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Model-based contact fatigue design of surface engineered titanium gears
Computational Materials Science 35 (2006) 447–457 www.elsevier.com/locate/commatsci
Model-based contact fatigue design of surface engineered titanium gears J. Luo, H. Dong *, T. Bell Department of Metallurgy and Materials, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 7 January 2005; accepted 15 March 2005
Abstract In order to fully realise the potential of advanced surface engineering in the automotive sector, a contact mechanics model has been developed based on modern theories of multi-layered surface contact, taking into account the multi-layered structure, real surface roughness and friction effects. With this model, the performance of surface engineered titanium automotive gears can be successfully predicted thus making possible the design of optimised surface engineering systems to meet particular engineering demands within the shortest possible time and with least cost. This paper presents the predictive contact fatigue design of Ti6Al4V based gears in racing car engines using the developed contact mechanics model for the real rough multilayered surface systems. Evaluation of the effectiveness of the predictive design is carried out through cam rig tests of real gears, simulating the in-service loading conditions. Good agreement between theoretical prediction and service performance has been achieved. By way of example, the present paper demonstrates a number of major steps towards designing dynamically loaded titanium automotive gears. 2005 Elsevier B.V. All rights reserved. PACS: 68.35.Ct; 82.20.Wt; 81.65.Mq; 81.40.Pq Keywords: Design; Modelling; Titanium gear; Surface engineering; Contact fatigue; Oxidation
1. Introduction Contact mechanics is concerned with the stresses and deformations which arise when the surfaces of two solid bodies are brought into contact [1]. Contact parameters such as pressure, stress and deformation are of great importance to the study of the mechanisms of friction, wear, mixed lubrication, load bearing capacity and fatigue. Surface engineering has been proved to be one of the most effective ways to improve the tribological performance of engineering components [2]. The computation of stresses in layered surfaces is of great analytical and practical importance in the design of surface engi*
Corresponding author. Tel.: +44 121 414 5197; fax: +44 121 414 7373. E-mail address: [email protected] (H. Dong). 0927-0256/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.03.013
neered systems where friction, wear and contact fatigue characteristics can be improved significantly by modifying the surfaces to generate hard wear resistant materials, such as nitrides, on to the surface. In view of the difference in the elastic and plastic properties between the surface layers and substrate materials, traditional contact mechanics approaches based on homogeneous materials can not be directly applied to find the solution for the design of layered surfaces. In the past decade, significant progress has been made in the study of the contact behaviour of layered surfaces. The underlying aims of such research have been to understand the basic mechanisms involved in various tribological processes of engineered surfaces, and to assist with the selection of an optimum coating/substrate combination system for a specific application and in the design of engineering components with modified surfaces.
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The first significant elastic contact analysis of surfaces was produced by Hertz in 1881 [1] when he presented a paper ‘‘on the contact of elastic pressure and subsurface stress distributions’’ which arise in a non-conforming contact. This classic work has formed the basis of the present theoretical knowledge of static and rolling contact mechanics. Hertzian theory is based on the following assumptions. (a) The bodies may be considered as elastic half-spaces for the purpose of calculating the local deformations. This implies firstly that the contacting bodies are elasticity homogeneous, that is, the elastic properties are constant throughout each body, and secondly that the significant dimensions of the contact area are small compared with the dimensions of each body and with the relative radii of curvature of the surfaces, and thirdly that strains are small and the material exhibits perfect linearly elastic behaviour. (b) The contacting surfaces are perfectly smooth and non-conforming. On a small scale this implies that there are no surface irregularities which could cause discontinuous contact, and on a large scale it means that the surfaces are continuous up to their second derivative in the contact region. (c) The surfaces are assumed to be frictionless so that only a normal pressure is transmitted between them. Hertzian theory shows that the highest maximum shear does not occur at the surface, but at certain distance below the surface, indicating that under Hertzian contact conditions, failure of materials is initiated below the surface where the highest maximum shear stress occurs. These assumptions are considered to be inappropriate when investigating the contact mechanics of layered surface [3,4]. Progress in the field of contact mechanics over the last few decades has been associated largely with the removal of Hertzian restrictions. Smith and Liu addressed [5] friction effects on subsurface stress distributions. For the subsurface stress distributions, if sliding is present and the coefficient of friction is not zero, then a tangential traction is also present on the surface, in addition to the normal loading. This tangential traction induces subsurface stresses in addition to those due to the normal load. Accordingly, it can be seen that the maximum shear stress is moved towards the surface with the introduction of friction. This movement will be increased as the coefficient of friction is increased and when the coefficient of friction reaches a critical value (about 0.3), the maximum shear stress will reach the surface. This has a significant effect on contact fatigue. Since real surfaces are rough on the microscopic scale, attempts have been made to remove the smooth surface assumption of the Hertzian theory. There are mainly three methods to treat rough surfaces. First, Greenwood and Williamson [6] assume some form for the statistical properties of the surface, combined with an asperity shape that allows the calculation of individual asperity load-displacement laws [7]. These models
have yielded important results about the average properties of the contact of rough surfaces, but information about the real pressure distribution and the deformed shape is lost because of the statistical nature of the approach. Second, the ÔdirectÕ method, as reviewed by Sayles [8], uses roughness data directly recorded from a stylus measuring instrument, and employs a numeric technique in the contact analysis. A two dimensional computer model for dry, frictionless contact of real engineered surfaces (i.e. non-perfectly smooth) has been presented by Webster and Sayles [9]. Asperity plasticity is also considered in this work and a significant effect of surface topography on the pressure distribution identified. Depending on the topographic parameters, the maximum pressure in rough contact can be many times higher than that in ÔideallyÕ smooth contact, and the pressure peaks arise from the highest asperities. Third, the most recent advances in the contact mechanics of a rough surface recognise the multi-scale geometry of the rough surface [7,10]. A fractal geometry characterisation of the multi-scale topography can be used to simulate the rough surface profile allowing it to be incorporated into contact mechanics models [11,12]. For an extensive review of the multi-scale features of rough surfaces, readers are referred to Majumdar and Bhushan [13] for both experimental and theoretical aspects of the subject. In the case of layered surface contact, significant progress has been achieved in the last decade. There are four major methods of considering layered surface contacts, viz., Fourier integral transform [14,15], finite/ boundary element method [16,17], the method of image charges [18–20] and a combination of the variational principle and fast Fourier integral transformation [21]. The Fourier integral transform is based on SneddonÕs theory [22] which solves the contact problems by using the Fourier integral transform of the Airy stress function. Of the four approaches for the multilayered contact mechanics analysis, perhaps the finite/boundary method is the most popular due to its flexibility in treating complicated three dimensional (3D) component geometries and the ready availability of commercial finite/boundary analysis packages. However, for rough surface contact problems with many asperities of arbitrary shape, the requirement of a large number of mesh elements makes the finite/boundary element approach unfeasible. Further, it suffers from a slow calculation speed. Although the method of image charges is advantageous in terms of its readiness for 3D analysis and fast calculation speed, it lacks the ability to cope with rough surfaces. By comparison, the Fourier integral transform and variational principle methods are both easy to incorporate the rough surface effects and provide fast calculation speed. The latter is also 3D based, but as a very new method, it can only treat single-layered surface system
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0.1
0.05
Microns
at this stage. In contrast, Fourier integral transform method is more appropriate for 2D problems and has been well established for at least two layered (exclusive of substrate) surface systems [14,15]. The Fourier integral transform method is most suitable therefore for modelling the line contact problem between gears. In the following section, the Fourier integral transform method for a multilayered surface structure and directly measured rough surface is extended to include an automatic algorithm to search the load bearing capacity of gears against contact fatigue and then is used for the design of surface engineered Ti alloy gears in the sports car industry.
449
0
-0.05
-0.1 0
500
1000
1500
2000
2500
3000
Microns
2. Contact stresses and modelling
Fig. 2. Digitised surface profile, sampling interval 1.0 lm, 3000 points.
