An analytical model for predicting residual stresses in progressively deposited coatings Part 1: Planar geometry

thin

JIIIH$ ELSEVIER

o

Thin Solid Films 306 (1997) 23-33

An analytical model for predicting residual stresses in progressively deposited coatings Part 1" Planar geometry Y.C. Tsui, T.W. Clyne * Department of Materials Science and Metallurgy, Unicersity of Cambridge, Pembroke Street, Cambridge CB2 3QZ UK

Received 13 December 1996: accepted 3 April 1997

Abstract

An analytical model has been developed to predict the residual stress distributions in progressively deposited coatings, such as those produced by thermal spraying. This is based on the concept of a misfit strain, caused by' either the deposition stress (e.g. due to quenching of splats in thermal spraying) or by differential thermal contraction between substrate and coating during cooling. The deposition stress is introduced as the coating is formed layer-by-layer, with a specified layer ti~[ckness, such that the misfit strain is accommodated after each layer addition (rather than for the coating as a whole). From a knowledge of deposition temperatures, material properties and specimen dimensions, residual stress distributions can be predicted. The model is straightforward. It can be implemented with a simple customised computer program or by using a standard spreadsheet. Comparisons are presented between predictions from this model and from a numerical model for three plasma sprayed systems. Good agreement is observed and the predictions also serve to highlight various features of typical sprayed systems. © 1997 Elsevier Science S.A. Ke3~,'ords: Stresses; Coatings

1. Introduction

Residual stresses are commonly generated during coating production. Coating performance indicators such as spallation resistance, thermal cycling life and fatigue properties are strongly influenced by residual stresses. A driving force for debonding is normally associated with the relaxation of these stresses [1-5]. Analytical modelling of residual stress generation in a bimaterial system due to temperature change is well developed. Closed-form solutions for metal/ceramic bonded strips have been derived for the elastic case [6,7] and for the elastic/plastic case with linear work hardening in the metal layer [8]. Suresh et al. [9] further extended their elastoplastic analysis for prediction of the change in residual stresses under thermal cycling. All of these models treat the stresses due to differential thermal contraction only. In most coating processes, however, residual stresses arise from two main sources. In

* Corresponding author. 0040-6090/97/$I7.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0040-6090(97)00 199-5

addition to the stresses caused by differential thermal contraction, there are the "intrinsic" or "quenching" stresses developed during the deposition process. Intrinsic stresses in coatings produced by atomistic processes, such as CVD o r PVI), can be tensile or compressive [10-12]. They are attributed to the incorporation of excess vacancies, the presence of impurities and bombardment by energetic particles ("atomic peening"). As an example, experimental measurements [13-15] of the intrinsic stress in diamond (CVD) films formed at around 1000-1200 K suggest tensile values in the range of 0 - 2 GPa. In thermal spraying, molten particles striking the substrate surface are flattened and quenched to the substrate temperature within a very short time ( ~ few ms). Thermal contraction of splats is constrained by the underlying solid, whichresults in tensile stress being developed. Kuroda and Clyne [16,17] termed this the quenching stress. %; it is always tensile within the deposit. The severity of the constraint depends on the nature of bonding and the geometry of the interface, while the degree to which it is resisted by the splat depends on the splat material and the substrate temperature. The linear contraction of a splat

24

KC. Tsui, Z W. Clyne/ Thin Solid Films 306 (]997) 23-33

during quenching is usually large (at least several millistrain); if this misfit strain were completely constrained, stresses much higher than the yield strength or the fracture strength would develop. In reality, however, there are a number of stress relaxation processes [16.17] which can operate to reduce the stress in the quenched splat below its maximum value, %0 ( = E e e q A T / ( 1 Vd))' Creep and yielding are the major stress relaxation mechanisms for metallic coatings, while microcracldng is usually responsible for stress reduction in ceramic coatings. A further complication with thermal spraying (and some other processes) is that the coatings may have a final thickness which is not negligible compared to that of the substrate. This means that, although the misfit strain ("quenching strain") is the same for each successive incremental layer of the deposit, this is imposed each time on a "substrate" of changing thickness. A succession of force and moment balances, therefore, determines the changing stress distribution (and specimen curvature). The final stress distribution will, for a thick deposit, differ significantly from that which would result if the coating were imposed on the substrate (with the same misfit strain) in a single operation. Although "'thick" deposits (i.e, coating thickness, h, and substrate thickness, H, such that h < -~ H ) are not very common in industrial practice, this problem is an important one in terms of experimental validation of models carried out with thin substrates in order to generate measurable curvature changes (see be-

experimental techniques of measuring residual stresses can be found in the literature [35,36]. In this paper, an analytical model is presented for prediction of the residual stress distributions in progressively deposited coating systems. It can be applied to many coating processes (see Section 3). It is relatively simple and incorporates the two main origins of residual stress. The effects of changing various parameters (e.g. material properties and intrinsic stress) can be examined, giving insights into possible approaches for controlling residual stress levels.

2. Formulation of the model

The following formulation relates to planar geometry; that for a cylindrical substrate is presented in Part 2. The stresses set up due to the deposition process are considered, followed by those due to CTE mismatch during final cooling to room temperature. It is assumed that the substrate is clamped only at one end and is free to bend during the process. A nomenclature listing is given in Appendix A.

