- Email: [email protected]
Validation of a simplified numerical contact model
ARTICLE IN PRESS
Tribology International 41 (2008) 926–933 www.elsevier.com/locate/triboint
Validation of a simplified numerical contact model Anders So¨derberg, Stefan Bjo¨rklund Department of Machine Design, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 1 February 2007; received in revised form 6 December 2007; accepted 20 February 2008 Available online 14 April 2008
Abstract Surface roughness tends to have a significant effect on how loads are transmitted at the contact interface between solid bodies. Most numerical contact models for analyzing rough surface contacts are computational demanding and more computationally efficient contact models are required. Depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. This paper discusses a simplified contact model called the elastic foundation model and its applicability to rough surfaces. The advantage of the model is that it is fast to evaluate, but its disadvantage is that it only gives an approximate solution to the contact problem. It is studied how surface roughness influences the errors in the elastic foundation solution in terms of predicted pressure distribution, real contact area, and normal and tangential contact stiffness. The results can be used to estimate the extent of error in the elastic foundation model, depending on the degree of surface roughness. The conclusion is that the elastic foundation model is not accurate enough to give a correct prediction of the actual contact stresses and contact areas, but it might be good enough for simulations where contact stiffness are of interest. r 2008 Elsevier Ltd. All rights reserved. Keywords: Contact calculations; Winkler; Surface roughness; Elastic foundation model
1. Introduction Real-life engineering surfaces are not perfectly smooth. Even highly polished surfaces possess some degree of roughness. It is well known that surface roughness has a significant effect on how loads are transmitted at the contact interface between solid bodies. Surface roughness causes high local pressures (of the same order of magnitude as the Vickers hardness) and significantly reduces the real contact area compared to the corresponding smooth case. Apart from causing high contact stresses, surface roughness is crucial with respect to the wear, friction, and lubrication properties of the contact. In the last two decades several numerical methods have been proposed for calculating the contact stresses and deformations in contacts between three-dimensional rough surfaces. Some of these methods assume frictionless contacts [1,2] and thus ignore tangential traction forces, while others offer a full solution to the contact problem Corresponding author. Tel.: +46 8 790 72 65; fax: +46 8 723 17 30.
E-mail address: [email protected] (A. So¨derberg). 0301-679X/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2008.02.013
[3–5]. All these methods share the assumption that the contacting bodies deform like elastic infinite half-spaces. Besides the elastic restriction, this assumption also requires the contact region to be small compared to the dimensions of the contacting solids. Most numerical methods of contact analysis work by replacing the continuous traction distributions on the surfaces with a discrete number of traction elements (‘‘traction’’ is here used to refer to all forces acting on the surfaces, i.e. both normal pressure and tangential shear stresses). The geometries of the bodies and the applied global displacements provide the boundary conditions. Knowing the relationships between tractions and deformations for the discrete elements, the unknown tractions can be solved. Numerical methods have been used for two–(line contacts) or three-dimensional problems. Since surface roughness is normally three-dimensional, except in the case of very anisotropic surfaces, three-dimensional methods are most widely used for surface roughness problems. In a discrete element, the tractions are approximated by a function of which the deformations are known. For three-dimensional problems with normal and tangential
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
Nomenclature a ah b c C1, C2 d E E* G G* h Kp Kq n
contact radius (m) Hertzian contact radius (m) radius of stick region (m) element width (m) non-dimensional stiffness constants (dimensionless) element length (m) Young’s modulus (Pa) combined Young’s modulus (Pa) shear modulus (Pa) combined shear modulus (Pa) depth of elastic foundation (m) elastic modulus of elastic foundation (Pa) shear modulus of elastic foundation (Pa) number of elements in contact (dimensionless)
tractions, this function can be either a concentrated force at the element center or a uniform traction over the element. A problem encountered when solving problems of rough surfaces is that the traction elements must be small enough to allow the effect from individual surface irregularities to be included in the solution. This means that when the nominal contact area is large compared to the surface features of interest, the necessary number of elements becomes too large for a reasonable computational effort. Hence, computationally efficient contact models are required, and depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. A fast, but approximate, method for calculating the pressure distribution, referred to as the Winkler elastic foundation model, is to ignore the mutual influence between different points in the contact. A similar approach, often called the brush model, can also be used for tangentially loaded contacts. The fundamentals of both methods are described by Johnson [6] and contact models based on these methods will here be referred to as elastic foundation models. Numerical elastic foundation models are less time consuming than more complex models, but they only provide an approximate solution to the contact problem. Nonetheless they can be efficiently used in wear simulations, where the need of repeated pressure calculations leads to a pay-off between required computation time and error [7,8]. To determine whether or not an elastic foundation model is adequate for a specific simulation purpose, it is important to be able to estimate the errors in the obtained solution. For smooth and quite simple contact geometries this can be done by comparing the elastic foundation solution with known analytical solutions [6–8], but for contacts between rough surfaces other approaches are needed. This paper discusses the application of numerical elastic foundation models to rough surface contact problems. The
927
q tangential traction (Pa) Q tangential traction force (N) p normal pressure (Pa) P normal force (N) maximum normal pressure (N) p0 ph maximum Hertzian normal pressure (Pa) r spatial cylindrical coordinate (m) Sa root mean square roughness (m) sx, sy, sz slip distances (m) u¯ x ; u¯ y ; u¯ z surface deformations (m) x,y,z spatial Cartesian coordinates (m) z1,z2 surface profiles (m) zrigid surface profile of rigid intender (m) dx,dy,dz global displacements (m) l autocorrelation length (m) m coefficient of friction (dimensionless) n Poisson’s ratio (dimensionless)
objective is to establish how surface roughness influences the errors in the elastic foundation solution. This is done by comparing the elastic foundation model’s results with those of a more accurate contact model that takes the mutual influence into account. First, the normal contact between a smooth sphere and a rough plane is investigated. Then, a tangential load is applied and the tangential traction in the contact is evaluated. The models are compared in terms of accuracy in predicted pressure distribution, real contact area, and normal and tangential contact stiffness. 2. The elastic foundation model 2.1. Pressure calculations In the elastic foundation model, the contact is regarded as a set of elastic bars deformed by a rigid indenter, as in Fig. 1. The geometry of the rigid indenter is given by the gap between the surfaces before deformation, zrigid ðx; yÞ ¼ z2 ðx; yÞ z1 ðx; yÞ
(1)
The shear forces between the elements of the foundation are ignored, and if the bodies are forced into contact so that the mutual approach of distant points in the two
Elastic foundation: Kp,K
Rigid intender: zrigid(x,y)= z2(x,y)- z1(x,y)
h
Fig. 1. Elastic foundation model.
