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Residual Stresses in Coated and Layered Systems
Residual Stresses in Coated and Layered Systems Residual stresses are present in most surface coatings. These can be of considerable significance, since they may influence characteristics such as the resistance of the coating to wear and fatigue crack propagation. Furthermore, there is often a danger that the presence of residual stresses may promote debonding and spallation of the coating. This becomes increasingly likely as the thickness of the coating is increased, since the release of stored elastic strain energy as the stresses become relaxed can drive this debonding and the quantity of energy released, per unit area of interface, normally rises more or less linearly with coating thickness. This article is not primarily concerned with such effects, but rather is aimed at providing an understanding of how residual stresses arise in surface coatings. It covers both the underlying mechanics involved and examples of the physical phenomena that can lead to stress generation. The treatment encompasses both thin coatings (usually produced by deposition from the vapor phase), in which the stress level is essentially uniform, and thicker ones (commonly formed by much faster processes such as droplet spraying), in which through-thickness variations of stress level can be significant. In both cases, however, the coatings can normally be taken as sufficiently thin to ensure that significant stresses do not build up in the direction normal to the plane of the coating. Therefore, at any depth in the system, neglecting edge effects and assuming an approximately planar specimen geometry, an equal biaxial stress state is established, characterized by a single stress value. Of course, more complex three-dimensional stress states can be set up with very thick coatings and shaped components, but an understanding of such cases, and numerical modeling to predict the stress state, would follow logically from the treatments described here. 1. Misfit Strains, Resultant Stresses, and Associated Curvatures While the main interest often lies in the stress levels that are generated within the surface coating (and the substrate), it is instructive to consider the situation in terms of misfit strains, i.e., relative differences between the stress-free dimensions of various layers. The simplest system is composed of just two layers, the coating and the substrate, but it may be appropriate (particularly for thick coatings) to consider them being deposited as a series of layers. A further general point is that it can be very instructive to note how the system will adopt a curvature as a result of the imposition of a misfit strain. This can be useful, not only in helping to understand the mechanics of stress generation, but also for using curvature monitoring to measure residual stress levels.
1.1 Force and Moment Balances Consider a pair of plates bonded together with a misfit strain ∆ε in the x-direction (see Fig. 1). The resultant stress distribution, σx( y ), and curvature, κ, can be obtained from simple beam bending theory. First, the misfit strain is removed by the application of two equal and opposite forces (kP and P). When the two plates are joined, this results in an unbalanced moment, M. Balancing this moment generates curvature of the composite plate. The moment is given by E
MlP F
hjH 2
G
(1) H
where h and H are the thicknesses of deposit and substrate, respectively. Now, the curvature of a beam, κ (equal to the through-thickness gradient of strain), can be expressed as the bending moment divided by the beam stiffness, Σ κl
M Σ
(2)
By calculating the beam stiffness of the composite beam (Clyne 1996), a general expression is obtained for the curvature arising from the imposition of a uniform misfit strain, ∆ε, such as would arise during a change in temperature (∆ε l ∆α∆T ) κl 6EdEs(hjH )hH∆ε Ed#h%j4EdEsh$Hj6EdEsh#H#j4EdEshH $jEs#H % (3) where E is Young’s modulus and the subscripts d and s refer to deposit (coating) and substrate, respectively. It may be noted that, for a given deposit\substrate thickness ratio (h\H ), the curvature is inversely proportional to the substrate thickness, H. This scale effect is very important in practical terms, since relatively thin substrates are essential if curvatures sufficiently large for accurate measurement are to be generated.
1.2 Biaxial Stresses and Bifurcation Equation (3) is valid provided the original imposed misfit strain is a uniform one (no through-thickness gradient) and the system remains elastic while the curvature is adopted. A modification arises, however, on consideration of in-plane stresses other than those in the x-direction. For an isotropic in-plane stress state, there is effectively another stress equal to σx in a 1
Residual Stresses in Coated and Layered Systems
y z b x De (e.g. DaDT) h
&
force balance b r(y) dy = 0 –H
–1
e (millistrain) 0 1
r(MPa) 0 20
y=h
–P
–P
–20
y=0 P
y = –H
P
–1
0
1
–20
0
20
–P
–P M
M
P
P
yc = 0
y=d
h–d
moment balance
1/j
&
b r(yc) yc dyc = 0 –H–d
Figure 1 Schematic depiction of the generation of curvature in a flat bi-material plate, as a result of the imposition of a uniform, linear misfit strain, ∆ε. The distributions of stress and strain shown are calculated, using Eqns. (7) and (8), for H l h, Es l 100 GPa, Ed\Es l 0.1, and ∆ε l 10−$.
direction at right angles to it (z-direction); this induces a Poisson strain in the x-direction. Assuming isotropic stiffness and negligible through-thickness stress (σy l 0), the net strain in the x-direction can be written εxE l σxkν(σyjσz) l σx(1kν)
(4)
so that the relationship between stress and strain in the x-direction can be expressed as σx E l l Eh εx 1kν 2
(5)
This modified form of Young’s modulus, Eh, is usually applicable in expressions referring to substrate\ coating systems having an equal biaxial stress state. A further point related to biaxial stresses concerns the possibility of multidirectional curvatures leading to mechanical instability. Under an equal biaxial stress state, an initially plane surface will tend to become convex or concave, with a curvature which should be equal in all directions (i.e., a spherical surface). However, high curvatures cannot be simultaneously accommodated in all in-plane directions. On increasing the curvature sufficiently, a bifurcation point will be reached, at which the curvature increases sharply in
Residual Stresses in Coated and Layered Systems directly to a deposit stress, σd (l Ed∆ε), so that the equation reduces to the form κl
6σd(1kνs)h E sH #
(6)
which is usually known as the Stoney equation (Stoney 1909) and is commonly used to relate stress to curvature for thin coatings. When the condition h H does not apply, then stresses and stress gradients are often significant in both constituents. Stress distributions are readily found for the simple misfit strain case outlined above using the following expressions: E
Figure 2 Predicted dependence (Clyne and Gill 1996) on specimen width\length ratio of the critical curvature at a bifurcation instability, for a residually stressed thin film on a substrate. The parameter Γ is the ratio of the shear modulus of the deposit to that of the substrate (Gd\Gs).
