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 ofthat 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