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
928
bodies is dz, the normal deformation of the foundation is ( dz zrigid ðx; yÞ; dz 4zrigid u¯ z ðx; yÞ ¼ (2) 0; dz pzrigid According to the laws of elasticity, the pressure at a point can be related to the normal deformation of the foundation, as follows: Kp u¯ z ðx; yÞ (3) h where Kp is the elastic modulus of the foundation and h is the depth of the elastic foundation. The critical question is the choice of Kp and h. For nonconformal contacts, the most straightforward way is to determine them by comparing the results of the elastic foundation model with known Hertzian solutions. According to Johnson [6], the results of the elastic foundation model correspond to the Hertzian solution for a given normal load if
pðx; yÞ ¼
Kp En K p En ¼ C1 ¼ C1 h ah h ah
(4)
where E n is the combined Young’s modulus of the surfaces, ah is the Hertzian contact radius, and C1 is a nondimensional constant. The combined Young’s modulus is defined as 1 1 u21 1 u22 þ n ¼ E E1 E2
(5)
where Ek and nk are the Young’s modulus and Poisson’s ratio for body k, respectively. Due to its approximate nature, the results of the elastic foundation model cannot be compatible with the Hertzian contact in all aspects simultaneously. Therefore, the nondimensional constant, C1, has to be chosen differently, depending on the type and aspect of the contact to be studied. In Fig. 2 different pressure distributions obtained with the elastic foundation model are compared with the Hertzian solution for a spherical point contact. In the case of a spherical point contact, one finds that when C1 ¼ 16/(3p), the elastic foundation model will give a
contact radius that is same as the Hertzian solution, though the compliance given will be only half and the maximum pressure 4/3 of the Hertzian solution. The maximum pressure given by the two methods will be the same if radius of the elastic foundation C1 ¼ 3/p. The contact p ffiffiffi solution will then be 2= 3 of the Hertzian contact radius and the compliance 2/3 of the Hertzian result. Equal compliance is obtained if C1 ¼ 4/(3p); in that pffiffiffi case, the contact radius will be larger by a factor of 2 than the Hertzian value and the maximum pressure only 2/3 of the Hertzian value Po˜dra [7] discusses the choice of C1 for both line and elliptical point contacts. 2.2. Traction calculations The elastic foundation model can be expanded so as to consider tangential loads and displacements. This kind of model is often referred to as a brush model, since the tangential behavior of the elements of the foundation can be likened to bristles that deform individually. The tangential surface deformations of the foundation due to applied tangential displacements dx and dy are ( u¯ x ðx; yÞ ¼ dx sx ðx; yÞ (6) u¯ y ðx; yÞ ¼ dy sy ðx; yÞ where sx and sy are the sliding distances. If sliding is prevented, the tangential tractions at a point are related to the deformations according to the laws of elasticity 8 Kq > > < qx ðx; yÞ ¼ u¯ x ðx; yÞ h (7) K q > > : qy ðx; yÞ ¼ u¯ y ðx; yÞ h where Kq is the shear modulus of the foundation. If Coulomb’s law of friction is assumed to be valid and unambiguous over the whole contact interface, the traction at a point is limited according to 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < qx ðx; yÞ2 þ qy ðx; yÞ2 ; qðx; yÞ ¼ : mpðx; yÞ; otherwise
if
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx ðx; yÞ2 þ qy ðx; yÞ2 pmpðx; yÞ
(8) Normal pressure
where m is the coefficient of friction. For the elastic foundation model to give an acceptable result, the foundation shear modulus, Kq, has to be chosen depending on contact geometry, type of motion, and the aspects of the contact to be studied. For non-conformal contacts, an approach similar to that used for the elastic modulus can be used. The ratio between the shear modulus and the width of the foundation is set to
p0 /p0h
1.5 Hertz C1= 4/3π C1= 3/π C1= 16/3π
1
0.5
0 -1.5
-1
-0.5
0 r/ah
0.5
1
1.5
Fig. 2. Hertzian pressure distribution in a spherical point contact compared with different solutions produced by the elastic foundation model.
Kq Gn ¼ C2 h ah
(9)
where C2 is a non-dimensional constant and G n is the combined shear modulus of the interacting surfaces,
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
1 2 u1 2 u2 þ n ¼ G1 G2 G
(10)
Here, Gk is the shear modulus of body k. As mentioned earlier for case where the load is acting normal to the surface, the results of the elastic foundation model cannot be compatible with the analytical solution at all aspects simultaneously. Johnson [6] discusses the choice of C2 regarding rolling and sliding contacts, by comparing the elastic foundation solution with the analytical solution for line contacts obtained by Carter [9]. For a tangentially loaded spherical point contact, the elastic foundation solution can be compared with the analytical solution obtained by Mindlin [10]. If the tangential force is less than the limiting friction, partial sliding will occur in the contact. The contact is divided into a slip region at the edge of the contact and a stick region at the center of the contact. According to Mindlin, the stick region is circular and concentric with the Hertzian contact circle with radius Q 1=3 b ¼ ah 1 (11) mP Here, Q is tangential force and P is the normal force that presses the two bodies into contact. Mindlin’s solution also states that, if the tangential force is taken to act parallel to the x-axis, the traction is radially symmetric in magnitude and everywhere parallel to the x-axis so that 8 1=2 > r2 > > mp 1 ; bprpah > < h a2h qx ðx; yÞ ¼ 1=2 1=2 > r2 mph b r2 > > > mp 1 1 ; rob : h ah a2h b2
using Mindlin’s solution. In Fig. 3 the tangential load–displacement curves obtained using Mindlin’s solution and the elastic foundation model are compared, while the tangential traction distributions for a given tangential displacement obtained using the same two methods are compared in Fig. 4. 2.3. Numerical formulation In a general 3D numerical elastic foundation model, the elements of the elastic foundation are rectangular (2c 2d) bars and the pressure and tractions are assumed to be constant over each element, see Fig. 5. Eqs. (2), (3), (6), and (7) can then be combined into an equation system that governs the tractiondisplacement relationships of the elements in the foundation 2 32 3 2 3 dx s x K qI 0 0 qx 16 7 6 7 6 K qI 0 5 4 dy s y 5 ¼ 4 qy 7 (16) 4 0 5 h dy z 0 0 K pI p where I is the identity matrix and sx, sy z, qx, qy, and p are vectors containing the slip distances, gap, tractions, and pressures for all elements. Since there is no coupling between individual elements, Eq. (16) can be solved fast Tangential Load-Displacement Curve
1.5
Mindlin Elastic Foundation
Qx / P
defined as
1
0.5
0 0
0.5
1
(12)
and consequentially the initial tangential contact stiffness is dQx ¼ 8Gn ah (14) ddx dx ¼0 Using Mindlin’s analytical solution as a reference, one finds that an equal initial tangential stiffness is obtained with the elastic foundation model if 8 ah 2 C2 ¼ (15) p a where a is the contact radius according to the elastic foundation model. The ratio ah/a is dependent on how C1 is chosen, but as long as C2 is set according to Eq. (15), the initial tangential stiffness obtained is same as that obtained
1.5
16ahGx / (3P) Fig. 3. Tangential load versus tangential displacement for a spherical point contact according to Mindlin’s analytical solution compared to the solution produced with the elastic foundation model calibrated to give the same initial stiffness as Mindlin’s solution.