one plane and decreases sharply in the plane normal to this (i.e., the shape becomes ellipsoidal and approaches that of a cylinder). This should be avoided, since it is difficult to relate a measured curvature to an internal stress distribution once a specimen has bifurcated. In general, it is a complex problem to predict the curvature at the bifurcation point for a substrate\ deposit system under residual stress. However, the case of a thin film on a thick, rectangular substrate (with a uniform stress in the film) has been analyzed (Salamon and Masters 1995). The critical curvature depends on the elastic constants, deposit\substrate thickness ratio (h\H ), substrate thickness\length ratio (H\L), substrate width\length ratio (b\L), and substrate length (L). Predicted behavior is shown in Fig. 2. The critical curvature decreases as the width\ length ratio increases. It is relatively insensitive to the elastic properties and to the deposit\substrate thickness ratio. It can be seen from the plot that there will be little or no danger of a bifurcation instability (i.e., κB 10 m−") if a long (L " 100 mm), fairly narrow (b\L 0.2) strip specimen is used. This is convenient, since such specimens are well suited to accurate measurement of curvature along the length of the strip.
1.3 The Effect of Coating Thickness and Use of the Stoney Equation A simplified form of Eqn. (3) results when the coating is much thinner than the substrate (h H ). Since the stress in the substrate then tends to become negligible, and that in the deposit will vary little as a result of curvature adoption, the misfit strain can be converted
σdQy = h lk∆ε F
E
σdQy = lk∆ε ! E
σsQy =−H l ∆ε F
F
F
jEdκ(hkδ) (7a) H
EdHEs hEdjHEs
EdhEs hEdjHEs E
σsQy = l ∆ε !
G
EdHEs hEdjHEs
G
kEdκδ
(7b)
kEs κ(Hjδ)
(8a)
H
G
H
EdhEs hEdjHEs H
G
kEs κδ
(8b)
The stress distributions shown in Fig. 1 are obtained using these equations. It can be seen that the adoption of curvature can effect substantial changes in stress levels and high through-thickness gradients can result. It may be noted from Eqns. (7) and (8) that (for a given value of h\H ) the stresses at y lkH, 0, and h do not depend on H, i.e., the stress distribution is independent of scale. The Stoney equation is accurate only in the limit where the film thickness, h, tends to zero. Unfortunately, in this limit the curvature, κ, must also tend to zero. It has been suggested (Brenner and Senderoff 1949) that the experimental error arising from the curvature being too small to measure accurately typically exceeds the error introduced via the approximation incorporated into the Stoney equation when the ratio of the thickness of the coating to that of the substrate, h\H, is less than about 5%. This is approximately correct, although the details are dependent on Young’s modulus and on the absolute thickness of the substrate. This is illustrated by the plots in Fig. 3, which are produced using Eqns. (3), (7), and (8). These equations refer to an elastic system exhibiting an equal biaxial stress state, resulting from the introduction of a uniform in-plane misfit strain, ∆ε, between the two constituents. Note that, for relatively thick coatings, the adoption of curvature not only changes the coating stress, but also introduces differences between the level at the free surface and that 3
Residual Stresses in Coated and Layered Systems will push the curvature up to more readily measurable levels, but such strains are not very common. In any event, it should be recognized that, in the regimes of small h\H and relatively large H, minor errors in measured curvature lead to major changes in deduced stress level.
E E H
1.4 Optimization of Curature Techniques for Stress Measurement E E
h/H
E E
E E E
h/H
Figure 3 Predicted dependence of (a) specimen curvature and (b) stress levels on the ratio of the thickness of the coating, h, to that of the substrate, H. The plots are obtained using the exact relationships represented by Eqns. (3), (7), and (8) and using the Stoney equation (Eqn. (6)). The Poisson ratios of substrate and deposit are both taken as 0.2.
at the interface. This is one reason why it is more rigorous to define a misfit strain than a coating stress, although through-thickness variations within the coating would not be significant for most thin coatings (produced by vapor deposition). The plots in Fig. 3 confirm that the Stoney equation is expected to be quite accurate for thickness ratios below a few percent or so, depending on the stiffnesses. Experimental studies with thin films often satisfy this condition, since a typical value for h is 1 µm and H ranges from about 100 µm to over 1 mm. However, the plots in Fig. 3(a) highlight the problem identified by Brenner and Senderoff for cases where the h\H ratio is very low. For a misfit strain of 10−$, many experimentally used combinations of H and h correspond to curvatures below 0.1 m−", or even below 0.01 m−" (100 m radius of curvature). Such curvatures are extremely difficult to measure with any real precision. Of course, the presence of much higher misfit strains 4
A more rational approach than that of conventional application of the Stoney equation, although surprisingly little used, is to generate relatively large curvatures by using high h\H ratios and to use Eqns. (3), (7), and (8) to deduce the misfit strains and hence the stress levels. Curvatures in the range 1–10 m−" are readily measured with high precision. Such curvatures can usually be generated by depositing relatively thick coatings. Unfortunately, this can lead to adhesion failure, since thicker coatings lead to higher strain energy release rates for interfacial debonding (see Drory et al. 1988). One solution is to use thinner substrates, although there is clearly a limit to this, since very thin substrates become difficult to handle and are prone to fracture or undergo plastic deformation. In practice, deposition stress can usually be measured accurately via curvature monitoring provided suitable control is exercised over substrate material, substrate thickness, and interfacial toughness. For example, the latter can often be increased by cleaning\roughening of the substrate surface, use of thin adherent interlayers, etc. It should be noted that the thermal expansivity of the substrate is also relevant, particularly for films deposited at high temperature, since large differential thermal contraction stresses will introduce errors in measurement of the deposition stress and may promote debonding during cooling. Measurement of curvature in situ during deposition, or at least during subsequent cooling, can alleviate these problems. Care should also be taken to avoid bifurcation (see Sect. 1.2), but this is usually straightforward provided suitable strip dimensions are chosen. 2. Stresses in Thin Coatings Formed by Vapor Deposition For a thin coating (h H ), the misfit strain is commonly assumed to be taken up entirely within it, so that the stress level is simply obtained on multiplying by the (biaxial) modulus. While the system may adopt a measurable curvature (see Sect. 1.3), the associated variations in stress level within the coating can usually be neglected. For all types of coating, the main sources of residual stress are (i) differential thermal contraction and (ii) phenomena occurring during deposition. It is common for these to be referred to, respectively, as extrinsic and intrinsic stresses. Other processes, such as phase transformations, plas-
Residual Stresses in Coated and Layered Systems
T
T
Figure 4 Thermal expansivity data (Peng and Clyne 1997) for diamond on tungsten, expressed as (a) αd(T ) and αs(T ) and (b) ∆εth(T ), obtained using Eqn. (10), with the polynomial expressions plotted in (a), for an ambient temperature, T , of ! 25 mC. #
tic flow, creep, etc., can also effectively generate a misfit strain. These can in principle be handled in much the same way as thermal stresses. 2.1 Differential Thermal Contraction Stresses Thermal stresses can readily be calculated from a knowledge of the thermal expansivities of the constituent materials. The associated misfit strain can, provided these expansivities are temperature independent, be written as ∆εth l ∆T(αskαd)
(9)
where ∆T is the temperature change. This strain, and the associated stress level in the coating at room temperature, will clearly be greater when deposition takes place at high temperature and when the expansivity mismatch is large. While this equation is commonly employed, it should be noted that neglect of the temperature dependence of the expansivities might be inaccurate. The misfit strain at ambient temperature, T , after cooling from the deposition temperature, T , ! # should be obtained from ∆εth l
&
T!
