Stress behaviour of surface-coated materials in concentrated sliding contact

Surface and Coatings Technology, 41(1990) 1

-

15

1

STRESS BEHAVIOUR OF SURFACE-COATED MATERIALS IN CONCENTRATED SLIDING CONTACT H. S. AHN and B. J. ROYLANCE Department of Mechanical Engineering, University of Wales, Swansea (U.K.) (Received December 5, 1988)

Summary The stress behaviour at the surface and in the near-surface region of layered (surface-coated) solids in concentrated sliding contact is investigated using the finite element method. Of particular interest is the behaviour of an elastic layer on an elastic substrate normally loaded by an elastic indenter and taking account of tangential loading at the interface. Stress contour plots are generated for “hard” and “soft” layers when expressed in terms of von Mises’ shear strain energy criterion normalized with respect to the maximum Hertz stress. The consequences of varying parameters such as friction coefficient, elastic modulus and layer thickness are examined. 1. Introduction Contact stresses, particularly those arising in non-conformal contacts, such as rolling element bearings, gear and cam systems, form one of the most important aspects in the design of such components. Stemming from Hertz’s analysis [1] attention in earlier years was directed to the simple normal loading condition by assuming a frictionless contact. In practice, contacts experience friction when relative motion is involved and it is therefore necessary to take into account the consequences of tangential as well as normal loading when establishing the stress behaviour in the contact region. Examples of earlier work which fully recognized these implications include studies by Poritsky [2], Hamilton and Goodman [3] and Smith and Liu [4]. The stresses involved when one or both contacting bodies are layered has only recently been evaluated. Solutions to the problem of an elastic layer joined to a rigid substrate and loaded by a rigid indenter have been presented in the literature [5], while Gupta and Walowit have obtained approximate numerical solutions to the generalized plain strain problem of contact between layered elastic solids [6]. The purpose of the present work is to provide a solution to the contact problem for an elastic layer on an elastic substrate loaded by an elastic 0257-8972/90/$3.50

© Elsevier Sequoia/Printed in The Netherlands

2

Fig. 1. General configuration for a layered solid under semi-elliptically distributed moving heat source and contact pressure: Po, hertzian maximum pressure; t, layer thickness; t’, sliding speed; a, contact halfwidth.

indenter and taking into account the effect of tangential loading. The finite element method (FEM) is utilized to obtain a solution which defines the stress behaviour in the surface and subsurface. The contact configuration for the layered solid is shown in Fig. 1.

2. Development of the finite element model 2.1. General procedure and discretization by finite elements The contact problem in general involves a set of stresses {u~,displacements fu~and known body forces per unit volume ~p}. The displacements on the part S~,of the bounding surface S are specified in terms of known values ~ Known applied loads or traction forces i1qe} are applied to the part S(,. Given this situation, the following conditions must be satisfied. 2.1.1. Compatibility Continuity and differentiability of ~u} to the necessary degree. Displacement boundary conditions: fu~=~zi}

onS~,

Strain—displacement law: [L]~u~

=

where [U is some linear operator.

2.1.2. Equilibrium + ~p}

~q~}

=

=

{L}T

+

=0

-[qeIl °‘~

where {S} [LIT is a linear operator that is the transpose of {L} (as defined above) and ~q~}represents internal traction forces.

3

2.1.3. Stress—strain {a}

=

[D]{e}

where [D] is a symmetric matrix with fixed terms that are functions of the elastic modulus E and Poisson’s ratio t.’. In the finite element displacement method, the displacement is assumed to have unknown values only at the nodal points, so that the variation within any element is described in terms of the nodal values by means of interpolation functions. Thus {u} = fN}T~ue} {c}

=

[B]~ue}

~u} = [D]{e} where {N}T is a set of shape functions, ~UC} is a vector of nodal displacement of element e, [B] is a strain matrix composed of derivations of the shape functions, and [D] is an elasticity matrix. For the contact problem, the governing equilibrium equations can be obtained by minimizing the total potential energy of the system. The potential energy can be expressed as PE

=

~

f{e}T{a}

dV

f{u}T~p}

dV~_f{u}T{q} dS

(1)

