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Stress analysis of a barb test for thermal barrier coatings
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Surface & Coatings Technology 202 (2008) 3413 – 3418 www.elsevier.com/locate/surfcoat
Stress analysis of a barb test for thermal barrier coatings Ning-Yuan Cao, Yutaka Kagawa ⁎, Yu-Fu Liu Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8409, Japan Received 16 May 2007; accepted in revised form 12 December 2007 Available online 17 December 2007
Abstract Stress distributions in a barb pullout specimen prepared from the thermal barrier coating (TBC)/substrate structure are analyzed by the finite element method. Contribution of thermal stresses and applied stresses is included in the computations. Interfacial stress singularity at the test support edge is evaluated and effects of various Young's modulus ratios between the TBC layer and substrate are shown. The results help to understand the mechanical behavior in the barb test. © 2008 Published by Elsevier B.V. Keywords: Thermal barrier coating; Interface; Stress distribution; FEM; Stress singularity
1. Introduction Thermal barrier coatings (TBCs) have been used for high temperature gas turbine components to protect superalloy substrates from severe environment [1–3]. Typically, TBC coatings under service exhibit undesirable delamination of the TBC layer from the substrate and then lose their excellent thermal insulating properties. Therefore, understanding of the delamination resistance has been an important research subject [4,5]. Measurement of the delamination resistance is one key issue and several methods have been proposed for this purpose [6,7]. A “Barb” test recently proposed in [8,9] seems attractive because it is relatively simple and could measure delamination resistance under shear loading mode. Fig. 1 shows a schematic of the test procedure. The method has already been applied to air plasma-sprayed and electron beam physical vapor deposited TBC systems [8], and it is known that steady state delamination appears in the test and the steady state interface toughness, Gss , could be easily derived from a simple equation [10]. Experimentally, it is sometimes difficult to introduce a notch or crack to the TBC-substrate interface because the TBC layer is quite thin. An unnotched smooth specimen, as shown in Fig. 1, ⁎ Corresponding author. Tel.: +81 3 5452 5086; fax: +81 3 5452 5087. E-mail address: [email protected] (Y. Kagawa). 0257-8972/$ - see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.surfcoat.2007.12.011
is thus needed for the barb test [8,9]. Stress fields due to temperature change induced in materials processing and the applied test loading provide most fundamental information to understand the mechanical behavior of the test, but are still lacking. In this study, numerical analysis of the stress distribution in the “Barb” test for a perfectly bonded TBCsubstrate system is conducted by the finite element method in order to understand the barb test thoroughly. 2. FEM analysis The commercial finite element analysis program MARCMENTAT [11] was used to analyze a half of the barb test model (Fig. 2), where the x–y coordinate system is also defined (Fig. 2b). Definition of an internal and a free edge is indicated in Fig. 2(a). The total number of elements are 4873 and the total number of nodes are 5088. The geometry of the model is Ltbc/htbc = 40 and htbc/hs = 1/15, to simulate the previous experimental dimensions [8]. The finite element model consists of arbitrary quadrilateral elements with four nodes and four integration points. Element sizes were selected by continuously refining the mesh until approximate convergence of the numerical solution (Fig. 2(b)). The minimum mesh size is 0.0005htbc, which corresponds to an actual dimension of ≈0.1 μm. A finer mesh model with about 4 times more elements gave a maximum stress difference of less than 3% at x = 0 and y/htbc = 10− 4. The current mesh size was
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Fig. 1. Barb pull-out test configuration of an TBC specimen.
