Research Papers: Contact Mechanics

Periodic Contact Problems in Plane Elasticity: The Fracture Mechanics Approach

[+] Author and Article Information
Yang Xu, Robert L. Jackson

Mechanical Engineering Department,
Auburn University,
Auburn, AL 36849

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received March 28, 2017; final manuscript received May 11, 2017; published online August 2, 2017. Assoc. Editor: James R. Barber.

J. Tribol 140(1), 011404 (Aug 02, 2017) (11 pages) Paper No: TRIB-17-1114; doi: 10.1115/1.4036920 History: Received March 28, 2017; Revised May 11, 2017

In this study, the concept of the fracture mechanics is used to solve the: (i) frictionless purely normal contact and (ii) the similar material contact under the mutual actions of the normal and tangential load. Considering the contact region is simply connected, the out-of-contact regions can be treated as periodic collinear cracks. Through evaluating the stress intensity factor (SIF), we are able to obtain the size and location of the contact/out-of-contact region. Then, the normal traction, shear traction and interfacial gap can be directly determined by the Green's function of the periodic collinear crack. In the case of frictionless purely normal contact, the new approach is applied to two classic problems, namely, the Westergaard problem (sinusoidal waviness punch) and the periodic flat-end punch problem. Then, the sinusoidal waviness contact pair in the full stick and the partial slip conditions under the mutual actions of the normal and tangential loads are solved by the newly developed approach.

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Grahic Jump Location
Fig. 1

Schematic representation of dissimilar material contact

Grahic Jump Location
Fig. 2

Schematic representation of an indentation of a half-plane by a rigid, periodic, punch (a) before and (b) after the loading. The amplitude of the wavy interface is exaggerated for the illustration purpose only.

Grahic Jump Location
Fig. 3

Schematic representation of the decomposition of (a) a complete contact and (b) a periodic, collinear, pressurized plane crack problem. Due to the symmetry, only the lower semi-infinite body with a lower crack surface is shown.

Grahic Jump Location
Fig. 4

Schematic representation of a sinusoidal waviness in partial contact with a rigid flat: (a) undeformed profile and (b) deformed profile under p¯

Grahic Jump Location
Fig. 5

(a) Normal traction distribution: p1(x), p2(x), p(x)=p1(x)+p2(x) and p(x) from Westergaard's solution [22,25] and (b) plots of h(x), g(x), uy(x,0)=h(x)+g(x), and uy(x,0) from Westergaard's solution [25]. Uniform normal compressive stress at far end is P/λ=0.5p*.

Grahic Jump Location
Fig. 6

Schematic representation of the indentation of a half-plane by a periodic flat-end punch

Grahic Jump Location
Fig. 7

Dimensionless traction, p(x)(λ/P), and bulge height, g(x)(πE*/2P). l/λ=0.25.

Grahic Jump Location
Fig. 8

Schematic representation of a periodic contact pair subjected to the mutual action of the normal and tangential loads

Grahic Jump Location
Fig. 9

Schematic representation of a periodic mode-II crack embedded in a continuum infinite medium. The crack surfaces are marked by the thick line.

Grahic Jump Location
Fig. 10

Schematic representation of a periodic mode-II crack embedded in a continuum infinite medium. The crack width is 2(l+c). Inside the slip zone, Ωslip=[−l−c,−l]∪[l,l+c], the upper/lower crack surfaces are subjected to q(x)=∓μp(x).

Grahic Jump Location
Fig. 11

The dimensionless shear stress, q(x)(λ/P), with one half contact region: x∈[l,λ] when the tangential boundary condition is (i) the Coulomb law, (ii) full stick, and (iii) partial slip. The constants are l/λ=0.25, μ=0.2, and Q/P=0.15.

Grahic Jump Location
Fig. 12

Schematic representation of one period of an infinite collinear plane crack problem under wedge loads. Only the normal and shear tractions on the left side of crack are shown.



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