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Stresses in hard coating due to a rigid spherical indenter on a layered elastic half-space
ARTICLE IN PRESS Tribology International 43 (2010) 1592–1601
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Stresses in hard coating due to a rigid spherical indenter on a layered elastic half-space Roman Kulchytsky-Zhyhailo , Gabriel Rogowski Bialystok University of Technology, Faculty of Mechanical Engineering, ul. Wiejska 45C, 15-351 Bialystok, Poland
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 November 2006 Received in revised form 25 January 2010 Accepted 2 March 2010 Available online 11 March 2010
The axisymmetrical contact problem of elasticity connected with an indentation of a rigid spherical indenter in an elastic semi-space covered by an elastic layer is considered. Stress tensor components in interior points of the non-homogeneous half-space by numerical calculation of some integrals was obtained. Detailed analysis of the maximal tensile stress distributions and Huber-von Mises reduced stress distributions produced by contact pressure is presented. The dependence between these stresses and the ratio between the layer thickness and contact area width is explored. The obtained results for stresses are compared with results obtained for half-space loaded by the Hertz pressure. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Spherical indentation Hard coatings Contact problem Stress
1. Introduction The progress of coating technology is the reason for wide employments of hard top layers for improvement of the tribological properties of sliding surfaces. This can lead to reduction of the friction coefficient and the wear quantity while the weight of the material is not changed. Fractures in the coating (cohesive failure) or delamination and spalling (adhesive failure) at the coating/substrate interface are the weak points of the top layers employed [1–4]. Therefore, the analysis of stress distributions produced by contact pressure is very important. Many authors during recent years take interest in the contact problems that a rigid sphere (axi-symmetrical problem) or a long rigid cylinder (two-dimensional problem) are pressured. The selected contact parameters such as penetration depth of the punch or radius of the contact area are recorded in an experiment [5–17], while stresses in the contact area are submit to theoretical consideration by solving a specified contact problem of the elasticity theory [1–3,13–34]. It is shown that fundamental factors responsible for destruction of layers are: maximum tensile stress [1–3,15,18–25,35], Huber–von Mises reduced stress [1,18–21,26–29,36,37] (or maximum shear stress [2,30,35]) and maximum shear stress at the coating/substrate interface [1,3,18,22,32,33]. The highest values of parameters specified above occur on the surface of non-homogeneous medium [21,23] or at the coating/surface interface [20,21,25]. In majority of studies
Corresponding author.
E-mail address: [email protected] (R. Kulchytsky-Zhyhailo). 0301-679X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2010.03.002
[1,20,21,23,25,29–31] the distribution of contact pressure as semipffiffiffiffiffiffiffiffiffiffiffi elliptical pressure is assumed pðrÞ ¼ pmax 1r 2 , where pmax— contact pressure in the centre of the contact area, r—dimensionless radius coordinate (related to radius of the contact zone—parameter a). This procedure is often correct and seems undisputed in case of relatively thick layers (h¼H/a41, where H is a thickness of the layer). When Hertz pressure distribution is assumed, the results are comparable (with a small deviation) with ones given in article [18] where h¼1/0.819E1.221. In the case of hard coating similar effect for ‘thin’ layers is observed [35]. The axi-symmetrical problem on the penetration of sphere into elastic half-space coated by harder layer is known to differ from Hertzian solution [38]. In the case that Young’s modulus of the layer differ considerably from Young’s modulus of the substrate, the maximum value of the contact pressure occurs in some distance from the centre of contact area [38] (about 0.8 of radius of the contact circle with E0/E1 ¼10, h¼0.3, n0 ¼ n1 ¼1/3, where E0, E1, n0, n1 are Young modules and Poisson ratios of the layer and the substrate, respectively). A similar result was observed in corresponding two-dimensional contact problem of elasticity [5,39]. The goal of the work is:
to compare the distribution of specify above stresses caused by true distribution of contact pressure and Hertzian distribution;
to study the ratio between maximal tensile stress and Huber–
von Mises reduced stress on the surface of the heterogeneous half-space and at the layer—half-space interface at different parameters; to explore the dependence between these stresses and the ratio between the layer thickness and contact area width.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
Nomenclature a aH
radius of the contact zone the Hertzian radius of contact for homogeneous halfspace with mechanical properties adequate for the layer E Young’s modulus h¼H/a dimensionless thickness of the layer related to actual radius of the contact zone h* ¼H/aH dimensionless thickness of the layer related to the radius aH H a thickness of the layer H(x) the Heaviside unit step function J0(r), J1(r)denotes the Bessel functions p contact pressure the Hertzian mean contact pressure for homogeneous pH half-space with mechanical properties adequate for the layer
Adequate calculations in the case of two-dimensional contact problem were considered in [40,41]. It is necessary to point out that calculations presented in this paper and [40,41] are consistent to what was given by [1,5,6,20,21,29,30,39,42,43]. We propose a simplified algorithm for solving the contact problem together with the classical algorithm. This algorithm allows one to obtain a formula for pressure calculation which simplifies the analysis of stress fields by finite element method or boundary element method without applying elliptical distribution (Hertz distribution).
contact pressure in the centre of the contact area mean contact pressure the total normal load the ratio of the contact pressure in the centre of the contact area to the actual mean contact pressure dimensionless cylindrical coordinates vector of the non-dimensional displacement referred to the radius of contact the penetration depth of the sphere the Laplace operator the voluminal strain Kirchhoff coefficient Poisson ratios the stress tensor the Huber-von Mises reduced stresses
pmax p0 P p^ r, j, z u
d D y
m n r sHM
where r, j, z are dimensionless cylindrical coordinates referenced to the radius of the area of the contact. The friction in the contact area is ignored. The ideal mechanical contact conditions between the layer and the half-space are assumed. All state functions denoted by the upper index 0 describe the state of displacement and stresses in the layer; the state functions for the elastic halfspace are denoted by the index 1. In this problem, the equations of the elasticity theory should be satisfied
DurðiÞ
urðiÞ 1 @yi ¼ 0, þ 12ni @r r2
DuzðiÞ þ
1 @yi ¼ 0, 12ni @z
i ¼ 0,1,
ð2Þ
and boundary conditions
2. Problem definition The problem of an elastic half-space covered by an elastic layer in which a rigid sphere is pressured, is considered (Fig. 1). We take into account that the radius of the sphere R is much larger than the radius of the area of contact. This approach permits to approximate the surface of the sphere by a paraboloid zðrÞ ¼ ar2 =2R,
1593
ð1Þ
ð1Þ uð0Þ r ¼ ur ,
ð1Þ uð0Þ z ¼ uz ,
ð1Þ ð0Þ ð1Þ sð0Þ rz ¼ srz , szz ¼ szz , z ¼ 0,
ð3Þ
ð0Þ sð0Þ rz ¼ 0, szz ¼ pðrÞHð1rÞ, z ¼ h,
ð4Þ
2 uð0Þ z ¼ ar =2Rd,
ð5Þ
uð1Þ -0,
r o 1,
z ¼ h,
r2 þ z2 -1,
and the equilibrium condition Z 1 p0 : rpðrÞ dx ¼ 2 0
ð6Þ
ð7Þ
where u is the vector of the non-dimensional displacement referred to the radius of contact, r is the stress tensor, p is an unknown contact pressure, p0 ¼P/(pa2) is the mean contact pressure, P is the total normal load, d is the penetration depth of the sphere, H(x) is the Heaviside unit step function, y is the voluminal strain and D is the Laplace operator.
3. The method of solution
Fig. 1. A scheme of problem.
The problem of an elasticity for the non-homogeneous halfspace loaded by p(r) pressure is considered first. The general solution of the problem presented in the Hankel transform space (see Appendix A). Eqs. (12)–(23) contain six unknown functions ai ðsÞ, i ¼ 1,2, . . . ,6. These functions are obtained as a solution to linear equations (see Appendix A), satisfying boundary conditions (3) and (4). The form of the solution is adequate to satisfy the regularity conditions in infinity (6). An unknown contact pressure distribution we can calculate satisfying boundary condition (5) and equilibrium condition (7).
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1.5
1.75
1 : E 0 E1 = 2 2 : E 0 E1 = 4
1
1.4
3 : E 0 E1 = 8
1.5
p(0)/p0
a/aH
3
ν 0 = ν 1 = 0.25
2
1.3
2 1.2
1 : E0 E1 = 2
1.1
3 : E 0 E1 = 8
2 : E 0 E1 = 4
1.25
3
1
ν 0 = ν 1 = 0.25
1
1
0
0.5
1
1.5
2
2.5
3
0
0.5
1
h
1.5
h
Fig. 2. (a) Non-dimensional radius of contact and (b) Pressure in the centre of the contact area as functions of the parameter E0/E1 (solution of integral Eq. (8)—black curves, approximate solution—gray curves, Hertz solution—dashed curve), h¼ H/a – dimensionless thickness of the layer related to actual radius of the contact zone.
1
1.5
1
1: E0/E1 = 2
2
1
2: E0/E1 = 4 0.75
3
2
3: E0/E1 = 8
1
h
p p0
3 0.5
0.5
1 : E0 E1 = 1
h = 0.4
2 : E0 E1 = 4
ν 0 = ν 1 = 0.25
0.25
3 : E0 E1 = 8
v0 = v1 = 0.25
0 0
0.25
0.5
0.75
1
r
0 0
Fig. 3. Distribution of contact pressure (solution of integral Eq. (8)—black curves, approximate solution—gray curves, Hertz solution—black curve No. 1).
We obtain the following integral equation: Z 1 3 p ðyÞKðr,yÞ dy ¼ pð1n0 Þa30 r, 0 rr r 1, 8 0
0.25
0.5
0.75
1
h* Fig. 4. Relationship between dimensionless thickness of the layer related to actual radius of the contact zone (h) and dimensionless thickness of the layer related to the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer (h*).
