《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_alumina-mullite-1

J.Am. Ceran.Sor,881879-1885(2005 DO:l0.lll1551-2916.2005.0037x urna Ceramic Composites with Three-Dimensional Architectures designed to Produce a Threshold Strength--Il Mechanical Observations Geoff E. Fair,*, T-f Ming Y. He, Robert M. McMeeking, and F. F. Lange* Materials Department, University of California, Santa Barbara. California 93106 Finite element modeling and linear elasti mechanics strated for laminate composites. The stress intensity function, are used to model the residual stresses and ramic composites consisting of polyhedral rounded by thin alumina/mullite layers in residual compression K=DaVia+ocia (3) his type of composite architecture is expected to exhibit iso- tropic threshold strength behavior, in which the strength of the was developed to explain the growth of a slit crack in a thick composite for a particular assumed fiaw will be constant and ndependent of the orientation of tensile loading. The results of layer through two thin, bounding compressive layers by an ap- plied tensile stress, Oa, and where 2a is the crack length, IA and he modeling indicate that the strengths of such architectures will be higher than those of laminates of similar architectural layers, respectively, and o is the biaxial cor in the compressive layers. Since the second term in Eq. ( 3) trength behavior for a particular fiaw type Flexural testing duces the stress intensity at the crack tip, the applied stress must of the polyhedral architectures reveals that failure is dominated by processing defects found at junctions between the polyhedra constantly increased in order for the crack to grow through he compressive layer in a stable manner. Namely, Eq (3)pre- Fractography revealed the interaction of these defects with the dicts that crack growth through the bounding compressive residual stresses in the compressive layers that separate the layers is stable, i.e., the crack exhibits an R-curve behavior; ex- perimental results have confirmed this prediction.2 Although the R-curve behavior predicted by Eq (3)and ob- L. Introducti rved for a number of laminar composites is interesting by it- self, of greater interest is the fact that the limits of Eq (3)can be WHsi laye st res ad enteri i thie the de together uses ive haoer s hat lashihai thresphosi s rent hing pei inc lumber of phenomena including differential thermal contrac- lure due to a particular type of defect(assumed or real) does tion during cooling from the fabrication temperature, a phase ot occur until a well-defined stress, o,hr, hereafter referred to as change during cooling, or a molar volume change due to a re- the threshold strength, is reached. Thus, as shown by Eq(3), action that forms one of the materials within the laminate. Using with increasing applied stress, the slit crack that spans the thick the example where the stresses arise due to differential therm layer will extend across the much thinner compressive layers ontraction, a biaxial compressive strain e develops in one layer til it reaches the next thick layer. At this applied stress, the because of its smaller thermal expansion coefficient. As reviewed crack tip is no longer shielded and the crack propagates across by Ho et al, the biaxial compressive stress within the thin layers the remaining layers to produce catastrophic failure. The ap- (material A)is given by plied stress needed to extend the slit crack across the compress- ive stresses can be determined by substituting 2a=/B+2IA and K= Ke into Eq. (3)and rearranging. The result is as follows: I+(IAEA/IBEB here E= El(1-v), E is the Youngs modulus, v is the Pois- thr sons ratio of the material, and ia and ib are the thicknesses of (1+) the thin and thick layers, respectively. The stress within the thick layers(material B)is given by IA\2 1-(1+ (4) IA Equation(4)shows that the threshold strength of the laminate is d on the ude of the residual compress It is clear that as (A/tg approaches zero, i.e., for the case of very ive stress in the thin layers, the fracture toughness of the thin compressive layers, the stress in the thick layers disappears. layers, and the thickness of both the thick and thin layer thin he concept of using compressive regions within brittle ma- Threshold strength behavior has been experimentally ob- terials to stop cracks and to provide an increasing resistance to rved for symmetric, periodic laminates consisting of nearly crack extension with increasing applied stress has been demon- stress-free thick layers(200-650 um)of alumina and thin layers (20-75 um)of mixtures of alumina and mullite(0.1-0.85 volume R. K. Bordiacontributing editor relative to the thick alumina layers. To demonstrate the Manuscript No. 10791. Received January 9, 200-4 approved December 31, 2004. threshold strength behavior of the laminates, indentation pre- I Research. under cracks of varying size (10-225 um) were placed in the center thick layer of the specimens. When the laminates were tested in 'Current address: Materials and Manufacturing Directorate, AFRL/MLLN, Air Force four-point bending with traverse loading with respect to the arch Laboratory, Wright-Paterson Air Force Base, OH 45433 layers, the strengths were observed to be independent of fil 1879