believed to be fine enough to compute the stress gradient near at the TBC ends with sufficiently high accuracy. To examine the effect of Young's modulus ratio between TBC layer and substrate on stress distribution, the Young's modulus ratio is defined as X¼
Etbc ; Es
ð1Þ
where E is Young's modulus with subscripts “tbc” and “s” referring to the TBC layer and substrate, respectively. In this
study, Es = 200 GPa [4–9], and the thermal expansion coefficients of a Ni-based superalloy substrate and a Y2O3–ZrO2 TBC layer are taken to be αs = 15 × 10− 6K− 1 and αtbc = 11 × 10− 5K− 1, respectively [1,2]. The modulus ratio, Ω, is changed to 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0 to take into account various thermal barrier coatings, e.g., in-plane Young's modulus of EB-PVD TBC is reported to be 40–60 GPa [1,2] and APS-TBC is reported to be 30–200 GPa [2]. Poisson's ratios of the TBC and substrate are assumed to be vtbc = 0.2 and vs = 0.3, respectively [4–9]. In addition, the TBC layer and substrate are assumed to be completely linear elastic materials undergoing plane strain deformation. The computation consists of two steps: (i) calculation of thermally induced stress in the entire specimen, and (ii) calculation of stress distribution under an applied stress σa = 90 MPa. This process is the same as reported previously by one of the authors [12]. To determine stresses at the interface, the stresses of TBC layer nodes nearest the interface were used [11]; detailed procedure of the definition of interfacial shear stress was reported elsewhere [12]. In the first step, thermal stresses are calculated under a reference temperature difference ΔT = 1000 °C in the present work. This ΔT value was used by assuming that a stressfree sample at an elevated temperature was cooled down uniformly to ambient. The temperature-dependent material properties such as thermal expansion coefficients are known to influence magnitude of the thermal stress, however, such temperature-dependent behavior is ignored. The symmetrical axis (x = −hs) of the model is fixed and all other nodes are allowed to move. This boundary condition means that the specimen could change its shape according to thermal mismatches. In the next step, the boundary conditions are changed to reproduce actual support conditions (Fig. 2(b)). The TBC layer is set to U = 0, V = 0
Fig. 2. FEM model of the barb test; (a) FEM mesh, boundary conditions and definition of coordinate system; (b) detail of A.
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Fig. 3. Thermal stress distribution of entire specimen: (a-1) stress contour of σxx of substrate and (a-2) of TBC layer; (b-1) σyy of substrate and (b-2) of TBC layer; (c-1) τxy of substrate and (c-2) of TBC layer.
at 0.005 ≤ x/htbc ≤ 1 and y = 0. Here, U and V are displacements along the x and y axes, respectively. This condition corresponds to a high frictional contact between the TBC layer and support, so that the contact area could not move. The other extreme situation is frictionless at the support, that is, V = 0 while U is free to move at 0.005 ≤ x/htbc ≤ 1 and y = 0. The difference between the two cases was examined under a typical material case of Ω = 0.2 with an arbitrary applied load. It was confirmed that the shear stress was nearly the same, while fairly large differences in the normal stresses occurred only within the range 0 b y/htbc b 0.02, e.g., the normal stress for the high friction case is about two times larger than that of frictionless case at y/htbc ≈ 0.01. Hereafter only the high friction boundary condition is considered. 3. Results 3.1. Thermal stress distribution Fig. 3 shows typical thermal stress distributions of the entire specimen with Young's modulus ratio Ω = 0.2. This Ω value is very close to the case of EB-PVD TBCs. The stresses σxx, σyy and τxy indicate normal and shear components of the defined coordinates respectively. The sign of τxy is defined in Fig. 4. These stresses are normalized by closed-form thermal stresses T σsT and σtbc (Appendix A), in the substrate and TBC layer, respectively. The thermal stress component σxx (Fig. 3(a)) in both TBC layer and substrate appears only in the coated region (y ≥ 0) and it does not exist in the uncoated region (x b 0, y b 0). This seems natural because there is no constraint in the range of x b 0 and y b 0. However, thermal stresses are not completely symmetric with respect to the free and internal edges. The distribution of the normal stress, σxx, in the substrate is almost symmetrical with respect to y ≈ Ltbc/2. Near the internal edge, σxx stress concentration occurs in the substrate, which only slight stress concentration in the σxx component appears at the edge of TBC
layer. The σyy stress component (Fig. 3(b)) shows different distribution from that of σxx component. In-plane compressive stress appears in the TBC layer and substrate vice versa. This seems quite reasonable because the relationship of the thermal expansion coefficient between TBC layer (αtbc) and substrate (αs) satisfies αtbc b αs. The stress component σyy in the substrate increases when approaching the interface, and the value approaches to ≈ σsT at a half of the width separated from the interface. On the other hand, the σyy component shows a T uniform distribution with a value of ≈ σtbc throughout the entire TBC layer. Fig. 3(c) shows the shear thermal stress distribution, τxy, in the substrate and TBC layer. Near the internal edges (y = 0), τxy stress concentration appears both in the substrate and the TBC layer, however, the sign of the shear stress is opposite at the free and internal edges (y = 0 and y = Ltbc, respectively). With separation from both the edges, shear components, τxy, completely disappear, this result suggests that the thermal shear stress concentration is limited only near the edge regions of the specimen. 3.2. Stress distribution change upon loading The stress changes upon loading are presented in Fig. 5. The normal stress distribution σxx is only slightly changed
Fig. 4. Definition of shear stress signs.
