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Thermal stresses in coated structures
ELSEVIER
Surfaceand Coatings Technology99 (1998) 125-131
Thermal stressesin coated structures S.Y. Hu a, Y.L. Li a,*3 D. Munz a*b,Y .Y. Yang b
Abstract Thermal stresses in coated structures were investigated by the boundary element method (BEM). It was found that the singular stress field near the free edge of the interface has a converging distribution n4th decreasing coating thickness. This converging distribution can be described by an asymptotic expression including one singular term and one constant term. In determining the effective range of the asymptotic expression, some quantities relating to a failure criterion of coated structures are presented. 0 1998 Elsevier Science S.A. Kqwwds:
Boundary elementmethod; Coated structure: Failure criterion; Singularstresses: Thermal stresses
1. Introduction
In order to utilise the characteristic properties of different materials, various coated structures have been developed. Examples include coatings for thermal protection, wear resistance, corrosion resistance and multilayer chips. Because of differences in the physical and mechanical properties of the bonded materials, stresses occur in coated systems during fabrication or during service loading. These stresses are especially high near the free edge of the interface between the coating and the substrate, and may cause delamination of the interface or failure in the coating or the substrate. These interfacial stresses have been evaluated in several theoretical studies. Suhir [ I] made an approximate calculation of the interf’dcial shear and peeling stresses applying the beam theory. Williams [2] and Bogy [3] showed that the stresses near the free edge of the interface are singular, and that the singularities depend on the Dundur parameters and on the joint geometry. Yang et al. [4-61 expressed the singular stresses as the sum of a singular term and a regular term and developed empirical relations between the stress intensity factor of the singular term, the material properties and the geometry of the joint. In Ref. [6] a joint with a thin coating was considered and general relations for the stress intensity factor of the singular term were presented. The relevant failure criterion for such a singular stress * Corresponding author. 0257-8972/98/$19.00 c 1998ElsevierScience B.V. All rightsreserved. PII SO257-8972(97)00418-O
field is an open problem. In this paper, additional results about the stress field in a coated structure are presented after a homogeneous change in temperature. obtained using the boundary element method (BEM). The extension of the singular stress field in relation to the coating thickness is evaluated. Finally, some ideas about a failure criterion are developed.
2. Numerical model
For the thermal stress analysis, the following assumptions were made for the structure shown in Fig. 1: (1) a homogeneous change in temperature, (2) perfect bonding at the interface, (3) plane strain and (4) both materials are linear elastic, homogeneous and isotropic. The material parameters are independent of the temperature. Three material combinations (A. B and C) are considered. Their Young’s modulus and Poisson’s ratios are listed in Table 1. For all combinations, the ratio of the
thermal expansion coefficients were identical. i.e. xJzl = 2. These three combinations were chosen to cover a wide range of the singular stress exponents of most bimaterial joints. The geometries with Hz/L= 1 and 0.01 I HI/H, I 1 are considered, which correspond to the normal coated structures. All calculated stresses are normalised by E,a,AT, where AT is the change in temperature.
126
S. Y. Hu et al. / Sqface
and Coatings
Teclwologq 0.04
99 (1998)
125-131
1
0.03 4
0.02 F a $
E, v2 c(* AT
0.01
: -.-.-.
H,/H,=0.033
-
H,iH,=0.025
~~
2 0.0
II 25
30
35
40
35
40
r/H,
,
2L.
(a)-0.02
Fig. 1, Geometry of the coated structure.
The boundary element program BEASY [7] was employed to calculate the thermal stresses. Considering the symmetry of the structure? only half of the specimen was modelled. In addition, the quadratic element with double precision was used and the mesh was made extremely fine near the singular point in order to simulate the singular stresses. The smallest distance between the nodes was about 10e4H,.
--.-'.
H,/H,=0.033
3. Results
3.1. General stressdistribution
The thermal stresses in the coated structure with material combinations A, B and C were calculated. From the numerical results it was found that the distribution character of the stresses along the interface in combination B is similar to those in combination A. Therefore, only stresses along the interface in combinations A and C are plotted; these are shown in Figs. 2 and 3. Note that the end of the curves are at r/L= r/H, = 1, i.e. at the symmetric plane of the structure. The stress components 0’)’ and u,.., are continuous across the interface, but B, has a jump. It can be seen from Figs. 2 and 3 that all the stress components converge monotonously with decreasing thickness of the coating HI/Hz. The general trends of the stresses are as follows: (1) at the free edge of the interface, all stresses
0
5
10
15
20
25
30
r/H,
(b)
0.9-l
csxin the coatirly
Table 1 Material parameters Combination
0.2 0
A B C
0.1
0.33
0.175 3.89
0.30 0.15
0.28 0.20 0.26
Cc)
I 5
I 10
I 15
t 20
I 25
I 30
I 35
r/H, Fig. 2. cij versus r/H, along the interface for combination A.
