《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_Si3N4-application

C 2006 The American Ceramic Society urna Mechanical Design for Accommodating thermal Expansion Mismatch in Multilayer Coatings for Environmental Protection at Ultrahigh Temperatures Jie Bai, Kurt Maute, Sandeep r Shah, and Rishi raj Ultrahigh Temperature Materials Laboratory, Department of Mechanical Engineering, University of Colorado, Boulder. colorado 8030 The design of coatings is like designing a system. Every coating The next generation gas turbines are slated to contain ceramic or more specific functions that determine the choice of components made from silicon nitrite(Si3 N4). However, silicon Is, and its architecture. In the case of environmental nitride suffers from erosion in the streaming humid environment atmosphere In high-temperature applications the stresses aris- the volatilization of the passivating silica scale. 3- Example of ing from thermal expansion mismatch between the topcoat and such a result from our laboratory given in Fig. l, which show 如出 strate must be ameliorated. In this article we consider the increasing weight loss with higher streaming velocity. These data of an intermediate layer of a multilayer coating system reach up to a relative velocity of 35 cm/s; the velocities en explicit objective of managing thermal expansion dif- ountered in the gas turbine are an order of magnitude higher ween the topcoat and the substrate. The design is which would be clearly intolerable for Si3 N4. The objective ed upon a columnar architecture where the columns serve as environmental barrier coatings(EBCs) is to protect the load cible beams to accommodate relative displacement withou bearing silicon-nitride structure from corrosive weight loss in the racture, The value of the maximum stresses in the beam and high velocity, high temperature, and humid environment of the the topcoat are calculated and used to develop a map with fail gas turbine and safe regimes. The safe region is defined by the prevention of The design of EBCs is constrained by at least three criteria: tracture in the beams, since their fracture would precipitate de- (a) the topcoat of the ebc must be able to survive in the gas- lamination of the topcoat. As a rule of thumb the topcoat thick- turbine environment, (b) the topcoat must be securely bonded to ness should be less than the width of the columns for safe the Si3 N4 substrate, and(c) the coating architecture must have operation(this condition changes somewhat with the aspect good thermal shock resistance atio of the columns). A larger aspect ratio of the columns The choice of the optimum material for the topcoat is often also promotes safe design. We further consider how the tractions juxtaposed against the issue of thermal shock, as matching ther- induced by the thermal stresses on the surface of the substrate mal expansion and, at the same time providing chemical dur- may influence the intrinsic fracture strength of the substrate ability, can pose a challenge to the materials engineer. Howeve The stresses in the coating are predicted to have an insignificant his approach has been successfully used to develop coating effect on the intrinsic fracture strength of the substrate. siliconcarbide-based ceramic structures. In the present work we consider another approach, one where the topcoat is chosen en L. Introduction tirely for its corrosion resistance, and then the thermal shock is managed by adding a compliant intermediate layer which ac- This h ase turbine ep itomines the significance and the need hr commodates the difference in the coefficients of thermal expa highest temperatures and the most severe corrosive environ- ments in the gas turbine are experienced by nozzles, linings, and ing it a good choice as the material for the topcoat. However,its most of all, by the rotating turbine blade. The push for higher combustion temperatures is creating a need for multifunctional thermal expansion is much larger than that of Si3 N4(10 ppm vs coatings that can withstand thermal shock, adhere well to the about 3.5 ppm/k) which would cause it to spall. A coating de- substrate, provide thermal insulation, and protect from envi- ign which can ameliorate the thermal stresses is illustrated in ronmental corrosion. The state-of-the-art blade materials are ig. 2. It consists of a topcoat, a compliant intermediate coat that accommodates thermal strains and a bond coat that se- metallic superalloys. Zirconia-based thermal barrier coatings cures the upper lavers to the substrate. The mechanical design of (TBCs) for superalloys have been in use for over a decade. the compliant intermediate coat is the main subject of this paper The TBCs were developed by intuition and experience, yet they The principal purpose of the above coating architecture is have laid the foundation for the conceptual design of high-tem- to prevent the high velocity humid environment from imping- oatings. As described in Strangman and Schiele- they g directly on to the surface of Si3 N4. The thermal expansio ased upon a reactive metallic bond coat for adherence of the topcoat can be expected to produce"periodically zirconia and the superalloy, a columnar, strain toleran spaced"cracks, which will allow the humid environment to zirconia layer, and a dense zirconia topcoat. seep into the coating. Therefore the function of this EBC is merely to subdue the velocity of the environment. The questio C.H. Hsuch--contnibuting editor hen arises whether oxidation of Si3 N4 under static humid environments can be acceptable. Work to be published in a companion paper by Shah and Raj shows that a special bond Manuscript No. 21601. Received March 17, 2006: approved August 24. 2006. coat that is made from polymer derived siliconcarbonitride is as supported by the MEANS program at the ce Office of Sci- effective against oxidation at high temperatures in static humid ty(but not under high-velocity conditions). Thus a combination CA P experimgtanl part if th ip reseaich wa s supported inner the Power and bnergy of the polymer-derived coating o and the thermal stress Author to whom correspondence should be addressed. e-mail: rishi. raja colorado.edu management approach described in this paper can be used

