Printed in Great Thermal residual stresses in plates and coatings composed of multi-layered and functionally graded materials on L. sh Department of Metallurgy and Materials Engineering, University of Connecticut Received 15 April 1997, accepted 12 September 1997) In this paper, thermal residual stresses in multi-layered and compositionally graded plates and coatings were analyzed. Systematic studies on effects of different material combinations, including the compositional gradient, the elastic modulus, the coefficient of thermal expansion, and the number of layers in the case of multi-layered materials on the magnitude and distribution of residual stresses were conducted. Geometry changes of plates and coatings due to these residual stresses were also investigated. It was found that the distribution and magnitude of thermal residual stresses within a plate can be adjusted by controlling the compositional gradient and selecting proper combination of ceramic and metal constituents. When the compositional gradient of a plate is such that rapid change in volume fraction and properties occurs near the ceramic face and a gradual change near the metal face, a residual compressive stress is produced on both faces of the plate. However, when the rapid change in volume fraction and properties occurs near the metal face and the gradual change near the ceramic face, a residual ensile stress is generated on both faces of the plate. Minimum residual stresses are obtained when the plate has a linear compositional gradient. As for compositionally graded coatings, the magnitude of the residual stresses on the surface of the coating cannot be decreased by introducing a compositional gradient. The gradient coating only alters the characteristics of the residual stress distribution, 1998 Elsevier Science Limited. All rights reserved (Keywords: thermal residual stresses; multilayers materials; functionally graded materials) INTRODUCTION example, the face with high ceramic content can provide superior wear-resistance, while the opposite face with high The concept of functionally graded materials(FGM)was metal content offers toughness and strength. Thus, such proposed in 1984 by Niino at the National Aerospace materials would be very desirable for tribological appli- Laboratory of Japan as a means of preparing thermal barrier cations where high wear resistance and toughness are materials. Since then, interest in FGM has grown rapidly required simultaneously. Ballistic applications of this class and about 200 possibilities of utilizing the FGM concept of materials are also very attractive. In this case, the ceramic have been proposed. Functionally graded materials are facing can provide functions to blunt, break-up and erode haracterized by a continuously changing property due to projectiles, while the high-metal-content face offers tough a continuous change in composition, in morphology, in ness and strength to maintain the integrity of the ceramic microstructure, or in crystal structure from one surface facing for as long as possible. Thus, the FGM can be used of the material to the other. The potential applications of for multiple impact applications with high ballistic effi FGM include thermal, structural, optical and electronic ciency, in contrast to the single impact applications for materials. Currently, most of the R&D activities in ceramic tiles structural applications have concentrated mainly on the Most of fgm have thermal residual stresses because of areas of joining and coating. However, great interest a continuously macroscopic change in composition, and also exists in using ceramic/metal FGM in a wide range of these residual stresses are bound to affect the mechanical potential application areas, including improved machine properties of FGM. Therefore, it is important to understand tools with high fracture toughness, high temperature and be able to predict the magnitude and distribution of aerospace and automotive components for improved wear thermal residual stresses in FGM and multi-layered resistance and/or oxidation resistance, and lightweight materials as a function of the compositional gradient,a armor materials with high ballistic efficiency. Many benefits combination of the constituents' properties, and the number re anticipated from using this class of the FGM. For of layers in the case of multi-layered materials. Many efforts
ELSEVIER PII: S1359-8368(97)00029-2 Composites Part B 29B (1998) 199-210 © 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00 Thermal residual stresses in plates and coatings composed of multi-layered and functionally graded materials Leon L. Shaw Department of Metallurgy and Materials Engineering, University of Connecticut, Storrs, CT 06269, USA (Received 15 April 1997; accepted 12 September 1997) In this paper, thermal residual stresses in multi-layered and compositionally graded plates and coatings were analyzed. Systematic studies on effects of different material combinations, including the compositional gradient, the elastic modulus, the coefficient of thermal expansion, and the number of layers in the case of multi-layered materials on the magnitude and distribution of residual stresses were conducted. Geometry changes of plates and coatings due to these residual stresses were also investigated. It was found that the distribution and magnitude of thermal residual stresses within a plate can be adjusted by controlling the compositional gradient and selecting a proper combination of ceramic and metal constituents. When the compositional gradient of a plate is such that a rapid change in volume fraction and properties occurs near the ceramic face and a gradual change near the metal face, a residual compressive stress is produced on both faces of the plate. However, when the rapid change in volume fraction and properties occurs near the metal face and the gradual change near the ceramic face, a residual tensile stress is generated on both faces of the plate. Minimum residual stresses are obtained when the plate has a linear compositional gradient. As for compositionally graded coatings, the magnitude of the residual stresses on the surface of the coating cannot be decreased by introducing a compositional gradient. The gradient coating only alters the characteristics of the residual stress distribution. © 1998 Elsevier Science Limited. All rights reserved (Keywords: thermal residual stresses; multilayers materials; functionally graded materials) INTRODUCTION The concept of functionally graded materials (FGM) was proposed in 1984 by Niino at the National Aerospace Laboratory of Japan as a means of preparing thermal barrier materials 1. Since then, interest in FGM has grown rapidly and about 200 possibilities of utilizing the FGM concept have been proposed t'2. Functionally graded materials are characterized by a continuously changing property due to a continuous change in composition, in morphology, in microstructure, or in crystal structure from one surface of the material to the other. The potential applications of FGM include thermal, structural, optical and electronic materials 1'2. Currently, most of the R&D activities in structural applications have concentrated mainly on the areas of joining and coating 3-5. However, great interest also exists in using ceramic/metal FGM in a wide range of potential application areas, including improved machine tools with high fracture toughness, high temperature aerospace and automotive components for improved wear resistance and/or oxidation resistance, and lightweight armor materials with high ballistic efficiency. Many benefits are anticipated from using this class of the FGM. For example, the face with high ceramic content can provide superior wear-resistance, while the opposite face with high metal content offers toughness and strength. Thus, such materials would be very desirable for tribological applications where high wear resistance and toughness are required simultaneously. Ballistic applications of this class of materials are also very attractive. In this case, the ceramic facing can provide functions to blunt, break-up and erode projectiles, while the high-metal-content face offers toughness and strength to maintain the integrity of the ceramic facing for as long as possible. Thus, the FGM can be used for multiple impact applications with high ballistic efficiency, in contrast to the single impact applications for ceramic tiles. Most of FGM have thermal residual stresses because of a continuously macroscopic change in composition, and these residual stresses are bound to affect the mechanical properties of FGM. Therefore, it is important to understand and be able to predict the magnitude and distribution of thermal residual stresses in FGM and multi-layered materials as a function of the compositional gradient, a combination of the constituents' properties, and the number of layers in the case of multi-layered materials. Many efforts 199