2.1. The Birmingham multilayered real rough surface contact mechanics model Fig. 1 shows the Birmingham model [14,15] where an elastic half-space, coated with n elastic layers, makes contact with an elastic cylinder. The layers are completely bonded to each other and to the elastic half-space they cover. The digitised surface profile as typically shown in Fig. 2 was obtained and is taken as the real surface roughness profile (in practice, it is better to take the roughness profile after shake-down of contact components for the calculation, i.e. after a period of run-in services perhaps in a lower loading condition than normal operation of machinery). GreenÕs function for a unit normal load and a unit tangential load for the generalised plane strain problem is then derived. Using the co-ordinate system and notation for the n + 1 component system shown in Fig. 1, the stresses rx, ry and sxy, in the x and y p directions respectively, in any layer ffiffiffiffiffiffiffi i are given by (j ¼ 1)
q(y) x1
2
x2
3
n n+1
y 1 y
1 2p
rxy i ¼
1 2p
Z
1
x2 Gi ejxy i dx
ð1Þ
1 1 2
1 Z 1
d Gi jxy i e dx d x2i jx
1
dGi jxy i e dx d xi
ð2Þ ð3Þ
where i is number of layers; x is Fourier transform integrating variable; Gi is the Fourier transform of the Airy stress function, and it is given by the following expression Gi ¼ ðAi þ Bi xi Þ ejxjxi þ ðC i þ Di xi Þ ejxjxi
ð4Þ
where the constants Ai, Bi, Ci, Di (the stress function coefficients) are general functions of x, and are determined from the boundary conditions. For the n layered structure shown in Fig. 1, the boundary conditions are as follows:
2
h1 h
2
x4
y
h
xn
y
hn
xn+1
ry i ¼
1 2p Z
• At the surface the normal and shear stresses must equal the applied pressures. • At every interface in the system, the normal and shear stresses and the normal and transverse deflections on each side of the interface must be equal. • The stresses fall to zero at a large distance from the load.
p(y) 1
rx i ¼
4 n
3
yn+1
Fig. 1. Co-ordinates and notations for multi-layered elastic solid.
Through examination of these boundary conditions, expressions for the stress function coefficients may be obtained [15]. Once the stress function coefficients are known, Gi is known and, by performing integrals on Gi and its derivatives, the stress and deflection at any point on and under the contact zone may be found. For any given rigid body displacement of the indentation cylinder, an iteration procedure is used to find the surface contact points and the pressure distribution. A detailed description of the solution procedure is given in the work of Mao and coworkers [14,15]. A typical
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of 0.5 lm for smooth contact. Whilst in smooth contact the maximum contact stress locates some distance below the surface, in rough contact the maximum stress appears at the surface arising from the highest asperities, which can be many times higher than that in smooth contact. The stress distribution varies with coefficient of friction, surface roughness, layer thickness as well as material properties of layers and substrate, which largely depends on the design of the surface engineering system. 2.2. Design of the surface engineered systems Fig. 3. Subsurface stress distribution (von Mises, GPa) for OD + TO treated Ti6Al4V alloy contacting with steel (friction coefficient l = 0.3) under an unit line load of 180 N/mm.
calculated von Mises (VMS) stress distribution for a surface engineered titanium alloy using a novel technique (i.e. 2 lm TiO2 and 200 lm oxygen diffusion zone on Ti6Al4V substrate) [23] is presented in Fig. 3 using the surface roughness profile shown in Fig. 2 and contact radius of 13.38 mm (cam gear contact condition, see Tables 1 and 2 in Section 3 for the description and calculation input parameters). It can be clearly seen that the stress distributions for a rough surface can be significantly different from the well established smooth case. The maximum von Mises stress for rough contact is 1.65 GPa at the surface, while it is 1.17 GPa at a depth Table 1 CoF, YoungÕs modulus (E), PoissonÕs ratio (m) of layers and substrate Materials
E (GPa)
m
CoF (with steel)
Ti6Al4V TO (TiO2) OD Steel
112 250 114 210
0.25 0.2 0.25 0.3
0.5 0.3 0.5 –
A recent succinct definition [24] of Surface Engineering states that Ôsurface engineering involves the use of traditional and innovative surface technologies in the design of a surface and substrate together to form a functionally graded system, which results in the cost effective performance enhancement of materials and componentsÕ. One of the key elements in the definition is design. In many cases, surface engineering design is conducted in the attempt to make sure that the designed surface system can bear an applied load, thus the term load bearing capacity. However, load bearing capacity is a concept which is loosely defined. Clear definition of the concept is difficult perhaps because, at least in part, load bearing capacity is closely related to the failure mode of a specific surface engineering system. In many tribological situations, failure of a hard coatingsubstrate system is seldom caused by conventional wear or yielding but rather by a fatigue failure process [25]. Therefore, the load bearing capacity of a surface system should be defined as the capacity to resist a specific kind of failure mode. Design of multi-layered surface systems therefore refers to the design activities that are carried out to choose the optimal layered surface structures for a given service condition and environment through the application of
Table 2 The gear specifications and the derived design parameters Parameters
Z T (N m) f (N/mm) w (mm) x (r.p.m.) Ri (mm) m (mm) h () Lubrication Dcyl (mm)
Alternator
Pump
Cam
Driver (Ti6Al4V)
Driven (steel)
Driver (steel)
Driven (Ti6Al4V)
Driver (steel)
Driven (Ti6Al4V)
32 0.33 2 6 6562 9.93
28 0.29 1.5 8 7500 8.69
17 2.43 24.8 5.8 17420 6.15
36 5.14 25.6 5.8 8250 13.03
38 30–40 86.3–115.0 9.2 8250 13.76
36 30–40 91.1–121.4 9.2 8708 13.03
1.81429 20 Oil splash 9.27
2.1167 20 Oil splash 8.36
2.1167 20 Oil splash 13.38
Note: Z: number of teeth; T: torque load; f: unit line load applied on pitch point; w: face width; x: rotation speed; Ri: the contact radii at pitch point, i stands for driver and driven; m: module; h: pressure angle; Dcyl: equivalent cylinder diameter when convert the gear contact as cylinder-half space contact.
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contact mechanics and material strength theories in the context of technical availability of surface treatment processes and their cost effectiveness, using material properties of the layered materials and their interfaces as input parameters. In the simplest case, surface engineering design refers to the prediction of the load bearing capacity against a certain failure mode for a given surface engineering system under a specific contact loading condition. The numerical model for the two-dimensional sliding contact of a cylinder with a real rough multilayered half space is extended in this paper to carry out surface engineered gear design against contact fatigue. The residual stress effect on the stress distributions can also be considered. In consideration of the multi-axial stress state under the contact zone, the octahedral shear stress theory, also termed von Mises theory, is used as the contact fatigue failure criterion: DrVMS P rFS
ð5Þ
where DrVMS is the cyclic range of the von Mises stress (GPa) and rSF is the fatigue strength (GPa). The minimum of the von Mises stress is taken as zero for the rolling-sliding contact of a gear. Thus for any depth beneath the surface, DrVMS equals the maximum von Mises stress at this depth, rVMS-max. The fatigue strength is changeable along the depth and is obtained by direct or indirect experiments (e.g. derived by a hardness depth profile). The critical load fc (the load bearing capacity,
451
represented by the unit line load along cylinder thickness, N/mm) is determined when the maximum von Mises stress at any depth overtakes the fatigue strength at that depth. The search procedure for fc is described in Fig. 4. An initial rigid displacement increment, Dd0, is applied to the indentation cylinder. The contact points and the surface pressure distribution are determined through an iteration procedure of the contact mechanics model. Then the stress field within the layers and substrate under contact can be calculated and superposed with the residual stresses if required. The fatigue strength depth profile can be determined (e.g., derived from hardness depth profile) and compared with the calculated maximum von Mises stress depth profile (rVMS-max = DrVMS). If the maximum von Mises stresses are lower than the fatigue strengths at all depth positions, a fixed displacement increment, Ddf, is added to the cylinder and the surface pressure and the stress fields under the contact zone are re-calculated and compared with the fatigue strength profile. This process is repeated until the calculated maximum von Mises stress reaches the fatigue strength at any depth. Then a further check is made to make sure that the relative difference between the fatigue strength and the maximum von Mises stress is within a precision requirement (e.g. 1%). If the difference is too large, an adjusted displacement increment, Dda, is calculated according to the difference and added to the cylinder. Again the surface pressure and stress field are re-calculated until the precision requirement is
Fig. 4. Flow chart for load bearing capacity prediction against contact fatigue.
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met. The final step sums up the surface pressure distribution to obtain the load bearing capacity, fc, of the surface system against contact fatigue. The real rough multilayered contact mechanics model and the contact design algorithm are programmed into a software package using the Fortran language.
Substituting Ti components for steel in industrial sectors, such as racing car engines, has been shown to be one of the most successful approaches to achieve high performance [26]. This is because titanium and its alloys possess the highest strength-to-weight ratio of all metallic materials. However, the reputation of titanium alloys for poor tribological behaviour has been the main barrier to their successful application. The poor tribological behaviour of the titanium is attributed to its inherent electron structure [26]. This problem may be overcome only by changing the nature of the surface. Some novel surface engineering technologies have been developed recently, which provide the necessary basis for realising the full potential of titanium alloys. One class of surface treatments of Ti alloys is based on oxygen thermal processing: i.e., ceramic conversion technology through thermal oxidation (TO) [27], oxygen boost diffusion technology (OD) [28] and the duplex treatment of OD in combination with TO [23,29]. The TO treatment creates an adherent thin rutile oxidation film (TiO2, about 2 lm thick) and a thin (about 10 lm) oxygen solution-hardened diffusion zone. The OD treatment creates a deep (about 200 lm) oxygen solution-hardened diffusion zone without the surface oxide layer and thus confers the high load bearing capacity needed for heavily loaded titanium components. However, it is found that the OD treatment cannot effectively improve the tribological properties of Ti alloys due to the fact that the metallurgical characteristics are hardly changed by the OD technology. Thus the duplex treatment of OD plus TO is necessary. The duplex treatment of OD + TO creates a similar layered surface structure as TO technology, but produces an oxygen solution-hardened diffusion zone beneath it with a depth of hardening similar in size to that produced by the OD treatment. In this paper, these three surface treatments are applied to Ti6Al4V to produce surface treated Ti alloy gears to be used in racing car engines. 3.1. Input mechanical parameters for the design To carry out the multi-layered contact behaviour simulations, the mechanical properties of the layers and substrate have to be determined first. The measurement of the surface hardness (H) and YoungÕs modulus (E) values of TiO2 and the oxygen diffusion hardened
Ti6Al4V
1100
TO
1000
OD
900
HV0.05
3. Case study—the design of surface engineered Ti gears
1200
OD + TO
800 700 600 500 400 300 200 0
50
100
150
200
250
300
Depth (microns) Fig. 5. Hardness (HV0.05) profiles for TO, OD, and OD + TO treated Ti6Al4V.