Mechanical Design for Accommodating Thermal Expansion Mismatch in Multilayer Coatings for Environmental Protection at Ultrahigh Temperatures Jie Bai, Kurt Maute, Sandeep R. Shah, and Rishi Rajw Ultrahigh Temperature Materials Laboratory, Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80302 The design of coatings is like designing a system. Every coating has one or more specific functions that determine the choice of materials, and its architecture. In the case of environmental barrier coatings the topcoat must be chemically inert to the atmosphere. In high-temperature applications the stresses aris￾ing from thermal expansion mismatch between the topcoat and the substrate must be ameliorated. In this article we consider the design of an intermediate layer of a multilayer coating system with the explicit objective of managing thermal expansion dif￾ference between the topcoat and the substrate. The design is based upon a columnar architecture where the columns serve as flexible beams to accommodate relative displacement without fracture. The value of the maximum stresses in the beam and in the topcoat are calculated and used to develop a map with fail and safe regimes. The safe region is defined by the prevention of fracture in the beams, since their fracture would precipitate de￾lamination of the topcoat. As a rule of thumb the topcoat thick￾ness should be less than the width of the columns for safe operation (this condition changes somewhat with the aspect ratio of the columns). A larger aspect ratio of the columns also promotes safe design. We further consider how the tractions induced by the thermal stresses on the surface of the substrate may influence the intrinsic fracture strength of the substrate. The stresses in the coating are predicted to have an insignificant effect on the intrinsic fracture strength of the substrate. I. Introduction THE gas turbine epitomizes the significance and the need for high temperature coatings for structural applications. The highest temperatures and the most severe corrosive environ￾ments in the gas turbine are experienced by nozzles, linings, and most of all, by the rotating turbine blade. The push for higher combustion temperatures is creating a need for multifunctional coatings that can withstand thermal shock, adhere well to the substrate, provide thermal insulation, and protect from envi￾ronmental corrosion. The state-of-the-art blade materials are metallic superalloys. Zirconia-based thermal barrier coatings (TBCs) for superalloys have been in use for over a decade. The TBCs were developed by intuition and experience,1 yet they have laid the foundation for the conceptual design of high-tem￾perature coatings. As described in Strangman and Schienle2 they were based upon a reactive metallic bond coat for adherence between zirconia and the superalloy, a columnar, strain tolerant zirconia layer, and a dense zirconia topcoat. The next generation gas turbines are slated to contain ceramic components made from silicon nitrite (Si3N4). However, silicon nitride suffers from erosion in the streaming humid environment of the gas turbine. The weight loss can be severe and is caused by the volatilization of the passivating silica scale.3–5 Example of such a result from our laboratory given in Fig. 1, which shows increasing weight loss with higher streaming velocity. These data reach up to a relative velocity of 35 cm/s6 ; the velocities en￾countered in the gas turbine are an order of magnitude higher which would be clearly intolerable for Si3N4. The objective of environmental barrier coatings (EBCs) is to protect the load bearing silicon-nitride structure from corrosive weight loss in the high velocity, high temperature, and humid environment of the gas turbine. The design of EBCs is constrained by at least three criteria: (a) the topcoat of the EBC must be able to survive in the gas￾turbine environment, (b) the topcoat must be securely bonded to the Si3N4 substrate, and (c) the coating architecture must have good thermal shock resistance. The choice of the optimum material for the topcoat is often juxtaposed against the issue of thermal shock, as matching ther￾mal expansion and, at the same time providing chemical dur￾ability, can pose a challenge to the materials engineer. However, this approach has been successfully used to develop coatings for siliconcarbide-based ceramic structures.7 In the present work we consider another approach, one where the topcoat is chosen en￾tirely for its corrosion resistance, and then the thermal shock is managed by adding a compliant intermediate layer which ac￾commodates the difference in the coefficients of thermal expan￾sion of the topcoat and the substrate. For example, zirconia has a proven record of durability as thermal barrier coatings, mak￾ing it a good choice as the material for the topcoat. However, its thermal expansion is much larger than that of Si3N4 (10 ppm vs about 3.5 ppm/K) which would cause it to spall. A coating de￾sign which can ameliorate the thermal stresses is illustrated in Fig. 2. It consists of a topcoat, a compliant intermediate coat that accommodates thermal strains, and a bond coat that se￾cures the upper layers to the substrate. The mechanical design of the compliant intermediate coat is the main subject of this paper. The principal purpose of the above coating architecture is to prevent the high velocity humid environment from imping￾ing directly on to the surface of Si3N4. The thermal expansion of the topcoat can be expected to produce ‘‘periodically spaced’’ cracks,8 which will allow the humid environment to seep into the coating. Therefore the function of this EBC is merely to subdue the velocity of the environment. The question then arises whether oxidation of Si3N4 under static humid environments can be acceptable.9 Work to be published in a companion paper by Shah and Raj10 shows that a special bond coat that is made from polymer derived siliconcarbonitride is effective against oxidation at high temperatures in static humid￾ity (but not under high-velocity conditions). Thus a combination of the polymer-derived coating10 and the thermal stress management approach described in this paper can be used to C.-H. Hsueh—contributing editor This research was supported by the MEANS program at the Air Force Office of Sci￾entific Research under the direction of Dr. Joan Fuller. The Grant number is F49620-01-1- 052. The experimental part of this research was supported under the Power and Energy CTA Program at Honeywell Inc., Phoenix, AZ, under the direction of Laura Lindberg. w Author to whom correspondence should be addressed. e-mail: rishi.raj@colorado.edu Manuscript No. 21601. Received March 17, 2006; approved August 24, 2006. Journal J. Am. Ceram. Soc., 90 [1] 170–176 (2007) DOI: 10.1111/j.1551-2916.2006.01354.x r 2006 The American Ceramic Society 170

