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Residual stress diagrams for electrodeposited metal coatings I: Plotting the residual stress diagrams during the anodic dissolution of metal coatings using the length change method
Surface Technology, 17 (1982) 321 - 327
321
RESIDUAL STRESS DIAGRAMS F O R ELECTRODEPOSITED METAL COATINGS I: PLOTTING THE R E S I D U A L STRESS DIAGRAMS D U R I N G THE ANODIC DISSOLUTION OF METAL COATINGS USING THE LENGTH CHANGE METHOD s. ARMYANOV and G. SOTIROVA Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1040 (Bulgaria)
(Received July 15, 1982)
Summary A technique is described for plotting residual stress diagrams by the method of length change of a strip cathode on the basis of data obtained during the anodic dissolution of metal coatings. The correlation between the different types of residual stresses is shown. Possible ways for the application of the various equations for residual stress evaluation are proposed.
1. Introduction Recent years have witnessed an ever-increasing interest in the application of the length change method at strip cathodes for measuring residual stress (RS). In the western literature the instrument for measuring RS based on this principle is known as the "IS meter". This method was first proposed by Popereka [1] in 1961. In 1971 Dvo~fik and Vrobel [2] described an instrument which after several improvements [3, 4] was widely applied in both industry and laboratory investigations [5, 6 ] . Other versions of similar devices have subsequently been produced; these are characterized by an increased sensitivity [ 7, 8]. Unfortunately, a unanimously accepted approach to the application of the various equations for the evaluation of RS by the length change method is still n o t established. Furthermore, different types of stresses are considered, b u t their correlation is not always clear cut. An adequate approach aimed at solving this problem would offer a possibility of finding strict criteria for the determination of the validity of a given equation. In our opinion, a similar approach is to derive accurately the relationships required for the plotting of the RS diagram (RSD), i.e. the stress distribution through the thickness of the coating. A m e t h o d for the plotting of the RSD on the basis of data obtained during the electroplating of metal coatings by the length change m e t h o d and 0376-4583/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
322
spiral c o n t r a c t o m e t e r techniques has been proposed by Wagner [ 9 ] . In our previous papers [10, 11] we have shown that, simultaneously with the formation o f the zinc coating, relaxation and RS redistribution processes occur in it, which may alter strongly the stressed state c o m p a r e d with that observed during the deposition. T he r ef or e it was proposed to plot the RSD by anodic dissolution after these processes had ceased, and the appropriate technique was described when the bent strip m e t h o d is used [ 12]. The latter is one of the mos t sensitive RS det er m i nat i on methods, but it must be m e n t i o n e d th at the equation for the evaluation of the RS in each layer with a given thickness At is quite complicated and the plotting of the RSD is labour consuming. The present work is aimed at (1) the devel opm ent of a technique for the plotting of RSDs by using the m e t h o d of length change of a strip cathode, on the basis of data for the change in the length of the electrode during anodic dissolution of the metal coatings, and (2) the accurate evalua~ tion o f the correlation between the different types of RS.
2. Derivation of the equation f or t he plotting of the residual stress diagram Let us imagine that the coating is f or m ed by num erous very thin layers parallel to the substrate (Fig. 1). In each of them a planar strained state is developed, i.e. we have two main stresses oi ~ and ai ~' that are equal to each other. The RSs in two adjacent layers, however, are n o t equal, and thus a distribution of the stresses along the thickness is developed, represented by the RSD. The relationship between the average stress 5 along the thickness of the coating and the stress ai in a given layer of the coating is given by the well-known equation O = --|o~ t J0
dt
(1)
Let us now consider in detail the way by which the equation for the evaluation of the RS a,~ in a given layer during the anodic dissolution of the coating is derived, i.e. we assume that the relaxation and RS redistribution processes (if any) are over. We consider the system, a c a t hode strip of thickness t o with a coating 2t in thickness deposited on both sides (Fig. 2). F o r the sake of simplicity, a uniform distribution o f the stresses along the thickness is assumed. It is well k n o wn that in the presence of a tensile stress the substrate is compressed c o m p a r e d with the initial state (Fig. 2(a)) and therefore X is negative. (All equations are derived for a > 0 f or the sake of clarity. The relationships are valid also for a ~ 0, but with the opposite d e f o r m a t i o n sign.) The balance of forces is easily expressed by the equation b S ( 2 t - - 2 At) + b o l 2 At + b a o t o = 0
(2)
323
(a)
(b)
i
Fig. 1. Schematic pattern of the RS distribution within the plane of the coating and along its thickness: 1, coating; 2, substrate. Fig. 2. Schematic presentation of the strained state of a strip cathode with a coating plated on both surfaces: (a) before dissolution; (b) after a layer with thickness At is stripped off. On the right-hand side the broken line shows the position of the free end of the electrode before plating: km= l m - lo is the change in the substrate length before dissolution of the mth layer compr~red with the state before plating: A~ m =lm+ 1 - - l m is the change in the length of the electrode due to the dissolution of the ruth layer only.