Thermal residual stresses in plates and coatings: LL.Shaw have been made to develop these understandings and The limitation of laminate theory is that the theory is abilities, For example, the research group in MIT led by valid only when the thickness of the plate is significantly Suresh has extensively investigated thermal residual smaller than the lateral dimensions. Furthermore, stress stresses in Ni-Al2O3 compositionally graded plates, distributions predicted from the theory represent the stresses including the consideration of plastic deformation induced away from the free edges of the plate. At the edges of the plate where the interfaces of bonded dissimilar materials Ino.12 Ga0. sAs-GaAs and carbon-epoxy systems subjected intersect the free surface, the tensile stress component and to temperature excursions have also been conducted by the shearing stress component perpendicular to the interface the same group". Residual stresses in multi-layered and are singular. 18.28. It is well established that both of these compositionally graded plates have also been investigated by stress components decrease to zero quickly away from al. for NiAl-Al2O3 systems, by Freund for the free edge. and the width of the region in which the graded semiconductor layers, by Tang et al. for PSZ-Mo stresses differ from those predicted by laminate theory is systems, by Ravichandran for Ni-Al2O3 systems, and by approximately equal to the thickness of the plate2 Itoh and Kashiwaya for material systems with different Furthermore, it has been shown that the stress singularities combinations of Youngs modulus and a linear compositional at the free edge are eliminated if the compositional gradient adient. In addition to plate geometries, coating and joint of the plate or coating are continuous rather than stepwise geometries have also been investigated. These include the Thus, the present analysis is only applicable to the stress work conducted by Lee and Erdogan for ZrO2 coatings on states of thin plates away from the free edges. Nevertheless, Ni-superalloy substrates, by Itoh and Kashiwaya for both stress analyses of this type are still highly desirable, since coatings and joints, and by Drake et al. and williamson ef in many cases it is these stresses that determine the behavior or joint geometries. Other geometries with graded of the material. One typical example of these would be compositions have also been investigated. These include a functionally graded armor tiles. In this case, when a high graded interface layer between the fiber and matrix analyzed speed projectile impacts the central portion of the tile, the by Rabzak et al. Pindera et al. and Salzar and Barton, a stress states at the central portion rather than at the free graded matrix bonded to a cylindrical fiber investigated by edges will play a critical role in the response of the tile to Duva et aL. and a graded fiber concentration in a matrix the studied by Aboudi et a1. 26 Although studies on the magnitude and effects of thermal sidual stresses in FGM with various geometries and DESCRIPTION OF PROBLEMS AND FORMULAE compositional gradients have been performed extensively as summarized in a recent review27, systematic examination of the effect of different material combinations has not The geometry of the residual stress problem for multi been explored. With this in mind, this work utilized layered plates is shown in Figure 1. Throughout this study, laminate theory to analyze thermal residual stresses in multi-layered and compositionally graded plates are con- multi-layered and compositionally graded plates and coat sidered to be composed of a ceramic and a metal, and are gs. A comprehensive study on the effects of different assumed to be infinite in the x- and y-directions. Further aterial combinations including the compositional gradi with no loss of generality, plates are assumed to have a unit ent. the elastic modulus. the coefficient of thermal ayers s are assume expansion(CTE), and the number of layers in the case of have equal thickness and the composition change from layer multi-layered materials on the magnitude and distribution to layer is stepwise. Moreover, the first and last layers are a of residual stresses was conducted. As shown later in this ceramic and a metal, respectively(Figure 1), and the multi paper, such a systematic approach was necessary, since it layers in between comprises metal-ceramic dual phase facilitated the development of a comprehensive under- microstructure with isotropic properties. The volume fraction standing of the magnitude and distribution of residual of the metal component in the kth layer, [Vm kk, is described by stresses as a function of the compositional gradient, a a power-law, similar to that proposed by Drake' combination of the constituents properties, and the number of layers in the case of multi-layered materials. Further, it [VmI led to the prediction of some interesting residual stress distributions, such as residual compressive stresses on both faces of the plate, a highly desirable residual stress state for Ceramic face any applications. Thus, desired states and magnitudes of residual stresses can be obtained through proper selection of material combinations and compositional gradients based h2(z=0.5 on the guidelines developed in this study In this paper, the terms 'compositionally graded materialsand composition, while multi-layered materials refers to materia Figure 1 Geometry of multi-layered plates
Thermal residual stresses in plates and coatings: L. L. Shaw have been made to develop these understandings and abilities. For example, the research group in MIT led by Suresh 6-1° has extensively investigated thermal residual stresses in Ni-A1203 compositionally graded plates,* including the consideration of plastic deformation induced by thermal cycling. Elastoplastic analyses for A1-Si, In0.12Ga0.ssAs-GaAs and carbon-epoxy systems subjected to temperature excursions have also been conducted by the same group 11. Residual stresses in multi-layered and compositionally graded plates have also been investigated by Lannutti et al. 12'13 for NiA1-A1203 systems, by Freund 14 for graded semiconductor layers, by Tang et al. 15 for PSZ-Mo systems, by Ravichandran t6 for Ni-A1203 systems, and by Itoh and Kashiwaya 17 for material systems with different combinations of Young' s modulus and a linear compositional gradient. In addition to plate geometries, coating and joint geometries have also been investigated. These include the work conducted by Lee and Erdogan ~s for ZrO2 coatings on Ni-superalloy substrates, by Itoh and Kashiwaya 17 for both coatings and joints, and by Drake et al. t9 and Williamson et al. 2°'21 for joint geometries. Other geometries with graded compositions have also been investigated. These include a graded interface layer between the fiber and matrix analyzed by Rabzak et al. 22, Pindera et al. 23 and Salzar and Barton 24, a graded matrix bonded to a cylindrical fiber investigated by Duva et al. 25, and a graded fiber concentration in a matrix studied by Aboudi et al. 26. Although studies on the magnitude and effects of thermal residual stresses in FGM with various geometries and compositional gradients have been performed extensively as summarized in a recent review 27, systematic examination of the effect of different material combinations has not been explored. With this in mind, this work utilized laminate theory to analyze thermal residual stresses in multi-layered and compositionally graded plates and coatings. A comprehensive study on the effects of different material combinations including the compositional gradient, the elastic modulus, the coefficient of thermal expansion (CTE), and the number of layers in the case of multi-layered materials on the magnitude and distribution of residual stresses was conducted. As shown later in this paper, such a systematic approach was necessary, since it facilitated the development of a comprehensive understanding of the magnitude and distribution of residual stresses as a function of the compositional gradient, a combination of the constituent' s properties, and the number of layers in the case of multi-layered materials. Further, it led to the prediction of some interesting residual stress distributions, such as residual compressive stresses on both faces of the plate, a highly desirable residual stress state for many applications. Thus, desired states and magnitudes of residual stresses can be obtained through proper selection of material combinations and compositional gradients based on the guidelines developed in this study. * In this paper, the terms 'compositionally graded materials' and 'functionally graded materials' are interchangeable and have a continuously changed composition, while 'multi-layered materials' refers to materials with a stepwise composition The limitation of laminate theory is that the theory is valid only when the thickness of the plate is significantly smaller than the lateral dimensions. Furthermore, stress distributions predicted from the theory represent the stresses away from the free edges of the plate. At the edges of the plate where the interfaces of bonded dissimilar materials intersect the free surface, the tensile stress component and the shearing stress component perpendicular to the interface are singular 17"18'28. It is well established that both of these stress components decrease to zero quickly away from the free edge 17'18 and the width of the region in which the stresses differ from those predicted by laminate theory is approximately equal to the thickness of the plate 28. Furthermore, it has been shown that the stress singularities at the free edge are eliminated if the compositional gradient of the plate or coating are continuous rather than stepwise Is. Thus, the present analysis is only applicable to the stress states of thin plates away from the free edges. Nevertheless, stress analyses of this type are still highly desirable, since in many cases it is these stresses that determine the behavior of the material. One typical example of these would be functionally graded armor tiles. In this case, when a high speed projectile impacts the central portion of the tile, the stress states at the central portion rather than at the free edges will play a critical role in the response of the tile to the impact. DESCRIPTION OF PROBLEMS AND FORMULAE Multi- laye red plates The geometry of the residual stress problem for multilayered plates is shown in Figure 1. Throughout this study, multi-layered and compositionally graded plates are considered to be composed of a ceramic and a metal, and are assumed to be infinite in the x- and y-directions. Further, with no loss of generality, plates are assumed to have a unit thickness. Layers within multi-layered plates are assumed to have equal thickness and the composition change from layer to layer is stepwise. Moreover, the first and last layers are a ceramic and a metal, respectively (Figure 1), and the multilayers in between comprises metal-ceramic dual phase microstructure with isotropic properties. The volume fraction of the metal component in the kth layer, [Vm]k, is described by a power-law, similar to that proposed by Draket9: [Vmlk= N- 1 N- 1 (1) Ceramic face ./ ( 1st la~.,~ -13/2 (z = -0. Nlayers f ~ OV/A~Jy" Metal face ~ J Z ~h/2 (z = 0.5) (last layer) y Figure 1 Geometry of multi-layered plates 200