January 2007 Mechanical Design for Accommodating Thermal Expa Si N iv) the relative density of the columnar interlayer, p, which Recession The elastic moduli of the two and the 13oo°C,p.ootm material used to construct the columns are written as Erc and is likely to be a good practice to use the same material for both in which case the ratio of the two -0.5 v:l I cm/s moduli becomes equal to unity. The difference between the coefficient of thermal expansion between the topcoat and the bstrate is written as Ao The difference in the thermal expansion between the topcoat d the substrate will lead to periodic cracks in the topcoat 35.4 v= velocity (assuming that the in-plane stress given by AoATErc is -1.5 greater than the fracture stress of the topcoat). The spacing 40 of these periodic cracks is likely to be similar to the description of the interfacial strength of thin films under uniform loading by shear-lag models. In the present instance the interfacial Fig 1. Influence of streaming water-vapor velocity on weight loss in stresses are accommodated by the flexure of the columns The mechanics of deformation may therefore be illustrated as in Fig. 3. The periodic crack spacing in the topcoat, A, is to a length produce effective EBCs for high-temperature, silicon-based column spacing. If the topcoat is sufficiently thin then the in-plane stress in the film may be assumed to be uniform in e z, or the out of plane direction. The stress in the topcoat for the geometrical structure of the columnar intermediate layer. then be symmetrically distributed about the center-line of the The performance goal is to prevent the delamination of the top- spacing between adjacent cracks in the topcoat, while the bend- coat--this condition is achieved if the maximum stress in the ing displacements of the columns, parallel to the interface are topcoat is greater than in the columnar structure, that is, if the antisymmetric. These bending displacements in the columns columns resist fracture better than the topcoat. The second are called u, where=0, n. The topcoat is also described by problem analyzed in this article pertains to the degradation in discrete elements, each of length L, such that they are synchron the fracture strength of the substrate as a result of the coating. ous with the columns. The stress in these elements are written a We consider increased loading on a surface flaw in the substrate OTC, where=l, n. The strains in these elements are described exerted by the forces of the thermal strains; a formal derivation by the difference between the displacements of the two columns shows this effect to be relatively inconsequential on either edge of the element (after compensating for the ther mal expansion strain). Thus the first element is stretched by (u1-o), and the jth element by(u-p-1)and so on. In this way Il. Mechanical Design of the Columnar Interlayer we have n elements in the topcoat, and (n+1)columns which flex nd the Topcoat to accommodate the strain in the topcoat. The objective of the (1) Analysis analysis is to solve for the shear displacements in the columns The problem is analyzed in two dimensions, that is, the sche- matic in Fig. 2 represents the cross-section of the eBC, which The analysis is based on the principle of minimum potential does not change in the normal direction. For volumetric quan- energy. The potential energy is the sum of the elastic strains in tities, e.g. the strain energy per column, we assume a unit depth the columns and in the elements of the topcoat. ( The substrate normal to the paper. The geometrical parameters of the mult being much thicker than the topcoat and the columnar layer, ver coating may be safely assumed to be a rigid body, as such the strain ( the thickness, or the height, of the topcoat and of the olumnar interlayer, given by hrc and hBo, respectively. (i the aspect ratio of the columns is A,=hBd/w, where W ( orc)m=△o△TErc is the width of the columns (ini the spacing between the columns, called L, and plane stress in the Column Aspect Ratio, Ar= hBc/W Barmer for Density, p= W/L Streaming H2O Top-Coat Compliant Bondcoat (Columnar Structure) W Substrate(Si3N44 Si3N4/SiC Substrate Fig. 2. Fig 3. The symmetrical in-plane stress in the topcoat, and the metric fiexure displacements in the columns between two ac tures in the topcoat. The model is conceptually equivalent to th lag models of interfacial tractions between thin films and rigid substrates

produce effective EBCs for high-temperature, silicon-based structural ceramics. The first objective of this paper is to develop design guidelines for the geometrical structure of the columnar intermediate layer. The performance goal is to prevent the delamination of the top￾coat—this condition is achieved if the maximum stress in the topcoat is greater than in the columnar structure, that is, if the columns resist fracture better than the topcoat. The second problem analyzed in this article pertains to the degradation in the fracture strength of the substrate as a result of the coating. We consider increased loading on a surface flaw in the substrate exerted by the forces of the thermal strains; a formal derivation shows this effect to be relatively inconsequential. II. Mechanical Design of the Columnar Interlayer and the Topcoat (1) Analysis The problem is analyzed in two dimensions, that is, the sche￾matic in Fig. 2 represents the cross-section of the EBC, which does not change in the normal direction. For volumetric quan￾tities, e.g. the strain energy per column, we assume a unit depth normal to the paper. The geometrical parameters of the multi￾layer coating are: (i) the thickness, or the height, of the topcoat and of the columnar interlayer, given by hTC and hBC, respectively, (ii) the aspect ratio of the columns is Ar 5 hBC/W, where W is the width of the columns, (iii) the spacing between the columns, called L, and (iv) the relative density of the columnar interlayer, r, which is simply r 5W/L. The elastic moduli of the two materials, the topcoat, and the material used to construct the columns are written as ETC and EBC. As we shall see later on it is likely to be a good practice to use the same material for both, in which case the ratio of the two moduli becomes equal to unity. The difference between the coefficient of thermal expansion between the topcoat and the substrate is written as Da. The difference in the thermal expansion between the topcoat and the substrate will lead to periodic cracks in the topcoat (assuming that the in-plane stress given by DaDTETC is greater than the fracture stress of the topcoat). The spacing of these periodic cracks is likely to be similar to the description of the interfacial strength of thin films under uniform loading by shear-lag models.8 In the present instance the interfacial stresses are accommodated by the flexure of the columns. The mechanics of deformation may therefore be illustrated as in Fig. 3. The periodic crack spacing in the topcoat, l, is equivalent to n columns, or to a length of nL as L is the column spacing. If the topcoat is sufficiently thin then the in-plane stress in the film may be assumed to be uniform in the z, or the out of plane direction. The stress in the topcoat will then be symmetrically distributed about the center-line of the spacing between adjacent cracks in the topcoat, while the bend￾ing displacements of the columns, parallel to the interface are antisymmetric. These bending displacements in the columns are called uj, where j 5 0, n. The topcoat is also described by discrete elements, each of length L, such that they are synchron￾ous with the columns. The stress in these elements are written as sTCj, where j 5 1, n. The strains in these elements are described by the difference between the displacements of the two columns on either edge of the element (after compensating for the ther￾mal expansion strain). Thus the first element is stretched by (u1
