Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 45 Finite element simulations of fretting contact systems G. Shi, D. Backman & N. Bellinger Structures and Materials Performance Laboratory, Institute for Aerospace Research, National Research Council, Canada Abstract This paper presents the results of finite element simulations of contact stresses in fretting fatigue tests. Finite element models were developed using MSC/MARC finite element software. Convergence studies with different mesh configurations were carried out to verify the model by comparing the results with theoretical solutions. To further validate the FE model, simple coupon tests were carried out. Both photoelastic and pressure film measurements were taken during these tests and the experimental test data was compared with the FE results. After the FE model was verified, various simulations were carried out to quantify the effect of varying the applied normal load. The stress distributions including normal, shear tractions and tangential stress were determined and compared with analytical solutions in four different load cases. The obtained FE results showed that the combination of the normal, shear and bulk forces have significant effects on the contact traction and stress distributions. Keywords: fretting contact, finite element simulation, contact tractions and stress, photoelasticity, pressure film, experimental stress analysis. Introduction Fretting can occur between two tight fitting surfaces that are subjected to small amplitude cyclic loads when there is relative motion between the two surfaces. These fretted regions are highly sensitive to fatigue cracking and this kind of damage has been observed between the faying surfaces in riveted lap joints, and is considered a crack nucleation mechanism. Due to the concerns for aircraft safety and integrity, a great deal of research has been carried out to determine the nature of fretting fatigue [-9].
46 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII A research project is currently underway to include fretting fatigue models into the holistic structural integrity design paradigm (HOLSIP). This design paradigm is capable of determining the life cycle of a structure from the early stages of damage formation to final failure. The aim of this study is to investigate the mechanism of crack nucleation and growth under the fretting fatigue condition. To accomplish this, the stress state in the contact region, including the contact normal pressure, the contact shear traction and tangential stress need to be determined using finite element techniques. In this research work, finite element models of a fretting contact problem were generated to verify the modelling techniques. Convergence studies with different mesh configurations were carried out to investigate the contact behaviour and the results were compared to analytical solutions. To further validate the finite element model, simple coupon tests were carried out with experimental strain and pressure measurements obtained using photoelastic strain analysis and pressure film measurement techniques, respectively. The contact normal pressure distributions obtained by FE analysis were compared to the pressure distribution measured experimentally using pressure sensitive film. The predicted shear strain distributions on the fretting pad and specimen were compared with experimental measurements obtained from photoelastic analysis. Reasonable agreement was found between the numerical predictions and the measured data. The sources of the discrepancies between the numerical predictions and the measured data were discussed. Using the computer model that resulted in the best agreement with the analytical solution, a simulation of a fretting fatigue test was performed. Finite element simulations were carried out by applying different load cases that included the normal, shear and bulk forces. The stress distributions including the contact normal, shear tractions and tangential stress were predicted and compared with analytical solutions under different load cases. 2 Problem description The geometrical configuration used to model the process of fretting fatigue is similar to that shown in Figure. Two fretting pads are loaded against a specimen of finite thickness by a normal force P and a cyclically varying tensile stress σ is induced into the specimen by some form of remote loading. A cyclically varying tangential force Q is also required when a single pad is used on each side of the specimen. The radius of the pad, R, was 27 mm (5 inch) and the thickness of the actual specimen, t, was 2.7 mm (.5 inch). In the actual test, pads and specimen were fabricated from 224-T3 bare aluminium: Young s Modulus (E), 68,9 Mpa (, ksi) and Poisson ratio (ν) of.3. Based on the Hertz contact theory, which was developed for the contact problem between a cylindrical (pad) and a flat body (specimen), the distributed contact pressure could be expressed as (Johnsion, [5]). 2 / 2 p ( x ) = p ( ( x / a ) ) () where p is the maximum pressure (Hertzian Peak Pressure) defined as