Ceramic Composites with Three-Dimensional Architectures Designed to Produce a Threshold Strength—II. Mechanical Observations Geoff E. Fair,* ,w,z Ming Y. He, Robert M. McMeeking, and F. F. Lange* Materials Department, University of California, Santa Barbara, California 93106 Finite element modeling and linear elastic fracture mechanics are used to model the residual stresses and failure stress of ce￾ramic composites consisting of polyhedral alumina cores sur￾rounded by thin alumina/mullite layers in residual compression. This type of composite architecture is expected to exhibit iso￾tropic threshold strength behavior, in which the strength of the composite for a particular assumed flaw will be constant and independent of the orientation of tensile loading. The results of the modeling indicate that the strengths of such architectures will be higher than those of laminates of similar architectural dimensions that were previously found to exhibit threshold strength behavior for a particular flaw type. Flexural testing of the polyhedral architectures reveals that failure is dominated by processing defects found at junctions between the polyhedra. Fractography revealed the interaction of these defects with the residual stresses in the compressive layers that separate the polyhedra. I. Introduction WHEN layers of dissimilar materials are bonded together, residual stresses may develop within the layers by a number of phenomena including differential thermal contrac￾tion during cooling from the fabrication temperature, a phase change during cooling, or a molar volume change due to a re￾action that forms one of the materials within the laminate. Using the example where the stresses arise due to differential thermal contraction, a biaxial compressive strain e develops in one layer because of its smaller thermal expansion coefficient. As reviewed by Ho et al.,1 the biaxial compressive stress within the thin layers (material A) is given by sA ¼ eE0 A 1 þ ðtAE0 A=tBE0 BÞ (1) where E0 i 5 Ei/(1
ni), E is the Young’s modulus, n is the Pois￾son’s ratio of the material, and tA and tB are the thicknesses of the thin and thick layers, respectively. The stress within the thick layers (material B) is given by sB ¼
sA tA tB (2) It is clear that as tA/tB approaches zero, i.e., for the case of very thin compressive layers, the stress in the thick layers disappears. The concept of using compressive regions within brittle ma￾terials to stop cracks and to provide an increasing resistance to crack extension with increasing applied stress has been demon￾strated for laminate composites.2,3 The stress intensity function, K ¼ sa ffiffiffiffiffi pa p þ sc ffiffiffiffiffi pa p 1 þ tA tB 2 p sin
1 tB 2a  
1   (3) was developed to explain the growth of a slit crack in a thick layer through two thin, bounding compressive layers by an ap￾plied tensile stress, sa, and where 2a is the crack length, tA and tB are the thickness of the thin compressive and thicker tensile layers, respectively, and sc is the biaxial compressive stress with￾in the compressive layers.3 Since the second term in Eq. (3) re￾duces the stress intensity at the crack tip, the applied stress must be constantly increased in order for the crack to grow through the compressive layer in a stable manner. Namely, Eq. (3) pre￾dicts that crack growth through the bounding compressive layers is stable, i.e., the crack exhibits an R-curve behavior; ex￾perimental results have confirmed this prediction.2 Although the R-curve behavior predicted by Eq. (3) and ob￾served for a number of laminar composites is interesting by it￾self, of greater interest is the fact that the limits of Eq. (3) can be used to show that laminar composites containing periodic com￾pressive layers can exhibit threshold strength behavior in which failure due to a particular type of defect (assumed or real) does not occur until a well-defined stress, sthr, hereafter referred to as the threshold strength, is reached. Thus, as shown by Eq. (3), with increasing applied stress, the slit crack that spans the thick layer will extend across the much thinner compressive layers until it reaches the next thick layer. At this applied stress, the crack tip is no longer shielded and the crack propagates across the remaining layers to produce catastrophic failure. The ap￾plied stress needed to extend the slit crack across the compress￾ive stresses can be determined by substituting 2a 5 tB12tA and K 5 Kc into Eq. (3) and rearranging. The result is as follows: sthr ¼ Kc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p tB 2 1 þ 2tA tB r   þ sc 1
1 þ tA tB 2 p sin
1 1 1 þ 2tA tB " # ! (4) Equation (4) shows that the threshold strength of the laminate is expected to depend on the magnitude of the residual compress￾ive stress in the thin layers, the fracture toughness of the thin layers, and the thickness of both the thick and thin layers. Threshold strength behavior has been experimentally ob￾served for symmetric, periodic laminates consisting of nearly stress-free thick layers (200–650 mm) of alumina and thin layers (20–75 mm) of mixtures of alumina and mullite (0.1–0.85 volume fraction), which is placed in residual compression during cooling from the densification temperature due to a lower average CTE relative to the thick alumina layers.2,3 To demonstrate the threshold strength behavior of the laminates, indentation pre￾cracks of varying size (10–225 mm) were placed in the center thick layer of the specimens. When the laminates were tested in four-point bending with traverse loading with respect to the layers, the strengths were observed to be independent of flaw 1879 Journal J. Am. Ceram. Soc., 88 [7] 1879–1885 (2005) DOI: 10.1111/j.1551-2916.2005.00377.x R. K. Bordia—contributing editor Supported by the Office of Naval Research, under contract N00014-03-1-0305. *Member, American Ceramic Society. w Author to whom correspondence should be addressed. e-mail: geoff.fair@wpafb.af.mil z Current address: Materials and Manufacturing Directorate, AFRL/MLLN, Air Force Research Laboratory, Wright-Paterson Air Force Base, OH 45433. Manuscript No. 10791. Received January 9, 2004; approved December 31, 2004.

