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Residual stresses in ceramic coatings as determined from the curvature of a coated strip
Materials Science and Engineering, A 150 ( 1992 ) 139-148
139
Residual stresses in ceramic coatings as determined from the curvature of a coated strip Chin-Chen Chiu Department of Metallur~', Mechanics, and Materials Science, Michigan State University, East Lansing, M148824 (USA~ (Received May 6, 1991 in revised form September 12, 1991 )
Abstract It is known that the curvature change developed in a coated strip is an index for determining the residual stresses in coatings. This paper uses beam theory to derive closed-form solutions for the stress analysis. In addition to theoretical derivation, experimental measurements are performed on TiC coating/graphite substrate composites. The experimental results reveal the presence of residual tensile stress in TiC coatings.
1. Introduction Ceramic coatings are commonly designed to protect substrates from erosive wear, chemical attack or hightemperature oxidation [ll. Because of the thermal expansion mismatch between the coatings and the substrates, the coated systems can be in a state of thermally induced residual stress; shear stress and peeling stress develop near the free edge and inplane normal stress occurs in the interior region [2-4]. The shear stress and peeling stress may delaminate the coatings from the free edge. The inplane normal stress may result in the coatings cracking, spalling or buckling [4-7]. Thus residual stress measurement is a basic study for characterizing the reliability and performance of coatings. (In this paper, our study focuses on the inplane normal stress measurement of coatings.) According to beam theory and the curvature developed in a coated strip, Stoney first in 1909 proposed an approximate formula for calculating the inplane normal stress of a coating [8]
o*: EJ' -
6rl~,
i l)
where the subscripts c and s refer to the properties of the coating and the substrate respectively. E and I are the elastic modulus and the thickness respectively, r is the radius of curvature of the substrate after coating. (Later investigators directly replaced the E~ in eqn. (1) by E~/( 1 - v~) in order to extend the one-dimensional solution to plane stress, where v is Poisson's ratio [9, I)921-5093/92/$5.00
101.) Stoney's formula is popular for the evaluation of residual stress of coatings [9-14], which may be attributed to its simplicity and convenience for calculation. As a result of approximations involved in the theoretical derivation [8, 10-12], eqn. (1) must be restricted to a condition that the ratio of the substrate thickness to the coating thickness is large. However, an analysis based on the propagation of errors [ 15, 16] reveals that an increase in the thickness ratio can increase the relative uncertainty of observed residual stress. Thus Stoney's formula should be analyzed for its reliance on the residual stress evaluation. In the present study, we derive formulas for the inplane normal stress calculation and analyze the deviation induced from Stoney's formula. Equations for the inplane elastic modulus calculation of a coating are also presented. In addition, the residual stresses and modulus of TiC thin coatings on graphite substrates are experimentally determined.
2. Theoretical background
2.1, Inplane normal stress of a coating When a coating/substrate composite is produced at elevated temperature and cooled down, the thermal expansion mismatch between the coating and the substrate results in thermal residual stresses. The stresses translate between the coating and the substrate by shear at the interface, causing the coated systems to contract, elongate, bend or twist. In this paper, we use © 1992 - Elsevier Sequoia. All rights reserved
140
C.-C. Chiu
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Residual stresses in ceramic coatings
satisfy dimensional match and to conform with coating adhesion, the coating layer and the substrate must be elongated and contracted respectively in the Z direction. Thus an axial tensile stress ac~ develops in the coating and an axial compressive stress a~j forms in the substrate (Fig. 1 (c)). The corresponding strains ec, and e~l are quantified by
beam theory to develop the relationship for determining the inplane normal stress which occurs in the internal region of a coated strip. When the coated.strip is long enough, end effects alter the stress significantly only near the end of the strip. Thus the beam theory can give sufficiently accurate evaluation of the inplane normal stress of a coating [2, 17]. Figure 1 illustrates the analytical framework of the internal stress equilibrium in a coated strip. Figure 1 (a) represents the cross-section of the coated strip which extends in the direction of the Z axis. The thicknesses of the coating and the substrate are l~ and l~ respectively. The distance from the neutral axis to the coating-substrate interface is I. If the coating has the same modulus as the substrate, the neutral axis will coincide with the centroid. For convenience, it is assumed that the coating has a larger elastic modulus than the substrate. The neutral axis then deviates from the centroid and approaches the interface. In general, coating procedures can be considered as forced adhesion at the fabrication temperature between the substrate and the coating. After cooling to room temperature, internal stress arises in the coated systems because of thermal expansion mismatch. For convenience, the thermal expansion coefficient of the coating a~ is assumed to be larger than that of the substrate a,. Thus the coating layer has a smaller room temperature length than the substrate (Fig. l(b)). To
(a)
A T ( a c - a~)= m = ec~ - e~l
(2a)
Ecec~ l~ + E~e~ [~= 0
(2b)
where A T is the temperature difference between the fabrication temperature (or annealing temperature) and room temperature. From the mechanics viewpoint, the system has to satisfy an internal stress equilibrium, namely ; o dA = 0
(3a)
faY dA = 0
(3b)
o is the normal stress in the direction of the Z axis. A is the cross-sectional area of the beam. Equations (3a) and (3b) represent the force and the bending moment equilibrium respectively. Since the stress state in Fig. 1(c) does not satisfy eqn. (3b) yet, the coated systems must be bent to some extent. The bending action results in another stress distribution in the system, which is shown in Fig. 1 (d). The corresponding surface stress and surface strain are oc_~,
Y Y
T
substrate
-f 2S
centroid
X,i~r
neutral axis interface
14~c --f
L.,-_-'~_ ......
J. - - -
-
z Y
C)
(d)
';s ~sl
+
(e)
%2
7
Y
2s
z neutral
q acl-
Ec~cl
/ ~ axis ~ l interface at2- Ec~c2
average residual stress Fig. 1. Analytical frameworks of the residual stresses in a coated strip: (a) cross-section; (b) dissection of a coated strip; (c) stress distribution as a result of axial contraction and elongation; (d) stress distribution due to pure bending moment; (e) equilibrium residual stress in a coated strip.
C.-C. Chiu / Residualstressesin ceramic"coatings 0~2, el22and e~2. According to stress superposition, the final state of the internal stress equilibrium is expressed in Fig. l(e). For a given coated strip, the residual stress distribution (Fig. 1 (e)) is unique and the self-equilibrium stress state must correspond to minimum elastic strain energy. The stress resolution reveals the existence and uniqueness of the pure bending moment (Fig. 1 (d)) and uniform axial stress (Fig. l(c)). Thus the elastic strain energy G, related only to the axial stress, also reaches the minimum value. (To satisfy the internal stress equilibrium, it is necessary and sufficient for G to reach the minimum value.) Therefore the first derivative of G with respect to the el2~is equal to zero: G-ll2E~e~*-+ l'£*e~'2
(4)
2
dG --=0 dgcl
(5)
The residual stress distribution in the coating layer is not constant (Fig. l(e)). Thus the average residual stresses o* are proposed and defined as _~ [" ) [o~.,(Y) + o~2(Y)] dA o* = ~/~
(11)
-t
f dA t
The average residual stress corresponds to the equilibrium stress in a real coated component (see Fig. l(e)), which is a function of the ratio R of substrate thickness to coating thickness. Combining eqns. (7d), (7e) and ( 11 ) gives
o* = El2ec, + -El2 - [ l 2 - ( l + l~)2] 2rl~.