Integrations are thus taken over the volume V of the structure and loaded area S. The first term on the right-hand side represents the initial strain energy; the second and third terms are, respectively, the contribution of the body forces and the distributed surface loads. Provided that the element shape functions have been chosen so that no singularities exist in the integrands of the functional, the total potential energy of the continuum will be the sum of the energy contributions arising from the individual elements PEe. The total potential energy of element e can be written as PEe

=

~ f{ue}T [W]T[D]EW]{ue}

_~~f{ue}T{Ne}T{q}

dV— f~ue}T~Ne}T{pe} dV

dS

(2)

where e denotes the individual element concerned. For the entire calculation domain

PE

=

~ flu}T[B]T[D1[Bffu}

dV— flu}T~N}T~p} dV~_fluHN}T{q} dS

4

Minimizing for the complete system gives dp

= [K]~u} du where

(3)

-~

=

f~N}T~pJ dV + flN}T~q}

=

JiLB]TIID] [B]

dS

system force vector

and [K]

dV

system stiffness matrix

Thus [KI~u~

(4)

=

The linear matrix equation set (eqn. (4)) can be solved in a reduced form by assembling all the relevant elements with the aid of the frontal solution method. The frontal elimination technique is preferred [7]because it provides minimum requirements of core storage, and it minimizes the number of arithmetic operations and use of peripheral equipment. The finite element stiffnesses and nodal forces are first assembled into a global stiffness matrix and load vector and solved for the unknown displacements by means of a gaussian elimination and back-substitution process. For the displacement method, the stresses are discontinuous because of the nature of the assumed displacement variation. To resolve this problem, the average of the nodal stresses of all the elements meeting at a common node is taken. However, this simple method can result in sizeable errors since the sizes of the adjacent elements are not taken into account. Hinton and Campbell [8]have shown that application of a local stress smoothing technique is appropriate for sampling finite element stresses in analyses using reduced integration (i.e. 2 X 2 gaussian integration). Thus smoothed corner node values may be obtained from Gauss point values by local stress smoothing. Hence, with local stress smoothing combined with nodal averaging, a complete and continuous stress field is defined over the whole domain in terms of the averaged smoothed stresses at the corner nodes, thus facilitating contour plotting. For the line contact problem, a plain strain condition is generally assumed since all values are almost independent of the coordinate perpendicular to the direction of motion on the surface. The matrices [B] and [D] are expressed as follows, assuming isotropic materials and employing quadratic isoparametric quadrilateral elements. [BI

=

where

[B1, B2,

...

B31

5

B~= aN1

0

ax

aN1

0

az

a~1 az

aN a.~

E(1—~)

1

(1+v)(1+2~)

11) 0

——-

0

1

0

1—v

0

1— 2v 2(1-—v)

The shape functions {N}T which are defined in a natural coordinate system (s, t) for quadratic isoparametric elements are given as {N}T

=

[N1, N2,

...

N8]

where N1(s, t)

-~(1—s)(1 t)(1 + s + t) 2)(1—t) N2(s, t)-~(1—s N 3(s, t)”-~.(1+s)(1—t)(s—t—1) 2) N4(s, t) =.~.(1+s)(1 t N 5(s, t) =-~(1+ s)(1 + t)(s + t—- 1) 2)(1+t) N6(s, t)~.(1—s N 7(s, t).~(1—s)(1+t)(—s+t-—1) 2 Ns(s, t)r~.~.(1—s)(1_t) =

—

---

--

2.2. Accuracy of model The discretization of the finite element model is shown in Fig. 2. To optimize the element size and number, use was made of the various combinations shown in Table 1, through which it was established that 480 quadratic isoparametric quadrilateral elements were the most appropriate for the analysis in the range —5a
6 C C

—~

C

In.

~-

m c~i

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C rC CI ~

C CI

,—~

C

CI C.

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C

C

CI

C-.I CI

C C

C

C C C

uird

C

a a ua~

I

C

C b

c~-CI

C C

C ‘..