(OD) layers was conducted using a NanoTest 600 machine, manufactured by Micro Materials Ltd., UK, with the typical resolutions of 0.1 nm in displacement and 100 nN in force. The hardness of the TiO2 film for TO and OD + TO treatments was measured as 11 GPa (roughly corresponds to 1100 HV) and is treated as a fixed value. The YoungÕs modulus (E) and the poissonÕs ratio (m) of the materials involved are listed in Table 1. The VickerÕs hardness (HV) profiles produced by these surface treatments and of the original Ti alloy (Ti6Al4V) are measured using a Leitz Mini Hardness tester and are shown in Fig. 5. Ideally the fatigue strength of layered materials and the substrate should be obtained experimentally. Unfortunately, the measurements of fatigue strength of thin layered materials are, if not impossible, very difficult. Therefore, in this work, the fatigue strength of materials is approximated by assuming an endurance ratio (fatigue limit/ultimate tensile strength) of 0.45 according to Ref. [30]. The tensile strengths of materials are in turn approximated by HillÕs theory [31] which states that the yield strength of material is the VickersÕ hardness divided by three. The fatigue strength derived in this way corresponds to the fatigue limit. 3.2. The target racing car engine gears Three racing car engine gears were targeted: alternator, pump and cam gears (their transmission torque load gradually increases). The specifications of these three involute spur gears are listed in Table 2. The counterparts of all these three gears are steel gears. Thus the YoungÕs modulus and PoissonÕs ratio of steel are also listed in Table 1. The coefficient of friction (CoF) between steel and TiO2 layer is from 0.1 to 0.3, while it is about 0.5 for untreated and OD treated Ti6Al4V
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
453
[27–29]. The CoF (with steel) used for the gear design is listed in Table 1. The gear specifications shown in Table 2 are used to derive contact geometries of the mating gears. The worst gear contact conditions should be simulated, i.e. highest contact load (at the pitch point) and highest sliding speed (at the start and recess points of gear meshing) although these two contact conditions can not happen at the same time in gear contacts [32,33]. For involute spur gears, the transmission torque load can be calculated by the unit line load (thickness), f, applied at the pitch point: mzwf cos h ð6Þ 2 where Z is the number of teeth, T torque load (N m), w face width (mm), m module (mm), h pressure angle (), f the unit line load (N/mm) applied on the pitch point. Vice versa, given the torque load, the unit line load f can be also calculated by Eq. (6). The calculated unit line loads applied on the three gears are shown in Table 2. The contact radii, Ri (i stands for driver and driven), at pitch point can be calculated according to gear theory and are shown in Table 2. To use the two dimensional contact mechanics model, the gear contact geometry is converted to an equivalent contact between a cylinder and a half space in terms of an equal maximum surface pressure. The diameter, Dcyl, of the equivalent cylinder can be worked out by the following equation: T ¼
Dcyl ¼
2 1 Rdriver
þ
1
Fig. 6. SEM micrograph showing interfacial fatigue origin (arrowed) and beach marks, nominal Hertzian stress, 650 MPa, cycles, 4.6 · 105.
ð7Þ
Rdriven
The calculated Dcyl value is also listed in Table 2. 3.3. The predictive design of the surface engineered Ti gears Initial calculations show that all three racing car engine Ti6Al4V gears are safe without root bending fatigue. Since our objective is concentrated on the contact fatigue load bearing capacity, details of the root bending fatigue design will not be described here. Initial rolling-sliding contact experiments for TO treated Ti6Al4V were carried out on an Amsler Wear Test Machine Type A135 with wheel-on-wheel configuration (the diameters of the TO treated and the steel counterpart wheels were both 48 mm) and reported elsewhere [34]. The laboratory study indicated that the existence of TiO2 after TO treatment (and OD + TO treatment) effectively improve the adhesive wear resistance of the untreated Ti6Al4V and the failure mode changes to contact fatigue. This is clearly shown in Fig. 6 for a Hertzian stress of 650 MPa (200 N/mm unit line load), as indicated by the bench marks taken from specimen running 4.6 · 105 cycles. As a comparison,
Fig. 7. Typical appearance of worn untreated Ti6Al4V.
Fig. 7 shows a typical morphology of untreated Ti6Al4V surface which shows an adhesive wear morphology characterised by adhesive spots and deep grooves resulting from interaction with transferred material. The contact fatigue load bearing capacity for the TO treated wheel is simulated following the methodology described in Section 2. Fig. 8 shows the fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distribution as a function of depth. The calculated contact fatigue load bearing capacity is 174.8 N/mm. It can be seen that the test load (200 N/mm) is above the contact fatigue load bearing capacity and the contact fatigue failure is therefore observed. The contact fatigue design methodology described in Section 2 is again applied to the target gears using the material and contact geometry parameters derived earlier. For all simulations, the surface roughness profile shown in Fig. 2 (which is a representative measurement after shake-down of the TO treated gears) is used. The simulated results are shown in Fig. 9 for the critical
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1.8
FS and maximum VMS (GPa)
1.6
FS(TO)
1.4
max VMS(TO, 174.8 N/mm)
1.2 1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
50
Depth (microns) Fig. 8. The fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distributions along depth and the corresponding contact fatigue load bearing capacity for TO treated Amsler wheel.
maximum von Mises stress depth profile and the corresponding load bearing capacity (here represented by the critical torque load, which is converted from the calculated fc by Eq. (6)), comparing to the fatigue strength profile of the surface engineered structures. The results are summarised in Table 3 which compares the service loadings with the calculated load bearing capacity of the Ti gears for different surface treatments in terms of both torque load and unit line load. For the lightly loaded alternator gear (Fig. 9a), the model predicts that even the untreated Ti6Al4V should have a contact fatigue load bearing capacity (2.44 N m torque load or 12.8 N/mm unit line load) higher than the applied load (0.33 N m torque load or 2 N/mm unit line load). The TO treatment provides sufficient load bearing capacity (16.5 N m torque load or 86.4 N/mm unit line load) for this gear (Fig. 9a). The question now is whether the TO treatment provides sufficient load bearing capacity for the pump gear. As shown in Fig. 9(b) and Table 3, the TO treated pump gear is safe because it provides a load bearing capacity of 16.1 N m in torque load or 80.3
1.8
2
1.6
FS and maximum VMS (GPa)
FS and maximum VMS (GPa)
FS(Ti6Al4V) FS(TO) Max VMS(Ti6Al4V, 2.44 Nm)
1.4
Max VMS(TO, 16.5 Nm)
1.2 1 0.8 0.6 0.4
Max VMS(TO, 37.4 Nm)
1.4 1.2 1 0.8 0.6 0.4
0 0
10
20
30
40
0
50
Depth (microns)
(a)
10
20
30
40
50
Depth (microns)
(c)
1.8
2 FS(TO)
1.6
FS and maximum VMS (GPa)
FS and maximum VMS (GPa)
FS(TO)
1.6
0.2
0.2 0
VMS(Rough TO, 16.1 Nm)
1.4 1.2 1 0.8 0.6 0.4
FS(OD) FS(OD and TO) Max VMS(OD, 45.2 Nm) Max VMS(OD and TO, 70.0 Nm)
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0.2 0 (b)
1.8
10
20
30
Depth (microns)
40
0
50 (d)
10
20
30
40
50
Depth (microns)
Fig. 9. The fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distributions along depth and the corresponding contact fatigue load bearing capacity for (a) untreated and TO treated alternator gears, (b) TO treated pump gear, (c) TO treated cam gear, and (d) OD and OD + TO treated cam gear.