2.1. Deposition stresses

low). For thermal spraying, several models [18-23] have been developed to predict the residual stress distribution after deposition. Most of these models consider only the stresses due to differential thermal contraction and neglect quenching stresses. This assumption is often totally unjustified. A further problem in many treatments concerns the correct thermal, physical and elastic properties, especially those of the deposits. The structure of sprayed coatings is such that these properties differ significantly from those of the corresponding bulk materials, so they should be evaluated (experimentally) before being incorporated into a model. Moreover, the temperature dependence of these properties is often significant, since large variations in specimen temperature may occur during deposition. The relative motion between specimen and sprayhead leads to variations in the heat flux arriving at a given point. The complexities thus introduced are best handled using numerical techniques. A one-dimensional, fully implicit finite difference model, which has been developed by Gill and Clyne [24,25], takes most of these important factors into account. Application of this model to predict residual stresses and their effect on debonding can be found elsewhere [4,26-32]. To validate the model experimentally, an in situ curvature monitoring technique [33,34] has been used. This has the advantages of being non-destructive and allowing stress development to be continuously monitored during spraying. Comments and comparisons on various

2.1.1. Deposition of the first layer Consider deposition of the first Iayer, with thickness w. The misfit strain is given by

E,, where % is the intrinsic (quenching) stress and Ea is the Young's modulus of the deposit. Imposition of the misfit strain sets up a pair of equal and opposite forces, Fa (see Fig. 1). The strain compatibility equation is

= cz - e, =

F1 bwEt

FI + -bHE~.

(2)

Rearranging, this becomes

F l = o'q bw

HE, + wE a

(3)

A tensile force with this magnitude acts on the deposit, while a compressive force of the same magnitude acts on the substrate. This pair of equal and opposite forces generates a bending moment, M t, given by M1 = FI ( H + w --5-- )

(4)

Y.C. Tsui, T. ~K Clyne / Thin Solid Fihns 306 (1997) 23-33

~ %(-o)

bottom of the substrate need to be established. These can be calculated from [37]

' Gx(= Gz )

~x?/"

25

--YI

O'~bI=G'[Y=-H= *w

layerl

,AH~

.

.

.

.

, .

(

y"\, \ K~

(negative into the substrate) ,7..2.8i

The neutral axis position, 8~, and the composite beam stiffness, .Svt, can be expressed as [3,37] -

(5)

w -- 31

Zl=bf

( W2

H ~,E(yc) v~dv=Edbw - - -~t'di + 8~) --

--

FI

~

"

1

3

¢

+ E~bH( - -3 + H 6 1 + 8~

mI ( w + H) F 1 Kl- K0= 2--~= 2X I

(7)

Normally, K0 would be set equal to zero (although the case of a substrate with an initial curvature could be treated). The curvature is defined to be positive if the layer of deposit is on the concave side (as in this case). An equal biaxial stress state ( G = oi), with negligible through thickness stress (0% =0), is assumed. The stress will induce a strain {n the x-direction, due to a Poisson effect. The net strain in the x-direction can thus be written as

G E = G - z,( o-y + o-z) = G ( 1 - v )

(8)

2

) -- ~l

'

(12)

Consider the next (second) layer impinging on the (coated) substrate (see Fig. 2). The magnitude of the misfit strain is the same as before. The strain compatibility equation (Eq. (2)) becomes F~ +

E2~b( H + w)

Edbw

-

(13)

In this equation, the force F~ is acting on a composite beam made up of a single layer of the deposit plus the substrate, having a neutral axis at 61 (see Fig. 1). This layer and the substrate are subjected to the same strain and E2~ is the equivalent Young's modulus for this composite beam, defined by Ref. [37]

(6)

Balancing the moment, Mr, induces a curvature change, K~ - K0, which is equal to

(w --EdK1

2,1.2. Deposition of the second laver

F2

w2Ed H2E, 2 ( wEd + HE,)

(11)

The stress value at the midpoint of the first layer is given by O"dl = O ' d [ y = w / 2 = -b' 'i~t

Fig. 1. Schematic depiction of the generation of a pair of equal and opposite forces and an unbalanced moment due to the deposition of the first layer of the deposit on the subbtrate surface.

~1 =

bH +E'~t81

G~=GI;'=°=

neutral axis of the composite beam

~-~'~s i ill 2 ........-~F l ~---- Ft

~

(10)

- F~

Ed

:: :'

M1 ( FI.~-- . . . . . F1-....--..1~~

bH +E't;a(H+6~)

~Ae=

E,~=

E, H + Ejw H+w

(14)

By substituting this into Eq. (13), the normal force, can be expressed as

Fz = crqbw( HE, + wEd ) HE~ + 2wE d

F,.

(15)

The force F 2 is shared between the first layer and the

Gq

w , .w • "i-£~---"X H

laver v A~ = - ~ ~ E. ~_..~---.~layer ~

~

,

M2

.

=

already solidined l of the

.o;_| ~ . . deposit neutral axis of the initial cum'ature of the ,composite b e a m composite b e a m = I<1 ,,," (negative into M2 y~ ~" the substrate/

so that the relation between stress and strain in the x-direction can be expressed as crx G

E - (l-v)

=e'

(9)

This effective Young's modulus value, E', should be used in the above and following equations. As only elastic processes are being considered, the variation of stress through the thickness of the substrate should be linear. Therefore, only the stresses at the top and

normal force, F2, acting along the neutral ads, 51, is shared between the substrate and the first layer, with the individual forces acting along their own neutral axes, and F2=F2w+ F2s 81 ,

_

_

h---*-'.--l~El- ~--h

F2w....~ . . . . . . rw. . . . . . . ~,__ F2w

F~ = - ~ ~ 2 - h s

Fig. 2. Schematic depiction of the generation of a pair of equal and opposite forces and an unbalanced moment due to the deposition of the second layer of the deposit on the coated substrate surface.

