Effects of Crystal Elasticity on Rolling Contact Fatigue Neil Paulson Ph.D. Research Assistant Title slide (option 2)

Outline Motivation and Background Crystal Structure Definitions Polycrystalline Material Model Steel Material Stiffness Model Hertzian Contact Modeling RCF Relative Life Study Future Work

Background and Motivation Material heterogeneity can play a role in rolling contact fatigue failure, Microstructure Topology Raje, Jalalahmadi, Slack, Weinzapfel, Warhadpande, Bomidi Voids or inclusions Microstructure anisotropy Microstructures are composed of many grains of multiple crystal phases The relation between stress and strain depend on how atoms are arranged in the crystal phase Grain Micrograph from electron backscatter diffraction (EBSD) scan showing grain orientations1 Objective Extend current RCF FE model to incorporate the effects of crystal elasticity on RCF 1“Bruker Quantax EBSD Analysis Functions” Bruker Corp., 2013

Homogenous & Isotropic Material Models Model for the bulk material behavior Material stiffness does not depend on the direction Infinite planes of symmetry Only two independent elastic constants are needed to define the stress strain response Stress-Strain Equations 𝝈 𝒙 𝝈 𝒚 𝝈 𝒛 𝝉 𝒙𝒚 𝝉 𝒙𝒛 𝝉 𝒚𝒛 = 𝑪 𝟏𝟏 𝑪 𝟏𝟐 𝑪 𝟏𝟐 𝟎 𝟎 𝟎 𝑪 𝟏𝟐 𝑪 𝟏𝟏 𝑪 𝟏𝟐 𝟎 𝟎 𝟎 𝑪 𝟏𝟐 𝑪 𝟏𝟐 𝑪 𝟏𝟏 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟏𝟏 − 𝑪 𝟏𝟐 𝟐 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟏𝟏 − 𝑪 𝟏𝟐 𝟐 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟏𝟏 − 𝑪 𝟏𝟐 𝟐 𝝐 𝒙 𝝐 𝒚 𝝐 𝒛 𝜸 𝒙𝒚 𝜸 𝒙𝒛 𝜸 𝒚𝒛 𝑪 𝟏𝟏 = 𝑬 𝟏−𝝂 (𝟏+𝝂)(𝟏−𝟐𝝂) 𝑪 𝟏𝟐 = 𝝂𝑬 (𝟏+𝝂)(𝟏−𝟐𝝂) 𝑪 𝟒𝟒 =𝑮= 𝑬 𝟐(𝟏+𝝂)

Cubic Crystal Structure Most widely used to incorporate crystal elasticity Elastic constants for many materials are available in literature Orientation of the crystal becomes important The shear modulus is decoupled from E and ν; otherwise the equations remain identical to isotropic material model 3 elastic constants are needed to define the stress strain response Stress-Strain Equations 𝝈 𝒙 𝝈 𝒚 𝝈 𝒛 𝝉 𝒙𝒚 𝝉 𝒙𝒛 𝝉 𝒚𝒛 = 𝑪 𝟏𝟏 𝑪 𝟏𝟐 𝑪 𝟏𝟐 𝟎 𝟎 𝟎 𝑪 𝟏𝟐 𝑪 𝟏𝟏 𝑪 𝟏𝟐 𝟎 𝟎 𝟎 𝑪 𝟏𝟐 𝑪 𝟏𝟐 𝑪 𝟏𝟏 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟒𝟒 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟒𝟒 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝑪 𝟒𝟒 𝝐 𝒙 𝝐 𝒚 𝝐 𝒛 𝜸 𝒙𝒚 𝜸 𝒙𝒛 𝜸 𝒚𝒛 𝑪 𝟏𝟏 = 𝑬 𝟏−𝝂 (𝟏+𝝂)(𝟏−𝟐𝝂) 𝑪 𝟏𝟐 = 𝝂𝑬 (𝟏+𝝂)(𝟏−𝟐𝝂) 𝐂 𝟒𝟒 =𝑮 Shear modulus is independent of E and ν