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 ele

size; 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

July 2005 Ceramic Composites with Three-Dimensional Architectures crack propagation through the composite architecture as dis- term is negative, i.e., it decreases the stress intensity factor pro- cussed below duced by the applied stress. Equation(8) lat the stress tensity decreases as the crack extends shell, that is, a greater applied stress must to maintain Ill. Fracture Mechanics Modeling of 3-D Architectures a constant value of Ke as the crack ex her into the ensity factor In the event that all proc defects are confined to the po compressive shell, where Ke is the critical yhedral cores, the failure stress of the composite will correspon of the compressive shell material. to the stress needed to propagate a crack from within the pol- Catastrophic crack extension occurs when the crack has Mhedral core outwards through the compressive layer formed grown through the compressive shell, i.e., when 2a=d+2r. Sub- by the second material separating the cores. Consequently, a stituting 2a= d+2t and K= Ke into Eq( 8)yields the maximum fracture mechanics model to predict the threshold strength of pplied stress that the crack can sustain before onset of cata- the 3-D composite architecture was developed using the super- strophic failure: this is the threshold stress. Oa=Othr, where the position of stress intensity factors in a similar manner as that stress intensity factor is given by used to derive Eq. 3) for the laminar composite In the laminar aterials, a slit crack, which was assumed to extend through the oppressive layers, was used to develop Eq. (3), and thus, z42 Eq (4). The crack within the polyhedral units that form the 3- D composite is assumed not to be larger than the size of the polyhedron. To estimate the stress intensity function, the poly |(om+o)(d+2)-(o+a)Vd+2n)2-2 hedron is assumed to be a sphere of diameter"d", containing a ydrostatic, residual tensile stress, ot, embedded within a spher (9) ical shell of diameter"d+2r", subjected to a residual hoop stress The re hip between the residual tensile stress within the of oc; these stresses are assumed to develop as a consequence of sphere and compressive hoop stress within the spherical shell sumed to contain a penny-shaped crack of diameter 2a. The result /erived using the thin-walled pressure vessel theory; the thermal mismatch between the two materials. The sphere is as- phere and the surrounding spherical shell are embedded in a continuous matrix of the same material that forms the embed- 41g ded sphere. It is assumed that the elastic properties of the sphere (10) spherical shell, and continuous matrix are identical to one an Substituting this result into Eq(9) and rearranging yields the and the continuous matrix are identical. but a different material function for the threshold stres forms the spherical shell. This system is shown on the left-hand side of Fig. 2 which illustrates the sphere con- taining a concentric, penny-shaped crack of diameter"2a"that Othr Kc V2(d+2) is acted upon by a stress, Oa, applied perpendicular to the plane of the crack he right-hand side of Fig. 2 shows that two states of stress (11) acting on the crack can be superimposed to produce the state of stress shown on the left-hand side. In the first, the crack only Figure 3 compares the expression for the threshold strength of exists in the matrix material and is subjected to the stress Ca-Oe. In the second state of stress shown on the far right, for the laminate architecture(Eq.(4). Figure 3 illustrates the nly acts over the central portion(diameter, d)of the crack. The rchitecture(given by Eq (4)and the 3-D architecture(given by stress intensity factor function for each of these two states of stress can be added together to yield% Eq (ID)as a function of residual compressive stress for the case where Ke=2 MPa. m, thick layers or cores are 600 um, and thin layers are one-tenth that dimension(or 60 um). For these conditions, in both Eqs. (4)and(D), the first term on the right-hand side of the equations becomes a constant; the second term becomes a constant multiplied by the residual stress in the It should be noted that for the case of zero thermal m Eq.(8)reduces to the Griffith equation for a penny crack in an isotropic material. The second term in Eq(8 exists when 2a d. Because the compressive stress within the 3D Architecture(Eq. 11) spherical shell"clamps "shuts the extending crack, the second Laminate(Eq 4) ↑↑↑↑↑↑↑个个个↑ 55 vol% mulli 5 vol% mullite Stresses ↓↓↓↓↓4↓ Residual Compressive Stress(MPa)