Thermal residual stresses in plates and coatings: L.L. Shaw where N'and M' are the thermal stress and moment resultants that are defined as IN1=L(o')dz T 0.504-0.3-0.2-0.100.10203040.5 o}=Q]{a}△T Location, z The A, B and D in eqn(4) are the stiffness matrices of the 后应改= [e]dz where k refers to the kth layer of the plate defined in Figure I, N is the total number of layers within the plate, and m is an arbitrary exponent that controls the composi- tional gradient, either linear or non-linear. Composition profiles for m values of 0. 1, 0. 2, 0.5, 1.0, 2.0 and 10.0 are =-M2Le12dz shown in Figure 2. As for the material properties of each layer within the plate, the rule of mixtures is applied and only elastic deformation is considered. Further, the interlayer bond is Plates with functionally graded compositions assumed to be perfect, and, therefore, there is no slippage between layers. Temperature is assumed to be uniform Compositionally graded plates have the same geometry throughout the plate and thermal residual stresses for a as the multi-layered plate shown in Figure The cooldown from 1000C(the stress-free state)to 25 C are compositional gradient is again described by a power-law studied with no consideration of stress relaxation during similar to eqn(I)but continuous in nature cooling. The thermal residual stresses at any position throug the thickness z, o, are calculated using laminate theory vm(x)=(x+05)m {a}=[]e}=[°)+x(k}-{a△T)(2) where Vm(z)is the volume fraction of the metal component where [@l is the stiffness matrix of material at position z, at the location of z, h is the thickness of the plate, and m is [e) the thermal residual strains at position z, e) the mid the same as in eqn (1). Comparing eqn (1) and eqn(8) plane strains, ( k) the plate curvatures, fa) the coefficients ndicates that when m is chosen to be the same for both of thermal expansion of material at position z, AT the tem equations, the compositional gradients will be the same perature change, and the notation with the only difference being a continuous gradient for dier he composition profiles (3) several m values can be found in Figure 2. Residual stresses in FGM are computed using eqns(2)-(7) directly with the only difference that material properties are continuous for with the subscript xy denoting the x, y, z coordinate system FGM. while they are stepwise for multi-layered plates sed. The same rule of the notation is applied to the thermal residual strains, the midplane strains, the plate curvatures, nd the coefficients of thermal expansion in eqn(2).For Mulri-layered and compositionally graded coatings plates with properties of each layer being isotropic as con- The geometry of the coating problem under consideration sidered in this paper, the only non-zero stress components is defined in Figure 3. In computation the coating and are o and o ,(=0). The midplane strains and curvatures substrate are assumed to be infinite at the x- and y-directions of the plate in eqn(2) are determined from (not shown in Figure 3). The thickness ratio of the substrate AB1「E° to the coating is fixed to be 100: 1 for both multi-layered (4) and compositionally graded coatings. The residual stresses are evaluated using eqns(2)-(7)
Thermal residual stresses in plates and coatings: L. L. Shaw _ I I m`v =i~:: .......... v~ .......... ri .......... : ~ ~ ~ ~- 0.8 ........ ........ o 0.6 i ........... i ......... 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Location, z Figure 2 Compositional gradients in functionally graded plates for several values of the gradient exponent m. Multi-layered plates with the same values of m as the functionally graded plates will have similar compositional gradients as shown, except that their gradients are stepwise in nature where k refers to the kth layer of the plate defined in Figure 1, N is the total number of layers within the plate, and m is an arbitrary exponent that controls the compositional gradient, either linear or non-linear. Composition profiles for m values of 0.1, 0.2, 0.5, 1.0, 2.0 and 10.0 are shown in Figure 2. As for the material properties of each layer within the plate, the rule of mixtures is applied and only elastic deformation is considered. Further, the interlayer bond is assumed to be perfect, and, therefore, there is no slippage between layers. Temperature is assumed to be uniform throughout the plate and thermal residual stresses for a cooldown from 1000°C (the stress-free state) to 25°C are studied with no consideration of stress relaxation during cooling. The thermal residual stresses at any position through the thickness z, { o}, are calculated using laminate theory 29: {o-} = [Q]{e} = [(Q]{{e °} +z{r} - {a}AT} (2) where [~)] is the stiffness matrix of material at position z, {e} the thermal residual strains at position z, {e °} the midplane strains, {K} the plate curvatures, {c~} the coefficients of thermal expansion of material at position z, AT the temperature change, and the notation {O} = O-y (3) with the subscript xy denoting the x, y, z coordinate system used. The same rule of the notation is applied to the thermal residual strains, the midplane strains, the plate curvatures, and the coefficients of thermal expansion in eqn (2). For plates with properties of each layer being isotropic as considered in this paper, the only non-zero stress components are o~ and o~ (= O-x). The midplane strains and curvatures of the plate in eqn (2) are determined from [N~] = [A B][ef] (4) where N r and M r are the thermal stress and moment resultants that are defined as f - h/2 {NT} ---- Jhl2 {0 "T} dz (5) f - h/2 {Mr} = Jh/2 {o-r}Z dz with { o r } given by {t7 T } = [QI{ot}AT (6) The A, B and D in eqn (4) are the stiffness matrices of the plate and are defined as (h/2 _ A = j_h/2[Q] dz (7) (h/2 _ B = J_ h/2 [Q]z dz Fh/2 D--- J_h/2[O_]Z2 dz Plates with functionally graded compositions Compositionally graded plates have the same geometry as the multi-layered plate shown in Figure 1. The compositional gradient is again described by a power-law, similar to eqn (1) but continuous in nature: where Vm(z) is the volume fraction of the metal component at the location of z, h is the thickness of the plate, and m is the same as in eqn (1). Comparing eqn (1) and eqn (8) indicates that when m is chosen to be the same for both equations, the compositional gradients will be the same with the only difference being a continuous gradient for compositionally graded plates and a stepwise gradient for multi-layered plates. The composition profiles for several m values can be found in Figure 2. Residual stresses in FGM are computed using eqns (2)-(7) directly with the only difference that material properties are continuous for FGM, while they are stepwise for multi-layered plates. Multi-layered and compositionally graded coatings The geometry of the coating problem under consideration is defined in Figure 3. In computation the coating and substrate are assumed to be infinite at the x- and y-directions (not shown in Figure 3). The thickness ratio of the substrate to the coating is fixed to be 100:1 for both multi-layered and compositionally graded coatings. The residual stresses are evaluated using eqns (2)-(7). 201
Thermal residual stresses in plates and coatings: L. L. Shaw (Table 1)with various compositional gradients are shown Material properties for each pair of the metal and ceramic in Figure 4. It is quite clear from Figure 4 that thermal forming multi-layered and compositionally graded plate residual stresses are a strong function of the compositional and coatings studied are summarized in Table 1. The gradient, Generally speaking, minimum residual stresses selection of material systems for study is dictated by are obtained when m=1.0(i.e for a linear compositional the following considerations: (1)to examine the effects of gradient). When m >1.0, both surface regions of the plate elastic moduli at constant CTE;(2)to study the effects (1.e. the high metal and the high ceramic regions) have of cte at constant elastic moduli: and to relate the residual tensile stresses, while for ms 1.0, both surfac results to real material systems whenever possible. As such. regions of the plate are in compression. Furthermore, as m becomes increasingly larger than 1. 0, residual stresses at metal/ceramic systems that have equivalent properties o both surface regions become more tensile. Similarly, as m the parameters evaluated are pointed out in the table, as shown within parentheses becomes increasingly smaller than 1.0, residual stresses at both face regions become more compressive. This is true for m within a range of 0. 2. However, when m becomes RESULTS extremely large(m= 10.0)or extremely small (m=0. 1) (Figure 4(d), the compressive or tensile stresses on one of the surface regions continue to increase, while on the other The in-plane residual stresses, O,(note that 0, =0),of urface region these stresses start to decrease. Finally, it is multi-layered plates with 51 layers for material system VI noted that since thermal residual stresses within the plate should be balanced among themselves, the interior of the plate will be in compression, when both surface regions are in tension. Similarly, when both surface regions are in compression, the interior of the plate will be in tension The midplane strains and curvatures of multi-layere plates(51 layers)as a function of the compositional gradient are shown in Figure 5. As expected, the midplane strains increase monotonically as m decreases. This is because Metal substrate decreasing m leads to an increase in the metal content in the multi-layered plate, thereby producing more shrinkage of Figure 3 Geometry of the coating problem the plate during cooldown. The maximum curvature of the able 1 Material properties of the metal and ceramic components forming FGM with m=0.5 and their calculated residual stresses, plate curvatures and midplane strains for AT=1000'C* Material: CTi/SIC) (Ni/TIO,) Em(GPa) 110 145 m(10-°C) 150 10.3 15.0 10.3 10 3.28 am+axy2(10°C 975 9.75 1195 l195 K(10-3) 8.880 8996 5226 4.969 -5.159 5.040 1.151 7450 947.5 10140 1254 face(om) E is the elastic modulus, a is the coefficient of thermal expansion(CTE), subscripts m and c refer to the metal and ceramic respecitvely, and Aa is the Cte mismatch. i.c. a t The material systems that have equivalent properties to the parameters evaluated are presented in parentheses Table 2 Plate curvatures, midplane strains, and residual stresses in the first and last layers of the multi-layered plates for material system VI with a linear omposition gradient (AT= 1000.C) Number of layers 6 11 5 e"(10-) -0.11869 0.11936 0.11946 0.11950 0.11950 0.11950 -0.11950 a, in the Ist layer 115 3. l86 -124 6.3 0.6 See the text for explanations for two stress values in each layer 02