26

K C, Tsui, T,W. Clyne/ Thin Solid Fihns 306 (1997) 23-33

substrate. The force acting on the substrate (see Fig. 2), Fx, is equal to

The stress at the midpoint of the first layer becomes O'dl = °'dJY =

w/2--

bw

EeK, ~- - 8 t

Fe, = Area X E, × strain = bHE~ E2~b(H + w) EdF2

-(

F2HE" ) HE, + wEa

(16)

The stress at the midpoint of the second layer can be calculated from

Similarly, the force, F 2,,,, acting on the first layer is

F>,. = HE, + wEd

(17)

Both F2~ and F2,,. are compressive and there is a tensile force, F 2, acting on the second layer. The pair of equal and opposite forces sets up a moment, M2, given by

(3 )

M 2 = F 2 ~t~.'-- 6 t

(24)

b( HE, + wed)

(18)

Balancing this moment induces a curvature change, K2 K,, which is equal to

g.12 = ~1>,=3w/2 - bw

(7)

Ed(Kz--Kt) - - - - 6

The above procedure is readily extended to treat impingement of the n ~ layer. The normal force, F~, the induced curvature change, K, -- K,_ ~, the composite beam stiffness (for n layers of the deposit plus the substrate), X~, and the neutral axis position from the interface. ~,,, can be expressed as follows.

& = o_bw(HE, + ("Z_1)2`% )

(26)

Kn

(27)

(19)

~2

where "~2 can be calculated from *Y2= Edb(2w) ( (2w)2 3

+

(H2

E, bH 5--

(25)

2.1.3. Deposition of the nth layer

HE~ + nwEa

/£2 -- K1 ~

2

+

(2w) 62 + 8~

H6~ + a~

-

-

Kn-l

:

~2 ~n

)

)

"~

(nw) 6. +

+ E~bH 7 - + Ha,,+ a?

(20)

mad 6 2 is determined by

3

( nw)- Ea - H-E, <3. = 2((nw) Ed + HE, )

( 2 w ) 2 E a -- H"Es

~2 = 2((2w) Eg + HE~)

(21)

(28)

(29)

The stresses at the top and bottom surfaces of the substrate are

The stresses on top and bottom surfaces of the substrate are given by

-F 1

o-.2=o-,I, = _ j = bH +e,~(H+a~) +E,(K 2-K1)(H+62)

-F l

-E, Fi b(HEs +(i-1)WEd)

+e,(~,- ~_,)(H+a,))

EsFz

b(HE, + wEd)

f( Cr'b"=~l:'=-H=i=l

(30)

(22) -E,F,

EsF2

~rz =crsly=o = bH + E~K16' - b(HE~ + WEd)

+E~(K2--K~)6 2

(23)

+E,(<- ~- 1) ai).

(31)

27

Y.C. Tsui, T.W. Clyne /-Thin Solid Films 306 (19971 2 3 - 3 3

layer n layer

~

...! . . . . ;

HI

~

,

! 0

Fig. 3. Schematic depiction of the stress distribution after depo~tion o}"n layers.

stresses can then be superimposed on those calculated as outlined in Section 2.1 to give the final stress distribution. Consider two bonded plates cooled from a stress-free state by AT (negative for cooling) so that a misfit strain, A 8 = ( a s -C~d)AT, is created. Imposition of this misfit strain results in equal and opposite forces, of magnitude ~CTZ~' being set up in both the deposit and the substrate. By a force balance argument in the x-direction (Fig. 4, which shows a case with AT negative and A oe positive), their relationship is given by

The values of Fi, ;
bhE

-

(<(r)

+ bilE----7 =

•(CTE)

=

b O"dJ = O"d]Y = (j--1/2)w

( E E, hH )

bw = A8

E d t7 +

(33)

E, H

- E.(,:-

These two forces create an unbalanced m o m e n t , MICTE). Balancing this moment generates a curvature change, Kc K~, of the composite beam. The moment is given by

-EdF, +

i =~j + i

b(HEs+(771)WEd)

-',),,,-,,))

( H + h I

(34)

(32)

where 1 < j < n. A typical resultant stress distribution is shown schematically in Fig. 3. In practice, there is no need to use a value of w equal to, for example, the actual splat thickness in thennal spraying. Any convenient fraction of the final deposit thickness could be employed. (Obviously the deposit thickness, h, must be equal to nw: a rational approach would be to establish w by fixing n at some convenient number, such as 10.) From the intrinsic stress, the specimen dimensions, the Poisson ratios and the Young's moduli, the stresses set up due to deposition can be determined from Eqs. (26)-(32). This can be done by writing a simple computer program or by using a spreadsheet. In either case, the procedure is straightforward and calculation time is extremely short.

The curvature change of the composite beam, % - %, can be related to this moment, M(CTE~, by ,~-

M(CTE)

K,,

v

(35)

~ec

where negative curvature change represents the deposit

Y ~-x h

'~

force balance b f ~(y) dy = 0 -I

2.2. Differential thermal contraction during cooling

F~c'~'<- ~ - - ~ ?

~

-

q v=h

'

;=0-7

0

,

y =-H -1

Mccre ~

1

F'crE'

~4

Stresses set up due to CTE mismatch, as the whole composite beam cools down to room temperature, can be calculated by following a similar procedure to that used for handling deposition stress. The final curvature after cooling down is %, while ~<,, is the curvature adopted after the last layer has been deposited, but before cooling starts. (Any thermal fluctuations during deposition are ignored.) Stresses due to CTE mismatch can be determined if the temperature drop, the specimen dimensions, the Young's moduli and the CTEs of the materials are known. These

-H (MPa)

e (mfllistraln)

1

-20

20

F

/ Mcret{c-~? " "/

h-a

/' 1/~: (-2 m)

moment balance b f ~(Yc) Y¢dy = 0 -n- 8

Fig. 4, Schematic depiction of the generation of curvature in a fiat bimateriaI plate, as a result of the imposition of a uniform, linear misfit strain, Ae. The distributions of stress and sUain shown, due to CTE

mismatch only, were calculated for h= H= 1 mm, Es= I00 GPa, Ed/E ~= 0.i and Ae = - l0 -3 (after Clyne [6]). In this case, Aa is positive and AT is negative.