where b is the width of the cathode and the coating, a0 is the RS in the substrate, ol is the RS in the upper layer and ~ is the average RS in the rest of the coating. After the most upper layer of thickness 2 At is dissolved, the substrate is relaxed and tends to recover to the initial state before plating. T herefore the change Ah 1 in the length of the substrate due to the dissolution of the first layer is positive, and the balance of forces is again restored (Fig. 2(b)): (~ + A o ) ( 2 t - - 2 At) + (o0 + Ao0)t0 = 0
(3)
Equation (2) is subtracted from eqn. (3) and after simple transformations we obtain o~
-
A~ lo
1 2 At (E(2t
2 At) + E0t0}
(4)
w h er e/~ = E/(1 -- v) and/~0 = E0/(1 -- Vo) are the reduced Young's moduli of the subtrate and the coating and v is Poisson's ratio. In a similar way, after dissolving the second layer, which is also 2 At in thickness, we obtain
324 02 -
AX2
1
lo
2 At
(/~(2t -- 4 At) +/~oto} --
AXl/~ lo
(5)
For each layer m (m < k, where k is the number of layers in the coating) we may write AX m 1 (l~(2t -- 2m At) +/~oto} - - . -l=m Ak./~l o 2 At n= 1 l0
O~rrl --
(6)
or
o,,,
=
~
A;%~
..... , , = ~ - ~ n = 1
(7)
/0
a,nm is the instantaneous stress, i.e. the stress which is removed during the dissolution of the layer m. The second term reflects the effect of the upper, already dissolved, layers. Thus by using eqn. (6) the strained state of the layer m is restored to that at the beginning of the dissolution. These considerations present a solution of a problem in the field of the strength of materials. Therefore the approach to this problem is almost similar to the one accepted by Wagner [9] but here the reverse case is treated, i.e. dissolution. Therefore it is of interest to juxtapose the results obtained by Wagner with our data and to draw some new conclusions. On the basis of data obtained during the deposition of the coating, Wagner [9] worked out the following equation for the RS in the ith layer of the coating, which consists of k layers: Aki
1
lo
2 At
O i --
{/~(i-- 1)2 At +/~0t0} + ~ AXj/~ j=i+l lo
(8)
or
k oi = oii +
Akj
E 1=i+1
F'
(9)
10
where A~, i = li - - l i z is the change in the length of the cathode after the deposition of the ith layer. Here again oii is the instantaneous stress, and the second term in eqn. (9) reflects the effect of the upper layers on the strained state of the ith layer after deposition is over. Equation (8) can be simplified in the following manner: o, = - - ( k , - k~_~ )l--~ A----t 2 {/~(i -- 1)2 At + £ o t o} + (kk - - k i ) lo
(10)
where k~ I i -- lo is the change in the length of the cathode, with respect to the state before deposition, caused by the deposition of layers I - i. The expression (10) coincides principally with the equation previously proposed by Popereka [1] (without stating how it was worked out, and n o t writing in detail the first term) for the determination of stress in the coating layer at given distance from the cathode surface. =
325
Equation (6) can be presented in a similar way: Om= (kin +, --hm)I--~ A-----t 2 {2/~(t--m At) +/~0t0) --(km --Xl)~.