ð8Þ
where Z
1
Kðr,yÞ ¼ y
s2 a 6 ðsÞJ0 ðsyÞJ1 ðsrÞ ds,
0
pðrÞ ¼ p0 p ðrÞ,
a ¼ aH a0 ,
where pi are unknown parameters, ai are points within interval [0,1] which are calculated from
~ a6 ðsÞ ¼ pðsÞa 6 ðsÞ=m0 , ð9Þ
aH is the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer, a3H ¼ 3PRð1n0 Þ=ð8m0 Þ. The solution to the integral equation is sought in the form qffiffiffiffiffiffiffiffiffiffiffiffiffi m X p ðrÞ ¼ pi a2i r2 Hðai rÞ, ð10Þ i¼1
ai ¼
ið2 þ cð2mi1ÞÞ , mð2 þ cðm1ÞÞ
i ¼ 0,1, . . . ,m:
ð11Þ
Eq. (11) is taken in such a form so that for c40 we obtain a partition of the interval [0,1], which condenses in the vicinity of the right end of the interval. Substituting Eq. (10) into integral (8) and comparing the left- and right-hand side in points rj ¼ (aj + aj 1)/2, j ¼1,2, ..., m, we obtain m linear algebraic equations containing (m+ 1) unknown parameters pi, i¼1,2, ..., m
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
1
1595
1.5
1 : E 0 E1 = 2 0.5
2 : E 0 E1 = 4
1
3 : E 0 E1 = 8
0 0
0.5
1
2 0.5
r
-1
σ rr p 0
σ rr p 0
-0.5
1.5
1 : E 0 E1 = 2
1
2 : E 0 E1 = 4
-1.5
3 : E 0 E1 = 8
2
-2
3 r
0 0
2
0.5
-0.5
ν 0 = ν 1 = 0.25
3
1
1.5
2
1.5
2
1
h = 0.2
-2.5
h = 0.2 ν 0 = ν 1 = 0.25
-1
-3 -3.5
-1.5
1
1.5
3
0.5
1
0 0
0.5
1
r
1.5
2
1 : E 0 E1 = 2
-1
2 : E 0 E1 = 4
1
-1.5
3 : E 0 E1 = 8
r 0 0
h = 0.4
2
-2
2
0.5
σ rr p 0
σ rr p 0
-0.5
1
-0.5
1 : E 0 E1 = 2
ν 0 = ν 1 = 0.25
-2.5
0.5
1
3
2 : E 0 E1 = 4
-1
h = 0.4
3 : E 0 E1 = 8
-3 -3.5
-1.5
1
1.5
ν 0 = ν 1 = 0.25
3
0.5 1 0 0
0.5
-1
0.5
1 : E 0 E1 = 2
1
2 : E 0 E1 = 4
2 -2.5
2
2
r
-1.5 -2
1.5
3
3 : E 0 E1 = 8 h = 0.8
1
σ rr p 0
σ rr p 0
-0.5
1
0
0.5
1
1.5
2
-0.5
1 : E 0 E1 = 2
ν 0 = ν 1 = 0.25 -1
2 : E 0 E1 = 4 3 : E 0 E1 = 8
-3 -3.5
r
0
h = 0.8
ν 0 = ν 1 = 0.25
-1.5
Fig. 5. Distribution of srr/p0 on the non-homogeneous half-space surface (a, b, c) and at the coating/substrate interface (a0 , b0 , c0 ): distribution of the contact pressure on the basis of integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
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and a0. The missing (m+ 1)th equation m X
pi a3i ¼
i¼1
3 2
ð12Þ
is found by satisfying equilibrium condition (7). The solution to the system of equations is obtained by making use of a numerical method. The results are compatible with those given by Chen and Engel [38].
non-homogeneous half-space are known from the boundary conditions. The formulas for calculating the stresses sð0Þ rr ðr,hÞ and sð0Þ jj ðr,hÞ are obtained taking into account the sixth equation of the system of (A.17)
sð0Þ rr ðr,hÞ ¼ pðrÞ Z
1
þ
1 r
Z
1
~ pðsÞJ 1 ðsrÞ ds 0
~ a 4 ðsÞpðsÞ 2d0 sJ0 ðsrÞð1 þd0 Þr 1 J1 ðsrÞ ds,
ð17Þ
0
4. A simplified algorithm for solving the contact problem The distribution of contact pressure is sought in the form pffiffiffiffiffiffiffiffiffiffiffi ^ 2 Þ 1r 2 , ð13Þ p ðrÞ ¼ ðp^ þ 1:25ð32pÞr
Z ~ pðsÞJ 1 ðsrÞ ds þ 0 þ ð1 þ d0 Þr 1 J1 ðsrÞ ds:
sð0Þ jj ðr,hÞ ¼
which satisfies equilibrium condition (7). It is necessary to point out that the parameter p^ is the ratio of the contact pressure in the centre of the contact area to the mean contact pressure. Contact boundary condition is replaced by the following conditions:
1
r 0
ð0Þ uð0Þ z ðr,hÞuz ð1,hÞ
a dr ¼ : 8R
i ¼ 1,2:
1
~ a 4 ðsÞpðsÞ ðd0 1ÞsJ0 ðsrÞ
0
ð18Þ
ν 0 = ν 1 = 0.25
1 : E 0 E1 = 2 2 : E 0 E1 = 4 3 : E 0 E1 = 8
1.4
ð15Þ 1.2
Substituting Eq. (13) into Eqs. (14) and (15), we obtain two equations for two unknown parameters ð1n0 Þa30 þAi p^ ¼ Bi ,
1
1.6
ð14Þ
ð16Þ
The forms Ai and Bi, i¼1,2 are presented in Appendix B. Figs. 2 and 3 show relationship between dimensionless radius of the contact zone a0 ¼a/aH (related to the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer) and contact pressure distribution to h and E0/E1 parameters. When hb1, the actual contact parameters insignificantly differ from equivalent parameters for Hertz problem. The larges differences between contact pressure distributions for our contact problem and Hertz’s pressure occurs when hA[0.2,0.6]. The ratio a/aH it is possible to use to calculation of the ratio between current mean contact pressure (p0) and the Hertzian mean contact pressure for homogeneous half-space with mechanical properties adequate for the layer (pH): p0/pH ¼(aH/a)2. It is necessary to point out that the parameter h is unknown. The relationship between this parameter and a priori known parameter h* ¼H/aH is described by the following expression: h* ¼ha/aH (Fig. 4). When h* b1 then hEh*. Figs. 2 and 3 show that the solution of to the system of Eqs. (16) well approximates integral Eq. (8). The largest differences among the solutions appear in the case of relatively thin layers. They increased together with an enlargement of the ratio of the layer and Young’s moduli of substrates. The error of estimation of the half-width of contact for h¼0.1 and E0/E1 ¼8 is equal to 2.9%. Similar differences between the approximate solution and integral equation solution for other combination of Poisson ratios is observed (n1 ¼ 0.45, n2 ¼0.05, n1 ¼0.05, n2 ¼0.45, n1 ¼0.05, n2 ¼0.05, n1 ¼0.45, n2 ¼0.45). As stated above, it is possible to use the simplification approach when hA[0.2;1.5] and E0/E1 o8 (when hb1 the solution approach the Hertz solution).
σ 1m ax p 0
Z
Z
1.8
1
3
0.8 0.6
2
0.4
1
0.2 0 0
0.2
0.4
0.6
0.8
1
h Fig. 6. Relationship between the maximum tensile stresses and non-dimensional layer thickness (distribution of the contact pressure on the basis of integral Eq. (8)—black curves; distribution of the contact pressure approximated by Eq. (13)—diamonds, distribution of pressure replaced by Hertz solution – triangles).
30
1 : E 0 E1 = 2 2 : E 0 E1 = 4
25
3 : E 0 E1 = 8
ν 0 = ν 1 = 0.25
20
ε1 , %
ð0Þ uð0Þ z ð0,hÞuz ð1,hÞ ¼ a=2R,
1 r
3
15
2 3
10
3 2
5
1
2
5. The calculation algorithm for stress tensor components 0
Stress tensor components in interior points of the nonhomogeneous half-space can be obtained by numerical calculation of integrals (A.2)–(A.4). The accuracy of calculation of these integrals is supported by the continuity of stresses for ð0Þ z-h. The stresses sð0Þ rz ðr,hÞ and szz ðr,hÞ on the surface of the
0
0.2
0.4
0.6
0.8
1
h Fig. 7. Error of calculation of the maximum tensile stresses (distribution of pressure replaced by Hertz solution—gray curves, contact pressure approximated by Eq. (13)—black curves).
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
We can calculate the first integrals from Eqs. (20) and (21) analytically and the second one numerically.
Asymptotic analysis of the solution to the system of Eqs. (A.1) for s-N shows that a 4 ðsÞ ¼ 1=d0 þa 4 ðsÞ,
lim a 4 ðsÞ ¼ 0:
ð19Þ
s-1
6. Numerical results and discussion
Taking into consideration Eq. (19), we can write expressions (17) and (18) in the following form: Z 12n0 1 ~ sð0Þ pðsÞJ 1 ðsrÞ ds rr ðr,hÞ ¼ pðrÞ þ r 0 Z 1 ~ þ a 4 ðsÞpðsÞ ð20Þ 2d0 sJ0 ðsrÞð1 þ d0 Þr 1 J1 ðsrÞ ds,
Upon the analysis of the relationship found for calculation of the dimensionless stresses sð0Þ ij =p0 , we conclude that they depend on four non-dimensional parameters: the ratio of the thickness of the top layer and the half-width of contact, the ratio of the layer and Young’s moduli of substrates and the layer and the substrate Poisson ratios. In the contact problem for homogeneous half-space, tensile stresses occur at the trailing edge of the unloaded half-space surface [44]. The maximum value of sð0Þ rr =p0 occurs at the edge area of contact (r ¼ 1,z¼h) and is equal to 0.5(1 2n0). An increment of the parameter E0/E1 causes that the maximum tensile stresses occur in a certain distance from the edge of the
0
Z 12n0 1 ~ pðsÞJ 1 ðsrÞ ds r 0 Z 1 ~ þ a 4 ðsÞpðsÞ ðd0 1ÞsJ0 ðsrÞ þ ð1 þd0 Þr 1 J1 ðsrÞ ds:
sð0Þ jj ðr,hÞ ¼ 2n0 pðrÞ 0
ð21Þ
3
3
1 : E 0 E1 = 2
1 : E 0 E1 = 2
2 : E 0 E1 = 4
2.5
h = 0.2
ν 0 = ν 1 = 0.25 3
1
3 : E 0 E1 = 8
3
2
σ HM p 0
σ HM p 0
2
2 : E 0 E1 = 4
2.5
3 : E 0 E1 = 8
1.5
1597
h = 0.4
ν 0 = ν 1 = 0.25 1.5
2
2
1
1
1 0.5
0.5
0
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
r
r 3
1 : E 0 E1 = 2 2 : E 0 E1 = 4
2.5
3 : E 0 E1 = 8
h = 0.8
σ HM p 0
2
3 1.5
ν 0 = ν 1 = 0.25
2
1
1 0.5
0 0
0.5
1
1.5
2
r Fig. 8. Distribution of the Huber–von Mises reduced stress at the coating/substrate interface: contact pressure distribution found from integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
contact area (Fig. 5a–c). This is more typical for relatively thick layers. When the thickness of the layer increases it leads to a decreasing level of dimensionless tensile stresses in the described area. The second area, in which tensile stress can occur, is the coating/substrate interface (Fig. 5a0 –c0 ). In the case of relatively thin layers, the tensile stresses in this area do not occur at all or their level is much lower with respect to tensile stresses appearing at the surface. Together with the increasing of thickness of the layer or E0/E1 parameter, tensile stresses at the interface also increase. For a certain layer thickness, the principal stress s1 (we make an assumption that s3 o s2 o s1) achieves the largest value at the interface. Fig. 6 shows that every curve consist of two sections. The first section (adequate for the smaller values of parameter h) is related with parameters for which the maximum tensile stress occurs on the surface of the nonhomogeneous half-space. The second section describes cases in which the maximum tensile stress at the interface is observed. In the case of relatively thin layers (h o0.2) or reasonable values of parameters E0/E1 (E0/E1 o 2) to calculation the maximum tensile stress we can replace the actual contact pressure by Hertz’s distribution (Fig. 7). Increasing of these parameters causes that the error of maximum tensile stress calculation also increase. The maximum values of tensile stresses at the interface calculated of the basis of Hertz contact pressure and the distribution of actual pressure are considerably differing. The distribution of contact pressure described by Eq. (13) permits one to calculate the maximum value of tensile stresses with satisfactory precision both on the surface and at the interface (black curves in Fig. 7). For large ratio of Young modules of the layer and the substrate (E0/E1 ¼8) and non-dimensional layer thickness h¼0.27, the largest value of the error for calculation of maximum tensile stress at the coating/substrate interface is observed. The error is equal to 10%, while the error for Hertz’s distribution in this case is equal to 60%. Calculations confirmed that the Huber-von Mises reduced stress received the maximal value at the coating/substrate interface in general. Relatively thin layers (h o0.2), relatively thick layers (h41) and layers, whose mechanical properties differ insignificantly from the substrate properties, are exception to this rule. For relatively thin layers the maximal value of these stresses on the surface of the non-homogeneous half-space we observed. 7
In the other cases described above, the Huber-von Mises reduced stress received the maximal value in the central part of top layer [29]. Typical distributions of the Huber-von Mises reduced stresses at the coating/substrate interface are shown in Fig. 8. The level of these stresses may be (with except relatively thin layers or very large values of ratio E0/E1) much higher than level of tensile stresses (Fig. 9). In the case when 0.2 oho0.7 considerable difference between the maximal value of Huber-von Mises reduced stress for the actual contact pressure and for Hertz’s distribution is observed (Fig. 10). Describing the contact pressure by Eq. (13), it permits one to calculate the largest value of the Huber-von Mises reduced stress with an accuracy of satisfying for engineering practice. A typical distribution of srz stress is shown in Fig. 11. Replacing of actual contact pressure by Hertz pressure does not cause considerable changes in the srz stress distribution at the coating/ substrate interface. The largest deviation of calculation of the maximum srz stress with an error up to 8% was observed. 30
1 : E 0 E1 = 2 2 : E 0 E1 = 4
25
3 : E 0 E1 = 8
2
3
20
ε2 ,%
1598
15
1 10
5
2
3
1
0 0
0.2
0.4
0.6
0.35
1 : E0 E1 = 2 2 : E0 E1 = 8 h = 0.4 ν 0 = ν 1 = 0.25
0.3
1 : E 0 E1 = 2 0.25
4
σ rz p0
m ax σ HM σ 1m ax
2 : E 0 E1 = 4
1
3
0.2 0.15
2
2
2
1
Fig. 10. Calculation error of the maximum sHM stresses (distribution of pressure replaced by Hertz solution—gray curves, contact pressure approximated by Eq. (13)—black curves).