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Fig. 5. Stress distribution after mechanical loading: (a-1) stress contour of σxx of substrate and (a-2) of TBC layer; (b-1) σyy of substrate and (b-2) of TBC layer; (c-1) τxy of substrate and (c-2) of TBC layer.
after loading except near both the internal and the free edges (Fig. 5(a)). On the contrary, the in-plane compressive stress in TBC layer, σyy, at the internal edge increases because both initial thermal and in-plane stress form by applied stresses to TBC layer are in compression (Fig. 5(b)). Tensile compressive stress in σyy component appears in the substrate near the internal edge of TBC layer. Another region of the substrate in y direction is in tension (σyy N 0), this result seems quite reasonable because pulling load is applied to the substrate. Upon loading, shear stress concentration becomes significant especially near internal edge (y ∼ 0) and it shows a positive stress concentration at the internal edge (y = 0), which is opposite with thermal stress distribution (c.f. Fig. 5(c)). However, the shear component in TBC layer shows a negative stress concentration at the internal edge (y = 0), and except for a small region near the edge, the stress distribution shows only slight change compared to the thermally-induced shear stress distribution. It should be noted that the direction of the thermal shear stress is opposite to that caused by the applied loading. 3.3. Stress distribution at the interface Fig. 6 show the interfacial normal (Fig. 6(a)) and shear (Fig. 6(b)) stress distributions along interface between TBC layer and substrate with Young's modulus ratio of Ω = 0.2 for pure thermal stress and with additional applied load. The stresses are normalized by ησa (η = htbc/hs). Prior to applying the mechanical loading, both normal and shear stresses show singular behavior near the two edges (y/Ltbc = 0 and 1), but quickly approach to zero at positions away from the two edges. Here, the singular behavior means that the stresses reach infinite mathematically as commonly defined. When the pullout loading σa is imposed, stress fields at the free edge (y/Ltbc = 1) remain unchanged. The total normal stress near the internal edge (y/ Ltbc = 0) moves to larger compressive stress, but the shear stress changes signs and are singular with an opposite direction.
It is well known that interfacial shear stress singularity appears at internal edge of dissimilar bonded materials by thermal mismatch strain [12,13]. In the present case, the effect of the modulus ratio Ω on the singularity is evaluated in the area y/Ltbc b 0.2, where strong singularity appears in the specimen. Here, it is assumed that the singularity is expressed as [13,14] sTxy ðrÞ ¼ sTxy rkðXÞ ;
ð2Þ
where λ(Ω) is a non-dimensional constant parameter which depends on the Young's modulus ratio, Ω. Fig. 7 is plot of normalized thermal stress components τxy(r) at interface versus distance from an internal edge, r(edge is taken as x = 0). The normalization is done using in-plane stress of TBC layer given by Eq. (A4). It is evident that the shear stress component, τxy(r), has nearly linear-logarithmic relation dependent of the distance r: this result means that r− λ(Ω) singularity clearly appears in the interfacial shear stress component. Assuming that the slope of the relation in Fig. 7 is given by a straight line, order of the singularity, λ, is given by slope of the line. The order of singularity, λ(Ω), is obtained using least squares fitting of the curve. The singularity order λ(Ω) slightly increases with the increase of Young's modulus ratio Ω. The curve fitting of Fig. 8 for 0.1 ≤ Ω ≤ 1.0 under a plane strain condition yields, kðXÞ ¼ 0:25 þ 0:36X 0:17X2 þ OðXÞ:
ð3Þ
where O(Ω) is the higher order term and could be assumed nearly zero (O(Ω) ≅ 0). Numerical fitting curve is also shown in Fig. 8 by a solid line. This equation is useful to obtain the initial shear stress distribution near an internal edge of the barb test specimen. After mechanical loading, the interfacial normal stress near the push end changes from tension to compression. However, the changed region is only in the range of 0 b y/Ltbc b 0.2, and the stress in the free edge region is not affected by the mechanical
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Fig. 8. Relationship between singularity order, λ, and Young's modulus ratio, Ω.
(0.2 b y/Ltbc b 0.8). This result seems natural became interface shear stress by applied load is concentrated only at the internal edge and the direction of shear stress at the free edge is opposite. The magnitude of stress concentration at the internal edge depends on the modulus ratio, Ω. Using superimpose principle of stress, the interfacial shear stress is expressed as sxy ð yÞ ¼ sTxy ð yÞ þ saxy ð yÞ;
ð4Þ
T a (y) is the shear stress by thermal misfit and τxy (y) is where, τxy the shear stress by the applied load.