crack propagation through the composite architecture as dis￾cussed below. III. Fracture Mechanics Modeling of 3-D Architectures In the event that all processing defects are confined to the pol￾yhedral cores, the failure stress of the composite will correspond to the stress needed to propagate a crack from within the pol￾yhedral core outwards through the compressive layer formed by the second material separating the cores. Consequently, a fracture mechanics model to predict the threshold strength of the 3-D composite architecture was developed using the super￾position of stress intensity factors in a similar manner as that used to derive Eq. (3) for the laminar composite. In the laminar materials, a slit crack, which was assumed to extend through the compressive layers, was used to develop Eq. (3), and thus, Eq. (4). The crack within the polyhedral units that form the 3- D composite is assumed not to be larger than the size of the polyhedron. To estimate the stress intensity function, the poly￾hedron is assumed to be a sphere of diameter ‘‘d ’’, containing a hydrostatic, residual tensile stress, st, embedded within a spher￾ical shell of diameter ‘‘d12t’’, subjected to a residual hoop stress of sc; these stresses are assumed to develop as a consequence of thermal mismatch between the two materials. The sphere is as￾sumed to contain a penny-shaped crack of diameter 2a. The sphere and the surrounding spherical shell are embedded in a continuous matrix of the same material that forms the embed￾ded sphere. It is assumed that the elastic properties of the sphere, spherical shell, and continuous matrix are identical to one an￾other; it is also assumed that the material that forms the sphere and the continuous matrix are identical, but a different material forms the spherical shell. This system is shown in cross section on the left-hand side of Fig. 2 which illustrates the sphere con￾taining a concentric, penny-shaped crack of diameter ‘‘2a’’ that is acted upon by a stress, sa, applied perpendicular to the plane of the crack. The right-hand side of Fig. 2 shows that two states of stress acting on the crack can be superimposed to produce the state of stress shown on the left-hand side. In the first, the crack only exists in the matrix material and is subjected to the stress, sa
sc. In the second state of stress shown on the far right, the same crack is subjected to a stress of magnitude sc1st that only acts over the central portion (diameter, d ) of the crack. The stress intensity factor function for each of these two states of stress can be added together to yield9 K ¼ 2 ffiffiffiffiffi pa p ðsa þ stÞa
ðsc þ stÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2
d2 4 " # r (8) It should be noted that for the case of zero thermal mismatch, Eq. (8) reduces to the Griffith equation for a penny-shaped crack in an isotropic material. The second term in Eq. (8) only exists when 2a d. Because the compressive stress within the spherical shell ‘‘clamps’’ shuts the extending crack, the second term is negative, i.e., it decreases the stress intensity factor pro￾duced by the applied stress. Equation (8) shows that the stress intensity decreases as the crack extends into the compressive shell, that is, a greater applied stress must be applied to maintain a constant value of Kc as the crack extends further into the compressive shell, where Kc is the critical stress intensity factor of the compressive shell material. Catastrophic crack extension occurs when the crack has grown through the compressive shell, i.e., when 2a 5 d12t. Sub￾stituting 2a 5 d12t and K 5 Kc into Eq. (8) yields the maximum applied stress that the crack can sustain before onset of cata￾strophic failure; this is the threshold stress, sa 5 sthr, where the stress intensity factor is given by Kc ¼ 1 ffiffiffiffiffiffiffiffiffiffiffi p dþ2t 2 q ð Þ sthr þ st ð Þ
d þ 2t ð Þ sc þ st ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ d þ 2t 2
d2   q (9) The relationship between the residual tensile stress within the sphere and compressive hoop stress within the spherical shell can be derived using the thin-walled pressure vessel theory; the result is st ¼ 4tsc d (10) Substituting this result into Eq. (9) and rearranging yields the function for the threshold stress sthr ¼ Kc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2ðd þ 2tÞ r þ sc 1 þ 4t d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
d2 ð Þ d þ 2t 2 s