I 40
S. Y, Hu et al. 1 Surface
and Coa?ings
0.2-
0.1 20.c
18-
-------.
H,/H,=0.05
---.
H,/H,=0.033
-
H,/H,=0.025
0.c l4-
2
99 (1998)
127
125-131
are singular, (2) from the free edge to the symmetric plane of the coated structure, the stress normal to the interface ~~ changes from a high value to nearly zero with a maximum and a minimum for combination A and with a minimum for combination C, (3) the stress parallel to the interface gX in the coating changes from high values to a constant value, with two minimums and one maximum for combination A and with one minimum for combination C, and (4) the shear stress gXY decreases from a high value to zero from the free edge to the symmetric plane of the structure. The singular stresses near the free edge of the interface will be discussed in Section 3.2. For the coated structure, the coating stress gx- along the surface of the coating is significant. Fig. 4 shows the variation of ox along the surface of the coating in the case of L/H, = 20, where the straight lines correspond to results from the beam theory [8]. They were calculated according to
......... H,/H,=O.l I
0.1 '6-
i= Q $
Technology
0. O(
10
15
25
20
30
35
40
r/H,
-0.0 14-0.0 18-
f (a) -0.1 0.8
E;dcldT(e-3h2-2h3)
I
0
5
I 10
I 15
H,/H,=O.I
-------.
H,/H,=0.05
---.
H,/H,=0.033
-
H,/H,=0.025
I 25
I 20
(1)
Gx=cZ+e(4h3+6iz2+4h)+h4
.‘.......
I 30
I
I
35
40
in which h = HI/Hz, e = E;/E;, Ei = Ei/( 1 - $) (i = 1,2), and Ar = M~(1 + v~)-cI~(1 + 11~).Because the geometry of the coated structure considered in this paper does not satisfy the assumption of the beam theory, the result of the beam theory is only valid in a small range near the centre of the structure. From Fig. 4 it can be seen that the coating stress cX reaches a limit value at the centre of the coating, which may determine the failure of the central part of the coating. The limit value calculated with BEM is in good agreement with the value determined from Eq. (1). This conclusion is also valid for other ratios of H,IL = HI/H,.
r/H,
('4
i= Q 2
0.8 I 0.6
’ 1’ !
' : I . . .. . .J. . . . . . .. . . ..-.......................................................................-. ______-----
__--
E ___.. _
z
“.”
I
0
L
-------.
H,/H,=0.05
---.
H,/H,=0.033
I
I
I
I
I
I
I
IO
15
20
25
30
35
40
r/H,
(4
Fig. 3. cij versus r/H,
along the interface for combination C.
Fig. 4. Comparison of the stress os along the surface of the coating from BEM and the beam theory (the straight lines are from the results of the beam theory).