4. Summery and concluding remarks Fig. 6. Distributions of interfacial stress caused by thermal mismatch and mechanical loading for Young's modulus ratio of Ω = 0.2: (a) normal stress component, σixx and (b) shear stress component, τixy.
loading. The shear stress originates by both thermal stress and mechanical loading, which only exist near a free edge and the value is zero in the middle part of the TBC coated region
Stress distributions in a barb test have been obtained by FEM analysis. The results show that the initial thermal stresses at the bonded interface edges are singular. Interfacial shear stresses at the support edge caused by the thermal stresses and applied stress have opposite signs. The order of the stress singularity at the internal edge of the barb test specimen shows strong dependence on the modulus ratio between the TBC layer and substrate, Ω. As demonstrated in Ref. [8], the as-received barb test specimen remains intact and delamination has been caused by subsequent mechanical loading. Experimentally, the very initiation of the interface delamination may be difficult to detect because fracture occurs suddenly in this case. From the current analysis, it can be said with confidence that the delamination must be started from the lower internal edge other than the upper free edge, although both edges show strong stress singularity. The underlying reason is that the applied loading changes the interfacial stress distribution only at the lower-side internal edge region of the specimen. The problem with a lowerside delamination crack will be addressed in a future work. Appendix A
Fig. 7. Stress singularity at interface for internal edge of specimen.
The analytical solution used here is based on the report of Kim et al. [7]. Thermal stress of two layer coating system
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where, Δα is the difference of thermal expansion coefficient given by Δα = α⁎s − α⁎tbc. Upon loading, the total stress of t , respectively, are obtained substrate, σst, and TBC layer, σtbc using a simple superimposition principle as rts ¼ ra þ
DaDT ; ⁎ g þ 1=Es⁎ 1=Etbc
1 DaDT rttbc ¼ ra ⁎ : g g=Es⁎ þ 1=Etbc
ðA5Þ
ðA6Þ
t are used to normalize the stress distributions The σst and σtbc obtained in Section 3.
References [1] [2] [3] [4]
Fig. A1. Schematic drawing of half of the barb test specimen.
(Fig. A1) can be obtained by static force balance at interface, given by E⁎ a⁎ mgE ⁎ a⁎ rs ¼ s s ⁎ tbc ⁎tbc DT ; 1mgEtbc =Es 0
0
rtbc ¼
Es⁎ a⁎s g
⁎ ⁎ þ gEtbc atbc ⁎ þ Etbc =Es⁎
DT ;
ðA1Þ
ðA2Þ
where, η = htbc/hs, Ei⁎ = Ei/(1 − vi2), and α⁎i = (1 + vi)αi (i = “tbc” and “s”). Subscripts “s” and “tbc”, respectively, indicate substrate and TBC layer. Then, thermal stress in the substrate and TBC layer are given by rTs ¼
DaDT ; ⁎ g þ 1=Es⁎ 1=Etbc
rTtbc ¼
DaDT ⁎ ; g=Es⁎ þ 1=Etbc
ðA3Þ
ðA4Þ
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
N.P. Padture, M. Gell, E.H. Jordan, Science 296 (2002) 280. J.C. Zhao, J.H. Westbrook, MRS Bull. 28 (2003) 622. J.R. Nicholls, MRS Bull. 28 (2003) 659. A.G. Evans, D.R. Mumm, J.W. Hutchinson, G.H. Meier, F.S. Pettit, Prog. Mater. Sci. 46 (2001) 505. K.W. Schlichting, N.P. Padture, E.H. Jordan, M. Gell, Mater. Sci. Eng. A342 (2003) 120. A. Vasinonta, J.L. Beuth, Eng. Fract. Mech. 68 (2001) 843. S.S. Kim, Y.F. Liu, Y. Kagawa, Acta Mater. 55 (2007) 3771. S.Q. Guo, D.R. Mumm, A.M. Karlsson, Y. Kagawa, Scr. Mater. 53 (2005) 1043. S.Q. Guo, Y. Tanaka, Y. Kagawa, J. Eur. Ceram. Soc. 27 (2007) 3425. Y.F. Liu, Y. Kagawa, A.G. Evans, Acta Mater. 56 (2008) 43. MSC Software Corporation, MSC. Marc, 2003 Volume A and B. K. Honda, Y. Kagawa, Acta Metall. Mater. 43 (1995) 1477. V.L. Hein, F. Erdogan, Int. J. Fract. Mech. 7 (1971) 317. D.B. Bogy, J. Appl. Mech. 38 (1971) 377.
