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172 of the American Ceramic Society-Bai et al Vol. 90. No. I ergy in the substrate can be ignored. The chemical bond coat parameter c, on the other hand, is extremely thin, relative to the topcoat and the columnar layer; therefore its volume is negligible in com- parison and the strain energy in it as well can be ignored )First, 12+c-1 we consider the latter. The strain energy in the jth element of the topcoat is given by I L=Erce △AT)hrcL (1) -12+c-1 net strain in the jth element of the topcoat. The elastic strain in this element is equal to the difference between displacement in where the element, divided by the length of the element L, and the 4a△A7 =0.n (9) thermal expansion strain. The strain energy being equal to the quare of the net displacement is insensitive to the sign of this Here u is the normalized value of the displa placement. as a difference. The quadratic term in Eq(1)when expanded gives fraction of the displacement to be expected from"free the following equation for the strain energy in the jth element of expansion. The results of the analysis can now be do terms of the non-dimensional parameter, c, which within it the elastic properties of the topcoat and the II=TETC( -u-1)2-ErchrcAuAT(u; -u-1) interlayer, and is given by +(△△T)2 ErchTcL L =1 (10) Equations in the matrix in(8)now are solved numerically for The strain e ins is simply related to the ii in terms of c and n. Note that n is a measure of the spacin bending displacement, u, to hBc and to the moment of inertia of between the cracks in the topcoat as i=nL. ory which gives It remains to write down the expressions for the stresses in the in the columns, which are explicitly related to uy The st the jth element of the topcoat, aTc, is normalized I to the thermal expansion stress, tha elaxed thermal stress in the topcoat, so that here=0, n. The total potential energy is equal to the sum of Eqs. (2)and (3) which leads to the following result after proper TO=Erc△△T Combining Eq.(II) with Eq(9)and recognizing that Ⅱ-∑“+∑气=9 ETe ∑ erche△n△r(4--1) (4) we obtain the following relationship between the stress in th topcoat elements and the displaceme (△△7)2EchL The stresses in the columns of the interlayer arise from bend- Using principle of minimum potential energy, we obtain the ing. We are interested in the maximum value of the stress arising governing set of equations for up from the bending; we call this stress o Bc and normalize it in the same way as the stress in the topcoat, as given by eq (11), and an 12EBcIuo, ErchTc (-41)+ erchE△△T denote it with a bar. Using the equation for maximum stress from beam theory, the following result is obtained 0 B=E7C△ AT hBc△ ATEC where= 0, n(13) on 12EBcluk ErchE Note that Eqs.(I1H13)describe the in plane stress in the BC ErchE topcoat, parallel to the interface, and the maximum stress in the 1×(k--)=0,k≠0,n beam produced by flexure, which is normal to the interface. This description deviates from the description of stresses in continu- ous films on substrates that are dealt by shear lag models; in an 12EBclum ErchE these models the shear stresses in the interface are equilibrated dr h3 L against the in-plane stress in the film echte△a△T=0 The calculation of the displacements from Eq(8)and sub- tituting them into eqs. (12)and (13)for obtaining the stress in the topcoat and in the columns completes the solution to the problem. However, Eq(8)must be solved numerically. The na- After some arrangement, the abo ture of the results depends on the non-dimensional parameter, c, be written in matrix form and in terms of one normalized given by Eq(10). Substituting for I in Eq(10), and recognizingenergy in the substrate can be ignored. The chemical bond coat, on the other hand, is extremely thin, relative to the topcoat and the columnar layer; therefore its volume is negligible in comparison and the strain energy in it as well can be ignored.) First, we consider the latter. The strain energy in the jth element of the topcoat is given by: YTC j ¼ 1 2 ETC uj uj1 L DaDT 2 hTCL (1) where j 5 1, n. The