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 47 and the contact length a 2 4 2 P p = (2) π a PR = (3) * π E Figure : Fretting contact configuration. in which 2 υ υ = + (4) * E E E 2 From Eqns (2)-(4), the effective Young s Modulus became E* = 37864 Mpa (5495.5 Ksi), the estimated contact semi-width was a =.55mm (.689 inch), and the maximum pressure was p = 23.54 Mpa (33.46 Ksi). 3 Validation of finite element model 3. Convergency study In this study, the finite element models were created using MSC/PATRAN (preand post processor) and analyzed using MSC/MARC. To verify the FE model and compare with coupon test measurements, the finite element model was created in such a way that only one pad and half of the specimens were modelled. The boundary conditions were applied to constrain the displacements on the bottom line of the specimen in the y-direction and the displacements along the centre vertical line of the pad and specimen in the x-direction. To prevent the rolling of the cylinder pad, all the nodes on the top of the cylinder were forced to keep consistent displacements using multiple point constraints (MPC) conditions in the vertical direction. The normal load was uniformly applied to the top surface, as shown in Figure 2. To perform a convergence study, three mesh configurations were generated. Each of them consisted of a different number of elements or different seed ratios in contact areas. Mesh configuration one (mesh-) was a uniform mesh case, in which the model consisted of 248 four-node quadrilateral elements associated 2 2
48 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII with 374 nodes resulting in 748 degrees of freedom. The second mesh configuration (mesh-2) consisted of a finer mesh with 84 four-node quadrilateral elements associated with 995 nodes resulting in 399 degrees of freedom. The mesh distribution was seeded with a :8 ratio in the contact area. The third mesh case (mesh-3) was created with the same number of elements as in the second case, but with a finer mesh seed ratio in the contact area of :, The FE model with the third mesh configuration is shown in Figure 2. Figure 2: Finite element model. A comparison between the normal contact pressures calculated using the Hertz theory and the FE analysis with three different mesh cases is shown in Figure 3. From these figures, it can be seen that the relatively coarse but uniform mesh configuration (mesh-) generated larger errors for both the contact pressure magnitude and contact length. The finer mesh with high seed ratio in contact area (mesh-3) provided more accurate contact pressure and contact length when compared to the Hertz solution..2 Normalized contact pressure.8.6.4 Hertz Theory FEM: Mesh- FEM: Mesh-2.2FEM: Mesh-3 -.5 - -.5.5.5 Normalized contact length Figure 3: Contact pressure comparison.
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 49 3.2 Comparisons with experimental test measurements The purpose of this experiment was to re-create the fretting fatigue setup in such a way as to allow experimental measurements to be made using both photoelasticity and pressure sensitive film. 3.2. Experimental setup A Baldwin electromechanical testing machine was selected to provide the necessary load steps. A test coupon and a 27 mm (5 inch) radius fretting pad that were previously machined were used to carry out this test, see Figure 4(a). Photoelastic coating.5mm thick was water-jet cut and bonded to the surface of both the fretting specimen and pad, Figure 4(b). Two finely ground platens were machined, one provided a smooth base for the fretting specimen and the other provided a uniformly distributed load for the fretting pad. An initial experiment was performed using pressure film to ensure that the loading surface and base were parallel to one another. Shim stock was used to adjust the parallelism of the system. (a) (b) Figure 4: (a) Photograph of the various coupons used in the test, (b) Close-up of test setup showing photoelastic coatings bonded to test coupon and pad. 3.2.2 Contact pressure comparison A pressure film stack consisting of film ranging from superlow to superhigh was used to measure the pressure distribution under a 7,7 N (,6 lbf) load. Loading profiles were taken in a direction parallel to the long axis of the specimen and the results are presented in Figure 5. Since the finite element results did not compare well to the results obtained from the pressure film for both the magnitude of the contact pressure as well as the contact area, it was decided to carry out another finite element analysis, which simulated the influence of the polyester film on the contact pressures. These results are compared to the pressure film results for two trials (trial and 2) in Figure 5. As can be seen from this figure, the inclusion of the pressure film stack into the finite element (FE) model significantly decreased the contact pressure between the fretting pad and coupon, while slightly increasing the contact area. The difference in the results for the new FE model and pressure film may be
