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