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
455
Table 3 The comparison of the service loadings and the calculated load bearing capacity of the Ti gears for different surface treatments in terms of both torque load and unit line load Gears
Alternator
Unit line load (N/mm) Applied Calculated capacity
2.0
25.6
Untreated TO OD TO + OD
12.8 86.4
80.3
Untreated TO OD TO + OD
2.44 16.5
Torque load (N m) Applied Calculated capacity
Pump
0.33
Cam 91.1–121.4 113.4 137.0 212.7
5.14
30–40
16.1
37.4 45.2 70.0
N/mm in unit line load, which is higher than the corresponding service loading (5.14 N m in torque load or 25.6 N/mm in unit line load). For the heavily loaded cam gear, following the similar comparison procedure, it is found that both the OD and OD + TO treatments are safe (Fig. 9d) but the TO treatment is unsafe (Fig. 9c). This indicates the necessity for a deeper hardened case and a duplex treatment for heavily loaded components.
Fig. 10. A TO treated alternator gear surviving from cam rig test.
4. Evaluation of the designed Ti gears A cam rig test was employed to evaluate the predictive design of the surface treated Ti gears. The cam rig comprises of a dummy crank-shaft driven by a 47 kW electric motor, complete heads and valve train, gear chest and full oil system, i.e. the top half of the engine. The cam rig is used for endurance running of new designs and batch testing of critical valve train components. The rig is programmed to simulate race conditions taken from a rigorous race lap (American Meadowlands circuit, peak rpm 14,750). The test results are summarised in Table 4. Table 4 The cam rig test results for Ti gears having different surface treatments
Untreated TO
OD OD + TO
Alternator gear
Pump gear
Cam gear
Failed shortly Survive for 2.5 race distance (500 miles, 3 · 106 cycles)
Survive for 1.25 race distance (250 miles, 1.5 · 106 cycles)
Failed
Failed shortly Survive for 1.5 race distance (300 miles, 1.8 · 106 cycles)
Fig. 11. A OD + TO treated cam gear surviving from the cam rig test.
The evaluation tests demonstrate the power of the predictive design summarised in Table 3. As predicted, the TO treatment is safe for the alternator and pump gears, but unsafe for the cam gear. A survival TO treated and tested alternator gear is shown in Fig. 10. Fig. 11 shows that the OD + TO treated cam gear which survived well from the cam rig test. These results agreed well with the predictions. However, exceptions are found for the untreated alternator (Fig. 12), and OD only treated cam gears (Fig. 13). For both cases, the design predicts their safe operations, but they failed early during the cam rig test. This is because without the existence of TiO2 film on the surface, the failure mode is, as mentioned earlier, adhesive wear (Fig. 7), thus the contact fatigue design concept cannot be applied in these two cases. The severe damage to the untreated alternator and OD treated cam gears under loads less than the predicted contact fatigue load bearing capacity can be attributed to the poor adhesive wear resistance which is determined by the electron structure of titanium alloys
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Fig. 12. Severe damage of cam rig tested, untreated alternator Drive Gear.
Fig. 13. Severe damage of cam rig tested, OD treated cam gear.
and the high coefficient of friction (0.5), rather than to the fatigue of the materials. It has been further confirmed by experimental results that when the OD layer was covered by a TiO2 layer through the duplex treatment of OD + TO, the potential of the increased capacity of the deepened OD layer against fatigue is realised.
5. Discussion An automatic search algorithm for the load bearing capacity of a surface engineering system has been incorporated into the existing Birmingham numerical model for the two-dimensional sliding contact of a cylinder with a real rough multilayered half space [15]. The model is then used in the design of surface engineered Ti gears against contact fatigue. For the three target racing car
engine gears (alternator, pump and cam gears), it has been shown that the TO treatment (which creates a relatively shallow surface hardened case) can provide sufficient load bearing capacity to the alternator and pump gears, while the relatively heavily loaded cam gear can only be supported by the duplex treatment of OD + TO (which creates a much deeper surface hardened case). The evaluation cam rig tests showed good agreement with the model prediction, thus demonstrating the power of the predictive design methodology employed. These results prove the importance of the design of duplex surface engineering systems [35] which involves the sequential application of two (or more) established surface technologies to produce a surface composite with combined properties which are unobtainable through any individual surface technology. It should be pointed out that the algorithms involved in surface engineering design, such as the automatic search of the contact fatigue load bearing capacity in this work, may produce a huge amount of computation load. Therefore, the faster contact mechanics modelling methodologies which were used have advantages over the popular finite element analysis (FEA) in this case. The model used in this work considers real rough surface by taking account of the direct measured surface roughness profile, and multilayered surface structure by a Fourier integral transform method. Apart from the fast computational speed, the advantages of this model over popular FEA method also include the easy incorporation of real surface roughness. The difficulties in fatigue design for multi-axial stress states encountered in contact fatigue have long been recognised in the fatigue community. Many multi-axial fatigue criteria have been developed and verified with experiments [36]. The fatigue failure modes are likely to be more complicated for a multilayered surface system. The outer surface treated layer usually consists of a hard but brittle material and may present normal cracking behaviour, while a shear/mixed cracking model is most likely to be more suitable for the ÔductileÕ substrate. Due to the difficulty in the experimental determination of the fatigue properties of thin film materials (which can be significantly different from the same bulk material), this paper has employed a quite simple fatigue criterion, the octahedral shear stress theory, for the contact fatigue design. In addition, fatigue strength has not been experimentally decided, but approximated from the hardness profile. Further studies should be carried out to investigate the fatigue failure modes of layered materials and choose more appropriate fatigue parameters and criteria for fatigue design.
6. Conclusion To-date, only the auto-sport industry has recognised the potential of surface engineered titanium automotive
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
gears. It is believed that, based on advanced surface engineering technologies and modelling-based predictive performance design, coupled with reduced material and manufacturing costs, the new millennium will see wide application of titanium in vehicles for the mass market. The agreement between the model prediction and the cam rig evaluation test demonstrates the power of the design methodology employed. It stresses the necessity for further systematic research into the fatigue behaviour of layered materials for the contact fatigue design under multi-axial stress conditions.
Acknowledgements The authors wish to acknowledge DTI/EPSRC for its support of the LINK project ÔAdvanced surface engineering of titanium alloy components (AdSurfEngTi, EPSRC/LINK GR/K70298), together with the financial support of BRITE Contract RI.1B.0151C(H), ÔSurface Engineering of Titanium ComponentsÕ. We would also like to extend many thanks to all the partners involved in the LINK project, especially Mercedes–Ilmor for their valuable contributions.
References [1] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1987. [2] T. Bell, Surface engineering, past, present and future, Surf. Eng. 6 (1990) 31–40. [3] P.K. Gupta, J.A. Walowit, Contact stressed between an elastic cylinder and a layered elastic solid, J. Lubricat. Technol., Trans. ASME (April) (1974) 251–256. [4] S.J. Cole, R.S. Sayles, A numerical model for the contact of layered elastic bodies with real rough surfaces, J. Tribol., Trans. ASME 114 (1992) 334–340. [5] J.O. Smith, C.K. Liu, Stresses due to tangential and normal loads on the elastic solid with application to some contact stress problems, J. Appl. Mech., Trans. ASME 75 (1953) 157–166. [6] J.A. Greenwood, J.B.P. Williamson, The contact of nominally flat surfaces, Proc. Royal Soc., London A295 (1966) 300–319. [7] G.G. Adams, M. Nosonovsky, Contact modeling-forces, Tribol. Int. 33 (2000) 431–442. [8] R.S. Sayles, Basic principles of rough surface contact analysis using numerical methods, Tribol. Int. 29 (1996) 639–650. [9] M.N. Webster, R.S. Sayles, A numerical model for the elastic frictionless contact of real rough surfaces, J. Tribol., Trans. ASME 108 (1986) 314–320. [10] J.R. Barber, M. Ciavarella, Contact mechanics, Int. J. Solids Struct. 37 (2000) 29–43. [11] Q. Tao, G.P. Lee, S.P. Lim, Contact mechanics of surfaces with various models of roughness descriptions, Wear 249 (2001) 539– 545.
