EVALUATION OF SHRINKAGE STRESSES AND ELASTIC
PROPERTIES IN A THIN COATING ON A STRIP-LIKE SUBSTRATE
D.A. Chelubeev, R.V. Goldstein, V.M. Kozintsev, D.A. Kurov, A.V. Podlesnyh and A.L. Popov
Laboratory on Mechanics of Strength and Fracture of Materials and Structures,
Institute for Problems in Mechanics of the Russian Academy of Sciences,
119526, prosp. Vernadskogo 101-1, Moscow, RUSSIA, [email protected]
ABSTRACT
Theoretical models and experimental schemes of deformation of a strip-like substrate at coating solidifying are considered. On
the basis of these models, the algorithm for evaluation of shrinkage stresses in a coating and its elastic properties is suggested. Similar studies for the case of the same elastic properties of a coating and a substrate have been fulfilled starting from
the classical paper by Stoney [1]. The method suggested in the present paper is applicable for the cases of coinciding and
different elastic properties of the coating and substrate materials. Moreover, the coating elastic properties can be unknown in
advance.
Let us assume that a coating of width δ <<h was deposited on a strip-like substrate of length l and rectangular cross-section of
width b and height h. The substrate is hinge supported at edges x=0, l (Figure 1).
Figure 1. The scheme for evaluation of shrinkage stresses in a coating
Coating shrinkage occurs after the film solidifying. It leads to generation of stresses in a substrate and coating and to substrate
bending.
At first the substrate and coating in the free disconnected state are considered as two independent strips of length l and l0.
Having accepted, that l > l0, lengthening Δl = l - l0 is determined Δl = Nl0/(E2F), where F=bδ is the area of cross-section, E2 is
the elastic modulus of a coating, N is the force which is necessary for coating stretching on size Δl. The forces N are assumed
to be acting at the substrate edges (Figure.1).
The appropriate bending moment equals M=Nh/2. On the other hand, the longitudinal force can be expressed through the
stress σ in the coating (N=σF) and the bending moment - through the maximal deflection of the hinge supported strip wmax,
bent by the edge moment M=8EJ wmax /l2, where E is the elastic modulus of the substrate material, J is the moment of inertia
of the substrate section.
Hence, having measured the maximal deflection of the strip, we can determine residual stresses in the coating by the formula
4 Eh 2 wmax
σ=
3l 2δ
(1)
If it is impossible to consider the coating being thin or its elastic characteristics exceed similar characteristics of the substrate it
is necessary to consider the additional factors influencing the value of shrinkage stress:
- the correction due to compression caused by the longitudinal forces [2], converting Eq. (1) to the following one
σ1 =
σ
(2)
1 + 16wmax ( р 2 h )
- a decreasing of the neutral line length under the action of the longitudinal forces as a result of which an additional (to the
value σ in Eq. (1)) stress in the coating will occur after extension of the neutral line up to its initial length:
σ Σ = σ + σ I = σ (1 +
δ E2
h E
(3)
)
The last correction will be essential, when the coating thickness tends to the substrate thickness and elastic moduli of joined
materials are close to Stoney [1];
- a decreasing of a length of the coating in the bended state; after substrate flattening up to the plane state an additional stress
occurs therefore the total stress in view of compression of the neutral line and bending will be equal to
σ Σ = σ (1 + 4
δ E2
h E
)
(4)
It is visible from here, that in the approximation δ << h the correction related to the support bending three times exceeds the
correction related to compression.
At δ ~ h and E2 ~ E also it is necessary to account for shear of a neutral axis. In this case
⎡
⎛
⎣
⎝
σ Σ = σ ⎢1 + δ * E2* ⎜⎜1 + 3
E
1 + δ * ⎞⎤
δ
⎟ , E2* = 2 , δ * =
* ⎟⎥
E
h
1 + δ * E2 ⎠ ⎦
(5)
Note, that in all aforementioned cases the stress state in the substrate was assumed to be uniaxial, i.e. shrinkage strains are
only developed along the substrate. At an equiaxial plane stress state in correction factors (5) one needs to add a factor 1/(1ν2), where ν2 is Poisson's ratio of the coating material [3].
Evaluation of the basic elastic characteristic of the coating - its elastic modulus – can also be made experimentally by the
interferometric measuring a substrate deflection under the action of known loads. For instance, from the maximal deflection
wmax of a cantilever beam under the action of the transverse force P on the free edge the stiffness of the substrate
3
EJ=Pl /(3 wmax ) is determined.
In some cases one can not observe the whole beam part from the clamping to the point of force application. Denote by l1
and l2 the distances from the clamping point to the boundaries of the observable interval, while the displacements difference
for these points, w(l1) - w(l2), will be denoted by d. Then from the equation of an elastic line we obtain
EJ =
P ( 3 l ( l 22 − l12 ) − ( l 23 − l13 ))
6d
(6)
Evaluation of the elastic modulus of a coating material was performed by comparing the stiffness values of samples with and
without coating, determined on the basis of the tests. Measurements were performed on the small-size electronic speckle interferometer. Its optical block with the cantilever type specimen fixed on its base is given in Figure. 2.
This possibility is illustrated in Figure 3 where interferograms are given of the bending samples with (top strip) and without the
coating by the same force. The difference of band numbers characterizes a difference in the maximal deflections.