Fig. 2 and Fig. 3 show that the stresses near the free edge of the interface are singular. These singular stresses can be expressed as [4]: Gij(r*O)
=
&A(‘)
+ GijCI(H) 1
where 1’and 0 are the polar coordinates defined in Fig. l? (o is the singular stress exponent, K is the stress intensity factor, J;j(a) are the angular functions, and ~ijo(B) is the regular stress term (in Cartesian coordinates, G).~= gD, crsO= 0 and cXYO=O). In Eq. (2) r is nomlalised by the height H, of the coating, so that the stress intensity factor K has the dimension of the stress. I’ could be normalised by Hz or L; if so, the value of K would be different. All the quantities in Eq. (2) depend on the elastic constants and can be calculated analytically (see Refs, [4,5]), with the exception of K. For thermal loading, we have: Go=
ATAx ET-1 -E*-’1
(3)
with ET =Ei/[l’i( 1 +vi)]: (i= 1, 2). Table 2 gives some of the parameters used in Eq. (2). The superscripts “ + ” and Lb- ” in Table 2 mean that the values at the interface are for the coating and the substrate, respectively. The quantity K depends on the loading, the elastic constants and the geometry (i.e. Hz/L and HI/H,), and has to be determined by a numerical method (e.g. BEM or FEM). From Eq. (2) it is known that log(crEF -aijo) has a linear relation to log(r;iH,). In Fig. 5 the stresses near the singular point, calculated by BEM, are plotted versus r/H, in the logarithm scale for combination B with H,/Hz=0.033 along U=O in the coating. Straight lines with a slope of --w = -0.0947 are also shown. From Fig. 5 it can be seen that log{“? -G’ijo) vs log(r/H,) is a straight line with a slope --w in the range -3,5
-2.5
-2.0
bdW) Fig. 5.log(gE - a,jo) versus log(r/H,) along the interface in the coating for combination 3 with H,/H,=0.033. Table 3 Stress intensity factor K/W,x,AT)
HI /Hz
for different HI/H2
A
B
c 14.54 14.57 14.55 14S 14.45
0.1
- 02686
-0.8915
0.05 0.033 0.025 Eqs. (4) and (5)
-0.2715 - 0.773 1 - 0.273 1 -0.2731
-0.8965 -0.8968 - E396S - 0.8769
Ref. [ 61. The limit value depends on the elastic constants. In Ref. ]6] a relation between the stress intensity factor K, the stress exponent o and the Poisson’s ratios is given (in Ref. [6] K is designated &!A,-). For a thin coating (HJH, ~0.05 and H,/L < 0.1 ), the relations are: -K/a0
=0.9919+0.1523v, -0.5966s;
-2.3825w-8.0247v,w
+ 14.5589v;o,+ 15.3373&
- 16.7054~~0~ -3.5281r:02 for l$ >E;, -K/a0
(4)
and
= 1.0137-0.186711, -2.864lw+
10.3654\1,0
+ 0.6783~: - 17.5983$o, + 2.3556~’ - 17.454~~ co2+ 36.2823~~0’ 1
(5)
Table 2 some parameters of Eq, (2) Combination
A B C
0.1864 0.0947 0.0403
0.3911 1.1209 - 15.522
- 0.6585 - 0.2942 0.0600
0.2776 0.1301 -0.1235
1.0 1.0 1.0
-0.2158 -0.1147 0.0545
S. Y. Hu et al. / Surfhce
and Coatings
Technology
99 (1998)
125-131
129
Combination A 0.3T
Combination
r’/H, Interface
Interface
0.1
0.15
0.1
0.05
0.05 i 0.1
B
0.15
6%us azAT)
(6 u, 0, AT)
1
(b)
(a)
Combination C 0.1T
0.05
U5 v, a, AT) 0.1
0.15
r’/H, Interface
Fig. 6. The effective range of Eq. (2) for combinations A, B and C with H,/H2=0.033.
130
S. I’. Ho et al. I Swfuce
rind Cwtings
Tccimology
for ET < Ez. The values of the last line in Table 3 were calculated from Eqs. (4) and (51, which are based on finite element results. There is good agreement between the results from BEM and Eqs. (4) and (5). Therefore, for a coated structure with HI/H2 ~0.05 and H,/L.
125-131
99 (1598)
0.06 0
0
00
0 a
0.05
0
0.04
A
3.3. E.utensionof the singzdmstress$ficld
A
AA
A
0
0
For the development of a failure criterion. the extension of the singular stress field is important. It is important to realise that a correct description of the singular stress field, even very close to the singular point, has to include the second term in Eq. (2) 141. The effective radius ? of the singular stress description can be obtained by comparing the stresses according to Eq. (2) with the results from BEM. The effective radius is defined at an accuracy of 5%, which means that the relative difference between the results from BEM and Eq. (2) for each stress component is not larger than 5%. In Fig. 6 the effective range is plotted for combinations A. B and C with H,/H,=O.O33. It can be seen that the effective radius P* depends strongly on the angle 8, and is larger in the material with the higher Young’s modulus. In Fig. 7 the ratio of 9/H, along the interface and the free edge of the coating is plotted versus log(H,/H,). For thin coatings, a limit value of P/H, is obtained. This limit value depends on the material properties and is in the range of 0.02
4. Failure criterion A failure criterion for a coated structure. in which the failure is initiated near the free edge of the interface, cannot be defined in a straightforward manner. The singular stress field in such a structure is an idealisation. similar to the tip of a crack in linear elastic fracture mechanics. In reality, a plastic zone or a process zone will develop near the singular point. Nevertheless, there is a unique relation between the elastic stress field and the state in the process zone as long as this zone is
DA
0 t
AB oc
-2.0
-1.5
-1.0
-0.5
log(H,/H,J
(ai 0.31
A
0.05
*
AAAA
Q
QQ
Q
0
0
0
o A
A
A
A
AB
0
0
0
QC
1
o.oyi-
__ -2.0
(b)
-1.5
-1 .o
-0.5
0.0
bN-W,)
Fig. 7. The effective radius r*/H, versus log(H,;HJ face and (b) along the free edge of the coating.