Surface & Coatings Technology 202 (2008) 3413 – 3418 www.elsevier.com/locate/surfcoat
Stress analysis of a barb test for thermal barrier coatings Ning-Yuan Cao, Yutaka Kagawa ⁎, Yu-Fu Liu Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8409, Japan Received 16 May 2007; accepted in revised form 12 December 2007 Available online 17 December 2007
Abstract Stress distributions in a barb pullout specimen prepared from the thermal barrier coating (TBC)/substrate structure are analyzed by the finite element method. Contribution of thermal stresses and applied stresses is included in the computations. Interfacial stress singularity at the test support edge is evaluated and effects of various Young's modulus ratios between the TBC layer and substrate are shown. The results help to understand the mechanical behavior in the barb test. © 2008 Published by Elsevier B.V. Keywords: Thermal barrier coating; Interface; Stress distribution; FEM; Stress singularity
1. Introduction Thermal barrier coatings (TBCs) have been used for high temperature gas turbine components to protect superalloy substrates from severe environment [1–3]. Typically, TBC coatings under service exhibit undesirable delamination of the TBC layer from the substrate and then lose their excellent thermal insulating properties. Therefore, understanding of the delamination resistance has been an important research subject [4,5]. Measurement of the delamination resistance is one key issue and several methods have been proposed for this purpose [6,7]. A “Barb” test recently proposed in [8,9] seems attractive because it is relatively simple and could measure delamination resistance under shear loading mode. Fig. 1 shows a schematic of the test procedure. The method has already been applied to air plasma-sprayed and electron beam physical vapor deposited TBC systems [8], and it is known that steady state delamination appears in the test and the steady state interface toughness, Gss , could be easily derived from a simple equation [10]. Experimentally, it is sometimes difficult to introduce a notch or crack to the TBC-substrate interface because the TBC layer is quite thin. An unnotched smooth specimen, as shown in Fig. 1, ⁎ Corresponding author. Tel.: +81 3 5452 5086; fax: +81 3 5452 5087. E-mail address: [email protected] (Y. Kagawa). 0257-8972/$ - see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.surfcoat.2007.12.011
is thus needed for the barb test [8,9]. Stress fields due to temperature change induced in materials processing and the applied test loading provide most fundamental information to understand the mechanical behavior of the test, but are still lacking. In this study, numerical analysis of the stress distribution in the “Barb” test for a perfectly bonded TBCsubstrate system is conducted by the finite element method in order to understand the barb test thoroughly. 2. FEM analysis The commercial finite element analysis program MARCMENTAT [11] was used to analyze a half of the barb test model (Fig. 2), where the x–y coordinate system is also defined (Fig. 2b). Definition of an internal and a free edge is indicated in Fig. 2(a). The total number of elements are 4873 and the total number of nodes are 5088. The geometry of the model is Ltbc/htbc = 40 and htbc/hs = 1/15, to simulate the previous experimental dimensions [8]. The finite element model consists of arbitrary quadrilateral elements with four nodes and four integration points. Element sizes were selected by continuously refining the mesh until approximate convergence of the numerical solution (Fig. 2(b)). The minimum mesh size is 0.0005htbc, which corresponds to an actual dimension of ≈0.1 μm. A finer mesh model with about 4 times more elements gave a maximum stress difference of less than 3% at x = 0 and y/htbc = 10− 4. The current mesh size was
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Fig. 1. Barb pull-out test configuration of an TBC specimen.
believed to be fine enough to compute the stress gradient near at the TBC ends with sufficiently high accuracy. To examine the effect of Young's modulus ratio between TBC layer and substrate on stress distribution, the Young's modulus ratio is defined as X¼
Etbc ; Es
ð1Þ
where E is Young's modulus with subscripts “tbc” and “s” referring to the TBC layer and substrate, respectively. In this
study, Es = 200 GPa [4–9], and the thermal expansion coefficients of a Ni-based superalloy substrate and a Y2O3–ZrO2 TBC layer are taken to be αs = 15 × 10− 6K− 1 and αtbc = 11 × 10− 5K− 1, respectively [1,2]. The modulus ratio, Ω, is changed to 0.1, 0.2, 0.4, 0.6, 0.8 and 1.0 to take into account various thermal barrier coatings, e.g., in-plane Young's modulus of EB-PVD TBC is reported to be 40–60 GPa [1,2] and APS-TBC is reported to be 30–200 GPa [2]. Poisson's ratios of the TBC and substrate are assumed to be vtbc = 0.2 and vs = 0.3, respectively [4–9]. In addition, the TBC layer and substrate are assumed to be completely linear elastic materials undergoing plane strain deformation. The computation consists of two steps: (i) calculation of thermally induced stress in the entire specimen, and (ii) calculation of stress distribution under an applied stress σa = 90 MPa. This process is the same as reported previously by one of the authors [12]. To determine stresses at the interface, the stresses of TBC layer nodes nearest the interface were used [11]; detailed procedure of the definition of interfacial shear stress was reported elsewhere [12]. In the first step, thermal stresses are calculated under a reference temperature difference ΔT = 1000 °C in the present work. This ΔT value was used by assuming that a stressfree sample at an elevated temperature was cooled down uniformly to ambient. The temperature-dependent material properties such as thermal expansion coefficients are known to influence magnitude of the thermal stress, however, such temperature-dependent behavior is ignored. The symmetrical axis (x = −hs) of the model is fixed and all other nodes are allowed to move. This boundary condition means that the specimen could change its shape according to thermal mismatches. In the next step, the boundary conditions are changed to reproduce actual support conditions (Fig. 2(b)). The TBC layer is set to U = 0, V = 0
Fig. 2. FEM model of the barb test; (a) FEM mesh, boundary conditions and definition of coordinate system; (b) detail of A.