(12)
Substituting eqn. (10) into eqn. (12) to eliminate r, and then combining with eqns. (6) and (8), we have
Combining eqns. (2a), (4) and (5) gives
e~, -
141
o* = AT(a~ - a,)Ec
1~E~ m
(6)
I~E~+I,E,
According to the force equilibrium (eqn. (3a)), the internal stress state in Fig. 1 (e) must satisfy I,-I
-I
f [o~,(Y)+o~2(Y)]dA+ f / where
[o~,(Y)+o~e(Y)]dA=O
-i-/~
E~R(El2 + E~R ~) 4(E~+E~R~)(E~+E~R)_3(EI2_E~R2) 2 where
R=I_,
(7a)
o<(Y)=E~e~,
(7b)
0~2(Y) = YEffr
(7c)
ol2,(Y)=El2el2,
(7d)
0122(Y) = YEl2/r
(7e)
r is the radius of curvature of the neutral axis. Substituting eqns. (2b), (7b), (7c), (7d) and (7e)into eqn. (7a) yields
(13a)
(13b)
1~ Equation (13a) expresses the relationship between the average residual stress development and the material parameters, o* is a function of R and the coating-tosubstrate elastic modulus ratio EJE, (Fig. 2). As R A =
C o
100
80
,,,j
E~I=-E~,I,. 2 l -
(8)
2Eft,+ 2E~(
According to the bending moment equilibrium (eqn. (3b)), the internal stress state in Fig. 1 (e) has to satisfy /, /
-/
f o~2(Y)YdA+ f I
-I
m
60
E c / E s : 30 w
"o
40
20
(D
o~2(Y)YdA=lc+l~El2e~,ll2 2 I~
(9)
n-
0.0
0
50
1()0
150
200
250
300
Thlckness Ratlo (R)
Substituting eqns. (7c) and (Te) into eqn. (9), we have
1 3
(ll2+l,)E~el2, l~
7 = 2 E~[(!,-t) ~ + ~3]+ E~[(I + t~)~ _/~]
(10)
Fig. 2. Variation of equilibrium residual stress with respect to Ec/E~ and R. The magnitude of equilibrium residual stress is expressed in terms of the percentage of saturation stress
EcAT(%-a,).
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Residual stresses in ceramic coatings
increases, o* monotonically increases and then converges to a saturation stress E c A T ( a ~ - a s ) . The greater ratio of E~/E~ will cause o* to converge slower to the saturation stress as~,. In practice, o* represents the equilibrium residual stress in a real coated component, which plays a role in the residual stress-induced damage. However, o~,t is only an index, showing the possibly maximum stress in components with the same coating and substrate materials. Substituting eqn. (10) into eqn. (12) to eliminate e~j, and then combining with eqns. (6) and (8), we get 1 + EJ: ' • r6(lc+l~)lc
o*-
(14)
Combining eqns. (13) and (14) yields as~r=EcAT(ac - as) 1 EcE~(lc+ I~)4+(E~-E~)(E~I~4-E~I~ 4) r
6E~I~!,(I~+!~)
(15)
'~
Equations (14) and (15) are closed-form solutions for calculating the equilibrium residual stress o* and the saturation stress a~,t respectively. If the substrate has a pre-existing curvature 1/r i before coating, the 1/r in eqns. (14) and (15) must be replaced by 1 / r - 1 / r i. Equation (15) is approximately the same as Brenner and Senderoff's eqn. (33)[11]. Because of the simplification involved in the moment of inertia, Brenner and Senderoff's eqn. (33) is just a simplified form of eqn. (15). The curvature change developed in a coated strip is an index for determining the residual stresses in coatings. For coated systems with the same coating and substrate materials, the variation of curvature with respect to R can be deduced according to eqn. (15).
1.5
When ls is constant, the normalized curvature I~/ [rA T ( a c - as)] decreases with increasing R, implying that the coated strip becomes fiat as R increases (Fig. 3(a)). When 1c is constant, the relationship between the normalized curvature l c / [ r A T ( a c - a ~ ) ] and R is expressed in Fig. 3(b). For a constant R, a greater ratio Ec/E~ results in a greater normalized curvature (Figs. 3(a) and 3(b)). It is clear that Stoney's formula is not equivalent to either eqn. (14) or (15). Although Stoney did not give a clear definition of residual stress~ Stoney's paper implicitly indicated that the stress calculated from eqn. (1) was the equilibrium residual stress [8]. As a result, some researchers [9, 18] used Stoney's formula to evaluate the equilibrium residual stress. In such an operation, the error induced from Stoney's formula, {[eqn. ( 1 ) - eqn. ( 14)]/eqn. ( 14)} x 100%, decreases with increasing R (Fig. 4(a)). However, some researchers [11, 19] considered Stoney's formula as an equation for evaluating the saturation stress. Thus the error induced from Stoney's formula, {[eqn. (1)-eqn. (15)]/eqn. (15)}× 100%, increases with increasing Ec/E~ (Fig. 4(b)). When R ~>1, the calculated result of eqn. (14) converges to that of eqn. (15). When R,> l, Stoney's approximate formula gives a significantly accurate stress evaluation in comparison with eqn. (15). Thus, to decrease the error induced from theoretical deviation, Stoney's formula should be kept under the condition that the ratio of the substrate thickness to the coating thickness is large. Strictly speaking, the error of observed residual stress, through the calculation of Stoney's formula, can originate from not only the theoretical deviation of Stoney's formula but also the experimental uncertainty of curvature measurement. From the theoretical viewpoint, the deviation induced from Stoney's formula can
0.04
(a)
(b)
0.03 0.02
1.0
I I~Ec/Es Z 0.5 N
0.0
o
lOO
: 30 z
2~o
Thlckness RaUo
:(R)
3oo
0
Thi100 ckness RaUo
200
300
(R)
Fig. 3. Non-dimensionalcurvature developed'in a coated strip: The normalized curvature is expressed in terms of (a)L/[rA T( at - a~)] and (b) l~/[rAT(a c - a~)l.
C-C, Chiu 20
/
Residual stresses in ceramic coatings 20-
\
(a)
(b) 0.0
1
10
1"f
-20 IIJ
UJ
-40
> rr0,)
Ec/E s = 30
0.0
143
/
10
-60
n-
~
/
Ec/E s = 30 -80
-10
0
2~0
40
(~0
80
-100
0
do
,'o
t()O
120
Thickness Ratio (R)
Thickness Ratio (R)
Fig. 4. Theoretical approximation-induced error in Stoney's formula when Stoney's formula is used for calculating (a) equilibrium residual stress and (b) saturation stress.
be decreased by increasing R. Unfortunately, an increase in R also decreases the rate of change of the curvature 1/r (Fig. 3). From the practical viewpoint, the absolute uncertainty of an instrument is a constant so that a decrease in the rate of change of 1/r can increase the relative uncertainty of observed residual stress [15, 16]. As a result, an extremely large R along with Stoney's formula seems to be impractical to improve the accuracy of residual stress measurement. For example, the theoretical deviation of Stoney's formula from eqn. (14) is eqn. ( 1 ) - eqn. (14) eqn. (14)
1 0 0 % - E r R : - E~ 100% ~ 100 % E~R ~+ E~ R (16a)
The full curve in Fig. 5 illustrates the theoretical deviation and the horizontal broken line represents the true value calculated from eqn. (14). Assume that the absolute uncertainty of the instrument for curvature measurement is +0.05 m-J. In the case of E c ~ E~, the relative uncertainty of curvature measurement approximates to 0.05 relative uncertainty = w7-, 100% l/r
51 (R 2 + 3 R )
%
6AT(a~-a~) (16b)
An analysis based on the propagation of errors indicates that, if the uncertainty in E~, lc and 1~ is insignificant, eqn. (16b) then accounts for the experimental uncertainty development in the observed residual stress [15, 16]. The relative uncertainty of observed stress, with respect to the full curve, is accordingly described in terms of the broken curves in Fig. 5. The observed stress must be located in the zone between
i ! \
\~~ffrom
"' 0 - \ . . . .
value calculated
equation (1)
~
z.- ..........
cl°sed'f°r m s°luti°n< um:g:r~len~ye~-t-- -- -~
Thickness Ratio (R)
Fig. 5. The propagation of errors through the calculation of Stoney's formula. The analysis superimposes theoretical approximation-induced error on experimental uncertaintyinduced error.
the two broken curves. An extremely large R along with Stoney's formula does not practically increase the accuracy of residual stress measurement. Consequently, a small R and closed-form solutions (eqns. (14) and (15)) are recommended for the residual stress measurement of a coating.