Thermal residual stresses in plates and coatings: L. L. Shaw Materials properties studied Material properties for each pair of the metal and ceramic forming multi-layered and compositionally graded plates and coatings studied are summarized in Table 1. The selection of material systems for study is dictated by the following considerations: (1) to examine the effects of elastic moduli at constant CTE; (2) to study the effects of CTE at constant elastic moduli; and (3) to relate the results to real material systems whenever possible. As such, metal/ceramic systems that have equivalent properties to the parameters evaluated are pointed out in the table, as shown within parentheses. RESULTS Multi-layered plates The in-plane residual stresses, ox (note that ay = ax), of multi-layered plates with 51 layers for material system VI -h/2 Ceranuc face ~ Multi-layered or (z = -50.5) ~ eompositionally 0 ~ x h/~0.5 )_ Metal substrate (z z Figure 3 Geometry of the coating problem (Table 1) with various compositional gradients are shown in Figure 4. It is quite clear from Figure 4 that thermal residual stresses are a strong function of the compositional gradient. Generally speaking, minimum residual stresses are obtained when m---- 1.0 (i.e. for a linear compositional gradient). When m > 1.0, both surface regions of the plate (i.e. the high metal and the high ceramic regions) have residual tensile stresses, while for m < 1.0, both surface regions of the plate are in compression. Furthermore, as m becomes increasingly larger than 1.0, residual stresses at both surface regions become more tensile. Similarly, as m becomes increasingly smaller than 1.0, residual stresses at both face regions become more compressive. This is true for m within a range of 0.2-5.0. However, when m becomes extremely large (m= 10.0) or extremely small (m =0.1) (Figure 4(d)), the compressive or tensile stresses on one of the surface regions continue to increase, while on the other surface region these stresses start to decrease. Finally, it is noted that since thermal residual stresses within the plate should be balanced among themselves, the interior of the plate will be in compression, when both surface regions are in tension. Similarly, when both surface regions are in compression, the interior of the plate will be in tension. The midplane strains and curvatures of multi-layered plates (51 layers) as a function of the compositional gradient are shown in Figure 5. As expected, the midplane strains increase monotonically as m decreases. This is because decreasing m leads to an increase in the metal content in the multi-layered plate, thereby producing more shrinkage of the plate during cooldown. The maximum curvature of the Table 1 Material properties of the metal and ceramic components forming FGM with m = 0.5 and their calculated residual stresses, plate curvatures and midplane strains for AT = 1000°C * t Material: I II III IV V VI VII VIII System 1D: (Ni/SiC) (Ti/SiC) (Ni/TiO 2) (Ti/Y20 3) Em (GPa) 145 110 110 110 145 145 110 145 Ec (GPa) 475 475 475 475 475 290 170 290 a m (10-6/°C) 15.0 15.0 15.0 10.3 15.0 15.0 10.3 10.3 oLc (10 6/~C) 4.5 4.5 8.9 4.5 8.9 8.9 8.3 8.3 Ec/Em 3.28 4.32 4.32 4.32 3.28 2.0 1.55 2.0 A~ (10-6/°C) 10.5 10.5 6.1 5.8 6.1 6.1 2.0 2.0 (~m + Ore)/2 (10 6/°C) 9.75 9.75 11.95 7.4 11.95 11.95 9.3 9.3 K (10 -3) --8.880 --8.996 --5.226 --4.969 --5.159 --5.040 --1.632 0 (10-2) -- 1.151 -- 1.153 -- 1.298 --0.838 -- 1.297 -- 1.296 --0.963 ax on the ceramic face (ac) --1745.0 --1715.0 --996.5 --947.5 -1014.0 --639.9 --125.4 ax on the metal face (am) --197.0 -160.8 -93.4 --88.8 --114.3 --100.3 -23.3 * E is the elastic modulus, ~ is the coefficient of thermal expansion (CTE), subscripts m and c refer to the metal and ceramic respecitvely, and Ac~ is the CTE mismatch, i.e. am - ac t The material systems that have equivalent properties to the parameters evaluated are presented in parentheses Table 2 Plate curvatures, midplane strains, and residual stresses in the first and last layers of the multi-layered plates for material system VI with a linear composition gradient (AT = 1000°C) Number of layers 3 6 11 51 101 201 991 (10 -2 ) -0.8062 -0.7106 -0.6652 -0.6219 -0.6160 -0.6130 -0.6106 0 (10-1) -0.11869 -0.11936 -0.11946 -0.11950 -0.11950 -0.11950 -0.11950 ox in the 1st layer* 439.7 214.1 115.8 24.8 12.5 6.29 1.3 -673.5 -276.5 - 134.7 -25.7 - 12.7 -6.34 - 1.3 ax in the last layer 370.2 144.0 68.0 12.9 6.4 3.2 0.6 - 186.5 - 101.3 -56.4 - 12.4 -6.3 -3.1 -0.6 * See the text for explanations for two stress values in each layer 202
Thermal residual stresses in plates and coatings: LL. Shaw (a) m=0.5 m=2.0 TTTT 05-04-03“02 3040.5 102030405 ocation. z 200 =0.2 m=0.1 1600 0.5-04-0.3-02 10.20.3 Location z cation, Z Figure 4 The in-plane residual es, a of multi-layered plates with 51 layers for material system VI. The compositional gradient exponents are(a) m=1.0,(b)m=0.5andm=2.0,(c)m=0.2andm=5.0,and(d)m=0. I and n=10.0 material system VI with a linear compositional gradient. Two stress values are listed for both the first layer and the -0.011 last layer since there is a stress gradient in every layer of the plate(see Figure 4)and the two values listed correspond -0,012 on the upper and lower faces of the .005 0.013 consideration It is noted that the absolute value of residual stresses decreases monotonically as the number of layers 0.014 increases, so does the warping of the plate, The midplane 0015 strain increases as the number of layers increases from 3 to 6081 51 layers and it levels off as the number of layers continues Log m to increase. The change in the midplane strain is believed to be associated with the amount of the metal content in the strain and the plate curvature as a function of the plate. For a plate with a linear compositional gradient, its positional gradient in a multi-layered plate with 51 layers for material system VI thereby resulting in an increase in the midplane strain However, when the number of layers is large enough(e. g ease in the plate is obtained when the compositional gradient is linear does not change significantly the metal content in the plate (. e. m=1.0). The warping of the plate decreases when and, therefore, the midplane strain becomes more or less m becomes increasingly larger or smaller than 1.0. This independent of the number of layers as the layer number sult can also be explained in terms of the change in exceeds 51 the compositional gradient and will be discussed in the When the compositional gradient is non-linear (Table 3), it is found that the rule found for the curvature of plates with Thermal residual stresses are found to be strongly m=1.0 still holds, that is, the warping of plates decreases dependent on the number of layers in the plate Table 2 as the number of layers increases. However, the midplane ummarizes the plate curvatures, the midplane strains, strain and residual stresses are found to vary differently residual stresses in the first layer(the ceramic face)and the from the case of m =1.0. As for the midplane strain, it is last layer(the metal face)of the multi-layered plate for found that when the number of layers increases, the
Thermal residual stresses in plates and coatings: L. L. Shaw #. ~E D ~E 100- (a)i : :: i n"l = 1 i O, -SO .................................................... i E -1oo ...... m , -O.S -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Location, z 1600-.ii!i;i .... .... I 12001 (c) 4 • 800--1_ ~ ~ ~ • i m=0.2 ,oo- ~i ¸ ....... i w~ o- .............. -400 ............... i': .... " 11i~i ........ ~i -800 -1200 - 1 600 -0.S -0.4 -0.3 -0.2 -0.1 0 0.1 Location, z ......... : ..... !m=5.0 ~ ~ :. : i : : { : ..... ! i ' I ' ' I ' ' I ' I ' 0.2 0.3 0.4 0.S ~E D ~E 800- 600~ 400 ~ 200- 0- -200 -400 -600 -800" i i i ..... d j il .... i ' m;0.si i i J ! ..................... • I ' ' I ' l " ' ' I ' ' ' -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Location, z 2000~ isoo~ : i Iooo~ ~ , i , soo: o~ -soo~ -1000" -1500 -2000 -0.S -0.4 -0.3 -0.2 -0.1 (d) : ......... " i i ...... i i l;i°,i ...... i .......... ~m=1°'°i~ .... ii " i~ ........ ' I ' I ' ' I ' I ' I ' ' 0.1 0.2 0.3 0.4 O.S Location, z Figure 4 The in-plane residual stresses, a x of multi-layered plates with 51 layers for material system VI. The compositional gradient exponents are (a) m= 1.0, (b)m = 0.5 and m= 2.0, (c) m=0.2 and m=5,0, and (d) m=0.1 and m= 10.0 -O.OOl -0.002 ~o -0.003 -0.004 -0.005 E -0.006 -0.007 • I ' I " I ' I ' l " I ' I ' I ' I ' 1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 Log m -0.009 -0.01 -0.011 -0.012 -0.013 -0.014 -0.015 g a~ Figure 5 The midplane strain and the plate curvature as a function of the compositional gradient in a multi-layered plate with 51 layers for material system VI plate is obtained when the compositional gradient is linear (i.e. m----1.0). The warping of the plate decreases when m becomes increasingly larger or smaller than 1.0. This result can also be explained in terms of the change in the compositional gradient and will be discussed in the Discussion section. Thermal residual stresses are found to be strongly dependent on the number of layers in the plate. Table 2 summarizes the plate curvatures, the midplane strains, residual stresses in the first layer (the ceramic face) and the last layer (the metal face) of the multi-layered plate for material system VI with a linear compositional gradient. Two stress values are listed for both the first layer and the last layer since there is a stress gradient in every layer of the plate (see Figure 4) and the two values listed correspond to stresses on the upper and lower faces of the layer under consideration. It is noted that the absolute value of residual stresses decreases monotonically as the number of layers increases, so does the warping of the plate. The midplane strain increases as the number of layers increases from 3 to 51 layers and it levels off as the number of layers continues to increase. The change in the midplane strain is believed to be associated with the amount of the metal content in the plate. For a plate with a linear compositional gradient, its metal content increases as the layer number increases, thereby resulting in an increase in the midplane strain. However, when the number of layers is large enough (e.g. 51 layers), a continuous increase in the number of layers does not change significantly the metal content in the plate and, therefore, the midplane strain becomes more or less independent of the number of layers as the layer number exceeds 51. When the compositional gradient is non-linear (Table 3), it is found that the rule found for the curvature of plates with m = 1.0 still holds, that is, the warping of plates decreases as the number of layers increases. However, the midplane strain and residual stresses are found to vary differently from the case of m = 1.0. As for the midplane strain, it is found that when the number of layers increases, the 203
Thermal residual stresses in plates and coatings: L.L. Shaw 导 ::g TTTTTTT MErv 79荡 m0.5 400 600 0504030.2-0.100.102030405 for material system VI. The compositional gradient exponents are(a) m=0.1,0.2,50and10.0,and(b)m=0.5.10and2.0 midplane strain increases monotonically for m=1.0, while I IN9 it decreases monotonically for m>1.0. These results are consistent with the change in the metal volume fraction in the plate, since for the plate with m 1.0, the more layers it has, the less metal is present. As for residual stresses on surface regions, it is found that residual stresses do not decrease monotonically as the number of layers increases. Instead, in most of the cases residual stresses increase on one face of the plate and 宁7源 decrease on the other face. Finally, it should be pointed out that a common feature for all multi-layered plates is that the stress gradient within each layer decreases as the number of layers increases, and that the stress difference between /F799 neighboring layers also becomes smaller as the number of layers increase cmeI graded plates for material system VI with various composi tional gradients are shown in Figure 6. It is noted that FGM ave very similar residual stress distributions and similar dependence of plate curvatures and midplane strains on the compositional gradient to those found with multi-layered 岩岩 plates. Specifically, minimum residual stresses are obtained when m=1.0. When m >1.0, both surface regions of the plate are in tension, 204
Thermal residual stresses in plates and coatings: L. L. Shaw LJ o F~ t- O . r. E t. r~ k~ .=. % == p E mb @. tt~ cJ tth (-q tt~ ~q =_ E z II II g~ ddM~ r( % D 1500 .............. ? .................................................... i ................................ - ...... (a) looo ......... ~ ................................................ ........................ ~ ...... .................................... ................. i i i -500"." -1000 i -I 500 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Location, z ,4 8°°-"(b)~ ......... ........ ........... ........ .......... ............ T ........ ............. 6oo-~ ......... ........... ; ....... i .......... ....... ..... ........... i ..... ! ............ ~ ...... I . ............ .......... ......... ..... .......... , 400"~ ....... "" 1.0 2.0 -6oo 1 .............. ........... " ....... ........ ........... i ............ -8OOl . • i ' ,' ; ' , . , • i • . , • -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Location, z Figure 6 The in-plane residual stresses, ax of compositionally graded plates for material system VI. The compositional gradient exponents are (a) m=0.1, 0.2, 5.0 and 10.0, and (b) m=0.5, 1.0 and 2.0 midplane strain increases monotonically for m = 1.0, while it decreases monotonically for m > 1.0. These results are consistent with the change in the metal volume fraction in the plate, since for the plate with m 1.0, the more layers it has, the less metal is present. As for residual stresses on surface regions, it is found that residual stresses do not decrease monotonically as the number of layers increases. Instead, in most of the cases residual stresses increase on one face of the plate and decrease on the other face. Finally, it should be pointed out that a common feature for all multi-layered plates is that the stress gradient within each layer decreases as the number of layers increases, and that the stress difference between neighboring layers also becomes smaller as the number of layers increases. Compositionally graded plates The in-plane residual stresses, ox, of compositionally graded plates for material system VI with various compositional gradients are shown in Figure 6. It is noted that FGM have very similar residual stress distributions and similar dependence of plate curvatures and midplane strains on the compositional gradient to those found with multi-layered plates. Specifically, minimum residual stresses are obtained when m ---- 1.0. When m > 1.0, both surface regions of the plate are in tension, whle they become in compression 204
Thermal residual stresses in plates and coatings: L.L. Shaw different cor -0002 Due to a smaller CtE than that of the metal substrate, every layer of the coating has a residual compressive stress with -0011 the largest stress in the top layer. In addition, the residual stresses within the substrate are very small. This is no -0.012 surprise, since the volume ratio of the substrate to the substrate and is very close to the shrinkage 0. of the substrate without any coatings. This large volume 0.007千 -0015 ratio also leads to an extremely small plate curvature, since 0806040.2002040.6081 the bending moment produced by the thermal mismatch between the coating and substrate is extremely small. Because of the extremely small plate curvature, the substrate are insensitive to the number of layers(not shown when m< 1.0. The midplane strain decreases with increas- in Figure 8). Furthermore, these residual stresses are found ing compositional exponent m, while the plate curvature to be primarily dependent on the CTE mismatch between displays a maximum at m=1.0(Figure 7). Furthermore, the substrate and the specific layer, and the effect of it is found that the magnitude of the residual stresses, the neighboring layers within the coating on these stresses is midplane strains, and the plate curvatures of FGM are similar very small. For example, the residual compressive stress in to that of multi-layered plates with 991 layers, suggesting the top layer of the coating(which is 100% ceramic)varies that FGM are the limiting cases for multi-layered plates from -2348 to-2497 MPa for m changing from 100to Residual stresses, plate curvatures and midplane strains 0. 1; that is -6% increase in the residual stress when the of fgm for m=0.5 and with different combinations of compositional gradient exponent, m, has changed 100 times material properties are summarized in Table 1. In the table. Similar results are obtained if a layer with -44% metal not only the coefficients of thermal expansion and the elastic is examined. In this case, the residual stress in the layer is moduli used for computation are listed, but the difference about -1000 MPa for different m values regardless where between the metal,s and ceramics cte and the modulus these layers are located and what the composition of its ratio of the ceramic to the metal are also included Several neighboring layer is(Figure 8) features have been noted: (a)as expected, for a constant As for compositionally graded coatings, the same results combination of elastic moduli. residual stresses increase as multi-layered coatings are obtained; these are: (1)the with increasing Aa; (b) for a constant Ao, residual stresses midplane strain is dominated by the substrate; (2)the plate increase with increasing elastic modulus of the ceramic curvature is very small; and (3) residual stresses within metal,or both;(c)as expected, the midplane strain is mainly the coating are insensitive to the compositional gradient and determined by the metals and ceramics CTE, and is nearl dependent of the elastic modulus of the metal and substrate's CtE and the coating materials CTE at each ceramic. In fact, it is found that the magnitude of the specific location within the coating. The only difference midplane strain can be ranked in terms of the order of is that residual stresses are continuously changed from the (ac+am)2. The larger the (a.