Ia~.

o L

a

,~,H

C

a__I N

C

-~:

CI

‘I

I ~ -~

~N

a

a~

..a

C

~

‘~

—

aa .~

all

C-

a

o

a

~

a

aa

CC io~r

C CI

CCC LCL’~IfD

CICI

—

CI~’ 100)

CCI

CI 0 CI

0 CI C

~ 0

a> ~a a 0 ~:a

Ce

~~

CC IL

~

—

~

CCC NCI ~

C C

CI

CC

C-

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CI 0 ,—~

C

CI

N>

a

C-

aCe

Ce -~ ‘~

~

a

CI CC LO

N>

a>

C-

o

0 a>

a

.~

aC-

aC-

a

a

a

o

—

a. a

O >e

0

Ce.~0 a>

~

Oaa

.~i>0~

a

—

a a>,

ae,

~

a

a.

0

a

E’L)

0.

a

..—---

a>

a

CC

a>

C.)

—

,—1

0

CC,~

~0-~

Oaa

Ce

a,a

7

1ff:~

[—LaYer

________________________________________

24 elements

I

Fig. 2. Finite element model.

given by Smith and Liu using the operating conditions and parameters listed in Table 2. The results are shown in Fig. 3 in terms of dimensionless stresses obtained by dividing the actual stresses by the maximum Hertz stress Po. It is evident that there is good agreement although the difference between the two solutions increases as z increases. However, it should be noted that the influence of the stresses in the deeper subsurface is relatively insignificant since the magnitude is much less than at the surface and in the near-surface region. The surface stress distributions confirm that a~is sensitive to the variation in tangential loading (expressed in terms of coefficient of friction); the higher it is, the higher is the magnitude of a,, and its distribution becomes skewed towards the trailing edges.

3. Behaviour of layered solids 3.1. Basic assumptions and conditions In order to perform the analysis, some general assumptions and limitations need to be recognized. First, it is assumed that there is perfect adherence between layer and substrate. Secondly, the pressure distribution and contact area are assumed to be unaffected by the friction behaviour. It is further assumed that the tangential load is linearly proportional to the normal load and that the constant of proportionality is the coefficient of friction. The tangential load q is expressed therefore as q(x) = fp(x); I is the friction coefficient. Finally, the elastic modulus and Poisson’s ratio for the indenter are taken to be the same as those of the substrate of the indented layered solid. This permits the adaptation of the contact pressure distribution and the contact halfwidth expressions obtained by Gupta and Walowit [6]. For the range of elastic modulus ratio 5 Ef/ES s~ 4, (f denotes layer, s denotes substrate) the contact pressure distribution does not differ significantly from that defined by conventional Hertz theory applied to unlayered solids. However, there will be a considerable change in the magnitude of the contact pressure and contact width. When Ef/ES> 1 (harder layer), -~

8 TABLE 2 A set of contact conditions for the model Fixed conditions (1) Running conditions Radius of cylindrical body Width of loaded surface perpendicular to direction of motion Ambient temperature of air surrounding contact system Boundary temperature

0.003 m 0.01 m 20 C 20 C

(2) Physical properties of bodies Properties Cylindrical body/substrate or semi-infinite body Elastic modulus 200 or 207 Poisson’s ratio 0.3 Thermal conductivity 40 (W m~ °C~) 3) 8100 Density (kg m Specific heat (J kg~ ~C’) 500 Thermal diffusivity 9.88 (X106 m2 s~) Thermal expansion 13.0 coefficient (X106 °C~1)

Less conduclive layer

More conduclive layer

400 0.3 10

69 0.3 200

8100 500 2.47

2700 900 82.3

3.3

22.0

(3) Physical properties of air surrounding the contact system Thermal conductivity (W m~ C~) (2.57 X 10~2±7.76 X 105)(Taf -— 20) Density (kg m3) 1.19 Specific heat (J kg~ ~C’) 500 Thermal diffusivity (X105 m2 s~’) 9.88 Dynamic viscosity (kgfm1 S ~) (1.812 X iO~ ±4.566 X 108)(Taf 20) Variable conditions Load, sliding speed, coefficient of friction

there is an increase in pressure and a corresponding decrease in contact width when compared with the unlayered solid for the same conditions. The reverse is true for the case where E~/E 5 < 1. The maximum pressure and contact halfwidth for a layered solid can be read from Fig. 4 where dimensionless maximum pressure Pm/P0 and width a/a0 are expressed as functions of the ratio t/a of the layer thickness to the contact width and of the elastic ratio Ef/E~.Figure 4 is derived from data in Table 4.1 of ref. 6. A similar result is reported by El-Sherbiney and Halling [9] for the contact radius obtained for a circular indenter and a layered body in the range 0 < t/a0 < 0.467. Leveson [10] also obtained comparable relationships between a/a0 and a0/t for soft layers subjected to cylindrical and also circular contacts.

