Journal of the American Ceramic Society-Fair et al. Vol. 88. No. 7 size: in contrast, the strength of monolithic alumina specimen was found to decrease with increasing indentation faw size ac- Table I. Material Properties for Alumina and Mullite used for Finite element calculations ding to the griffith relationship. Similar results have recen been obtained using laminates of similar architectural dimen- x(×10-°O E(GPa) ons, in which the compressive stress within the thin layers was developed using the tetragonal-to-monoclinic phase transforma- Alumina 8.30 tion of zirconia. These results demonstrate that a threshold Mullite 5.30 157 strength is obtained for a particular flaw type, namely a surface flaw in the thick layer approximating a through-thickness slit when the laminates are loaded in a particular orientation with given by espect to the layers in the exact manner as predicted by the above fracture mechanics argument. In the current work, the mechanical properties of ceramic composites containing three-dimensional (3-D)architectures of thin compressive layers of an alumina/mullite mixture surround ing larger polyhedral regions of alumina are examined. The fab- rication of these unusual architectures was reported in the first where Vi is the volume fraction of the minor phase. The thermal expansion coefficient of the lumina mixture was calcu- paper of this series. Here, we report the preliminary observa- lated using tions concerning the mechanical properties of these materials. Results of finite element analysis are presented to illustrate the unusual stresses in the thin layers surrounding the polyhedra a,K/I+aK,v2 regions. In addition, a fracture mechanics analysis is presente KIVI+k,? that derives a stress intensity function that is analogous to eq (3), but for the extension of an assumed penny-shaped crack that where a Kp and V, are the thermal expansion coefficient, bulk ould extend within one of the polyhedra. Finally, fractograph- modulus, and volume fraction of each phase, respectively. The of specimens failed in bending reveals the in Poissons ratio of the mullite-alumina layers was calculated us action of processing defects with the residual stresses existing in ing a simple rule of mixtures. The thickness of the layers was the composite architectures. fixed to either one-tenth or one-twentieth of the core diameter as measured between parallel faces of the prisms. While the results f the 2-D finite element analysis for the hexagonal prisms do Il. Modeling of residual Stresses in 3-D Architectures not apply directly to the 3-D architectures produced by consol- dating coated, spherical agglomerates, the trends in residual Unlike the well-studied laminate system, where the differential stress with varying layer thickness and composition are expected strains perpendicular to the layers are not constrained, simple to be similar biaxial stresses do not exist within the layer of material between Figure I shows a cross section through an array of coated the polyhedra cores in the 3-D architecture. Instead, as shown hexagonal prisms as well as the average values of principal stress below, a triaxial stress state exists within the" compressive ma- within the composite at various locations(core, coating, and terial";in addition, the stresses change from position to posi- triple point). All stresses shown act within the plane in the in- tion. The nature of the triaxial stresses that exist within the dicated direction; for instance, oAx represents the stress at point composite formed with polyhedra can be visualized by consid- A acting in the x direction. As shown, only tensile stresses exist ering a sphere of one material surrounded by a shell of a second within the prismatic rods. Except for positions near the junction material that shrinks less during cooling from a processing tem- where the tensile stresses within the rods can be large, the tensile perature. It is well know that the sphere will contain hydrostatic stress within the rods is relatively uniform. Very large compress- tensile stresses, while the shell will contain compressive hoop ive stresses exist within the layer separating the prisms, and act (tangential)stresses, and radial tensile stresses. Thus, unlike the parallel to the faces of the prisms; tensile stresses of much lower laminate, the layer surrounding the sphere will contain both magnitude act normal to the faces of the prisms. Thus, for most compressive stresses and tensile stresses. The magnitude of the locations, the state of stress within the layer material is relatively compressive stresses will be much larger than that of the tensile uniform, namely, tensile stresses perpendicular to the interface. stresses when the thickness of the shell is much smaller than the and compressive stress parallel to the interface. It can also be diameter of the core seen that the tensile stress within the layers is less than the tensile To increase the understanding of the residual stresses present stress within the rods. As shown, only compressive stresses exist In the composite architecture studied here and to estimate the within the layer material near the junction of the three adjacent threshold strength, a two-dimensional (2-D)finite element anal rods, i.e., only biaxial compressive stresses exist at the locations. sis(ABaQUS) was used to study the residual stresses devel As shown in Fig. I, the tensile stress within the compressive oped in an array of hexagonal prisms, surrounded and separated layers increases with both increasing mullite content, namely from one another by a material in which compressive stresses larger differential thermal contraction during cooling, and in arise upon cooling from 1000 C. The cores of the prisms were creasing layer thickness. This result is expected to infiuence assigned the properties of alumina, whereas the compressive layer material between the hexagonal prisms were assigned the properties of either 25 vol% mullite/75 vol% alumina, or 5 vol% mullite/45 vol% mullite using the values of the properties shown in Table i for mullite and alumina. The elastic modul of the mixture was calculated using E=CEiE2+ E2)(+e-E+ EeZ (cE1+E2)(1+c) 80180 Fig 1. Average valt pal stress(in MPa) at various locations in a composite arch coated hexagonal prisms as a function of nd mullite content determined by a two- estimate given by ravichandran, in which the parameter c is dimensional finite elesize; in contrast, the strength of monolithic alumina specimens was found to decrease with increasing indentation flaw size ac￾cording to the Griffith relationship. Similar results have recently been obtained using laminates of similar architectural dimen￾sions, in which