28

Y.C, Tsui, T.W. Clyne/Thin Solid Films 306 (1997) 23-33

surface being more convex (or less concave), as expected if a, > a d (Aoe positive)..Z~ is the composite beam stiffness and is given by

,,cd}=bEdh

-/-/- &

+bEtH

o-,:,=o-,I.,,=_u=

-h6+62

(.2

+ E,( Ki - Ki_,)( H + 6i))

(36)

- -F{CTE) -bH +E,(K~- ~°)(H+ 8.)

(41)

and -E,,F,

b(HE, + ( - 7 7 1 ) w E a )

° " = °7~ ~'=°= ~

-

(37)

+e,(~,-~_,)a,)
From Eq. (34) and Eq. (35), F(CTE~ can be expressed as,

+ E,( K,. - K.) 6.

2(K~- K~)~c h + H

(3S)

Combining Eq. (33) and Eqs. (36)-(38), the curvature change arising from the misfit strain, A e, can be expressed

(42)

For the deposit, the stress at the midpoint of jth layer is, % = o-al, = ~ - ~/2>,,.

6

as, KC

b(HE~+-(7--1)wt%)

--f- 2t-t"l~ "4-6 2

Ec~h2 Es H2 2( Eah + E,H)

F~CTE~ =

i=L

)

i~ is the distance [3] from the neutral axis ()% = 0) to the interface (y = 0) and is equal to 6~. It is positive when it lies within the deposit and its value is given by 6=

contributions represented by Eq. (30) and Eq. (40e) or Eq. (31) and Eq. (40d),

1

E.(~,- ~,_,)((j- ~)w- %)

-bw -- K n

6EdE,hH(h + H)ae

=

-

(39)

+ i = jL+

EZh 4 +4EeE, h3H +6E~E~h2H e +4EdE~hH 3 + E~H 4

I

e.F~

b(HE,+(i_I)wEd )

Therefore, the stresses due to CTE mismatch can be calculated from [37] o'.l,,=h =

F(CTE)

b--~ --Ed(%- K n ) ( h - 6) F, CTE ~

o'dl

y= (j-

1/2~,,)

--

- -

bh

Ed( ~%-- %,)( ( J - l w ) - 6 ) (40b)

C%ly=0 cr']Y=°

F(CTE)

b~-- + Ed(Kc-- K.)6

-- F(CTE )

°'*["=-H =

b------~ + E,~( % - K~)~

-- F(CTE )

b-h~ + E,(%- K~)(H+ &)

+ F~CTE'-------2~ -- Ed( % - K~)

(40a)

bh

((')

j -- 7 W - 6, ~

)

(43)

where 1 _
=

O'dl:= ~,, -

i/2),,.

(40c)

+

1)w

(40d)

(40e)

Incorporation of the effect of cooling is therefore a simple extension to the procedure outlined for handling coating deposition.

2,3. Final stress levels

3. U s i n g t h e m o d e l

The final residual stress levels at the bottom face and top face of the substrate are obtained by adding the

Implementation of the model is straightforward, once the property and processing data have been identified.

Iz. C. Tsui, T.W. CI)'ne / Thin Solid Films 306 (1997) 23-33 150

I(30

g

....

J ....

t ....

i

. . . .

i

. . . .

i

. . . .

,

....

:'L

a4

-'

5o

-~ a

-50

~

,

~

m '~ -I00

[ . . . . . . . . . progressive (n=5) block (n 1)

1

-150 0 0 ~~ , r ' . . . . .2__~.0 ' -1,5

] ........ ~ ,0 -0,5

I .... 0.0

] .... 0.5

• r 1.0

29

[24,25] is essential (one such case is considered below). Analysis of the effect of metal yielding on residual stresses generated due to temperature change for a bimaterial system can be found in the literature [8,9]. Moreover, the current model does not consider any variation of material properties with temperature, except the coefficients of thermal expansion. This could affect the accuracy of the predictions if the deposition temperature were high and the properties varied considerably with temperature.

: 1.5

Distance from the interface (mm) Fig. 5. Predicted stress distributions due to quenching only, for APS NiCrAIY on PK33, by treating the deposition as a block process or as a progressive process with n, the number of layers, equal to 5 and 100.