u0), and the jth element by (uj
uj
1) and so on. In this way we have n elements in the topcoat, and (n11) columns which flex to accommodate the strain in the topcoat. The objective of the analysis is to solve for the shear displacements in the columns, i.e. uj, j 5 0, n. The analysis is based on the principle of minimum potential energy. The potential energy is the sum of the elastic strains in the columns and in the elements of the topcoat. (The substrate being much thicker than the topcoat and the columnar layer, may be safely assumed to be a rigid body; as such the strain Fig. 2. The three elements of the architecture of an environmental bar￾rier coatings. The topcoat provides environmental protection, the com￾pliant columnar interlayer accommodates the thermal strains, and the chemical bond coat helps adherence of the upper layers to the substrate. Fig. 3. The symmetrical in-plane stress in the topcoat, and the antisym￾metric flexure displacements in the columns between two adjacent frac￾tures in the topcoat. The model is conceptually equivalent to the shear￾lag models of interfacial tractions between thin films and rigid substrates. Fig. 1. Influence of streaming water-vapor velocity on weight loss in coated silicon nitride at 13001C. January 2007 Mechanical Design for Accommodating Thermal Expansion Mismatch 171

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 recognizing

energy 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 com￾parison 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
uj
1 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
uj
1Þ 2
ETChTCDaDTðuj
uj
1Þ 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
uj
1Þ 2
Xn j¼1 ETChTCDaDTðuj
uj
1Þ 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
uk
1Þ ¼ 0; k ¼6 0; n (6) qP qun ¼ 12EBCIun h3 BC þ ETChTC L ðun
un
1Þ
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 u
0 : : : : : : u
n 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 u
j ¼ uj LDaDT j ¼ 0; n (9) Here u
j 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 u
j 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 u
j. The stress in the jth element of the topcoat, sTCj, is normalized with respect to the thermal expansion stress, that is the un￾relaxed thermal stress in the topcoat, so that s
TCj ¼ sTCj ETCDaDT (11) Combining Eq. (11) with Eq. (9) and recognizing that: sTCj ¼ ETC uj
uj
1 L
DaDT
we obtain the following relationship between the stress in the topcoat elements and the displacements: s
TCj ¼ u
jþ1
u
j
1 for j ¼ 1; n (12) The stresses in the columns of the interlayer arise from bend￾ing. 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: s
BCj 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 continu￾ous 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 sub￾stituting 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￾ture 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

January 2007 Mechanical Design for Accommodating Thermal Expa that the aspect ratio of the columns, A,= hBc/w, gives for safe design. therefore. is that the fracture in the te should accommodate the thermal strains, in preference to a EBc(L/hTc) ed by the following equation. This Note that c pulls together the principal materials parameters the elastic moduli of the top coat and the columnar layer, as well s the geometrical parameters of the columnar structure, and the thickness of the topcoat. The stresses and displacements in the where the hat signifies the maximum value of the stresses in the EBC can be described in terms of this non-dimensional parameter topcoat, OTc, and in the columnar beams, aBc. The maximum dding to the generality of the results. Later we shall find that ress in the topcoat is simply given by ErcAoAT, while dBc is A,3 lies in the transition region for safe design of the EBC. As- given by [o Bcl_n. From Eq. (13), since, as seen in Fig. 3the suming the elastic moduli of the topcoat and the columns to be first beam at the edge of the crack in the topcoat suffers the nearly equal, and the spacing of the columns, L, to be about th greatest elastic strain. Substituting these expressions into eq same as the thickness of the topcoat, hTc, we note that this con- (15)gives the following result for"safe"design dition corresponds to c0.05. This overview gives the range of the he anal- E where tio is the bending displacement in the first column. The value for io was computed various values of c, which varies with bution of Stresses and Displacements hBcw. For simplicity, it was assumed that EBcErc= l. with The physically interesting result that can be obtained from the this assumption the map depends on three geometrical param- analysis is how the stresses in the topcoat and the bending dis- eters: the thickness of the topcoat relative to the width of the placements in the columns vary from the free edge of a crack in columns, hrd w, the aspect ratio of the columns, A and the topcoat, that es vary Ior elative density of the columnar structure which is given by and so on. I he stresses are symmetric and the displacements P=WL. the fail (where the to Both quantities of course will depend on c minate), and the safe region(where the topcoat will develop The stresses and displacements are plotted in Fig 4 for two eriodic cracks but will remain attached to the substrate via the values of c=0.04 and 0. 1. The spacing between the adjacent olumnar structure)is given in Fig. 6. The design space is charted cracks in the topcoat is held constant(at n=60) to show how a field described by hrc/w and Ar The "fail"and"safe"re- the decay of these quantities depends on c. A smaller value of mes for two values of p=0.25 and 0.5 are shown. A higher leads to a more gradual decay, that is, the stresses are spread density of the columnar interlayer enlarges the safe region, and it over a larger distance. This result can be qualitatively under ecomes, therefore, more forgiving. This observation can be phys- tood from Eq . (14), as a smaller value of c is obtained for ically explained by the greater load bearing capacity of the col- ect ratio of the columns. A larger aspect ratio means umns without increasing the maximum stress experienced by the that the columns are more compliant and therefore the relax bending ation of the stress in the topcoat is spread over a larger distance The map shows two regimes for safe design, one to the left of The variation of the decay length of the stress from a free the minimum in the boundary separating the two regions, and dge in the topcoat, with respect to c, is plotted in Fig. 5. The the other to the right of the minimum. The left hand reg decay length, expressed in terms of the number of units, n, of bears theoretical uncertainty as the present analysis is based column spacing varies from about 50 at c=0.01 to approxi pon Bernoulli beam analysis where the edge effects of the mately 20 at c=0.I and n=10 at c=0.3 beam are neglected. For small a ratio this assur would be inaccurate. In any event the left hand region is likely (2) Map for Safe Design to be narrow and therefore may not offer the same degree of latitude in design and manufacturing as the safe regime on the The objective of the mechanical design is that the topcoat should right-hand side. On this side we find that a larger aspect ratio of not delaminate under the influence of thermal strains. The cri- the columns is safer, as is a thin topcoat relative to the width of 8s%3E8 Distance, j(column number) 1.0 Fig 4. Results for the in-plane stress in the topcoat and the f displacements in the columns for two values of the non-dimer dal C Fig. 5. The decay distance of the in-plane stress next to a free edge of