4t d " # (11) Figure 3 compares the expression for the threshold strength of the 3-D composite given by Eq. (11) with that previously derived for the laminate architecture (Eq. (4)). Figure 3 illustrates the variation in predicted threshold strength for both the laminate architecture (given by Eq. (4)) and the 3-D architecture (given by Eq. (11)) as a function of residual compressive stress for the case where Kc 5 2 MPa  m1/2, thick layers or cores are 600 mm, and thin layers are one-tenth that dimension (or 60 mm). For these conditions, in both Eqs. (4) and (11), the first term on the right-hand side of the equations becomes a constant; the second term becomes a constant multiplied by the residual stress in the = a c − a a t c 2a t c+ t c+ t d d+2t c+ t + σ σ σ σ c+ t σ σ σ σ σ a c σ σ − σ Fig. 2. Schematic of superposition model used to derive the expression for threshold strength of three-dimensional architecture. 0 500 1000 1500 0 200 400 600 25 vol% mullite Predicted Threshold Strength (MPa) Residual Compressive Stress (MPa) 3D Architecture (Eq. 11) Laminate (Eq. 4) 55 vol% mullite Predicted Compressive Stresses Fig. 3. Predicted threshold strength as a function of residual compress￾ive stress for laminate and three-dimensional composite architectures. July 2005 Ceramic Composites with Three-Dimensional Architectures 1881

Journal of the American Ceramic Society-Fair et al. Vol. 88. No. 7 thin layers. Hence, for the case examined in Fig 3, one line ap- holed to room temperature at 5.C/min. The dimensions of the pears for the 3-D architecture and one line al densified bars were approximately 3 mm x3 mm x 35 mm. inate architecture. Next the values of residual stress achieved in In addition to the composite bars bars were also machined ach architecture are highlighted on their corresponding lines in from monoliths produced by slip-casting a 95 vol% alumina the figure. For the laminate architecture, the values of residual 5 vol% zirconia slurry and consolidating uncoated agglom stress for both 25 and 55 vol% mullite layers are calculated us- ates. For the slip-cast monolith, a 30 vol% solids slurry was ing Eq. (1)in the text and the values of modulus expected for the repared by adding alumina(AKP-15, Sumitomo, Tokyo, Ja- layers given by Eq(5). For the 3-D architecture, the values of pan)and yttria-stabilized zirconia(TZ-3YS, Tosoh, Tokyo, Ja esidual stress highlighted are the range of stresses calculated by pan) to deionized water at pH 3. The powders were stirred into the finite element routine for compressive layers comprised of the acidified water while sonicating with an ultrasonic horn both 25 and 55 vol% mullite, the average value of which appears(Fisher Scientific Sonic Dismembrator Model 550, Fairlawn, the table of NJ). Following addition of all solids, the slurry was attrition Figure 3 shows that for the same critical stress intensity fact milled(Union Process Szegvari Attritor)at 375 rpm for 1.5 h and similar architecture dimensions, the residual compressive The slurry was then slip cast on a plaster block. After drying in tresses are significantly larger in the 3-D architecture for the air for 3 days, the slip-cast monolith was presintered to impart two different layer compositions. Thus, as shown in Fig 3 for trength for subsequent machining by heating to 800C at 5C/ the two different materials(see arrows), one would expect much min with a 2-h hold at temperature followed by cooling to room greater compressive stress and therefore higher threshold temperature at 5.C/min. Bars were then machined from the trengths for the 3-D composites with similar architectural di- monolith using the method described above. The bars were then the compressive stress is the same for the laminate and the 3-d by cooling to room temperature at 5C/min. Specimens were composite, the reason why the threshold stress is larger for the 3- also produced using uncoated, spherical agglomerates produced D laminate is because the k function for the slit crack(used to sing the same consolidation, machining, and firing procedures determine the function for the laminate)is larger than the k described above for the composite bars function for the penny-shaped crack defined by the diameter of the polyhedral units in the 3-D composite (2) Mechanical Testing and fractography Mechanical testing of the composite architectures was per I. Experimental Procedure ormed only to assess the effectiveness of the processing pro dures in producing defect-free architectures suitable for studies (1) Mechanical Test Specimens of the threshold strength behavior of the architecture. i.e. free of Spherical powder agglomerates of composition 95 vol% alumi- ross ng-related defects: a second oal was to eluci- na/5 vol% zirconia coated with mullite-alumina were produced ate the interaction of cracks with the residual stresses present in as detailed in prior work, and in Part I of this series.,10The the composites. All bars were tested in four-point flexure (inner mullite-alumina coatings were either thin (15 um)or thick(30 span 13.05 mm, outer span 30.2 mm)using a screw-driven test um) and were either 25 or 55 vol% mullite. To produce com ng machine (Instron 8562, Norwood, MA). The machine was operated in displacement-control mode with a crosshead speed agglomerates were loaded into a bar-shaped die and submerged of 0. 0l mm/ min. Load-displacement data were collected using a in 0. 