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[12] M. Borri-Brunetto, A. Carpinteri, B. Chiaia, Lacunarity of the contact domain between elastic bodies with rough boundaries, in: G. Frantziskonis (Ed.), PROBAMAT-21st Century: Probabilities and Materials, Kluwer, Dordrecht, 1998, pp. 45–64. [13] A. Majumdar, B. Bhushan, Characterization and modelling of surface roughness and contact mechanics, in: Handbook of Micro/Nano Tribology, CRC Press, New York, 1995, pp. 109– 165. [14] K. Mao, T. Bell, Y. Sun, Effect of sliding friction on contact stresses for multi-layered elastic bodies with rough surfaces, J. Tribol., Trans. ASME 119 (1997) 476–480. [15] K. Mao, Y. Sun, T. Bell, A numerical model for the dry sliding contact of layered elastic bodies with rough surfaces, Tribol. Trans. STLE 39 (1996) 416–421. [16] Y. Sun, T. Bell, S. Zheng, Finite element analysis of the critical ratio of coating thickness to indentation depth for coating property measurements by nanoindentation, Thin Solid Films 258 (1995) 198–204. [17] K. Komvopoulos, Elastic–plastic finite element analysis of indented layered media, J. Tribol., Trans. ASME 111 (1989) 430–439. [18] N. Schwarzer, Coating design due to analytical modelling of mechanical contact problems on multilayer systems, Surf. Coat. Technol. 133–134 (2000) 397–402. [19] N. Schwarzer, F. Richter, G. Hecht, The elastic field in a coated half-space under Hertzian pressure distribution, Surf. Coat. Technol. 114 (1999) 292–304. [20] N. Schwarzer, Arbitrary load distribution on a layered half space, J. Tribol., Trans. ASME 122 (2000) 672–681. [21] W. Peng, B. Bhushan, J. Tribol., Trans. ASME 124 (2002) 46–61. [22] I.N. Sneddon, Fourier Transforms, McGraw-Hill, 1951. [23] H. Dong, A. Bloyce, P.M. Morton, T. Bell, Methods of case hardening of titanium alloys, European Patent EP1000180, 17 November 2001. [24] T. Bell, Towards a universal surface engineering road map, Surf. Eng. 16 (2000) 89–90. [25] J.W. He et al., Interfacial fatigue limit as measure of cyclic bonding strength of hard coatings, Surf. Eng. 12 (1996) 49–54. [26] H. Dong, T. Bell, A. Mynott, Sports Engng. 2 (1999) 213–219. [27] H. Dong, A. Bloyce, P.M. Morton, T. Bell, Surface oxidation of a titanium or titanium alloy article, Patent number WO9802595, 22 January 1998. [28] H. Dong, X.Y. Li, Oxygen boost diffusion for the deep-case hardening of titanium alloys, Mater. Sci. Eng. A 280 (2000) 303– 310. [29] H. Dong, A. Bloyce, T. Bell, Oxygen thermochemical treatment combining with DLC coating for enhanced load-bearing capacity of Ti6Al4V, Surf. Eng. 14 (1998) 505–512. [30] R. Boyer, G. Welsch, E.W. Collings, Materials Properties Handbook: Titanium Alloys, ASM International, 1998. [31] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, 1967. [32] R.J. Drago, Fundamentals of Gear Design, Butterworths, Boston, 1988. [33] D.W. Dudley, Handbook of Practical Gear Design, McGraw-Hill, New-York, 1984. [34] H. Dong, T. Bell, Enhanced wear resistance of titanium surfaces by a new thermal oxidation treatment, Wear 238 (2000) 131–137. [35] T. Bell, H. Dong, Y. Sun, Realising the potential of duplex surface engineering, Tribol. Int. 31 (1998) 127–137. [36] B.-R. You, S.-B. Lee, Int J. Fatigue 18 (1996) 235–244.
Model-based contact fatigue design of surface engineered titanium gears J. Luo, H. Dong *, T. Bell Department of Metallurgy and Materials, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 7 January 2005; accepted 15 March 2005
Abstract In order to fully realise the potential of advanced surface engineering in the automotive sector, a contact mechanics model has been developed based on modern theories of multi-layered surface contact, taking into account the multi-layered structure, real surface roughness and friction effects. With this model, the performance of surface engineered titanium automotive gears can be successfully predicted thus making possible the design of optimised surface engineering systems to meet particular engineering demands within the shortest possible time and with least cost. This paper presents the predictive contact fatigue design of Ti6Al4V based gears in racing car engines using the developed contact mechanics model for the real rough multilayered surface systems. Evaluation of the effectiveness of the predictive design is carried out through cam rig tests of real gears, simulating the in-service loading conditions. Good agreement between theoretical prediction and service performance has been achieved. By way of example, the present paper demonstrates a number of major steps towards designing dynamically loaded titanium automotive gears. 2005 Elsevier B.V. All rights reserved. PACS: 68.35.Ct; 82.20.Wt; 81.65.Mq; 81.40.Pq Keywords: Design; Modelling; Titanium gear; Surface engineering; Contact fatigue; Oxidation
1. Introduction Contact mechanics is concerned with the stresses and deformations which arise when the surfaces of two solid bodies are brought into contact [1]. Contact parameters such as pressure, stress and deformation are of great importance to the study of the mechanisms of friction, wear, mixed lubrication, load bearing capacity and fatigue. Surface engineering has been proved to be one of the most effective ways to improve the tribological performance of engineering components [2]. The computation of stresses in layered surfaces is of great analytical and practical importance in the design of surface engi*
Corresponding author. Tel.: +44 121 414 5197; fax: +44 121 414 7373. E-mail address: [email protected] (H. Dong). 0927-0256/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.03.013
neered systems where friction, wear and contact fatigue characteristics can be improved significantly by modifying the surfaces to generate hard wear resistant materials, such as nitrides, on to the surface. In view of the difference in the elastic and plastic properties between the surface layers and substrate materials, traditional contact mechanics approaches based on homogeneous materials can not be directly applied to find the solution for the design of layered surfaces. In the past decade, significant progress has been made in the study of the contact behaviour of layered surfaces. The underlying aims of such research have been to understand the basic mechanisms involved in various tribological processes of engineered surfaces, and to assist with the selection of an optimum coating/substrate combination system for a specific application and in the design of engineering components with modified surfaces.
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The first significant elastic contact analysis of surfaces was produced by Hertz in 1881 [1] when he presented a paper ‘‘on the contact of elastic pressure and subsurface stress distributions’’ which arise in a non-conforming contact. This classic work has formed the basis of the present theoretical knowledge of static and rolling contact mechanics. Hertzian theory is based on the following assumptions. (a) The bodies may be considered as elastic half-spaces for the purpose of calculating the local deformations. This implies firstly that the contacting bodies are elasticity homogeneous, that is, the elastic properties are constant throughout each body, and secondly that the significant dimensions of the contact area are small compared with the dimensions of each body and with the relative radii of curvature of the surfaces, and thirdly that strains are small and the material exhibits perfect linearly elastic behaviour. (b) The contacting surfaces are perfectly smooth and non-conforming. On a small scale this implies that there are no surface irregularities which could cause discontinuous contact, and on a large scale it means that the surfaces are continuous up to their second derivative in the contact region. (c) The surfaces are assumed to be frictionless so that only a normal pressure is transmitted between them. Hertzian theory shows that the highest maximum shear does not occur at the surface, but at certain distance below the surface, indicating that under Hertzian contact conditions, failure of materials is initiated below the surface where the highest maximum shear stress occurs. These assumptions are considered to be inappropriate when investigating the contact mechanics of layered surface [3,4]. Progress in the field of contact mechanics over the last few decades has been associated largely with the removal of Hertzian restrictions. Smith and Liu addressed [5] friction effects on subsurface stress distributions. For the subsurface stress distributions, if sliding is present and the coefficient of friction is not zero, then a tangential traction is also present on the surface, in addition to the normal loading. This tangential traction induces subsurface stresses in addition to those due to the normal load. Accordingly, it can be seen that the maximum shear stress is moved towards the surface with the introduction of friction. This movement will be increased as the coefficient of friction is increased and when the coefficient of friction reaches a critical value (about 0.3), the maximum shear stress will reach the surface. This has a significant effect on contact fatigue. Since real surfaces are rough on the microscopic scale, attempts have been made to remove the smooth surface assumption of the Hertzian theory. There are mainly three methods to treat rough surfaces. First, Greenwood and Williamson [6] assume some form for the statistical properties of the surface, combined with an asperity shape that allows the calculation of individual asperity load-displacement laws [7]. These models
have yielded important results about the average properties of the contact of rough surfaces, but information about the real pressure distribution and the deformed shape is lost because of the statistical nature of the approach. Second, the ÔdirectÕ method, as reviewed by Sayles [8], uses roughness data directly recorded from a stylus measuring instrument, and employs a numeric technique in the contact analysis. A two dimensional computer model for dry, frictionless contact of real engineered surfaces (i.e. non-perfectly smooth) has been presented by Webster and Sayles [9]. Asperity plasticity is also considered in this work and a significant effect of surface topography on the pressure distribution identified. Depending on the topographic parameters, the maximum pressure in rough contact can be many times higher than that in ÔideallyÕ smooth contact, and the pressure peaks arise from the highest asperities. Third, the most recent advances in the contact mechanics of a rough surface recognise the multi-scale geometry of the rough surface [7,10]. A fractal geometry characterisation of the multi-scale topography can be used to simulate the rough surface profile allowing it to be incorporated into contact mechanics models [11,12]. For an extensive review of the multi-scale features of rough surfaces, readers are referred to Majumdar and Bhushan [13] for both experimental and theoretical aspects of the subject. In the case of layered surface contact, significant progress has been achieved in the last decade. There are four major methods of considering layered surface contacts, viz., Fourier integral transform [14,15], finite/ boundary element method [16,17], the method of image charges [18–20] and a combination of the variational principle and fast Fourier integral transformation [21]. The Fourier integral transform is based on SneddonÕs theory [22] which solves the contact problems by using the Fourier integral transform of the Airy stress function. Of the four approaches for the multilayered contact mechanics analysis, perhaps the finite/boundary method is the most popular due to its flexibility in treating complicated three dimensional (3D) component geometries and the ready availability of commercial finite/boundary analysis packages. However, for rough surface contact problems with many asperities of arbitrary shape, the requirement of a large number of mesh elements makes the finite/boundary element approach unfeasible. Further, it suffers from a slow calculation speed. Although the method of image charges is advantageous in terms of its readiness for 3D analysis and fast calculation speed, it lacks the ability to cope with rough surfaces. By comparison, the Fourier integral transform and variational principle methods are both easy to incorporate the rough surface effects and provide fast calculation speed. The latter is also 3D based, but as a very new method, it can only treat single-layered surface system
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
0.1
0.05
Microns
at this stage. In contrast, Fourier integral transform method is more appropriate for 2D problems and has been well established for at least two layered (exclusive of substrate) surface systems [14,15]. The Fourier integral transform method is most suitable therefore for modelling the line contact problem between gears. In the following section, the Fourier integral transform method for a multilayered surface structure and directly measured rough surface is extended to include an automatic algorithm to search the load bearing capacity of gears against contact fatigue and then is used for the design of surface engineered Ti alloy gears in the sports car industry.