that the aspect ratio of the columns, Ar 5 hBC/W, gives: c ¼ EBC ETC ðL=hTCÞ A3 r (14) Note that c pulls together the principal materials parameters, the elastic moduli of the top coat and the columnar layer, as well as the geometrical parameters of the columnar structure, and the thickness of the topcoat. The stresses and displacements in the EBC can be described in terms of this non-dimensional parameter, adding to the generality of the results. Later we shall find that Ar3 lies in the transition region for safe design of the EBC. As￾suming the elastic moduli of the topcoat and the columns to be nearly equal, and the spacing of the columns, L, to be about the same as the thickness of the topcoat, hTC, we note that this con￾dition corresponds to c0.05. This overview gives the range of the values for c that should be explored in the results from the anal￾ysis. III. Results (1) Distribution of Stresses and Displacements The physically interesting result that can be obtained from the analysis is how the stresses in the topcoat and the bending dis￾placements in the columns vary from the free edge of a crack in the topcoat, that is, how these quantities vary for j 5 0, 1, 2, 3. y and so on. The stresses are symmetric and the displacements antisymmetric between two neighboring cracks in the topcoat. Both quantities of course will depend on c. The stresses and displacements are plotted in Fig. 4 for two values of c 5 0.04 and 0.1. The spacing between the adjacent cracks in the topcoat is held constant (at n 5 60) to show how the decay of these quantities depends on c. A smaller value of c leads to a more gradual decay, that is, the stresses are spread over a larger distance. This result can be qualitatively under￾stood from Eq. (14), as a smaller value of c is obtained for a larger aspect ratio of the columns. A larger aspect ratio means that the columns are more compliant and therefore the relax￾ation of the stress in the topcoat is spread over a larger distance. The variation of the decay length of the stress from a free edge in the topcoat, with respect to c, is plotted in Fig. 5. The decay length, expressed in terms of the number of units, n, of column spacing varies from about 50 at c 5 0.01 to approxi￾mately 20 at c 5 0.1 and n 5 10 at c 5 0.3. (2) Map for Safe Design The objective of the mechanical design is that the topcoat should not delaminate under the influence of thermal strains. The cri￾terion for safe design, therefore, is that the fracture in the top￾coat should accommodate the thermal strains, in preference to a fracture in the columnar beams. This ‘‘safe’’ criterion is expressed by the following equation: s^BC s^TC < 1 (15) where the hat signifies the maximum value of the stresses in the topcoat, s^TC, and in the columnar beams, s^BC. The maximum stress in the topcoat is simply given by ETCDaDT, while s^BC is given by sBCj  j¼0. From Eq. (13), since, as seen in Fig. 3 the first beam at the edge of the crack in the topcoat suffers the greatest elastic strain. Substituting these expressions into Eq. (15) gives the following result for ‘‘safe’’ design: 3 EBC ETC u
0 A2 rr < 1 (16) where u
0 is the bending displacement in the first column. The value for u
0 was computed various values of c, which varies with hBC/W. For simplicity, it was assumed that EBC/ETC 5 1. With this assumption the map depends on three geometrical param￾eters: the thickness of the topcoat relative to the width of the columns, hTC/W, the aspect ratio of the columns, Ar, and the relative density of the columnar structure which is given by r 5W/L. The map showing the fail (where the topcoat is likely to de￾laminate), and the safe region (where the topcoat will develop periodic cracks but will remain attached to the substrate via the columnar structure) is given in Fig. 6. The design space is charted in a field described by hTC/W and Ar. The ‘‘fail’’ and ‘‘safe’’ re￾gimes for two values of r 5 0.25 and 0.5 are shown. A higher density of the columnar interlayer enlarges the safe region, and it becomes, therefore, more forgiving. This observation can be phys￾ically explained by the greater load bearing capacity of the col￾umns without increasing the maximum stress experienced by the bending. The map shows two regimes for safe design, one to the left of the minimum in the boundary separating the two regions, and the other to the right of the minimum. The left hand regime bears theoretical uncertainty as the present analysis is based upon Bernoulli beam analysis11 where the edge effects of the beam are neglected. For small aspect ratio this assumption would be inaccurate. In any event the left hand region is likely to be narrow and therefore may not offer the same degree of latitude in design and manufacturing as the safe regime on the right-hand side. On this side we find that a larger aspect ratio of the columns is safer, as is a thin topcoat relative to the width of Fig. 4. Results for the in-plane stress in the topcoat and the flexure displacements in the columns for two values of the non-dimensional parameter, c. Fig. 5. The decay distance of the in-plane stress next to a free edge of the topcoat as a function of c. January 2007 Mechanical Design for Accommodating Thermal Expansion Mismatch 173

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