(αskαf) dT
(10)
T#
It can be seen from the data presented in Fig. 4, which relates to deposition of diamond films on a tungsten substrate, that use of invariant expansivity values (which would give a linear plot of misfit strain against temperature) can introduce large errors. 2.2 Deposition (Intrinsic) Stresses Stresses can arise in several ways during molecular deposition from the vapor phase. For example, mol-
ecular species arriving with high energies can become implanted within the deposit, where they may occupy interstitial sites (or vacant lattice sites which are thermodynamically stable) and hence generate compressive stress. Bombardment with energetic nondepositing species can also promote such site occupancy and hence have a similar effect, which is often termed ‘‘atomic peening.’’ Processes can also take place during deposition which generate excess vacancies and hence tensile stresses. For example, preferential removal of a depositing molecular species via etching, immediately after deposition, can lead to this effect. A further point to note is that there will be a tendency for both excess vacancies and interstitial\ vacancy depletion to be removed by short-range (surface) diffusion, provided there is sufficient thermal energy available. Both substrate temperature and the short duration thermal energy injected by bombarding or implanting species are important in determining the degree to which such annealing processes occur. Deposition of diamond films requires relatively high substrate temperatures and also leads to generation of excess vacancies, and associated tensile stresses, as etching occurs during deposition. The deposition (intrinsic) stress in diamond films deposited onto tungsten is plotted against methane content and deposition temperature in Fig. 5. Preferential etching by atomic hydrogen of sp# carbon during diamond deposition is a primary source of excess vacancies. (The etching rate of sp# carbon is more than 20 times that of sp$ carbon.) Incorporation of sp# carbon into the film, however, tends to produce compressive stress, since its specific volume is 1.5 times that of sp$ carbon. At low methane levels, there is little initial formation of sp# carbon and hence the excess vacancy concentration is low. As the methane level rises, a higher sp# carbon is initially formed, but most of this is etched away by the atomic hydrogen. This gives rise to a high (tensile) intrinsic stress. At still higher methane con5
Residual Stresses in Coated and Layered Systems
Figure 5 Measured intrinsic (deposition) stresses (Peng and Clyne 1997) in diamond films on tungsten substrates, obtained by curvature measurement at room temperature and subtraction of the thermal stress, as a function of (a) methane concentration, at a temperature of 825 mC, and (b) temperature, at 1.0% methane concentration.
E
Figure 6 Comparison between the intrinsic stress levels measured experimentally, for DLC films deposited at 10 Pa, with various negative bias voltages, and the predicted curve obtained using the Davis model (after Peng and Clyne 1998).
tents, however, a significant amount of sp# carbon survives the etching by atomic hydrogen and becomes incorporated into the film. This makes the stress more compressive (and also leads to a reduction in Young’s modulus and changes in film morphology). A higher substrate temperature raises the recombination length for atomic hydrogen, promoting the etching of sp# carbon and making the stress more tensile (at least for methane levels where there is substantial deposition of sp# carbon). However, higher temperatures also enhance diffusion and thus tend to reduce the excess vacancy concentration. This effect, which decreases the tensile stress, tends to become significant only at relatively high temperatures (Fig. 5(b)). During deposition of diamond-like carbon (DLC), however, intrinsic stresses are normally compressive 6
and arise by quite different mechanisms from those operating with diamond. Deposition temperatures are lower than during diamond formation and there is much more bombardment by energetic species. Measured intrinsic stress values are plotted in Fig. 6 as a function of negative bias voltage, Vb. Initially, the stress rapidly becomes strongly compressive as Vb rises. This is the result of pronounced implantation of bombarding carbon ions, once they have enough energy to penetrate the structure. Further increases in bombardment energies lead to intensive local heating (thermal spike) and consequent reduction in the compressive stress as the structure undergoes thermal relaxation. Also shown in Fig. 6 is a predicted curve based on a simple model (Davis 1993) describing the variation of compressive stress in bombarded thin films, based on the competing effects of implantation and relaxation. Details are given elsewhere (Peng and Clyne 1998). The level of agreement, obtained using physically reasonable values for the parameters in the model, suggests that it does encapsulate the most important features of the process. 3. Stresses in Thick Coatings Produced by Droplet Spraying 3.1 Effects with Coatings of Significant Thickness For a relatively thick coating, it may become inaccurate to assume that a misfit strain is accommodated entirely within it. This applies to misfit strains arising from differential thermal contraction, the deposition process or inelastic deformation such as might arise from creep, plastic flow, microcracking, etc. Furthermore, while the misfit strain from postdeposition differential thermal contraction will be uniform within the coating, changes in curvature during deposition may mean that it should be treated as if it took place by formation of a series of discrete layers. In fact, an analytical model (Tsui and Clyne
Residual Stresses in Coated and Layered Systems
Ni–20Cr Ni–20Cr Ni–20Cr
Homologous temperature
Figure 7 Experimental quenching stress data (Kuroda and Clyne 1991) for plasma spraying in air (APS) or in vacuum (VPS). The values are plotted against the ratio of the specimen temperature to the melting temperature of the deposit. The data are obtained from specimen curvature measurements made during spraying.