N.-Y. Cao et al. / Surface & Coatings Technology 202 (2008) 3413–3418
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Fig. 3. Thermal stress distribution of entire specimen: (a-1) stress contour of σxx of substrate and (a-2) of TBC layer; (b-1) σyy of substrate and (b-2) of TBC layer; (c-1) τxy of substrate and (c-2) of TBC layer.
at 0.005 ≤ x/htbc ≤ 1 and y = 0. Here, U and V are displacements along the x and y axes, respectively. This condition corresponds to a high frictional contact between the TBC layer and support, so that the contact area could not move. The other extreme situation is frictionless at the support, that is, V = 0 while U is free to move at 0.005 ≤ x/htbc ≤ 1 and y = 0. The difference between the two cases was examined under a typical material case of Ω = 0.2 with an arbitrary applied load. It was confirmed that the shear stress was nearly the same, while fairly large differences in the normal stresses occurred only within the range 0 b y/htbc b 0.02, e.g., the normal stress for the high friction case is about two times larger than that of frictionless case at y/htbc ≈ 0.01. Hereafter only the high friction boundary condition is considered. 3. Results 3.1. Thermal stress distribution Fig. 3 shows typical thermal stress distributions of the entire specimen with Young's modulus ratio Ω = 0.2. This Ω value is very close to the case of EB-PVD TBCs. The stresses σxx, σyy and τxy indicate normal and shear components of the defined coordinates respectively. The sign of τxy is defined in Fig. 4. These stresses are normalized by closed-form thermal stresses T σsT and σtbc (Appendix A), in the substrate and TBC layer, respectively. The thermal stress component σxx (Fig. 3(a)) in both TBC layer and substrate appears only in the coated region (y ≥ 0) and it does not exist in the uncoated region (x b 0, y b 0). This seems natural because there is no constraint in the range of x b 0 and y b 0. However, thermal stresses are not completely symmetric with respect to the free and internal edges. The distribution of the normal stress, σxx, in the substrate is almost symmetrical with respect to y ≈ Ltbc/2. Near the internal edge, σxx stress concentration occurs in the substrate, which only slight stress concentration in the σxx component appears at the edge of TBC
layer. The σyy stress component (Fig. 3(b)) shows different distribution from that of σxx component. In-plane compressive stress appears in the TBC layer and substrate vice versa. This seems quite reasonable because the relationship of the thermal expansion coefficient between TBC layer (αtbc) and substrate (αs) satisfies αtbc b αs. The stress component σyy in the substrate increases when approaching the interface, and the value approaches to ≈ σsT at a half of the width separated from the interface. On the other hand, the σyy component shows a T uniform distribution with a value of ≈ σtbc throughout the entire TBC layer. Fig. 3(c) shows the shear thermal stress distribution, τxy, in the substrate and TBC layer. Near the internal edges (y = 0), τxy stress concentration appears both in the substrate and the TBC layer, however, the sign of the shear stress is opposite at the free and internal edges (y = 0 and y = Ltbc, respectively). With separation from both the edges, shear components, τxy, completely disappear, this result suggests that the thermal shear stress concentration is limited only near the edge regions of the specimen. 3.2. Stress distribution change upon loading The stress changes upon loading are presented in Fig. 5. The normal stress distribution σxx is only slightly changed
Fig. 4. Definition of shear stress signs.