January 2007 Mechanical Design for Accommodating Thermal ex 17 for P into Eq. (20) for AK an 1.1 sion into(21)gives the following final result for the degradation in the fracture strength due to thermal stresses induced by the L EBC 07 ErcπA3Kc A special case of Eq.(24)can be reduced to a physically nportant result. For this purpose assume tha = tates that the elastic modulus for the topcoat and the Fig8. A plot of fc), as given by Eq(28)as a function of c, for materials is the same and that the periodic spacing of the estimating the degradation in the fracture strength is about equal to the faw size(which is approximately 10 um). In this case Eq. (24)reduces to the follow expression slurry of zirconia which is spun coated on to the substrate. This procedure results in a type of structure shown schematically in to the dia (26) latex spheres, while width of the columns is varied by changing geometrical analysis gives the following relationship Note that =s(ny) as it follows from Eq(19)that y n+l with j Equation(26)can be expressed in another form by substituting for A, from Eq. (14). Assuming again that EBcR Erc and that (Lhro)≈1, we obtain where vr is the volume fraction of the latex spheres. The result btained with latex spheres having a diameter of 25 um and olume fraction v=0. 5 is shown in the micrograph in Fig9. As on (30) this constitution gives an aspect ratio A NI and wa25 um In this example the columnar structure is made from cles of zirconia while the continuou made from electron-beam physical-vapor deposited HfO (=u According to the design map in Fig. 4, the aspect ratio of one falls to the left hand side of the minimum. Since in the present case pa0.5, we note that for"safe"operation of the coating, it radation in strength, which is given by Eq(1 aso'the deg- is necessary that (hrd/W<0.4. Since Wx25 um, this condition Note that since f(c), in Eq(28), depends only on means that hrc should be less than about 10 primarily on c, except for the additional effect of Kic. Therefore, The results from two EBCs, one with hrc 40 um and the the degradation in strength can be estimated by plotting f(e), as other with hrc 10 um after exposing to streaming humid en- a function of c. This plot is given in Fig. 8. The result is that f(c) vironment at 1250C for 30 h are shown in Fig. 10. while the is of the order of unity, varying from approximately 0.3 to 1.1 itely 0.3 to l 1, thick topcoat delaminates, the thin topcoat, which meets the safe over a wide range of values for c. Taking the near-highest value of this range that is c= l, and assuming that Kic a 2 as dis- cussed earlier (for silicon nitride), we have that the degradation As Prepared After Exposure in strength is approximately equal to af 24r (29) that is. less than 2%. The conclusion to be drawn from the above analysis is that the columnar structure of the bond coat Si3N4 will reduce the influence of the topcoat on the fracture strength of the substrate to just a few percent 50 Detailed experiments for the survivability EBCs prepared ac- ording to the guidelines developed in this paper are being re- o ported separately, where the processing of the multilayer coatings on silicon-nitride is also described. Here we report on the concept for creating the columnar structure, and study whether or not the design-map in Fig. 6 gives a credible predic tion. Note that the safe design space depends on the thickness of the topcoat relative to the width of the columns, and upon the pect ratio of the columns. The method for controlling the above design parameters illustrated in Fig 9. The effective width of the columns and the Fig9. An experimental strategy for creating a columnar structure for pect ratio is controlled by introducing latex spheres into a accommodating the thermal strain in the topcoat

tion for Pj into Eq. (20) for DK and again inserting this expres￾sion into (21) gives the following final result for the degradation in the fracture strength due to thermal stresses induced by the coating: s0 f sf ¼ 1
L a EBC ETC 1 pA3 rK
IC Xn j¼0 u
jOj (24) A special case of Eq. (24) can be reduced to a physically important result. For this purpose assume that EBC ETC 1; and; L a 1 (25) which states that the elastic modulus for the topcoat and the column materials is the same and that the periodic spacing of the columns is about equal to the flaw size (which is approximately equal to 10 mm). In this case Eq. (24) reduces to the following expression: s0 f sf ¼ 1
1 pA3 rK
IC Xn j¼0 u
jOðnjÞ (26) Note that Oj 5 O(nj) as it follows from Eq. (19) that Yj 5 nj where nj 5 1,2, y, n11 with j 5 0, 1,y, n when La. Equation (26) can be expressed in another form by substituting for A3 r from Eq. (14). Assuming again that EBCETC and that (L/hTC)1, we obtain Dsf sf ¼ sf
s0 f sf ¼ fðcÞ 12pK
IC (27) where fðcÞ ¼ c Xn j¼0 m
jOðnjÞ (28) Note that since f(c), in Eq. (28), depends only on c, the deg￾radation in strength, which is given by Eq. (27), also depends primarily on c, except for the additional effect of K
IC. Therefore, the degradation in strength can be estimated by plotting f(c), as a function of c. This plot is given in Fig. 8. The result is that f(c) is of the order of unity, varying from approximately 0.3 to 1.1, over a wide range of values for c. Taking the near-highest value of this range, that is c 5 1, and assuming that K
IC 2 as dis￾cussed earlier (for silicon nitride), we have that the degradation in strength is approximately equal to: Dsf sf : 1 24p (29) that is, less than 2%. The conclusion to be drawn from the above analysis is that the columnar structure of the bond coat will reduce the influence of the topcoat on the fracture strength of the substrate to just a few percent. V. Experiments Detailed experiments for the survivability EBCs prepared ac￾cording to the guidelines developed in this paper are being re￾ported separately,10 where the processing of the multilayer coatings on silicon-nitride is also described. Here we report on the concept for creating the columnar structure, and study whether or not the design-map in Fig. 6 gives a credible predic￾tion. Note that the safe design space depends on the thickness of the topcoat relative to the width of the columns, and upon the aspect ratio of the columns. The method for controlling the above design parameters is illustrated in Fig. 9. The effective width of the columns and the aspect ratio is controlled by introducing latex spheres into a slurry of zirconia which is spun coated on to the substrate. This procedure results in a type of structure shown schematically in Fig. 9. The height of the columns is equal to the diameter of the latex spheres, while width of the columns is varied by changing the volume fraction of the latex spheres in the slurry. Simplified geometrical analysis gives the following relationship: Ar ¼ hBC W ¼ ffiffiffiffiffiffiffi 6vf p r (30) where vf is the volume fraction of the latex spheres. The result obtained with latex spheres having a diameter of 25 mm and volume fraction vf 5 0.5 is shown in the micrograph in Fig. 9. As expected from Eq. (30) this constitution gives an aspect ratio Ar1 and W25 mm. In this example the columnar structure is made from particles of zirconia while the continuous topcoat is made from electron-beam physical-vapor deposited HfO2. According to the design map in Fig. 4, the aspect ratio of one falls to the left hand side of the minimum. Since in the present case r0.5, we note that for ‘‘safe’’ operation of the coating, it is necessary that (hTC/W)r0.4. Since W25 mm, this condition means that hTC should be less than about 10 mm. The results from two EBCs, one with hTC40 mm and the other with hTC10 mm after exposing to streaming humid en￾vironment at 12501C for 30 h are shown in Fig. 10. While the thick topcoat delaminates, the thin topcoat, which meets the safe Fig. 8. A plot of f(c), as given by Eq. (28) as a function of c, for estimating the degradation in the fracture strength. Fig. 9. An experimental strategy for creating a columnar structure for accommodating the thermal strain in the topcoat. January 2007 Mechanical Design for Accommodating Thermal Expansion Mismatch 175