1997) is available to treat deposition processes in this way, although it does involve repeated calculations, which are most conveniently carried out using a computer program. A multilayer approach within a full numerical process model may also be advisable in some cases, particularly if the heat flow taking place during spraying, and the effects of the associated through-thickness thermal gradients, are to be incorporated. This also allows prediction of the curvature changes taking place during deposition and subsequent cooling. Comparison between measured and predicted curvature histories provides a powerful method of validating the input data and boundary conditions employed (Gill and Clyne 1994).
3.2 The Quenching Stress When a droplet impinges on a substrate (or predeposited coating layer) during spray deposition, it will tend to spread into a pancake-like splat, which will then quickly solidify and cool. During this cooling, the thermal contraction of the splat will be inhibited by the underlying material, so that a tensile stress will be set up in the splat. Since splats are typically a few micrometers in thickness, it can always be assumed that they are on a massive substrate and the misfit strain associated with the contraction is entirely accommodated within the splat. The so-called quenching stress (Kuroda and Clyne 1991) is therefore given by the product of the misfit strain and the (biaxial) modulus of the deposit. However, while it is always tensile, the value of the quenching stress can vary over a wide range. The nominal value of the misfit strain is
simply the product of the coating expansivity and the temperature drop on cooling from the melting point to the substrate temperature. Corresponding stress levels are commonly in the gigapascal region. However, quenching stress values are normally well below these levels: in fact, they usually range from a few megapascals to a few hundred negapascals. There are two reasons for this. First, the stiffness of sprayed material is often appreciably lower than its handbook value, as a consequence of the presence of porosity, microcracks, etc. Second, it is common for stress relaxation processes to come into operation during splat quenching. These may include microcracking, plastic flow, creep, interfacial sliding, etc. Some such effects are apparent in the data presented in Fig. 7, which shows measured quenching stress values as a function of the homologous temperature. At lower temperatures, all materials tend to show gradual increases with temperature, which is attributed to improvements in intersplat bonding. Microcracking is common with many ceramics, leading to low quenching stress values. For metals, depending on their strength, a reduction in quenching stress tends to occur on moving to relatively high temperatures, as creep and plastic flow during quenching become pronounced. (Also, of course, the nominal misfit strain tends to zero as the homologous temperature approaches unity.)
3.3 Stress Distributions in Thermally Sprayed Systems As an example of a layer-by-layer simulation of a thermal spray coating process, data are presented in 7
Residual Stresses in Coated and Layered Systems
Top coat
Top coat
Bond coat
Bond coat
Figure 8 Comparison between (a) an experimental curvature history obtained during spraying and (b) the corresponding prediction from the numerical process model for a specimen composed of a CoNiCrAlY bond coat and a ZrO –8 wt.% Y O top # # $ coat, deposited onto a mild steel substrate. Substrate
Bond coat
Top coat
Modeled
Z
Figure 9 Comparison between stress distributions obtained experimentally (using neutron diffraction) and by numerical process modeling, for the specimen referred to in Fig. 8.
Fig. 8, which refer to a thermal barrier system. These are commonly composed of an oxidation-resistant metallic bond coat (an MCrAlY of some sort) and a ceramic top coat (usually zirconia) offering a large thermal resistance. The substrates most commonly protected in this way are nickel-based superalloys (in aeroengines) or various steels (in land-based turbines or internal combustion engines). In Fig. 8 a comparison is shown between measured and simulated specimen curvature changes during spraying, with both bond coat and top coat being divided into about 100 volume elements for the simulation. The periodicity in these plots is associated with the movement of the spray gun, which involves a cycle of passes over the specimen, with some cooling occurring between cycles. 8
(Good agreement is also obtained between measured and modeled thermal histories.) For this specimen, through-thickness stress distributions are measured by neutron diffraction, in order to provide an independent validation of the process model (see Fig. 9). The agreement is clearly quite good (although the resolution of the neutron diffraction measurements is inadequate to confirm the presence of the high stress gradient through the bond coat). Incidentally, this gradient arises mainly as a consequence of the relatively large curvature changes occurring during the process, which are deliberately stimulated by having a thin substrate. In practice, with a thicker (and perhaps nonplanar) substrate, curvature changes would be much smaller, but of course the (validated) model could equally well be used to predict stress distributions in such cases. Finally, it may be noted that the low stress levels in the top coat are largely attributable to its low stiffness (owing to microcracks, etc), which confers a high strain tolerance. Under service conditions, however, which involve the free surface of the top coat reaching high temperatures, this stiffness may rise as a consequence of sintering phenomena, leading to higher stresses and thus to a greater danger of debonding.