Selection of n (hence w) is arbitrary. Treating the coating deposition as a progressive process will, in many cases, be essential if the stress distribution is to be accurately predicted. This is illustrated by Fig. 5, which shows the effect of varying n (hence w) on the predicted residual stress distribution due to quenching of splats for the APS NiCrAIY on PK33 system (see Section 4.1). It should be noted that the standard procedure described in the analytical model monitors the stress at the midpoint of each layer only. However, in this figure, the stresses at the top and bottom faces of each layer are also evaluated. For the case when n is equal to 1 (i.e. block deposition), the stress level and the stress gradient in the deposit are significantly different to those when n is equal to 5 or 100. This is due to the fact that, in the progressive deposition process, the tensile component of stress in each layer is progressively reduced by deposition of successive layers on top of it (see Eq. (43)). Therefore, the stress at the deposit surface should be more tensile compared to that at the interface (i.e. a positive stress gradient). However, this is not the case if the misfit strain is applied to the deposit as a block, in which case a negative stress gradient is predicted (as a result of the curvature adopted by the composite beam). It may be noted that such a negative stress gradient is, in fact, generated within each layer: this is apparent for the n = 5 plot. However, as n becomes large, this effect becomes insignificant and the real trend is that of the stress rising from the bottom to the top face of the deposit. In practice, using a suitable large value for n of, say, 10 (and taking the stress as uniform within each layer), will give a fairly accurate indication of the stress distribution. A flow chart giving the sequence of calculation is shown in Fig. 6. For the case when the composite beam is not allowed to bend, the final residual stress distribution can be obtained simply by neglecting any stresses set up due to curvature changes. The model takes account of elastic processes only. If inelastic processes such as creep [29] or yielding are likely to occur, numerical modelling

4. C o m p a r i s o n s with predictions f r o m a numerical model Three systems are considered in order to illustrate the use of this analytical model. They are: (a) Ni(22 wt.%)Cr(10 wt.%)Al(1 wt.%)Y on PK33 (nickel-based superalloy) deposited by APS, (b) B4C on T i - 6 A I - 4 V deposited by VPS and (c) ZRO2-8% Y203 on PK33 deposited by APS. The properties and parameters used in the numerical model can be found elsewhere [26,29-32]. Those properties required for the analytical model are listed in Table 1 (see also Table 2). It is important to note that the temperature dependence of the Young's modulus is only incorporated in the numerical model. For the analytical model, the values at room temperature are used.

Input the dimensions, the Young's moduli, the } Poisson's ratios, the CTEs, the deposition strew,s, ] the number of layers and the initial curvarare,

START~ , -

J

Ii

//

",

"-, ,, -: Is deposition" \, i,,e s t'~o in progress? / .

\.

[

[

I--

" i Deposition ofi th layer ]

('<~n) / • / ¥ fmal

' L_ coo~g

] ]

,

Determine the neutral axis position. ] 5 i, the beam stiffness. Z i, the curvature I change, :'q'N-t and the normal force, F~ ! (equations (26)-(29/).

[

+ Determine the curvature change, Kc-Kn and the normal force, F{,CTEr developed due to differential thermal contraction (equations (38) and (39)). V Determine the stresses set lip in the , substrate and at the mid-points of all the deposited layers and add the~e to the existing stresse~ (equations (4I)-(43)).

f Determine the su:esses set up in the substtute, at the mid-points of the previous deposited i-1 layers and the outermost layer in this proces~ and sum t h e e to the stresses developed from the previoub processes (equations (30)-(32)).

r

i/Outputresuits,[

ii=i+l

I •Output+results.

I

~-'-"~ END i

Fig. 6. Flow chart showing the algorithm for the analytical model.

30

Y.C. Tsui, T.W. Ctyne / Thin Solid Fihns 306 (1997) 23-33

Table 1 Material properties used in the analytical and the numerical models Property Material

CTE (MK -~ )

]34C (VPS)

Zr02-8% Y203 (APS)

NiCrA1Y (APS)

PK33

Ti-6A1-4V

4.3 (293 K) 5.1 (973 K)

7.9 (283 K) 10.1 (81i K)

12.0 (293 K) 13.0 (1273 K)

12.1 (294 K) I3.1 (811 K)

8.6 (293 K) I3.5 (t073 K)

i162 (293 K) 39.5 (973 K) 0. t 9 300 (293-1000 K) 30 (723-923 K)

I2.2 (1367 K) 2.1 (293 K) 1.4 (1073 K) 0,30 50 (293-1000 K) 2 (673-873 K)

11.5 (1 t44 K)

Young's modulus (GPa)

Poisson's ratio Elastic limit (MPa) Quenching Stress (MPa)

4.1. A P S N i C r A I Y on P K 3 3

16.2 (1144 K)

34.1 (293 K) 22.2 (1073 K) 0.31 148 (293- I000 K) 65 (673-873 K)

167.0 (293 K) 113.5 (i073 K)

106.0 (293 K) 62.1 (1073 K)

0.31 366 (293 K) 70 (1173 K) -

0.31 1100 (293-973 K) -

4.2. V P S B 4 C on T i - 6 A I - 4 V

This system has a large quenching stress value and a small CTE mismatch. As shown in Fig. 7(a), the stresses arising from quenching of splats account for most of the final stress levels. Notice that, using the analytical model, the predicted average shape of the curve for the deposit stresses is very similar to that from the numerical process model. Large errors would result in this case if the final stress levels were estimated only from the CTE mismatch. The temperature during deposition was assumed to be constant in the analytical model, whereas in fact it varied somewhat. A constant spraying temperature of 420°C was assumed in the analytical model and room temperature was assumed to be 20°C. The effect of assuming higher or lower values (covering the actual range during spraying) is shown in Fig. 7(b). The final stress levels are not very sensitive to this temperature, since the CTE mismatch is so low. Since no inelastic processes are predicted to occur in this case, the temperature dependence of the Young's modulus (incorporated only in the numerical model) is entirely responsible for the differences between the predictions. The temperature variations during spraying account for the fluctuation of the deposit stresses about their mean value in the predictions of the numerical model.