6
3 : E 0 E1 = 8
0.8
h
ν 0 = ν 1 = 0.25
5
ν 0 = ν 1 = 0.25
3
0.1
3
1
1
0.05 0 0
0.2
0.4
0.6
0.8
1
h max Fig. 9. Relationship between smax and the non-dimensional layer thickness HM =s1 (distribution of the contact pressure on the basis of integral Eq. (8)—black curves; distribution of the contact pressure approximated by Eq. (13)—diamonds).
0 0
0.5
1
1.5
2
Fig. 11. Typical distribution of srz/p0 stresses at the coating/substrate interface: contact pressure distribution found from integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
1
1.5
ν 0 = ν 1 = 0.25
ν 0 = ν 1 = 0.25 1
1.4
3 max max σ HM σ HM, 0
0.9
max p0 σ HM
1599
2 0.8
2
1.3
1 1.2
1 : E 0 E1 = 2
1 : E 0 E1 = 2
0.7
2 : E 0 E1 = 4
3
2 : E 0 E1 = 4
1.1
3 : E 0 E1 = 8
3 : E 0 E1 = 8 1
0.6 0
0.2
0.4
0.6
0
0.2
h
0.4
0.6
h
Fig. 12. (a) Relationship between the maximum Huber-von Mises reduced stress in the substrate and non-dimensional layer thickness and (b) Relationship between max max smax HM =sHM,0 and the non-dimensional layer thickness (sHM,0 —maximum Huber-von Mises reduced stress for homogeneous half-space with mechanical properties of substrate).
The level of stresses in half-space (mark ‘‘1’’) we will evaluate on the basis of distribution of Huber-von Mises reduced stress. When the layer thickness is higher than specific value the maximum value of this stress appear on the z-axis at the coating/surface interface. This value is checked off by cross at Fig. 12a. At small thickness of cover layer the maximum value of sHM stress is placed in small distance of the centre of the area of contact in comparison with proper distance for homogeneous half-space (0.46 of the radius of the contact area when n1 ¼0.25). The value of smax HM =p0 parameter in the case of half-space covered by layer is smallest than proper value for homogeneous half-space is equal to 0.965 (n1 ¼0.25). In Fig. 12a this value is checked off by ‘diamond’. However, it is necessary to point out that incrementation of mean contact pressure when appear with half-space is covered exceed the reduce of smax HM =p0 parameter. It is causes that the higher level of Huber-von Mises reduced stress occur in the substrate in comparison with homogeneous halfspace case.
7. Conclusions The contact problem of elasticity connected with an indentation of a rigid sphere in an elastic semi-space covered by an elastic layer has been considered. The paper presents detailed analysis of stress distribution produced by contact pressure. The analysis of stress distribution in the hard top layer proved that this layer can be divided into three sections: relatively thin layers (h o0.2), relatively thick layers (h40.7) and those in between. In the case of relatively thin layers, the maximum tensile stress occurs at the unloaded half-space surface. Another area where the tensile stress may occur is the coating/substrate interface. An increase of layer thickness or E0/E1 parameter is also accompanied by an increase of tensile stresses at the interface. For a certain layer thickness the tensile stresses at the coating/substrate interface are greater than the ones on the surface of the nonhomogeneous half-space. In the case of relatively thick layers, the predominant stresses are mostly tensile stresses at the coating/ substrate interface. When 0.2 oho0.7, the maximum tensile
stress occurs either at the unloaded half-space surface or at the coating/substrate interface. In the case of a ‘reasonable’ thickness of layer (ho1) the maximum value of the Huber-von Mises reduced stress occurs at the coating/substrate interface (with an exception of relatively thin layers or when the mechanical properties of the layer differ insignificantly from the substrate properties). For relatively thin layers the maximum sHM stress appears on the surface of nonhomogeneous half-space. The level of the Huber-von Mises reduced stresses may be much higher (more than twice) compared to the largest value of the tensile stress. The replacement of the actual contact pressure with Hertz pressure makes it possible to calculate the maximum tensile stresses on the surface of non-homogeneous half-space and the maximum shear stress srz at the coating/substrate interface with a satisfactory precision for engineering calculations. The maximum tensile stresses at the coating/surface interface and the Huber-von Mises reduced stresses considered in view of actual contact pressure and Hertz distribution differs considerably (with exception of relatively thick layers). In calculations, we can describe distribution of actual pressure by approximate relation (13).
Acknowledgements The investigation described in this paper is a part of the project S/WM/2/08, funded by the Polish State Committee for Scientific Research and realized at Bialystok University of Technology.
Appendix A The general solution of the problem can be written in the form Z 1 Z 1 UrðiÞ ðs,zÞJ1 ðsrÞds, uzðiÞ ðr,zÞ ¼ UzðiÞ ðs,zÞJ0 ðsrÞds, i ¼ 0,1, urðiÞ ðr,zÞ ¼ 0
0
ðA:1Þ
srrðiÞ ðr,zÞ ¼ mi
Z
1 0
ðiÞ sSrr ðs,zÞJ0 ðsrÞds
2 r
Z
1 0
UrðiÞ ðs,zÞJ1 ðsrÞds,
i ¼ 0,1,
ðA:2Þ
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sðiÞ 2ni jj ðr,zÞ ¼ mi 12ni
Z
1 0
sSðiÞ jj ðs,zÞJ0 ðsrÞds þ
2 r
Z
1 0
UrðiÞ ðs,zÞJ1 ðsrÞds,
i ¼ 0,1,
ðA:3Þ ðiÞ szz ðr,zÞ ¼ mi
Z
1 0
sSðiÞ zz ðs,zÞJ0 ðsrÞds,
ðiÞ srz ðr,zÞ ¼ mi
Z
1 0
sSðiÞ rz ðs,zÞJ1 ðsrÞds,
ðA:4Þ
where 2Urð0Þ ðs,zÞ ¼ a3 ðsÞfð2 þd0 ÞsinhðsðhzÞÞ þ d0 ðhzÞs coshðsðhzÞÞg þa4 ðsÞfð2 þ d0 ÞcoshðsðhzÞÞ þ d0 ðhzÞs sinhðsðhzÞÞg þ2a5 ðsÞs coshðsðhzÞÞ þ2a6 ðsÞs sinhðsðhzÞÞ, 0 r z rh,
A16 ¼ 2s sinhðshÞ,A22 ¼ 2s, A23 ¼ d0 sh sinhðshÞ, A24 ¼ d0 sh coshðshÞ, A25 ¼ 2s sinhðshÞ, A26 ¼ 2s coshðshÞ, A31 ¼ m ð1 þd1 Þ, A32 ¼ 2m s, A33 ¼ ð1 þd0 ÞcoshðshÞd0 sh sinðshÞ, A34 ¼ ð1 þ d0 ÞsinhðshÞd0 sh coshðshÞ, A35 ¼ 2s sinhðshÞ, A36 ¼ 2s coshðshÞ,A41 ¼ m , A42 ¼ 2m s, A43 ¼ sinhðshÞ þd0 sh coshðshÞ,A44 ¼ coshðshÞ þ d0 sh sinhðshÞ, A45 ¼ 2s coshðshÞ, A46 ¼ 2s sinhðshÞ,A53 ¼ 1 þd0 , A56 ¼ 2s, A64 ¼ 1, A65 ¼ 2s, m ¼ m1 =m0 , Z 1 ~ ¼ ~ m0 g, pðsÞ rpðrÞJ0 ðsrÞdr: b ¼ f0,0,0,0,0,pðsÞ= 0
ðA:5Þ Appendix B 2Urð1Þ ðs,zÞ ¼ a1 ðsÞð2 þ d1 þ d1 szÞexpðszÞ2a2 ðsÞs expðszÞ,
z r0, ðA:6Þ
2Uzð0Þ ðs,zÞ ¼ d0 ðhzÞsa3 ðsÞsinhðsðhzÞÞ þ d0 ðhzÞsa4 ðsÞcoshðsðhzÞÞ þ2a5 ðsÞs sinhðsðhzÞÞ þ 2a6 ðsÞs coshðsðhzÞÞ, 0 r z rh,
2Uzð1Þ ðs,zÞ ¼ d1 a1 ðsÞsz expðszÞ þ2a2 ðsÞs expðszÞ,
z r 0,
Formulas for calculating parameters Ai and Bi, i¼1, 2 are Z 1 Z 1 8 16 sa 6 ðsÞF1 ðsÞG1 ðsÞ ds, A2 ¼ pffiffiffiffiffiffi sa 6 ðsÞF1 ðsÞG2 ðsÞ ds, A1 ¼ pffiffiffiffiffiffi 2p 0 2p 0
ðA:7Þ
20 B1 ¼ pffiffiffiffiffiffi 2p
ðA:8Þ
F1 ðsÞ ¼ 5
Sð0Þ rr ðs,zÞ ¼ a3 ðsÞfð2d0 þ1ÞsinhðsðhzÞÞ þ d0 ðhzÞscoshðsðhzÞÞg þ a4 ðsÞfð2d0 þ 1ÞcoshðsðhzÞÞ þ d0 ðhzÞssinhðsðhzÞÞg þ 2a5 ðsÞscoshðsðhzÞÞ þ2a6 ðsÞssinhðsðhzÞÞ, 0 rz r h,
Z
1
sa 6 ðsÞF2 ðsÞG1 ðsÞ ds, 0
40 B2 ¼ pffiffiffiffiffiffi 2p
Z
1
sa 6 ðsÞF2 ðsÞG2 ðsÞ ds, 0
J5=2 ðsÞ J3=2 ðsÞ J5=2 ðsÞ J3=2 ðsÞ 3=2 , F2 ðsÞ ¼ 3 5=2 3=2 , s5=2 s s s G1 ðsÞ ¼ 1J0 ðsÞ, G2 ðsÞ ¼ J2 ðsÞ:
References
ðA:9Þ
Sð1Þ rr ðs,zÞ ¼ ð2d1 þ 1 þ d1 szÞa1 ðsÞexpðszÞ2a2 ðsÞs expðszÞ,
z r0, ðA:10Þ
Sð0Þ jj ðs,zÞ ¼ a3 ðsÞsinhðsðhzÞÞ þa4 ðsÞcoshðsðhzÞÞ, Sð1Þ jj ðs,zÞ ¼ a1 ðsÞexpðszÞ,
0 r z rh,
z r0,
ðA:11Þ ðA:12Þ
Sð0Þ zz ðs,zÞ ¼ a3 ðsÞfsinhðsðhzÞÞ þ d0 ðhzÞs coshðsðhzÞÞg a4 ðsÞfcoshðsðhzÞÞ þ d0 ðhzÞs sinhðsðhzÞÞg 2a5 ðsÞs coshðsðhzÞÞ2a6 ðsÞs sinhðsðhzÞÞ, 0 r z rh, ðA:13Þ