4. Concluding Points In all cases, stresses arise primarily as a result of (i) the material being deposited initially in a nonequilibrium state and (ii) differential thermal contraction occurring between the coating and the substrate, during postdeposition temperature changes. Both mechanisms can generate either tensile or compressive stress in the coating. While differential thermal expansion stresses are quite easy to predict, the mechanisms determining the deposition stresses can be more complex. These can, however, be measured and in many cases they can
Residual Stresses in Coated and Layered Systems be rationalized and predicted, at least in a semiquantitative manner. Further work is required in order to understand fully how deposition stresses arise in different systems. The effect of residual stresses on the thermomechanical stability of coatings is also an area requiring concerted research effort. See also: Thin Films: Stresses; Thin Films: Stress Measurement Techniques; Coatings for Corrosion Protection: An Overview
Bibliography Brenner A, Senderoff S 1949 Calculation of stress in electrodeposits from the curvature of a plated strip. J. Res. Nat. Bur. Stand. 42, 105–23 Clyne T W 1996 Residual stresses in surface coatings and their effects on interfacial debonding. Key Eng. Mater. 116/7, 307–30 Clyne T W, Gill S C 1996 Residual stresses in thermally sprayed coatings and their effect on interfacial adhesion—a review of recent work. J. Thermal Spray Technol. 5, 1–18 Davis C A 1993 A simple model for the formation of compressive stress in thin film by ion bombardment. Thin Solid Films 226, 30–4
Drory M D, Thouless M D, Evans A G 1988 On the decohesion of residually stressed thin films. Acta Metall. 36, 2019–28 Gill S C, Clyne T W 1994 Investigation of residual stress generation during thermal spraying by continuous curvature measurement. Thin Solid Films 250, 172–80 Kuroda S, Clyne T W 1991 The quenching stress in thermally sprayed coatings. Thin Solid Films 200, 49–66 Matejicek J, Sampath S, Brand P C, Prask H J 1999 Quenching, thermal and residual stress in plasma sprayed deposits: NiCrAlY and YSZ coatings. Acta Mater. 49, 607–17 Mortensen A, Suresh S 1995 Functionally graded metals and metal–ceramic composites: I. Processing. Int. Mater. Re. 40, 239–65 Peng X L, Clyne T W 1997 Formation and adhesion of hot filament CVD diamond films on titanium substrates. Thin Solid Films 293, 261–9 Peng X L, Clyne T W 1998 Mechanical stability of DLC films on metallic substrates. Thin Solid Films 312, 207–18 Salamon N J, Masters C B 1995 Bifurcation in isotropic thin film\substrate plates. Int. J. Solids Structures 32, 473–81 Stoney G G 1909 The tension of metallic films deposited by electrolysis. Proc. R. Soc. A82, 172–5 Tsui Y C, Clyne T W 1997 An analytical model for predicting residual stresses in progressively deposited coatings. Thin Solid Films 306, 23–61
T. W. Clyne
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 8126–8134 9
1.1 Force and Moment Balances Consider a pair of plates bonded together with a misfit strain ∆ε in the x-direction (see Fig. 1). The resultant stress distribution, σx( y ), and curvature, κ, can be obtained from simple beam bending theory. First, the misfit strain is removed by the application of two equal and opposite forces (kP and P). When the two plates are joined, this results in an unbalanced moment, M. Balancing this moment generates curvature of the composite plate. The moment is given by E
MlP F
hjH 2
G
(1) H
where h and H are the thicknesses of deposit and substrate, respectively. Now, the curvature of a beam, κ (equal to the through-thickness gradient of strain), can be expressed as the bending moment divided by the beam stiffness, Σ κl
M Σ
(2)
By calculating the beam stiffness of the composite beam (Clyne 1996), a general expression is obtained for the curvature arising from the imposition of a uniform misfit strain, ∆ε, such as would arise during a change in temperature (∆ε l ∆α∆T ) κl 6EdEs(hjH )hH∆ε Ed#h%j4EdEsh$Hj6EdEsh#H#j4EdEshH $jEs#H % (3) where E is Young’s modulus and the subscripts d and s refer to deposit (coating) and substrate, respectively. It may be noted that, for a given deposit\substrate thickness ratio (h\H ), the curvature is inversely proportional to the substrate thickness, H. This scale effect is very important in practical terms, since relatively thin substrates are essential if curvatures sufficiently large for accurate measurement are to be generated.
1.2 Biaxial Stresses and Bifurcation Equation (3) is valid provided the original imposed misfit strain is a uniform one (no through-thickness gradient) and the system remains elastic while the curvature is adopted. A modification arises, however, on consideration of in-plane stresses other than those in the x-direction. For an isotropic in-plane stress state, there is effectively another stress equal to σx in a 1
Residual Stresses in Coated and Layered Systems
y z b x De (e.g. DaDT) h
&
force balance b r(y) dy = 0 –H
–1
e (millistrain) 0 1
r(MPa) 0 20
y=h
–P
–P
–20
y=0 P
y = –H
P
–1
0
1
–20
0
20
–P
–P M
M
P
P
yc = 0
y=d
h–d
moment balance
1/j
&
b r(yc) yc dyc = 0 –H–d
Figure 1 Schematic depiction of the generation of curvature in a flat bi-material plate, as a result of the imposition of a uniform, linear misfit strain, ∆ε. The distributions of stress and strain shown are calculated, using Eqns. (7) and (8), for H l h, Es l 100 GPa, Ed\Es l 0.1, and ∆ε l 10−$.
direction at right angles to it (z-direction); this induces a Poisson strain in the x-direction. Assuming isotropic stiffness and negligible through-thickness stress (σy l 0), the net strain in the x-direction can be written εxE l σxkν(σyjσz) l σx(1kν)
(4)
so that the relationship between stress and strain in the x-direction can be expressed as σx E l l Eh εx 1kν 2
(5)
This modified form of Young’s modulus, Eh, is usually applicable in expressions referring to substrate\ coating systems having an equal biaxial stress state. A further point related to biaxial stresses concerns the possibility of multidirectional curvatures leading to mechanical instability. Under an equal biaxial stress state, an initially plane surface will tend to become convex or concave, with a curvature which should be equal in all directions (i.e., a spherical surface). However, high curvatures cannot be simultaneously accommodated in all in-plane directions. On increasing the curvature sufficiently, a bifurcation point will be reached, at which the curvature increases sharply in
Residual Stresses in Coated and Layered Systems directly to a deposit stress, σd (l Ed∆ε), so that the equation reduces to the form κl
6σd(1kνs)h E sH #
(6)
which is usually known as the Stoney equation (Stoney 1909) and is commonly used to relate stress to curvature for thin coatings. When the condition h H does not apply, then stresses and stress gradients are often significant in both constituents. Stress distributions are readily found for the simple misfit strain case outlined above using the following expressions: E
Figure 2 Predicted dependence (Clyne and Gill 1996) on specimen width\length ratio of the critical curvature at a bifurcation instability, for a residually stressed thin film on a substrate. The parameter Γ is the ratio of the shear modulus of the deposit to that of the substrate (Gd\Gs).
one plane and decreases sharply in the plane normal to this (i.e., the shape becomes ellipsoidal and approaches that of a cylinder). This should be avoided, since it is difficult to relate a measured curvature to an internal stress distribution once a specimen has bifurcated. In general, it is a complex problem to predict the curvature at the bifurcation point for a substrate\ deposit system under residual stress. However, the case of a thin film on a thick, rectangular substrate (with a uniform stress in the film) has been analyzed (Salamon and Masters 1995). The critical curvature depends on the elastic constants, deposit\substrate thickness ratio (h\H ), substrate thickness\length ratio (H\L), substrate width\length ratio (b\L), and substrate length (L). Predicted behavior is shown in Fig. 2. The critical curvature decreases as the width\ length ratio increases. It is relatively insensitive to the elastic properties and to the deposit\substrate thickness ratio. It can be seen from the plot that there will be little or no danger of a bifurcation instability (i.e., κB 10 m−") if a long (L " 100 mm), fairly narrow (b\L 0.2) strip specimen is used. This is convenient, since such specimens are well suited to accurate measurement of curvature along the length of the strip.