The experimental ratio of the stiffnesses of the substrate with and without coating (EJ)c and (EJ)0, respectively, is expressed
through number of the interference bands mc, m0 observed in both cases:
( EJ )с m0
=
,
( EJ )0 mс
( EJ ) 0 =
bE1h3
12
(7)
1
2
3
4
Figure 2. Electronic speckle interferometer LIMON-TV: 1 – video camera, 2 – Model specimen of the
cantilever type, 3 - Division system for laser beam (two semitransparent mirrors and ground glass), 4 - Laser
Hence, for evaluation of the modulus of the substrate material there is enough to measure coating, δ, and substrate, h, thicknesses.
Figure 3. Interferograms of bending of the samples with
(above) and without coating
Figure 4. Steel and cardboard strips after the bending made
by the sprayed paint
In the case of a multilayered coating of the thickness less as compared to the support one the bending stiffness of the support
can be neglected. Hence, the separate layers of the coating will have a resistance to tension and compression. Then the
bending of the substrate with the coating can be presented as a bending of a homogeneous body with an equivalent modulus
which is determined by the formula, analogous to those which is used to evaluate an effective elasticity modulus at tension of
several rods of the same length
n
⎛Eh
⎞
E = ⎜ 1 1 + ∑ Ei hi ⎟
i =2
⎝ 2
⎠
n
∑h
i =1
i
(8)
where Ei, hi (i=1,…,n) are moduli and thicknesses of separate layers, including the substrate, n is the number of layers. It is
accounted for in Eq. (8), that the substrate bending is equivalent to tension-compression of a half-thickness body.
Usage of the effective modulus allows to account for various alternatives of the changing of the properties from one layer to
another in the coating: periodic, monotonic, etc, and also reduces the problem of a substrate bending with a coating to one
equation of the classical theory of a plate bending. The Young moduli of separate layers can also be determined. It is enough
for this purpose, to know a substrate deflection at least in one point and the Young modulus variation from one layer to another.
An influence of the same coating is essentially different on the substrates of different materials. In Figure 4 steel and cardboard strips are shown after the bending made by the solidifying of sprayed aerosol double acrylic concentrated metallic paint,
put on these strips on the one side. The sizes and elastic characteristics of strips, as well as the measured sizes of their
maximal deflections under the action of shrinkage stresses and values of these stresses are given in Table 1.
TABLE 1. The characteristics of substrates and shrinkage stresses in coating
Material
Sizes / mm
wmax / mm
E / GPa
E2 / GPa
Shrinkage stresses / МPa
Steel
30 × 6 × 0,15
0,55
200
24
2,8
Cardboard
17,5 × 8 × 0,1
3
2,5
24
1,3
After evaluation of the values
E2
and
σ Σ , the shrinkage can be calculated from formula (5):
⎛
⎛ 1 + δ * ⎞ ⎞ ⎫⎪
4 E wh 2 ⎧⎪
⎟⎟ ,
ε = 1 2 ⎨1 + δ * E2* ⎜⎜1 + 3⎜⎜
* ⎟ ⎟⎬
+
E2 3δl ⎪⎩
1
δ
E
* 2 ⎠ ⎠⎪
⎝
⎝
⎭
(9)
or assuming equiaxial stress condition,
⎛ 1 + δ * ⎞ ⎞ ⎫⎪
4 E1wh 2 (1 − ν 2 ) ⎧⎪ δ * E2* ⎛
⎜1 + 3⎜⎜
⎟⎟
ε=
⎨1 +
2
* ⎟ ⎟⎬
E2 3δl
1
+
δ
E
⎪⎩ 1 − ν 2 ⎜⎝
* 2 ⎠ ⎠⎪
⎝
⎭
where
(10)
w = wmax .
The Poisson Ratio of the Coating Material. In the last case, the Poisson ratio of the coating material should be known. Its
evaluation can be done from the experiment by the following conditions: after selection the geometry and material of the substrate for the condition δ * E 2
ing approximate formula
*
>> 1 , ε is evaluated from the experiment with equiaxial plane stress condition using the follow-
ε0 =
As
⎛ 1 + δ* ⎞ ⎞
4 wh ⎛
⎜1 + 3⎜⎜
⎟⎟ ,
2 ⎜
* ⎟⎟
3l ⎝
⎝ 1 + δ * E2 ⎠ ⎠
ε = ε 0 , the Poisson ratio is determined from Eq. (10) after performing another experiment with the same material as in the
previous test, but with the condition
δ * E2* << 1 :
ν2 = 1−
3ε 0 E2*δ *l 2
4 wh
(11)
Thus, the described method enables to evaluate shrinkage stresses at coating solidifying as well as unknown elastic characteristics of the coating and substrate. Values of these characteristics are determined experimentally by measuring the substrate form changing. In the case of coating with unknown history of solidifying the problem on shrinkage stresses evaluation is
similar to the problem on evaluation of residual stresses, for example, in a welded joints by drilling of blind holes. In this case it
may be rational to evaluate the shrinkage stresses in coating by means of an appropriate scratch-test.
Acknowledgments
The study was supported by the Russian Foundation for the Basic Research (Projects 06-08-01017 and 05-01-08017) and the
Grant of the RF President for supporting the leading scientific schools of Russia (N Sh - 4472.2006.1).
References
1.
2.
3.
Stoney G.G. The tension of metallic films, deposited by electrolysis // Proc. Roy. Soc. London.. Ser. A. 1909 V. 82. № 553.
P. 172–175.
Strength. Stability. Oscillations. Handbook. Eds. I.A.Birger and J.G.Panovko. V. 1. Moscow: Publ. “Mashinostroenie”,
1968. 831 p. (in Russian).
Fokin V.G. Residual stress and warping at coating deposition // Bull. Samara State Univ. Ser. Tech. Sciences 2000. №10.
P. 40-45. (in Russian).