Inelastic constitutive equation of plasma-sprayed ceramic thermal barrier coatings

Transcription

1 Available online at Acta Metall. Sin.(Engl. Lett.Vol.4 No. pp April 011 Inelastic constitutive equation of plasma-sprayed ceramic thermal barrier coatings Masayuki ARAI Materials Science Research Laboratory, Central Research Institute of Electric Power Industry, -6-1, Nagasaka, Yokosuka-shi, Kanagawa-ken , Japan Manuscript received 1 December 010; in revised form 3 March 011 Ceramic thermal barrier coatings (TBCs are a very important technology for protecting the hot parts of gas turbines (GTs from a high-temperature environment. The coating stress generated in the operation of GTs brings cracking and peeling damage to the TBCs. Thus, it is necessary to evaluate precisely such coating stress in a TBC system. We have obtained a stress-strain curve for a freestanding ceramic coat specimen peeled from a TBC coated substrate by conducting the bending test. The test results have revealed that the ceramic coating deforms nonlinearly with the applied loading. In this study, an inelastic constitutive equation for the ceramic thermal barrier coatings deposited by APS is developed. The obtained results are as follows: (1 the micromechanics-based constitutive equation was formulated with micro crack density formed at splat boundary, and ( it was shown that the numerical results for a nonlinearly deformed beam simulated by the developed constitutive equation agreed with the experimental results obtained by cantilever bending tests. KEY WORDS Inelastic constitutive equation; Thermal barrier coating; Gas turbine; Micromechanics 1 Introduction Ceramic thermal barrier coatings (TBCs are a very important technology for protecting the high-temperature components of gas turbines from an aggressive environment [1]. A TBC is usually deposited by a plasma-spraying process that is based on a deposition technique impacting particles molten into a plasma flow onto the underlying substrate []. The process parameters, e.g., particle velocity, particle temperature, substrate temperature and coating thickness, affect the mechanical properties of the ceramic coating. The stress-strain curve in one of the mechanical properties is a fundamental one. The authors [3] have examined the influence of particle velocity and coating thickness on the stress-strain curve of a freestanding ceramic coating specimen that was obtained by peeling from a TBC-coated sample. Obvious nonlinear behavior was observed in the stress-strain curve, and the nonlinear degree increased with decreasing particle velocity. Microstructure observation using a scanning electron microscope (SEM has revealed that the nonlinear Corresponding author. Senior Research Scientist, PhD; Tel: ; Fax: address: marai@criepi.denken.or.jp (Masayuki ARAI

2 16 deformation of the ceramic coating progressed with initiation of a micro crack and a slip along the splat boundary. The micro crack initiation originated from the coating stress such as residual stress after the deposition process and mechanical loading from SEM in situ observation. Such nonlinear degree enables us to reduce the coating stress in service, which certifies a safety margin when applying a TBC system to gas turbine components. The aim of this study is to develop a constitutive equation suite for ceramic coating in a TBC system. To do so, we attend to a variation of micro crack density including in the microstructure a mechanical loading, viz., define damage parameter as the micro crack density. In order to formulate the constitutive equation under this idealization, an inelastic strain tensor is related to the volume change by micro crack opening and micro crack density changing with average stress. Herein, micro crack propagation and interaction between micro cracks is not considered in the formulation. For showing validity of our constitutive equation, nonlinear deformation analysis for the cantilever beam of the freestanding ceramic coating is conducted, and then the analysis results are compared with our previous experimental data [3]. Microstructure of Plasma Sprayed Ceramic Coating Plasma spray technology is the simple deposition process in which molten particles are impacted with a high velocity onto the target surface (see Fig.1. The deposited particles spread along the target surface (the flattened particle is called a splat, and shrink rapidly by being cooled on the target as a heat sink. This shrinkage generates a large tensile thermal stress, which leads to many Fig.1 Schematic illustration of deposition process of a single splat. tortoise-like cracks in the splat. The microstructure of plasma-sprayed ceramic coating thus includes many micro defects into the microstructure. The other aspect of defects is globular pores, resulting from trapping gas into the splat boundary during the deposition process. The mechanical property of the plasma-sprayed ceramic coating is affected strongly by both micro defects (see Fig.a. Fig. Model for the microstructure of ceramic coating by penny-shaped micro cracks: (a microstructure of ceramic coating; (b microcracks distributed in an elastic solid.

