174 Journal of the American Ceramic Society--Bai et Vol. 90. No. I 0.32 05 P=025 Po P1 P2 020 afe Substrate 0.16 Ar Fig. 7. The tractions exerted by the environ barrier coatings on a Bg 6. The design-map for choosing the aspect ratio of the columns the thickness of the preclude delamination of where the columns(which typically be less than about 1/3). In summary, the des suggests that a thinner topcoat, a 97的|3-0支) larger aspect ratio of the columns and a higher packing density in the columnar interlayer, favor the probability of avoiding 5/4 delamination of the topcoat due to thermal strains. +Y2 IV. Influence of the ebc on the fracture strength of a brittle substrate ere a thermally strained coating exerts surface tractions on the sub- trate. These tractions can increase the loading on the flaws near the surface of the substrate material thereby having a negat Y (19) EBCs made from BAS for silicon carbide ceramics. The ques The total increase in the loading of the crack due to shear tion arises to what extent the compliant interlayer architecture tractions induced by the coating is given by summing over of the present EBC can influence the fracture behavior of the all AKp The problem is approached analytically by ass worst-case scenario where the location of the faw in the sub- △K=∑√∑9 strate coincides with the position where the shear trac. tion is exerted on the substrate by the bending of the beams These tractions are highest near the free edge of a crack in the The fracture strength decreases because the critical stress in- topcoat, that is next to j=0, 1, 2, 3... and so on. The analytical tensity factor given in Eq (17) is effectively reduced by Ak given procedure then, as illustrated in Fig. 7, is to calculate the addi- above in Eq.(20). The ratio of the fracture strength with the coating,or, with respect to the intrinsic fracture strength of the tional loading on a flaw of size a placed near j=0, by adding up substrate, or is then given by the incremental stress-intensity produced at the crack tip by the periodically occurring shear tractions exerted by the bending of the columns at j=0, 1, 2... on one side and j=n,(n-1) n-2).... on the other side. Being antisymmetric both sets of KI tractions increase the loading on the faw The intrinsic flaw size in the substrate is estimated from the We normalize the intrinsic fracture toughness, Kic, in the fracture toughness, Kic and the fracture strength of the mate- following way rial, using the equation (17) △a△T where o is the fracture strength and a is the flaw size To get an order of magnitude for Kic following values for the numbers for the fracture toughness and the fracture strength of terms in the denominator: ETc= 300 GPa, Am=7x10, and Si3N are 5 MPa- m/and espectively, which yields a these values the denominator becomes at si ie of appraximate te stl ak exerted by the coating s MPa m/2 we note Assuming Kic for silicon-nitride of 5.1 ve note that Kic: 2 calculated by considering a point force acting from each of the The expression for P, in Eq(18)is rel lated to the displac columns on the crack and summing up the effect from all col- ments in the columns parallel to the interface, u, by the well umns. Forces from the(n+1) columns, from each side of the known force-displacement equation for the constrained bending crack, will exert a pull force on the crack of size a. The stress of a beam with rectangular cross-section, which translates into intensity exerted by one of these symmetrically placed pair of forces, each of strength P, at a distance y, from the crack, as ically in Fig. 7, is give P1= △K 92 (18) above equation according to Eq. 9), and substituting this equa-the columns (which should typically be less than about 1/3). In summary, the design map suggests that a thinner topcoat, a larger aspect ratio of the columns and a higher packing density in the columnar interlayer, favor the probability of avoiding delamination of the topcoat due to thermal strains. IV. Influence of the EBC on the Fracture Strength of a Brittle Substrate A thermally strained coating exerts surface tractions on the sub￾strate. These tractions can increase the loading on the flaws near the surface of the substrate material thereby having a negative impact on its fracture strength. Such effect has been reported for EBCs made from BAS for silicon carbide ceramics.12 The ques￾tion arises to what extent the compliant interlayer architecture of the present EBC can influence the fracture behavior of the substrate. The problem is approached analytically by assuming the worst-case scenario where the location of the flaw in the sub￾strate coincides with the position where the highest shear trac￾tion is exerted on the substrate by the bending of the beams. These tractions are highest near the free edge of a crack in the topcoat, that is next to j 5 0, 1, 2, 3 y and so on. The analytical procedure then, as illustrated in Fig. 7, is to calculate the addi￾tional loading on a flaw of size a placed near j 5 0, by adding up the incremental stress-intensity produced at the crack tip by the periodically occurring shear tractions exerted by the bending of the columns at j 5 0, 1, 2y on one side and j 5 n, (n
1), (n
2),y on the other side. Being antisymmetric both sets of tractions increase the loading on the flaw. The intrinsic flaw size in the substrate is estimated from the fracture toughness, KIC and the fracture strength of the mate￾rial, using the equation: KIC ¼ sf ffiffiffiffiffi pa p (17) where sf is the fracture strength and a is the flaw size. Typical numbers for the fracture toughness and the fracture strength of Si3N4 are 5 MPa  m1/2 and 1 GPa, respectively, which yields a flaw size of approximately 10 mm. The incremental stress intensity, DK, exerted by the coating is calculated by considering a point force acting from each of the columns on the crack and summing up the effect from all col￾umns. Forces from the (n11) columns, from each side of the crack, will exert a pull force on the crack of size a. The stress intensity exerted by one of these symmetrically placed pair of forces, each of strength Pj at a distance yj from the crack, as shown schematically in Fig. 7, is given by13 DKj ¼ 2Pj ffiffiffiffiffi pa p Oj (18) where Oj ¼ 1 þ 2Y2 j ð1 þ Y2 j Þ 1:5 1:3
0:3 Yj ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Y2 j q 0 B@ 1 CA 2 6 4 5=4 0:665
0:267 Yj ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Y2 j q 0 B@ 1 CA 5=4 Yj ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Y2 j q
0:73 0 B@ 1 CA 0 B@ 1 CA 3 7 5 Here Yj ¼ yj a (19) The total increase in the loading of the crack due to shear tractions induced by the coating is given by summing over all DKj : DK ¼ Xn j¼0 DKj ¼ 2 ffiffiffiffiffi pa p Xn j¼0 PjOj (20) The fracture strength decreases because the critical stress in￾tensity factor given in Eq. (17) is effectively reduced by DK given above in Eq. (20). The ratio of the fracture strength with the coating, s0 f , with respect to the intrinsic fracture strength of the substrate, sf is then given by s0 f sf ¼ 1
DK KIC (21) We normalize the intrinsic fracture toughness, KIC, in the following way: K
IC ¼ KIC ETCDaDT ffiffiffiffiffi pa p (22) To get an order of magnitude for K
IC following values for the terms in the denominator: ETC 5 300 GPa, Da 5 7 10
6 , and a 5 10 mm. Substituting these values the denominator becomes equal to 3.7 MPa  m1/2. Assuming KIC for silicon-nitride of 5.1 MPa  m1/2 we note that K
IC : 2. The expression for Pj in Eq. (18) is related to the displace￾ments in the columns parallel to the interface, uj, by the well￾known force–displacement equation for the constrained bending of a beam with rectangular cross-section, which translates into the following relationship with the present nomenclature11: Pj ¼ uj EBC 2A3 r (23) Introducing the normalization for the displacements uj in the above equation according to Eq. (9), and substituting this equa￾Fig. 7. The tractions exerted by the environmental barrier coatings on a Fig. 6. The design-map for choosing the aspect ratio of the columns flaw in the substrate. and the thickness of the topcoat in order to preclude delamination of the topcoat. 174 Journal of the American Ceramic Society—Bai et al. Vol. 90, No. 1