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TECHNICAL PAPERS

Flaw Tolerance in a Thin Strip Under Tension

[+] Author and Article Information
Huajian Gao, Shaohua Chen

Max Planck Institute for Metals Research, Heisenbergstrasse 3, D-70569 Stuttgart, Germany

J. Appl. Mech 72(5), 732-737 (Jan 15, 2005) (6 pages) doi:10.1115/1.1988348 History: Received September 09, 2004; Revised January 15, 2005

Abstract

Recent studies on hard and tough biological materials have led to a concept called flaw tolerance which is defined as a state of material in which pre-existing cracks do not propagate even as the material is stretched to failure near its limiting strength. In this process, the material around the crack fails not by crack propagation, but by uniform rupture at the limiting strength. At the failure point, the classical singular stress field is replaced by a uniform stress distribution with no stress concentration near the crack tip. This concept provides an important analogy between the known phenomena and concepts in fracture mechanics, such as notch insensitivity, fracture size effects and large scale yielding or bridging, and new studies on failure mechanisms in nanostructures and biological systems. In this paper, we discuss the essential concept for the model problem of an interior center crack and two symmetric edge cracks in a thin strip under tension. A simple analysis based on the Griffith model and the Dugdale-Barenblatt model is used to show that flaw tolerance is achieved when the dimensionless number $Λft=ΓE∕(S2H)$ is on the order of 1, where $Γ$ is the fracture energy, $E$ is the Young’s modulus, $S$ is the strength, and $H$ is the characteristic size of the material. The concept of flaw tolerance emphasizes the capability of a material to tolerate cracklike flaws of all sizes.

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Figures

Figure 1

Flaw tolerance in a center cracked strip under tension. (a) The crack configuration and (b) the state of flaw tolerance. The stress around the crack uniformly reaches the theoretical strength of the material with no stress concentration near the crack tip.

Figure 2

Flaw tolerance of a double edge cracked strip under tension. (a) The double edge crack configuration and (b) the corresponding flaw tolerance solution.

Figure 3

The normalized critical size for a strip of width 2H to tolerate a crack of length 2a. The minimum value of such curves corresponds to the critical size for the strip to tolerate cracks of all sizes. Calculations based on (a) the Griffith model and (b) the Dugdale model.

Figure 4

The flaw tolerance solution based on the Dugdale model of the center cracked strip. (a) A Dugdale interaction law is assumed in the plane of the crack. The condition of flaw tolerance is equivalent to requiring δtip not to exceed δ0 for any crack size a. In this case, the opening displacement in the plane of the crack outside the crack region should lie within the range of cohesive interaction δ0. (b) The reduced elastic problem of a semi-infinite strip subjected to a uniform stress S over the top surface outside the crack region.

Figure 5

A periodic crack problem used to derive an approximate solution for the strip cracks in Figs.  12. Symmetry conditions along the dotted lines suggest that the periodic crack problem can be used as an approximate solution to the strip crack problems. In the mode III case, the period crack solution becomes the exact solutions for the strip cracks. (a) The period crack configuration and (b) the flaw tolerant solution.

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