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Fig. 5. Stress distribution after mechanical loading: (a-1) stress contour of σxx of substrate and (a-2) of TBC layer; (b-1) σyy of substrate and (b-2) of TBC layer; (c-1) τxy of substrate and (c-2) of TBC layer.
after loading except near both the internal and the free edges (Fig. 5(a)). On the contrary, the in-plane compressive stress in TBC layer, σyy, at the internal edge increases because both initial thermal and in-plane stress form by applied stresses to TBC layer are in compression (Fig. 5(b)). Tensile compressive stress in σyy component appears in the substrate near the internal edge of TBC layer. Another region of the substrate in y direction is in tension (σyy N 0), this result seems quite reasonable because pulling load is applied to the substrate. Upon loading, shear stress concentration becomes significant especially near internal edge (y ∼ 0) and it shows a positive stress concentration at the internal edge (y = 0), which is opposite with thermal stress distribution (c.f. Fig. 5(c)). However, the shear component in TBC layer shows a negative stress concentration at the internal edge (y = 0), and except for a small region near the edge, the stress distribution shows only slight change compared to the thermally-induced shear stress distribution. It should be noted that the direction of the thermal shear stress is opposite to that caused by the applied loading. 3.3. Stress distribution at the interface Fig. 6 show the interfacial normal (Fig. 6(a)) and shear (Fig. 6(b)) stress distributions along interface between TBC layer and substrate with Young's modulus ratio of Ω = 0.2 for pure thermal stress and with additional applied load. The stresses are normalized by ησa (η = htbc/hs). Prior to applying the mechanical loading, both normal and shear stresses show singular behavior near the two edges (y/Ltbc = 0 and 1), but quickly approach to zero at positions away from the two edges. Here, the singular behavior means that the stresses reach infinite mathematically as commonly defined. When the pullout loading σa is imposed, stress fields at the free edge (y/Ltbc = 1) remain unchanged. The total normal stress near the internal edge (y/ Ltbc = 0) moves to larger compressive stress, but the shear stress changes signs and are singular with an opposite direction.
It is well known that interfacial shear stress singularity appears at internal edge of dissimilar bonded materials by thermal mismatch strain [12,13]. In the present case, the effect of the modulus ratio Ω on the singularity is evaluated in the area y/Ltbc b 0.2, where strong singularity appears in the specimen. Here, it is assumed that the singularity is expressed as [13,14] sTxy ðrÞ ¼ sTxy rkðXÞ ;
ð2Þ
where λ(Ω) is a non-dimensional constant parameter which depends on the Young's modulus ratio, Ω. Fig. 7 is plot of normalized thermal stress components τxy(r) at interface versus distance from an internal edge, r(edge is taken as x = 0). The normalization is done using in-plane stress of TBC layer given by Eq. (A4). It is evident that the shear stress component, τxy(r), has nearly linear-logarithmic relation dependent of the distance r: this result means that r− λ(Ω) singularity clearly appears in the interfacial shear stress component. Assuming that the slope of the relation in Fig. 7 is given by a straight line, order of the singularity, λ, is given by slope of the line. The order of singularity, λ(Ω), is obtained using least squares fitting of the curve. The singularity order λ(Ω) slightly increases with the increase of Young's modulus ratio Ω. The curve fitting of Fig. 8 for 0.1 ≤ Ω ≤ 1.0 under a plane strain condition yields, kðXÞ ¼ 0:25 þ 0:36X 0:17X2 þ OðXÞ:
ð3Þ
where O(Ω) is the higher order term and could be assumed nearly zero (O(Ω) ≅ 0). Numerical fitting curve is also shown in Fig. 8 by a solid line. This equation is useful to obtain the initial shear stress distribution near an internal edge of the barb test specimen. After mechanical loading, the interfacial normal stress near the push end changes from tension to compression. However, the changed region is only in the range of 0 b y/Ltbc b 0.2, and the stress in the free edge region is not affected by the mechanical
N.-Y. Cao et al. / Surface & Coatings Technology 202 (2008) 3413–3418
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Fig. 8. Relationship between singularity order, λ, and Young's modulus ratio, Ω.
(0.2 b y/Ltbc b 0.8). This result seems natural became interface shear stress by applied load is concentrated only at the internal edge and the direction of shear stress at the free edge is opposite. The magnitude of stress concentration at the internal edge depends on the modulus ratio, Ω. Using superimpose principle of stress, the interfacial shear stress is expressed as sxy ð yÞ ¼ sTxy ð yÞ þ saxy ð yÞ;
ð4Þ
T a (y) is the shear stress by thermal misfit and τxy (y) is where, τxy the shear stress by the applied load.
