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Special Issue Honoring Professor Fazil Erdogan’s Contributions to Mixed Boundary Value Problems of Inhomogeneous and Functionally Graded Materials

On the Singularities in Fracture and Contact Mechanics

[+] Author and Article Information
Fazil Erdogan, Murat Ozturk

Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015

This may not be very clear from Fig. 1. However, by expressing k(b) as a function of the crack tip location x=a, it can be shown that lima0ddak(b)= and, as expected the function k(b), at x=a=0, is ill defined.

Some of the techniques to be developed for dealing with three-dimensional mixed boundary value problems may include a 3D theory equivalent to the conventional two-dimensional complex function theory and two-dimensional singular integral equations.

J. Appl. Mech 75(5), 051111 (Jul 11, 2008) (12 pages) doi:10.1115/1.2936241 History: Received August 31, 2007; Revised January 10, 2008; Published July 11, 2008

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials.

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Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 13

Semi-infinite homogeneous (a) and graded (b) coating bonded to a homogeneous quarter plane

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Figure 5

Variation of the singularity α with the system parameters λ and θ0 (see Eq. 44)

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Figure 6

Coupling of singularities in an elastic wedge and a rigid stamp

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Figure 7

Notation for a plane crack in a nonhomogeneous medium

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Figure 8

Geometry and notation for a plane crack

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Figure 9

Geometry of the problem for a penny-shaped crack

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Figure 10

Normalized strain energy release rates for a penny-shaped crack in graded interfacial zone (Fig. 9)

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Figure 11

Geometry of the crack terminating at the interface

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Figure 12

Normalized stress intensity factors for a plane crack in a homogeneous medium bonded to a nonhomogeneous half space, Fig. 8, a=0, μ1(x)=μ0exp(βx), d=c=b∕2, k(a)=k1(a)∕σ0c, ν=0.3, k(b)=k1(b)∕σ0c, σ0=−σyy(x,0), and 0<x<b

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Figure 14

Typical stamp geometries pressed upon a graded substrate

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Figure 1

(a) and (b) the initial crack geometry, (c) the qualitative crack geometry after some subcritical crack growth under Mode I conditions, (d) the qualitative crack geometry after some subcritical crack growth under Mode II/Mode III dominated loading conditions, (e) thumbnail, and (f) reverse thumbnail crack fronts in a plate with constant thickness

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Figure 2

Crack perpendicular to a bimaterial interface

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Figure 3

Contact problem for a membrane stiffener with variable thickness

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Figure 4

The sketch of end point behavior of the stiffener with thickness h(x)=A(x−a)γ

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