(a) along the inter-
small. For a crack. the stresses always change with the square root of the distance from the crack tip. Therefore, the stress intensity factor can describe the stress situation completely and failure occurs at a critical value, i.e. the fracture toughness. In a two-material coated structure, the stress field is determined by the quantities entering into Eq. (2)$ i.e. the stress intensity factor K, the stress exponent LO, the functions fij and the constant stress term Go. All these quantities depend on the elastic constants, as shown in Fig. 8, where 1’1= v,=O.3 and _f,, is calculated at the interface: i.e. O=O. Not only R or o, but the cor~?bincztiorz of all tht: quantities determines the stress field, and thus the failure behaviour. For the assessment of the stresses in the coated structures with different material combinations, a value is proposed which is obtained by calculating the average stress along
S. Y. Hu et al. / 0.4
Surfanceand
Coatings
I25-131
H,,/(E,c(,AT)
\
\
\
\
\
I
I
I
I
-3
-2
-1
1
I
I
2
3
log(E,/E,)
(4
-0.4 J K
~‘o, H,, H,,Y(E,a,AT)
‘O-
I I I I I
I ; _
\
,/
,,--
HY
,I ..-----------
,/‘
\ 1,
,/’
I
-2
-1
‘\
\ \ \ I
131
along the interface
Combination
ffx,I(~,~,AT)
A B
0.1709 0.1718
C
0.9900
.A ‘\
-3
99 (1998)
Table 4
1
0 -‘-------
Technology
0
2
3
bX3W c _ _, ,_, _... - ..- .-...-....
no
,
ff,,/E@C -0.4010 -0.3773 2.6474
Hs along the interface are listed in Table 4 for the three material combinations considered, where (i-/H,), = 0.01 is chosen within the effective range of Eq. (2). To show in which way the different parameters in Eq. (2) influence Hij, the quantity Hij is also plotted along the interface versus E2/E1 for the case of \ll =v2 =0.3 in Fig. 8. From Fig. 8 it can be seen that the material combination with higher Q does not always correspond to larger values of H, and H.yy. By comparing the values of Hij from different material combinations, a favourable coating material can be found for a given substrate.
5. Conclusions
The stresses near the free edge of the interface in a coated structure after a change in temperature can be described by Eq. (2), where all the parameters are independent of the thickness of the coating with exception of the stress intensity factor K. Below a critical thickness ratio HI/H,, K is constant. Eq. (2) is valid within r I P, where I’* depends on the elastic properties of the coating and the substrate and on the angle 0. Applying Eq. (2), a failure criterion can be developed on the basis of an average normal or shear stress along a given distance from the free edge.
Acknowledgement
(b) Fig. 8. The parameters in Eq. (2) and H,, versus r,=v,=0.3.
log(E,,‘E,)
for
a small distance from the singular point as follows [5,9]:
E=f,jCn)
= (1 -o)(r/PI,)r”
fcrijO(l))
(6)
This value depends on the chosen length (r/H,),, the considered stress component and the angle U. For the coating, either the shear stress G.~~or the normal stress Go is responsible for delamination. Therefore. H,, and
Y .L. Li gratefully acknowledges financial support provided by the Alexander von Humboldt Foundation. References [I] [2] [3] [4] [j] [6] [7]
E. Suhir, Int. J. Soiids Struct. 27 (1991) 1025. M.L. Williams. J. Appl. Mech. 19 (1952) 526. D.B. Bogy, J. Appl. Mech. 38 (1971) 377. D. Munz, Y.Y. Yang, J. Appl. Mech. 59 (1992) 557. D. Munz, Y.Y. Yang, J. Eur. Ceram. Sot. 13 ( 1994) 453. M. Tilscher. D. Munz, Y.Y. Yang, J. Adhesion 49 (1995) 1. BEASY version 5.0, Computational Mechanics BEASY Ltd., Southampton, 1994. [S] S. Timoshenko, J. Opt. Sot. Am. 11 (1925) 233. [9] F. Kroupa, Z. Knesl, J. Zemankova. in: Proceedings of the International Conference on Engineering Ceramics, Smolnice Castle, ed. M. Haviar. 1992, p. 102.