Sð1Þ zz ðs,zÞ ¼ a1 ðsÞð1 þ d1 szÞexpðszÞ þ 2a2 ðsÞs expðszÞ,
z r 0,
ðA:14Þ
Sð0Þ rz ðs,zÞ ¼ a3 ðsÞfð1 þd0 ÞcoshðsðhzÞÞ þ d0 ðhzÞssinhðsðhzÞÞg a4 ðsÞfð1þ d0 ÞsinhðsðhzÞÞ þd0 ðhzÞs coshðsðhzÞÞg 2a5 ðsÞs sinhðsðhzÞÞ2a6 ðsÞs coshðsðhzÞÞ, 0 r z rh, ðA:15Þ
Sð1Þ rz ðs,zÞ ¼ a1 ðsÞð1 þ d1 þ d1 szÞexpðszÞ2a2 ðsÞs expðszÞ,
z r 0, ðA:16Þ
where di ¼1/(1 2ni), i¼0,1, J0(r), J1(r) denotes the Bessel functions and m0, m1 are Kirchhoff coefficients of the layer and the substrate, respectively. A system of linear equations for determination of the functions ai ðsÞ, i ¼ 1,2, . . . ,6 is Aa ¼ b, where the non-zero elements of the matrix A are A11 ¼ 2 þ d1 , A12 ¼ 2s, A13 ¼ ð2 þ d0 ÞsinhðshÞ þd0 sh coshðhÞ, A14 ¼ ð2 þ d0 ÞcoshðshÞ þ d0 sh sinhðshÞ, A15 ¼ 2s coshðshÞ,
ðA:17Þ
[1] Houmid Bennani H, Takadoum J. Finite element model of elastic stresses in thin coatings submitted to applied forces. Surf Coat Technol 1999;111:80–5. [2] Bhomwick S, Kale AN, Jayaram V, Biswas SK. Contact damage in TiN coatings steel. Thin Solid Films 2003;436:250–8. [3] Xie C, Tong W. Cracking and decohesion of a thin Al2O3 film on a ductile Al-5%Mg substrate. Acta Mater 2005;53:477–85. [4] Holmberg K, Matthews A, Ronkainen H. Coatings tribology—contact mechanisms and surface design. Tribol Int 1998;31:107–20. [5] Gupta PK, Walowit JA. Contact stresses between an elastic cylinder and a layered elastic solid. Trans ASME, J Lubr Technol 1974;96:250–7. [6] Chen WT, Engel P. Impact and contact stress analysis in multilayered media. Int J Solid Struct 1972;8:1257–81. [7] Swain MV, Mencik J. Mechanical property characterization of thin films using spherical tipped indenters. Thin Solid Films 1994;253:204–11. [8] Gan L, Ben-Nissan B, Ben-David A. Modelling and finite element analysis of ultra-microhardness indentation of thin films. Thin Solid Films 1996; 290–291:362–6. [9] Schwarzer N, Whittling M, Swain M, Richter F. The analytical solution of the contact problem of spherical indenters on layered materials: application for the investigation of TiN films on silicon. Thin Solid Films 1995;270:371–5. [10] Chudoba T, Schwarzer N, Richter F. New possibilities of mechanical surface characterization with spherical indenters by comparison of experimental and theoretical results. Thin Solid Films 1999;355–356:284–9. [11] Mishina H. Microscopic real time observation of failure, wear and deformation on coating/subsurface. Tribol Int 1999;32:137–44. [12] Su YL, Kao WH. Tribological behaviour and wear mechanism of MoS2– Cr coatings sliding against various counterbody. Tribol Int 2003;36:11–23. [13] Wang DF, Kato K. Effect of friction cycles on wear particle generation of carbon nitride coating against a spherical diamond. Tribol Int 2000;33: 115–22. [14] Wang DF, Kato K. Critical load for wear particle generation in carbon nitride coatings by single sliding against a spherical diamond. Tribol Int 2001;34:407–18. [15] Oliveira SAG, Bower AF. An analysis of fracture and delamination in thin coatings subjected to contact loading. Wear 1996;198:15–32. [16] Cole SJ, Sayles RS. A numerical model for the contact of layered elastic bodies with real rough surface. ASME J Tribol 1992;114:334–40. [17] Chiu YP, Hartnett MJ. A numerical solution for layered solid contact problems with application to bearings. ASME J Lubr Technol 1983;105:585–90. [18] O’Sullivan TC, King RB. Sliding contact stress field due to a spherical indenter on a layered elastic half-space. ASME J Tribol 1989;110:235–40. [19] Anderson A, Collins IF. Plane strain stress distributions in descrete and blended coated solids under normal and sliding contact. Wear 1995;185: 23–33. [20] Schwarzer N, Richter F, Hecht G. The elastic field in a coated half-space under Hertzian pressure distribution. Surf Coat Technol 1999;114:292–304. [21] Schwarzer N. Coating due to analytical modelling of mechanical contact problems on multilayer systems. Surf Coat Technol 2000;133–134:397–402.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
[22] Ramalingam S, Zheng L. Film-substrate interface stresses and their role in the tribological performance of surface coatings. Tribol Int 1995;28(3): 145–61. [23] Diao DF, Kato K, Hayashi K. The maximum tensile stress on a hard coating under sliding friction. Tribol Int 1994;27(4):267–72. [24] Kato K. Wear in relation of friction—a review. Wear 2000;24:151–7. [25] Bargallini GM, Cavatorta MP, Sainsot P. Coated contact: a strain approach. Tribol Int 2003;36:935–41. [26] Elsharkawy AA. Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solid. Int J Solids Struct 1999;36:3903–15. [27] Diao DF. Finite element analysis on local yield map and critical maximum contact pressure for yielding in hard coating with an interlayer under sliding contact. Tribol Int 1999;32:25–32. [28] Schwarzer N, Chudoba T, Richter F. Investigation of ultra thin coatings using nanoindentation. Surf Coat Technol 2006;200:5566–80. [29] Kouitat Nijwa R, Consiglio R, von Stebut J. Boundary element modeling of coated materials in static and sliding ball-flat elastic contact. Surf Coat Technol 1998;102:148–53. [30] Kouitat Nijwa R, von Stebut J. Boundary element numerical modelling as a surface engineering tool: application to very thin coatings. Surf Coat Technol 1999;116–119:573–9. [31] Shodja HM, Ghahremaninejad A. An FGM coated elastic solid under thermomechanical loading: a two dimensional linear elastic approach. Surf Coat Technol 2006;200:4050–64. [32] Abdul-Baqi A, Van der Giessen E. Numerical analysis of indentation-induced cracking of brittle coatings on ductile substrates. Int J Solids Struct 2002;39: 1427–42.
1601
[33] Wong SK, Kapoor A. Effect of hard and stiff overlay coatings on the strength of surfaces in repeated sliding. Tribol Int 1996;29(8):695–702. [34] Plumet S, Dubourg MC. A 3-D model for a multilayered body loaded normally and tangentially against a rigid body: application to specific coatings. ASME J Tribol 1998;120:668–76. [35] Kuo CH, Keer LM. Contact stress analysis of a layered transversely isotropic half-space. ASME J Tribol 1992;114:253–62. [36] Peng W, Bushan B. Three-dimensional contact analysis of layered elastic/ plastic solids with rough surfaces. Wear 2001;249:741–60. [37] Nogi T, Kato T. Influence of hard surface layer on the limit of elastic contact – Part I: analysis using a real surface model. ASME J Tribol 1997;119:493–500. [38] Chen WT, Engel PA. Impact and contact stress analysis in multilayer media. Int J Solids Struct 1972;8:1257–81. [39] Gupta PK, Walowit JA, Finkin EF. Stress distributions in plane strain layered elastic solids subjected to arbitrary boundary loading. Trans ASME J Lubr Technol 1973:427–33. [40] Kulchytsky-Zhyhailo R, Rogowski G. Tensile stresses in solid coating in the two-dimensional problem for contact with friction. J Frict Wear 2006;27:30–8. [41] Kulchytsky-Zhyhailo R, Rogowski G. Stresses of hard coating under sliding contact. J Theor Appl Mech 2007;45:753–71. [42] King RB, O’Sullivan TC. Sliding contact stresses in a two-dimensional layered elastic half-space. Int J Solid Struct 1987;23:581–97. [43] Elsharkawy AA, Hamrock BJ. A numerical solution for dry sliding line contact of multilayered elastic bodies. J Tribol 1993;115:237–45. [44] Timoshenko S, Goodier JN. Theory of elasticity. New York: McGraw-Hill Book Company; 1951.