449
0
-0.05
-0.1 0
500
1000
1500
2000
2500
3000
Microns
2. Contact stresses and modelling
Fig. 2. Digitised surface profile, sampling interval 1.0 lm, 3000 points.
2.1. The Birmingham multilayered real rough surface contact mechanics model Fig. 1 shows the Birmingham model [14,15] where an elastic half-space, coated with n elastic layers, makes contact with an elastic cylinder. The layers are completely bonded to each other and to the elastic half-space they cover. The digitised surface profile as typically shown in Fig. 2 was obtained and is taken as the real surface roughness profile (in practice, it is better to take the roughness profile after shake-down of contact components for the calculation, i.e. after a period of run-in services perhaps in a lower loading condition than normal operation of machinery). GreenÕs function for a unit normal load and a unit tangential load for the generalised plane strain problem is then derived. Using the co-ordinate system and notation for the n + 1 component system shown in Fig. 1, the stresses rx, ry and sxy, in the x and y p directions respectively, in any layer ffiffiffiffiffiffiffi i are given by (j ¼ 1)
q(y) x1
2
x2
3
n n+1
y 1 y
1 2p
rxy i ¼
1 2p
Z
1
x2 Gi ejxy i dx
ð1Þ
1 1 2
1 Z 1
d Gi jxy i e dx d x2i jx
1
dGi jxy i e dx d xi
ð2Þ ð3Þ
where i is number of layers; x is Fourier transform integrating variable; Gi is the Fourier transform of the Airy stress function, and it is given by the following expression Gi ¼ ðAi þ Bi xi Þ ejxjxi þ ðC i þ Di xi Þ ejxjxi
ð4Þ
where the constants Ai, Bi, Ci, Di (the stress function coefficients) are general functions of x, and are determined from the boundary conditions. For the n layered structure shown in Fig. 1, the boundary conditions are as follows:
2
h1 h
2
x4
y
h
xn
y
hn
xn+1
ry i ¼
1 2p Z
• At the surface the normal and shear stresses must equal the applied pressures. • At every interface in the system, the normal and shear stresses and the normal and transverse deflections on each side of the interface must be equal. • The stresses fall to zero at a large distance from the load.
p(y) 1
rx i ¼
4 n
3
yn+1
Fig. 1. Co-ordinates and notations for multi-layered elastic solid.
Through examination of these boundary conditions, expressions for the stress function coefficients may be obtained [15]. Once the stress function coefficients are known, Gi is known and, by performing integrals on Gi and its derivatives, the stress and deflection at any point on and under the contact zone may be found. For any given rigid body displacement of the indentation cylinder, an iteration procedure is used to find the surface contact points and the pressure distribution. A detailed description of the solution procedure is given in the work of Mao and coworkers [14,15]. A typical
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of 0.5 lm for smooth contact. Whilst in smooth contact the maximum contact stress locates some distance below the surface, in rough contact the maximum stress appears at the surface arising from the highest asperities, which can be many times higher than that in smooth contact. The stress distribution varies with coefficient of friction, surface roughness, layer thickness as well as material properties of layers and substrate, which largely depends on the design of the surface engineering system. 2.2. Design of the surface engineered systems Fig. 3. Subsurface stress distribution (von Mises, GPa) for OD + TO treated Ti6Al4V alloy contacting with steel (friction coefficient l = 0.3) under an unit line load of 180 N/mm.
calculated von Mises (VMS) stress distribution for a surface engineered titanium alloy using a novel technique (i.e. 2 lm TiO2 and 200 lm oxygen diffusion zone on Ti6Al4V substrate) [23] is presented in Fig. 3 using the surface roughness profile shown in Fig. 2 and contact radius of 13.38 mm (cam gear contact condition, see Tables 1 and 2 in Section 3 for the description and calculation input parameters). It can be clearly seen that the stress distributions for a rough surface can be significantly different from the well established smooth case. The maximum von Mises stress for rough contact is 1.65 GPa at the surface, while it is 1.17 GPa at a depth Table 1 CoF, YoungÕs modulus (E), PoissonÕs ratio (m) of layers and substrate Materials
E (GPa)
m
CoF (with steel)
Ti6Al4V TO (TiO2) OD Steel
112 250 114 210
0.25 0.2 0.25 0.3
0.5 0.3 0.5 –
A recent succinct definition [24] of Surface Engineering states that Ôsurface engineering involves the use of traditional and innovative surface technologies in the design of a surface and substrate together to form a functionally graded system, which results in the cost effective performance enhancement of materials and componentsÕ. One of the key elements in the definition is design. In many cases, surface engineering design is conducted in the attempt to make sure that the designed surface system can bear an applied load, thus the term load bearing capacity. However, load bearing capacity is a concept which is loosely defined. Clear definition of the concept is difficult perhaps because, at least in part, load bearing capacity is closely related to the failure mode of a specific surface engineering system. In many tribological situations, failure of a hard coatingsubstrate system is seldom caused by conventional wear or yielding but rather by a fatigue failure process [25]. Therefore, the load bearing capacity of a surface system should be defined as the capacity to resist a specific kind of failure mode. Design of multi-layered surface systems therefore refers to the design activities that are carried out to choose the optimal layered surface structures for a given service condition and environment through the application of
Table 2 The gear specifications and the derived design parameters Parameters
Z T (N m) f (N/mm) w (mm) x (r.p.m.) Ri (mm) m (mm) h () Lubrication Dcyl (mm)
Alternator
Pump
Cam
Driver (Ti6Al4V)
Driven (steel)
Driver (steel)
Driven (Ti6Al4V)
Driver (steel)
Driven (Ti6Al4V)
32 0.33 2 6 6562 9.93
28 0.29 1.5 8 7500 8.69
17 2.43 24.8 5.8 17420 6.15
36 5.14 25.6 5.8 8250 13.03
38 30–40 86.3–115.0 9.2 8250 13.76
36 30–40 91.1–121.4 9.2 8708 13.03
1.81429 20 Oil splash 9.27
2.1167 20 Oil splash 8.36
2.1167 20 Oil splash 13.38
Note: Z: number of teeth; T: torque load; f: unit line load applied on pitch point; w: face width; x: rotation speed; Ri: the contact radii at pitch point, i stands for driver and driven; m: module; h: pressure angle; Dcyl: equivalent cylinder diameter when convert the gear contact as cylinder-half space contact.
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contact mechanics and material strength theories in the context of technical availability of surface treatment processes and their cost effectiveness, using material properties of the layered materials and their interfaces as input parameters. In the simplest case, surface engineering design refers to the prediction of the load bearing capacity against a certain failure mode for a given surface engineering system under a specific contact loading condition. The numerical model for the two-dimensional sliding contact of a cylinder with a real rough multilayered half space is extended in this paper to carry out surface engineered gear design against contact fatigue. The residual stress effect on the stress distributions can also be considered. In consideration of the multi-axial stress state under the contact zone, the octahedral shear stress theory, also termed von Mises theory, is used as the contact fatigue failure criterion: DrVMS P rFS
ð5Þ
where DrVMS is the cyclic range of the von Mises stress (GPa) and rSF is the fatigue strength (GPa). The minimum of the von Mises stress is taken as zero for the rolling-sliding contact of a gear. Thus for any depth beneath the surface, DrVMS equals the maximum von Mises stress at this depth, rVMS-max. The fatigue strength is changeable along the depth and is obtained by direct or indirect experiments (e.g. derived by a hardness depth profile). The critical load fc (the load bearing capacity,
451
represented by the unit line load along cylinder thickness, N/mm) is determined when the maximum von Mises stress at any depth overtakes the fatigue strength at that depth. The search procedure for fc is described in Fig. 4. An initial rigid displacement increment, Dd0, is applied to the indentation cylinder. The contact points and the surface pressure distribution are determined through an iteration procedure of the contact mechanics model. Then the stress field within the layers and substrate under contact can be calculated and superposed with the residual stresses if required. The fatigue strength depth profile can be determined (e.g., derived from hardness depth profile) and compared with the calculated maximum von Mises stress depth profile (rVMS-max = DrVMS). If the maximum von Mises stresses are lower than the fatigue strengths at all depth positions, a fixed displacement increment, Ddf, is added to the cylinder and the surface pressure and the stress fields under the contact zone are re-calculated and compared with the fatigue strength profile. This process is repeated until the calculated maximum von Mises stress reaches the fatigue strength at any depth. Then a further check is made to make sure that the relative difference between the fatigue strength and the maximum von Mises stress is within a precision requirement (e.g. 1%). If the difference is too large, an adjusted displacement increment, Dda, is calculated according to the difference and added to the cylinder. Again the surface pressure and stress field are re-calculated until the precision requirement is
Fig. 4. Flow chart for load bearing capacity prediction against contact fatigue.