(11)
Equations (10) and (11) can also be presented in the following manner: O'i ---- (~ki--1 - am = ( X m + l -
Xi){(i -- 1)K, + K2} -- (Xi -- Xk)K, Xm)((k--re)K1
+K2}
--(~km --X,)K1
(12) (13)
where K1 = E / l o a n d K 2 = Eot0/2 At 10. Let us assume that the deposition and dissolution of a given layer proceeds with an identical step At. Then it can be proved that, provided that a complete s y m m e t r y exists between the deposition and dissolution processes, and no relaxation or redistribution of the RS is present, eqns. (12) and (13) when applied to the given layer of the coating are identical. It must be taken into consideration that i increases from the substrate to the last layer k, while m increases in the reverse direction; therefore i + m = k + 1. This means t h a t under the above-mentioned conditions Xm+, -- Xm = Xi-, -- X~
(14)
~kt'l - - ~1 ---- ~ i - - ~kh
(15)
These circumstances prove the correctness of the approach for deriving eqns. (6) and (8). Various equations have been proposed for the evaluation of the average RS through the thickness of the coating [3, 5, 13, 14]. By using the relationship between o i and {~ (eqn. (1)) and eqns. (8) and (6) for the determination of the stress in a layer with thickness At worked o u t by Wagner and us, we could check which of the miscellaneous equations for the evaluation of the average stress is correct. If we introduce eqn. (10) in eqn. (1), and replace integration by summation, we obtain Eot_____~o k E2 At k -- Xi)(i -- 1) -- (X; -- ~k)} = 2tlo ~I(Xi - 1 - - }ki) + --2tl ~-~ ( ( X i - 1 i= ---o ~= 1
(16)
It is readily shown that k
(17)
~(Xi_ 1 - X/); --Xh i=1 h
h
~ ( X i _ l - - Xi)(i-- 1) = ~ (Xi --Xk) i=1
i=l
= X , + X2 + . . . + X k ( h - - 1 )
(18)
Thus the second term in eqn. (16) is reduced to zero and we obtain for
/~oto X 2t
Io
(19)
326
which is the only c or r e c t equation for the average RS along the thickness. It can be shown that the relationship proposed by Perakh [13] for the average stresses
/~0to+/~2t X
0 -
10
(20)
2t
is n o t correct. The plating stress o¢i determined by Wagner [9] coincides with the stress d e n o t e d previously by Perakh [13] as "real". It is somewhat different from t h a t considered up to now in our paper, and represents the stress in a given layer before its interaction with the layers u n d e r n e a t h and the substrate. However, it is n o t c o r r e c t to shift from stress in a given layer to average stress within the entire coating by simple substitution of AX/2 At by
X/2t. If we evaluate the average plating stress 1 oei = -- ~ Oei
(21)
rl i = l
where, following Wagner [ 9 ] , eei -
AXi
1
li
2 At
(i2 At/~ +/~oto)
(22)
we will obtain:
g 0el
--
-
-
n2 At l
(Z~.
1
2 At + AX2 4 At + ... + AX, 2n At) +
Eoto -
-
n2 At l
~k n
(23)
Obviously eqn. (23) does n o t agree with eqn. (20), regardless of the fact that the appropriate relationships for separate layers of the coating are identical. The physical cause of this discrepancy is the lack of an approach which considers each lower layer as a substrate for the next.
3. Conclusions These investigations show clearly t hat a strict correlation exists between the various equations for the de t e r m i na t i on of the RS. Which equation is to be selected f o r the actual case depends on the viewpoint from which we are approaching the RS. Thus, for example, if we try to relate the value of the stresses to the electrocrystallization processes it is necessary to use Wagner's [9] equation for plating stress oei in the appropriate layer. This is the stress in the layer before its interaction with the substrate and with layers that have been already deposited. The oe; values are classified by Wagner as specific for the process, i.e. depending on the bath f or m ul at i on and deposition conditions.
