Transcript Effects of Crystal Elasticity on Rolling Contact Fatigue Neil Paulson Ph.D. Research Assistant Mechanical Engineering Tribology Laboratory (METL) November 14, 2013

Effects of Crystal Elasticity
on Rolling Contact Fatigue
Neil Paulson
Ph.D. Research Assistant
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
2
Outline
•
•
•
•
•
•
•
Motivation and Background
Crystal Structure Definitions
Polycrystalline Material Model
Steel Material Stiffness Model
Hertzian Contact Modeling
RCF Relative Life Study
Future Work
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
3
Background and Motivation
•
Material heterogeneity can play a role in rolling
contact fatigue failure,
o Microstructure Topology

•
•
Raje, Jalalahmadi, Slack, Weinzapfel,
Warhadpande, Bomidi
o Voids or inclusions
o 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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 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
𝝈𝒙
𝝈𝒚
𝝈𝒛
𝝉𝒙𝒚
𝝉𝒙𝒛
𝝉𝒚𝒛
=
𝑪𝟏𝟏
𝑪𝟏𝟐
𝑪𝟏𝟐
𝑪𝟏𝟐
𝑪𝟏𝟏
𝑪𝟏𝟐
𝑪𝟏𝟐
𝑪𝟏𝟐
𝑪𝟏𝟏
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝑪𝟏𝟏 − 𝑪𝟏𝟐
𝟐
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝝐𝒙
𝝐𝒚
𝝐𝒛
𝟎
𝟎
𝜸𝒙𝒚
𝟎
𝜸𝒙𝒛
𝟎
𝟎
𝟎
𝟎
𝑪𝟏𝟏 − 𝑪𝟏𝟐
𝟐
𝟎
𝟎
𝟎
𝟎
𝟎
𝑪𝟏𝟏 − 𝑪𝟏𝟐 𝜸𝒚𝒛
𝟐
𝑬 𝟏−𝝂
(𝟏 + 𝝂)(𝟏 − 𝟐𝝂)
𝝂𝑬
𝑪𝟏𝟐 =
(𝟏 + 𝝂)(𝟏 − 𝟐𝝂)
𝑬
𝑪𝟒𝟒 = 𝑮 =
𝟐(𝟏 + 𝝂)
𝑪𝟏𝟏 =
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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
𝝈𝒙
𝑪𝟏𝟏
𝝈𝒚
𝑪𝟏𝟐
𝝈𝒛
𝑪𝟏𝟐
=
𝝉𝒙𝒚
𝟎
𝝉𝒙𝒛
𝟎
𝝉𝒚𝒛
𝟎
𝑪𝟏𝟐
𝑪𝟏𝟏
𝑪𝟏𝟐
𝟎
𝟎
𝟎
𝑪𝟏𝟐
𝑪𝟏𝟐
𝑪𝟏𝟏
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝑪𝟒𝟒
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝑪𝟒𝟒
𝟎
𝝐𝒙
𝟎
𝝐𝒚
𝟎
𝝐𝒛
𝟎
𝟎 𝜸𝒙𝒚
𝟎 𝜸𝒙𝒛
𝑪𝟒𝟒 𝜸𝒚𝒛
𝑬 𝟏−𝝂
(𝟏 + 𝝂)(𝟏 − 𝟐𝝂)
𝝂𝑬
=
(𝟏 + 𝝂)(𝟏 − 𝟐𝝂)
𝑪𝟏𝟏 =
𝑪𝟏𝟐
𝐂𝟒𝟒 = 𝑮
Shear modulus is
independent of E and ν
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
6
Modeling Polycrystalline Aggregates
Each individual crystal has a unique orientation
Cubic Stiffness Matrix
Isotopic 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
𝑪𝒈 = 𝑹𝒛" (𝑹𝒙′ (𝑹𝒛 𝑪𝑹𝑻𝒛 )𝑹𝑻𝒙′ )𝑹𝑻𝒛"
𝟐𝟔𝟗 𝟏𝟏𝟓 𝟏𝟏𝟓 𝟎
𝟎
𝟏𝟏𝟓 𝟐𝟔𝟗 𝟏𝟏𝟓 𝟎
𝟎
𝟏𝟏𝟓 𝟏𝟏𝟓 𝟐𝟔𝟗 𝟎
𝟎
𝟎
𝟎
𝟎
𝟕𝟕 𝟎
𝟎
𝟎
𝟎
𝟎 𝟕𝟕
𝟎
𝟎
𝟎
𝟎
𝟎
Isotropic stiffness matrix is
identical after rotation
𝟎
𝟎
𝟎
𝟎
𝟎
𝟕𝟕
𝟐𝟗𝟓
𝟏𝟐𝟓
𝟔𝟎
𝟒
𝟐𝟒
𝟐𝟓
𝟏𝟐𝟓 𝟔𝟎
𝟒
𝟐𝟒
𝟐𝟓
𝟐𝟓𝟎 𝟏𝟎𝟓 −𝟏𝟒 −𝟏𝟔 −𝟒𝟔
𝟏𝟎𝟓 𝟑𝟏𝟓 𝟏𝟎
−𝟖
𝟐𝟏
−𝟏𝟒 𝟎𝟏𝟎 𝟏𝟏𝟑 𝟐𝟓 −𝟏𝟔
−𝟏𝟔 −𝟖
𝟐𝟓
𝟒𝟗
𝟏𝟎
−𝟒𝟔 𝟐𝟏 −𝟏𝟔 𝟏𝟎
𝟗𝟒
Cubic stiffness matrix becomes
fully anisotropic after rotation
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
7
Material & Model Verification
𝜹𝒙 = 𝟎
• 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
𝜹𝒙 = 𝒄
𝒚
𝒙
• Uniaxial Strain was applied
• Reaction forces were measured 𝑭𝒙
𝒚
𝒙
𝜹𝒚 = 𝟎
•
Global material properties of the model were evaluated1:
𝑬𝒃 =
𝝈 𝒙 + 𝟐𝝈 𝒚 𝝈 𝒙 − 𝝈 𝒚
𝝐𝒙 (𝝈𝒙 + 𝝈𝒚 )
𝝂𝒃 =
𝑭𝒚
𝝈𝒚
𝝈𝒙 + 𝝈𝒚
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.