This system has a moderate deposit quenching stress value and a large CTE mismatch. The stresses from CTE mismatch account for most of the final stress levels, as shown in Fig. 8(a). Fairly good agreement is obtained between the numerical model and the analytical model if a constant spraying temperature of 500°C is assumed. As only elastic processes are considered in the analytical model, the effect of creep [291 in the T i - 6 A I - 4 V substrates on the residual stress distributions cannot be predicted in the analytical model. However, the average values are still in good agreement, particularly if only the residual stress distributions relaxed on debonding are of interest [29] (see Part 3). Since the CTE mismatch is large, the final stress levels depend on the spraying temperature used in the analytical model. This is shown in Fig. 8(b) in which two other temperatures, the maximum and the minimum temperature during the actual spray run, were used. 4.3. A P S Z r O 2 on P K 3 3

This system has a small quenching stress and a moderate CTE mismatch. The average spraying temperature used was 500°C. The sign of the gradient of stress in the substrate is different for the predictions between the nu-

Table 2 Specimen dimensions used in predicting residual stresses for three plasma sprayed system~ System

Substrate thickness (ram)

Deposit thickness (ram)

Specimen width (ram)

APS ZRO2-8%YzO3/PK33 APS NiCrA1Y/PK33 VPS B.,C/Ti-6AI-4V

1.58 1.62 2.01

1.29 1.04 0.95

20.28 20.30 21.10

Y.C. Tsui, T.W. Clyne / T h i n Solid Fihns 306 (1997) 2 3 - 3 3 150

3o[i

~" 20

e~

50 0

~ p.

~

-50

-150

,,,

....

, ....

, .... [

L

,,,,

,,

-----'-~um-er~cal

,_

model

"

-

.......analytical model: [ quer[ching stress only [ -- - - analytical model: [ final stresses

/

lo L

I ....... ical model ~. - . . . . . . analytical model: "¢~ I quenching stress only X~X~. | -analytical model: ~ J final stresses

-1(30

ai

31

J

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(a)

.~

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,. . . . .

, ....

~ ....

t

....

t

~ ....

~ -2o

, , , I , , ,

,'

....

30

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.

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,

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........ i

.

.

,

i ....

i

i

20

} 50 ~,,

_~ -50

~

/

~ :

o e¢

d

-i5o -200

,1

:b)

.25%_.. . . .

, ....

-i.5

, ....

, ....

, . . . . . . . . . .

-1.0 -0.5 0.0 0.5 '1.0' Distance from the interface (ram)

1.5

,

i

,

,

I

,

~

i

,

i

.

....

oo

•

[ . . . . . . . anaiyncal model: [ using 360~C I . . . . . analytk:al model: using 480°C

"X~, "~,, ~'~,

-100

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i

- numencal model . . . . . . . analytical model: using 400°C - . . . . analytical model: using 600°C

/•

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,

,

i

i

r

i

i

r

I

-l,0 -05 0.0 0.5 1.0 Distance from the interface (nun)

,

,

1.5

Fig. 9. C o m p a r i s o n s b e t w e e n predictions f r o m the n u m e r i c a l and analytiFig. 7. C o m p a r i s o n s b e t w e e n predictions f r o m the n u m e r i c a l and analytical ( n = 100) m o d e l s for A P S N i C r A 1 Y on P K 3 3 . Plots for the analytical m o d e l are s h o w n (a) with and without the final c o o i - d o w n ( f r o m 4 2 0 ° C ) and (b) with c o o l - d o w n f r o m 3 6 0 ° C and 480°C.

.....

~ ....

~ ....

t

.

,

,

2oo (a)

. . . .

....

t

/"

/ -

.

./2."

0

-100

.]

/

k#10o

"~

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i

-9"

"

-200 ,

r

r

i

,

i

,

i

,

4

,

,

,

,

,

i

200 - (b)

,

~

i

i

i

,•"/"

I00 m

'3 -100

~'. " •

.......

:

-200

ng 50 C i

-2.0

-I.5

j

r

:

cal ( n = 100) m o d e l s for A P S Z r O ; - 8 ( w t . % ) Y 2 0 3 on P K 3 3 . Plots for the analytical m o d e l are s h o w n (a) with and without the final c o o l - d o w n ( f r o m 5 0 0 ° C ) and (b) with c o o l - d o w n f r o m 4 0 0 ° C and 600°C,

merical model and the analytical model (see Fig. 9(a)). The main reason for this is that the stresses due to CTE mismatch are overestimated in the latter. This arises because, at high temperatures, the Young's moduli are lower than the corresponding values at low temperatures (see Table 1). This has not been considered in the analytical model. For the same reason, the average stress in the deposit was predicted to be compressive instead of tensile, as predicted by the numerical model. Of course, it would be possible to use temperature-averaged Young's modulus values and this is an example of a case where this would be worthwhile. However, in terms of magnitude, the difference is not large, since the residual stress levels are low, mainly because the Young's modulus of the deposit and the quenching stress are low. The final stress levels predicted by the analytical model varied considerably as different spraying temperatures were used, as shown in Fig. 9(b), mainly because the stresses due to CTE mismatch contribute more to the final stress states than that due to the quenching of splats.

-~ /

i

-1.0 -0.5 0.0 0.5 Distance from the interface (ram)

1.0

Fig. 8. C o m p a r i s o n s b e t w e e n predictions f r o m the n u m e r i c a l and analytical ( n = 100) m o d e l s for V P S B a C on T i - 6 A I - 4 V . Plots for the analytical m o d e l are s h o w n (a) w i t h and without the final c o o i - d o w n ( f r o m 500°C) a n d (b) w i t h c o o l - d o w n f r o m 3 5 0 ° C and 650°C.