327 I f we w a n t t o find o u t t h e reasons f o r early c r a c k i n g d u r i n g the deposit i o n process, we should be i n t e r e s t e d m a i n l y in t h e i n s t a n t a n e o u s stress o,-i. This is t h e stress in t h e freshly d e p o s i t e d layer, w h i c h has already i n t e r a c t e d with t h e l o w e r layers a n d t h e substrate. F o r practical a p p l i c a t i o n s , h o w e v e r , it is m o r e i m p o r t a n t to k n o w the e f f e c t e x e r t e d b y t h e RS o n s o m e p h y s i c o m e c h a n i c a l , p r o t e c t i v e - d e c o r a t i v e or o t h e r p r o p e r t i e s w h i c h are r e l a t e d t o t h e d e s i g n a t i o n o f the coatings. In this case t h e strained s t a t e d u r i n g t h e f o r m a t i o n o f the coatings is n o t v e r y i m p o r t a n t , w h e r e a s t h e s t e a d y s t a t e stress is o f m a j o r c o n c e r n . T o evaluate this it is necessary to d e t e r m i n e first o f all t h e average stress ~ by using eqn. (19) b u t t a k i n g into c o n s i d e r a t i o n t h e p o s t - d e p o s i t i o n alterations. I f a m o r e detailed investigation of the e f f e c t e x e r t e d b y the RS on the o p e r a t i n g characteristics o f the c o a t i n g is r e q u i r e d , t h e n t h e R S D (am) m u s t be p l o t t e d on the basis o f d a t a o b t a i n e d d u r i n g the dissolution o f the c o a t i n g (eqn. (13)). T h e a p p l i c a t i o n s o f this a p p r o a c h to the p r o b l e m s dealing with the RS, a n d a b o v e all the use o f eqns. (12), (13) and (19), are p r e s e n t e d in the accompanying paper [ 15].
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
M. Ya Popereka, Zavod. Lab., 27 (1961) 1135. A. Dvo~fik and L. Vrobel, Trans. Inst. Met. Finish., 49 (1971) 153. A. Dvo~fik, Y. Prusek and L. Vrobel, Metalloberfliiche, 27 (1973} 284. A. Dvo}fik, Metalloberfla'che, 31 (1977) 59. W. H. Gleghorn, K. S. A. Gnakasekaren and D. Y. Hall, Met. Finish., 18 (1972) 92. G. G. Weiler and Th. Blech,Metalloberfliiche, 29 (1975) 552. Y. B. Kushner, Plating, 60 (1973) 1246. S. A. Armyanov and R. Well, Plat. Surf. Finish., 63 (1976) 49. E. Wagner, Z. Werkstofftech., 6 (1975) 95. S. A. Armyanov and G. S. Sotirova, Surf. Technol., 8 (1979) 311. S. A. Armyanov and G. S. Sotirova, Mater. Sci. Eng., 36 (1978) 253. S. A. Armyanov and G. S. Sotirova, Surf. Technol., 8 (1979) 319. M. Perakh, Surf. Technol., 4 (1976) 527. M. Perakh, Surf. Technol., 4 (1976) 538. S. A. Armyanov and G. S. Sotirova, Surf. Technol., 17 (1982) 329.
321
RESIDUAL STRESS DIAGRAMS F O R ELECTRODEPOSITED METAL COATINGS I: PLOTTING THE R E S I D U A L STRESS DIAGRAMS D U R I N G THE ANODIC DISSOLUTION OF METAL COATINGS USING THE LENGTH CHANGE METHOD s. ARMYANOV and G. SOTIROVA Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1040 (Bulgaria)
(Received July 15, 1982)
Summary A technique is described for plotting residual stress diagrams by the method of length change of a strip cathode on the basis of data obtained during the anodic dissolution of metal coatings. The correlation between the different types of residual stresses is shown. Possible ways for the application of the various equations for residual stress evaluation are proposed.