+am)/2 is the more substrate to the top surface of the coating for composition- midplane strain a FGM has; and (d) as expected, the ally graded coatings, while these stresses are stepwise curvature of the plate is affected by Aa; the larger Aa is, multi-layered coatings the more warping the plate has. The warping of the plate is also influenced by the elastic modulus of the ceramic and IMPLICATIONS AND DISCUSSION metal. At constant Ao, the warping increases as the ratio of the ceramic's modulus to the metal's modulus increases. Control of thermal residual stresses in plates Finally, it should be pointed out that most of the behavior described above for Fgm with different combinations of One of the important results of the present analysis is that material properties for m=0. 5 also holds for m=2.0 and the distribution and magnitude of thermal residual stresses m=1.0(not shown in Table D). The only difference for within a plate can be adjusted by controlling the compo- m=2.0 is that residual stresses on the metal and ceramic sitional gradient and selecting a proper combination of es are both tensile, while for m=l o residual stresses on ceramic and metal constituents. In many applications a the both faces are nearly zero no matter what combinations multi-layered or a compositionally graded plate with of material properties are a compressive residual stress on both faces of the plate is highly desirable, while in other cases a minimum residual Multi-layered and compositionally graded coatings stress in the plate is desired. This study clear indicates that the former can be achieved by designing a plate with a The in-plane residual stresses, or, in the coating and the compositional gradient exponent m 1.0, whereas the latter substrate for multi-layered coatings(10 layers)with several can be obtained by selecting a plate with a linear 205
Thermal residual stresses in plates and coatings: L. L. Shaw -0.001 -0.002 -0.003 m -0.004 ,,.,9 -0.005 -0.006 -0.007 -0.009 -0.01 -0.011 -0.o12 -o.o13 _a -0.014 -o.015 1 -0.8-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Log m Figure 7 The midplane strain and the plate curvature as a function of the compositional gradient in a functionally graded plate for material system VI when m < 1.0. The midplane strain decreases with increasing compositional exponent m, while the plate curvature displays a maximum at m = 1.0 (Figure 7). Furthermore, it is found that the magnitude of the residual stresses, the midplane strains, and the plate curvatures of FGM are similar to that of multi-layered plates with 991 layers, suggesting that FGM are the limiting cases for multi-layered plates. Residual stresses, plate curvatures and midplane strains of FGM for m = 0.5 and with different combinations of material properties are summarized in Table 1. In the table, not only the coefficients of thermal expansion and the elastic moduli used for computation are listed, but the difference between the metal's and ceramic's CTE and the modulus ratio of the ceramic to the metal are also included. Several features have been noted: (a) as expected, for a constant combination of elastic moduli, residual stresses increase with increasing Ac~; (b) for a constant Ac~, residual stresses increase with increasing elastic modulus of the ceramic, metal, or both; (c) as expected, the midplane strain is mainly determined by the metal's and ceramic's CTE, and is nearly independent of the elastic modulus of the metal and ceramic. In fact, it is found that the magnitude of the midplane strain can be ranked in terms of the order of (C~c+am)/2. The larger the (O~c+Otm)/2 is, the more midplane strain a FGM has; and (d) as expected, the curvature of the plate is affected by As; the larger As is, the more warping the plate has. The warping of the plate is also influenced by the elastic modulus of the ceramic and metal. At constant Ac~, the warping increases as the ratio of the ceramic's modulus to the metal's modulus increases. Finally, it should be pointed out that most of the behavior described above for FGM with different combinations of material properties for m = 0.5 also holds for m = 2.0 and m = 1.0 (not shown in Table 1). The only difference for m = 2.0 is that residual stresses on the metal and ceramic faces are both tensile, while for m = 1.0 residual stresses on the both faces are nearly zero no matter what combinations of material properties are. Multi-layered and compositionally graded coatings The in-plane residual stresses, ax, in the coating and the substrate for multi-layered coatings (10 layers) with several different compositional gradients are shown in Figure 8. Due to a smaller CTE than that of the metal substrate, every layer of the coating has a residual compressive stress with the largest stress in the top layer. In addition, the residual stresses within the substrate are very small. This is no surprise, since the volume ratio of the substrate to the coating is very large. Because of this, the midplane strain is dominated by the substrate and is very close to the shrinkage of the substrate without any coatings. This large volume ratio also leads to an extremely small plate curvature, since the bending moment produced by the thermal expansion mismatch between the coating and substrate is extremely small. Because of the extremely small plate curvature, the magnitudes of the residual compressive stresses in each layer of the coating and the residual stresses within the substrate are insensitive to the number of layers (not shown in Figure 8). Furthermore, these residual stresses are found to be primarily dependent on the CTE mismatch between the substrate and the specific layer, and the effect of neighboring layers within the coating on these stresses is very small. For example, the residual compressive stress in the top layer of the coating (which is 100% ceramic) varies from -2348 to -2497 MPa for m changing from 10.0 to 0.1; that is --6% increase in the residual stress when the compositional gradient exponent, m, has changed 100 times. Similar results are obtained if a layer with --44% metal is examined. In this case, the residual stress in the layer is about -1000 MPa for different m values regardless where these layers are located and what the composition of its neighboring layer is (Figure 8). As for compositionally graded coatings, the same results as multi-layered coatings are obtained; these are: (1) the midplane strain is dominated by the substrate; (2) the plate curvature is very small; and (3) residual stresses within the coating are insensitive to the compositional gradient and are primarily determined by the mismatch between the substrate's CTE and the coating material's CTE at each specific location within the coating. The only difference is that residual stresses are continuously changed from the substrate to the top surface of the coating for compositionally graded coatings, while these stresses are stepwise in multi-layered coatings. IMPLICATIONS AND DISCUSSION Control of thermal residual stresses in plates One of the important results of the present analysis is that the distribution and magnitude of thermal residual stresses within a plate can be adjusted by controlling the compositional gradient and selecting a proper combination of ceramic and metal constituents. In many applications a multi-layered or a compositionally graded plate with a compressive residual stress on both faces of the plate is highly desirable, while in other cases a minimum residual stress in the plate is desired. This study clear indicates that the former can be achieved by designing a plate with a compositional gradient exponent m < 1.0, whereas the latter can be obtained by selecting a plate with a linear 205
Thermal residual stresses in plates and coatings: L. L. Shaw 2000 2500 49.5 Location, z (b) 1500 m=0.5 in the coating m= 2.0 in the coating -2500 50.5 50.5 -49.5 Location. z ocation. z Figure 8 The in-plane residual stresses, ox, in the coating and the substrate for multi-layered coatings(10 layers) with(a)m=l0. (b)m=0.5, and(c) 2.0. Note that only a small portion of the substrate is show Signs of thermal residual stresses in plates 军+官 Minimum The distribution of thermal residual stresses, the curva- ture and midplane strain of the plate can be understood through a schematic shown in Fig →巨目 changes of multi- layered plates for three different composi tional gradients during cooldown with and without con- straints from neighboring layers are sketched. For a linear compositional gradient (Figure 9(a)), if there were no e metal face having the largest shrinkage and the ceramic face the smallest shrinkage. When the constraints from neighboring layers are present, the different shrinkage tendencies are accommodated through warping of the plate. oution of thermal residual stresses. Dashed lines represent the As such, very small in-plane residual stresses are present, especially when the number of layers is large. When m<1.0, i.e. more metal content in the plate(Figure 9(b)), the shrinkage tendency of each layer without the constraints from neighboring layers will lead to a situation that a large compositional gradient, 1. e. m=1.0. In fact, to minimize different shrinkage tendency is present at the ceramic rich residual stresses, not only a linear compositional gradient region. Thus, for a bonded plate a residual compressive should be chosen, but the layers within the plate should stress is created on the ceramic face since the neighboring also be as many as possible if the plate is multi-layered. high metal content layers will apply a compressive stress Finally, it would be useful to point out that m<1.0 to the high ceramic content region and leave themselves in corresponds to the situation where a more rapid change tension. Another result from this compositional gradient is in volume fraction and properties occurs near the a larger midplane strain than that of the plate in Figure 9(a), ceramic face, and more gradual changes occur near since more me tal content is present in the plate with the metal face. m <1.0. The plate curvature is also expected to become 206
