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Journal of Tribology | Volume 131 | Issue 2 | Research Paper
Research Papers: Other (Seals, Manufacturing)

A Voronoi Finite Element Study of Fatigue Life Scatter in Rolling Contacts

[+] Author and Article Information
Behrooz Jalalahmadi

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907bjalalah@purdue.edu

Farshid Sadeghi

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907sadeghi@ecn.purdue.edu

J. Tribol 131(2), 022203 (Mar 04, 2009) (15 pages) doi:10.1115/1.3063818 History: Received June 19, 2008; Revised September 30, 2008; Published March 04, 2009
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Microlevel material failure has been recognized as one of the main modes of failure for rolling contact fatigue (RCF) of bearing. Therefore, microlevel features of materials will be of significant importance to RCF investigation. At the microlevel, materials consist of randomly shaped and sized grains, which cannot be properly analyzed using the classical and commercially available finite element method. Hence, in this investigation, a Voronoi finite element method (VFEM) was developed to simulate the microstructure of bearing materials. The VFEM was then used to investigate the effects of microstructure randomness on rolling contact fatigue. Here two different types of randomness are considered: (i) randomness in the microstructure due to random shapes and sizes of the material grains, and (ii) the randomness in the material properties considering a normally (Gaussian) distributed elastic modulus. In this investigation, in order to determine the fatigue life, the model proposed by Raje (“A Numerical Model for Life Scatter in Rolling Element Bearings,” ASME J. Tribol., 130, pp. 011011-1–011011-10), which is based on the Lundberg–Palmgren theory (“Dynamic Capacity of Rolling Bearings,” Acta Polytech. Scand., Mech. Eng. Ser., 1(3), pp. 7–53), is used. This model relates fatigue life to a critical stress quantity and its corresponding depth, but instead of explicitly assuming a Weibull distribution of fatigue lives, the life distribution is obtained as an outcome of numerical simulations. We consider the maximum range of orthogonal shear stress and the maximum shear stress as the critical stress quantities. Forty domains are considered to study the effects of microstructure on the fatigue life of bearings. It is observed that the Weibull slope calculated for the obtained fatigue lives is in good agreement with previous experimental studies and analytical results. Introduction of inhomogeneous elastic modulus and initial flaws within the material domain increases the average critical stresses and decreases the Weibull slope.
Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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+(a) Discretizing a domain with the Voronoi elements, (b) dividing a Voronoi element into finer triangular elements, and (c) linear triangular element
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(a) Discretizing a domain with the Voronoi elements, (b) dividing a Voronoi element into finer triangular elements, and (c) linear triangular element
Figure 1

(a) Discretizing a domain with the Voronoi elements, (b) dividing a Voronoi element into finer triangular elements, and (c) linear triangular element

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+The two approaches for storing the global stiffness matrix (K): (a) a single sparse matrix and (b) two much smaller condensed matrices
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The two approaches for storing the global stiffness matrix (K): (a) a single sparse matrix and (b) two much smaller condensed matrices
Figure 2

The two approaches for storing the global stiffness matrix (K): (a) a single sparse matrix and (b) two much smaller condensed matrices

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+The theoretical internal stress distribution below the surface for a cylinder in contact with a semi-infinite domain
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The theoretical internal stress distribution below the surface for a cylinder in contact with a semi-infinite domain
Figure 3

The theoretical internal stress distribution below the surface for a cylinder in contact with a semi-infinite domain

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+(a) The dimensions and grain sizes of the domain simulating the semi-infinite domain forming the bearing line contact, and (b) the zoomed view
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(a) The dimensions and grain sizes of the domain simulating the semi-infinite domain forming the bearing line contact, and (b) the zoomed view
Figure 4

(a) The dimensions and grain sizes of the domain simulating the semi-infinite domain forming the bearing line contact, and (b) the zoomed view

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+Statistical distribution of elements in one of the generated domains: (a) distribution of number of sides of Voronoi elements in the domain and (b) distribution of the elements’ area
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Statistical distribution of elements in one of the generated domains: (a) distribution of number of sides of Voronoi elements in the domain and (b) distribution of the elements’ area
Figure 5

Statistical distribution of elements in one of the generated domains: (a) distribution of number of sides of Voronoi elements in the domain and (b) distribution of the elements’ area

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+The stress contours obtained for a randomly generated domain under the static Hertzian loading: (a) τxy and (b) σy
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The stress contours obtained for a randomly generated domain under the static Hertzian loading: (a) τxy and (b) σy
Figure 6

The stress contours obtained for a randomly generated domain under the static Hertzian loading: (a) τxy and (b) σy

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+Variations in normal stress in the y-direction (σy) and orthogonal shear stress (τxy) along the depth for four randomly generated domains
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Variations in normal stress in the y-direction (σy) and orthogonal shear stress (τxy) along the depth for four randomly generated domains
Figure 7