1. Introduction Recent years have witnessed an ever-increasing interest in the application of the length change method at strip cathodes for measuring residual stress (RS). In the western literature the instrument for measuring RS based on this principle is known as the "IS meter". This method was first proposed by Popereka [1] in 1961. In 1971 Dvo~fik and Vrobel [2] described an instrument which after several improvements [3, 4] was widely applied in both industry and laboratory investigations [5, 6 ] . Other versions of similar devices have subsequently been produced; these are characterized by an increased sensitivity [ 7, 8]. Unfortunately, a unanimously accepted approach to the application of the various equations for the evaluation of RS by the length change method is still n o t established. Furthermore, different types of stresses are considered, b u t their correlation is not always clear cut. An adequate approach aimed at solving this problem would offer a possibility of finding strict criteria for the determination of the validity of a given equation. In our opinion, a similar approach is to derive accurately the relationships required for the plotting of the RS diagram (RSD), i.e. the stress distribution through the thickness of the coating. A m e t h o d for the plotting of the RSD on the basis of data obtained during the electroplating of metal coatings by the length change m e t h o d and 0376-4583/82/0000-0000/$02.75
© Elsevier Sequoia/Printed in The Netherlands
322
spiral c o n t r a c t o m e t e r techniques has been proposed by Wagner [ 9 ] . In our previous papers [10, 11] we have shown that, simultaneously with the formation o f the zinc coating, relaxation and RS redistribution processes occur in it, which may alter strongly the stressed state c o m p a r e d with that observed during the deposition. T he r ef or e it was proposed to plot the RSD by anodic dissolution after these processes had ceased, and the appropriate technique was described when the bent strip m e t h o d is used [ 12]. The latter is one of the mos t sensitive RS det er m i nat i on methods, but it must be m e n t i o n e d th at the equation for the evaluation of the RS in each layer with a given thickness At is quite complicated and the plotting of the RSD is labour consuming. The present work is aimed at (1) the devel opm ent of a technique for the plotting of RSDs by using the m e t h o d of length change of a strip cathode, on the basis of data for the change in the length of the electrode during anodic dissolution of the metal coatings, and (2) the accurate evalua~ tion o f the correlation between the different types of RS.
2. Derivation of the equation f or t he plotting of the residual stress diagram Let us imagine that the coating is f or m ed by num erous very thin layers parallel to the substrate (Fig. 1). In each of them a planar strained state is developed, i.e. we have two main stresses oi ~ and ai ~' that are equal to each other. The RSs in two adjacent layers, however, are n o t equal, and thus a distribution of the stresses along the thickness is developed, represented by the RSD. The relationship between the average stress 5 along the thickness of the coating and the stress ai in a given layer of the coating is given by the well-known equation O = --|o~ t J0
dt
(1)
Let us now consider in detail the way by which the equation for the evaluation of the RS a,~ in a given layer during the anodic dissolution of the coating is derived, i.e. we assume that the relaxation and RS redistribution processes (if any) are over. We consider the system, a c a t hode strip of thickness t o with a coating 2t in thickness deposited on both sides (Fig. 2). F o r the sake of simplicity, a uniform distribution o f the stresses along the thickness is assumed. It is well k n o wn that in the presence of a tensile stress the substrate is compressed c o m p a r e d with the initial state (Fig. 2(a)) and therefore X is negative. (All equations are derived for a > 0 f or the sake of clarity. The relationships are valid also for a ~ 0, but with the opposite d e f o r m a t i o n sign.) The balance of forces is easily expressed by the equation b S ( 2 t - - 2 At) + b o l 2 At + b a o t o = 0
(2)
323
(a)
(b)
i
Fig. 1. Schematic pattern of the RS distribution within the plane of the coating and along its thickness: 1, coating; 2, substrate. Fig. 2. Schematic presentation of the strained state of a strip cathode with a coating plated on both surfaces: (a) before dissolution; (b) after a layer with thickness At is stripped off. On the right-hand side the broken line shows the position of the free end of the electrode before plating: km= l m - lo is the change in the substrate length before dissolution of the mth layer compr~red with the state before plating: A~ m =lm+ 1 - - l m is the change in the length of the electrode due to the dissolution of the ruth layer only.