3 163 The existence of a micro crack contributes to reducing thermal conductivity and provides a good insulator for a high-temperature environment. On the other hand, the increased number of micro cracks with applied load, in addition to the defects, brings lower elastic modulus, a stronger nonlinear degree in the stress-strain curve and lower fracture strength. The importance of stress analysis of plasma-sprayed TBC systems is increasing recently, because precise evaluation of cracking and peeling off of the ceramic coating in TBCs is needed by TBC users, such as electric power companies. Nowadays, a conventional constitutive equation in assumption of modeling a ceramic coating as viscous plastic isotropic solid has been assembled into a finite element code [4]. However, this assumption should be discarded because the mechanical properties indicate strong anisotropy by the microstructure including the oriented micro cracks. Thus, it is necessary to develop an alternative model for the coating in consideration of those defects. 3 Formulation 3.1 Strain field of elastic solid with single penny-shaped crack In this study, we treat the plasma-sprayed ceramic coating material as a solid with many penny-shaped microcracks (see Fig.b. Firstly, consider a region bound by volume V R 3 including a single penny-shaped crack in infinite elastic media. Average stress and strain tensor for the region is expressed by σ ij and ε ij. We introduce the strain tensor α ij formed by the crack opening (it is called additional strain tensor. By combination of elastic strain distributed in elastic media without the crack and the additional strain induced by dilatation of volume when the crack opens, we have the following constitutive equation for elastic media with a single penny-shaped crack in space ( x i R 3. ε ij = S 0 ijklσ kl + α ij (1 where Sijkl 0 is elastic compliance tensor of infinite elastic media without a crack. compliance tensor is given by, Sijkl 0 = 1 + v (δ ik δ jl v E 1 + v δ ijδ kl where E is Young s modulus and v Poisson s ratio. Additional strain tensor can be given by [5], α ij = 1 1 V (b in j + b j n i (3 S V where S V is the crack surface, n i S is the vector normal to the crack surface and b i is a discontinuous displacement vector across the crack. The global coordinate system (x i R 3 fixed in the elastic media and the local coordinate system (x i R3 fixed at the crack surface are defined individually. The geometrical relation between the global and local coordinates is connected with Euler angles (θ, φ as shown in Fig.3. The transformation matrix [g ij ] R 3 3 from the global to local coordinates can be expressed by, [g ij ] = cosθcosφ cosθsinφ sinφ sinφ cosφ 0 sinθcosφ sinθsinφ cosθ This ( (4

4 164 where the angles are varied over the range: 0<θ π, 0<φ π. The base vectors of local coordinate e i and global coordinate e i are related via Eq.(4 as follows: or the inverse form: e i = g ij e j (5 e i = g ji e j (6 Thus, the vector n i can be also transformed according to Eq.(6, n i = g ji n j (7 Substituting the vector components n 1 =0, n =0, and n 3 =1 suite to our problem into Eq.(7, we have, n i = g 3i (8 Fig.3 Definition of coordinate systems. The displacement vector b i can be related to the local coordinate as well as the vector normal to the crack surface. b i = g ji b j (9 or the inverse of Eq.(9 becomes, Stress tensor can be related as follows: b i = g ij b j (10 σ ij = g ik g jl σ kl (11 The crack surface lies on the (x 1, x S plane in the local coordinate (the vector normal to the crack surface coincides with x 3. Thus, the crack opening displacement vector observed when we are in the local coordinate could take the form [6] : b i = B ii (a r 1/ σ 3i (1 where r is the measure form origin in the coordinate, and coefficients B ii are, B 11 = B = 16(1 v πe( v, B 33 = 8(1 v πe (13 It should be noticed that the stress components contributing to the crack opening are σ 33, σ 31 and σ 3. Substituting Eq.(11 into Eq.(1, we have, b i = B ii (a r 1/ g 3k g il σ kl (14

