Asymptotic Analysis of a Mode III Stationary Crack in a Ductile Functionally Graded Material

[+] Author and Article Information
Dhirendra V. Kubair1

Aerospace Engineering, University of Illinois at Urbana-Champaign, 306 Talbot Laboratory, 104 South Wright Street, Urbana, IL 61801

Philippe H. Geubelle

Aerospace Engineering, University of Illinois at Urbana-Champaign, 306 Talbot Laboratory, 104 South Wright Street, Urbana, IL 61801

John Lambros2

Aerospace Engineering, University of Illinois at Urbana-Champaign, 306 Talbot Laboratory, 104 South Wright Street, Urbana, IL 61801lambros@uiuc.edu


Current address: Assistant Professor, Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560 012, India.


Author to whom correspondence should be addressed.

J. Appl. Mech 72(4), 461-467 (Oct 20, 2004) (7 pages) doi:10.1115/1.1876434 History: Received December 08, 2003; Revised October 20, 2004

The dominant and higher-order asymptotic stress and displacement fields surrounding a stationary crack embedded in a ductile functionally graded material subjected to antiplane shear loading are derived. The plastic material gradient is assumed to be in the radial direction only and elastic effects are neglected. As in the elastic case, the leading (most singular) term in the asymptotic expansion is the same in the graded material as in the homogeneous one with the properties evaluated at the crack tip location. Assuming a power law for the plastic strains and another power law for the material spatial gradient, we derive the next term in the asymptotic expansion for the near-tip fields. The second term in the series may or may not differ from that of the homogeneous case depending on the particular material property variation. This result is a consequence of the interaction between the plasticity effects associated with a loading dependent length scale (the plastic zone size) and the inhomogeneity effects, which are also characterized by a separate length scale (the property gradient variation).

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 7

Radial variation of σθ ahead of the crack, showing the effect of γ on the region of dominance of the most singular term (denoted by the solid curve). For comparison, the two-term approximation is also shown in the homogeneous case (obtained for B∕A=0.3). The curves have been obtained for n=8 and, in the FGM case, c=0.4.

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Figure 4

Angular variation of the second asymptotic term for the graded case for (a) n=8(cmax=0.520) and (b) n=20(cmax=0.283), and for various values of c

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Figure 3

Variation of m with respect to the material exponents n(1⩽n⩽10) and c[0<c⩽cmax(n)]

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Figure 2

Variation with respect to n of the exponent of the first (p, dotted curve) and second (q, symbols) asymptotic terms, including the purely plastic and elastic-plastic solutions obtained by (13)

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Figure 1

Influence of property gradient exponent c on the variation of plastic material properties surrounding the crack tip located at r∕rF=0

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Figure 6

Stress contour plots (similar to those shown in Figs. 5) obtained with the two-term approximation for the FGM case with n=8, c=0.4, and γ=−1 (a), γ=0.1 (b), and γ=1 (c).

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Figure 5

Contour plot of the two near-tip stress components (normalized by G0) in a circular domain of radius equal to 0.8rP and centered at the crack tip. The top half of each circle corresponds to σr (which is odd in θ) and the bottom half to σθ (even in θ). (a) and (b), respectively, correspond to the one- and two-term approximations for the homogeneous case with n=8, while (c) shows the one-term solution for n=1000.



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