5. Conclusions An analytical model based on force and moment balances has been developed. This model takes into account the two main residual stress generation mechanisms (intrinsic stress during deposition and differential thermal

32

Y.C. rsui, Z W. Clyne/Thin Solid Fihns 306 (1997) 23-33

contraction), allowing their relative contributions to the final residual stress state to be determined. This model is relatively simple, compared to most numerical models, but provides a clear illustration of how stresses build up during the coating process. With this model, residual stress distributions can be determined directly by knowing the specimen dimensions, material properties and the intrinsic stress. There is no modelling of the heat flow or account taken of inelastic processes. In most cases, this does not give rise to large errors, although the (small) fluctuations in stress level associated with passes of the spray gun (in the case

of thermal spraying) cannot be predicted. The effect of varying different parameters can be predicted easily, so the model provides a potentially useful tool for controlling residual stress levels.

Acknowledgements Financial support for one of us (YCT) has been supplied by Commonwealth Scholarship Commission in the United Kingdom and by Howmedica.

Appendix A A. 1. Nomenclature a

K -1

b

m

(~i

m

AT

K or °C

E

Pa

E'

Pa Pa N N N N m m

m

ge Fczz

Fiw H h Kc

Ki

MCTE

v,

m-1

m-1 Nm Nm

n o"

O-da

X

o-q % % % w x

Y Y~ Z

Pa Pa Pa m 4 Pa Pa Pa Pa m m

m m m

coefficient of thermal expansion width of specimen neutral axis position relative to the interface (positive inside the deposit) for a composite beam consisting the substrate and i layers of deposit misfit strain difference between the deposition temperature and the room temperature (negative when temperature drops) Young' s modulus elastic strain effective stiffness in plane stress effective Young's modulus of a composite beam with the substrate and j - 1 layers of the deposit force set up due to differential thermal contraction force set up due to the deposition of the i th layer force acting on the substrate alone due to the deposition of the i ~h layer force acting on each layer (except the outermost layer) due to the deposition of the i th layer thickness of the substrate thickness of the deposit curvature of the composite beam after cooling down to room temperature curvature after deposition of i layers of deposit (K 0 iS the initial curvature before deposition) bending moment set up due to differential thermal contraction bending moment set up due to the deposition of the ith layer Poisson ratio number of layers stress

stress at the midpoint of the jm layer of the deposit flexural stiffness of a composite beam with the substrate and i layers of deposit intrinsic stress quenching stress predicted by linear elastic behaviour as splat cools to substrate temperature stress at the back of the substrate (~bi iS the stress after deposition of the ith layer of the deposit) stress at the top of the substrate ( % , is the stress after deposition of the ith layer of the deposit) layer thickness displacement along the length of the beam displacement, relative to the interface, through the thickness of the beam displacement, relative to the neutral axis, through the thickness of the beam displacement through the width of the beam

Y.C. Tsui, T.W. CIyne /Thin Solid Films 306 (1997) 23-33

A.2. Subscripts x, y, z d, s, c

orthogonal directions (see Fig. 4) deposit, substrate and composite parts of the beam respectively

[20] [21]

References [1] M.Y. He, A.G. Evans, The Strength and Fracture of Metal/Ceramic Bonds, Acta MetalI. Mater. 39 (1991) 1587-1593. [2] A. Bartlett, A.G. Evans, M. Ruhle, Residual Stress Cracking of Metal/Ceramic Bonds, Acta Metall. Mater. 39 (I99i) 1579-1585. [3] S.J. Howard, Y.C. Tsui, T.W. Clyne, The Effect of Residual Stresses on the Debonding of Coatings Part I: A Model for Delamination at a Bimaterial Interface, Acta Metall. Mater. 42 (1994) 2823-2836. [4] Y.C. Tsui, S.J. Howard, T.W. Clyne, The Effect of Residual Stresses on the Debonding of Coatings Part II: An Experimental Study of a Thermally Sprayed System, Acta MetalI. Mater. 42 (1994) 28372844. [5] A.G. Evans, G.B. Crumley, R.E. Demaray, On the Mechanical Behaviour of Brittle Coatings and Layers, Oxid. Met. 20 (1983) 193-216. [6] T.W. Clyne, Residual Stresses in Surface Coatings and Their Effects on Interfacial Debonding, "'Interfacial Effects in Particulate, Fibrous and Layered Composite Materials", Key Eng. Mats. i 16-117 (i 996) 307-330. [7] A. Brenner, S. Senderoff, Calculation of Stress in Electrodeposits from the Curvature of a Plated Strip, J. Res. Natl. Bur. Stand. 42 (1949) 105-123. [8] C.H. Hsueh, A.G. Evans, Residual Stresses in Metal/Ceramic Bonded Strips, J. Am. Ceram. Soc. 68 (1985) 241-248. [9] S. Suresh, A.E. Giannakopoulos, M. Olsson, Elastoplastic Analysis of Thermal Cycling: Layered Materials with Sharp Interfaces, J. Mech. Phys. Solids 42 (1994) 979-I018. [i0] D.W. Hoffman, Film Stress Diagnostic in the Sputter Deposition of Metals, in: Proc. 7th ICVM (Ed.), Iron and Steel Inst. Japan, 1982, 145-I57. [i1] D.W. Hoffman, Stresses in Thin Films: The Relevance of Grain Boundaries and Impurities, Thin Solid Films 34 (1976) i85-190. [12] J.A. Thornton, D.W. Hoffman, Stress-Related Effects in Thin Films, Thin Solid Films 17I (I989) 5-3i. [i3] X.L. Peng, Y.C. Tsui and T.W. Clyne, Stiffness, Residual Stresses and Interfacial Toughness of Diamond Films on Titanium; submitted to Diad. Relat. Mater., i996. [14] D. Schwarzbach, R. Haubner, B. Lux, Internal Stress& in CVD Diamond Layers, Diad. Relat. Mater. 3 (1994) 757-764. [15] H. Windischmann, G.F. Epps, Y. Chong, R.W. Collins, Intrinsic Stress in Diamond Films Prepared by Microwave Plasma CVD, J. AppI. Phys. 68 (1991) 2231-2237. [16] S. Kuroda and T.W. Clyne, The Origin and Quantification of the Quenching Stress Associated with Splat Cooling during Spray Deposition, in: H. Eschenauer, P. Huber, A.R. Nicoll and S. Sandmeier (Eds.), 2nd Plasma Technik Symposium, Plasma Technik, 3, 1991, 273-284. [17] S. Kuroda, T.W. Clyne, The Quenching Stress in Thermally Sprayed Coatings, Thin Solid Films 200 (1991) 49-66. [i8] R. Elsing, O. Knotek, U. Baiting, Calculation of Residual Thermal Stress in Plasma-Sprayed Coatings, Surf. Coat. Technol. 43/44 (1990) 416-425. [19] Y.C. Kim, T. Terasaki and T.H. North, A Method of Measuring The