9 14

12

—io

Smith &Liu F EM

N

I

12 — — —

b

Smith&Liu friC Coef ~0 FE~M friC Coef = 0

~TT\.~TJ!~kL1

05 10

15

25 2.530 35 40

(a)

05

10

15 2025

30 35 40

(b)

~\\

~-os

10 ~_05

~10r -

05

Smith & Liu f = 0 F E,M f= 0 Smith 0. Liu f — 0,5 FEll f=05

10

—10

—05

Smith & Liu F.E.M

— —

(c)

05

10

— — —

(d)

Fig. 3. Stress distributions, showing a comparison between the FEM solution and that of Smith and Liu [4]: (a) a~/povs. z/a along z axis; (b) U~/Povs. z/a along z axis; (c) a~/po vs. x/a on surface; (d) a~/povs. x/a on surface.

__________________________________________

E~E,/E

_____

-~

06

(a)

____________

06~

io~

~

10°

(b)

10.1

100

Fig. 4. Variations in maximum pressure Pm and contact halfwidth a with layer thickness for elastic indenter and elastic substrate: (a) PIU/PO vs. tia; (b) a/ao vs. f/a.

3.2. Stress fields The finite element program facilitates the generation of contour plots for various stresses including von Mises’ equivalent stress and the maximum shear stress; von Mises’ shear strain energy criterion is expressed as the square root of the second invariant of the stress deviator tensor:

10

+u~)2~+r~3,2+r2+y

~

2=k2= Y2/3 (5)

where k and Y denote the values of yield stress of the material in simple shear and simple tension (or compression) respectively. It is convenient to normalize the relationship as J 2/p() and examine the consequences of varying the friction coefficient. Figure 5 shows the stress fields for unlayered solids for [=0, 0.25, 0.5 and 0.75. These are compared with plots for hard and soft layers in Figs. 6 and 7. It is evident that whereas the presence of a hard layer increases the stress condition in the layer, the converse is the case for the softer layer. The figures also depict the effect of tangential loading on the magnitude and location of the maximum stress condition in the surface and subsurface. The effect of varying the elastic modulus of the layer, relative to that of the substrate, on the surface normal stresses a~and o~is illustrated in Fig. 8 in which the stresses are normalized in terms of the maximum hertzian contact pressure p0 for an unlayered solid. For a~,the stress distribution and contact width differ noticeably such that for the hard layer, the maximum stress is larger and the contact width is reduced, while for the soft layer the reverse is the case. Figure 8 also shows the effect of the friction coefficient on a,, for hard and soft layers. The results demonstrate that the presence of a hard layer increases both the maximum tensile and maximum compressive stresses on the surface. The reverse is true for the soft layer case.

200

--~—-

-

,-—

——

.

(a)

(I))

(c)

(d)

---—

—-

.

-

-

--

Fig. 5. von Mises’ equivalent stress contours for unlayered solid. (a) f = 0; data, 0.008 68 0.33; contour interval, 0.05. (b) f 0.25; data, 0.03 - 0.351; contour interval, 0.05. (c) f= 0.50; data, 0.0493 - 0.513; contour interval, 0.05. (d) f= 0.75; data, 0.0551 - 0.728; contour interval, 0.1.

11 2 00 1 75 1 50

025 __________

_______

1.75 1.50

___________ ,~ -~

025

(a)000

~

_______ _________

___________ ________

___________

125 100 2 00

0 75 050

~ .