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Stresses in hard coating due to a rigid spherical indenter on a layered elastic half-space Roman Kulchytsky-Zhyhailo , Gabriel Rogowski Bialystok University of Technology, Faculty of Mechanical Engineering, ul. Wiejska 45C, 15-351 Bialystok, Poland
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 November 2006 Received in revised form 25 January 2010 Accepted 2 March 2010 Available online 11 March 2010
The axisymmetrical contact problem of elasticity connected with an indentation of a rigid spherical indenter in an elastic semi-space covered by an elastic layer is considered. Stress tensor components in interior points of the non-homogeneous half-space by numerical calculation of some integrals was obtained. Detailed analysis of the maximal tensile stress distributions and Huber-von Mises reduced stress distributions produced by contact pressure is presented. The dependence between these stresses and the ratio between the layer thickness and contact area width is explored. The obtained results for stresses are compared with results obtained for half-space loaded by the Hertz pressure. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Spherical indentation Hard coatings Contact problem Stress
1. Introduction The progress of coating technology is the reason for wide employments of hard top layers for improvement of the tribological properties of sliding surfaces. This can lead to reduction of the friction coefficient and the wear quantity while the weight of the material is not changed. Fractures in the coating (cohesive failure) or delamination and spalling (adhesive failure) at the coating/substrate interface are the weak points of the top layers employed [1–4]. Therefore, the analysis of stress distributions produced by contact pressure is very important. Many authors during recent years take interest in the contact problems that a rigid sphere (axi-symmetrical problem) or a long rigid cylinder (two-dimensional problem) are pressured. The selected contact parameters such as penetration depth of the punch or radius of the contact area are recorded in an experiment [5–17], while stresses in the contact area are submit to theoretical consideration by solving a specified contact problem of the elasticity theory [1–3,13–34]. It is shown that fundamental factors responsible for destruction of layers are: maximum tensile stress [1–3,15,18–25,35], Huber–von Mises reduced stress [1,18–21,26–29,36,37] (or maximum shear stress [2,30,35]) and maximum shear stress at the coating/substrate interface [1,3,18,22,32,33]. The highest values of parameters specified above occur on the surface of non-homogeneous medium [21,23] or at the coating/surface interface [20,21,25]. In majority of studies
Corresponding author.
E-mail address: [email protected] (R. Kulchytsky-Zhyhailo). 0301-679X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2010.03.002
[1,20,21,23,25,29–31] the distribution of contact pressure as semipffiffiffiffiffiffiffiffiffiffiffi elliptical pressure is assumed pðrÞ ¼ pmax 1r 2 , where pmax— contact pressure in the centre of the contact area, r—dimensionless radius coordinate (related to radius of the contact zone—parameter a). This procedure is often correct and seems undisputed in case of relatively thick layers (h¼H/a41, where H is a thickness of the layer). When Hertz pressure distribution is assumed, the results are comparable (with a small deviation) with ones given in article [18] where h¼1/0.819E1.221. In the case of hard coating similar effect for ‘thin’ layers is observed [35]. The axi-symmetrical problem on the penetration of sphere into elastic half-space coated by harder layer is known to differ from Hertzian solution [38]. In the case that Young’s modulus of the layer differ considerably from Young’s modulus of the substrate, the maximum value of the contact pressure occurs in some distance from the centre of contact area [38] (about 0.8 of radius of the contact circle with E0/E1 ¼10, h¼0.3, n0 ¼ n1 ¼1/3, where E0, E1, n0, n1 are Young modules and Poisson ratios of the layer and the substrate, respectively). A similar result was observed in corresponding two-dimensional contact problem of elasticity [5,39]. The goal of the work is:
to compare the distribution of specify above stresses caused by true distribution of contact pressure and Hertzian distribution;
to study the ratio between maximal tensile stress and Huber–
von Mises reduced stress on the surface of the heterogeneous half-space and at the layer—half-space interface at different parameters; to explore the dependence between these stresses and the ratio between the layer thickness and contact area width.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
Nomenclature a aH
radius of the contact zone the Hertzian radius of contact for homogeneous halfspace with mechanical properties adequate for the layer E Young’s modulus h¼H/a dimensionless thickness of the layer related to actual radius of the contact zone h* ¼H/aH dimensionless thickness of the layer related to the radius aH H a thickness of the layer H(x) the Heaviside unit step function J0(r), J1(r)denotes the Bessel functions p contact pressure the Hertzian mean contact pressure for homogeneous pH half-space with mechanical properties adequate for the layer
Adequate calculations in the case of two-dimensional contact problem were considered in [40,41]. It is necessary to point out that calculations presented in this paper and [40,41] are consistent to what was given by [1,5,6,20,21,29,30,39,42,43]. We propose a simplified algorithm for solving the contact problem together with the classical algorithm. This algorithm allows one to obtain a formula for pressure calculation which simplifies the analysis of stress fields by finite element method or boundary element method without applying elliptical distribution (Hertz distribution).
contact pressure in the centre of the contact area mean contact pressure the total normal load the ratio of the contact pressure in the centre of the contact area to the actual mean contact pressure dimensionless cylindrical coordinates vector of the non-dimensional displacement referred to the radius of contact the penetration depth of the sphere the Laplace operator the voluminal strain Kirchhoff coefficient Poisson ratios the stress tensor the Huber-von Mises reduced stresses
pmax p0 P p^ r, j, z u
d D y
m n r sHM
where r, j, z are dimensionless cylindrical coordinates referenced to the radius of the area of the contact. The friction in the contact area is ignored. The ideal mechanical contact conditions between the layer and the half-space are assumed. All state functions denoted by the upper index 0 describe the state of displacement and stresses in the layer; the state functions for the elastic halfspace are denoted by the index 1. In this problem, the equations of the elasticity theory should be satisfied
DurðiÞ
urðiÞ 1 @yi ¼ 0, þ 12ni @r r2
DuzðiÞ þ
1 @yi ¼ 0, 12ni @z
i ¼ 0,1,
ð2Þ
and boundary conditions
2. Problem definition The problem of an elastic half-space covered by an elastic layer in which a rigid sphere is pressured, is considered (Fig. 1). We take into account that the radius of the sphere R is much larger than the radius of the area of contact. This approach permits to approximate the surface of the sphere by a paraboloid zðrÞ ¼ ar2 =2R,
1593
ð1Þ
ð1Þ uð0Þ r ¼ ur ,
ð1Þ uð0Þ z ¼ uz ,
ð1Þ ð0Þ ð1Þ sð0Þ rz ¼ srz , szz ¼ szz , z ¼ 0,
ð3Þ
ð0Þ sð0Þ rz ¼ 0, szz ¼ pðrÞHð1rÞ, z ¼ h,
ð4Þ
2 uð0Þ z ¼ ar =2Rd,
ð5Þ
uð1Þ -0,
r o 1,
z ¼ h,
r2 þ z2 -1,
and the equilibrium condition Z 1 p0 : rpðrÞ dx ¼ 2 0
ð6Þ
ð7Þ
where u is the vector of the non-dimensional displacement referred to the radius of contact, r is the stress tensor, p is an unknown contact pressure, p0 ¼P/(pa2) is the mean contact pressure, P is the total normal load, d is the penetration depth of the sphere, H(x) is the Heaviside unit step function, y is the voluminal strain and D is the Laplace operator.
3. The method of solution
Fig. 1. A scheme of problem.
The problem of an elasticity for the non-homogeneous halfspace loaded by p(r) pressure is considered first. The general solution of the problem presented in the Hankel transform space (see Appendix A). Eqs. (12)–(23) contain six unknown functions ai ðsÞ, i ¼ 1,2, . . . ,6. These functions are obtained as a solution to linear equations (see Appendix A), satisfying boundary conditions (3) and (4). The form of the solution is adequate to satisfy the regularity conditions in infinity (6). An unknown contact pressure distribution we can calculate satisfying boundary condition (5) and equilibrium condition (7).
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1.5
1.75
1 : E 0 E1 = 2 2 : E 0 E1 = 4
1
1.4
3 : E 0 E1 = 8
1.5
p(0)/p0
a/aH
3
ν 0 = ν 1 = 0.25
2
1.3
2 1.2
1 : E0 E1 = 2
1.1
3 : E 0 E1 = 8
2 : E 0 E1 = 4
1.25
3
1
ν 0 = ν 1 = 0.25
1
1
0
0.5
1
1.5
2
2.5
3
0
0.5
1
h
1.5
h
Fig. 2. (a) Non-dimensional radius of contact and (b) Pressure in the centre of the contact area as functions of the parameter E0/E1 (solution of integral Eq. (8)—black curves, approximate solution—gray curves, Hertz solution—dashed curve), h¼ H/a – dimensionless thickness of the layer related to actual radius of the contact zone.
1
1.5
1
1: E0/E1 = 2
2
1
2: E0/E1 = 4 0.75
3
2
3: E0/E1 = 8
1
h
p p0
3 0.5
0.5
1 : E0 E1 = 1
h = 0.4
2 : E0 E1 = 4
ν 0 = ν 1 = 0.25
0.25
3 : E0 E1 = 8
v0 = v1 = 0.25
0 0
0.25
0.5
0.75
1
r
0 0
Fig. 3. Distribution of contact pressure (solution of integral Eq. (8)—black curves, approximate solution—gray curves, Hertz solution—black curve No. 1).
We obtain the following integral equation: Z 1 3 p ðyÞKðr,yÞ dy ¼ pð1n0 Þa30 r, 0 rr r 1, 8 0
0.25
0.5
0.75
1
h* Fig. 4. Relationship between dimensionless thickness of the layer related to actual radius of the contact zone (h) and dimensionless thickness of the layer related to the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer (h*).