[22]

[23]

[24]

[25]

[26] [27]

[28]

[29]

[30]

[31] [32i [33]

[34]

[35]

[36] [37]

33

Through-Thickness Residual Stress in a Thermally-Sprayed Coating, in: T.F. Bernecki (Ed.), Proceedings of The Fourth National Thermal Spray Conference, The Materials Information Society, 1991, 221227. F. Kroupa. Stresses in Coatings on Cylindrical Surfaces, Acta Techn. CSAV 39 (1994) 243-274. R.L. Mullen, R.C. Hendricks, G. McDonald, Finite Element Analysis of Residual Stress in Plasma Sprayed Ceramic, Ceram. Eng. Sci. Proc. 6 (1985) 871-879. S. Takeuchi, M. Ito, K. Takeda, Modelling of Residual Stress in Plasma-Sprayed Coatings: Effect of Substrate Temperature, Surf. Coat. Technol. 43/44 (1990) 426-435. D. Stover, D.A. Jager and H.G. Schutz, Residual Stresses in Low Pressure Plasma Sprayed Chromia Coatings, in: T.F. Bernecki (Ed.), Proceedings of The Fourth National Thermal Spray Conference, The Materials Information Society, 1991, 215-219. S.C. Gill and T.W. Clyne, ThermomechanicaI Modelling of the Development of Residual Stress during Thermal Spraying, in: H. Eschenauer, P. Huber, A.R. NicoI1 and S. Sandmeier (Ed.), 2nd Plasma Technik Symposium, Plasma Technik. 3, I991, 227-238. S.C. Gill and T.W. CIyne, Property Data Evaluation for the Modelling of Residual Stress Development during Vacuum Plasma Spray Deposition, in: H. Exner (Ed.), 1st European Conf. on Adv. Mats. and Procs. (Euromat "89), Deutsch. Geselh f. IVletallk., 1, 1990, I221-1230. Y.C. Tsui, Adhesion of Plasma Sprayed Coatings, Ph.D. Thesis, University of Cambridge, 1996. A. Itoh and T.W. Clyne, Initiation and Propagation of InterfaciaI Cracks during Spontaneous Debonding of Thermally Sprayed Coatings, in: C.C. Berndt and S. Sampath (Eds.), Advances in Thermal Spray Science and Technology, ASM, Materials Park, Ohio, 1995, 425-431. A. Itoh, S.C. Gill and T.W. Clyne, The Effect of Cooling Conditions on the Spontaneous Debonding of Thermally Sprayed Coatings, in: P. Vincenzini (Ed.), Advances in Inorganic Films and Coatings, Techna Srl., 1995, 451-458. Y.C. Tsui, S.C. Gill and T.W. Clyne, The Effect of Substrate Creep on Residual Stress Development during Spraying of Boron Carbide onto Titanium, in: C.C. Berndt and S. Sampath (Eds.), Thermal Spray Industrial Applications, ASM, ivlaterials Park, Ohio, 1994, 669-674. Y.C. Tsui, S.J. Howard and T.W. Clyne, Application of a NIodel for the Effect of Residual Stresses on Debonding of Coatings under Applied Loads, in: P. Vincenzini (Ed.), Advances in Inorganic Films and Coatings, Techna Srl., 1995, 19-26. Y.C, Tsui and T.W. Clyne, Mechanical and Environmental Stability of Thermal Barrier Coatings, in preparation, 1996. Y.C. Tsui and T.W. Clyne, Characterisation of Hydroxyapatite Coatings on Orthopaedic Implants, in preparation, 1996. S.C. Gill, T.W. Clyne, Investigation of Residual Stress Generation during Thermal Spraying by Continuous Curvature Measurement, Thin Solid Ffirns 250 (i994) 172-180. S.C. Gill and T.W. Clyne, Monitoring of Residual Stress Generation during Thermal Spraying by Curvature Measurements, in: C.C. Berndt and S. Sampath (Eds.), Thermal Spray Industrial Applications, ASM, Materials Park, Ohio, 1994, 581-586. T.W. Clyne, S.C. Gill, Residual Stresses in Thermally Sprayed Coatings and their Effect on InterfaciaI Adhesion - - A Review of Recent Work, J. Thermal Spray Technol. 5 (4) (1996) 1-I8. A.J. Perry, J.A. Sue, P.J. Martin, Practical Measurement of the Residual Stress in Coatings, Surf. Coat. Technol. 8i (1996) 17-28. S.P. Timoshenko and J.M. Gere, Mechanics of Materials, D. Van Nostrand Company, 1972,