2.2. lnplane elastic modulus of a coating qb determine the inplane normal stress o* or o~t, the elastic moduli of coating and substrate have to be known in advance. E~ can be measured before the coating is applied. E c can be determined through the deflection measurement of one end of a loaded cantilever beam. Figure 6(a) illustrates the schematic of a cantilever beam subjected to a force P at point B. The deflection d~ is calculated by [20] d~ -
P£
3EI
(17)
C.-C. Chiu
144
/
Residual stresses in ceramic coatings P
P
V" (a)
S
4
L
.I "1
(b)
Fig. 6. Schematics of the loading deflection techniques for determining a coating's modulus: (a) a cantilever beam subjected to a loading at one end; and (b) a simple beam in a three-point bending test.
A
B
I
/
/
I I
10/2//
/
/,~
/ /
.- /
/
/ i
4
it,/
X
0 (a)
C
3_
(b)
Fig. 7. Curvature determination of a coated strip: (a) trigonometric relations among the beam's curvature, length and deflection; (b) illustration of measuring the deflection of one end of a coated strip.
where L is the beam's length, extending from point A to point B. I is the second moment of inertia of the beam with respect to the neutral axis. In the case of a coated strip (see Fig. 1 (a)) L-1
EI=E~ f -z
the deflection d 2 at the midpoint is [201 ps 3
d2 = 48E---~
(20)
-i
where S is the span of a three-point bending fixture. Combining eqns. (8), (18) and (20) yields
Y2 dA +E c f y2 dA ~l-lc
b
= ~ {E,[(/~ -/)3 + 13] + Ec[(l + lc)3 _/3]}
(18)
where b is the beam's width. Combining eqns. (8), (17) and (18) gives a formula for calculating the modulus of thecoating:
E _ 1 [PL ~3(E~I2-E~Ic2) 2] - 73 E~l~3 + 1~ [dlb 4 E~/~+~/~ J
(19)
In addition to the deflection of a loaded cantilever beam, the beam deflection in a three-point bending test is also useful for the determination of E~ (see Fig. 6(b)). The relationship between the external loading P and
Ec=/~,3 [16deb
Esl"3+4
Efls+Eclc J
(21)
Equations (19) and (21) are formulae for calculating the inplane elastic modulus of the coating. Although both equations are in implicit form, E c can be solved using a numerical iteration method [21]. An analysis based on the propagation of errors indicates that an accurate l¢ and a small R are helpful for an accurate measurement of Ec [15, 16].
2.3. Curvature measurement As a result of thermal residual stress, a coated strip develops a curvature after coating. Figure 7(a) outlines
C.-C. Chiu
/
Residual stresses in ceramic coatings
the trigonometric relations among the strip's length, curvature and deflection. Assume that the coated strip is anchored at point A. The straight line A-B and arc A-C, having the same length L, represent the strip before coating and after coating respectively. The broken lines are auxiliary lines for helping the trigonometric analysis. The radius of curvature of the arc A-C, having point O as the circle center, can be determined from: (a) measuring the deflection h and (b) measuring the midpoint deflection f. Since the triangle AOE is similar to the triangle ACG, the curvature 1/r is calculated by 1
r
-
1 r
2h 2 g 8f g2 +4f2
(22a)
(22b)
where g is the straight-line distance from point A to point C. Since the angle AOC is calculated as 0 = L/r, we further have 1 - 1 h[ 1 - c ° s ( L ) ] r
(23a)
1 _ 1 1-cos
(23b)
rf
For a coated strip with a small curvature, the magnitude of g is close to L. Thus 1/r calculated from 1 2h -r "~ L 2 1=
r
~ 8f L-+4f 2
(24a)
145
ing thickness was measured by scanning electron microscopy. The curvature r in a coated strip was determined from measuring the deflection of one end of the coated strip. First, one end of the coated strip (point A) was anchored on a horizontal stage made of flat glass plates (Fig. 7(b)). Because of internal stress, the coated strip itself behaved as a curved beam. The deflection h was then measured using a micrometer which was vertically fixed on a large magnet. (The horizontal line A-B was obtained and calibrated by pressing a flat glass plate on the stage.) After measuring the deflection h, we evaluated an approximate curvature through eqn. (24a). Using the approximate curvature as an initial value, we further calculated the exact solution through eqn. (23a) by means of a numerical iteration method. The equilibrium residual stresses a* and saturation stress o~,~ were then calculated using eqn. (14) and eqn. (15) respectively. Monolithic graphite specimens 4.9 cm × 0.8 cm × 0.15 cm (without coatings) were also cut from the asreceived billets for elastic modulus measurement. The moduli were measured using a standing wave resonance method which was described elsewhere [22, 23]. The inplane elastic modulus of TiC coating was determined from measuring the deflection of one end of the coated strip which was subjected to a concentrated loading (see Fig. 6(a)). The coated strip was first anchored on a horizontal stage. The end of the beam was then hung with a weight, ranging from 1 to 10 gf. The deflection dl was measured using a micrometer and E c was calculated using eqn. (19). For comparison, E c was also statistically inferred from the curvatures in coated strips (see the Appendix).
(24b)
is a good approximation, in comparison with the exact solution calculated from eqns. (23a) and (23b).
3. Experimental procedures
TiC coating/graphite substrate composites prepared by the chemical vapor deposition technique (The Carbon/Graphite Group, Dallas, TX) were used for this experiment. Using a diamond saw, as-received billets were cut into coated strips about 4.9 cm × 0.8 cm × 0.1 cm with coating on a single 4.9 cm × 0.8 cm surface. The uncoated surfaces of graphite substrates were further polished using 600 grit SiC polishing paper to prepare coated strips with various substrate thicknesses. The specimen's thickness was determined to within + 0.002 cm using a micrometer and the coat-
4. Results and discussion
Figure 8 is a scanning electron micrograph of the fracture surface of a TiC coating/graphite composite, showing that the coating thickness is around 5.2/am. In this study, the coating thickness of tested specimens is approximately constant but the substrate thickness varies. Figure 9 illustrates the relationship between the curvature and the ratio of the substrate thickness to the coating thickness. The curvature of coated strips decreases with increasing R, which agrees with the theoretical prediction in Fig. 3. The inplane elastic moduli of TiC coatings were determined from: (1) the statistical inference of the curvatures in coated strips (see the Appendix), and (2) the loading deflection measurement of one end of a coated strip. The statistical inference yields Ec=391 GPa. However, the loading deflection method gives a mean modulus Ec = 428 GPa with a standard deviation of 57
C.-C. Chiu
146
/
Residual stresses in ceramic coatings 400o. ; f.
300 o
200
: , .s o /,f,
0 m
,: ~,o~ .~.o ~''~ •
.3 !00 o"
i
LU
0
100
2()0
300
400
Thickness Ratio (R)
Fig. 10. Equilibrium residual stress developed in TiC coating/ graphite substrate composites. Fig. 8. A scanning electron micrograph of a TiC coating/graphite substrate composite.
~2'
\
0
3 E
\
\
EsR(E c+ E~R 3) o*= 2844(Ec + E~R3)(Ec + E~R)_3(Ec_ E~R2)2 (25)
-i
u
o\
0
0
1()0
200 3()0 Thickness Ratio (R)
The saturation stress in TiC coating/graphite substrate composites is 284 MPa with a standard deviation of 33 MPa. Since eqn. (A5) originates from eqn. (15), both equations give the same result. Generally speaking, substituting the saturation stress E~AT(a~-a~) into eqn. (13a) can yield a useful equation for predicting the equilibrium residual stress in TiC coatings. For example
400
As a result, a* in a coated strip with Ic = 0.2/~m and ls = 100 ~m can be inferred to be 216 MPa, without a direct experimental measurement.
Fig. 9. Curvature developed in TiC coating/graphite substrate composites, as a function of the ratio of substrate thickness to coating thickness.