176 Journal of the American Ceramic Society-Bai et al Vol. 90. No. I ments that are apparently in agreement with the design map in these results are presented in Figs. 9 and 10 ly, the concern that stresses in the EBC can degrade the intrinsic fracture strength of the substrate is addressed quanti- tatively. The general result is given by Eqs.(27)and(28), while the approximate result, which should suffice for most applica tions, is given by Eq (29). The most notable feature of Eq. (29) is that the columnar structure of the bond coat reduces the frac ture-stress penalty to less than 5% VIL. Conclusions 20um A general architecture for an environmental barrier coating is presented. It consists of a topcoat material which resists corro- sion, a columnar bond coat that is strain tolerant and the chem- ical bond coat that helps the adhesion of the columnar structure and the topcoat to the substrate. The function of the compliant bond coat is to accommodate the mismatch between the thermal expansion coefficients of the topcoat and the substrate. The stresses and displacements in the topcoat and columnar aminates while a thin topc structure are analyzed. The fracture criterion in the columns is ap in Fig. 6. related to the maximum bending stress produced in them. The safe "design of the coating is based upon the maximum stress design criterion, does indeed survive the exposure. These obser oat, as this would prevent delamination of the coating. The vations are in approximate agreement with the design guidelines nalysis then leads to the design map shown in Fig. 6. Expe ments reported in Figs. 9 and 10 are agreement with the pre- developed in this article. r\ cannot be expected to provide hermetic isolation between the environment and the substrate. The topcoat will necessarily de- High-temperature EBCs are multifunctional; they not only pro- velop periodic cracks if the thermal stress in it is greater than its tect the substrate from environmental attack but also must be fracture strength; however, the spacing between such cracks in designed to withstand thermal shock. In general, silicon con- the top coat can be controlled by the design of the columnar taining ceramics are unstable in streaming water-vapor env structure(the half spacing between the cracks in the topcoat will onment at high temperatures due to volatilization of the silica al to the decay distance shown ald scale which otherwise is protective in static oxidation conditions. The main purpose of the topcoat is to subdue the velocity of the environment at the substrate interface convert to hydroxides in humid conditions, such as zirconia or Finally, formal analysis shows that the tractions exerted by hafnia, are a natural choice for the topcoat in EBCs. However, the above architecture of the coating on the substrate surface these oxides also have a much larger coefficient of thermal ex pansion relative to silicon-based ceramics. The accommodation strength of the substanc at influence on the intrinsic fracture will have an insignificant of this thermal expansion strains by employing a compliant interlayer is the main topic of this article The compliant interlayer is assumed to be constructed from a columnar structure of beams "which can flex to accommodate References the columnar structure and the dense topcoat deposited on to it, Patent No. 4,321,331 an, ""Columnar Grain Ceramic Thermal Barrier Coatings" U.S. lies in the condition that the maximum value of the stress in the -T.E. Strangman and J. L. Schiele, "Tailoring Zirconia Coatings for Per olumns, due to fiexure, must be less than the stress in the top- farine Gas Turbine Environment. " J. Engin. Gas Turbines Power. coat. In this case, if both the columns and the topcoat are made l12.531-5(1990) of the Oxidation Rate of Silicon Nitride with Vapor Pres- from the same material. then the columns would not fracture sure,J. Am. Ceram. Soc., 82[3]625-36(1999) thereby precluding delamination of the topcoat N. S. Jacobson."Corrosion of silicon-Based Ceramics in Combustion envir- The analysis in the article focuses on the analysis of the stresses in the topcoat and the fiexure displacements in the columns. The 133-134. 1-7(2000) approach is conceptually similar to the shear lag models for the B. Sudhir and R. Raj, Effect of Steam Velocity on the Hydrothermal Oxida- interplay between the in-plane stress in thin films and the shear tion/Volatilization of Silicon Nitride, "J. Am. Ceram. Soc., 89[91 1380-7(2006). tractions at the interface when the film is mechanically loaded by Evaluation of Environmental Barier coatings for Silicon Nitride" United tech- shear tractions in the interfacial layer are borne by the flexure of TN, November 18, 20 in array of discreetly distributed columns. The analysis is formulated in terms of a non-dimensional arer.37126570(1989) K. More, unpublished work, parameter, c, given by Eq(14). The displacements in the col- Ige National Laboratory, Knoxville. TN Design and Evaluation of a High Term- umns are given by the set of difference equations in Eqs. (8)and perature Environmental Coating for Si-Based Ceramics, "J Am. Ceram. Soc. (9). These equations lead to the full solution to the probler Figure 4 gives the stress distribution in the topcoat, and the dis- IF. P. Beer, E. R. Jr. Johnston, and J. T. De Wolf (eds). Mechanics of Ma HilL. New York 2006 placements in the columns, as a function of the distance from a free edge of a crack in the topcoat. The effective decay distan of the displacements is plotted as a function of c in Fig. 5. The on /nnovative Processing and Synthesis of Ceramics, Glasses, and Composites at the criterion for safe design leads to the map in Fig. 6. The map shows two"safe"regimes, one at low aspect ratio of the columns IH. Tada, P. C. Paris, and G. R. Irwin (eds), The Stress Analys nd the other for the high aspect ratio of the columns. Experi Cracks Handhook. 3rd edition. ASME Press. New York. 2000