terms within the brackets represent the net strain in the jth element of the topcoat. The elastic strain in this element is equal to the difference between displacement in the element, divided by the length of the element L, and the thermal expansion strain. The strain energy being equal to the square of the net displacement is insensitive to the sign of this difference. The quadratic term in Eq. (1) when expanded gives the following equation for the strain energy in the jth element of the topcoat: YTC j ¼ hTCETC 2L ðuj uj1Þ 2 ETChTCDaDTðuj uj1Þ þ 1 2ðDaDTÞ 2 ETChTCL (2) The strain energy in the columns is simply related to the bending displacement, uj, to hBC and to the moment of inertia of the columns, I, by beam theory which gives11: Ybeam j ¼ 6EBCIu2 j h3 BC (3) where j 5 0, n. The total potential energy is equal to the sum of Eqs. (2) and (3) which leads to the following result after proper summation: Y ¼ Xn j¼0 6EBCIu2 j h3 BC þXn j¼1 hTCETC 2L ðuj uj1Þ 2 Xn j¼1 ETChTCDaDTðuj uj1Þ þ n 2 ðDaDTÞ 2 ETChTCL (4) Using principle of minimum potential energy, we obtain the governing set of equations for uj: qP qu0 ¼ 12EBCIu0 h3 BC þ ETChTC L ðu0 u1Þ þ ETChTCDaDT ¼ 0 (5) qP quk ¼ 12EBCIuk h3 BC þ ETChTC L ðuk ukþ1Þ þ ETChTC L ðuk uk1Þ ¼ 0; k ¼6 0; n (6) qP qun ¼ 12EBCIun h3 BC þ ETChTC L ðun un1Þ ETChTCDaDT ¼ 0 (7) After some arrangement, the above equations can be written in matrix form and in terms of one normalized parameter c, 1þ c 1 1 2þ c 1 : : : : 1 2þ c 1 1 1þ c 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 u0 : : : : : : un 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ¼ 1 0 0 : : : 0 1 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 (8) where uj ¼ uj LDaDT j ¼ 0; n (9) Here uj is the normalized value of the displacement, as a fraction of the displacement to be expected from ‘‘free’’ thermal expansion. The results of the analysis can now be described in terms of the non-dimensional parameter, c, which embodies within it the elastic properties of the topcoat and the compliant interlayer, and is given by c ¼ 12EBC ETC IL hTCh3 BC (10) Equations in the matrix in (8) now are solved numerically for uj in terms of c and n. Note that n is a measure of the spacing between the cracks in the topcoat as l 5 nL. It remains to write down the expressions for the stresses in the topcoat and in the columns, which are explicitly related to uj. The stress in the jth element of the topcoat, sTCj, is normalized with respect to the thermal expansion stress, that is the unrelaxed thermal stress in the topcoat, so that sTCj ¼ sTCj ETCDaDT (11) Combining Eq. (11) with Eq. (9) and recognizing that: sTCj ¼ ETC uj uj1 L DaDT we obtain the following relationship between the stress in the topcoat elements and the displacements: sTCj ¼ ujþ1 uj 1 for j ¼ 1; n (12) The stresses in the columns of the interlayer arise from bending. We are interested in the maximum value of the stress arising from the bending; we call this stress sBCj and normalize it in the same way as the stress in the topcoat, as given by Eq. (11), and denote it with a bar. Using the equation for maximum stress from beam theory, the following result is obtained11: sBCj ¼ sBCj ETCDaDT ¼ 3EBCWuj h2 BCDaDTETC where j ¼ 0; n (13) Note that Eqs. (11)–(13) describe the in plane stress in the topcoat, parallel to the interface, and the maximum stress in the beam produced by flexure, which is normal to the interface. This description deviates from the description of stresses in continuous films on substrates that are dealt by shear lag models; in these models the shear stresses in the interface are equilibrated against the in-plane stress in the film.8 The calculation of the displacements from Eq. (8) and substituting them into Eqs. (12) and (13) for obtaining the stress in the topcoat and in the columns completes the solution to the problem. However, Eq. (8) must be solved numerically. The nature of the results depends on the non-dimensional parameter, c, given by Eq. (10). Substituting for I in Eq. (10), and recognizing 172 Journal of the American Ceramic Society—Bai et al. Vol. 90, No. 1