Tangential traction
1.5
qx / p0h
Furthermore, the magnitude of the tangential force is related to the relative tangential displacement between the two bodies according to 3=2 16ah G n Qx ¼ mP 1 1 dx (13) 3mP
929
Mindlin C1= 4/3π, C2= 4/π C1= 3/π, C2= 6/π C1= 16/3π, C2= 8/π
1
0.5
0 -1.5
-1
-0.5
0 r/ah
0.5
1
1.5
Fig. 4. Example of traction distributions at a spherical point contact exposed to a tangential displacement according to Mindlin’s analytical solution compared to different solutions produced by the elastic foundation model.
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
930
2c
2d
z
y x h
Fig. 5. Three-dimensional elastic foundation deformed by a spherical rigid intender.
and directly without iterations. The solution strategy for a given displacement and initial geometry is thus straightforward. First, Eq. (16) is solved assuming that slip is prevented in all elements, i.e. sx ¼ sy ¼ 0. The surface of the elastic foundation is larger than the actual contact zone and to keep the whole surface of the foundation in contact with the rigid intender the pressure will have to be negative at some elements. These elements fall outside the contact and their pressure is set to zero. Next step is to control if the condition given by Eq. (8) is violated at any element. If so, the tractions at these elements are corrected and the corresponding slip distances are computed. The total forces carried by the contact are X X X P ¼ 4cd p; Qx ¼ 4cd qx ; Qy ¼ 4cd qy (17) and the area of contact is A ¼ 4cdn
(18)
where n is the number of elements for which p40 MPa. 3. Investigation of validity for contact between rough surfaces So far only the applicability of the elastic foundation model to problems of contact between smooth surfaces has been discussed. To establish whether or not the elastic foundation model is suitable for the numerical analysis of contact between rough surfaces, the contact between a smooth sphere and a rough plane is investigated. Both bodies are assumed to be made of steel with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. The coefficient of friction is set to 0.1. The diameter of the sphere is 4 mm and the surface roughness of the plane is generated using the numerical method of Patir and Cheng [11]. Eighteen different classes of isotropic surfaces with Gaussian height distributions of varying degrees of root mean square roughness (Sa ¼ 0.1, 0.2, 0.4, 0.6, 0.8, and 1 mm) and autocorrelation lengths (l ¼ 12, 20, and 40 mm) are created. The autocorrelation length is defined as the length at which the profile autocorrelation function reduces to 50% of its initial value. Each class contains 10 different surfaces. Most times when generating surface roughness by
numerical means much attention is devoted to make sure that the generated surface is realistic. In this case, however, the goal is to study the characteristics of the contact model in itself; the actual surface geometry is of secondary importance and will therefore not be discussed in more detail. Nevertheless, the roughness parameters are chosen to reflect realistic engineering situations. To solve the problem numerically, the contact is divided into a mesh of square-shaped elements each side of which is 6 mm in length. The error in the solution given by the elastic foundation model is established by comparing the solution to the more accurate solution obtained using the numerical method described by Bjo¨rklund and Andersson [5]. This reference model provides a full solution to the contact problem, i.e. the only error in the result is due to the numerical discretization. The same mesh is used in both the elastic foundation and reference models. How well the model handles a purely normally loaded contact is studied by varying the normal load between 5 and 40 N. The upper load limit corresponds to a maximum Hertzian pressure of 3 GPa, whereas the lower limit is the load for which the area of a single contact element is 0.5% of the Hertzian contact area. For lower loads, the mesh is regarded as too coarse to give an accurate result. The behavior of the model under tangential loads is studied by applying different tangential displacements from zero up to 0.5 mm. It has already been pointed out that the solution obtained using the elastic foundation model is dependent on the choice of foundation moduli. Here the elastic modulus of the foundation is chosen so that agreement is obtained between the elastic foundation and the corresponding Hertzian solution in terms of maximal pressure, contact radius, and compliance, and the shear modulus of the foundation is set so that the elastic foundation model gives the same initial tangential stiffness as Mindlin’s analytical solution does. Other choices of foundation moduli are not taken into consideration. 4. Results 4.1. Normal loading The result for rough surfaces shows that the results of the elastic foundation model do not agree very well with those of the more accurate reference model, as the maximum pressure given is lower and the pressure is distributed more uniformly, see Fig. 6. This can be related to the fact that the real area of contact is overestimated by the elastic foundation model. The real area of contact computed using the elastic foundation model is smaller than the corresponding Hertzian result, especially at low loads, but increases more rapidly with increasing load than is found with the reference model. Fig. 7 shows the mean value and standard deviation of the relative error of maximal pressure if the elastic foundation is calibrated so as to give the same maximal
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
931
Fig. 6. Example of pressure distributions generated with the elastic foundation model (C1 ¼ 16/(3p)) and the reference model (P ¼ 20 N, Sa ¼ 0.4 mm, l ¼ 20 mm).