Surfaceand Coatings Technology99 (1998) 125-131
Thermal stressesin coated structures S.Y. Hu a, Y.L. Li a,*3 D. Munz a*b,Y .Y. Yang b
Abstract Thermal stresses in coated structures were investigated by the boundary element method (BEM). It was found that the singular stress field near the free edge of the interface has a converging distribution n4th decreasing coating thickness. This converging distribution can be described by an asymptotic expression including one singular term and one constant term. In determining the effective range of the asymptotic expression, some quantities relating to a failure criterion of coated structures are presented. 0 1998 Elsevier Science S.A. Kqwwds:
Boundary elementmethod; Coated structure: Failure criterion; Singularstresses: Thermal stresses
1. Introduction
In order to utilise the characteristic properties of different materials, various coated structures have been developed. Examples include coatings for thermal protection, wear resistance, corrosion resistance and multilayer chips. Because of differences in the physical and mechanical properties of the bonded materials, stresses occur in coated systems during fabrication or during service loading. These stresses are especially high near the free edge of the interface between the coating and the substrate, and may cause delamination of the interface or failure in the coating or the substrate. These interfacial stresses have been evaluated in several theoretical studies. Suhir [ I] made an approximate calculation of the interf’dcial shear and peeling stresses applying the beam theory. Williams [2] and Bogy [3] showed that the stresses near the free edge of the interface are singular, and that the singularities depend on the Dundur parameters and on the joint geometry. Yang et al. [4-61 expressed the singular stresses as the sum of a singular term and a regular term and developed empirical relations between the stress intensity factor of the singular term, the material properties and the geometry of the joint. In Ref. [6] a joint with a thin coating was considered and general relations for the stress intensity factor of the singular term were presented. The relevant failure criterion for such a singular stress * Corresponding author. 0257-8972/98/$19.00 c 1998ElsevierScience B.V. All rightsreserved. PII SO257-8972(97)00418-O
field is an open problem. In this paper, additional results about the stress field in a coated structure are presented after a homogeneous change in temperature. obtained using the boundary element method (BEM). The extension of the singular stress field in relation to the coating thickness is evaluated. Finally, some ideas about a failure criterion are developed.
2. Numerical model
For the thermal stress analysis, the following assumptions were made for the structure shown in Fig. 1: (1) a homogeneous change in temperature, (2) perfect bonding at the interface, (3) plane strain and (4) both materials are linear elastic, homogeneous and isotropic. The material parameters are independent of the temperature. Three material combinations (A. B and C) are considered. Their Young’s modulus and Poisson’s ratios are listed in Table 1. For all combinations, the ratio of the
thermal expansion coefficients were identical. i.e. xJzl = 2. These three combinations were chosen to cover a wide range of the singular stress exponents of most bimaterial joints. The geometries with Hz/L= 1 and 0.01 I HI/H, I 1 are considered, which correspond to the normal coated structures. All calculated stresses are normalised by E,a,AT, where AT is the change in temperature.
126
S. Y. Hu et al. / Sqface
and Coatings
Teclwologq 0.04
99 (1998)
125-131
1
0.03 4
0.02 F a $
E, v2 c(* AT
0.01
: -.-.-.
H,/H,=0.033
-
H,iH,=0.025
~~
2 0.0
II 25
30
35
40
35
40
r/H,
,
2L.
(a)-0.02
Fig. 1, Geometry of the coated structure.
The boundary element program BEASY [7] was employed to calculate the thermal stresses. Considering the symmetry of the structure? only half of the specimen was modelled. In addition, the quadratic element with double precision was used and the mesh was made extremely fine near the singular point in order to simulate the singular stresses. The smallest distance between the nodes was about 10e4H,.
--.-'.
H,/H,=0.033
3. Results
3.1. General stressdistribution
The thermal stresses in the coated structure with material combinations A, B and C were calculated. From the numerical results it was found that the distribution character of the stresses along the interface in combination B is similar to those in combination A. Therefore, only stresses along the interface in combinations A and C are plotted; these are shown in Figs. 2 and 3. Note that the end of the curves are at r/L= r/H, = 1, i.e. at the symmetric plane of the structure. The stress components 0’)’ and u,.., are continuous across the interface, but B, has a jump. It can be seen from Figs. 2 and 3 that all the stress components converge monotonously with decreasing thickness of the coating HI/Hz. The general trends of the stresses are as follows: (1) at the free edge of the interface, all stresses
0
5
10
15
20
25
30
r/H,
(b)
0.9-l
csxin the coatirly
Table 1 Material parameters Combination
0.2 0
A B C
0.1
0.33
0.175 3.89
0.30 0.15
0.28 0.20 0.26
Cc)
I 5
I 10
I 15
t 20
I 25
I 30
I 35
r/H, Fig. 2. cij versus r/H, along the interface for combination A.
