Composites with a Threshold Strength; I: Processing, Microstructure and Residual Stresses;’’ to be published. 12A. J. Monkowski and G. E. Beltz, ‘‘Suppression of Edge Cracking in Lay￾ered Ceramic Composites by Edge Coating,’’ Int. J. Solids Struct., 42, 581–90 (2005). & July 2005 Ceramic Composites with Three-Dimensional Architectures 1885