ð8Þ
where Z
1
Kðr,yÞ ¼ y
s2 a 6 ðsÞJ0 ðsyÞJ1 ðsrÞ ds,
0
pðrÞ ¼ p0 p ðrÞ,
a ¼ aH a0 ,
where pi are unknown parameters, ai are points within interval [0,1] which are calculated from
~ a6 ðsÞ ¼ pðsÞa 6 ðsÞ=m0 , ð9Þ
aH is the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer, a3H ¼ 3PRð1n0 Þ=ð8m0 Þ. The solution to the integral equation is sought in the form qffiffiffiffiffiffiffiffiffiffiffiffiffi m X p ðrÞ ¼ pi a2i r2 Hðai rÞ, ð10Þ i¼1
ai ¼
ið2 þ cð2mi1ÞÞ , mð2 þ cðm1ÞÞ
i ¼ 0,1, . . . ,m:
ð11Þ
Eq. (11) is taken in such a form so that for c40 we obtain a partition of the interval [0,1], which condenses in the vicinity of the right end of the interval. Substituting Eq. (10) into integral (8) and comparing the left- and right-hand side in points rj ¼ (aj + aj 1)/2, j ¼1,2, ..., m, we obtain m linear algebraic equations containing (m+ 1) unknown parameters pi, i¼1,2, ..., m
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
1
1595
1.5
1 : E 0 E1 = 2 0.5
2 : E 0 E1 = 4
1
3 : E 0 E1 = 8
0 0
0.5
1
2 0.5
r
-1
σ rr p 0
σ rr p 0
-0.5
1.5
1 : E 0 E1 = 2
1
2 : E 0 E1 = 4
-1.5
3 : E 0 E1 = 8
2
-2
3 r
0 0
2
0.5
-0.5
ν 0 = ν 1 = 0.25
3
1
1.5
2
1.5
2
1
h = 0.2
-2.5
h = 0.2 ν 0 = ν 1 = 0.25
-1
-3 -3.5
-1.5
1
1.5
3
0.5
1
0 0
0.5
1
r
1.5
2
1 : E 0 E1 = 2
-1
2 : E 0 E1 = 4
1
-1.5
3 : E 0 E1 = 8
r 0 0
h = 0.4
2
-2
2
0.5
σ rr p 0
σ rr p 0
-0.5
1
-0.5
1 : E 0 E1 = 2
ν 0 = ν 1 = 0.25
-2.5
0.5
1
3
2 : E 0 E1 = 4
-1
h = 0.4
3 : E 0 E1 = 8
-3 -3.5
-1.5
1
1.5
ν 0 = ν 1 = 0.25
3
0.5 1 0 0
0.5
-1
0.5
1 : E 0 E1 = 2
1
2 : E 0 E1 = 4
2 -2.5
2
2
r
-1.5 -2
1.5
3
3 : E 0 E1 = 8 h = 0.8
1
σ rr p 0
σ rr p 0
-0.5
1
0
0.5
1
1.5
2
-0.5
1 : E 0 E1 = 2
ν 0 = ν 1 = 0.25 -1
2 : E 0 E1 = 4 3 : E 0 E1 = 8
-3 -3.5
r
0
h = 0.8
ν 0 = ν 1 = 0.25
-1.5
Fig. 5. Distribution of srr/p0 on the non-homogeneous half-space surface (a, b, c) and at the coating/substrate interface (a0 , b0 , c0 ): distribution of the contact pressure on the basis of integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
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R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
and a0. The missing (m+ 1)th equation m X
pi a3i ¼
i¼1
3 2
ð12Þ
is found by satisfying equilibrium condition (7). The solution to the system of equations is obtained by making use of a numerical method. The results are compatible with those given by Chen and Engel [38].
non-homogeneous half-space are known from the boundary conditions. The formulas for calculating the stresses sð0Þ rr ðr,hÞ and sð0Þ jj ðr,hÞ are obtained taking into account the sixth equation of the system of (A.17)
sð0Þ rr ðr,hÞ ¼ pðrÞ Z
1
þ
1 r
Z
1
~ pðsÞJ 1 ðsrÞ ds 0
~ a 4 ðsÞpðsÞ 2d0 sJ0 ðsrÞð1 þd0 Þr 1 J1 ðsrÞ ds,
ð17Þ
0
4. A simplified algorithm for solving the contact problem The distribution of contact pressure is sought in the form pffiffiffiffiffiffiffiffiffiffiffi ^ 2 Þ 1r 2 , ð13Þ p ðrÞ ¼ ðp^ þ 1:25ð32pÞr
Z ~ pðsÞJ 1 ðsrÞ ds þ 0 þ ð1 þ d0 Þr 1 J1 ðsrÞ ds:
sð0Þ jj ðr,hÞ ¼
which satisfies equilibrium condition (7). It is necessary to point out that the parameter p^ is the ratio of the contact pressure in the centre of the contact area to the mean contact pressure. Contact boundary condition is replaced by the following conditions:
1
r 0
ð0Þ uð0Þ z ðr,hÞuz ð1,hÞ
a dr ¼ : 8R
i ¼ 1,2:
1
~ a 4 ðsÞpðsÞ ðd0 1ÞsJ0 ðsrÞ
0
ð18Þ
ν 0 = ν 1 = 0.25
1 : E 0 E1 = 2 2 : E 0 E1 = 4 3 : E 0 E1 = 8
1.4
ð15Þ 1.2
Substituting Eq. (13) into Eqs. (14) and (15), we obtain two equations for two unknown parameters ð1n0 Þa30 þAi p^ ¼ Bi ,
1
1.6
ð14Þ
ð16Þ
The forms Ai and Bi, i¼1,2 are presented in Appendix B. Figs. 2 and 3 show relationship between dimensionless radius of the contact zone a0 ¼a/aH (related to the Hertzian radius of contact for homogeneous half-space with mechanical properties adequate for the layer) and contact pressure distribution to h and E0/E1 parameters. When hb1, the actual contact parameters insignificantly differ from equivalent parameters for Hertz problem. The larges differences between contact pressure distributions for our contact problem and Hertz’s pressure occurs when hA[0.2,0.6]. The ratio a/aH it is possible to use to calculation of the ratio between current mean contact pressure (p0) and the Hertzian mean contact pressure for homogeneous half-space with mechanical properties adequate for the layer (pH): p0/pH ¼(aH/a)2. It is necessary to point out that the parameter h is unknown. The relationship between this parameter and a priori known parameter h* ¼H/aH is described by the following expression: h* ¼ha/aH (Fig. 4). When h* b1 then hEh*. Figs. 2 and 3 show that the solution of to the system of Eqs. (16) well approximates integral Eq. (8). The largest differences among the solutions appear in the case of relatively thin layers. They increased together with an enlargement of the ratio of the layer and Young’s moduli of substrates. The error of estimation of the half-width of contact for h¼0.1 and E0/E1 ¼8 is equal to 2.9%. Similar differences between the approximate solution and integral equation solution for other combination of Poisson ratios is observed (n1 ¼ 0.45, n2 ¼0.05, n1 ¼0.05, n2 ¼0.45, n1 ¼0.05, n2 ¼0.05, n1 ¼0.45, n2 ¼0.45). As stated above, it is possible to use the simplification approach when hA[0.2;1.5] and E0/E1 o8 (when hb1 the solution approach the Hertz solution).
σ 1m ax p 0
Z
Z
1.8
1
3
0.8 0.6
2
0.4
1
0.2 0 0
0.2
0.4
0.6
0.8
1
h Fig. 6. Relationship between the maximum tensile stresses and non-dimensional layer thickness (distribution of the contact pressure on the basis of integral Eq. (8)—black curves; distribution of the contact pressure approximated by Eq. (13)—diamonds, distribution of pressure replaced by Hertz solution – triangles).
30
1 : E 0 E1 = 2 2 : E 0 E1 = 4
25
3 : E 0 E1 = 8
ν 0 = ν 1 = 0.25
20
ε1 , %
ð0Þ uð0Þ z ð0,hÞuz ð1,hÞ ¼ a=2R,
1 r
3
15
2 3
10
3 2
5
1
2
5. The calculation algorithm for stress tensor components 0
Stress tensor components in interior points of the nonhomogeneous half-space can be obtained by numerical calculation of integrals (A.2)–(A.4). The accuracy of calculation of these integrals is supported by the continuity of stresses for ð0Þ z-h. The stresses sð0Þ rz ðr,hÞ and szz ðr,hÞ on the surface of the
0
0.2
0.4
0.6
0.8
1
h Fig. 7. Error of calculation of the maximum tensile stresses (distribution of pressure replaced by Hertz solution—gray curves, contact pressure approximated by Eq. (13)—black curves).
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
We can calculate the first integrals from Eqs. (20) and (21) analytically and the second one numerically.
Asymptotic analysis of the solution to the system of Eqs. (A.1) for s-N shows that a 4 ðsÞ ¼ 1=d0 þa 4 ðsÞ,
lim a 4 ðsÞ ¼ 0:
ð19Þ
s-1
6. Numerical results and discussion
Taking into consideration Eq. (19), we can write expressions (17) and (18) in the following form: Z 12n0 1 ~ sð0Þ pðsÞJ 1 ðsrÞ ds rr ðr,hÞ ¼ pðrÞ þ r 0 Z 1 ~ þ a 4 ðsÞpðsÞ ð20Þ 2d0 sJ0 ðsrÞð1 þ d0 Þr 1 J1 ðsrÞ ds,
Upon the analysis of the relationship found for calculation of the dimensionless stresses sð0Þ ij =p0 , we conclude that they depend on four non-dimensional parameters: the ratio of the thickness of the top layer and the half-width of contact, the ratio of the layer and Young’s moduli of substrates and the layer and the substrate Poisson ratios. In the contact problem for homogeneous half-space, tensile stresses occur at the trailing edge of the unloaded half-space surface [44]. The maximum value of sð0Þ rr =p0 occurs at the edge area of contact (r ¼ 1,z¼h) and is equal to 0.5(1 2n0). An increment of the parameter E0/E1 causes that the maximum tensile stresses occur in a certain distance from the edge of the
0
Z 12n0 1 ~ pðsÞJ 1 ðsrÞ ds r 0 Z 1 ~ þ a 4 ðsÞpðsÞ ðd0 1ÞsJ0 ðsrÞ þ ð1 þd0 Þr 1 J1 ðsrÞ ds:
sð0Þ jj ðr,hÞ ¼ 2n0 pðrÞ 0
ð21Þ
3
3
1 : E 0 E1 = 2
1 : E 0 E1 = 2
2 : E 0 E1 = 4
2.5
h = 0.2
ν 0 = ν 1 = 0.25 3
1
3 : E 0 E1 = 8
3
2
σ HM p 0
σ HM p 0
2
2 : E 0 E1 = 4
2.5
3 : E 0 E1 = 8
1.5
1597
h = 0.4
ν 0 = ν 1 = 0.25 1.5
2
2
1
1
1 0.5
0.5
0
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
r
r 3
1 : E 0 E1 = 2 2 : E 0 E1 = 4
2.5
3 : E 0 E1 = 8
h = 0.8
σ HM p 0
2
3 1.5
ν 0 = ν 1 = 0.25
2
1
1 0.5
0 0
0.5
1
1.5
2
r Fig. 8. Distribution of the Huber–von Mises reduced stress at the coating/substrate interface: contact pressure distribution found from integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
contact area (Fig. 5a–c). This is more typical for relatively thick layers. When the thickness of the layer increases it leads to a decreasing level of dimensionless tensile stresses in the described area. The second area, in which tensile stress can occur, is the coating/substrate interface (Fig. 5a0 –c0 ). In the case of relatively thin layers, the tensile stresses in this area do not occur at all or their level is much lower with respect to tensile stresses appearing at the surface. Together with the increasing of thickness of the layer or E0/E1 parameter, tensile stresses at the interface also increase. For a certain layer thickness, the principal stress s1 (we make an assumption that s3 o s2 o s1) achieves the largest value at the interface. Fig. 6 shows that every curve consist of two sections. The first section (adequate for the smaller values of parameter h) is related with parameters for which the maximum tensile stress occurs on the surface of the nonhomogeneous half-space. The second section describes cases in which the maximum tensile stress at the interface is observed. In the case of relatively thin layers (h o0.2) or reasonable values of parameters E0/E1 (E0/E1 o 2) to calculation the maximum tensile stress we can replace the actual contact pressure by Hertz’s distribution (Fig. 7). Increasing of these parameters causes that the error of maximum tensile stress calculation also increase. The maximum values of tensile stresses at the interface calculated of the basis of Hertz contact pressure and the distribution of actual pressure are considerably differing. The distribution of contact pressure described by Eq. (13) permits one to calculate the maximum value of tensile stresses with satisfactory precision both on the surface and at the interface (black curves in Fig. 7). For large ratio of Young modules of the layer and the substrate (E0/E1 ¼8) and non-dimensional layer thickness h¼0.27, the largest value of the error for calculation of maximum tensile stress at the coating/substrate interface is observed. The error is equal to 10%, while the error for Hertz’s distribution in this case is equal to 60%. Calculations confirmed that the Huber-von Mises reduced stress received the maximal value at the coating/substrate interface in general. Relatively thin layers (h o0.2), relatively thick layers (h41) and layers, whose mechanical properties differ insignificantly from the substrate properties, are exception to this rule. For relatively thin layers the maximal value of these stresses on the surface of the non-homogeneous half-space we observed. 7
In the other cases described above, the Huber-von Mises reduced stress received the maximal value in the central part of top layer [29]. Typical distributions of the Huber-von Mises reduced stresses at the coating/substrate interface are shown in Fig. 8. The level of these stresses may be (with except relatively thin layers or very large values of ratio E0/E1) much higher than level of tensile stresses (Fig. 9). In the case when 0.2 oho0.7 considerable difference between the maximal value of Huber-von Mises reduced stress for the actual contact pressure and for Hertz’s distribution is observed (Fig. 10). Describing the contact pressure by Eq. (13), it permits one to calculate the largest value of the Huber-von Mises reduced stress with an accuracy of satisfying for engineering practice. A typical distribution of srz stress is shown in Fig. 11. Replacing of actual contact pressure by Hertz pressure does not cause considerable changes in the srz stress distribution at the coating/ substrate interface. The largest deviation of calculation of the maximum srz stress with an error up to 8% was observed. 30
1 : E 0 E1 = 2 2 : E 0 E1 = 4
25
3 : E 0 E1 = 8
2
3
20
ε2 ,%
1598
15
1 10
5
2
3
1
0 0
0.2
0.4
0.6
0.35
1 : E0 E1 = 2 2 : E0 E1 = 8 h = 0.4 ν 0 = ν 1 = 0.25
0.3
1 : E 0 E1 = 2 0.25
4
σ rz p0
m ax σ HM σ 1m ax
2 : E 0 E1 = 4
1
3
0.2 0.15
2
2
2
1
Fig. 10. Calculation error of the maximum sHM stresses (distribution of pressure replaced by Hertz solution—gray curves, contact pressure approximated by Eq. (13)—black curves).