Modeling Polycrystalline Aggregates Each individual crystal has a unique orientation Isotopic Stiffness Matrix Cubic Stiffness Matrix 𝟐𝟔𝟗 𝟏𝟏𝟓 𝟏𝟏𝟓 𝟎 𝟎 𝟎 𝟏𝟏𝟓 𝟐𝟔𝟗 𝟏𝟏𝟓 𝟎 𝟎 𝟎 𝟏𝟏𝟓 𝟏𝟏𝟓 𝟐𝟔𝟗 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟐𝟎𝟓 𝟏𝟑𝟖 𝟏𝟑𝟖 𝟎 𝟎 𝟎 𝟏𝟑𝟖 𝟐𝟎𝟓 𝟏𝟑𝟖 𝟎 𝟎 𝟎 𝟏𝟑𝟖 𝟏𝟑𝟖 𝟐𝟎𝟓 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟏𝟐𝟔 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟏𝟐𝟔 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟏𝟐𝟔 Euler Angles rotate the local stiffness matrix into the global coordinate frame 𝑪 𝒈 = 𝑹 𝒛" (𝑹 𝒙′ (𝑹 𝒛 𝑪 𝑹 𝒛 𝑻 ) 𝑹 𝒙 ′ 𝑻 ) 𝑹 𝒛" 𝑻 𝟐𝟔𝟗 𝟏𝟏𝟓 𝟏𝟏𝟓 𝟎 𝟎 𝟎 𝟏𝟏𝟓 𝟐𝟔𝟗 𝟏𝟏𝟓 𝟎 𝟎 𝟎 𝟏𝟏𝟓 𝟏𝟏𝟓 𝟐𝟔𝟗 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟕𝟕 𝟐𝟗𝟓 𝟏𝟐𝟓 𝟔𝟎 𝟒 𝟐𝟒 𝟐𝟓 𝟏𝟐𝟓 𝟐𝟓𝟎 𝟏𝟎𝟓 −𝟏𝟒 −𝟏𝟔 −𝟒𝟔 𝟔𝟎 𝟏𝟎𝟓 𝟑𝟏𝟓 𝟏𝟎 −𝟖 𝟐𝟏 𝟒 −𝟏𝟒 𝟎𝟏𝟎 𝟏𝟏𝟑 𝟐𝟓 −𝟏𝟔 𝟐𝟒 −𝟏𝟔 −𝟖 𝟐𝟓 𝟒𝟗 𝟏𝟎 𝟐𝟓 −𝟒𝟔 𝟐𝟏 −𝟏𝟔 𝟏𝟎 𝟗𝟒 Isotropic stiffness matrix is identical after rotation Cubic stiffness matrix becomes fully anisotropic after rotation

Material & Model Verification A representative model of polycrystalline material was developed using Voronoi cells to represent individual grains The stiffness matrix of each grain was rotated to the global coordinates 𝜹𝒚=𝟎 𝜹𝒙=𝟎 𝜹𝒙=𝒄 𝒙 𝒚 Uniaxial Strain was applied Reaction forces were measured 𝑭 𝒙 𝑭 𝒚 𝒙 𝒚 Global material properties of the model were evaluated1: 𝑬 𝒃 = 𝝈 𝒙 + 𝟐 𝝈 𝒚 𝝈 𝒙 − 𝝈 𝒚 𝝐 𝒙 ( 𝝈 𝒙 + 𝝈 𝒚 ) 𝝂 𝒃 = 𝝈 𝒚 𝝈 𝒙 + 𝝈 𝒚 1 Toonder, J, Dommelen, J, Baaijens, F. The relation between single crystal elasticity and the effective elastic behaviour of polycrystalline materials: theory, measurement and computation, Modelling Simul. Mater. Sci. Eng.