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met. The final step sums up the surface pressure distribution to obtain the load bearing capacity, fc, of the surface system against contact fatigue. The real rough multilayered contact mechanics model and the contact design algorithm are programmed into a software package using the Fortran language.
Substituting Ti components for steel in industrial sectors, such as racing car engines, has been shown to be one of the most successful approaches to achieve high performance [26]. This is because titanium and its alloys possess the highest strength-to-weight ratio of all metallic materials. However, the reputation of titanium alloys for poor tribological behaviour has been the main barrier to their successful application. The poor tribological behaviour of the titanium is attributed to its inherent electron structure [26]. This problem may be overcome only by changing the nature of the surface. Some novel surface engineering technologies have been developed recently, which provide the necessary basis for realising the full potential of titanium alloys. One class of surface treatments of Ti alloys is based on oxygen thermal processing: i.e., ceramic conversion technology through thermal oxidation (TO) [27], oxygen boost diffusion technology (OD) [28] and the duplex treatment of OD in combination with TO [23,29]. The TO treatment creates an adherent thin rutile oxidation film (TiO2, about 2 lm thick) and a thin (about 10 lm) oxygen solution-hardened diffusion zone. The OD treatment creates a deep (about 200 lm) oxygen solution-hardened diffusion zone without the surface oxide layer and thus confers the high load bearing capacity needed for heavily loaded titanium components. However, it is found that the OD treatment cannot effectively improve the tribological properties of Ti alloys due to the fact that the metallurgical characteristics are hardly changed by the OD technology. Thus the duplex treatment of OD plus TO is necessary. The duplex treatment of OD + TO creates a similar layered surface structure as TO technology, but produces an oxygen solution-hardened diffusion zone beneath it with a depth of hardening similar in size to that produced by the OD treatment. In this paper, these three surface treatments are applied to Ti6Al4V to produce surface treated Ti alloy gears to be used in racing car engines. 3.1. Input mechanical parameters for the design To carry out the multi-layered contact behaviour simulations, the mechanical properties of the layers and substrate have to be determined first. The measurement of the surface hardness (H) and YoungÕs modulus (E) values of TiO2 and the oxygen diffusion hardened
Ti6Al4V
1100
TO
1000
OD
900
HV0.05
3. Case study—the design of surface engineered Ti gears
1200
OD + TO
800 700 600 500 400 300 200 0
50
100
150
200
250
300
Depth (microns) Fig. 5. Hardness (HV0.05) profiles for TO, OD, and OD + TO treated Ti6Al4V.
(OD) layers was conducted using a NanoTest 600 machine, manufactured by Micro Materials Ltd., UK, with the typical resolutions of 0.1 nm in displacement and 100 nN in force. The hardness of the TiO2 film for TO and OD + TO treatments was measured as 11 GPa (roughly corresponds to 1100 HV) and is treated as a fixed value. The YoungÕs modulus (E) and the poissonÕs ratio (m) of the materials involved are listed in Table 1. The VickerÕs hardness (HV) profiles produced by these surface treatments and of the original Ti alloy (Ti6Al4V) are measured using a Leitz Mini Hardness tester and are shown in Fig. 5. Ideally the fatigue strength of layered materials and the substrate should be obtained experimentally. Unfortunately, the measurements of fatigue strength of thin layered materials are, if not impossible, very difficult. Therefore, in this work, the fatigue strength of materials is approximated by assuming an endurance ratio (fatigue limit/ultimate tensile strength) of 0.45 according to Ref. [30]. The tensile strengths of materials are in turn approximated by HillÕs theory [31] which states that the yield strength of material is the VickersÕ hardness divided by three. The fatigue strength derived in this way corresponds to the fatigue limit. 3.2. The target racing car engine gears Three racing car engine gears were targeted: alternator, pump and cam gears (their transmission torque load gradually increases). The specifications of these three involute spur gears are listed in Table 2. The counterparts of all these three gears are steel gears. Thus the YoungÕs modulus and PoissonÕs ratio of steel are also listed in Table 1. The coefficient of friction (CoF) between steel and TiO2 layer is from 0.1 to 0.3, while it is about 0.5 for untreated and OD treated Ti6Al4V
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453
[27–29]. The CoF (with steel) used for the gear design is listed in Table 1. The gear specifications shown in Table 2 are used to derive contact geometries of the mating gears. The worst gear contact conditions should be simulated, i.e. highest contact load (at the pitch point) and highest sliding speed (at the start and recess points of gear meshing) although these two contact conditions can not happen at the same time in gear contacts [32,33]. For involute spur gears, the transmission torque load can be calculated by the unit line load (thickness), f, applied at the pitch point: mzwf cos h ð6Þ 2 where Z is the number of teeth, T torque load (N m), w face width (mm), m module (mm), h pressure angle (), f the unit line load (N/mm) applied on the pitch point. Vice versa, given the torque load, the unit line load f can be also calculated by Eq. (6). The calculated unit line loads applied on the three gears are shown in Table 2. The contact radii, Ri (i stands for driver and driven), at pitch point can be calculated according to gear theory and are shown in Table 2. To use the two dimensional contact mechanics model, the gear contact geometry is converted to an equivalent contact between a cylinder and a half space in terms of an equal maximum surface pressure. The diameter, Dcyl, of the equivalent cylinder can be worked out by the following equation: T ¼
Dcyl ¼
2 1 Rdriver
þ
1
Fig. 6. SEM micrograph showing interfacial fatigue origin (arrowed) and beach marks, nominal Hertzian stress, 650 MPa, cycles, 4.6 · 105.
ð7Þ
Rdriven
The calculated Dcyl value is also listed in Table 2. 3.3. The predictive design of the surface engineered Ti gears Initial calculations show that all three racing car engine Ti6Al4V gears are safe without root bending fatigue. Since our objective is concentrated on the contact fatigue load bearing capacity, details of the root bending fatigue design will not be described here. Initial rolling-sliding contact experiments for TO treated Ti6Al4V were carried out on an Amsler Wear Test Machine Type A135 with wheel-on-wheel configuration (the diameters of the TO treated and the steel counterpart wheels were both 48 mm) and reported elsewhere [34]. The laboratory study indicated that the existence of TiO2 after TO treatment (and OD + TO treatment) effectively improve the adhesive wear resistance of the untreated Ti6Al4V and the failure mode changes to contact fatigue. This is clearly shown in Fig. 6 for a Hertzian stress of 650 MPa (200 N/mm unit line load), as indicated by the bench marks taken from specimen running 4.6 · 105 cycles. As a comparison,
Fig. 7. Typical appearance of worn untreated Ti6Al4V.