6
3 : E 0 E1 = 8
0.8
h
ν 0 = ν 1 = 0.25
5
ν 0 = ν 1 = 0.25
3
0.1
3
1
1
0.05 0 0
0.2
0.4
0.6
0.8
1
h max Fig. 9. Relationship between smax and the non-dimensional layer thickness HM =s1 (distribution of the contact pressure on the basis of integral Eq. (8)—black curves; distribution of the contact pressure approximated by Eq. (13)—diamonds).
0 0
0.5
1
1.5
2
Fig. 11. Typical distribution of srz/p0 stresses at the coating/substrate interface: contact pressure distribution found from integral Eq. (8)—black curves, distribution of actual contact pressure replaced by Hertz solution—gray curves.
ARTICLE IN PRESS R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
1
1.5
ν 0 = ν 1 = 0.25
ν 0 = ν 1 = 0.25 1
1.4
3 max max σ HM σ HM, 0
0.9
max p0 σ HM
1599
2 0.8
2
1.3
1 1.2
1 : E 0 E1 = 2
1 : E 0 E1 = 2
0.7
2 : E 0 E1 = 4
3
2 : E 0 E1 = 4
1.1
3 : E 0 E1 = 8
3 : E 0 E1 = 8 1
0.6 0
0.2
0.4
0.6
0
0.2
h
0.4
0.6
h
Fig. 12. (a) Relationship between the maximum Huber-von Mises reduced stress in the substrate and non-dimensional layer thickness and (b) Relationship between max max smax HM =sHM,0 and the non-dimensional layer thickness (sHM,0 —maximum Huber-von Mises reduced stress for homogeneous half-space with mechanical properties of substrate).
The level of stresses in half-space (mark ‘‘1’’) we will evaluate on the basis of distribution of Huber-von Mises reduced stress. When the layer thickness is higher than specific value the maximum value of this stress appear on the z-axis at the coating/surface interface. This value is checked off by cross at Fig. 12a. At small thickness of cover layer the maximum value of sHM stress is placed in small distance of the centre of the area of contact in comparison with proper distance for homogeneous half-space (0.46 of the radius of the contact area when n1 ¼0.25). The value of smax HM =p0 parameter in the case of half-space covered by layer is smallest than proper value for homogeneous half-space is equal to 0.965 (n1 ¼0.25). In Fig. 12a this value is checked off by ‘diamond’. However, it is necessary to point out that incrementation of mean contact pressure when appear with half-space is covered exceed the reduce of smax HM =p0 parameter. It is causes that the higher level of Huber-von Mises reduced stress occur in the substrate in comparison with homogeneous halfspace case.
7. Conclusions The contact problem of elasticity connected with an indentation of a rigid sphere in an elastic semi-space covered by an elastic layer has been considered. The paper presents detailed analysis of stress distribution produced by contact pressure. The analysis of stress distribution in the hard top layer proved that this layer can be divided into three sections: relatively thin layers (h o0.2), relatively thick layers (h40.7) and those in between. In the case of relatively thin layers, the maximum tensile stress occurs at the unloaded half-space surface. Another area where the tensile stress may occur is the coating/substrate interface. An increase of layer thickness or E0/E1 parameter is also accompanied by an increase of tensile stresses at the interface. For a certain layer thickness the tensile stresses at the coating/substrate interface are greater than the ones on the surface of the nonhomogeneous half-space. In the case of relatively thick layers, the predominant stresses are mostly tensile stresses at the coating/ substrate interface. When 0.2 oho0.7, the maximum tensile
stress occurs either at the unloaded half-space surface or at the coating/substrate interface. In the case of a ‘reasonable’ thickness of layer (ho1) the maximum value of the Huber-von Mises reduced stress occurs at the coating/substrate interface (with an exception of relatively thin layers or when the mechanical properties of the layer differ insignificantly from the substrate properties). For relatively thin layers the maximum sHM stress appears on the surface of nonhomogeneous half-space. The level of the Huber-von Mises reduced stresses may be much higher (more than twice) compared to the largest value of the tensile stress. The replacement of the actual contact pressure with Hertz pressure makes it possible to calculate the maximum tensile stresses on the surface of non-homogeneous half-space and the maximum shear stress srz at the coating/substrate interface with a satisfactory precision for engineering calculations. The maximum tensile stresses at the coating/surface interface and the Huber-von Mises reduced stresses considered in view of actual contact pressure and Hertz distribution differs considerably (with exception of relatively thick layers). In calculations, we can describe distribution of actual pressure by approximate relation (13).
Acknowledgements The investigation described in this paper is a part of the project S/WM/2/08, funded by the Polish State Committee for Scientific Research and realized at Bialystok University of Technology.
Appendix A The general solution of the problem can be written in the form Z 1 Z 1 UrðiÞ ðs,zÞJ1 ðsrÞds, uzðiÞ ðr,zÞ ¼ UzðiÞ ðs,zÞJ0 ðsrÞds, i ¼ 0,1, urðiÞ ðr,zÞ ¼ 0
0
ðA:1Þ
srrðiÞ ðr,zÞ ¼ mi
Z
1 0
ðiÞ sSrr ðs,zÞJ0 ðsrÞds
2 r
Z
1 0
UrðiÞ ðs,zÞJ1 ðsrÞds,
i ¼ 0,1,
ðA:2Þ
ARTICLE IN PRESS 1600
R. Kulchytsky-Zhyhailo, G. Rogowski / Tribology International 43 (2010) 1592–1601
sðiÞ 2ni jj ðr,zÞ ¼ mi 12ni
Z
1 0
sSðiÞ jj ðs,zÞJ0 ðsrÞds þ
2 r
Z
1 0
UrðiÞ ðs,zÞJ1 ðsrÞds,
i ¼ 0,1,
ðA:3Þ ðiÞ szz ðr,zÞ ¼ mi
Z
1 0
sSðiÞ zz ðs,zÞJ0 ðsrÞds,
ðiÞ srz ðr,zÞ ¼ mi
Z
1 0
sSðiÞ rz ðs,zÞJ1 ðsrÞds,
ðA:4Þ
where 2Urð0Þ ðs,zÞ ¼ a3 ðsÞfð2 þd0 ÞsinhðsðhzÞÞ þ d0 ðhzÞs coshðsðhzÞÞg þa4 ðsÞfð2 þ d0 ÞcoshðsðhzÞÞ þ d0 ðhzÞs sinhðsðhzÞÞg þ2a5 ðsÞs coshðsðhzÞÞ þ2a6 ðsÞs sinhðsðhzÞÞ, 0 r z rh,
A16 ¼ 2s sinhðshÞ,A22 ¼ 2s, A23 ¼ d0 sh sinhðshÞ, A24 ¼ d0 sh coshðshÞ, A25 ¼ 2s sinhðshÞ, A26 ¼ 2s coshðshÞ, A31 ¼ m ð1 þd1 Þ, A32 ¼ 2m s, A33 ¼ ð1 þd0 ÞcoshðshÞd0 sh sinðshÞ, A34 ¼ ð1 þ d0 ÞsinhðshÞd0 sh coshðshÞ, A35 ¼ 2s sinhðshÞ, A36 ¼ 2s coshðshÞ,A41 ¼ m , A42 ¼ 2m s, A43 ¼ sinhðshÞ þd0 sh coshðshÞ,A44 ¼ coshðshÞ þ d0 sh sinhðshÞ, A45 ¼ 2s coshðshÞ, A46 ¼ 2s sinhðshÞ,A53 ¼ 1 þd0 , A56 ¼ 2s, A64 ¼ 1, A65 ¼ 2s, m ¼ m1 =m0 , Z 1 ~ ¼ ~ m0 g, pðsÞ rpðrÞJ0 ðsrÞdr: b ¼ f0,0,0,0,0,pðsÞ= 0
ðA:5Þ Appendix B 2Urð1Þ ðs,zÞ ¼ a1 ðsÞð2 þ d1 þ d1 szÞexpðszÞ2a2 ðsÞs expðszÞ,
z r0, ðA:6Þ
2Uzð0Þ ðs,zÞ ¼ d0 ðhzÞsa3 ðsÞsinhðsðhzÞÞ þ d0 ðhzÞsa4 ðsÞcoshðsðhzÞÞ þ2a5 ðsÞs sinhðsðhzÞÞ þ 2a6 ðsÞs coshðsðhzÞÞ, 0 r z rh,
2Uzð1Þ ðs,zÞ ¼ d1 a1 ðsÞsz expðszÞ þ2a2 ðsÞs expðszÞ,
z r 0,
Formulas for calculating parameters Ai and Bi, i¼1, 2 are Z 1 Z 1 8 16 sa 6 ðsÞF1 ðsÞG1 ðsÞ ds, A2 ¼ pffiffiffiffiffiffi sa 6 ðsÞF1 ðsÞG2 ðsÞ ds, A1 ¼ pffiffiffiffiffiffi 2p 0 2p 0
ðA:7Þ
20 B1 ¼ pffiffiffiffiffiffi 2p
ðA:8Þ
F1 ðsÞ ¼ 5
Sð0Þ rr ðs,zÞ ¼ a3 ðsÞfð2d0 þ1ÞsinhðsðhzÞÞ þ d0 ðhzÞscoshðsðhzÞÞg þ a4 ðsÞfð2d0 þ 1ÞcoshðsðhzÞÞ þ d0 ðhzÞssinhðsðhzÞÞg þ 2a5 ðsÞscoshðsðhzÞÞ þ2a6 ðsÞssinhðsðhzÞÞ, 0 rz r h,
Z
1
sa 6 ðsÞF2 ðsÞG1 ðsÞ ds, 0
40 B2 ¼ pffiffiffiffiffiffi 2p
Z
1
sa 6 ðsÞF2 ðsÞG2 ðsÞ ds, 0
J5=2 ðsÞ J3=2 ðsÞ J5=2 ðsÞ J3=2 ðsÞ 3=2 , F2 ðsÞ ¼ 3 5=2 3=2 , s5=2 s s s G1 ðsÞ ¼ 1J0 ðsÞ, G2 ðsÞ ¼ J2 ðsÞ:
References
ðA:9Þ
Sð1Þ rr ðs,zÞ ¼ ð2d1 þ 1 þ d1 szÞa1 ðsÞexpðszÞ2a2 ðsÞs expðszÞ,
z r0, ðA:10Þ
Sð0Þ jj ðs,zÞ ¼ a3 ðsÞsinhðsðhzÞÞ þa4 ðsÞcoshðsðhzÞÞ, Sð1Þ jj ðs,zÞ ¼ a1 ðsÞexpðszÞ,
0 r z rh,
z r0,
ðA:11Þ ðA:12Þ
Sð0Þ zz ðs,zÞ ¼ a3 ðsÞfsinhðsðhzÞÞ þ d0 ðhzÞs coshðsðhzÞÞg a4 ðsÞfcoshðsðhzÞÞ þ d0 ðhzÞs sinhðsðhzÞÞg 2a5 ðsÞs coshðsðhzÞÞ2a6 ðsÞs sinhðsðhzÞÞ, 0 r z rh, ðA:13Þ