where b is the width of the cathode and the coating, a0 is the RS in the substrate, ol is the RS in the upper layer and ~ is the average RS in the rest of the coating. After the most upper layer of thickness 2 At is dissolved, the substrate is relaxed and tends to recover to the initial state before plating. T herefore the change Ah 1 in the length of the substrate due to the dissolution of the first layer is positive, and the balance of forces is again restored (Fig. 2(b)): (~ + A o ) ( 2 t - - 2 At) + (o0 + Ao0)t0 = 0
(3)
Equation (2) is subtracted from eqn. (3) and after simple transformations we obtain o~
-
A~ lo
1 2 At (E(2t
2 At) + E0t0}
(4)
w h er e/~ = E/(1 -- v) and/~0 = E0/(1 -- Vo) are the reduced Young's moduli of the subtrate and the coating and v is Poisson's ratio. In a similar way, after dissolving the second layer, which is also 2 At in thickness, we obtain
324 02 -
AX2
1
lo
2 At
(/~(2t -- 4 At) +/~oto} --
AXl/~ lo
(5)
For each layer m (m < k, where k is the number of layers in the coating) we may write AX m 1 (l~(2t -- 2m At) +/~oto} - - . -l=m Ak./~l o 2 At n= 1 l0
O~rrl --
(6)
or
o,,,
=
~
A;%~
..... , , = ~ - ~ n = 1
(7)
/0
a,nm is the instantaneous stress, i.e. the stress which is removed during the dissolution of the layer m. The second term reflects the effect of the upper, already dissolved, layers. Thus by using eqn. (6) the strained state of the layer m is restored to that at the beginning of the dissolution. These considerations present a solution of a problem in the field of the strength of materials. Therefore the approach to this problem is almost similar to the one accepted by Wagner [9] but here the reverse case is treated, i.e. dissolution. Therefore it is of interest to juxtapose the results obtained by Wagner with our data and to draw some new conclusions. On the basis of data obtained during the deposition of the coating, Wagner [9] worked out the following equation for the RS in the ith layer of the coating, which consists of k layers: Aki
1
lo
2 At
O i --
{/~(i-- 1)2 At +/~0t0} + ~ AXj/~ j=i+l lo
(8)
or
k oi = oii +
Akj
E 1=i+1
F'
(9)
10
where A~, i = li - - l i z is the change in the length of the cathode after the deposition of the ith layer. Here again oii is the instantaneous stress, and the second term in eqn. (9) reflects the effect of the upper layers on the strained state of the ith layer after deposition is over. Equation (8) can be simplified in the following manner: o, = - - ( k , - k~_~ )l--~ A----t 2 {/~(i -- 1)2 At + £ o t o} + (kk - - k i ) lo
(10)
where k~ I i -- lo is the change in the length of the cathode, with respect to the state before deposition, caused by the deposition of layers I - i. The expression (10) coincides principally with the equation previously proposed by Popereka [1] (without stating how it was worked out, and n o t writing in detail the first term) for the determination of stress in the coating layer at given distance from the cathode surface. =
325
Equation (6) can be presented in a similar way: Om= (kin +, --hm)I--~ A-----t 2 {2/~(t--m At) +/~0t0) --(km --Xl)~.
(11)
Equations (10) and (11) can also be presented in the following manner: O'i ---- (~ki--1 - am = ( X m + l -
Xi){(i -- 1)K, + K2} -- (Xi -- Xk)K, Xm)((k--re)K1
+K2}
--(~km --X,)K1
(12) (13)
where K1 = E / l o a n d K 2 = Eot0/2 At 10. Let us assume that the deposition and dissolution of a given layer proceeds with an identical step At. Then it can be proved that, provided that a complete s y m m e t r y exists between the deposition and dissolution processes, and no relaxation or redistribution of the RS is present, eqns. (12) and (13) when applied to the given layer of the coating are identical. It must be taken into consideration that i increases from the substrate to the last layer k, while m increases in the reverse direction; therefore i + m = k + 1. This means t h a t under the above-mentioned conditions Xm+, -- Xm = Xi-, -- X~
(14)
~kt'l - - ~1 ---- ~ i - - ~kh
(15)
