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

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

3. 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

4. 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)

5. 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

6. 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

7. 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.

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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

15. 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
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

16. 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

17. 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)

18. 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

19. 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

20. 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