Relative error of real area of contact (λ=40 μm, C1=16/(3π))
Relative error of maximum pressure (λ=40 μm, C1=3/π) 20
200
0 -20
150
Sa =0.6m Sa =0.8m Sa =1m
Sa =0.2m Sa =0.4m
relative error [%]
relative error [%]
Sa =0m Sa =0.1m
-40 -60
100
50
0
-80
Sa =0m
-100 0
10
20
30
40
50
P [N] Fig. 7. Mean value of relative error of maximum pressure between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same maximum pressure as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
pressure as is given by the Hertzian solution (i.e. C1 ¼ p/3) and the surfaces all have an autocorrelation length of 40 mm. The relative error increases with both increasing root mean square roughness and decreasing autocorrelation length. Fig. 8 presents the mean value and standard deviation of the relative error of the real area of contact, if the elastic foundation is calibrated so as to give the same contact radius as is given by the Hertzian solution (i.e. C1 ¼ 16/ (3p)) and if the surfaces all have an autocorrelation length of 40 m. The relative error of the real area of contact increases with both increasing root mean square roughness and decreasing autocorrelation length. Finally, Fig. 9 presents the mean value and standard deviation of the relative error of compliance, if the elastic foundation is calibrated so as to give the same compliance as is given by the Hertzian solution (i.e. C1 ¼ 4/(3p)) and if
-50 0
10
Sa =0.6m
Sa =0.2m Sa =0.4m Sa =1m
Sa =0.1m
20
30
Sa =0.8m
40
50
P [N] Fig. 8. Mean value of relative error of real area of contact between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same contact radius as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
the surface has an autocorrelation length of 40 mm. The relative error of compliance increases with both decreasing load and increasing root mean square roughness of the surface. The autocorrelation length has less influence on the error than the other parameters do, but the overall trend is that relative error increases with decreasing autocorrelation length. The results presented in Figs. 7–9 can be seen as bestcase scenarios, since the error increases for surfaces of other autocorrelation lengths or if other standard values are used for parameter C1. 4.2. Tangential loading As for the normally loaded case, the tangential solutions obtained using the elastic foundation model do not agree well with those given by the reference model. The elastic
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
932
foundation model is more affected by the surface roughness than the reference model is, and displays lower tangential contact stiffness. Fig. 10 shows how the obtained tangential traction distribution may differ between the models, while Fig. 11 shows how the surface roughness affects the load–displacement curve. The mean value and standard deviation of the relative error of the initial tangential stiffness are presented in Fig. 12. The error increases with decreasing normal load and increasing root mean square roughness; however, the autocorrelation length has less influence on the error. In both figures the elastic foundation model is calibrated so as to give a contact radius equal to that given by the Hertzian solution and an initial tangential stiffness equal to that given by Mindlin’s analytical solution.
Relative error of compliance (λ=40 μm, C1=4/3π) 40 Sa =0m Sa =0.1m Sa =0.2m Sa =0.4m
Sa =0.8m Sa =1m
Numerical elastic foundation models provide a fast and simplified method of contact analysis. Since the mutual influence between surrounding points is ignored, an elastic foundation model gives only an approximate solution to the contact problem. As long as smooth surfaces are considered, the model in most cases can be calibrated by comparing its results with analytical solutions for known standard cases. By its inherent nature, the results of the model cannot correspond to all aspects of the reference solution at the same time, but its errors can be estimated and if they are found to be acceptable the model can be used. This paper discusses the application of the elastic foundation model to rough surface contact problems, and how the error in the elastic foundation model is estimated by comparing its results with those of a more accurate numerical model. The first problem that occurs with rough surfaces is that there exists no analytical solution against which to calibrate the elastic foundation model. This is handled by choosing the parameters in the model based on their correspondence with those of analytical solutions, assuming smooth surfaces. The second
20
Tangential Load-Displacement Curve
1.5
Mindlin Reference model Elastic foundation
10
Qx / P
relative error [%]
30
Sa =0.6m
5. Discussion
0
-10 0
10
20
30
40
50
P [N] Fig. 9. Mean value of relative error of compliance between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same compliance as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
1
0.5
0 0
0.5
1
1.5
16ahGx /(3P) Fig. 11. Example of tangential load-displacement curves generated with the elastic foundation model (C1 ¼ 16/(3p) and C1 ¼ 8/p) and the reference model. The standard deviation is indicated by error bars (P ¼ 20 N, Sa ¼ 0.4 mm, l ¼ 20 mm).
Fig. 10. Example of tangential traction distributions generated with the elastic foundation model (C1 ¼ 16/(3p) and C1 ¼ 8/p) and the reference model (P ¼ 20 N, x ¼ 0.12 mm, Sa ¼ 0.4 mm, l ¼ 20 mm).
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
0.2 mm for the estimated relative error to be less than 50%. The fact that the estimated relative errors of the compliance and initial tangential stiffness are considerably lower indicate that the model is more applicable for the prediction of contact stiffness. The ultimate question is, however, whether the degrees of error are acceptable. From an engineering point of view, the model should be as simple as possible and as accurate as necessary. The advantage of the model is that it is fast and simple to implement; whether or not this outweighs the degrees of error and uncertainty found depends, of course, on the application and must be determined from case to case.
Relative error of initial tangential contact stiffness (λ=40 μm, C1=16/(3π)μm, C2=8/π) 10
relative error [%]
0
-10
-20 Sa =0m Sa =0.1m
-30
Sa =0.2m Sa =0.4m
Sa =0.6m Sa =0.8m Sa =1m
Acknowledgment
-40 0
10
20
30
933
40
50
P [N] Fig. 12. Mean value of relative error of the initial tangential contact stiffness between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give a contact radius equal to that given by the Hertzian solution and an initial tangential contact stiffness equal to that given by Mindlin’s analytical solution. The standard deviation for each class of surfaces is indicated by error bars.
problem is that there exists no parameter that fully describes surface roughness. Although two surfaces may have the same root mean square roughness and autocorrelation length, they can display very different contact characteristics. This lack is handled by studying not one, but a group of surfaces with the same parameters. The results presented here can be used to estimate the extent of the error in the elastic foundation model that depends on the surface roughness. The main conclusion is that the results of the elastic foundation model do not agree well with those of the reference model. The general trend is that the elastic foundation model gives a larger area of contact, lower and more evenly distributed pressure, and lower tangential contact stiffness than the reference model does. The elastic foundation model seems to be particularly misleading regarding the prediction of real area of contact and maximum pressure, where the root mean square roughness of the surface must be below
This research was performed as part of the INTEFACE project, financially supported by the Swedish Foundation for Strategic Research via the ProViking program.
References [1] Liang X, Linqing Z. A numerical model for the elastic contact of three-dimensional real rough surfaces. Wear 1991;148:91–100. [2] Ren N, Lee SC. Contact simulation of three-dimensional rough surfaces using moving grid method. J Tribol 1993;115:597–601. [3] Kalker JJ. Three-dimensional elastic bodies in rolling contact. Dordrecht: Kluwer Academic Publishers; 1990. [4] Jaeger J. Elastic impact with friction. Doctoral thesis, University of Delft, Delft, The Netherlands, 1992. [5] Bjo¨rklund S, Andersson S. A numerical method for real elastic contacts subjected to normal and tangential loading. Wear 1994;179: 117–22. [6] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1992. [7] Po˜dra P, Andersson S. Wear simulation with the Winkler surface model. Wear 1997;207:79–85. [8] Spiegelberg C, Andersson S. Simulation of friction and wear in the contact between the valve bridge and rocker arm pad in a cam mechanism. Wear 2006;261:58–67. [9] Carter FW. On the action of a locomotive driving wheel. Proc Royal Soc A 1926;112:151–7. [10] Mindlin RD. Compliance of elastic bodies in contact. J Appl Mech 1949;259–268. [11] Patir N, Cheng HS. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. ASME J Lubr Technol 1978;100:12–7.