5. Summary
GPa. From the viewpoint of statistics, the experimental results do not contradict each other, since E¢=391 GPa is still within the interval E c = 428 + 57 GPa. As a result, both methods should be identical for the inplane elastic modulus determination of the coating. The standing wave resonance method reveals that the modulus of graphite substrate is E~ = 10.0 GPa. In the present study, E c = 391 GPa and E, = 10.0 GPa are adopted for the residual stress calculation of coating. T h e curvature developed in a coated strip is an index from which the residual stress can be determined. Since all coated strips exhibited concave curvature on the coated side, the residual stresses in TiC coatings are in tension. The experimental data in Fig. 10 illustrate the variation of the equilibrium residual stresses in TiC coatings with respect to R. The tensile stress is a function of R, which agrees with the theoretical prediction (Fig. 2).
According to the beam theory and the curvature developed in a coated strip, formulae have been derived for the determination of the inplane residual stress and elastic modulus of a coating. Theoretical analyses indicate that: (1) equilibrium residual stress in coatings is a function of the ratio R of the substrate thickness to the coating thickness, (2) the equilibrium residual stress increases with increasing R until a saturation stress is reached, (3) the saturation stress is a constant for the coated systems with the same coating and substrate materials, (4) the curvature developed in a coated strip increases with decreasing R, (5) a small R is advantageous for the determination of residual stress and elastic modulus, and (6) a coating's modulus plays a significant role in determining the residual stress when R is small. The TiC coating/graphite substrate composite is representative of a soft substrate deposited with a hard coating.' The experimental results show that: (1) accur-
C.-C. Chiu
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Residual stresses in ceramic coatings
ate m e a s u r e m e n t s of coating thickness and curvature are basic necessities to evaluate the elastic m o d u l u s and residual stress in the coating, (2) the loading deflection m e t h o d and the curvature inference m e t h o d have identical functions in determining the elastic m o d u l u s of a coating, (3) the m o d u l u s of T i C coating is 391 GPa, (4) the residual stresses in T i C coatings are in tension, and (5) the saturation stress in T i C coating/ graphite systems is 284 MPa.
Acknowledgments T h e a u t h o r gratefully acknowledges Burl M. M o o n ( T h e C a r b o n / G r a p h i t e G r o u p , Dallas, TX) for providing T i C coating/graphite substrate specimens.
References I R. F. Bunshah, Deposition Technologies for Hlms and Coat.ings, Noyes Publications, Park Ridge, N J, 1982, pp. 158. 2 E. Suhir, Stress in bi-metal thermostats, J. Appl. Mech., 53 (9)(1986)657-660. 3 A. Blech and A. A. Levi, Comments on Aleck's stress distribution in clamped plates, J. Appl. Mech., 48 (6) (1981) 442-445. 4 J. C. Lambropoulos and S. M. Wan, Stress concentration along interfaces of elastic-plastic thin films, Mater. Sci. Eng. A, 107(1989)169-175. 5 R. W. Hoffman, Overview of the solid-solid interface: mechanical stability, Mater. Sci. Eng., 53 (1982) 37-46. 6 A. G. Evans and J. W. Hutchinson, On the mechanics of delamination and spalling in compressive films, Int. J. So6ds Struct., 20 (5) (1984) 455-466. 7 A. S. Argon, V. Gupta, H. S. Landis and J. A. Cornie, Intrinsic toughness of interfaces between SiC coatings and substrates of Si or C fiber, J. Mater. Sci., 24 (1989) 1207-1218. 8 G. G. Stoney, The tension of metallic films deposited by electrolysis, Proc. R. Soc. London, Ser. A, 82 (1909) 172-175. 9 R. J. Jaccodine and W. A. Schlegel, Measurement of strains at Si-SiO, interface, J. Appl. Phys., 37 (6) (1966) 2429-2436. l I) R. W. Hoffman, The mechanical properties of thin condensed films, in Physics of Thin Films, Vol, 3, Academic Press, New York, NY, 1966, pp. 211-273. 11 A. Brenner and S. Senderoff, Calculation of stress in electrodeposits from the curvature of a plated strip, J. Res. Nat. Bur. Stand., 42 (2)(1949) 105-123. 12 P. A. Flinn, D. S. Gardner and W. D. Nix, Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history, IEEE Trans. Electron Devices, 34 (3) (1987) 689-699. 13 K. Roll, Analysis of stress and strain distribution in thin films and substrates, J. Appl. Phys., 47(7) (1976) 3224-3229. 14 P. H. Townsend and D. W. Barnett, Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate, J. Appl. Phys., 62 (11)(1987) 4438-4444.
147
15 J. R. Taylor, An Introduction to Error Analysis, University Science Books, Mill Valley, CA, 1982, 40-74. 16 P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, NY, 1969, p. 3. 17 S. P. Timoshenko, Analysis of bi-metal thermostats, J, Opt. Soc. Am., II (1925) 233-255. 18 M. F. Gruniner, B. R. Lawn and E. N. Farabaugh, Measurement of residual stresses in coatings on brittle substrates by indentation fracture, J. Am. Ceram. Sot., 70 (5) (1987) 344-348. 19 E. M. Corcoran, Determining stresses in organic coatings using plates beam deflection, J. Paint Technol., 41 (538) (1969) 630-640. 20 S. P. Timoshenko and J. M. Gere, Mechanics of Materials, Van Nostrand Reinhold, New York, NY, 1972. 21 C. F. Gerald, Applied Numerical Analysis, Addison-Wesley, Menlo Park, CA, 1970, 1-20. 22 C.-C. Chiu and E. D. Case, Elastic modulus determination of coating layers as applied to layered ceramic composites, Mater. Sci. Eng. A, 132 ( 1991 ) 39-47. 23 Standard test method for Young's modulus, shear modulus, and Poisson's ratio for glass and glass-ceramics by resonance, ASTM Standard, C023-71, American Society for Testing and Materials, Reapproved 1981.
Appendix: Determination of a coating's elastic modulus T h e E c determination f r o m the loading deflection m e t h o d s described a b o v e d e p e n d s o n an action of external loading. W i t h o u t the help of external loading, E c also can be statistically inferred f r o m the curvatures d e v e l o p e d in a g r o u p of c o a t e d strips. T h e inference is p e r f o r m e d a c c o r d i n g to the least-squares m e t h o d along with eqn. (15). For example, eqn. ( 1 5 ) i s rewritten in terms of a simple f o r m 1 EcE~(I~+ l~)4+ (E~ _ E~)(E~I~ 4 - Eft.,a) O'sa [ =
--
r
6Eflcl~(lc+l~)
= [A]
(A1)
For coated systems with the same coating and substrate materials, the saturation stress a~, is a constant. In addition, the relationships a m o n g the saturation stress, curvature and material p a r a m e t e r s have to o b e y equation (A1). T h u s let us define the sum of squares for e r r o r SSE as /t
SSE = Z (o~,,t-[A],) ~
(A2)
i=|
where n is the n u m b e r of experimental data points. [A] i represents the value o b t a i n e d by substituting the ith data point (E~, l~, Ic, r) i into [A]. SSE is a function of O~a, and Ec. A necessary condition for SSE to be a mini-
148
C.-C. Chiu
/
Residual stresses in ceramic coatings tt
mum is a(SSE)
o~,a,= - Z [A]i
(A3a)
-0
(AS)
F/i= I
where O(SSE) OE~
0
(A3b)
Substituting eqn. (A2) into eqns. (A3a) and (A3b) yields n
tt
tl
Z [A]~ ~, [B]~-n ~ [A]~[B]~=0 i=I
i=1
i=1
(A4)
[B] =
2Ec/c 4 + E~[(/c +/~)4 _ lc 4 _ L4)]
6rE~lcl~(lc + l~)
Equation (A4) is a quadratic equation and only a function of E c. Equation (A4) can be solved using a numerical iteration method. After E c is obtained, asat is calculated using eqn. (A5).