I 40
S. Y, Hu et al. 1 Surface
and Coa?ings
0.2-
0.1 20.c
18-
-------.
H,/H,=0.05
---.
H,/H,=0.033
-
H,/H,=0.025
0.c l4-
2
99 (1998)
127
125-131
are singular, (2) from the free edge to the symmetric plane of the coated structure, the stress normal to the interface ~~ changes from a high value to nearly zero with a maximum and a minimum for combination A and with a minimum for combination C, (3) the stress parallel to the interface gX in the coating changes from high values to a constant value, with two minimums and one maximum for combination A and with one minimum for combination C, and (4) the shear stress gXY decreases from a high value to zero from the free edge to the symmetric plane of the structure. The singular stresses near the free edge of the interface will be discussed in Section 3.2. For the coated structure, the coating stress gx- along the surface of the coating is significant. Fig. 4 shows the variation of ox along the surface of the coating in the case of L/H, = 20, where the straight lines correspond to results from the beam theory [8]. They were calculated according to
......... H,/H,=O.l I
0.1 '6-
i= Q $
Technology
0. O(
10
15
25
20
30
35
40
r/H,
-0.0 14-0.0 18-
f (a) -0.1 0.8
E;dcldT(e-3h2-2h3)
I
0
5
I 10
I 15
H,/H,=O.I
-------.
H,/H,=0.05
---.
H,/H,=0.033
-
H,/H,=0.025
I 25
I 20
(1)
Gx=cZ+e(4h3+6iz2+4h)+h4
.‘.......
I 30
I
I
35
40
in which h = HI/Hz, e = E;/E;, Ei = Ei/( 1 - $) (i = 1,2), and Ar = M~(1 + v~)-cI~(1 + 11~).Because the geometry of the coated structure considered in this paper does not satisfy the assumption of the beam theory, the result of the beam theory is only valid in a small range near the centre of the structure. From Fig. 4 it can be seen that the coating stress cX reaches a limit value at the centre of the coating, which may determine the failure of the central part of the coating. The limit value calculated with BEM is in good agreement with the value determined from Eq. (1). This conclusion is also valid for other ratios of H,IL = HI/H,.
r/H,
('4
i= Q 2
0.8 I 0.6
’ 1’ !
' : I . . .. . .J. . . . . . .. . . ..-.......................................................................-. ______-----
__--
E ___.. _
z
“.”
I
0
L
-------.
H,/H,=0.05
---.
H,/H,=0.033
I
I
I
I
I
I
I
IO
15
20
25
30
35
40
r/H,
(4
Fig. 3. cij versus r/H,
along the interface for combination C.
Fig. 4. Comparison of the stress os along the surface of the coating from BEM and the beam theory (the straight lines are from the results of the beam theory).
Fig. 2 and Fig. 3 show that the stresses near the free edge of the interface are singular. These singular stresses can be expressed as [4]: Gij(r*O)
=
&A(‘)
+ GijCI(H) 1
where 1’and 0 are the polar coordinates defined in Fig. l? (o is the singular stress exponent, K is the stress intensity factor, J;j(a) are the angular functions, and ~ijo(B) is the regular stress term (in Cartesian coordinates, G).~= gD, crsO= 0 and cXYO=O). In Eq. (2) r is nomlalised by the height H, of the coating, so that the stress intensity factor K has the dimension of the stress. I’ could be normalised by Hz or L; if so, the value of K would be different. All the quantities in Eq. (2) depend on the elastic constants and can be calculated analytically (see Refs, [4,5]), with the exception of K. For thermal loading, we have: Go=
ATAx ET-1 -E*-’1
(3)
with ET =Ei/[l’i( 1 +vi)]: (i= 1, 2). Table 2 gives some of the parameters used in Eq. (2). The superscripts “ + ” and Lb- ” in Table 2 mean that the values at the interface are for the coating and the substrate, respectively. The quantity K depends on the loading, the elastic constants and the geometry (i.e. Hz/L and HI/H,), and has to be determined by a numerical method (e.g. BEM or FEM). From Eq. (2) it is known that log(crEF -aijo) has a linear relation to log(r;iH,). In Fig. 5 the stresses near the singular point, calculated by BEM, are plotted versus r/H, in the logarithm scale for combination B with H,/Hz=0.033 along U=O in the coating. Straight lines with a slope of --w = -0.0947 are also shown. From Fig. 5 it can be seen that log{“? -G’ijo) vs log(r/H,) is a straight line with a slope --w in the range -3,5