Tribology International 41 (2008) 926–933 www.elsevier.com/locate/triboint
Validation of a simplified numerical contact model Anders So¨derberg, Stefan Bjo¨rklund Department of Machine Design, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden Received 1 February 2007; received in revised form 6 December 2007; accepted 20 February 2008 Available online 14 April 2008
Abstract Surface roughness tends to have a significant effect on how loads are transmitted at the contact interface between solid bodies. Most numerical contact models for analyzing rough surface contacts are computational demanding and more computationally efficient contact models are required. Depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. This paper discusses a simplified contact model called the elastic foundation model and its applicability to rough surfaces. The advantage of the model is that it is fast to evaluate, but its disadvantage is that it only gives an approximate solution to the contact problem. It is studied how surface roughness influences the errors in the elastic foundation solution in terms of predicted pressure distribution, real contact area, and normal and tangential contact stiffness. The results can be used to estimate the extent of error in the elastic foundation model, depending on the degree of surface roughness. The conclusion is that the elastic foundation model is not accurate enough to give a correct prediction of the actual contact stresses and contact areas, but it might be good enough for simulations where contact stiffness are of interest. r 2008 Elsevier Ltd. All rights reserved. Keywords: Contact calculations; Winkler; Surface roughness; Elastic foundation model
1. Introduction Real-life engineering surfaces are not perfectly smooth. Even highly polished surfaces possess some degree of roughness. It is well known that surface roughness has a significant effect on how loads are transmitted at the contact interface between solid bodies. Surface roughness causes high local pressures (of the same order of magnitude as the Vickers hardness) and significantly reduces the real contact area compared to the corresponding smooth case. Apart from causing high contact stresses, surface roughness is crucial with respect to the wear, friction, and lubrication properties of the contact. In the last two decades several numerical methods have been proposed for calculating the contact stresses and deformations in contacts between three-dimensional rough surfaces. Some of these methods assume frictionless contacts [1,2] and thus ignore tangential traction forces, while others offer a full solution to the contact problem Corresponding author. Tel.: +46 8 790 72 65; fax: +46 8 723 17 30.
E-mail address: [email protected] (A. So¨derberg). 0301-679X/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2008.02.013
[3–5]. All these methods share the assumption that the contacting bodies deform like elastic infinite half-spaces. Besides the elastic restriction, this assumption also requires the contact region to be small compared to the dimensions of the contacting solids. Most numerical methods of contact analysis work by replacing the continuous traction distributions on the surfaces with a discrete number of traction elements (‘‘traction’’ is here used to refer to all forces acting on the surfaces, i.e. both normal pressure and tangential shear stresses). The geometries of the bodies and the applied global displacements provide the boundary conditions. Knowing the relationships between tractions and deformations for the discrete elements, the unknown tractions can be solved. Numerical methods have been used for two–(line contacts) or three-dimensional problems. Since surface roughness is normally three-dimensional, except in the case of very anisotropic surfaces, three-dimensional methods are most widely used for surface roughness problems. In a discrete element, the tractions are approximated by a function of which the deformations are known. For three-dimensional problems with normal and tangential
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
Nomenclature a ah b c C1, C2 d E E* G G* h Kp Kq n
contact radius (m) Hertzian contact radius (m) radius of stick region (m) element width (m) non-dimensional stiffness constants (dimensionless) element length (m) Young’s modulus (Pa) combined Young’s modulus (Pa) shear modulus (Pa) combined shear modulus (Pa) depth of elastic foundation (m) elastic modulus of elastic foundation (Pa) shear modulus of elastic foundation (Pa) number of elements in contact (dimensionless)
tractions, this function can be either a concentrated force at the element center or a uniform traction over the element. A problem encountered when solving problems of rough surfaces is that the traction elements must be small enough to allow the effect from individual surface irregularities to be included in the solution. This means that when the nominal contact area is large compared to the surface features of interest, the necessary number of elements becomes too large for a reasonable computational effort. Hence, computationally efficient contact models are required, and depending on the purpose of the simulation, simplified and less accurate models can be preferable to more accurate, but also more complex, models. A fast, but approximate, method for calculating the pressure distribution, referred to as the Winkler elastic foundation model, is to ignore the mutual influence between different points in the contact. A similar approach, often called the brush model, can also be used for tangentially loaded contacts. The fundamentals of both methods are described by Johnson [6] and contact models based on these methods will here be referred to as elastic foundation models. Numerical elastic foundation models are less time consuming than more complex models, but they only provide an approximate solution to the contact problem. Nonetheless they can be efficiently used in wear simulations, where the need of repeated pressure calculations leads to a pay-off between required computation time and error [7,8]. To determine whether or not an elastic foundation model is adequate for a specific simulation purpose, it is important to be able to estimate the errors in the obtained solution. For smooth and quite simple contact geometries this can be done by comparing the elastic foundation solution with known analytical solutions [6–8], but for contacts between rough surfaces other approaches are needed. This paper discusses the application of numerical elastic foundation models to rough surface contact problems. The
927
q tangential traction (Pa) Q tangential traction force (N) p normal pressure (Pa) P normal force (N) maximum normal pressure (N) p0 ph maximum Hertzian normal pressure (Pa) r spatial cylindrical coordinate (m) Sa root mean square roughness (m) sx, sy, sz slip distances (m) u¯ x ; u¯ y ; u¯ z surface deformations (m) x,y,z spatial Cartesian coordinates (m) z1,z2 surface profiles (m) zrigid surface profile of rigid intender (m) dx,dy,dz global displacements (m) l autocorrelation length (m) m coefficient of friction (dimensionless) n Poisson’s ratio (dimensionless)
objective is to establish how surface roughness influences the errors in the elastic foundation solution. This is done by comparing the elastic foundation model’s results with those of a more accurate contact model that takes the mutual influence into account. First, the normal contact between a smooth sphere and a rough plane is investigated. Then, a tangential load is applied and the tangential traction in the contact is evaluated. The models are compared in terms of accuracy in predicted pressure distribution, real contact area, and normal and tangential contact stiffness. 2. The elastic foundation model 2.1. Pressure calculations In the elastic foundation model, the contact is regarded as a set of elastic bars deformed by a rigid indenter, as in Fig. 1. The geometry of the rigid indenter is given by the gap between the surfaces before deformation, zrigid ðx; yÞ ¼ z2 ðx; yÞ z1 ðx; yÞ
(1)
The shear forces between the elements of the foundation are ignored, and if the bodies are forced into contact so that the mutual approach of distant points in the two
Elastic foundation: Kp,K
Rigid intender: zrigid(x,y)= z2(x,y)- z1(x,y)
h
Fig. 1. Elastic foundation model.