Variations in normal stress in the y-direction (σy) and orthogonal shear stress (τxy) along the depth for four randomly generated domains

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+The stress contours obtained from ANSYS using the square elements for a domain under the static Hertzian loading
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The stress contours obtained from ANSYS using the square elements for a domain under the static Hertzian loading
Figure 8

The stress contours obtained from ANSYS using the square elements for a domain under the static Hertzian loading

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+The stress contours obtained from ANSYS using the linear triangular elements for a domain under the static Hertzian loading
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The stress contours obtained from ANSYS using the linear triangular elements for a domain under the static Hertzian loading
Figure 9

The stress contours obtained from ANSYS using the linear triangular elements for a domain under the static Hertzian loading

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+(a) The values and (b) the critical depths of the maximum range of orthogonal shear stress (Δτxy) and the maximum shear stress (τmax) for the considered 40 domains with different microstructures
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(a) The values and (b) the critical depths of the maximum range of orthogonal shear stress (Δτxy) and the maximum shear stress (τmax) for the considered 40 domains with different microstructures
Figure 10

(a) The values and (b) the critical depths of the maximum range of orthogonal shear stress (Δτxy) and the maximum shear stress (τmax) for the considered 40 domains with different microstructures

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+Two-parameter Weibull plot for the critical depths of the 40 considered domains using Δτxy and τmax as the critical stress
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Two-parameter Weibull plot for the critical depths of the 40 considered domains using Δτxy and τmax as the critical stress
Figure 11

Two-parameter Weibull plot for the critical depths of the 40 considered domains using Δτxy and τmax as the critical stress

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+Two-parameter Weibull plot for the obtained lives of the 40 considered domains using Δτxy and τmax as the critical stress
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Two-parameter Weibull plot for the obtained lives of the 40 considered domains using Δτxy and τmax as the critical stress
Figure 12

Two-parameter Weibull plot for the obtained lives of the 40 considered domains using Δτxy and τmax as the critical stress

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+The two different normal Gaussian distributions of the elastic modulus employed for Domain 1 with a 10% variation in the average value of 200 GPa
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The two different normal Gaussian distributions of the elastic modulus employed for Domain 1 with a 10% variation in the average value of 200 GPa
Figure 13

The two different normal Gaussian distributions of the elastic modulus employed for Domain 1 with a 10% variation in the average value of 200 GPa

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+(a) The values and (b) the critical depths of Δτxy and τmax obtained for the 40 domains using the homogeneous elastic modulus and its normal distribution
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(a) The values and (b) the critical depths of Δτxy and τmax obtained for the 40 domains using the homogeneous elastic modulus and its normal distribution
Figure 14

(a) The values and (b) the critical depths of Δτxy and τmax obtained for the 40 domains using the homogeneous elastic modulus and its normal distribution

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+The comparison of the Weibull life distribution for cases of the uniform elastic modulus and its normal distribution using (a) Δτxy as the critical stress and (b) τmax as the critical stress
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The comparison of the Weibull life distribution for cases of the uniform elastic modulus and its normal distribution using (a) Δτxy as the critical stress and (b) τmax as the critical stress
Figure 15

The comparison of the Weibull life distribution for cases of the uniform elastic modulus and its normal distribution using (a) Δτxy as the critical stress and (b) τmax as the critical stress

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+A sample simulated RVE and the five initial flaws introduced in it
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A sample simulated RVE and the five initial flaws introduced in it
Figure 16

A sample simulated RVE and the five initial flaws introduced in it

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+Variation in (a) the values and (b) the depths of Δτxy and τmax with the introduction of five initial flaws in the domains
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Variation in (a) the values and (b) the depths of Δτxy and τmax with the introduction of five initial flaws in the domains
Figure 17

Variation in (a) the values and (b) the depths of Δτxy and τmax with the introduction of five initial flaws in the domains

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+The comparison of the Weibull life distribution between the domains without initial flaw and the domains including five initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress
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The comparison of the Weibull life distribution between the domains without initial flaw and the domains including five initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress
Figure 18

The comparison of the Weibull life distribution between the domains without initial flaw and the domains including five initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress

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+The comparison of the Weibull life distribution between the domains having different numbers of initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress
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The comparison of the Weibull life distribution between the domains having different numbers of initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress
Figure 19

The comparison of the Weibull life distribution between the domains having different numbers of initial flaws using (a) Δτxy as the critical stress and (b) τmax as the critical stress

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+The variation in the Weibull slope in terms of numbers of initial flaws in the domains using (a) Δτxy as the critical stress and (b) τmax as the critical stress
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The variation in the Weibull slope in terms of numbers of initial flaws in the domains using (a) Δτxy as the critical stress and (b) τmax as the critical stress
Figure 20

The variation in the Weibull slope in terms of numbers of initial flaws in the domains using (a) Δτxy as the critical stress and (b) τmax as the critical stress

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