Evaluation of Stresses Around Inclusions in Hertzian Contacts Using the Discrete Element Method

[+] Author and Article Information
Nihar Raje, Farshid Sadeghi

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906

Richard G. Rateick

 Engines, Systems & Services, Honeywell Aerospace, South Bend, IN 46620

Michael R. Hoeprich

 Timken Research, Canton, OH 44706

J. Tribol 129(2), 283-291 (Dec 05, 2006) (9 pages) doi:10.1115/1.2464132 History: Received January 18, 2006; Revised December 05, 2006

Inclusions are the primary sites for subsurface fatigue crack initiation in bearing contacts. To understand the mechanisms of subsurface crack nucleation under contact loading, a detailed description of the stress field around these inclusions is necessary. This paper presents a new approach to computing stresses in an inhomogeneous medium where inclusions are treated as inhomogeneities in a homogeneous material matrix. The approach is based on the Discrete Element (DE) Method in which the material continuum is replaced by a set of rigid discrete interacting elements. The elements are connected to each other along their sides through springs and dampers to form the macro-continuum and undergo relative displacements in accordance with Newton’s laws of motion under the action of external loading. The spring properties are derived in terms of the overall elastic properties of the continuum. The relative motion between elements gives rise to contact forces due to stretching or compression of the inter-element springs. These forces are evaluated at each time-step and the corresponding equations of motion are solved for each element. Stresses are calculated from the inter-element joint forces. A Hertzian line contact case, with and without the presence of subsurface inclusions, is analyzed using the DE model. The DE model was used to determine stresses for an inclusion-free medium that compares well with that obtained from the continuum elasticity models. Parametric studies are then carried out to investigate the effects of size, location, orientation, and elastic properties of inclusions on the subsurface stress field. Both inclusions that are stiffer and/or softer than the base material are seen to give rise to stress concentrations. For inclusions that are stiffer than the base material (semi-infinite domain), the stress concentration effect increases with their elastic modulus. The stress concentration effect of a softer inclusion is higher than that of a stiffer inclusion. Inclusions that are oriented perpendicular to the surface give rise to much higher von Mises stresses than the ones that are oriented parallel to the surface. There is little change in the maximum von Mises stress for inclusions that are located deep within the surface.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

A macro-continuum formed by an assemblage of discrete elements

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Figure 2

A schematic of the discrete element scheme

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Figure 3

(a) Joint model and (b) fiber model

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Figure 4

(a) Fiber displacement, (b) fiber spring force, and (c) fiber damping force

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Figure 5

(a) Resultant fiber force and (b) resultant joint force and moment

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Figure 6

The resultant force and moment on the center of gravity of the element

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Figure 7

The equivalent spring stiffnesses and the beam element

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Figure 8

The semi-infinite domain formed by discrete elements

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Figure 9

The normalized stresses σxx, σzz, τxz, and σvm with no inclusions

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Figure 10

Inclusion geometry

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Figure 11

The normalized von Mises stress field with a single inclusion located at x=0, z=0.7b (Lx=80μm, Lz=20μm, Ei=2Em, νi=νm=0.3)

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Figure 12

Variation of von Mises stress field with inclusion elastic modulus (inclusion location: x=0, z=0.7b, inclusion size: Lx=80μm, Lz=20μm)

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Figure 13

Variation of von Mises stress field with inclusion size (inclusion location: x=0, z=0.7b, inclusion elastic modulus: Ei=400GPa, material elastic modulus: Em=200GPa)

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Figure 14

Variation of von Mises stress field with inclusion orientation (inclusion location: x=0, z=0.7b, inclusion elastic modulus: Ei=400GPa, material elastic modulus: Em=200GPa)

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Figure 15

Variation of von Mises stress field with inclusion depth (inclusion size: Lx=80μm, Lz=20μm, inclusion elastic modulus: Ei=400GPa, material elastic modulus: Em=200GPa)




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