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
928
bodies is dz, the normal deformation of the foundation is ( dz zrigid ðx; yÞ; dz 4zrigid u¯ z ðx; yÞ ¼ (2) 0; dz pzrigid According to the laws of elasticity, the pressure at a point can be related to the normal deformation of the foundation, as follows: Kp u¯ z ðx; yÞ (3) h where Kp is the elastic modulus of the foundation and h is the depth of the elastic foundation. The critical question is the choice of Kp and h. For nonconformal contacts, the most straightforward way is to determine them by comparing the results of the elastic foundation model with known Hertzian solutions. According to Johnson [6], the results of the elastic foundation model correspond to the Hertzian solution for a given normal load if
pðx; yÞ ¼
Kp En K p En ¼ C1 ¼ C1 h ah h ah
(4)
where E n is the combined Young’s modulus of the surfaces, ah is the Hertzian contact radius, and C1 is a nondimensional constant. The combined Young’s modulus is defined as 1 1 u21 1 u22 þ n ¼ E E1 E2
(5)
where Ek and nk are the Young’s modulus and Poisson’s ratio for body k, respectively. Due to its approximate nature, the results of the elastic foundation model cannot be compatible with the Hertzian contact in all aspects simultaneously. Therefore, the nondimensional constant, C1, has to be chosen differently, depending on the type and aspect of the contact to be studied. In Fig. 2 different pressure distributions obtained with the elastic foundation model are compared with the Hertzian solution for a spherical point contact. In the case of a spherical point contact, one finds that when C1 ¼ 16/(3p), the elastic foundation model will give a
contact radius that is same as the Hertzian solution, though the compliance given will be only half and the maximum pressure 4/3 of the Hertzian solution. The maximum pressure given by the two methods will be the same if radius of the elastic foundation C1 ¼ 3/p. The contact p ffiffiffi solution will then be 2= 3 of the Hertzian contact radius and the compliance 2/3 of the Hertzian result. Equal compliance is obtained if C1 ¼ 4/(3p); in that pffiffiffi case, the contact radius will be larger by a factor of 2 than the Hertzian value and the maximum pressure only 2/3 of the Hertzian value Po˜dra [7] discusses the choice of C1 for both line and elliptical point contacts. 2.2. Traction calculations The elastic foundation model can be expanded so as to consider tangential loads and displacements. This kind of model is often referred to as a brush model, since the tangential behavior of the elements of the foundation can be likened to bristles that deform individually. The tangential surface deformations of the foundation due to applied tangential displacements dx and dy are ( u¯ x ðx; yÞ ¼ dx sx ðx; yÞ (6) u¯ y ðx; yÞ ¼ dy sy ðx; yÞ where sx and sy are the sliding distances. If sliding is prevented, the tangential tractions at a point are related to the deformations according to the laws of elasticity 8 Kq > > < qx ðx; yÞ ¼ u¯ x ðx; yÞ h (7) K q > > : qy ðx; yÞ ¼ u¯ y ðx; yÞ h where Kq is the shear modulus of the foundation. If Coulomb’s law of friction is assumed to be valid and unambiguous over the whole contact interface, the traction at a point is limited according to 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < qx ðx; yÞ2 þ qy ðx; yÞ2 ; qðx; yÞ ¼ : mpðx; yÞ; otherwise
if
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx ðx; yÞ2 þ qy ðx; yÞ2 pmpðx; yÞ
(8) Normal pressure
where m is the coefficient of friction. For the elastic foundation model to give an acceptable result, the foundation shear modulus, Kq, has to be chosen depending on contact geometry, type of motion, and the aspects of the contact to be studied. For non-conformal contacts, an approach similar to that used for the elastic modulus can be used. The ratio between the shear modulus and the width of the foundation is set to
p0 /p0h
1.5 Hertz C1= 4/3π C1= 3/π C1= 16/3π
1
0.5
0 -1.5
-1
-0.5
0 r/ah
0.5
1
1.5
Fig. 2. Hertzian pressure distribution in a spherical point contact compared with different solutions produced by the elastic foundation model.
Kq Gn ¼ C2 h ah
(9)
where C2 is a non-dimensional constant and G n is the combined shear modulus of the interacting surfaces,
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
1 2 u1 2 u2 þ n ¼ G1 G2 G
(10)
Here, Gk is the shear modulus of body k. As mentioned earlier for case where the load is acting normal to the surface, the results of the elastic foundation model cannot be compatible with the analytical solution at all aspects simultaneously. Johnson [6] discusses the choice of C2 regarding rolling and sliding contacts, by comparing the elastic foundation solution with the analytical solution for line contacts obtained by Carter [9]. For a tangentially loaded spherical point contact, the elastic foundation solution can be compared with the analytical solution obtained by Mindlin [10]. If the tangential force is less than the limiting friction, partial sliding will occur in the contact. The contact is divided into a slip region at the edge of the contact and a stick region at the center of the contact. According to Mindlin, the stick region is circular and concentric with the Hertzian contact circle with radius Q 1=3 b ¼ ah 1 (11) mP Here, Q is tangential force and P is the normal force that presses the two bodies into contact. Mindlin’s solution also states that, if the tangential force is taken to act parallel to the x-axis, the traction is radially symmetric in magnitude and everywhere parallel to the x-axis so that 8 1=2 > r2 > > mp 1 ; bprpah > < h a2h qx ðx; yÞ ¼ 1=2 1=2 > r2 mph b r2 > > > mp 1 1 ; rob : h ah a2h b2
using Mindlin’s solution. In Fig. 3 the tangential load–displacement curves obtained using Mindlin’s solution and the elastic foundation model are compared, while the tangential traction distributions for a given tangential displacement obtained using the same two methods are compared in Fig. 4. 2.3. Numerical formulation In a general 3D numerical elastic foundation model, the elements of the elastic foundation are rectangular (2c 2d) bars and the pressure and tractions are assumed to be constant over each element, see Fig. 5. Eqs. (2), (3), (6), and (7) can then be combined into an equation system that governs the tractiondisplacement relationships of the elements in the foundation 2 32 3 2 3 dx s x K qI 0 0 qx 16 7 6 7 6 K qI 0 5 4 dy s y 5 ¼ 4 qy 7 (16) 4 0 5 h dy z 0 0 K pI p where I is the identity matrix and sx, sy z, qx, qy, and p are vectors containing the slip distances, gap, tractions, and pressures for all elements. Since there is no coupling between individual elements, Eq. (16) can be solved fast Tangential Load-Displacement Curve
1.5
Mindlin Elastic Foundation
Qx / P
defined as
1
0.5
0 0
0.5
1
(12)
and consequentially the initial tangential contact stiffness is dQx ¼ 8Gn ah (14) ddx dx ¼0 Using Mindlin’s analytical solution as a reference, one finds that an equal initial tangential stiffness is obtained with the elastic foundation model if 8 ah 2 C2 ¼ (15) p a where a is the contact radius according to the elastic foundation model. The ratio ah/a is dependent on how C1 is chosen, but as long as C2 is set according to Eq. (15), the initial tangential stiffness obtained is same as that obtained
1.5
16ahGx / (3P) Fig. 3. Tangential load versus tangential displacement for a spherical point contact according to Mindlin’s analytical solution compared to the solution produced with the elastic foundation model calibrated to give the same initial stiffness as Mindlin’s solution.