1.3 The Effect of Coating Thickness and Use of the Stoney Equation A simplified form of Eqn. (3) results when the coating is much thinner than the substrate (h H ). Since the stress in the substrate then tends to become negligible, and that in the deposit will vary little as a result of curvature adoption, the misfit strain can be converted
σdQy = h lk∆ε F
E
σdQy = lk∆ε ! E
σsQy =−H l ∆ε F
F
F
jEdκ(hkδ) (7a) H
EdHEs hEdjHEs
EdhEs hEdjHEs E
σsQy = l ∆ε !
G
EdHEs hEdjHEs
G
kEdκδ
(7b)
kEs κ(Hjδ)
(8a)
H
G
H
EdhEs hEdjHEs H
G
kEs κδ
(8b)
The stress distributions shown in Fig. 1 are obtained using these equations. It can be seen that the adoption of curvature can effect substantial changes in stress levels and high through-thickness gradients can result. It may be noted from Eqns. (7) and (8) that (for a given value of h\H ) the stresses at y lkH, 0, and h do not depend on H, i.e., the stress distribution is independent of scale. The Stoney equation is accurate only in the limit where the film thickness, h, tends to zero. Unfortunately, in this limit the curvature, κ, must also tend to zero. It has been suggested (Brenner and Senderoff 1949) that the experimental error arising from the curvature being too small to measure accurately typically exceeds the error introduced via the approximation incorporated into the Stoney equation when the ratio of the thickness of the coating to that of the substrate, h\H, is less than about 5%. This is approximately correct, although the details are dependent on Young’s modulus and on the absolute thickness of the substrate. This is illustrated by the plots in Fig. 3, which are produced using Eqns. (3), (7), and (8). These equations refer to an elastic system exhibiting an equal biaxial stress state, resulting from the introduction of a uniform in-plane misfit strain, ∆ε, between the two constituents. Note that, for relatively thick coatings, the adoption of curvature not only changes the coating stress, but also introduces differences between the level at the free surface and that 3
Residual Stresses in Coated and Layered Systems will push the curvature up to more readily measurable levels, but such strains are not very common. In any event, it should be recognized that, in the regimes of small h\H and relatively large H, minor errors in measured curvature lead to major changes in deduced stress level.
E E H
1.4 Optimization of Curature Techniques for Stress Measurement E E
h/H
E E
E E E
h/H
Figure 3 Predicted dependence of (a) specimen curvature and (b) stress levels on the ratio of the thickness of the coating, h, to that of the substrate, H. The plots are obtained using the exact relationships represented by Eqns. (3), (7), and (8) and using the Stoney equation (Eqn. (6)). The Poisson ratios of substrate and deposit are both taken as 0.2.
at the interface. This is one reason why it is more rigorous to define a misfit strain than a coating stress, although through-thickness variations within the coating would not be significant for most thin coatings (produced by vapor deposition). The plots in Fig. 3 confirm that the Stoney equation is expected to be quite accurate for thickness ratios below a few percent or so, depending on the stiffnesses. Experimental studies with thin films often satisfy this condition, since a typical value for h is 1 µm and H ranges from about 100 µm to over 1 mm. However, the plots in Fig. 3(a) highlight the problem identified by Brenner and Senderoff for cases where the h\H ratio is very low. For a misfit strain of 10−$, many experimentally used combinations of H and h correspond to curvatures below 0.1 m−", or even below 0.01 m−" (100 m radius of curvature). Such curvatures are extremely difficult to measure with any real precision. Of course, the presence of much higher misfit strains 4
A more rational approach than that of conventional application of the Stoney equation, although surprisingly little used, is to generate relatively large curvatures by using high h\H ratios and to use Eqns. (3), (7), and (8) to deduce the misfit strains and hence the stress levels. Curvatures in the range 1–10 m−" are readily measured with high precision. Such curvatures can usually be generated by depositing relatively thick coatings. Unfortunately, this can lead to adhesion failure, since thicker coatings lead to higher strain energy release rates for interfacial debonding (see Drory et al. 1988). One solution is to use thinner substrates, although there is clearly a limit to this, since very thin substrates become difficult to handle and are prone to fracture or undergo plastic deformation. In practice, deposition stress can usually be measured accurately via curvature monitoring provided suitable control is exercised over substrate material, substrate thickness, and interfacial toughness. For example, the latter can often be increased by cleaning\roughening of the substrate surface, use of thin adherent interlayers, etc. It should be noted that the thermal expansivity of the substrate is also relevant, particularly for films deposited at high temperature, since large differential thermal contraction stresses will introduce errors in measurement of the deposition stress and may promote debonding during cooling. Measurement of curvature in situ during deposition, or at least during subsequent cooling, can alleviate these problems. Care should also be taken to avoid bifurcation (see Sect. 1.2), but this is usually straightforward provided suitable strip dimensions are chosen. 2. Stresses in Thin Coatings Formed by Vapor Deposition For a thin coating (h H ), the misfit strain is commonly assumed to be taken up entirely within it, so that the stress level is simply obtained on multiplying by the (biaxial) modulus. While the system may adopt a measurable curvature (see Sect. 1.3), the associated variations in stress level within the coating can usually be neglected. For all types of coating, the main sources of residual stress are (i) differential thermal contraction and (ii) phenomena occurring during deposition. It is common for these to be referred to, respectively, as extrinsic and intrinsic stresses. Other processes, such as phase transformations, plas-
Residual Stresses in Coated and Layered Systems
T
T
Figure 4 Thermal expansivity data (Peng and Clyne 1997) for diamond on tungsten, expressed as (a) αd(T ) and αs(T ) and (b) ∆εth(T ), obtained using Eqn. (10), with the polynomial expressions plotted in (a), for an ambient temperature, T , of ! 25 mC. #
tic flow, creep, etc., can also effectively generate a misfit strain. These can in principle be handled in much the same way as thermal stresses. 2.1 Differential Thermal Contraction Stresses Thermal stresses can readily be calculated from a knowledge of the thermal expansivities of the constituent materials. The associated misfit strain can, provided these expansivities are temperature independent, be written as ∆εth l ∆T(αskαd)
(9)
where ∆T is the temperature change. This strain, and the associated stress level in the coating at room temperature, will clearly be greater when deposition takes place at high temperature and when the expansivity mismatch is large. While this equation is commonly employed, it should be noted that neglect of the temperature dependence of the expansivities might be inaccurate. The misfit strain at ambient temperature, T , after cooling from the deposition temperature, T , ! # should be obtained from ∆εth l
&
T!