139
Residual stresses in ceramic coatings as determined from the curvature of a coated strip Chin-Chen Chiu Department of Metallur~', Mechanics, and Materials Science, Michigan State University, East Lansing, M148824 (USA~ (Received May 6, 1991 in revised form September 12, 1991 )
Abstract It is known that the curvature change developed in a coated strip is an index for determining the residual stresses in coatings. This paper uses beam theory to derive closed-form solutions for the stress analysis. In addition to theoretical derivation, experimental measurements are performed on TiC coating/graphite substrate composites. The experimental results reveal the presence of residual tensile stress in TiC coatings.
1. Introduction Ceramic coatings are commonly designed to protect substrates from erosive wear, chemical attack or hightemperature oxidation [ll. Because of the thermal expansion mismatch between the coatings and the substrates, the coated systems can be in a state of thermally induced residual stress; shear stress and peeling stress develop near the free edge and inplane normal stress occurs in the interior region [2-4]. The shear stress and peeling stress may delaminate the coatings from the free edge. The inplane normal stress may result in the coatings cracking, spalling or buckling [4-7]. Thus residual stress measurement is a basic study for characterizing the reliability and performance of coatings. (In this paper, our study focuses on the inplane normal stress measurement of coatings.) According to beam theory and the curvature developed in a coated strip, Stoney first in 1909 proposed an approximate formula for calculating the inplane normal stress of a coating [8]
o*: EJ' -
6rl~,
i l)
where the subscripts c and s refer to the properties of the coating and the substrate respectively. E and I are the elastic modulus and the thickness respectively, r is the radius of curvature of the substrate after coating. (Later investigators directly replaced the E~ in eqn. (1) by E~/( 1 - v~) in order to extend the one-dimensional solution to plane stress, where v is Poisson's ratio [9, I)921-5093/92/$5.00
101.) Stoney's formula is popular for the evaluation of residual stress of coatings [9-14], which may be attributed to its simplicity and convenience for calculation. As a result of approximations involved in the theoretical derivation [8, 10-12], eqn. (1) must be restricted to a condition that the ratio of the substrate thickness to the coating thickness is large. However, an analysis based on the propagation of errors [ 15, 16] reveals that an increase in the thickness ratio can increase the relative uncertainty of observed residual stress. Thus Stoney's formula should be analyzed for its reliance on the residual stress evaluation. In the present study, we derive formulas for the inplane normal stress calculation and analyze the deviation induced from Stoney's formula. Equations for the inplane elastic modulus calculation of a coating are also presented. In addition, the residual stresses and modulus of TiC thin coatings on graphite substrates are experimentally determined.
2. Theoretical background
2.1, Inplane normal stress of a coating When a coating/substrate composite is produced at elevated temperature and cooled down, the thermal expansion mismatch between the coating and the substrate results in thermal residual stresses. The stresses translate between the coating and the substrate by shear at the interface, causing the coated systems to contract, elongate, bend or twist. In this paper, we use © 1992 - Elsevier Sequoia. All rights reserved
140
C.-C. Chiu
/
Residual stresses in ceramic coatings
satisfy dimensional match and to conform with coating adhesion, the coating layer and the substrate must be elongated and contracted respectively in the Z direction. Thus an axial tensile stress ac~ develops in the coating and an axial compressive stress a~j forms in the substrate (Fig. 1 (c)). The corresponding strains ec, and e~l are quantified by
beam theory to develop the relationship for determining the inplane normal stress which occurs in the internal region of a coated strip. When the coated.strip is long enough, end effects alter the stress significantly only near the end of the strip. Thus the beam theory can give sufficiently accurate evaluation of the inplane normal stress of a coating [2, 17]. Figure 1 illustrates the analytical framework of the internal stress equilibrium in a coated strip. Figure 1 (a) represents the cross-section of the coated strip which extends in the direction of the Z axis. The thicknesses of the coating and the substrate are l~ and l~ respectively. The distance from the neutral axis to the coating-substrate interface is I. If the coating has the same modulus as the substrate, the neutral axis will coincide with the centroid. For convenience, it is assumed that the coating has a larger elastic modulus than the substrate. The neutral axis then deviates from the centroid and approaches the interface. In general, coating procedures can be considered as forced adhesion at the fabrication temperature between the substrate and the coating. After cooling to room temperature, internal stress arises in the coated systems because of thermal expansion mismatch. For convenience, the thermal expansion coefficient of the coating a~ is assumed to be larger than that of the substrate a,. Thus the coating layer has a smaller room temperature length than the substrate (Fig. l(b)). To
(a)
A T ( a c - a~)= m = ec~ - e~l
(2a)
Ecec~ l~ + E~e~ [~= 0
(2b)
where A T is the temperature difference between the fabrication temperature (or annealing temperature) and room temperature. From the mechanics viewpoint, the system has to satisfy an internal stress equilibrium, namely ; o dA = 0
(3a)
faY dA = 0
(3b)
o is the normal stress in the direction of the Z axis. A is the cross-sectional area of the beam. Equations (3a) and (3b) represent the force and the bending moment equilibrium respectively. Since the stress state in Fig. 1(c) does not satisfy eqn. (3b) yet, the coated systems must be bent to some extent. The bending action results in another stress distribution in the system, which is shown in Fig. 1 (d). The corresponding surface stress and surface strain are oc_~,
Y Y
T
substrate
-f 2S
centroid
X,i~r
neutral axis interface
14~c --f
L.,-_-'~_ ......
J. - - -
-
z Y
C)
(d)
';s ~sl
+
(e)
%2
7
Y
2s
z neutral
q acl-
Ec~cl
/ ~ axis ~ l interface at2- Ec~c2
average residual stress Fig. 1. Analytical frameworks of the residual stresses in a coated strip: (a) cross-section; (b) dissection of a coated strip; (c) stress distribution as a result of axial contraction and elongation; (d) stress distribution due to pure bending moment; (e) equilibrium residual stress in a coated strip.
C.-C. Chiu / Residualstressesin ceramic"coatings 0~2, el22and e~2. According to stress superposition, the final state of the internal stress equilibrium is expressed in Fig. l(e). For a given coated strip, the residual stress distribution (Fig. 1 (e)) is unique and the self-equilibrium stress state must correspond to minimum elastic strain energy. The stress resolution reveals the existence and uniqueness of the pure bending moment (Fig. 1 (d)) and uniform axial stress (Fig. l(c)). Thus the elastic strain energy G, related only to the axial stress, also reaches the minimum value. (To satisfy the internal stress equilibrium, it is necessary and sufficient for G to reach the minimum value.) Therefore the first derivative of G with respect to the el2~is equal to zero: G-ll2E~e~*-+ l'£*e~'2
(4)
2
dG --=0 dgcl
(5)
The residual stress distribution in the coating layer is not constant (Fig. l(e)). Thus the average residual stresses o* are proposed and defined as _~ [" ) [o~.,(Y) + o~2(Y)] dA o* = ~/~
(11)
-t
f dA t
The average residual stress corresponds to the equilibrium stress in a real coated component (see Fig. l(e)), which is a function of the ratio R of substrate thickness to coating thickness. Combining eqns. (7d), (7e) and ( 11 ) gives
o* = El2ec, + -El2 - [ l 2 - ( l + l~)2] 2rl~.