Thermal residual stresses in plates and coatings: L. L. Shaw SO0- 0- -S00" -1000" -1500' -2000' -2500 -50.5 44~me~al [ ~ )] i ] L ))i i i i 'i -50 -49.5 -49 Location, z ID sooo" -500' -1000' - 1 SO0' -2000 -2SO0 -50.5 -SO -49.5 -49 Location, z ,001 , [ ............ 0 .i i i i i ~ ~ : ~ ~ ~ ~ i :: i.! ~.~ ~.-~.+..~.÷--! ~ :: :: i iThelayer with i i ~ i i i i i i i i :: i i i i i i 44%metal i i I i i ) i ) i i :, ': i i ! -SO0" ~ ! ! iiii i i :iiii 'f~~ i i i ! i ! ": i °.,ooo, i ..... i iii i .......... i....... i - 1 SO0 ~ i--i-J -2000.2500 ] i i i i ~ ~'~ i i i...] mi=i~.0i".th~¢°'tlingl i] i ! -50,5 "SO -49.5 -49 Location, z Figure 8 The in-plane residual stresses, a~, in the coating and the substrate for multi-layered coatings (10 layers) with (a) m= 1.0, (b) m=0.5, and (c) m = 2.0. Note that only a small portion of the substrate is shown (a) (b) Cersn~c face ['."_ I .... ] ~ -- ~'~" stresses Metal face : ." [" I I . _. - ~ Compressiv( I .... 4 "~.,~-~" stresses :.'.1 I .... -, ~ Tensile ,_ .... I I ...... With no constraints from With constraants from neighbonng layers neighboring layers Figure 9 Schematic showing the effect of the compositional gradient on the distribution of thermal residual stresses. Dashed lines represent the dimensions of each layer before cooldown. (a) m = 1.0, (b) m 1.0 compositional gradient, i.e. m = 1.0. In fact, to minimize residual stresses, not only a linear compositional gradient should be chosen, but the layers within the plate should also be as many as possible if the plate is multi-layered. Finally, it would be useful to point out that m < 1.0 corresponds to the situation where a more rapid change in volume fraction and properties occurs near the ceramic face, and more gradual changes occur near the metal face. Signs of thermal residual stresses in plates The distribution of thermal residual stresses, the curvature and midplane strain of the plate can be understood through a schematic shown in Figure 9. In this figure the changes of multi-layered plates for three different compositional gradients during cooldown with and without constraints from neighboring layers are sketched. For a linear compositional gradient (Figure 9(a)), if there were no constraints from neighboring layers, the amount of shrinkage during cooldown for each layer would vary gradually with the metal face having the largest shrinkage and the ceramic face the smallest shrinkage. When the constraints from neighboring layers are present, the different shrinkage tendencies are accommodated through warping of the plate. As such, very small in-plane residual stresses are present, especially when the number of layers is large. When m < 1.0, i.e. more metal content in the plate (Figure 9(b)), the shrinkage tendency of each layer without the constraints from neighboring layers will lead to a situation that a large different shrinkage tendency is present at the ceramic rich region. Thus, for a bonded plate a residual compressive stress is created on the ceramic face since the neighboring high metal content layers will apply a compressive stress to the high ceramic content region and leave themselves in tension. Another result from this compositional gradient is a larger midplane strain than that of the plate in Figure 9(a), since more metal content is present in the plate with m < 1.0. The plate curvature is also expected to become 206
Thermal residual stresses in plates and coatings: L. L. Shaw 1500 TTTTTTTTTTTTT 2000千 E△a△T|,MPa E△a&T|.MPa 5 5 Figure 10 Plots for compositionally graded plates with m=0.5, showing(a)a linear relationship between the in-plane residual stress on the metal face om and Em Aa AT. (b)a linear relationship between the in-plane residual stress on the ceramic face ae and Ec Ao AT, and (c)a linear relationship between ae/o, and Ec/Er smaller in comparison with the plate in Figure 9(a), since sketches shown in Figure 9 suggest that plates have residual most of the plate is made of one component (i. e the metal tensile stresses on both faces of plates for m> 1.0, have in this case)and, therefore, the bending moment is small. residual compressive stresses on both faces for m 1.0(Figure 9(c). On pressive region on the ceramic face changes from 10.9 to and near the metal face, residual tensile stresses will be 14.7%o of the plate thickness when m changes from 0.1 produced due to the large mismatch between the high metal to 0.8. Changes in the depth caused by different elasti content face and the high ceramic content neighboring moduli of the constituents are even smaller. For instance, for layers. Finally, on and near the ceramic face residual tensile functionally graded plates with m=0.5, the depth of the stresses will be induced due to the superimposition of the residual compressive region on the ceramic face is 11.6, tensile strain from the plate curvature to the midplane strain 12.0, 12.9 and 13.8%of the plate thickness for the plate of the plate. In summary, thermal residual stresses are due systems with a modulus ratio of the metal to the ceramic of to the Cte mismatch between the metal and ceramic, and 110/475, 145/475, 145/290 and 110/170, respectively
Thermal residual stresses in plates and coatings: L. L. Shaw -5O" ~E e" -loo' -150 -200 0.2 .... T ........... i . , , • . , • ," • ' • I " ' " • ' ' I " " I 0.4 0.6 0.8 1 1.2 1.4 1.6 IE Aa&TI, MPa R o- O- -500" - 1000 -I 500 -2000 • ' ' ' ' ' ' ' ' • " " I ' " " ' I " ' " " 3 4 ] Ec&ct AT I, MPa a C Om 11 10 9' 8' 7' 6 5 1.5 '''' '''' ''''1'''' '''' '''" 2 2.5 3 3.5 4.5 Ec /Era Figure 10 Plots for compositionally graded plates with rn = 0.5, showing (a) a linear relationship between the in-plane residual stress on the metal face G m and Em hcx z%T, (b) a linear relationship between the in-plane residual stress on the ceramic face ac and E c z~ot A~T, and (c) a linear relationship between at~am and Ec/E m smaller in comparison with the plate in Figure 9(a), since most of the plate is made of one component (i.e. the metal in this case) and, therefore, the bending moment is small. Finally, a residual compressive stress is developed on and near the metal face. This can be understood by the superimposition of two processes: (1) the metal face can shrink with neighboring layers to the amount of the midplane strain without much constraint since the neighboring layers also have high metal contents; and (2) to the midplane strain another compressive strain induced by the plate curvature is superimposed. Thus, a residual compressive stress results on and near the metal face, while the stress state in the center of the plate is nearly unaffected by the plate curvature. In short, the combination of the midplane strain, the plate curvature and the constraint by the neighboring layers results in compressive stresses on both the metal and ceramic faces and tensile stresses in the central portion of the plate. Using the same reasoning as for m 1.0 (Figure 9(c)). On and near the metal face, residual tensile stresses will be produced due to the large mismatch between the high metal content face and the high ceramic content neighboring layers. Finally, on and near the ceramic face residual tensile stresses will be induced due to the superimposition of the tensile strain from the plate curvature to the midplane strain of the plate. In summary, thermal residual stresses are due to the CTE mismatch between the metal and ceramic, and sketches shown in Figure 9 suggest that plates have residual tensile stresses on both faces of plates for m > 1.0, have residual compressive stresses on both faces for m < 1.0, and minimum residual stresses for m = 1.0. These are consistent with the results from the computation of laminated theory, as presented in the Results section. In many instances it is useful to know how deep the surface region with a residual compressive (or tensile) stress is. Based on the materials systems studied (Table 1), it is found that the depth of the surface region either compressive or tensile is nearly independent of the CTE of the metal and ceramic constituents, and dependent on the compositional gradient and the elastic modulus of the metal and ceramic. Even so, this depth does not change much with the compositional gradient and the elastic modulus of the constituents in the range we studied. For example, for a combination of a metal (E m = 145 GPa and Oem=15 × 10-6/°C) and a ceramic (Ec = 290GPa and o~ c = 8.9 × 10 6°C), the depth of the residual compressive region on the ceramic face changes from 10.9 to 14.7% of the plate thickness when m changes from 0.1 to 0.8. Changes in the depth caused by different elastic moduli of the constituents are even smaller. For instance, for functionally graded plates with m = 0.5, the depth of the residual compressive region on the ceramic face is 11.6, 12.0, 12.9 and 13.8% of the plate thickness for the plate systems with a modulus ratio of the metal to the ceramic of 110/475, 145/475, 145/290 and 110/170, respectively. 207