FEA results match isotropic constants Steel Material Model FEA Model Bulk Properties Eavg 199.9 200 Estd 8.91 νavg 0.291 0.30 νstd 0.009 FEA results match isotropic constants Example Cases

Rolling Contact Fatigue Domain Strong stress gradients inside grains require modifications to FE domain Isotropic Domain Anisotropic Domain Voronoi Centroid Discretization Fixed Element Area Discretization Anisotropic Isotropic Linear Strain Elements Constant Strain Elements

Anisotropic Hertzian Contact Isotropic Material Hertzian Centerline Stresses Anisotropic Material Anisotropic stress profiles deviate from isotropic stresses Stress concentrations occur at grain boundaries due to orientation change

Rolling Contact Fatigue Life Equations Microstructures models simulate randomness from experimental testing Lundberg-Palmgren equation can be reduced for constant survivability and volume: 𝑵~ 𝒛 𝒉 𝝉 𝒄 𝒄=𝟐.𝟏𝟏 𝒉=𝟏𝟎.𝟑𝟑 Three different numerical models have been proposed with Isotropic Voronoi Element microstructure 2D Discrete Element Model 2D Finite Element Model 3D Finite Element Model 𝝉=𝒈𝒓𝒂𝒊𝒏 𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆 𝒔𝒉𝒆𝒂𝒓 𝒔𝒕𝒓𝒆𝒔𝒔 𝒓𝒆𝒗𝒆𝒓𝒔𝒂𝒍 𝝉=𝒊𝒏𝒑𝒍𝒂𝒏𝒆 𝒔𝒉𝒆𝒂𝒓 𝒔𝒕𝒓𝒆𝒔𝒔 𝒓𝒆𝒗𝒆𝒓𝒔𝒂𝒍 𝝉=𝒈𝒓𝒂𝒊𝒏 𝒊𝒏𝒕𝒆𝒓𝒇𝒂𝒄𝒆 𝒓𝒆𝒔𝒐𝒍𝒗𝒆𝒅 𝒔𝒉𝒆𝒂𝒓 𝒓𝒆𝒗𝒆𝒓𝒔𝒂𝒍 𝑾𝒆𝒊𝒃𝒖𝒍𝒍 𝑺𝒍𝒐𝒑𝒆 3.36 𝑾𝒆𝒊𝒃𝒖𝒍𝒍 𝑺𝒍𝒐𝒑𝒆 2.65 𝑾𝒆𝒊𝒃𝒖𝒍𝒍 𝑺𝒍𝒐𝒑𝒆 4.55 𝐈𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 𝐩𝐫𝐨𝐩𝐞𝐫𝐭𝐢𝐞𝐬 𝐫𝐞𝐬𝐮𝐥𝐭 𝐢𝐧 𝐰𝐞𝐢𝐛𝐮𝐥𝐥 𝐬𝐥𝐨𝐩𝐞𝐬 𝐨𝐯𝐞𝐫 𝐝𝐨𝐮𝐛𝐥𝐞 the experiments of Lundberg and Palmgren

Modeling Rolling Contact Hertzian Line Contact Load 21 Loading Steps Load Transverses Anisotropic Region 𝝉 𝒙𝒚, 𝒎𝒂𝒙 𝝉 𝒙𝒚, 𝒎𝒊𝒏 𝚫𝝉 𝒙𝒚 𝚫𝝉 𝒙𝒚 was evaluated for each element Maximum 𝚫𝝉 𝒙𝒚 value and location recorded

Shear Stress Reversal Results 33 crystal orientation maps were run for a given topological model Isotropic shear stress matches theory Anisotropic shear stress increased by orientation mismatch Experimentally Observed μ-crack Bounds1 Isotropic Shear Stress independent of grain boundaries Maximum Shear Stress on Voronoi Boundaries 1 Chen, Q., Shao, E., Zhao, D., Guo, J., & Fan, Z. (1991). Measurement of the critical size of inclusions initiating contact fatigue cracks and its application in bearing steel. Wear, 147, 285–294.

RCF Relative Life Relative life equation was used to determine bearing fatigue life 𝑵~ 𝝉 𝒄 𝒛 𝒉 𝒄=𝟐.𝟏𝟏 𝒉=𝟏𝟎.𝟑𝟑 Shear Stress results from crystal orientations were used to create Weibull plot of RCF life 33 Different topological microstructures