(12)
Substituting eqn. (10) into eqn. (12) to eliminate r, and then combining with eqns. (6) and (8), we have
Combining eqns. (2a), (4) and (5) gives
e~, -
141
o* = AT(a~ - a,)Ec
1~E~ m
(6)
I~E~+I,E,
According to the force equilibrium (eqn. (3a)), the internal stress state in Fig. 1 (e) must satisfy I,-I
-I
f [o~,(Y)+o~2(Y)]dA+ f / where
[o~,(Y)+o~e(Y)]dA=O
-i-/~
E~R(El2 + E~R ~) 4(E~+E~R~)(E~+E~R)_3(EI2_E~R2) 2 where
R=I_,
(7a)
o<(Y)=E~e~,
(7b)
0~2(Y) = YEffr
(7c)
ol2,(Y)=El2el2,
(7d)
0122(Y) = YEl2/r
(7e)
r is the radius of curvature of the neutral axis. Substituting eqns. (2b), (7b), (7c), (7d) and (7e)into eqn. (7a) yields
(13a)
(13b)
1~ Equation (13a) expresses the relationship between the average residual stress development and the material parameters, o* is a function of R and the coating-tosubstrate elastic modulus ratio EJE, (Fig. 2). As R A =
C o
100
80
,,,j
E~I=-E~,I,. 2 l -
(8)
2Eft,+ 2E~(
According to the bending moment equilibrium (eqn. (3b)), the internal stress state in Fig. 1 (e) has to satisfy /, /
-/
f o~2(Y)YdA+ f I
-I
m
60
E c / E s : 30 w
"o
40
20
(D
o~2(Y)YdA=lc+l~El2e~,ll2 2 I~
(9)
n-
0.0
0
50
1()0
150
200
250
300
Thlckness Ratlo (R)
Substituting eqns. (7c) and (Te) into eqn. (9), we have
1 3
(ll2+l,)E~el2, l~
7 = 2 E~[(!,-t) ~ + ~3]+ E~[(I + t~)~ _/~]
(10)
Fig. 2. Variation of equilibrium residual stress with respect to Ec/E~ and R. The magnitude of equilibrium residual stress is expressed in terms of the percentage of saturation stress
EcAT(%-a,).
142
C-C. Chiu
/
Residual stresses in ceramic coatings
increases, o* monotonically increases and then converges to a saturation stress E c A T ( a ~ - a s ) . The greater ratio of E~/E~ will cause o* to converge slower to the saturation stress as~,. In practice, o* represents the equilibrium residual stress in a real coated component, which plays a role in the residual stress-induced damage. However, o~,t is only an index, showing the possibly maximum stress in components with the same coating and substrate materials. Substituting eqn. (10) into eqn. (12) to eliminate e~j, and then combining with eqns. (6) and (8), we get 1 + EJ: ' • r6(lc+l~)lc
o*-
(14)
Combining eqns. (13) and (14) yields as~r=EcAT(ac - as) 1 EcE~(lc+ I~)4+(E~-E~)(E~I~4-E~I~ 4) r
6E~I~!,(I~+!~)
(15)
'~
Equations (14) and (15) are closed-form solutions for calculating the equilibrium residual stress o* and the saturation stress a~,t respectively. If the substrate has a pre-existing curvature 1/r i before coating, the 1/r in eqns. (14) and (15) must be replaced by 1 / r - 1 / r i. Equation (15) is approximately the same as Brenner and Senderoff's eqn. (33)[11]. Because of the simplification involved in the moment of inertia, Brenner and Senderoff's eqn. (33) is just a simplified form of eqn. (15). The curvature change developed in a coated strip is an index for determining the residual stresses in coatings. For coated systems with the same coating and substrate materials, the variation of curvature with respect to R can be deduced according to eqn. (15).
1.5
When ls is constant, the normalized curvature I~/ [rA T ( a c - as)] decreases with increasing R, implying that the coated strip becomes fiat as R increases (Fig. 3(a)). When 1c is constant, the relationship between the normalized curvature l c / [ r A T ( a c - a ~ ) ] and R is expressed in Fig. 3(b). For a constant R, a greater ratio Ec/E~ results in a greater normalized curvature (Figs. 3(a) and 3(b)). It is clear that Stoney's formula is not equivalent to either eqn. (14) or (15). Although Stoney did not give a clear definition of residual stress~ Stoney's paper implicitly indicated that the stress calculated from eqn. (1) was the equilibrium residual stress [8]. As a result, some researchers [9, 18] used Stoney's formula to evaluate the equilibrium residual stress. In such an operation, the error induced from Stoney's formula, {[eqn. ( 1 ) - eqn. ( 14)]/eqn. ( 14)} x 100%, decreases with increasing R (Fig. 4(a)). However, some researchers [11, 19] considered Stoney's formula as an equation for evaluating the saturation stress. Thus the error induced from Stoney's formula, {[eqn. (1)-eqn. (15)]/eqn. (15)}× 100%, increases with increasing Ec/E~ (Fig. 4(b)). When R ~>1, the calculated result of eqn. (14) converges to that of eqn. (15). When R,> l, Stoney's approximate formula gives a significantly accurate stress evaluation in comparison with eqn. (15). Thus, to decrease the error induced from theoretical deviation, Stoney's formula should be kept under the condition that the ratio of the substrate thickness to the coating thickness is large. Strictly speaking, the error of observed residual stress, through the calculation of Stoney's formula, can originate from not only the theoretical deviation of Stoney's formula but also the experimental uncertainty of curvature measurement. From the theoretical viewpoint, the deviation induced from Stoney's formula can
0.04
(a)
(b)
0.03 0.02
1.0
I I~Ec/Es Z 0.5 N
0.0
o
lOO
: 30 z
2~o
Thlckness RaUo
:(R)
3oo
0
Thi100 ckness RaUo
200
300
(R)
Fig. 3. Non-dimensionalcurvature developed'in a coated strip: The normalized curvature is expressed in terms of (a)L/[rA T( at - a~)] and (b) l~/[rAT(a c - a~)l.
C-C, Chiu 20
/
Residual stresses in ceramic coatings 20-
\
(a)
(b) 0.0
1
10
1"f
-20 IIJ
UJ
-40
> rr0,)
Ec/E s = 30
0.0
143
/
10
-60
n-
~
/
Ec/E s = 30 -80
-10
0
2~0
40
(~0
80
-100
0
do
,'o
t()O
120
Thickness Ratio (R)
Thickness Ratio (R)
Fig. 4. Theoretical approximation-induced error in Stoney's formula when Stoney's formula is used for calculating (a) equilibrium residual stress and (b) saturation stress.
be decreased by increasing R. Unfortunately, an increase in R also decreases the rate of change of the curvature 1/r (Fig. 3). From the practical viewpoint, the absolute uncertainty of an instrument is a constant so that a decrease in the rate of change of 1/r can increase the relative uncertainty of observed residual stress [15, 16]. As a result, an extremely large R along with Stoney's formula seems to be impractical to improve the accuracy of residual stress measurement. For example, the theoretical deviation of Stoney's formula from eqn. (14) is eqn. ( 1 ) - eqn. (14) eqn. (14)
1 0 0 % - E r R : - E~ 100% ~ 100 % E~R ~+ E~ R (16a)
The full curve in Fig. 5 illustrates the theoretical deviation and the horizontal broken line represents the true value calculated from eqn. (14). Assume that the absolute uncertainty of the instrument for curvature measurement is +0.05 m-J. In the case of E c ~ E~, the relative uncertainty of curvature measurement approximates to 0.05 relative uncertainty = w7-, 100% l/r
51 (R 2 + 3 R )
%
6AT(a~-a~) (16b)
An analysis based on the propagation of errors indicates that, if the uncertainty in E~, lc and 1~ is insignificant, eqn. (16b) then accounts for the experimental uncertainty development in the observed residual stress [15, 16]. The relative uncertainty of observed stress, with respect to the full curve, is accordingly described in terms of the broken curves in Fig. 5. The observed stress must be located in the zone between
i ! \
\~~ffrom
"' 0 - \ . . . .
value calculated
equation (1)
~
z.- ..........
cl°sed'f°r m s°luti°n< um:g:r~len~ye~-t-- -- -~
Thickness Ratio (R)
Fig. 5. The propagation of errors through the calculation of Stoney's formula. The analysis superimposes theoretical approximation-induced error on experimental uncertaintyinduced error.
the two broken curves. An extremely large R along with Stoney's formula does not practically increase the accuracy of residual stress measurement. Consequently, a small R and closed-form solutions (eqns. (14) and (15)) are recommended for the residual stress measurement of a coating.
