An Experimental Study and Fatigue Damage Model for Fretting Fatigue Aditya A. Walvekar Ph.D. Research Assistant

Outline Motivation Objective Fretting Fatigue Test Rig Experimental results Fatigue Damage Model Fretting Fatigue Life Predictions Summary Future work

Fretting Fatigue Test configuration* Motivation Fretting is associated with the small amplitude relative oscillatory motion between two solid surfaces in contact Fretting fatigue is a damage mechanism observed in a machine components subjected to fretting in tandem with fluctuating bulk stresses If the material is concurrently subjected to partial slip fretting and fluctuating bulk loading, stress concentration at the contact region results in premature nucleation and acceleration of crack growth when compared to fatigue situations without fretting Fretting Fatigue Test configuration* * ASTM E2789-10 : Standard Guide for Fretting Fatigue Testing

Objective Experimental investigation of the fretting fatigue behavior of AISI 4140 vs. Ti-6-4 in a cylinder-on-flat contact configuration Analyze the effect of bulk stress on the fretting fatigue life at a fixed normal load Analyze the crack propagation i.e. crack length vs. number of cycles Develop a model based on damage mechanics to analytically investigate fretting fatigue Incorporate Voronoi tessellation to account for the randomness of the material microstructure and conduct life variability studies

Fretting Fatigue Test Rig 𝑭 𝒃 − 𝑭 𝒖 =𝟐 𝑭 𝑻 Fretting fatigue fixture mounted on MTS machine Schematic of fretting fatigue test rig A fretting test fixture was designed and developed which was coupled with an MTS machine to impose the fretting fatigue damage

Picture of contact pads and specimen assembled in the test rig Experimental Results Picture of contact pads and specimen assembled in the test rig Fretting and bulk stress vs. life Fretting fatigue tests were conducted in a cylinder-on-flat contact configuration under completely reversed constant-amplitude axial load control conditions (R = -1) at 5 Hz frequency The amplitude of the axial bulk stress was varied from 100 MPa to 600 MPa while the normal force was held constant at 11 kN (peak Hertzian pressure of 3 GPa) Fretting stress ( 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 ) at the trailing edge of the contact is calculated using – 𝝈 𝒇𝒓𝒆𝒕𝒕𝒊𝒏𝒈 = 𝝈 𝟎 +𝟐 𝒑 𝒉 𝛍 𝑭 𝑻 𝑭 𝑵

Determination of estimated crack initiation (Bulk Stress = 348 MPa) Estimated crack initiation life – 34000 cycles First visible crack observed at 33420 cycles with a length of 765 microns Pictures of the crack growth taken as the test is running (Bulk Stress = 348 MPa) Crack length vs. life cycles (Bulk Stress = 348 MPa)

Coefficient of Friction Measurement A fretting test was performed in the gross slip regime to determine the coefficient of friction The specimen was only held with the bottom grip while the top end of specimen was free Completely-reversed sinusoidal displacement at a frequency of 1 Hz was applied to the specimen 𝐶𝑂𝐹= 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 𝑎𝑡 𝑔𝑟𝑜𝑠𝑠 𝑠𝑙𝑖𝑝 𝑁𝑜𝑟𝑚𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 Fretting wear test at gross slip (displacement amplitude = 150 μm) Disp. Amp. (micron) Normal force (N) Contact stress (MPa) Average tangential load at gross slip (N) COF 150 417.2 585.45 250 0.60

Finite Element Model Randomness of material microstructure topology is simulated using Voronoi tessellation to account for the variability in fretting fatigue life The sinusoidal reaction stress with amplitude “σreaction” is applied on the left edge of the lower body in phase with the bulk stress to model FT 𝝈 𝒓𝒆𝒂𝒄𝒕𝒊𝒐𝒏 = 𝝈 𝒐 − 𝑭 𝑻 𝑭 𝑵 𝒍𝒊𝒏𝒆 𝒇𝒊𝒕 ∗ 𝑭 𝑵 𝑨 𝒔 Finite element mesh using Voronoi Tessellation The geometry and the applied loading conditions (a = 365 μm) FT/FN obtained from experiments and FE model

Model Validation To validate the stress distribution obtained from the FE model, shear and tangential stress distribution on the contact surface were compared with the analytical solution Comparison of shear stress and normalized tangential stress distribution on the contact surface at the positive peak of the fretting cycle obtained using FE model and analytical solution. (Bulk Stress = 400 MPa)

