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