4. Summery and concluding remarks Fig. 6. Distributions of interfacial stress caused by thermal mismatch and mechanical loading for Young's modulus ratio of Ω = 0.2: (a) normal stress component, σixx and (b) shear stress component, τixy.
loading. The shear stress originates by both thermal stress and mechanical loading, which only exist near a free edge and the value is zero in the middle part of the TBC coated region
Stress distributions in a barb test have been obtained by FEM analysis. The results show that the initial thermal stresses at the bonded interface edges are singular. Interfacial shear stresses at the support edge caused by the thermal stresses and applied stress have opposite signs. The order of the stress singularity at the internal edge of the barb test specimen shows strong dependence on the modulus ratio between the TBC layer and substrate, Ω. As demonstrated in Ref. [8], the as-received barb test specimen remains intact and delamination has been caused by subsequent mechanical loading. Experimentally, the very initiation of the interface delamination may be difficult to detect because fracture occurs suddenly in this case. From the current analysis, it can be said with confidence that the delamination must be started from the lower internal edge other than the upper free edge, although both edges show strong stress singularity. The underlying reason is that the applied loading changes the interfacial stress distribution only at the lower-side internal edge region of the specimen. The problem with a lowerside delamination crack will be addressed in a future work. Appendix A
Fig. 7. Stress singularity at interface for internal edge of specimen.
The analytical solution used here is based on the report of Kim et al. [7]. Thermal stress of two layer coating system
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where, Δα is the difference of thermal expansion coefficient given by Δα = α⁎s − α⁎tbc. Upon loading, the total stress of t , respectively, are obtained substrate, σst, and TBC layer, σtbc using a simple superimposition principle as rts ¼ ra þ
DaDT ; ⁎ g þ 1=Es⁎ 1=Etbc
1 DaDT rttbc ¼ ra ⁎ : g g=Es⁎ þ 1=Etbc
ðA5Þ
ðA6Þ
t are used to normalize the stress distributions The σst and σtbc obtained in Section 3.
References [1] [2] [3] [4]
Fig. A1. Schematic drawing of half of the barb test specimen.
(Fig. A1) can be obtained by static force balance at interface, given by E⁎ a⁎ mgE ⁎ a⁎ rs ¼ s s ⁎ tbc ⁎tbc DT ; 1mgEtbc =Es 0
0
rtbc ¼
Es⁎ a⁎s g
⁎ ⁎ þ gEtbc atbc ⁎ þ Etbc =Es⁎
DT ;
ðA1Þ
ðA2Þ
where, η = htbc/hs, Ei⁎ = Ei/(1 − vi2), and α⁎i = (1 + vi)αi (i = “tbc” and “s”). Subscripts “s” and “tbc”, respectively, indicate substrate and TBC layer. Then, thermal stress in the substrate and TBC layer are given by rTs ¼
DaDT ; ⁎ g þ 1=Es⁎ 1=Etbc
rTtbc ¼
DaDT ⁎ ; g=Es⁎ þ 1=Etbc
ðA3Þ
ðA4Þ
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
N.P. Padture, M. Gell, E.H. Jordan, Science 296 (2002) 280. J.C. Zhao, J.H. Westbrook, MRS Bull. 28 (2003) 622. J.R. Nicholls, MRS Bull. 28 (2003) 659. A.G. Evans, D.R. Mumm, J.W. Hutchinson, G.H. Meier, F.S. Pettit, Prog. Mater. Sci. 46 (2001) 505. K.W. Schlichting, N.P. Padture, E.H. Jordan, M. Gell, Mater. Sci. Eng. A342 (2003) 120. A. Vasinonta, J.L. Beuth, Eng. Fract. Mech. 68 (2001) 843. S.S. Kim, Y.F. Liu, Y. Kagawa, Acta Mater. 55 (2007) 3771. S.Q. Guo, D.R. Mumm, A.M. Karlsson, Y. Kagawa, Scr. Mater. 53 (2005) 1043. S.Q. Guo, Y. Tanaka, Y. Kagawa, J. Eur. Ceram. Soc. 27 (2007) 3425. Y.F. Liu, Y. Kagawa, A.G. Evans, Acta Mater. 56 (2008) 43. MSC Software Corporation, MSC. Marc, 2003 Volume A and B. K. Honda, Y. Kagawa, Acta Metall. Mater. 43 (1995) 1477. V.L. Hein, F. Erdogan, Int. J. Fract. Mech. 7 (1971) 317. D.B. Bogy, J. Appl. Mech. 38 (1971) 377.