(αskαf) dT
(10)
T#
It can be seen from the data presented in Fig. 4, which relates to deposition of diamond films on a tungsten substrate, that use of invariant expansivity values (which would give a linear plot of misfit strain against temperature) can introduce large errors. 2.2 Deposition (Intrinsic) Stresses Stresses can arise in several ways during molecular deposition from the vapor phase. For example, mol-
ecular species arriving with high energies can become implanted within the deposit, where they may occupy interstitial sites (or vacant lattice sites which are thermodynamically stable) and hence generate compressive stress. Bombardment with energetic nondepositing species can also promote such site occupancy and hence have a similar effect, which is often termed ‘‘atomic peening.’’ Processes can also take place during deposition which generate excess vacancies and hence tensile stresses. For example, preferential removal of a depositing molecular species via etching, immediately after deposition, can lead to this effect. A further point to note is that there will be a tendency for both excess vacancies and interstitial\ vacancy depletion to be removed by short-range (surface) diffusion, provided there is sufficient thermal energy available. Both substrate temperature and the short duration thermal energy injected by bombarding or implanting species are important in determining the degree to which such annealing processes occur. Deposition of diamond films requires relatively high substrate temperatures and also leads to generation of excess vacancies, and associated tensile stresses, as etching occurs during deposition. The deposition (intrinsic) stress in diamond films deposited onto tungsten is plotted against methane content and deposition temperature in Fig. 5. Preferential etching by atomic hydrogen of sp# carbon during diamond deposition is a primary source of excess vacancies. (The etching rate of sp# carbon is more than 20 times that of sp$ carbon.) Incorporation of sp# carbon into the film, however, tends to produce compressive stress, since its specific volume is 1.5 times that of sp$ carbon. At low methane levels, there is little initial formation of sp# carbon and hence the excess vacancy concentration is low. As the methane level rises, a higher sp# carbon is initially formed, but most of this is etched away by the atomic hydrogen. This gives rise to a high (tensile) intrinsic stress. At still higher methane con5
Residual Stresses in Coated and Layered Systems
Figure 5 Measured intrinsic (deposition) stresses (Peng and Clyne 1997) in diamond films on tungsten substrates, obtained by curvature measurement at room temperature and subtraction of the thermal stress, as a function of (a) methane concentration, at a temperature of 825 mC, and (b) temperature, at 1.0% methane concentration.
E
Figure 6 Comparison between the intrinsic stress levels measured experimentally, for DLC films deposited at 10 Pa, with various negative bias voltages, and the predicted curve obtained using the Davis model (after Peng and Clyne 1998).
tents, however, a significant amount of sp# carbon survives the etching by atomic hydrogen and becomes incorporated into the film. This makes the stress more compressive (and also leads to a reduction in Young’s modulus and changes in film morphology). A higher substrate temperature raises the recombination length for atomic hydrogen, promoting the etching of sp# carbon and making the stress more tensile (at least for methane levels where there is substantial deposition of sp# carbon). However, higher temperatures also enhance diffusion and thus tend to reduce the excess vacancy concentration. This effect, which decreases the tensile stress, tends to become significant only at relatively high temperatures (Fig. 5(b)). During deposition of diamond-like carbon (DLC), however, intrinsic stresses are normally compressive 6
and arise by quite different mechanisms from those operating with diamond. Deposition temperatures are lower than during diamond formation and there is much more bombardment by energetic species. Measured intrinsic stress values are plotted in Fig. 6 as a function of negative bias voltage, Vb. Initially, the stress rapidly becomes strongly compressive as Vb rises. This is the result of pronounced implantation of bombarding carbon ions, once they have enough energy to penetrate the structure. Further increases in bombardment energies lead to intensive local heating (thermal spike) and consequent reduction in the compressive stress as the structure undergoes thermal relaxation. Also shown in Fig. 6 is a predicted curve based on a simple model (Davis 1993) describing the variation of compressive stress in bombarded thin films, based on the competing effects of implantation and relaxation. Details are given elsewhere (Peng and Clyne 1998). The level of agreement, obtained using physically reasonable values for the parameters in the model, suggests that it does encapsulate the most important features of the process. 3. Stresses in Thick Coatings Produced by Droplet Spraying 3.1 Effects with Coatings of Significant Thickness For a relatively thick coating, it may become inaccurate to assume that a misfit strain is accommodated entirely within it. This applies to misfit strains arising from differential thermal contraction, the deposition process or inelastic deformation such as might arise from creep, plastic flow, microcracking, etc. Furthermore, while the misfit strain from postdeposition differential thermal contraction will be uniform within the coating, changes in curvature during deposition may mean that it should be treated as if it took place by formation of a series of discrete layers. In fact, an analytical model (Tsui and Clyne
Residual Stresses in Coated and Layered Systems
Ni–20Cr Ni–20Cr Ni–20Cr
Homologous temperature
Figure 7 Experimental quenching stress data (Kuroda and Clyne 1991) for plasma spraying in air (APS) or in vacuum (VPS). The values are plotted against the ratio of the specimen temperature to the melting temperature of the deposit. The data are obtained from specimen curvature measurements made during spraying.
1997) is available to treat deposition processes in this way, although it does involve repeated calculations, which are most conveniently carried out using a computer program. A multilayer approach within a full numerical process model may also be advisable in some cases, particularly if the heat flow taking place during spraying, and the effects of the associated through-thickness thermal gradients, are to be incorporated. This also allows prediction of the curvature changes taking place during deposition and subsequent cooling. Comparison between measured and predicted curvature histories provides a powerful method of validating the input data and boundary conditions employed (Gill and Clyne 1994).