Fig. 7 shows a typical morphology of untreated Ti6Al4V surface which shows an adhesive wear morphology characterised by adhesive spots and deep grooves resulting from interaction with transferred material. The contact fatigue load bearing capacity for the TO treated wheel is simulated following the methodology described in Section 2. Fig. 8 shows the fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distribution as a function of depth. The calculated contact fatigue load bearing capacity is 174.8 N/mm. It can be seen that the test load (200 N/mm) is above the contact fatigue load bearing capacity and the contact fatigue failure is therefore observed. The contact fatigue design methodology described in Section 2 is again applied to the target gears using the material and contact geometry parameters derived earlier. For all simulations, the surface roughness profile shown in Fig. 2 (which is a representative measurement after shake-down of the TO treated gears) is used. The simulated results are shown in Fig. 9 for the critical
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1.8
FS and maximum VMS (GPa)
1.6
FS(TO)
1.4
max VMS(TO, 174.8 N/mm)
1.2 1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
50
Depth (microns) Fig. 8. The fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distributions along depth and the corresponding contact fatigue load bearing capacity for TO treated Amsler wheel.
maximum von Mises stress depth profile and the corresponding load bearing capacity (here represented by the critical torque load, which is converted from the calculated fc by Eq. (6)), comparing to the fatigue strength profile of the surface engineered structures. The results are summarised in Table 3 which compares the service loadings with the calculated load bearing capacity of the Ti gears for different surface treatments in terms of both torque load and unit line load. For the lightly loaded alternator gear (Fig. 9a), the model predicts that even the untreated Ti6Al4V should have a contact fatigue load bearing capacity (2.44 N m torque load or 12.8 N/mm unit line load) higher than the applied load (0.33 N m torque load or 2 N/mm unit line load). The TO treatment provides sufficient load bearing capacity (16.5 N m torque load or 86.4 N/mm unit line load) for this gear (Fig. 9a). The question now is whether the TO treatment provides sufficient load bearing capacity for the pump gear. As shown in Fig. 9(b) and Table 3, the TO treated pump gear is safe because it provides a load bearing capacity of 16.1 N m in torque load or 80.3
1.8
2
1.6
FS and maximum VMS (GPa)
FS and maximum VMS (GPa)
FS(Ti6Al4V) FS(TO) Max VMS(Ti6Al4V, 2.44 Nm)
1.4
Max VMS(TO, 16.5 Nm)
1.2 1 0.8 0.6 0.4
Max VMS(TO, 37.4 Nm)
1.4 1.2 1 0.8 0.6 0.4
0 0
10
20
30
40
0
50
Depth (microns)
(a)
10
20
30
40
50
Depth (microns)
(c)
1.8
2 FS(TO)
1.6
FS and maximum VMS (GPa)
FS and maximum VMS (GPa)
FS(TO)
1.6
0.2
0.2 0
VMS(Rough TO, 16.1 Nm)
1.4 1.2 1 0.8 0.6 0.4
FS(OD) FS(OD and TO) Max VMS(OD, 45.2 Nm) Max VMS(OD and TO, 70.0 Nm)
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
0.2 0 (b)
1.8
10
20
30
Depth (microns)
40
0
50 (d)
10
20
30
40
50
Depth (microns)
Fig. 9. The fatigue strength (FS) profile and the critical maximum von Mises (VMS) stress distributions along depth and the corresponding contact fatigue load bearing capacity for (a) untreated and TO treated alternator gears, (b) TO treated pump gear, (c) TO treated cam gear, and (d) OD and OD + TO treated cam gear.
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
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Table 3 The comparison of the service loadings and the calculated load bearing capacity of the Ti gears for different surface treatments in terms of both torque load and unit line load Gears
Alternator
Unit line load (N/mm) Applied Calculated capacity
2.0
25.6
Untreated TO OD TO + OD
12.8 86.4
80.3
Untreated TO OD TO + OD
2.44 16.5
Torque load (N m) Applied Calculated capacity
Pump
0.33
Cam 91.1–121.4 113.4 137.0 212.7
5.14
30–40
16.1
37.4 45.2 70.0
N/mm in unit line load, which is higher than the corresponding service loading (5.14 N m in torque load or 25.6 N/mm in unit line load). For the heavily loaded cam gear, following the similar comparison procedure, it is found that both the OD and OD + TO treatments are safe (Fig. 9d) but the TO treatment is unsafe (Fig. 9c). This indicates the necessity for a deeper hardened case and a duplex treatment for heavily loaded components.
Fig. 10. A TO treated alternator gear surviving from cam rig test.
4. Evaluation of the designed Ti gears A cam rig test was employed to evaluate the predictive design of the surface treated Ti gears. The cam rig comprises of a dummy crank-shaft driven by a 47 kW electric motor, complete heads and valve train, gear chest and full oil system, i.e. the top half of the engine. The cam rig is used for endurance running of new designs and batch testing of critical valve train components. The rig is programmed to simulate race conditions taken from a rigorous race lap (American Meadowlands circuit, peak rpm 14,750). The test results are summarised in Table 4. Table 4 The cam rig test results for Ti gears having different surface treatments
Untreated TO
OD OD + TO
Alternator gear
Pump gear
Cam gear
Failed shortly Survive for 2.5 race distance (500 miles, 3 · 106 cycles)
Survive for 1.25 race distance (250 miles, 1.5 · 106 cycles)
Failed
Failed shortly Survive for 1.5 race distance (300 miles, 1.8 · 106 cycles)
Fig. 11. A OD + TO treated cam gear surviving from the cam rig test.
The evaluation tests demonstrate the power of the predictive design summarised in Table 3. As predicted, the TO treatment is safe for the alternator and pump gears, but unsafe for the cam gear. A survival TO treated and tested alternator gear is shown in Fig. 10. Fig. 11 shows that the OD + TO treated cam gear which survived well from the cam rig test. These results agreed well with the predictions. However, exceptions are found for the untreated alternator (Fig. 12), and OD only treated cam gears (Fig. 13). For both cases, the design predicts their safe operations, but they failed early during the cam rig test. This is because without the existence of TiO2 film on the surface, the failure mode is, as mentioned earlier, adhesive wear (Fig. 7), thus the contact fatigue design concept cannot be applied in these two cases. The severe damage to the untreated alternator and OD treated cam gears under loads less than the predicted contact fatigue load bearing capacity can be attributed to the poor adhesive wear resistance which is determined by the electron structure of titanium alloys
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Fig. 12. Severe damage of cam rig tested, untreated alternator Drive Gear.
Fig. 13. Severe damage of cam rig tested, OD treated cam gear.
and the high coefficient of friction (0.5), rather than to the fatigue of the materials. It has been further confirmed by experimental results that when the OD layer was covered by a TiO2 layer through the duplex treatment of OD + TO, the potential of the increased capacity of the deepened OD layer against fatigue is realised.
5. Discussion An automatic search algorithm for the load bearing capacity of a surface engineering system has been incorporated into the existing Birmingham numerical model for the two-dimensional sliding contact of a cylinder with a real rough multilayered half space [15]. The model is then used in the design of surface engineered Ti gears against contact fatigue. For the three target racing car
engine gears (alternator, pump and cam gears), it has been shown that the TO treatment (which creates a relatively shallow surface hardened case) can provide sufficient load bearing capacity to the alternator and pump gears, while the relatively heavily loaded cam gear can only be supported by the duplex treatment of OD + TO (which creates a much deeper surface hardened case). The evaluation cam rig tests showed good agreement with the model prediction, thus demonstrating the power of the predictive design methodology employed. These results prove the importance of the design of duplex surface engineering systems [35] which involves the sequential application of two (or more) established surface technologies to produce a surface composite with combined properties which are unobtainable through any individual surface technology. It should be pointed out that the algorithms involved in surface engineering design, such as the automatic search of the contact fatigue load bearing capacity in this work, may produce a huge amount of computation load. Therefore, the faster contact mechanics modelling methodologies which were used have advantages over the popular finite element analysis (FEA) in this case. The model used in this work considers real rough surface by taking account of the direct measured surface roughness profile, and multilayered surface structure by a Fourier integral transform method. Apart from the fast computational speed, the advantages of this model over popular FEA method also include the easy incorporation of real surface roughness. The difficulties in fatigue design for multi-axial stress states encountered in contact fatigue have long been recognised in the fatigue community. Many multi-axial fatigue criteria have been developed and verified with experiments [36]. The fatigue failure modes are likely to be more complicated for a multilayered surface system. The outer surface treated layer usually consists of a hard but brittle material and may present normal cracking behaviour, while a shear/mixed cracking model is most likely to be more suitable for the ÔductileÕ substrate. Due to the difficulty in the experimental determination of the fatigue properties of thin film materials (which can be significantly different from the same bulk material), this paper has employed a quite simple fatigue criterion, the octahedral shear stress theory, for the contact fatigue design. In addition, fatigue strength has not been experimentally decided, but approximated from the hardness profile. Further studies should be carried out to investigate the fatigue failure modes of layered materials and choose more appropriate fatigue parameters and criteria for fatigue design.
6. Conclusion To-date, only the auto-sport industry has recognised the potential of surface engineered titanium automotive
J. Luo et al. / Computational Materials Science 35 (2006) 447–457
gears. It is believed that, based on advanced surface engineering technologies and modelling-based predictive performance design, coupled with reduced material and manufacturing costs, the new millennium will see wide application of titanium in vehicles for the mass market. The agreement between the model prediction and the cam rig evaluation test demonstrates the power of the design methodology employed. It stresses the necessity for further systematic research into the fatigue behaviour of layered materials for the contact fatigue design under multi-axial stress conditions.
Acknowledgements The authors wish to acknowledge DTI/EPSRC for its support of the LINK project ÔAdvanced surface engineering of titanium alloy components (AdSurfEngTi, EPSRC/LINK GR/K70298), together with the financial support of BRITE Contract RI.1B.0151C(H), ÔSurface Engineering of Titanium ComponentsÕ. We would also like to extend many thanks to all the partners involved in the LINK project, especially Mercedes–Ilmor for their valuable contributions.
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