5 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII related to the material properties that were used for the film stack in the numerical model. Also since the individual films were very thin, only the full thickness of the film stack was modelled, which could also have affected the results. Contact pressure (Mpa) 25 2 5 5 P = 6 lbf P = 6 lbf, film Trail (Pressure Film) Trail 2 (Pressure Film) -6-4 -2 2 4 6 Contact length (mm) Figure 5: Comparison between pressure film and FEA. 3.2.3 Maximum shear strain comparison A fretting specimen and pad with a photoelastic coating were tested. The maximum shear strain distribution were measured and compared to those obtained from the finite element analysis. The results at a load level of 4,448 N (,lbf) are shown in Figure 6. The shear strains were extracted along the line at the centre of the specimen and pad from the bottom to the top of the specimen and pad. 4 Simulation of fretting contact systems To predict the stress distributions that would be present in the actual fretting test specimen, the fretting test configuration and loading conditions shown in Figure were used in the FE analysis. Two-dimensional plane strain finite element models were generated and four cases were examined using the same loads as those from the experiments at IAR/NRC. The cases studied were: Load case I: only normal load on the pad, P=28 N/mm (3,2 lbf/in); Load case II: Same normal load on the pad, add σ =93 MPa (28 Ksi) bulk stress on the specimen; Load case III: Same normal load on the pad, shear load 5% of normal load, Q =.5P = 4 N/mm (,6 lbf/in). No bulk stress; Load case IV: Same normal load on the pad, shear load 5% of normal load, Q =.5P = 4 N/mm (,6 lbf/in). Add σ = 93 Mpa (28 Ksi) bulk stress on the specimen.
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 5 Max. shear strain (microstrain) 4 FEA 2 Photoelastic 8 6 4 On specimen 2 On pad -5-2 -9-6 -3 3 6 9 2 Distance from contact surface (mm) Figure 6: Shear strain comparison between photoelastic and FEA. Due to symmetry, only one-half of the specimen and pad were modelled, as shown in Figure 7, and the thickness used for the model was b = t/2 or 6.35 mm (.25 inch). Symmetric boundary conditions were applied along the bottom surface of the model to constrain all the displacements in the y direction. The displacements on the left end of the specimen were restricted in the x direction. To prevent rolling of the cylinder pad, all the nodes on the top surface of the pad were forced to keep consistent displacements by multiple point constraints (MPC) conditions in the y-direction. The normal, shear and bulk forces were uniformly applied to the corresponding surfaces. The coefficient of friction for the specimen and pad was assumed to be.65. P Q d x = F/2 d y = y x Figure 7: Symmetric FE model.
52 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 4. Contact pressure In the following figures, the contact pressure, friction stress and stress component S xx was normalized by p = 23.54 Mpa (33.46 Ksi). Figure 8 shows the comparisons of the normal contact tractions calculated by the Hertz theory without friction and the FE analyses with a friction coefficient of.65. The FE results were obtained using the four different load combinations. From this figure, it can be seen that the bulk and shear forces had very little effect on the contact normal tractions. Contact Pressure p(x)/p.2.8.6 Hertz.4Theory FEM - Normal only FEM -Normal + Bulk FEM.2 -Normal + Shear FEM -Normal + Shear + Bulk -.5 - -.5.5.5 x/a Figure 8: 4.2 Contact friction stress Contact normal pressure. Figure 9 shows a comparison in variations of the contact friction stresses, as determined from an analytical formulation (Nowell [9]) and the FE analysis with a friction coefficient of.65. The Mindlin solution represents the analytical results under normal and shear forces, but without bulk tension. With bulk tension, the Mindlin solution shows a non-symmetric distribution of the friction stress due to the large eccentricity. The FE results were obtained using the four different load combinations. From Figure 9, it can be seen that the bulk stress generated larger contact shear tractions than the normal-only load case, but there was no eccentricity in this case. The shear force generated even larger shear tractions on the contact surface and led to an eccentricity in the shear traction in the contact region. The combination of bulk and shear forces produced the largest shear tractions with an eccentricity. The eccentricity was e = aσ / (4µ p ) =.58 mm (.2 inch), which is relatively large compared to the contact semi-length, a =.6 inch. 4.3 Stress component Sxx on contact region The variation in the stress component S xx along the contact surface (y=b) using the four load cases are shown in Figure. It can be seen that under the normal-
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 53 only load case, the stress component was symmetric and compressive within the contact area - x/a. However, for the normal and bulk load case, the stress component S xx increased in magnitude and became tensile in the area of contact -.5<x/a>.5 but remained symmetrical. For the normal-shear load case, the stress component S xx was non-symmetrical and became tensile around the contact area x/a=, while the stress component S xx was compressive in the area x/a< and a larger compressive stress occurred at the contact edge, x/a=-. Under the combination of normal, shear and bulk forces, the stress component S xx became non-symmetrical and a large tensile stress occurred around the contact area x/a=. Mindlin solution.4 Mindlin solution + Bulk Friction Stress q /µp Normal only Normal + Shear.2.8.6 VC.4.2 Normal + Bulk Normal + Shear + Bulk -.5 - -.5.5.5 x/a Figure 9: Friction stress comparison. 2 Sxx/p -4. -3. -2. -... 2. 3. 4. - Normal only Normal + Bulk Normal + Shear Normal + Shear + Bulk -2 x/a Figure : Variation of stress component S xx along y = b.