~~-4°’ I

-

025

-

0 25

000 (b)

-~

2 00 1.50

0.20

J

1 25 175 —~ C-

1.00 075

0 50

0.50

0 25 0 00 _________________________________________________

0 25 0 00

(d)

(c)

__________ _________ ______________

—1.5

—1.0

_______________________________

—0.5

0.00

05

1.0

1 5

0/a

Fig. 6. von Mises’ equivalent stress contours for a hard-layered solid (Ej/E~= 2). (a) f = 0; data, 0.012 32 - 0.4092; contour interval, 0.05. (b) f 0.25; data, 0.073 48 - 0.4147; contour interval, 0.05. (c) f=0.50; data, 0.096 91 -0.65; contour interval, 0.1. (d)f= 0.75; data, 0.1 - 0.9; contour interval, 0.1. 200 1.75 200 125 07E

,,

150 1.75 1.25

~

100

___________

__________________ ___________

/

05

________________

_____

___________ ___ _______________________

0.75

050 025 _______

050 0 25 0.00

______________ __

000

(a)

(b)

2.00

2.00

(

1.50 1.75

_______________ ________________

0.75 125

___________ __

0.50

_______________

025 000

(c)

____ _______________________ ___ _______________________ ___

_______________ __ ____________

Z ~

~‘

0.30

1 50 1.75

~C- 100 0.75 125

0.25

0 50

~~~iii 1 0

025 -~ 000 15

(d)

-

-05

0.0

s/a Fig. 7. von Mises’ equivalent stress contours for a soft-layered solid (E 0/E5

0 S

=

1.0

1.5

1/2).

The stress distributions at various depths below the contact surface are depicted in Fig. 9 for the unlayered, hard and soft layer conditions, respectively. The effect of layer thickness is also assessed in terms of a~and a~.The results are shown in Fig. 10 in which a0 denotes the contact halfwidth for an unlayered solid while a1 and a2 represent the same parameter for layered solids. With a hard layer, the magnitude of a>, within the layer is higher than for the unlayered case but it is lower in the region of the substrate; the variation increases with an increase in layer thickness. For the

12 Unlayered

01

(a)

-15 C)

1 0

1

05

——a-— ~‘

Unlayered Ef/Es=2

Et/Es=

~

~

(b)

Ef/Es=2 Ef/Es=

—-

—e--— __._

½

101 -‘

1.’O.

~

Unlayered Ef/Es=2 Ef/Es=1/~

05~

1.5

-15

-1.5 ~

-201

-20

f=o25

—.

f=05

Fig. 8. Stress distributions for various elastic modular ratios E

1/E5: (a) u~/povs.x Ia; (b)

a,,/p0vs. x/a.

soft layer situation, a reverse condition exists. The effect of the layer thickness on a~ is less marked, although a marked change within the layer is evident. The change in a~below z = a is also noticeable although it does not affect significantly the overall stress field since its magnitude decreases rapidly below the surface. Figure 11 shows contour lots of von Mises’ equivalent stress for the variation in hard layer thickness as characterized by the ratio t/a. It is evident that the maximum has shifted to the surface. However, its magnitude decreases as the layer thickness increases such that the plot closely resembles that of an unlayered solid when the layer is comparatively thick (Fig. 11(d)) and thus tends to behave as an unlayered solid having the layer’s properties. In contrast with the hard layer condition, the use of a soft layer (Fig. 12) causes a reduction in the stress condition and the maximum value moves into the subsurface as the layer thickness increases.

4. Closure It is evident that using a relatively thin layer of coating material has a marked effect on the stress behaviour which is also a function of the tangential load and the relative elastic moduli of the materials involved. The

13 CI

a Ca a

o

5.> Cr>

CrC

I.:

II

I

~I/~/) )

10

CI

m’O~0mm

~

a C.

Lf~

=‘

C

~IIII~

~