3.2 The Quenching Stress When a droplet impinges on a substrate (or predeposited coating layer) during spray deposition, it will tend to spread into a pancake-like splat, which will then quickly solidify and cool. During this cooling, the thermal contraction of the splat will be inhibited by the underlying material, so that a tensile stress will be set up in the splat. Since splats are typically a few micrometers in thickness, it can always be assumed that they are on a massive substrate and the misfit strain associated with the contraction is entirely accommodated within the splat. The so-called quenching stress (Kuroda and Clyne 1991) is therefore given by the product of the misfit strain and the (biaxial) modulus of the deposit. However, while it is always tensile, the value of the quenching stress can vary over a wide range. The nominal value of the misfit strain is
simply the product of the coating expansivity and the temperature drop on cooling from the melting point to the substrate temperature. Corresponding stress levels are commonly in the gigapascal region. However, quenching stress values are normally well below these levels: in fact, they usually range from a few megapascals to a few hundred negapascals. There are two reasons for this. First, the stiffness of sprayed material is often appreciably lower than its handbook value, as a consequence of the presence of porosity, microcracks, etc. Second, it is common for stress relaxation processes to come into operation during splat quenching. These may include microcracking, plastic flow, creep, interfacial sliding, etc. Some such effects are apparent in the data presented in Fig. 7, which shows measured quenching stress values as a function of the homologous temperature. At lower temperatures, all materials tend to show gradual increases with temperature, which is attributed to improvements in intersplat bonding. Microcracking is common with many ceramics, leading to low quenching stress values. For metals, depending on their strength, a reduction in quenching stress tends to occur on moving to relatively high temperatures, as creep and plastic flow during quenching become pronounced. (Also, of course, the nominal misfit strain tends to zero as the homologous temperature approaches unity.)
3.3 Stress Distributions in Thermally Sprayed Systems As an example of a layer-by-layer simulation of a thermal spray coating process, data are presented in 7
Residual Stresses in Coated and Layered Systems
Top coat
Top coat
Bond coat
Bond coat
Figure 8 Comparison between (a) an experimental curvature history obtained during spraying and (b) the corresponding prediction from the numerical process model for a specimen composed of a CoNiCrAlY bond coat and a ZrO –8 wt.% Y O top # # $ coat, deposited onto a mild steel substrate. Substrate
Bond coat
Top coat
Modeled
Z
Figure 9 Comparison between stress distributions obtained experimentally (using neutron diffraction) and by numerical process modeling, for the specimen referred to in Fig. 8.
Fig. 8, which refer to a thermal barrier system. These are commonly composed of an oxidation-resistant metallic bond coat (an MCrAlY of some sort) and a ceramic top coat (usually zirconia) offering a large thermal resistance. The substrates most commonly protected in this way are nickel-based superalloys (in aeroengines) or various steels (in land-based turbines or internal combustion engines). In Fig. 8 a comparison is shown between measured and simulated specimen curvature changes during spraying, with both bond coat and top coat being divided into about 100 volume elements for the simulation. The periodicity in these plots is associated with the movement of the spray gun, which involves a cycle of passes over the specimen, with some cooling occurring between cycles. 8
(Good agreement is also obtained between measured and modeled thermal histories.) For this specimen, through-thickness stress distributions are measured by neutron diffraction, in order to provide an independent validation of the process model (see Fig. 9). The agreement is clearly quite good (although the resolution of the neutron diffraction measurements is inadequate to confirm the presence of the high stress gradient through the bond coat). Incidentally, this gradient arises mainly as a consequence of the relatively large curvature changes occurring during the process, which are deliberately stimulated by having a thin substrate. In practice, with a thicker (and perhaps nonplanar) substrate, curvature changes would be much smaller, but of course the (validated) model could equally well be used to predict stress distributions in such cases. Finally, it may be noted that the low stress levels in the top coat are largely attributable to its low stiffness (owing to microcracks, etc), which confers a high strain tolerance. Under service conditions, however, which involve the free surface of the top coat reaching high temperatures, this stiffness may rise as a consequence of sintering phenomena, leading to higher stresses and thus to a greater danger of debonding.
4. Concluding Points In all cases, stresses arise primarily as a result of (i) the material being deposited initially in a nonequilibrium state and (ii) differential thermal contraction occurring between the coating and the substrate, during postdeposition temperature changes. Both mechanisms can generate either tensile or compressive stress in the coating. While differential thermal expansion stresses are quite easy to predict, the mechanisms determining the deposition stresses can be more complex. These can, however, be measured and in many cases they can
Residual Stresses in Coated and Layered Systems be rationalized and predicted, at least in a semiquantitative manner. Further work is required in order to understand fully how deposition stresses arise in different systems. The effect of residual stresses on the thermomechanical stability of coatings is also an area requiring concerted research effort. See also: Thin Films: Stresses; Thin Films: Stress Measurement Techniques; Coatings for Corrosion Protection: An Overview
Bibliography Brenner A, Senderoff S 1949 Calculation of stress in electrodeposits from the curvature of a plated strip. J. Res. Nat. Bur. Stand. 42, 105–23 Clyne T W 1996 Residual stresses in surface coatings and their effects on interfacial debonding. Key Eng. Mater. 116/7, 307–30 Clyne T W, Gill S C 1996 Residual stresses in thermally sprayed coatings and their effect on interfacial adhesion—a review of recent work. J. Thermal Spray Technol. 5, 1–18 Davis C A 1993 A simple model for the formation of compressive stress in thin film by ion bombardment. Thin Solid Films 226, 30–4
Drory M D, Thouless M D, Evans A G 1988 On the decohesion of residually stressed thin films. Acta Metall. 36, 2019–28 Gill S C, Clyne T W 1994 Investigation of residual stress generation during thermal spraying by continuous curvature measurement. Thin Solid Films 250, 172–80 Kuroda S, Clyne T W 1991 The quenching stress in thermally sprayed coatings. Thin Solid Films 200, 49–66 Matejicek J, Sampath S, Brand P C, Prask H J 1999 Quenching, thermal and residual stress in plasma sprayed deposits: NiCrAlY and YSZ coatings. Acta Mater. 49, 607–17 Mortensen A, Suresh S 1995 Functionally graded metals and metal–ceramic composites: I. Processing. Int. Mater. Re. 40, 239–65 Peng X L, Clyne T W 1997 Formation and adhesion of hot filament CVD diamond films on titanium substrates. Thin Solid Films 293, 261–9 Peng X L, Clyne T W 1998 Mechanical stability of DLC films on metallic substrates. Thin Solid Films 312, 207–18 Salamon N J, Masters C B 1995 Bifurcation in isotropic thin film\substrate plates. Int. J. Solids Structures 32, 473–81 Stoney G G 1909 The tension of metallic films deposited by electrolysis. Proc. R. Soc. A82, 172–5 Tsui Y C, Clyne T W 1997 An analytical model for predicting residual stresses in progressively deposited coatings. Thin Solid Films 306, 23–61
T. W. Clyne
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 8126–8134 9