-2.5
-2.0
bdW) Fig. 5.log(gE - a,jo) versus log(r/H,) along the interface in the coating for combination 3 with H,/H,=0.033. Table 3 Stress intensity factor K/W,x,AT)
HI /Hz
for different HI/H2
A
B
c 14.54 14.57 14.55 14S 14.45
0.1
- 02686
-0.8915
0.05 0.033 0.025 Eqs. (4) and (5)
-0.2715 - 0.773 1 - 0.273 1 -0.2731
-0.8965 -0.8968 - E396S - 0.8769
Ref. [ 61. The limit value depends on the elastic constants. In Ref. ]6] a relation between the stress intensity factor K, the stress exponent o and the Poisson’s ratios is given (in Ref. [6] K is designated &!A,-). For a thin coating (HJH, ~0.05 and H,/L < 0.1 ), the relations are: -K/a0
=0.9919+0.1523v, -0.5966s;
-2.3825w-8.0247v,w
+ 14.5589v;o,+ 15.3373&
- 16.7054~~0~ -3.5281r:02 for l$ >E;, -K/a0
(4)
and
= 1.0137-0.186711, -2.864lw+
10.3654\1,0
+ 0.6783~: - 17.5983$o, + 2.3556~’ - 17.454~~ co2+ 36.2823~~0’ 1
(5)
Table 2 some parameters of Eq, (2) Combination
A B C
0.1864 0.0947 0.0403
0.3911 1.1209 - 15.522
- 0.6585 - 0.2942 0.0600
0.2776 0.1301 -0.1235
1.0 1.0 1.0
-0.2158 -0.1147 0.0545
S. Y. Hu et al. / Surfhce
and Coatings
Technology
99 (1998)
125-131
129
Combination A 0.3T
Combination
r’/H, Interface
Interface
0.1
0.15
0.1
0.05
0.05 i 0.1
B
0.15
6%us azAT)
(6 u, 0, AT)
1
(b)
(a)
Combination C 0.1T
0.05
U5 v, a, AT) 0.1
0.15
r’/H, Interface
Fig. 6. The effective range of Eq. (2) for combinations A, B and C with H,/H2=0.033.
130
S. I’. Ho et al. I Swfuce
rind Cwtings
Tccimology
for ET < Ez. The values of the last line in Table 3 were calculated from Eqs. (4) and (51, which are based on finite element results. There is good agreement between the results from BEM and Eqs. (4) and (5). Therefore, for a coated structure with HI/H2 ~0.05 and H,/L.
125-131
99 (1598)
0.06 0
0
00
0 a
0.05
0
0.04
A
3.3. E.utensionof the singzdmstress$ficld
A
AA
A
0
0
For the development of a failure criterion. the extension of the singular stress field is important. It is important to realise that a correct description of the singular stress field, even very close to the singular point, has to include the second term in Eq. (2) 141. The effective radius ? of the singular stress description can be obtained by comparing the stresses according to Eq. (2) with the results from BEM. The effective radius is defined at an accuracy of 5%, which means that the relative difference between the results from BEM and Eq. (2) for each stress component is not larger than 5%. In Fig. 6 the effective range is plotted for combinations A. B and C with H,/H,=O.O33. It can be seen that the effective radius P* depends strongly on the angle 8, and is larger in the material with the higher Young’s modulus. In Fig. 7 the ratio of 9/H, along the interface and the free edge of the coating is plotted versus log(H,/H,). For thin coatings, a limit value of P/H, is obtained. This limit value depends on the material properties and is in the range of 0.02
4. Failure criterion A failure criterion for a coated structure. in which the failure is initiated near the free edge of the interface, cannot be defined in a straightforward manner. The singular stress field in such a structure is an idealisation. similar to the tip of a crack in linear elastic fracture mechanics. In reality, a plastic zone or a process zone will develop near the singular point. Nevertheless, there is a unique relation between the elastic stress field and the state in the process zone as long as this zone is
DA
0 t
AB oc
-2.0
-1.5
-1.0
-0.5
log(H,/H,J
(ai 0.31
A
0.05
*
AAAA
Q
Q
0
0
0
o A
A
A
A
AB
0
0
0
QC
1
o.oyi-
__ -2.0
(b)
-1.5
-1 .o
-0.5
0.0
bN-W,)
Fig. 7. The effective radius r*/H, versus log(H,;HJ face and (b) along the free edge of the coating.