hermal residual stresses in plates and coatings: L.L. Shaw Nevertheless, it can be concluded that the depth of the modulus of the material on the one residual compressive or tensile region on the ceramic face face has little influence on the magnitude of residual stresses creases, while the depth of the compressive or tensile on the other face region on the metal face decreases with increasing m Changes in the depth with the elastic moduli of the constituents is very small, and the depth is nearly Geometry changes of plates independent of the cte of the constituents In this paper the geometry change of plates refers to the midplane strain and the plate curvature. The results from determined by the compositional exponent m, the number Based on the discussion above, the magnitude of thermal of layers, and the constituents properties (i.e. (ae +am )2) residual stresses on the ceramic and metal faces can also be estimated quickly without resorting to laminate theory All of these can be attributed to the change in the metal Since thermal residual stresses are predominantly due to content (or the ceramic content) and the constituents the CTE mismatch between the metal and ceramic, the properties. If the metal has a higher CTE than the ceramic, magnitude of residual stresses on the ceramic face and metal variables which can increase the metal content and the CTE roportional to the Cte mismatch,△ax,ie of the metal and ceramic. Specifically, the midplane strain increases when(1)m decreases, (2) the number of layers o|∝|E△a△Tl (9) increases, and (3) the cte of both the metal and ceramic om|∝En△a△Tl (10) As discussed previously, the curvature of a plate is where Aa=am-a. E is the elastic modulus of the con- determined by the resultant thermal bending moment which stituent, and subscripts c and m refer to the ceramic and in turn depends on the CTE mismatch, the modulus ratio metal,respectively. The effect of the compositional expo. of the ceramic to the metal, and the compositional gradient nent, m, can be included into eqn(9)and eqn(10)througl in the plate. Increasing Aa leads to an increase in the two unknown F(m) and Fm(m) functions with Fc( bending moment and, therefore, an describing residual stresses on the ceramic face and Fm(m) curvature. Increasing Ec/Em also has the same effects as describing residual stresses on the metal face. Thus, eqn(9) increasing Ao. Finally, the plate curvature decreases when and egn(10)can be rewritten m becomes increasingly larger or smaller than 1.0. This is because when m becomes increasingly larger or smaller ∝E△a△TF(m) (11) than 1.0, the plate becomes predominantly composed of one lom|x|Em△a△TFn(m) constituent(Figure 2), thereby a decrease in the bending (12) moment and the plate curvature. Based on eqn(I1)and eqn(12), we would expect that at a constant compositional gradient residual stresses on the Residual stresses and geometry changes in coatings ceramic face and on the metal face are a linear function of EΔαΔ T and e△a△T, respectively. Furthermore,the The current analysis indicates that the distribution and ratio of o to om should be proportional to the ratio of Ec magnitude of thermal residual stresses for coatings are theory. Using the date in table 1, Figure 10 is generated, the coating and substrate, provided that the substrate is and it does indicate that oe and om are proportional to substantially thicker than the coating. This is because of E△a△ T and e△aΔT, respectively, and the ratio of o the dominant effect of the substrate due to its larger volume to om is proportional to the ratio of Ec and Em. It is further in comparison with the coating's volume. Because of this noted that at a constant compositional gradient oe is primar the midplane strain is also primarily dependent on the CtE ily dependent of Ec and the effect of Em is hardly discern- of the metal substrate, and the curvature of the coating and ible. Similarly, om is predominately determined by Em and substrate extremely small. The graded composition the effect of Ec is negligible. Examination of the residual in the coating only leads to a gradual change of residual stresses in compositionally graded plates with m=0.2 also stresses from the substrate to the top face of the coating with leads to the same conclusions. Thus it can be concluded that no decrease in the magnitude of the residual stresses on the eqn(11)and eqn(12)can be used as a tool to guide the ceramic face of the coating. This conclusion is consistent lection of materials for multi-layered or compositionally with other studies. ince the coating and substrate graded plates with desired residual stresses. Specifically, if a curvature is extremely small, eqn(9)can be modified as high residual stress(either tensile or compressive)is desired follows in the ceramic face, a ceramic with a high elastic modulus and a large CTE mismatch with the metal should be 0∝Ec△a△T (13 selected. Similarly, a high residual stress on the metal When the two-dimensional stress-strain relationship is face results if a metal with a high elastic mo taken into consideration, eqn(13)can be further modified as a large CTE mismatch with the ceramic is chosen ore,at a constant compositional gradient ≈E2△a△7(1+p)(1-p2) (14)
Thermal residual stresses in plates and coatings: L. L. Shaw Nevertheless, it can be concluded that the depth of the residual compressive or tensile region on the ceramic face increases, while the depth of the compressive or tensile region on the metal face decreases with increasing m. Changes in the depth with the elastic moduli of the constituents is very small, and the depth is nearly independent of the CTE of the constituents. Magnitudes of thermal residual stresses in plates Based on the discussion above, the magnitude of thermal residual stresses on the ceramic and metal faces can also be estimated quickly without resorting to laminate theory. Since thermal residual stresses are predominantly due to the CTE mismatch between the metal and ceramic, the magnitude of residual stresses on the ceramic face and metal face, ~r c and am, should be proportional to the CTE mismatch, Ac~, i.e. loci ~ [Ec Ac~ AT] (9) Icrml ~ IEm Ac~ ATI (10) where As = o~ m -- OQ, E is the elastic modulus of the constituent, and subscripts c and m refer to the ceramic and metal, respectively. The effect of the compositional exponent, m, can be included into eqn (9) and eqn (10) through two unknown Fc(m ) and Fm(m ) functions with Fc(m) describing residual stresses on the ceramic face and Fm(m) describing residual stresses on the metal face. Thus, eqn (9) and eqn (10) can be rewritten as: lacl ~ IEc Ao~ AT Fc(m){ (11) IOm[ ~ IEm AO~ AT Fm(m)l (12) Based on eqn (11) and eqn (12), we would expect that at a constant compositional gradient residual stresses on the ceramic face and on the metal face are a linear function of Ec Ace AT and Em Ace AT, respectively. Furthermore, the ratio of ~r~ to a m should be proportional to the ratio of E~ and Em. These are confirmed by the result from laminate theory. Using the date in Table 1, Figure 10 is generated, and it does indicate that cr c and ffm are proportional to E c Ace AT and E m Aot AT, respectively, and the ratio of o c to o" m is proportional to the ratio of E~ and Em. It is further noted that at a constant compositional gradient o c is primarily dependent of Ec and the effect of E m is hardly discernible. Similarly, a m is predominately determined by E m and the effect of E c is negligible. Examination of the residual stresses in compositionally graded plates with m = 0.2 also leads to the same conclusions. Thus, it can be concluded that eqn (11) and eqn (12) can be used as a tool to guide the selection of materials for multi-layered or compositionally graded plates with desired residual stresses. Specifically, if a high residual stress (either tensile or compressive) is desired in the ceramic face, a ceramic with a high elastic modulus and a large CTE mismatch with the metal should be selected. Similarly, a high residual stress on the metal face results if a metal with a high elastic modulus and a large CTE mismatch with the ceramic is chosen. Furthermore, at a constant compositional gradient a change in the elastic modulus of the material on the one face has little influence on the magnitude of residual stresses on the other face. Geometry changes of plates In this paper the geometry change of plates refers to the midplane strain and the plate curvature. The results from laminate theory indicate that the midplane strain is determined by the compositional exponent m, the number of layers, and the constituent's properties (i.e. (% + Otm)/2 ). All of these can be attributed to the change in the metal content (or the ceramic content) and the constituents' properties. If the metal has a higher CTE than the ceramic, then the midplane strain can be increased through any variables which can increase the metal content and the CTE of the metal and ceramic. Specifically, the midplane strain increases when (1) m decreases, (2) the number of layers increases, and (3) the CTE of both the metal and ceramic increases. As discussed previously, the curvature of a plate is determined by the resultant thermal bending moment which in turn depends on the CTE mismatch, the modulus ratio of the ceramic to the metal, and the compositional gradient in the plate. Increasing Ac~ leads to an increase in the bending moment and, therefore, an increase in the plate curvature. Increasing Ec/E m also has the same effects as increasing Ac~. Finally, the plate curvature decreases when m becomes increasingly larger or smaller than 1.0. This is because when m becomes increasingly larger or smaller than 1.0, the plate becomes predominantly composed of one constituent (Figure 2), thereby a decrease in the bending moment and the plate curvature. Residual stresses and geometry changes in coatings The current analysis indicates that the distribution and magnitude of thermal residual stresses for coatings are almost solely determined by the CTE mismatch between the coating and substrate, provided that the substrate is substantially thicker than the coating. This is because of the dominant effect of the substrate due to its larger volume in comparison with the coating's volume. Because of this, the midplane strain is also primarily dependent on the CTE of the metal substrate, and the curvature of the coating and substrate is also extremely small. The graded composition in the coating only leads to a gradual change of residual stresses from the substrate to the top face of the coating with no decrease in the magnitude of the residual stresses on the ceramic face of the coating. This conclusion is consistent with other studies ~8. Since the coating and substrate curvature is extremely small, eqn (9) can be modified as follows: a c ~ E c As AT (13) When the two-dimensional stress-strain relationship is taken into consideration, eqn (13) can be further modified as cr c ~ E c As AT(1 + v)/(1 -- v 2) (14) 208