25 M NH4 CI at pH 3.5 for I h. The salt solution was for mulated to have the same ph and salt concentration as the slur ts was recorded directly from the user interface of the Instron ry used to produce the agglomerates in order to re-establish the and was subsequently used to calculate the failure strength of interparticle potentials within the spheres as those present in the the specimens. The failure strengths of the specimens, or,were nitial slurry. After I h, the excess solution was drained and determined with the agglomerates were uniaxially consolidated and filter pressed at 150 MPa for 10 min Following pressing, the compacts were removed from the die and stored in a plastic bag containing a crs 3P(so-s) damp paper towel overnight. The compacts were then wrapped in filter paper, packed within 40 g dry alumina powder in a where P is the failure load si and so are the inner and outer plastic bag, and isostatically pressed at 350 MPa for 10 min. The spans, respectively, w is the specimen width, and h is the spec compacts were then allowed to dry for one day in air prior to men height. The fracture surfaces of the failed composites were examined using an SEM ( JEOL 6300, Peabody, MA)with EDX Bars were machined from dried, unfired comp apability. Fracture surfaces were closely examined for possible abrasive wheel (Dreml tool). Two bars were machined from Failure origins edX was used to determine the location within each compact. Prior to beginning the machining process, the the composite architecture of the failure origins and fracture bottom and side of each unfired specimen were flattened with 400 grit SiC sandpaper. Following machining, each face of the bar was lightly sanded with 400 and 800 grit SiC paper. Ma chined bars were fired by heating to 600C at 2.C/min with a 2-h V. Results hold at temperature to allow for binder burn-out, followed by All composite architectures tested in this work had the same heating to 1550C at 5.C/min with a 2-h hold; the bars were ther microstructural and architectural features as those produced Table Il. Average Failure Stresses and Range of Strengths Obtained for Specimens Tested in This Study Specimen type (number of specimens Range of stresses(MPa) Slipcast monolith(95% Al2O3/5% ZrO2) 417.7(4) 084526.7 Monolith from uncoated spheres(95% Al2O3/5%ZrO2) 340.8(4) 286.0-407 Composite: thick 25 vol% mullite layers 1430(4) 127.3-151.7 Composite: thin 25 vol% mullite layers 159.7(3 1582-1624 Composite: thick 55 vol% mullite layers l10.7(4) 1055-116.1 Composite: thin 55 vol% mullite layers 139.7(4) 136.2-147.2

thin layers. Hence, for the case examined in Fig. 3, one line ap￾pears for the 3-D architecture and one line appears for the lam￾inate architecture. Next, the values of residual stress achieved in each architecture are highlighted on their corresponding lines in the figure. For the laminate architecture, the values of residual stress for both 25 and 55 vol% mullite layers are calculated us￾ing Eq. (1) in the text and the values of modulus expected for the layers given by Eq. (5). For the 3-D architecture, the values of residual stress highlighted are the range of stresses calculated by the finite element routine for compressive layers comprised of both 25 and 55 vol% mullite, the average value of which appears in the table of Fig. 1. Figure 3 shows that for the same critical stress intensity factor and similar architecture dimensions, the residual compressive stresses are significantly larger in the 3-D architecture for the two different layer compositions. Thus, as shown in Fig. 3 for the two different materials (see arrows), one would expect much greater compressive stress and therefore higher threshold strengths for the 3-D composites with similar architectural di￾mensions relative to the laminate composites. For the case where the compressive stress is the same for the laminate and the 3-D composite, the reason why the threshold stress is larger for the 3- D laminate is because the K function for the slit crack (used to determine the function for the laminate) is larger than the K function for the penny-shaped crack defined by the diameter of the polyhedral units in the 3-D composite. IV. Experimental Procedure (1) Mechanical Test Specimens Spherical powder agglomerates of composition 95 vol% alumi￾na/5 vol% zirconia coated with mullite–alumina were produced as detailed in prior work, and in Part I of this series.5,10 The mullite–alumina coatings were either thin (B15 mm) or thick (B30 mm) and were either 25 or 55 vol% mullite. To produce com￾posite bars used for flexural strength measurements, the coated agglomerates were loaded into a bar-shaped die and submerged in 0.25 M NH4Cl at pH 3.5 for 1 h. The salt solution was for￾mulated to have the same pH and salt concentration as the slur￾ry used to produce the agglomerates