𝐀𝐧𝐢𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 𝐦𝐨𝐝𝐞𝐥 𝐦𝐚𝐭𝐜𝐡𝐞𝐬 𝐞𝐱𝐩𝐞𝐫𝐢𝐦𝐞𝐧𝐭𝐚𝐥 𝐫𝐞𝐬𝐮𝐥𝐭𝐬 𝐞𝐱𝐭𝐫𝐞𝐦𝐞𝐥𝐲 𝐰𝐞𝐥𝐥 RCF Relative Life All topological domain results combined into one RCF relative life plot Weibull Distribution Function 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐹𝑎𝑖𝑙𝑢𝑟𝑒=1− 𝑒 − 𝑁−𝛼 𝛽 𝜖 𝛼=𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 (𝑚𝑖𝑛. 𝑙𝑖𝑓𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑) 𝛽=𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ (𝑙𝑖𝑓𝑒) 𝜖=𝑠𝑙𝑜𝑝𝑒 (𝑠𝑐𝑎𝑡𝑡𝑒𝑟) 2-Parameter Weibull 𝜷=𝟕𝟕.𝟓 𝝐=𝟏.𝟏𝟎𝟒 3-Parameter Weibull 𝜶=𝟑.𝟐𝟒 Model Model Type 2-Weibull Slope Lundberg-Palmgren Experimental 1.125 Harris and Kotzalas Experimental Bounds 0.7-3.5 Raje 2D DEM 3.36 Jalalahmadi 2D FEA 2.65 Weinzapfel 3D FEA 4.55 Current Model Anisotropy 1.174 𝐀𝐧𝐢𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 𝐦𝐨𝐝𝐞𝐥 𝐦𝐚𝐭𝐜𝐡𝐞𝐬 𝐞𝐱𝐩𝐞𝐫𝐢𝐦𝐞𝐧𝐭𝐚𝐥 𝐫𝐞𝐬𝐮𝐥𝐭𝐬 𝐞𝐱𝐭𝐫𝐞𝐦𝐞𝐥𝐲 𝐰𝐞𝐥𝐥

Current Model Development Implement damage mechanics coupled with crystal elasticity to model both crack initiation and propagation Develop a multi-phase representative model for bearing steel microstructure Model a nonuniform distribution of crystal orientations (texture) 𝜶-phase 𝜷-phase

Measurement of Skidding in Cam and Roller Follower Skidding in Cam and Followers causes wear and premature failure A test rig has been developed to study the causes of skidding An analytical model is under development to model the causes of skidding Title slide (option 2)

Cam and Follower Test Rig Shaft Coupling – rigidly connects the driven shaft and camshaft Test Cell – Assembly contains the camshaft, tappet, lubrication pathways and speeds sensor - Tappets are machined to hold optical sensor - Rollers are laser etched with 20-60 divisions - Time between divisions is measured with optical sensor - Sensor is sealed from environment with sleeve Speed Measurement Flywheel – 700 mm flywheel to maintain constant shaft speed under alternating load conditions One way Clutch Allows deceleration under flywheel inertia for Stribeck curve data Drive Motor – 55 HP provides power to camshaft with speeds up to 1800 RPM

Test Rig Roller Skidding Skidding created with modified roller follower Skidding only apparent at low loads Results show a transition from skidding into a pure rolling regime Short skidding regions are seen after the transition to rolling regime Cam shows initial stages of skidding wear Rolling Region Skidding Region Intermittent Skid

Analytical Roller Skidding Model ωcam I ∙ αroller W ∙ μaxle W ∙ μcam W 2D roller skidding model under development EHL of cam and roller interface ( 𝜇 𝑐𝑎𝑚 ) HL of roller and pin joint ( 𝜇 𝑎𝑥𝑙𝑒 ) Kinematics of cam and follower ( 𝐼 𝑟𝑜𝑙𝑙𝑒𝑟 , 𝜔 𝑐𝑎𝑚 ) Torque Balance to find angular velocity ( 𝛼 𝑟𝑜𝑙𝑙𝑒𝑟 ) EHL & Mixed EHL Cam Kinematics Torque Balance HL 𝝁 𝒄𝒂𝒎 𝜶 𝒓𝒐𝒍𝒍𝒆𝒓 𝝁 𝒂𝒙𝒍𝒆 𝝎 𝒄𝒂𝒎 𝝎 𝒓𝒐𝒍𝒍𝒆𝒓