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Research Papers: Contact Mechanics

Fully Coupled Resolution of Heterogeneous Elastic–Plastic Contact Problem

[+] Author and Article Information
Kwassi Vilevo Amuzuga

INSA-Lyon,
Université de Lyon,
LaMCoS UMR CNRS 5259,
Villeurbanne F69621, France;
SKF Aerospace France,
29 Chemin de Thabor,
Valence 26000, France

Thibaut Chaise, Arnaud Duval

INSA-Lyon,
Université de Lyon,
LaMCoS UMR CNRS 5259,
Villeurbanne F69621, France

Daniel Nelias

INSA-Lyon,
Université de Lyon,
LaMCoS UMR CNRS 5259,
Villeurbanne F69621, France
e-mail: daniel.nelias@insa-lyon.fr

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received May 29, 2015; final manuscript received November 10, 2015; published online February 15, 2016. Assoc. Editor: James R. Barber.

J. Tribol 138(2), 021403 (Feb 15, 2016) (22 pages) Paper No: TRIB-15-1172; doi: 10.1115/1.4032072 History: Received May 29, 2015; Revised November 10, 2015

The recent development of semi-analytical methods (SAM) has led to numerous improvements in their capabilities in terms of phenomena that can be accounted for and numerical efficiency. They now allow to perform fast and robust simulations of contact between inelastic—with either elastic–plastic or viscoelastic behavior—and anisotropic or heterogeneous materials. All effects may be combined, with either coating, inclusions, cavities, or fibers as inhomogeneities. The coupling between local and global scales remains numerically difficult. A framework is proposed here for contact problems considering the effect of elastic heterogeneities within an elastic–plastic matrix. The mutual interactions among heterogeneities and their surrounding plastic zone as well as the interactions between them and the contact surface through which the load is transmitted should be accounted for. These couplings are outside the validity domain of the Eshelby’s equivalent inclusion method (EIM) that assumes a uniform stress field in an infinite space far from the inhomogeneity. In the presence of heterogeneities close to the surface or located at the Hertzian depth, the yield stress can be reached locally due to the additional stress it generates, whereas the stress and strain state would remain purely elastic for a matrix without inclusion. It is well known that for rolling element bearing and gear applications, the ruin of components is often linked to cracks initiated in the vicinity of large or hard inclusions that act as stress raisers. It turned out that plastic strains tend to reduce the stress generated by the contact pressure while hard heterogeneities will increase it. As plastic strain accumulation can provide the basis for fatigue damage criteria, the second half of the paper will illustrate how the method can be used to identify and rank geometrical and material parameters that influence the location and magnitude of the maximal plastic strain.

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Figures

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Fig. 1

Elastic contact problem

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Fig. 2

Subsurface contribution for solving an HEPC problem

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Fig. 3

Single heterogeneity transformation into inclusion in the sense of Eshelby and subsequent eigenstress

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Fig. 4

EIM decomposition method for a half-space

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Fig. 5

Algorithm of HEPC problem

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Fig. 7

Finite element (FE) model used for the validation

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Fig. 8

Comparison of the contact pressure profiles by SAM and FEM: (a) homogeneous half-space and (b) half-space containing one single spherical heterogeneous inclusion

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Fig. 10

Comparison of the contact pressure profiles for cuboidal and spherical heterogeneities

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Fig. 9

Contact pressure profiles between a spherical indenter and HEP body: (a) heterogeneity stiffer than the matrix and (b) heterogeneity softer than the matrix

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Fig. 11

Plastic strain distribution around a single stiff (γ = 3) and spherical heterogeneity: (a) elastic–plastic matrix without heterogeneity; heterogeneity centered at (b) zi=0.35×a; (c) zi=0.7×a; and (d) zi=0.9×a. The heterogeneities have the same size of 0.2a.

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Fig. 12

Plastic strain according to heterogeneity nature: (a) heterogeneity softer than the matrix and (b) heterogeneity harder than the matrix

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Fig. 13

Local plastic strain concentration produced around two spherical heterogeneous inclusions, the first one of small size located in the center of the contact and at the Hertzian depth, the second one being an outer heterogeneity of varying size and located at the Hertzian depth at the border of the contact (r = a): elastic–plastic matrix without heterogeneity (a); with an outer heterogeneity of radius ri=0.05×a (b); ri=0.2×a (c); ri=0.3×a (d); and ri=0.4×a (e)

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Fig. 14

Plastic strain according to the heterogeneity stringer orientation: (a) elastic–plastic matrix without heterogeneity; (b) horizontal orientation; (c) tilted stringer of orientation θ=π/6; (d) tilted stringer of orientation θ=π/4; (e) tilted stringer of orientation θ=π/3; and (f) vertical orientation

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Fig. 15

Contact pressure profiles and related plastic strain fields: (a) pressure distribution; (b) plastic strain within the homogeneous body; and (c) plastic strain within the heterogeneous body

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Fig. 16

Decomposition of the total residual stress (von Mises) observed after unloading: (a) residual stress due to plasticity only without heterogeneity within the EP matrix; (b) eigen-residual stress due to the presence of the heterogeneity and residual stresses resulting from the surrounding plasticity; (c) residual stress due to plasticity only in the presence of the heterogeneity; and (d) total balanced residual stress

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Fig. 17

Decomposition of the total residual stress (hydrostatic pressure) observed after unloading: (a) residual stress due to plasticity only without heterogeneity within the EP matrix; (b) eigen-residual stress due to the presence of the heterogeneity and residual stresses resulting from the surrounding plasticity; (c) residual stress due to plasticity only in the presence of the heterogeneity; and (d) total balanced residual stress

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Fig. 18

Principal stress I and principal direction I: (a) elastic–plastic matrix and (b) heterogeneous elastic–plastic body

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Fig. 19

Zoom on the highest principal stress location and orientation: prediction of crack initiation direction that should be perpendicular to the principal stress direction for mode I

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Fig. 20

Effect of a cubic heterogeneity on the contact pressure distribution, according to (a) its depth (for β=0.2, γ = 3); (b) its size (for α=0.3, γ = 3); and (c) elastic modulus (for α=0.3, β=0.2)

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Fig. 21

Effect of a cubic heterogeneity on the equivalent plastic strain profile versus depth, according to (a) its depth (for β=0.2, γ = 3); (b) its size (for α=0.3, γ = 3); and (c) elastic modulus (for α=0.3, β=0.2)

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