(a) along the inter-
small. For a crack. the stresses always change with the square root of the distance from the crack tip. Therefore, the stress intensity factor can describe the stress situation completely and failure occurs at a critical value, i.e. the fracture toughness. In a two-material coated structure, the stress field is determined by the quantities entering into Eq. (2)$ i.e. the stress intensity factor K, the stress exponent LO, the functions fij and the constant stress term Go. All these quantities depend on the elastic constants, as shown in Fig. 8, where 1’1= v,=O.3 and _f,, is calculated at the interface: i.e. O=O. Not only R or o, but the cor~?bincztiorz of all tht: quantities determines the stress field, and thus the failure behaviour. For the assessment of the stresses in the coated structures with different material combinations, a value is proposed which is obtained by calculating the average stress along
S. Y. Hu et al. / 0.4
Surfanceand
Coatings
I25-131
H,,/(E,c(,AT)
\
\
\
\
\
I
I
I
I
-3
-2
-1
1
I
I
2
3
log(E,/E,)
(4
-0.4 J K
~‘o, H,, H,,Y(E,a,AT)
‘O-
I I I I I
I ; _
\
,/
,,--
HY
,I ..-----------
,/‘
\ 1,
,/’
I
-2
-1
‘\
\ \ \ I
131
along the interface
Combination
ffx,I(~,~,AT)
A B
0.1709 0.1718
C
0.9900
.A ‘\
-3
99 (1998)
Table 4
1
0 -‘-------
Technology
0
2
3
bX3W c _ _, ,_, _... - ..- .-...-....
no
,
ff,,/E@C -0.4010 -0.3773 2.6474
Hs along the interface are listed in Table 4 for the three material combinations considered, where (i-/H,), = 0.01 is chosen within the effective range of Eq. (2). To show in which way the different parameters in Eq. (2) influence Hij, the quantity Hij is also plotted along the interface versus E2/E1 for the case of \ll =v2 =0.3 in Fig. 8. From Fig. 8 it can be seen that the material combination with higher Q does not always correspond to larger values of H, and H.yy. By comparing the values of Hij from different material combinations, a favourable coating material can be found for a given substrate.
5. Conclusions
The stresses near the free edge of the interface in a coated structure after a change in temperature can be described by Eq. (2), where all the parameters are independent of the thickness of the coating with exception of the stress intensity factor K. Below a critical thickness ratio HI/H,, K is constant. Eq. (2) is valid within r I P, where I’* depends on the elastic properties of the coating and the substrate and on the angle 0. Applying Eq. (2), a failure criterion can be developed on the basis of an average normal or shear stress along a given distance from the free edge.
Acknowledgement
(b) Fig. 8. The parameters in Eq. (2) and H,, versus r,=v,=0.3.
log(E,,‘E,)
for
a small distance from the singular point as follows [5,9]:
E=f,jCn)
= (1 -o)(r/PI,)r”
fcrijO(l))
(6)
This value depends on the chosen length (r/H,),, the considered stress component and the angle U. For the coating, either the shear stress G.~~or the normal stress Go is responsible for delamination. Therefore. H,, and
Y .L. Li gratefully acknowledges financial support provided by the Alexander von Humboldt Foundation. References [I] [2] [3] [4] [j] [6] [7]
E. Suhir, Int. J. Soiids Struct. 27 (1991) 1025. M.L. Williams. J. Appl. Mech. 19 (1952) 526. D.B. Bogy, J. Appl. Mech. 38 (1971) 377. D. Munz, Y.Y. Yang, J. Appl. Mech. 59 (1992) 557. D. Munz, Y.Y. Yang, J. Eur. Ceram. Sot. 13 ( 1994) 453. M. Tilscher. D. Munz, Y.Y. Yang, J. Adhesion 49 (1995) 1. BEASY version 5.0, Computational Mechanics BEASY Ltd., Southampton, 1994. [S] S. Timoshenko, J. Opt. Sot. Am. 11 (1925) 233. [9] F. Kroupa, Z. Knesl, J. Zemankova. in: Proceedings of the International Conference on Engineering Ceramics, Smolnice Castle, ed. M. Haviar. 1992, p. 102.