the compressive stress within the thin layers was developed using the tetragonal-to-monoclinic phase transforma￾tion of zirconia.4 These results demonstrate that a threshold strength is obtained for a particular flaw type, namely a surface flaw in the thick layer approximating a through-thickness slit, when the laminates are loaded in a particular orientation with respect to the layers in the exact manner as predicted by the above fracture mechanics argument. In the current work, the mechanical properties of ceramic composites containing three-dimensional (3-D) architectures of thin compressive layers of an alumina/mullite mixture surround￾ing larger polyhedral regions of alumina are examined. The fab￾rication of these unusual architectures was reported in the first paper of this series.5 Here, we report the preliminary observa￾tions concerning the mechanical properties of these materials. Results of finite element analysis are presented to illustrate the unusual stresses in the thin layers surrounding the polyhedra regions. In addition, a fracture mechanics analysis is presented that derives a stress intensity function that is analogous to Eq. (3), but for the extension of an assumed penny-shaped crack that would extend within one of the polyhedra. Finally, fractograph￾ic examination of specimens failed in bending reveals the inter￾action of processing defects with the residual stresses existing in the composite architectures. II. Modeling of Residual Stresses in 3-D Architectures Unlike the well-studied laminate system, where the differential strains perpendicular to the layers are not constrained, simple biaxial stresses do not exist within the layer of material between the polyhedra cores in the 3-D architecture. Instead, as shown below, a triaxial stress state exists within the ‘‘compressive ma￾terial’’; in addition, the stresses change from position to posi￾tion. The nature of the triaxial stresses that exist within the composite formed with polyhedra can be visualized by consid￾ering a sphere of one material surrounded by a shell of a second material that shrinks less during cooling from a processing tem￾perature. It is well know that the sphere will contain hydrostatic tensile stresses, while the shell will contain compressive hoop (tangential) stresses, and radial tensile stresses.6 Thus, unlike the laminate, the layer surrounding the sphere will contain both compressive stresses and tensile stresses. The magnitude of the compressive stresses will be much larger than that of the tensile stresses when the thickness of the shell is much smaller than the diameter of the core. To increase the understanding of the residual stresses present in the composite architecture studied here and to estimate the threshold strength, a two-dimensional (2-D) finite element anal￾ysis (ABAQUS) was used to study the residual stresses devel￾oped in an array of hexagonal prisms, surrounded and separated from one another by a material in which compressive stresses arise upon cooling from 10001C. The cores of the prisms were assigned the properties of alumina, whereas the compressive layer material between the hexagonal prisms were assigned the properties of either 25 vol% mullite/75 vol% alumina, or 55 vol% mullite/45 vol% mullite using the values of the properties shown in Table I for mullite and alumina.7 The elastic modulus of the mixture was calculated using E ¼ ðcE1E2 þ E2 2 Þ ð1 þ cÞ 2
E2 2 þ E1E2 ðcE1 þ E2Þ ð1 þ cÞ 2 (5) where E1 and E2 are the elastic moduli of the minor and major phases in the layer, respectively. This relation is the lower bound estimate given by Ravichandran,8 in which the parameter c is given by c ¼ 1 V1 1=3
1 (6) where V1 is the volume fraction of the minor phase. The thermal expansion coefficient of the mullite–alumina mixture was calcu￾lated using a ¼ a1K1V1 þ a2K2V2 K1V1 þ K2V2 (7) where ai, Ki, and Vi are the thermal expansion coefficient, bulk modulus, and volume fraction of each phase, respectively.6 The Poisson’s ratio of the mullite–alumina layers was calculated us￾ing a simple rule of mixtures. The thickness of the layers was fixed to either one-tenth or one-twentieth of the core diameter as measured between parallel faces of the prisms. While the results of the 2-D finite element analysis for the hexagonal prisms do not apply directly to the 3-D architectures produced by consol￾idating coated, spherical agglomerates, the trends in residual stress with varying layer thickness and composition are expected to be similar. Figure 1 shows a cross section through an array of coated hexagonal prisms as well as the average values of principal stress within the composite at various locations (core, coating, and triple point). All stresses shown act within the plane in the in￾dicated direction; for instance, sAx represents the stress at point A acting in the x direction. As shown, only tensile stresses exist within the prismatic rods. Except for positions near the junction where the tensile stresses within the rods can be large, the tensile stress within the rods is relatively uniform. Very large compress￾ive stresses exist within the layer separating the prisms, and act parallel to the faces of the prisms; tensile stresses of much lower magnitude act normal to the faces of the prisms. Thus, for most locations, the state of stress within the layer material is relatively uniform, namely, tensile stresses perpendicular to the interface, and compressive stress parallel to the interface. It can also be seen that the tensile stress within the layers is less than the tensile stress within the rods. As shown, only compressive stresses exist within the layer material near the junction of the three adjacent rods, i.e., only biaxial compressive stresses exist at the locations. As shown in Fig. 1, the tensile stress within the compressive layers increases with both increasing mullite content, namely, larger differential thermal contraction during cooling, and in￾creasing layer thickness. This result is expected to influence 180 180 50 50 50 30 30 10 10, 55 − 1800 20, 55 30 10, 25 50 50 30 − 125 − 117 − 190 − 190 − 175 − 175 − 1000 − 117 − 600 − 125 − 500 20, 25 d/t, vol%mullite/ A B C x y d t A B C x y d t Fig. 1. Average values of principal stress (in MPa) at various locations in a composite architecture of coated hexagonal prisms as a function of compressive layer thickness and mullite content determined by a two￾dimensional finite element analysis. Table I. Material Properties for Alumina and Mullite used for Finite Element Calculations7 Material a ( 10
6 /1C) E (GPa) n K (GPa) Alumina 8.30 401 0.22 166 Mullite 5.30 220 0.27 157 1880 Journal of the American Ceramic Society—Fair et al. Vol. 88, No. 7