2.2. lnplane elastic modulus of a coating qb determine the inplane normal stress o* or o~t, the elastic moduli of coating and substrate have to be known in advance. E~ can be measured before the coating is applied. E c can be determined through the deflection measurement of one end of a loaded cantilever beam. Figure 6(a) illustrates the schematic of a cantilever beam subjected to a force P at point B. The deflection d~ is calculated by [20] d~ -
P£
3EI
(17)
C.-C. Chiu
144
/
Residual stresses in ceramic coatings P
P
V" (a)
S
4
L
.I "1
(b)
Fig. 6. Schematics of the loading deflection techniques for determining a coating's modulus: (a) a cantilever beam subjected to a loading at one end; and (b) a simple beam in a three-point bending test.
A
B
I
/
/
I I
10/2//
/
/,~
/ /
.- /
/
/ i
4
it,/
X
0 (a)
C
3_
(b)
Fig. 7. Curvature determination of a coated strip: (a) trigonometric relations among the beam's curvature, length and deflection; (b) illustration of measuring the deflection of one end of a coated strip.
where L is the beam's length, extending from point A to point B. I is the second moment of inertia of the beam with respect to the neutral axis. In the case of a coated strip (see Fig. 1 (a)) L-1
EI=E~ f -z
the deflection d 2 at the midpoint is [201 ps 3
d2 = 48E---~
(20)
-i
where S is the span of a three-point bending fixture. Combining eqns. (8), (18) and (20) yields
Y2 dA +E c f y2 dA ~l-lc
b
= ~ {E,[(/~ -/)3 + 13] + Ec[(l + lc)3 _/3]}
(18)
where b is the beam's width. Combining eqns. (8), (17) and (18) gives a formula for calculating the modulus of thecoating:
E _ 1 [PL ~3(E~I2-E~Ic2) 2] - 73 E~l~3 + 1~ [dlb 4 E~/~+~/~ J
(19)
In addition to the deflection of a loaded cantilever beam, the beam deflection in a three-point bending test is also useful for the determination of E~ (see Fig. 6(b)). The relationship between the external loading P and
Ec=/~,3 [16deb
Esl"3+4
Efls+Eclc J
(21)
Equations (19) and (21) are formulae for calculating the inplane elastic modulus of the coating. Although both equations are in implicit form, E c can be solved using a numerical iteration method [21]. An analysis based on the propagation of errors indicates that an accurate l¢ and a small R are helpful for an accurate measurement of Ec [15, 16].
2.3. Curvature measurement As a result of thermal residual stress, a coated strip develops a curvature after coating. Figure 7(a) outlines
C.-C. Chiu
/
Residual stresses in ceramic coatings
the trigonometric relations among the strip's length, curvature and deflection. Assume that the coated strip is anchored at point A. The straight line A-B and arc A-C, having the same length L, represent the strip before coating and after coating respectively. The broken lines are auxiliary lines for helping the trigonometric analysis. The radius of curvature of the arc A-C, having point O as the circle center, can be determined from: (a) measuring the deflection h and (b) measuring the midpoint deflection f. Since the triangle AOE is similar to the triangle ACG, the curvature 1/r is calculated by 1
r
-
1 r
2h 2 g 8f g2 +4f2
(22a)
(22b)
where g is the straight-line distance from point A to point C. Since the angle AOC is calculated as 0 = L/r, we further have 1 - 1 h[ 1 - c ° s ( L ) ] r
(23a)
1 _ 1 1-cos
(23b)
rf
For a coated strip with a small curvature, the magnitude of g is close to L. Thus 1/r calculated from 1 2h -r "~ L 2 1=
r
~ 8f L-+4f 2
(24a)
145
ing thickness was measured by scanning electron microscopy. The curvature r in a coated strip was determined from measuring the deflection of one end of the coated strip. First, one end of the coated strip (point A) was anchored on a horizontal stage made of flat glass plates (Fig. 7(b)). Because of internal stress, the coated strip itself behaved as a curved beam. The deflection h was then measured using a micrometer which was vertically fixed on a large magnet. (The horizontal line A-B was obtained and calibrated by pressing a flat glass plate on the stage.) After measuring the deflection h, we evaluated an approximate curvature through eqn. (24a). Using the approximate curvature as an initial value, we further calculated the exact solution through eqn. (23a) by means of a numerical iteration method. The equilibrium residual stresses a* and saturation stress o~,~ were then calculated using eqn. (14) and eqn. (15) respectively. Monolithic graphite specimens 4.9 cm × 0.8 cm × 0.15 cm (without coatings) were also cut from the asreceived billets for elastic modulus measurement. The moduli were measured using a standing wave resonance method which was described elsewhere [22, 23]. The inplane elastic modulus of TiC coating was determined from measuring the deflection of one end of the coated strip which was subjected to a concentrated loading (see Fig. 6(a)). The coated strip was first anchored on a horizontal stage. The end of the beam was then hung with a weight, ranging from 1 to 10 gf. The deflection dl was measured using a micrometer and E c was calculated using eqn. (19). For comparison, E c was also statistically inferred from the curvatures in coated strips (see the Appendix).
(24b)
is a good approximation, in comparison with the exact solution calculated from eqns. (23a) and (23b).
3. Experimental procedures
TiC coating/graphite substrate composites prepared by the chemical vapor deposition technique (The Carbon/Graphite Group, Dallas, TX) were used for this experiment. Using a diamond saw, as-received billets were cut into coated strips about 4.9 cm × 0.8 cm × 0.1 cm with coating on a single 4.9 cm × 0.8 cm surface. The uncoated surfaces of graphite substrates were further polished using 600 grit SiC polishing paper to prepare coated strips with various substrate thicknesses. The specimen's thickness was determined to within + 0.002 cm using a micrometer and the coat-
4. Results and discussion
Figure 8 is a scanning electron micrograph of the fracture surface of a TiC coating/graphite composite, showing that the coating thickness is around 5.2/am. In this study, the coating thickness of tested specimens is approximately constant but the substrate thickness varies. Figure 9 illustrates the relationship between the curvature and the ratio of the substrate thickness to the coating thickness. The curvature of coated strips decreases with increasing R, which agrees with the theoretical prediction in Fig. 3. The inplane elastic moduli of TiC coatings were determined from: (1) the statistical inference of the curvatures in coated strips (see the Appendix), and (2) the loading deflection measurement of one end of a coated strip. The statistical inference yields Ec=391 GPa. However, the loading deflection method gives a mean modulus Ec = 428 GPa with a standard deviation of 57
C.-C. Chiu
146
/
Residual stresses in ceramic coatings 400o. ; f.
300 o
200
: , .s o /,f,
0 m
,: ~,o~ .~.o ~''~ •
.3 !00 o"
i
LU
0
100
2()0
300
400
Thickness Ratio (R)
Fig. 10. Equilibrium residual stress developed in TiC coating/ graphite substrate composites. Fig. 8. A scanning electron micrograph of a TiC coating/graphite substrate composite.
~2'
\
0
3 E
\
\
EsR(E c+ E~R 3) o*= 2844(Ec + E~R3)(Ec + E~R)_3(Ec_ E~R2)2 (25)
-i
u
o\
0
0
1()0
200 3()0 Thickness Ratio (R)
The saturation stress in TiC coating/graphite substrate composites is 284 MPa with a standard deviation of 33 MPa. Since eqn. (A5) originates from eqn. (15), both equations give the same result. Generally speaking, substituting the saturation stress E~AT(a~-a~) into eqn. (13a) can yield a useful equation for predicting the equilibrium residual stress in TiC coatings. For example
400
As a result, a* in a coated strip with Ic = 0.2/~m and ls = 100 ~m can be inferred to be 216 MPa, without a direct experimental measurement.
Fig. 9. Curvature developed in TiC coating/graphite substrate composites, as a function of the ratio of substrate thickness to coating thickness.