Tangential traction
1.5
qx / p0h
Furthermore, the magnitude of the tangential force is related to the relative tangential displacement between the two bodies according to 3=2 16ah G n Qx ¼ mP 1 1 dx (13) 3mP
929
Mindlin C1= 4/3π, C2= 4/π C1= 3/π, C2= 6/π C1= 16/3π, C2= 8/π
1
0.5
0 -1.5
-1
-0.5
0 r/ah
0.5
1
1.5
Fig. 4. Example of traction distributions at a spherical point contact exposed to a tangential displacement according to Mindlin’s analytical solution compared to different solutions produced by the elastic foundation model.
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
930
2c
2d
z
y x h
Fig. 5. Three-dimensional elastic foundation deformed by a spherical rigid intender.
and directly without iterations. The solution strategy for a given displacement and initial geometry is thus straightforward. First, Eq. (16) is solved assuming that slip is prevented in all elements, i.e. sx ¼ sy ¼ 0. The surface of the elastic foundation is larger than the actual contact zone and to keep the whole surface of the foundation in contact with the rigid intender the pressure will have to be negative at some elements. These elements fall outside the contact and their pressure is set to zero. Next step is to control if the condition given by Eq. (8) is violated at any element. If so, the tractions at these elements are corrected and the corresponding slip distances are computed. The total forces carried by the contact are X X X P ¼ 4cd p; Qx ¼ 4cd qx ; Qy ¼ 4cd qy (17) and the area of contact is A ¼ 4cdn
(18)
where n is the number of elements for which p40 MPa. 3. Investigation of validity for contact between rough surfaces So far only the applicability of the elastic foundation model to problems of contact between smooth surfaces has been discussed. To establish whether or not the elastic foundation model is suitable for the numerical analysis of contact between rough surfaces, the contact between a smooth sphere and a rough plane is investigated. Both bodies are assumed to be made of steel with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. The coefficient of friction is set to 0.1. The diameter of the sphere is 4 mm and the surface roughness of the plane is generated using the numerical method of Patir and Cheng [11]. Eighteen different classes of isotropic surfaces with Gaussian height distributions of varying degrees of root mean square roughness (Sa ¼ 0.1, 0.2, 0.4, 0.6, 0.8, and 1 mm) and autocorrelation lengths (l ¼ 12, 20, and 40 mm) are created. The autocorrelation length is defined as the length at which the profile autocorrelation function reduces to 50% of its initial value. Each class contains 10 different surfaces. Most times when generating surface roughness by
numerical means much attention is devoted to make sure that the generated surface is realistic. In this case, however, the goal is to study the characteristics of the contact model in itself; the actual surface geometry is of secondary importance and will therefore not be discussed in more detail. Nevertheless, the roughness parameters are chosen to reflect realistic engineering situations. To solve the problem numerically, the contact is divided into a mesh of square-shaped elements each side of which is 6 mm in length. The error in the solution given by the elastic foundation model is established by comparing the solution to the more accurate solution obtained using the numerical method described by Bjo¨rklund and Andersson [5]. This reference model provides a full solution to the contact problem, i.e. the only error in the result is due to the numerical discretization. The same mesh is used in both the elastic foundation and reference models. How well the model handles a purely normally loaded contact is studied by varying the normal load between 5 and 40 N. The upper load limit corresponds to a maximum Hertzian pressure of 3 GPa, whereas the lower limit is the load for which the area of a single contact element is 0.5% of the Hertzian contact area. For lower loads, the mesh is regarded as too coarse to give an accurate result. The behavior of the model under tangential loads is studied by applying different tangential displacements from zero up to 0.5 mm. It has already been pointed out that the solution obtained using the elastic foundation model is dependent on the choice of foundation moduli. Here the elastic modulus of the foundation is chosen so that agreement is obtained between the elastic foundation and the corresponding Hertzian solution in terms of maximal pressure, contact radius, and compliance, and the shear modulus of the foundation is set so that the elastic foundation model gives the same initial tangential stiffness as Mindlin’s analytical solution does. Other choices of foundation moduli are not taken into consideration. 4. Results 4.1. Normal loading The result for rough surfaces shows that the results of the elastic foundation model do not agree very well with those of the more accurate reference model, as the maximum pressure given is lower and the pressure is distributed more uniformly, see Fig. 6. This can be related to the fact that the real area of contact is overestimated by the elastic foundation model. The real area of contact computed using the elastic foundation model is smaller than the corresponding Hertzian result, especially at low loads, but increases more rapidly with increasing load than is found with the reference model. Fig. 7 shows the mean value and standard deviation of the relative error of maximal pressure if the elastic foundation is calibrated so as to give the same maximal
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
931
Fig. 6. Example of pressure distributions generated with the elastic foundation model (C1 ¼ 16/(3p)) and the reference model (P ¼ 20 N, Sa ¼ 0.4 mm, l ¼ 20 mm).
Relative error of real area of contact (λ=40 μm, C1=16/(3π))
Relative error of maximum pressure (λ=40 μm, C1=3/π) 20
200
0 -20
150
Sa =0.6m Sa =0.8m Sa =1m
Sa =0.2m Sa =0.4m
relative error [%]
relative error [%]
Sa =0m Sa =0.1m
-40 -60
100
50
0
-80
Sa =0m
-100 0
10
20
30
40
50
P [N] Fig. 7. Mean value of relative error of maximum pressure between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same maximum pressure as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
pressure as is given by the Hertzian solution (i.e. C1 ¼ p/3) and the surfaces all have an autocorrelation length of 40 mm. The relative error increases with both increasing root mean square roughness and decreasing autocorrelation length. Fig. 8 presents the mean value and standard deviation of the relative error of the real area of contact, if the elastic foundation is calibrated so as to give the same contact radius as is given by the Hertzian solution (i.e. C1 ¼ 16/ (3p)) and if the surfaces all have an autocorrelation length of 40 m. The relative error of the real area of contact increases with both increasing root mean square roughness and decreasing autocorrelation length. Finally, Fig. 9 presents the mean value and standard deviation of the relative error of compliance, if the elastic foundation is calibrated so as to give the same compliance as is given by the Hertzian solution (i.e. C1 ¼ 4/(3p)) and if
-50 0
10
Sa =0.6m
Sa =0.2m Sa =0.4m Sa =1m
Sa =0.1m
20
30
Sa =0.8m
40
50
P [N] Fig. 8. Mean value of relative error of real area of contact between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same contact radius as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
the surface has an autocorrelation length of 40 mm. The relative error of compliance increases with both decreasing load and increasing root mean square roughness of the surface. The autocorrelation length has less influence on the error than the other parameters do, but the overall trend is that relative error increases with decreasing autocorrelation length. The results presented in Figs. 7–9 can be seen as bestcase scenarios, since the error increases for surfaces of other autocorrelation lengths or if other standard values are used for parameter C1. 4.2. Tangential loading As for the normally loaded case, the tangential solutions obtained using the elastic foundation model do not agree well with those given by the reference model. The elastic
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
932
foundation model is more affected by the surface roughness than the reference model is, and displays lower tangential contact stiffness. Fig. 10 shows how the obtained tangential traction distribution may differ between the models, while Fig. 11 shows how the surface roughness affects the load–displacement curve. The mean value and standard deviation of the relative error of the initial tangential stiffness are presented in Fig. 12. The error increases with decreasing normal load and increasing root mean square roughness; however, the autocorrelation length has less influence on the error. In both figures the elastic foundation model is calibrated so as to give a contact radius equal to that given by the Hertzian solution and an initial tangential stiffness equal to that given by Mindlin’s analytical solution.