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
Steel Material Model
300
200
150
Bulk
Properties
0.45
Eavg
199.9
200
0.4
Estd
8.91
νavg
0.291
νstd
0.009
0
Voigt Bound
Reuss Bound
FEM
0.35
0.30
100
50
0.5
FEA results
match isotropic
constants
Poisson's Ratio
Young's Modulus (GPa)
250
FEA
Model
Voigt Bound
Reuss Bound
FEM
0.3
0.25
0.2
0.15
0.1
0.05
0
Example Cases
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
11
Rolling Contact Fatigue Life Equations
Microstructures models simulate randomness from experimental testing
Lundberg-Palmgren equation can be reduced for constant survivability and volume:
𝒛𝒉
𝑵~ 𝒄
𝝉
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
12
Modeling Rolling Contact
• Hertzian Line Contact Load
• 21 Loading Steps
• Load Transverses Anisotropic
Region
𝝉𝒙𝒚,𝒎𝒂𝒙
𝚫𝝉𝒙𝒚
• 𝚫𝝉𝒙𝒚 was evaluated for each element
•
Maximum 𝚫𝝉𝒙𝒚 value and location recorded
𝝉𝒙𝒚,𝒎𝒊𝒏
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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.
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
14
RCF Relative Life
•
Relative life equation was used to
determine bearing fatigue life
𝝉𝒄
𝑵~ 𝒉
𝒛
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
15
RCF Relative Life
All topological domain results combined into one RCF relative life plot
Weibull Distribution Function
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐹𝑎𝑖𝑙𝑢𝑟𝑒
=1−𝑒
−
𝑁−𝛼 𝜖
𝛽
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
𝜷 = 𝟕𝟕. 𝟓
𝝐 = 𝟏. 𝟏𝟎𝟒
Model
Model Type
2-Weibull
Slope
LundbergPalmgren
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
2D FEA
Anisotropy
1.174
𝐀𝐧𝐢𝐬𝐨𝐭𝐫𝐨𝐩𝐢𝐜 𝐦𝐨𝐝𝐞𝐥 𝐦𝐚𝐭𝐜𝐡𝐞𝐬
𝐞𝐱𝐩𝐞𝐫𝐢𝐦𝐞𝐧𝐭𝐚𝐥 𝐫𝐞𝐬𝐮𝐥𝐭𝐬 𝐞𝐱𝐭𝐫𝐞𝐦𝐞𝐥𝐲 𝐰𝐞𝐥𝐥
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
18
Cam and Follower Test Rig
Shaft Coupling – rigidly
connects the driven shaft
and camshaft
Flywheel – 700 mm flywheel to
maintain constant shaft speed
under alternating load conditions
Test Cell – Assembly
contains the camshaft,
tappet, lubrication
pathways and speeds
sensor
Speed Measurement
- 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
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
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013
19
Test Rig Roller Skidding
Roller Velocity (RPM)
• 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
300
250
200
Cam shows initial stages of
skidding wear
150
100
Skidding Region
0
300
250
150
2000
100
Intermittent
Skid
50
0
0
3
6
9
12
15
1000
18
0
3000
Time (s)
Rolling Region
Mechanical Engineering Tribology Laboratory (METL)
Normal Load (N)
3000
200
2000
al Load (N)
Roller Velocity (RPM)
50
November 14, 2013
20
Analytical Roller Skidding Model
I ∙ αroller
W ∙ μaxle
2D roller skidding model under development
–
–
–
–
W ∙ μcam
W
EHL of cam and roller interface (𝜇𝑐𝑎𝑚 )
HL of roller and pin joint (𝜇𝑎𝑥𝑙𝑒 )
Kinematics of cam and follower (𝐼𝑟𝑜𝑙𝑙𝑒𝑟 , 𝜔𝑐𝑎𝑚 )
Torque Balance to find angular velocity (𝛼𝑟𝑜𝑙𝑙𝑒𝑟 )
ωcam
Cam Kinematics
𝝎𝒄𝒂𝒎
𝝎𝒓𝒐𝒍𝒍𝒆𝒓
EHL &
Mixed EHL
𝝁𝒄𝒂𝒎
𝝁𝒂𝒙𝒍𝒆
Torque
Balance
HL
𝜶𝒓𝒐𝒍𝒍𝒆𝒓
Mechanical Engineering Tribology Laboratory (METL)
November 14, 2013