5. Summary
GPa. From the viewpoint of statistics, the experimental results do not contradict each other, since E¢=391 GPa is still within the interval E c = 428 + 57 GPa. As a result, both methods should be identical for the inplane elastic modulus determination of the coating. The standing wave resonance method reveals that the modulus of graphite substrate is E~ = 10.0 GPa. In the present study, E c = 391 GPa and E, = 10.0 GPa are adopted for the residual stress calculation of coating. T h e curvature developed in a coated strip is an index from which the residual stress can be determined. Since all coated strips exhibited concave curvature on the coated side, the residual stresses in TiC coatings are in tension. The experimental data in Fig. 10 illustrate the variation of the equilibrium residual stresses in TiC coatings with respect to R. The tensile stress is a function of R, which agrees with the theoretical prediction (Fig. 2).
According to the beam theory and the curvature developed in a coated strip, formulae have been derived for the determination of the inplane residual stress and elastic modulus of a coating. Theoretical analyses indicate that: (1) equilibrium residual stress in coatings is a function of the ratio R of the substrate thickness to the coating thickness, (2) the equilibrium residual stress increases with increasing R until a saturation stress is reached, (3) the saturation stress is a constant for the coated systems with the same coating and substrate materials, (4) the curvature developed in a coated strip increases with decreasing R, (5) a small R is advantageous for the determination of residual stress and elastic modulus, and (6) a coating's modulus plays a significant role in determining the residual stress when R is small. The TiC coating/graphite substrate composite is representative of a soft substrate deposited with a hard coating.' The experimental results show that: (1) accur-
C.-C. Chiu
/
Residual stresses in ceramic coatings
ate m e a s u r e m e n t s of coating thickness and curvature are basic necessities to evaluate the elastic m o d u l u s and residual stress in the coating, (2) the loading deflection m e t h o d and the curvature inference m e t h o d have identical functions in determining the elastic m o d u l u s of a coating, (3) the m o d u l u s of T i C coating is 391 GPa, (4) the residual stresses in T i C coatings are in tension, and (5) the saturation stress in T i C coating/ graphite systems is 284 MPa.
Acknowledgments T h e a u t h o r gratefully acknowledges Burl M. M o o n ( T h e C a r b o n / G r a p h i t e G r o u p , Dallas, TX) for providing T i C coating/graphite substrate specimens.
References I R. F. Bunshah, Deposition Technologies for Hlms and Coat.ings, Noyes Publications, Park Ridge, N J, 1982, pp. 158. 2 E. Suhir, Stress in bi-metal thermostats, J. Appl. Mech., 53 (9)(1986)657-660. 3 A. Blech and A. A. Levi, Comments on Aleck's stress distribution in clamped plates, J. Appl. Mech., 48 (6) (1981) 442-445. 4 J. C. Lambropoulos and S. M. Wan, Stress concentration along interfaces of elastic-plastic thin films, Mater. Sci. Eng. A, 107(1989)169-175. 5 R. W. Hoffman, Overview of the solid-solid interface: mechanical stability, Mater. Sci. Eng., 53 (1982) 37-46. 6 A. G. Evans and J. W. Hutchinson, On the mechanics of delamination and spalling in compressive films, Int. J. So6ds Struct., 20 (5) (1984) 455-466. 7 A. S. Argon, V. Gupta, H. S. Landis and J. A. Cornie, Intrinsic toughness of interfaces between SiC coatings and substrates of Si or C fiber, J. Mater. Sci., 24 (1989) 1207-1218. 8 G. G. Stoney, The tension of metallic films deposited by electrolysis, Proc. R. Soc. London, Ser. A, 82 (1909) 172-175. 9 R. J. Jaccodine and W. A. Schlegel, Measurement of strains at Si-SiO, interface, J. Appl. Phys., 37 (6) (1966) 2429-2436. l I) R. W. Hoffman, The mechanical properties of thin condensed films, in Physics of Thin Films, Vol, 3, Academic Press, New York, NY, 1966, pp. 211-273. 11 A. Brenner and S. Senderoff, Calculation of stress in electrodeposits from the curvature of a plated strip, J. Res. Nat. Bur. Stand., 42 (2)(1949) 105-123. 12 P. A. Flinn, D. S. Gardner and W. D. Nix, Measurement and interpretation of stress in aluminum-based metallization as a function of thermal history, IEEE Trans. Electron Devices, 34 (3) (1987) 689-699. 13 K. Roll, Analysis of stress and strain distribution in thin films and substrates, J. Appl. Phys., 47(7) (1976) 3224-3229. 14 P. H. Townsend and D. W. Barnett, Elastic relationships in layered composite media with approximation for the case of thin films on a thick substrate, J. Appl. Phys., 62 (11)(1987) 4438-4444.
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15 J. R. Taylor, An Introduction to Error Analysis, University Science Books, Mill Valley, CA, 1982, 40-74. 16 P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, NY, 1969, p. 3. 17 S. P. Timoshenko, Analysis of bi-metal thermostats, J, Opt. Soc. Am., II (1925) 233-255. 18 M. F. Gruniner, B. R. Lawn and E. N. Farabaugh, Measurement of residual stresses in coatings on brittle substrates by indentation fracture, J. Am. Ceram. Sot., 70 (5) (1987) 344-348. 19 E. M. Corcoran, Determining stresses in organic coatings using plates beam deflection, J. Paint Technol., 41 (538) (1969) 630-640. 20 S. P. Timoshenko and J. M. Gere, Mechanics of Materials, Van Nostrand Reinhold, New York, NY, 1972. 21 C. F. Gerald, Applied Numerical Analysis, Addison-Wesley, Menlo Park, CA, 1970, 1-20. 22 C.-C. Chiu and E. D. Case, Elastic modulus determination of coating layers as applied to layered ceramic composites, Mater. Sci. Eng. A, 132 ( 1991 ) 39-47. 23 Standard test method for Young's modulus, shear modulus, and Poisson's ratio for glass and glass-ceramics by resonance, ASTM Standard, C023-71, American Society for Testing and Materials, Reapproved 1981.
Appendix: Determination of a coating's elastic modulus T h e E c determination f r o m the loading deflection m e t h o d s described a b o v e d e p e n d s o n an action of external loading. W i t h o u t the help of external loading, E c also can be statistically inferred f r o m the curvatures d e v e l o p e d in a g r o u p of c o a t e d strips. T h e inference is p e r f o r m e d a c c o r d i n g to the least-squares m e t h o d along with eqn. (15). For example, eqn. ( 1 5 ) i s rewritten in terms of a simple f o r m 1 EcE~(I~+ l~)4+ (E~ _ E~)(E~I~ 4 - Eft.,a) O'sa [ =
--
r
6Eflcl~(lc+l~)
= [A]
(A1)
For coated systems with the same coating and substrate materials, the saturation stress a~, is a constant. In addition, the relationships a m o n g the saturation stress, curvature and material p a r a m e t e r s have to o b e y equation (A1). T h u s let us define the sum of squares for e r r o r SSE as /t
SSE = Z (o~,,t-[A],) ~
(A2)
i=|
where n is the n u m b e r of experimental data points. [A] i represents the value o b t a i n e d by substituting the ith data point (E~, l~, Ic, r) i into [A]. SSE is a function of O~a, and Ec. A necessary condition for SSE to be a mini-
148
C.-C. Chiu
/
Residual stresses in ceramic coatings tt
mum is a(SSE)
o~,a,= - Z [A]i
(A3a)
-0
(AS)
F/i= I
where O(SSE) OE~
0
(A3b)
Substituting eqn. (A2) into eqns. (A3a) and (A3b) yields n
tt
tl
Z [A]~ ~, [B]~-n ~ [A]~[B]~=0 i=I
i=1
i=1
(A4)
[B] =
2Ec/c 4 + E~[(/c +/~)4 _ lc 4 _ L4)]
6rE~lcl~(lc + l~)
Equation (A4) is a quadratic equation and only a function of E c. Equation (A4) can be solved using a numerical iteration method. After E c is obtained, asat is calculated using eqn. (A5).