These circumstances prove the correctness of the approach for deriving eqns. (6) and (8). Various equations have been proposed for the evaluation of the average RS through the thickness of the coating [3, 5, 13, 14]. By using the relationship between o i and {~ (eqn. (1)) and eqns. (8) and (6) for the determination of the stress in a layer with thickness At worked o u t by Wagner and us, we could check which of the miscellaneous equations for the evaluation of the average stress is correct. If we introduce eqn. (10) in eqn. (1), and replace integration by summation, we obtain Eot_____~o k E2 At k -- Xi)(i -- 1) -- (X; -- ~k)} = 2tlo ~I(Xi - 1 - - }ki) + --2tl ~-~ ( ( X i - 1 i= ---o ~= 1
(16)
It is readily shown that k
(17)
~(Xi_ 1 - X/); --Xh i=1 h
h
~ ( X i _ l - - Xi)(i-- 1) = ~ (Xi --Xk) i=1
i=l
= X , + X2 + . . . + X k ( h - - 1 )
(18)
Thus the second term in eqn. (16) is reduced to zero and we obtain for
/~oto X 2t
Io
(19)
326
which is the only c or r e c t equation for the average RS along the thickness. It can be shown that the relationship proposed by Perakh [13] for the average stresses
/~0to+/~2t X
0 -
10
(20)
2t
is n o t correct. The plating stress o¢i determined by Wagner [9] coincides with the stress d e n o t e d previously by Perakh [13] as "real". It is somewhat different from t h a t considered up to now in our paper, and represents the stress in a given layer before its interaction with the layers u n d e r n e a t h and the substrate. However, it is n o t c o r r e c t to shift from stress in a given layer to average stress within the entire coating by simple substitution of AX/2 At by
X/2t. If we evaluate the average plating stress 1 oei = -- ~ Oei
(21)
rl i = l
where, following Wagner [ 9 ] , eei -
AXi
1
li
2 At
(i2 At/~ +/~oto)
(22)
we will obtain:
g 0el
--
-
-
n2 At l
(Z~.
1
2 At + AX2 4 At + ... + AX, 2n At) +
Eoto -
-
n2 At l
~k n
(23)
Obviously eqn. (23) does n o t agree with eqn. (20), regardless of the fact that the appropriate relationships for separate layers of the coating are identical. The physical cause of this discrepancy is the lack of an approach which considers each lower layer as a substrate for the next.
3. Conclusions These investigations show clearly t hat a strict correlation exists between the various equations for the de t e r m i na t i on of the RS. Which equation is to be selected f o r the actual case depends on the viewpoint from which we are approaching the RS. Thus, for example, if we try to relate the value of the stresses to the electrocrystallization processes it is necessary to use Wagner's [9] equation for plating stress oei in the appropriate layer. This is the stress in the layer before its interaction with the substrate and with layers that have been already deposited. The oe; values are classified by Wagner as specific for the process, i.e. depending on the bath f or m ul at i on and deposition conditions.
327 I f we w a n t t o find o u t t h e reasons f o r early c r a c k i n g d u r i n g the deposit i o n process, we should be i n t e r e s t e d m a i n l y in t h e i n s t a n t a n e o u s stress o,-i. This is t h e stress in t h e freshly d e p o s i t e d layer, w h i c h has already i n t e r a c t e d with t h e l o w e r layers a n d t h e substrate. F o r practical a p p l i c a t i o n s , h o w e v e r , it is m o r e i m p o r t a n t to k n o w the e f f e c t e x e r t e d b y t h e RS o n s o m e p h y s i c o m e c h a n i c a l , p r o t e c t i v e - d e c o r a t i v e or o t h e r p r o p e r t i e s w h i c h are r e l a t e d t o t h e d e s i g n a t i o n o f the coatings. In this case t h e strained s t a t e d u r i n g t h e f o r m a t i o n o f the coatings is n o t v e r y i m p o r t a n t , w h e r e a s t h e s t e a d y s t a t e stress is o f m a j o r c o n c e r n . T o evaluate this it is necessary to d e t e r m i n e first o f all t h e average stress ~ by using eqn. (19) b u t t a k i n g into c o n s i d e r a t i o n t h e p o s t - d e p o s i t i o n alterations. I f a m o r e detailed investigation of the e f f e c t e x e r t e d b y the RS on the o p e r a t i n g characteristics o f the c o a t i n g is r e q u i r e d , t h e n t h e R S D (am) m u s t be p l o t t e d on the basis o f d a t a o b t a i n e d d u r i n g the dissolution o f the c o a t i n g (eqn. (13)). T h e a p p l i c a t i o n s o f this a p p r o a c h to the p r o b l e m s dealing with the RS, a n d a b o v e all the use o f eqns. (12), (13) and (19), are p r e s e n t e d in the accompanying paper [ 15].
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