5 165 Substitution of Eqs.(14 into Eq.(9 leads to, b j = B ss (a r 1/ g 3k g sl g sj σ kl (15 in the global coordinate system. Consequently, substituting Eqs.(8 and (15 into Eq.(3, the additional strain tensor can be reduced to, α ij = 1 V S ds(a r 1/ B ss G 3k g sl (g si g 3j + g sj g 3i σ kl (16 Performing the integration over the crack surface S V, we can derive the full form as follows: α ij = πa3 3V B ssg 3k g sl (g si g 3j + g sj g 3i σ kl (17 We approximate the crack orientation angle θ 1, in the sense of the microstructure observation in Fig.. Actually, it can be recognized that the splat boundary as the site generating the micro crack almost spreads along a plane parallel to the surface of the substrate. This approximation brings us the following simple forms for the matrix [g ij ]. [g ij ] cosφ sinφ θ sinφ cosφ 0 θcosφ θsinφ 1 (18 3. Two-dimensional constitutive equation Here, the constitutive equation in two-dimensional plane (x, x 3 R is formulated below. It will be easy to extend the results shown here to the three-dimensional case. In this study, the plane stress problem under the following restriction is considered. σ 1i = 0 (i = 1,, 3 V (19 The key to obtaining the associated constitutive equation is to calculate additional strain tensor(eq.(17. Expanding the tensor with the condition (19, Eqs.(13 and (18, thoseinterest components are, α (θ, φ = 16a3 3V E α 33 (θ, φ = 16a3 3V E α 3 (θ, φ = 16a3 3V E v v {θ sin φσ vθ sin φσ 33 + θsinφσ 3 } + O(θ 3 {(θ + ( vσ 33 vθ sin φσ + (1 vθsinφσ 3 } + O(θ 3 {θsinφσ + (1 vθsinφσ 33 + (1 + (1 vθ sin φσ 3 } + O(θ 3 v (0 Next, we try to apply the single crack problem to a multiple cracks problem involving N R + cracks in a unit volume. The stress interaction between cracks is ignored for simplification of mathematical treatment. This concept allows us to apply the superposition

6 166 Before doing all this, we introduce the following non- principle of the solutions (0. dimensional parameter D R +, D = Na 3 (1 which indicates the occupancy rate of the micro crack in a unit volume and can be regarded as the damage parameter in damage mechanics, as is well known. Furthermore, we apply the statistical function F (φ for reflecting the statistical distribution of micro cracks to our model. It should be noted that this function implies the following condition: π dφf (φ = 1 ( 0 The additional strain tensor α ij is rewritten with consideration of statistically distributed cracks. For instance, if we write down the component α, it is, α (θ = 16 π dφf (φd(φ {θ sin φσ vθ sin φσ 33 + θsinφσ 3 } (3 3E v 0 in which the perturbed terms smaller than θ 3 were neglected. If damage parameter D varied in φ space has isotropy, the integration can be simply performed: α (θ = 16 3E v α 33 (θ = 16 3E v α 3 (θ = 16 3E v ( σ v σ 33 θ D {(θ + ( vσ 33 v θ σ } D {1 + (1 vθ }σ 3 D (4 Taking account of the contribution of crack opening on compliance of the cracked solid, we put the damage parameter D by D σ 33, where is the bracket symbol: x = x (1 + x (5 x and σ 33 = π 0 1 dφf (φ g 3k g 3l σ kl θ σ + σ 33 Eventually, combination of Eqs.(1 and (0 leads to our constitutive equation for the two-dimensional problem. ε = 1 E {( v ( θ D σ 33 σ v v } θ D σ 33 σ 33 (6 ε 33 = 1 E {( v ε 3 = 1 + v E ( (θ + ( v D σ 33 σ 33 v ( v v } θ D σ 33 σ (1 + (1 vθ D σ 33 σ 3 (7