Stresses resolved along the Voronoi grain boundaries Fatigue Damage Model In order to introduce randomness into the life predictions, the alternating normal stress ( 𝜎 𝑛 𝑎 ) acting along the Voronoi grain boundary during the fretting cycle is assumed to cause damage Damage evolution rate equation – Alternating Normal Stress – 𝒅𝑫 𝒅𝑵 = 𝝈 𝒏 𝒂 𝝈 𝑹 (𝟏−𝐃) 𝒎 𝝈 𝒏 𝒂 = 𝝈 𝒏 𝒎𝒂𝒙 − 𝝈 𝒏 𝒎𝒊𝒏 𝟐 σ n max and σ n min are the maximum and the minimum normal stresses acting on the Voronoi grain boundary during a fretting cycle Stresses resolved along the Voronoi grain boundaries

Variation of Elasticity Modulus Increase in the internal damage as the fatigue cycles progress, manifests as the reduction in the modulus of elasticity. Elastic modulus of the damaged element - 𝑬 =𝑬 𝟏−𝑫 Rearranging, 𝑫=𝟏− 𝑬 𝑬 Accurate strain measurements are important for measuring elasticity modulus so a strain gauge was installed on in the constant cross sectional area region of the specimen Iteration No. E (1-D) (GPa) D 1 176.0 2 167.1 0.051 3 160.6 0.088 4 157.3 0.106 5 155.8 0.115 6 155.2 0.119 7 154.5 0.123 8 0.122 Stress vs. strain plot at various cycles for the variation of elasticity modulus test

Evaluation of Damage Parameters The peak in the tensile stress at the trailing edge of the contact ( 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 ) drives the crack initiation in fretting fatigue. The critical stress component causing the damage is assumed to be the fretting stress The damage parameters σR and m were evaluated using the maximum fretting stress 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 and fretting fatigue life data from experiments. Applying a power law curve fit to the data: 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 =𝐴 𝑁 𝑏 where, 𝐴=11440 𝑀𝑃𝑎, 𝑏=−0.13 𝑑𝐷 𝑑𝑁 = 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 𝜎 𝐑 (1−𝐷) 𝑚 Integrating, 0 𝑁 𝑓 𝑑𝑁 = 0 𝐷 𝑐𝑟𝑖𝑡 𝜎 𝑅 (1−𝐷) 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 𝑚 𝑑𝐷 𝑁 𝑓 = 𝜎 𝑅 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 𝑚 1 𝑚+1 − 1− 𝐷 𝑐𝑟𝑖𝑡 𝑚+1 𝑚+1 Rearranging, 𝜎 𝑓𝑟𝑒𝑡𝑡𝑖𝑛𝑔 = 𝜎 𝑅 1 𝑚+1 − 1− 𝐷 𝑐𝑟𝑖𝑡 𝑚+1 𝑚+1 1 𝑚 𝑁 𝑓 − 1 𝑚 Comparing Coefficients – 𝑚=− 1 𝑏 , 𝜎 𝑅 =𝐴 1 − 1 𝑏 +1 − 1− 𝐷 𝑐𝑟𝑖𝑡 − 1 𝑏 +1 − 1 𝑏 +1 𝑏 𝒎=𝟕.𝟓, 𝝈 𝑹 = 𝟏𝟔𝟎𝟔𝟏 𝑴𝑷𝒂

Fretting Fatigue Life Predictions Fatigue Damage model was used for predicting fretting fatigue life of 30 randomly generated microstructure domains for four different loading conditions Degree of scatter is quantified using two-parameter Weibull probability distribution Material properties used in the analysis Comparison between the fretting fatigue lives from model and experiments Loading conditions applied and predicted Weibull slope and strength parameters Weibull probability plot for fretting fatigue lives

Summary A fretting fixture was designed, built and used with an MTS 810 machine simulating the fretting fatigue in a cylinder-on-flat configuration For a fixed contact pressure, the fretting fatigue life decreased with increasing bulk stress A fatigue damage finite element model was proposed to replicate the fretting fatigue experiments and numerically estimate the fretting fatigue life The fretting fatigue lives predicted by the fatigue damage model are in good agreement with the experimental results The predicted fatigue life data displayed a larger degree of scatter for the lower bulk stress when the contact pressure is fixed

Future Work Modify the fatigue damage model to include crack propagation Evaluate the effects of shot-peening, residual stress on Fretting Fatigue behavior Analyze the effects of inclusions and voids on the fretting fatigue life Incorporate plasticity in the fatigue damage model