Relative error of compliance (λ=40 μm, C1=4/3π) 40 Sa =0m Sa =0.1m Sa =0.2m Sa =0.4m
Sa =0.8m Sa =1m
Numerical elastic foundation models provide a fast and simplified method of contact analysis. Since the mutual influence between surrounding points is ignored, an elastic foundation model gives only an approximate solution to the contact problem. As long as smooth surfaces are considered, the model in most cases can be calibrated by comparing its results with analytical solutions for known standard cases. By its inherent nature, the results of the model cannot correspond to all aspects of the reference solution at the same time, but its errors can be estimated and if they are found to be acceptable the model can be used. This paper discusses the application of the elastic foundation model to rough surface contact problems, and how the error in the elastic foundation model is estimated by comparing its results with those of a more accurate numerical model. The first problem that occurs with rough surfaces is that there exists no analytical solution against which to calibrate the elastic foundation model. This is handled by choosing the parameters in the model based on their correspondence with those of analytical solutions, assuming smooth surfaces. The second
20
Tangential Load-Displacement Curve
1.5
Mindlin Reference model Elastic foundation
10
Qx / P
relative error [%]
30
Sa =0.6m
5. Discussion
0
-10 0
10
20
30
40
50
P [N] Fig. 9. Mean value of relative error of compliance between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give the same compliance as is given by the Hertzian solution. The standard deviation for each class of surfaces is indicated by error bars.
1
0.5
0 0
0.5
1
1.5
16ahGx /(3P) Fig. 11. Example of tangential load-displacement curves generated with the elastic foundation model (C1 ¼ 16/(3p) and C1 ¼ 8/p) and the reference model. The standard deviation is indicated by error bars (P ¼ 20 N, Sa ¼ 0.4 mm, l ¼ 20 mm).
Fig. 10. Example of tangential traction distributions generated with the elastic foundation model (C1 ¼ 16/(3p) and C1 ¼ 8/p) and the reference model (P ¼ 20 N, x ¼ 0.12 mm, Sa ¼ 0.4 mm, l ¼ 20 mm).
ARTICLE IN PRESS A. So¨derberg, S. Bjo¨rklund / Tribology International 41 (2008) 926–933
0.2 mm for the estimated relative error to be less than 50%. The fact that the estimated relative errors of the compliance and initial tangential stiffness are considerably lower indicate that the model is more applicable for the prediction of contact stiffness. The ultimate question is, however, whether the degrees of error are acceptable. From an engineering point of view, the model should be as simple as possible and as accurate as necessary. The advantage of the model is that it is fast and simple to implement; whether or not this outweighs the degrees of error and uncertainty found depends, of course, on the application and must be determined from case to case.
Relative error of initial tangential contact stiffness (λ=40 μm, C1=16/(3π)μm, C2=8/π) 10
relative error [%]
0
-10
-20 Sa =0m Sa =0.1m
-30
Sa =0.2m Sa =0.4m
Sa =0.6m Sa =0.8m Sa =1m
Acknowledgment
-40 0
10
20
30
933
40
50
P [N] Fig. 12. Mean value of relative error of the initial tangential contact stiffness between the elastic foundation and reference models for surfaces with an autocorrelation length of 40 mm if the elastic foundation model is calibrated so as to give a contact radius equal to that given by the Hertzian solution and an initial tangential contact stiffness equal to that given by Mindlin’s analytical solution. The standard deviation for each class of surfaces is indicated by error bars.
problem is that there exists no parameter that fully describes surface roughness. Although two surfaces may have the same root mean square roughness and autocorrelation length, they can display very different contact characteristics. This lack is handled by studying not one, but a group of surfaces with the same parameters. The results presented here can be used to estimate the extent of the error in the elastic foundation model that depends on the surface roughness. The main conclusion is that the results of the elastic foundation model do not agree well with those of the reference model. The general trend is that the elastic foundation model gives a larger area of contact, lower and more evenly distributed pressure, and lower tangential contact stiffness than the reference model does. The elastic foundation model seems to be particularly misleading regarding the prediction of real area of contact and maximum pressure, where the root mean square roughness of the surface must be below
This research was performed as part of the INTEFACE project, financially supported by the Swedish Foundation for Strategic Research via the ProViking program.
References [1] Liang X, Linqing Z. A numerical model for the elastic contact of three-dimensional real rough surfaces. Wear 1991;148:91–100. [2] Ren N, Lee SC. Contact simulation of three-dimensional rough surfaces using moving grid method. J Tribol 1993;115:597–601. [3] Kalker JJ. Three-dimensional elastic bodies in rolling contact. Dordrecht: Kluwer Academic Publishers; 1990. [4] Jaeger J. Elastic impact with friction. Doctoral thesis, University of Delft, Delft, The Netherlands, 1992. [5] Bjo¨rklund S, Andersson S. A numerical method for real elastic contacts subjected to normal and tangential loading. Wear 1994;179: 117–22. [6] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1992. [7] Po˜dra P, Andersson S. Wear simulation with the Winkler surface model. Wear 1997;207:79–85. [8] Spiegelberg C, Andersson S. Simulation of friction and wear in the contact between the valve bridge and rocker arm pad in a cam mechanism. Wear 2006;261:58–67. [9] Carter FW. On the action of a locomotive driving wheel. Proc Royal Soc A 1926;112:151–7. [10] Mindlin RD. Compliance of elastic bodies in contact. J Appl Mech 1949;259–268. [11] Patir N, Cheng HS. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. ASME J Lubr Technol 1978;100:12–7.