I

: /fj___

d,

a

/ 4//,y~

2

I

~ ~uII~I

—j±~” ~o

~)d/.o

T~t

~ 22

E

-.-..

:~i2~

I ~10

14 1

~

1

——~— —~

12

~

25~0

Unlayered F/a I/a 2

= =

14

04 20

125

0510 12 ~

(lj)

.2505

Unlayened

Uniayered

—~—

I/a, I/a>

—~

15202.53.03040 z/a —~--—

~

04

=

20

152025303540

z/a

p

‘

—

07510 12515 17520

=

Unlayeed t/a S 04

0250.5075 10 12515 175 20 z/a

Fig. 10. Effect of layer thickness on the stress distribution beneath the contact surface. (a) x/o = 0, E1/E5 = 2, a1/ao = 0.94, a2/a0 = 0.88 and Ey/Jo]5 = 1/2; a1/a0 1.09, 02/017 = 1.20; (b) .v/a = 0.4, Ef/E5 = 2 and E0/E5 = 1/2.

20

(a)

,_-

~°°1~ .

(h)

.z (c)

_________

(d)

—15

-10

-05

______________

x/s000

05

10

Fig. 11. Stress contours for a hard-layered solid (E0/E5 = 2) for various layer thicknesses (f> 0.3). (a) Unlayered; data, 0.0333 - 0.36; contour interval, 0.05. (b) f/a = 0.4; data, 0.058 19 - 0.4785; contour interval, 0.05. (c) 1/a = 1.0; data, 0.072 71 - 0.4565; contour interval, 0.05. (d) 1/a = 4.0; data, 0.033 88 - 0.4; contour interval, 0.05.

5

Ut_____

-15

-10

—05

_____

00

(a)

05

10

1.5

-15

n/a

(b) =

-

-05

0.0

05

10

15

x/a

Fig. 12. Stress contours for a soft-layered solid (Ef/E5 (f 0.3). (a) t/a 0.4; data, 0.026 364 6 0.335 232; 4.0; data, 0.034 570 8 0.32; contour interval, 0.05. =

—10

1/2) for various layer thicknesses contour interval. 0.05. (h) t/a =

-

thermal behaviour of such contacts is also markedly [11] affected and it is necessary therefore to examine further the implications of combined thermal and mechanical stresses for these conditions in relation to possible surface failure mechanisms [12]. Acknowledgment The financial support of the British Council for one of the authors (H.S.A.) during his period of study in Swansea is deeply appreciated. References 1 K. L. Johnson, One hundred years of Hertz contact, Proc. Inst. Mech. Eng., 196 (1982) 363 - 378. 2 H. Poritsky, Stresses and deflections of cylindrical bodies in contact, J. Appi. Mech., 17 (1950) 191 - 201. 3 G. M. Hamilton and L. E. Goodman, The stress field created by a circular sliding contact, J. Appi. Mech., 33 (1966) 371 - 376. 4 J. 0. Smith and C. K. Liu, Stresses due to tangential and normal loads on an elastic solid with application to some contact stress problems, J. Appi. Mech., 20 (1953) 157 - 166. 5 M. Hannah, Contact stress and deformation in a thin elastic layer, Q. J. Mech. Appi. Math., 4 (1951) 94 - 105. 6 P. K. Gupta and J. A. Walowit, Contact stresses between an elastic cylinder and a layered elastic solid, J. Lubr. Technol., 96 (1974) 250 - 257. 7 B. M. Irons, A frontal solution program, mt. J. Num. Meth. Eng., 2 (1970) 5 - 32. 8 E. Hinton and J. S. Campbell, Local and global smoothing of discontinuous finite element function using a least-square method, mt. J. Num. Meth. Eng., 8 (1974) 461 - 480. 9 J. HaIling and M. G. D. El-Sherbiney, The hertzian contact of surfaces covered with metallic films, Wear, 40 (1976) 325 - 337. 10 R. C. Leveson, The mechanics of elastic contact with film-covered surfaces, J. Appi. Phys., 45 (1974) 1041 - 1043. 11 H. S. Ahn and B. J. Roylance, The thermal behaviour of surface-coated materials in concentrated sliding contacts: I — fast-moving heat sources, to be published. 12 B. J. Roylance, S. W. Sui and D. A. Vaughan, Thermally-related access behaviour in concentrated contacts and the implications for scuffing behaviour, Proc. 12th Leeds— Lyon Symp. on Mechanisms and Surface Distress, 1985, Butterworths, London, 1985, pp. 117 - 127.