54 Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII Figure shows the variation in the stress component S xx along the vertical path through the thickness (y=b to y=.) of the specimen at x=a under the four load cases. From this figure, it can be clearly seen that the bulk load played a significant role in determining the magnitude of the stress. For those load cases that contained the bulk load, the stress component S xx at the layer close to the contact surface (y=b) was higher than the stress close to the bottom surface, which was much higher than the stress resulting from those load cases that did contain a bulk load. For the load case that contained the shear force, the stress component also increased at the layer close to the contact surface..6.2 Normal only Normal + Bulk Normal + Shear Normal + Shear + Bulk Sxx/p.8.4 -.4 5 Summary 2 3 4 Depth, (b-y)/a Figure : Variation of stress component S xx along x = a. Finite element simulations of fretting contact systems were carried out using MSC/Marc finite element software. The FE model and simulation results were compared and verified with analytical solutions and test measurements. The contact tractions and tangential stress distributions in the contact region were predicted under different load cases. The effects that the shear force and bulk load had on the stress distributions were demonstrated. Some conclusions that can be drawn from this investigation are: The contact tractions and stress distributions in the specimen are highly dependent on the combination of load conditions. There is no eccentricity that occurs under the load case of normal plus bulk forces (without shear force). An eccentricity occurred under the combination of normal, shear and bulk forces, and the amount of the eccentricity were dependent on the ratio of the bulk stress and the maximum normal pressure.
Computer Methods and Experimental Measurements for Surface Effects and Contact Mechanics VII 55 The largest tensile stress occurred at the contact edge in the specimen, which is the driving force for crack formation and growth. Acknowledgements This work has been carried out under IAR Program 33, Aerospace Structures, Project 46_QJ_39, IAR/APES Collaboration on fretting fatigue Program. Thanks to Dr. Scott Prost-Domasky of APES Inc., USA for providing useful information and valuable discussions. References [] McVeigh, P.A. and Farris, T.N., Finite element analysis of fretting stresses, Journal of Tribology, Vol. 9, Oct. 997. [2] Hills, D.A. and Mugadu, A., An overview of progress in the study of fretting fatigue, Journal of Strain Analysis, Vol. 37, No.6, 22. [3] Nicholas, T., Hutson, A., John, R., and Olson, S., A fracture mechanics methodology assessment for fretting fatigue, International Journal of Fatigue, 25, 23. [4] Saez, R., Mugadu, A., Fuenmayor, J and Hills, D.A., Frictional shakedown in a complete contact, Journal of Strain Analysis, Vol. 38, No. 4, 23. [5] Johnson, K.L., Contact mechanics, Cambridge University Press, 987 [6] Suresh, S., Fatigue of materials, 2 nd Edition, 998. [7] Hills, D.A., Nowell, D. and O Connor, J. J., On the mechanics of fretting fatigue, Wear, Vol. 25, 988. [8] Nowell, D. and Hills, D.A., Crack initiation criteria in fretting fatigue, Wear, Vol.36, 99. [9] Nowell, D. and Hills, D.A., Mechanics of fretting fatigue tests, International Journal of Mechanical Science, Vol. 29, No. 5, 987.