in order to re-establish the interparticle potentials within the spheres as those present in the initial slurry. After 1 h, the excess solution was drained and the agglomerates were uniaxially consolidated and filter pressed at 150 MPa for 10 min. Following pressing, the compacts were removed from the die and stored in a plastic bag containing a damp paper towel overnight. The compacts were then wrapped in filter paper, packed within 40 g dry alumina powder in a plastic bag, and isostatically pressed at 350 MPa for 10 min. The compacts were then allowed to dry for one day in air prior to green machining. Bars were machined from dried, unfired compacts using an abrasive wheel (Dreml tool). Two bars were machined from each compact. Prior to beginning the machining process, the bottom and side of each unfired specimen were flattened with 400 grit SiC sandpaper. Following machining, each face of the bar was lightly sanded with 400 and 800 grit SiC paper. Ma￾chined bars were fired by heating to 6001C at 21C/min with a 2-h hold at temperature to allow for binder burn-out, followed by heating to 15501C at 51C/min with a 2-h hold; the bars were then cooled to room temperature at 51C/min. The dimensions of the densified bars were approximately 3 mm 3 mm 35 mm. In addition to the composite bars, bars were also machined from monoliths produced by slip-casting a 95 vol% alumina/ 5 vol% zirconia slurry and consolidating uncoated agglomer￾ates. For the slip-cast monolith, a 30 vol% solids slurry was prepared by adding alumina (AKP-15, Sumitomo, Tokyo, Ja￾pan) and yttria-stabilized zirconia (TZ-3YS, Tosoh, Tokyo, Ja￾pan) to deionized water at pH 3. The powders were stirred into the acidified water while sonicating with an ultrasonic horn (Fisher Scientific Sonic Dismembrator Model 550, Fairlawn, NJ). Following addition of all solids, the slurry was attrition milled (Union Process Szegvari Attritor) at 375 rpm for 1.5 h. The slurry was then slip cast on a plaster block. After drying in air for 3 days, the slip-cast monolith was presintered to impart strength for subsequent machining by heating to 8001C at 51C/ min with a 2-h hold at temperature followed by cooling to room temperature at 51C/min. Bars were then machined from the monolith using the method described above. The bars were then fired by heating to 15501C at 51C/min with a 2-h hold, followed by cooling to room temperature at 51C/min. Specimens were also produced using uncoated, spherical agglomerates produced using the same consolidation, machining, and firing procedures described above for the composite bars. (2) Mechanical Testing and Fractography Mechanical testing of the composite architectures was per￾formed only to assess the effectiveness of the processing proce￾dures in producing defect-free architectures suitable for studies of the threshold strength behavior of the architecture, i.e., free of gross processing-related defects; a secondary goal was to eluci￾date the interaction of cracks with the residual stresses present in the composites. All bars were tested in four-point flexure (inner span 13.05 mm, outer span 30.2 mm) using a screw-driven test￾ing machine (Instron 8562, Norwood, MA). The machine was operated in displacement-control mode with a crosshead speed of 0.01 mm/min. Load–displacement data were collected using a personal computer. The maximum load achieved during the tests was recorded directly from the user interface of the Instron and was subsequently used to calculate the failure strength of the specimens. The failure strengths of the specimens,sf, were determined with sf ¼ 3P sð Þ o
si 2wh2 (12) where P is the failure load, si and so are the inner and outer spans, respectively, w is the specimen width, and h is the spec￾imen height. The fracture surfaces of the failed composites were examined using an SEM (JEOL 6300, Peabody, MA) with EDX capability. Fracture surfaces were closely examined for possible failure origins. EDX was used to determine the location within the composite architecture of the failure origins and fracture path. V. Results All composite architectures tested in this work had the same microstructural and architectural features as those produced in Table II. Average Failure Stresses and Range of Strengths Obtained for Specimens Tested in This Study Specimen type Average failure stress (MPa) (number of specimens) Range of stresses (MPa) Slipcast monolith (95% Al2O3/5% ZrO2) 417.7 (4) 308.4–526.7 Monolith from uncoated spheres (95% Al2O3/5% ZrO2) 340.8 (4) 286.0–407.0 Composite: thick 25 vol% mullite layers 143.0 (4) 127.3–151.7 Composite: thin 25 vol% mullite layers 159.7 (3) 158.2–162.4 Composite: thick 55 vol% mullite layers 110.7 (4) 105.5–116.1 Composite: thin 55 vol% mullite layers 139.7 (4) 136.2–147.2 1882 Journal of the American Ceramic Society—Fair et al. Vol. 88, No. 7