Sð1Þ zz ðs,zÞ ¼ a1 ðsÞð1 þ d1 szÞexpðszÞ þ 2a2 ðsÞs expðszÞ,
z r 0,
ðA:14Þ
Sð0Þ rz ðs,zÞ ¼ a3 ðsÞfð1 þd0 ÞcoshðsðhzÞÞ þ d0 ðhzÞssinhðsðhzÞÞg a4 ðsÞfð1þ d0 ÞsinhðsðhzÞÞ þd0 ðhzÞs coshðsðhzÞÞg 2a5 ðsÞs sinhðsðhzÞÞ2a6 ðsÞs coshðsðhzÞÞ, 0 r z rh, ðA:15Þ
Sð1Þ rz ðs,zÞ ¼ a1 ðsÞð1 þ d1 þ d1 szÞexpðszÞ2a2 ðsÞs expðszÞ,
z r 0, ðA:16Þ
where di ¼1/(1 2ni), i¼0,1, J0(r), J1(r) denotes the Bessel functions and m0, m1 are Kirchhoff coefficients of the layer and the substrate, respectively. A system of linear equations for determination of the functions ai ðsÞ, i ¼ 1,2, . . . ,6 is Aa ¼ b, where the non-zero elements of the matrix A are A11 ¼ 2 þ d1 , A12 ¼ 2s, A13 ¼ ð2 þ d0 ÞsinhðshÞ þd0 sh coshðhÞ, A14 ¼ ð2 þ d0 ÞcoshðshÞ þ d0 sh sinhðshÞ, A15 ¼ 2s coshðshÞ,
ðA:17Þ
[1] Houmid Bennani H, Takadoum J. Finite element model of elastic stresses in thin coatings submitted to applied forces. Surf Coat Technol 1999;111:80–5. [2] Bhomwick S, Kale AN, Jayaram V, Biswas SK. Contact damage in TiN coatings steel. Thin Solid Films 2003;436:250–8. [3] Xie C, Tong W. Cracking and decohesion of a thin Al2O3 film on a ductile Al-5%Mg substrate. Acta Mater 2005;53:477–85. [4] Holmberg K, Matthews A, Ronkainen H. Coatings tribology—contact mechanisms and surface design. Tribol Int 1998;31:107–20. [5] Gupta PK, Walowit JA. Contact stresses between an elastic cylinder and a layered elastic solid. Trans ASME, J Lubr Technol 1974;96:250–7. [6] Chen WT, Engel P. Impact and contact stress analysis in multilayered media. Int J Solid Struct 1972;8:1257–81. [7] Swain MV, Mencik J. Mechanical property characterization of thin films using spherical tipped indenters. Thin Solid Films 1994;253:204–11. [8] Gan L, Ben-Nissan B, Ben-David A. Modelling and finite element analysis of ultra-microhardness indentation of thin films. Thin Solid Films 1996; 290–291:362–6. [9] Schwarzer N, Whittling M, Swain M, Richter F. The analytical solution of the contact problem of spherical indenters on layered materials: application for the investigation of TiN films on silicon. Thin Solid Films 1995;270:371–5. [10] Chudoba T, Schwarzer N, Richter F. New possibilities of mechanical surface characterization with spherical indenters by comparison of experimental and theoretical results. Thin Solid Films 1999;355–356:284–9. [11] Mishina H. Microscopic real time observation of failure, wear and deformation on coating/subsurface. Tribol Int 1999;32:137–44. [12] Su YL, Kao WH. Tribological behaviour and wear mechanism of MoS2– Cr coatings sliding against various counterbody. Tribol Int 2003;36:11–23. [13] Wang DF, Kato K. Effect of friction cycles on wear particle generation of carbon nitride coating against a spherical diamond. Tribol Int 2000;33: 115–22. [14] Wang DF, Kato K. Critical load for wear particle generation in carbon nitride coatings by single sliding against a spherical diamond. Tribol Int 2001;34:407–18. [15] Oliveira SAG, Bower AF. An analysis of fracture and delamination in thin coatings subjected to contact loading. Wear 1996;198:15–32. [16] Cole SJ, Sayles RS. A numerical model for the contact of layered elastic bodies with real rough surface. ASME J Tribol 1992;114:334–40. [17] Chiu YP, Hartnett MJ. A numerical solution for layered solid contact problems with application to bearings. ASME J Lubr Technol 1983;105:585–90. [18] O’Sullivan TC, King RB. Sliding contact stress field due to a spherical indenter on a layered elastic half-space. ASME J Tribol 1989;110:235–40. [19] Anderson A, Collins IF. Plane strain stress distributions in descrete and blended coated solids under normal and sliding contact. Wear 1995;185: 23–33. [20] Schwarzer N, Richter F, Hecht G. The elastic field in a coated half-space under Hertzian pressure distribution. Surf Coat Technol 1999;114:292–304. [21] Schwarzer N. Coating due to analytical modelling of mechanical contact problems on multilayer systems. Surf Coat Technol 2000;133–134:397–402.
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[22] Ramalingam S, Zheng L. Film-substrate interface stresses and their role in the tribological performance of surface coatings. Tribol Int 1995;28(3): 145–61. [23] Diao DF, Kato K, Hayashi K. The maximum tensile stress on a hard coating under sliding friction. Tribol Int 1994;27(4):267–72. [24] Kato K. Wear in relation of friction—a review. Wear 2000;24:151–7. [25] Bargallini GM, Cavatorta MP, Sainsot P. Coated contact: a strain approach. Tribol Int 2003;36:935–41. [26] Elsharkawy AA. Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solid. Int J Solids Struct 1999;36:3903–15. [27] Diao DF. Finite element analysis on local yield map and critical maximum contact pressure for yielding in hard coating with an interlayer under sliding contact. Tribol Int 1999;32:25–32. [28] Schwarzer N, Chudoba T, Richter F. Investigation of ultra thin coatings using nanoindentation. Surf Coat Technol 2006;200:5566–80. [29] Kouitat Nijwa R, Consiglio R, von Stebut J. Boundary element modeling of coated materials in static and sliding ball-flat elastic contact. Surf Coat Technol 1998;102:148–53. [30] Kouitat Nijwa R, von Stebut J. Boundary element numerical modelling as a surface engineering tool: application to very thin coatings. Surf Coat Technol 1999;116–119:573–9. [31] Shodja HM, Ghahremaninejad A. An FGM coated elastic solid under thermomechanical loading: a two dimensional linear elastic approach. Surf Coat Technol 2006;200:4050–64. [32] Abdul-Baqi A, Van der Giessen E. Numerical analysis of indentation-induced cracking of brittle coatings on ductile substrates. Int J Solids Struct 2002;39: 1427–42.
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[33] Wong SK, Kapoor A. Effect of hard and stiff overlay coatings on the strength of surfaces in repeated sliding. Tribol Int 1996;29(8):695–702. [34] Plumet S, Dubourg MC. A 3-D model for a multilayered body loaded normally and tangentially against a rigid body: application to specific coatings. ASME J Tribol 1998;120:668–76. [35] Kuo CH, Keer LM. Contact stress analysis of a layered transversely isotropic half-space. ASME J Tribol 1992;114:253–62. [36] Peng W, Bushan B. Three-dimensional contact analysis of layered elastic/ plastic solids with rough surfaces. Wear 2001;249:741–60. [37] Nogi T, Kato T. Influence of hard surface layer on the limit of elastic contact – Part I: analysis using a real surface model. ASME J Tribol 1997;119:493–500. [38] Chen WT, Engel PA. Impact and contact stress analysis in multilayer media. Int J Solids Struct 1972;8:1257–81. [39] Gupta PK, Walowit JA, Finkin EF. Stress distributions in plane strain layered elastic solids subjected to arbitrary boundary loading. Trans ASME J Lubr Technol 1973:427–33. [40] Kulchytsky-Zhyhailo R, Rogowski G. Tensile stresses in solid coating in the two-dimensional problem for contact with friction. J Frict Wear 2006;27:30–8. [41] Kulchytsky-Zhyhailo R, Rogowski G. Stresses of hard coating under sliding contact. J Theor Appl Mech 2007;45:753–71. [42] King RB, O’Sullivan TC. Sliding contact stresses in a two-dimensional layered elastic half-space. Int J Solid Struct 1987;23:581–97. [43] Elsharkawy AA, Hamrock BJ. A numerical solution for dry sliding line contact of multilayered elastic bodies. J Tribol 1993;115:237–45. [44] Timoshenko S, Goodier JN. Theory of elasticity. New York: McGraw-Hill Book Company; 1951.