7 Damage evolution Consider the mechanical situation of pulling out the single splat deposited on the target surface. If the splat is subjected to both traction p normal to and traction q parallel to the target surface, the energy Γ needed to pull it out will depend on a loading direction, of course. Thus, the damage parameter associated with cracking occurring as the splat is pulled out across the splat boundary should be affected by the loading condition. Taking account of this loading dependency for the damage, the evolution relation in the damage parameter varied with a loading condition could be assumed as follows: D = 1 [( p n ( q m ] D 0 + (8 p 0 q 0 where D 0 is a damage constant, p 0 and q 0 are stress constants and m and n are exponents describing how the damage develops in a stress space (σ ij. In the case of traction p only, it is expressed with the following in a single cracked solid. p = σ 33 = θ sin φσ + θsinφσ 3 + σ 33 (9 Integrating with function F for considering the statistical distribution, we have, In the case of traction q only, it is expressed as where the stress components included in Eq.(31 are, p := 1 θ σ + σ 33 (30 q = ((σ 13 + (σ 3 1/ (31 σ 13 = θ(σ 33 sin φσ + (1 θ sinφσ 3 σ 3 = θsinφcosφσ + cosφσ 3 (3 in a single cracked solid. Thus, substituting Eq.(3 into Eq.(31, it can be reduced to, q θ σ 33 + θ sin φσ + (1 θ sin φσ 3 θ sin φσ σ 33 + θsinφσ 3 (σ σ 33 (33 Considering the statistical distribution F for Eq.(33, we have, q σ 3 + θ {σ σ σ 3 σ σ 33 } (34 4 Nonlinear Deformation Analysis of Freestanding Ceramic Coating 4.1 Numerical procedure The bending test for a freestanding ceramic coat is simulated with the incremental procedure using the Newton-Raphson method. The nonlinear beam model as shown in Fig.4 has geometry of height H and width w. This beam is also subjected to a pure bending moment M. The incremental calculation was performed to decide the neutral line that was gradually changed with the increase of the moment. Material constant is utilized with the following: elastic modulus E of ceramic coating is given by, E = ( {sinh 1 γηv p sinh( 1 γηv p sinh(γηv p + sinh(γηv p } E c (35

8 168 where v p is particle velocity in the process parameter, E c is elastic modulus for bulk zirconia ceramic, Poisson s ratio v=0.3 and γ = [ 5a E c ] 1/, η = L h c In this study, we put those values as, 1 a = , E c = 00 GPa, ( L = h c Fig.4 Nonlinear beam geometry used in this study. For some damage parameters, it was assumed that the damage constant D 0 =0.3, the orientation of microcrack θ=π/10 and the stress constants p 0 =q 0 =1.0. The exponent was given by function of a particle velocity. n = v p v p 4. Numerical results and discussion Fig.5 shows the stress-strain curve in comparison with analysis and experimental results. Here, we displayed the apparent stress along the vertical axis, because the stress obtained by the experimental data was Fig.5 Comparison between apparent stressstrain curve obtained by the analysis based on the nonlinear constitutive equation. transferred directly from a load cell based on classical beam theory for an isotropic beam. Thus, stress obtained from a bending moment and strain on the surface beam in our numerical result were related. This comparison provides us good agreement with our numerical results, which indicates the validity of our proposed model. 5 Conclusions (1 The micromechanics-based constitutive equation was formulated with micro crack density formed at splat boundary. ( The numerical results for nonlinear deformed beam simulated by the developed constitutive equation agreed with the experimental results obtained by cantilever bending tests. REFERENCES [1] G.W. Goward, Surf Coat Technol ( [] M. Arai, E. Wada and K. Kishimoto, J Solid Mech Mater Eng 1(19 ( [3] M. Arai, X.H. Wu and K. Fujimoto, J Solid Mech Mater Eng 4( ( [4] W. Xie, J.M. Jordan and M.Gell, Mater Sci Eng A419 ( [5] L.G. Margolin, Int J Fract ( [6] B. Budiansky and R.J. O Connel, Int J Solid Struct 1 (