July 2005 Ceramic Composites with Three-Dimensional Architectures 1883 1 aKU (c) Fig 4. SEM micrographs of representative fracture surfaces for composite architectures considered in this study.(a)thin( 30 um)25 vol% mullite /75 vol% alumina layers, (b)thick(60 um)25 vol% mullite/75 vol% alumina layers, (c)thin( 30 um)55 vol% mullite/45 vol% alumina layers, and (d)thick(-60 um)55 vol% mullite/45 vol% alumina layers the first paper of this series; consequently and for reasons de- plete consolidation of the agglomerates described in Part I of cribed below, cracks running down the center of the compress ive layers intersecting the surface of the specimens(edge cracks) Figure 7 summarizes the results of an extensive EDX explo- were observed for the composites containing both thick and thin ation of mating fracture surfaces of the thick 55 vol% mullite compressive layer composite just beneath the tensile surfac Table II lists the average strength for the monoliths and com- posite specimens as well as the range of strengths obtained for each architecture. As shown the strengths of the monoliths were larger than those of the composites, but their values exhibited larger scatter in values. The strengths of the composites decrease ith increasing layer thickness and larger apparent compressive composite architectures considered. The fracture surfaces in- crease in roughness with both increasing layer t hickness and in- m asing mullite content Figures 5 and 6 show surfaces within the composites that encompass a crack-like void for thick compressive laye taining 25 and 55 vol% mullite, respectively. These surfaces could be identified as voids because the surfaces are dentical to those of an external surface of a dense poly ne body, where grooves are present wherever a grain boundary ntercepts an external surface. They are easily d from a fracture surface due to the rounded ap 2 m EDX analysis of the non-bonded regions on matin surfaces confirmed the presence of Si, namely, the Inous of mullite. on both surfaces indicating that the void bonded lie entirely within the compressive layer and result from incom- 60 uregion on the fracture surface of a composite with thick 25 vol% mullite/75 vol% alumina lay

the first paper of this series;5 consequently and for reasons de￾scribed below, cracks running down the center of the compress￾ive layers intersecting the surface of the specimens (edge cracks) were observed for the composites containing both thick and thin layers formulated with 55 vol% mullite. Table II lists the average strength for the monoliths and com￾posite specimens as well as the range of strengths obtained for each architecture. As shown, the strengths of the monoliths were larger than those of the composites, but their values exhibited a larger scatter in values. The strengths of the composites decrease with increasing layer thickness and larger apparent compressive stress (increasing mullite content). The range of strengths ob￾served for the composites was relatively small (o710 MPa). Figure 4 shows representative fracture surfaces of the four composite architectures considered. The fracture surfaces in￾crease in roughness with both increasing layer thickness and in￾creasing mullite content. Figures 5 and 6 show surfaces within the composites that encompass a crack-like void for thick compressive layers containing 25 and 55 vol% mullite, respectively. These surfaces could be identified as voids because the surfaces are identical to those of an external surface of a dense polycrystal￾line body, where grooves are present wherever a grain boundary intercepts an external surface. They are easily distinguished from a fracture surface due to the rounded appearance of the grains. EDX analysis of the non-bonded regions on mating fracture surfaces confirmed the presence of Si, namely, the presence of mullite, on both surfaces, indicating that the voids lie entirely within the compressive layer and result from incom￾plete consolidation of the agglomerates described in Part I of this series. Figure 7 summarizes the results of an extensive EDX explo￾ration of mating fracture surfaces of the thick 55 vol% mullite compressive layer composite just beneath the tensile surface. Fig. 4. SEM micrographs of representative fracture surfaces for composite architectures considered in this study. (a) thin (B30 mm) 25 vol% mullite /75 vol% alumina layers, (b) thick (B60 mm) 25 vol% mullite/75 vol% alumina layers, (c) thin (B30 mm) 55 vol% mullite/45 vol% alumina layers, and (d) thick (B60 mm) 55 vol% mullite/45 vol% alumina layers. Fig. 5. SEM micrographs at various magnifications showing a non￾bonded region on the fracture surface of a composite with thick (B60 mm) 25 vol% mullite/75 vol% alumina layers. July 2005 Ceramic Composites with Three-Dimensional Architectures 1883