design criterion, does indeed survive the exposure. These obser￾vations are in approximate agreement with the design guidelines developed in this article. VI. Summary High-temperature EBCs are multifunctional; they not only pro￾tect the substrate from environmental attack but also must be designed to withstand thermal shock. In general, silicon con￾taining ceramics are unstable in streaming water-vapor envir￾onment at high temperatures due to volatilization of the silica scale which otherwise is protective in static oxidation conditions. It is therefore likely that oxides, especially those that do not convert to hydroxides in humid conditions, such as zirconia or hafnia, are a natural choice for the topcoat in EBCs. However, these oxides also have a much larger coefficient of thermal ex￾pansion relative to silicon-based ceramics. The accommodation of this thermal expansion strains by employing a compliant interlayer is the main topic of this article. The compliant interlayer is assumed to be constructed from a columnar structure of ‘‘beams’’ which can flex to accommodate the thermal strain, without fracture. Indeed the safe design of the columnar structure and the dense topcoat deposited on to it, lies in the condition that the maximum value of the stress in the columns, due to flexure, must be less than the stress in the top￾coat. In this case, if both the columns and the topcoat are made from the same material, then the columns would not fracture, thereby precluding delamination of the topcoat. The analysis in the article focuses on the analysis of the stresses in the topcoat and the flexure displacements in the columns. The approach is conceptually similar to the shear lag models for the interplay between the in-plane stress in thin films and the shear tractions at the interface when the film is mechanically loaded by thermal strain. The difference in the present problem is that the shear tractions in the interfacial layer are borne by the flexure of an array of discreetly distributed columns. The analysis is formulated in terms of a non-dimensional parameter, c, given by Eq. (14). The displacements in the col￾umns are given by the set of difference equations in Eqs. (8) and (9). These equations lead to the full solution to the problem. Figure 4 gives the stress distribution in the topcoat, and the dis￾placements in the columns, as a function of the distance from a free edge of a crack in the topcoat. The effective decay distance of the displacements is plotted as a function of c in Fig. 5. The criterion for safe design leads to the map in Fig. 6. The map shows two ‘‘safe’’ regimes, one at low aspect ratio of the columns and the other for the high aspect ratio of the columns. Experi￾ments that are apparently in agreement with the design map in Fig. 6 are described; these results are presented in Figs. 9 and 10. Finally, the concern that stresses in the EBC can degrade the intrinsic fracture strength of the substrate is addressed quanti￾tatively. The general result is given by Eqs. (27) and (28), while the approximate result, which should suffice for most applica￾tions, is given by Eq. (29). The most notable feature of Eq. (29) is that the columnar structure of the bond coat reduces the frac￾ture-stress penalty to less than 5%. VII. Conclusions A general architecture for an environmental barrier coating is presented. It consists of a topcoat material which resists corro￾sion, a columnar bond coat that is strain tolerant, and the chem￾ical bond coat that helps the adhesion of the columnar structure and the topcoat to the substrate. The function of the compliant bond coat is to accommodate the mismatch between the thermal expansion coefficients of the topcoat and the substrate. The stresses and displacements in the topcoat and columnar structure are analyzed. The fracture criterion in the columns is related to the maximum bending stress produced in them. The ‘‘safe’’ design of the coating is based upon the maximum stress in the columns being less than the maximum stress in the top￾coat, as this would prevent delamination of the coating. The analysis then leads to the design map shown in Fig. 6. Experi￾ments reported in Figs. 9 and 10 are agreement with the pre￾diction from this map. It should be noted that the topcoat cannot be expected to provide hermetic isolation between the environment and the substrate. The topcoat will necessarily de￾velop periodic cracks if the thermal stress in it is greater than its fracture strength; however, the spacing between such cracks in the top coat can be controlled by the design of the columnar structure (the half spacing between the cracks in the topcoat will be equal to the decay distance shown along the y-axis in Fig. 5). The main purpose of the topcoat is to subdue the velocity of the environment at the substrate interface. Finally, formal analysis shows that the tractions exerted by the above architecture of the coating on the substrate surface will have an insignificant influence on the intrinsic fracture strength of the substrate. References 1 T. E. Strangman, ‘‘Columnar Grain Ceramic Thermal Barrier Coatings’’; U.S. Patent No. 4,321,331, 1982. 2 T. E. Strangman and J. L. Schienle, ‘‘Tailoring Zirconia Coatings for Per￾formance in Marine Gas Turbine Environment,’’ J. Engin. Gas Turbines Power, 112, 531–5 (1990). 3 J. Opila, ‘‘Variation of the Oxidation Rate of Silicon Nitride with Vapor Pres￾sure,’’ J. Am. Ceram. Soc., 82 [3] 625–36 (1999). 4 N. S. Jacobson, ‘‘Corrosion of Silicon-Based Ceramics in Combustion Envir￾onment,’’ J. Am. Ceram. Soc., 76 [1] 3–28 (1993). 5 K. N. Lee, ‘‘Current status of EBCs for Si Based Ceramics,’’ Surf. Coat. Tech., 133–134, 1–7 (2000). 6 B. Sudhir and R. Raj, ‘‘Effect of Steam Velocity on the Hydrothermal Oxida￾tion/Volatilization of Silicon Nitride,’’ J. Am. Ceram. Soc., 89 [9] 1380–7 (2006). 7 T. Bhatia, H. Eaton, J. Holowczak, E. Sun, and V. Vedula, ‘‘Development and Evaluation of Environmental Barrier Coatings for Silicon Nitride’’; United Tech￾nologies Research Center, East Hartford, CT, DOE-EBC Workshop, Nashville, TN, November 18, 2003. 8 D. C. Agrawal and R. Raj, ‘‘Measurement of the Ultimate Shear Strength of a Metal Ceramic Interface,’’ Acta Met. Mater., 37 [4] 1265–70 (1989). 9 K. More, unpublished work, Oak Ridge National Laboratory, Knoxville, TN. 10S. R. Shah and R. Raj, ‘‘Multilayer Design and Evaluation of a High Tem￾perature Environmental Coating for Si-Based Ceramics,’’ J. Am. Ceram. Soc., (2006), in press. 11F. P. Beer, E. R. , Jr. Johnston, and J. T. DeWolf (eds), Mechanics of Ma￾terials, 4th edition, McGraw Hill, New York 2006. 12K. Sharma, P. S. Shankar, and J. P. Singh, ‘‘Mechanical Behavior of Si3N4 Substrates with Environmental Barrier Coatings’’; Proceedings of the Symposium on Innovative Processing and Synthesis of Ceramics, Glasses, and Composites at the 105th American Ceramic Society Annual Meeting and Exposition, Nashville, TN, April 27–30, 2003. 13H. Tada, P. C. Paris, and G. R. Irwin (eds), The Stress Analysis of Cracks Handbook, 3rd edition, ASME Press, New York, 2000. & Fig. 10. A thick topcoat delaminates while a thin topcoat does not, in agreement with the design-map in Fig. 6. 176 Journal of the American Ceramic Society—Bai et al. Vol. 90, No. 1

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