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Research Papers

Shakedown Fatigue Limits for Materials With Minute Porosity

[+] Author and Article Information
Jehuda Tirosh, Sharon Peles

 Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel

J. Appl. Mech 76(3), 031016 (Mar 13, 2009) (9 pages) doi:10.1115/1.3005961 History: Received December 11, 2007; Revised July 10, 2008; Published March 13, 2009

The intention of this study is to predict the fatigue-safe long life behavior of elastoplastic porous materials subjected to zero-tension fluctuating load. It is assumed that the materials contain a dilute amount of voids (less than 5%) and obey Gurson’s model of plastic yielding. The question to be answered is what would be the highest allowable stress amplitude that a porous material can endure (the “endurance limit”) when undergoing an infinite number of loading/unloading cycles. To reach the answer we employ the two shakedown theorems: (a) Melan’s static shakedown theorem (“elastic shakedown”) for establishing the lower bound to fatigue limit and (b) Koiter’s kinematic shakedown theorem (“plastic shakedown”) for establishing its upper bound. The two bounds are formulated rigorously but solved with some numerical assistance, mainly due to the nonlinear pressure dependency of the material behavior and the complex description of the plastic flow near stress-free voids. Both bounds (“dual bounds”) are adjusted to capture Gurson-like porous materials with noninteractive voids. General residual stresses (either real or virtual) are presented in the analysis. They are assumed to be time-independent as generated, say, by permanent temperature gradient between void surfaces and remote material boundaries. Such a situation is common, for instance, in ordinary porous sleeves (used in space industry and alike). A few experiments agree satisfactorily with the shakedown bounding concept.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic illustration of shakedown bounds to fatigue limit (endurance limit).

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Figure 2

(a) A view of typical material with voids subjected to unidirectional cyclic loading. (b) Elastic stress distribution around a spherical void caused by remote unidirectional traction σ∞. The various curves are the radial distributions around the void at various angles (the zero angle is collinear with the load direction).

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Figure 3

An admissible residual stress distribution along the radial distance ρ=r/r0 (solved by Eq. 6) normalized with its magnitude p(p=αEΔT/(1−ν))

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Figure 4

The restricted permissible magnitude of the residual stress, pmax, versus the porosity level of the solid as solved by Eq. 7

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Figure 5

The loci of Melan’s (1) elastic shakedown conditions (the limit load for ratcheting-free material response at a given void fraction f). The area inside each ellipse is the domain of fatigue-safe stress amplitudes.

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Figure 6

Stream lines of smooth plastic flow around a spherical void in a structure subjected to unidirectional remote load (in vertical direction)

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Figure 7

The plastic shakedown condition (the equality sign of Eq. 10) for a safe fluctuating stress in materials with porosity fractions of f=0.01, f=0.05, and f=0.1

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Figure 8

(a) Dual bounds for the ordinary porous materials (with zero residual stresses). The experimental data are from Sanderow (27) and Katsushi (28). (b) Dual shakedown bounds for porous materials with a prescribed amount of residual stresses of magnitude p=±0.3σ0. The experimental data are taken from Sanderow (27) and Katsushi (28).

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Figure 9

Porous material with uniform array of spherical voids

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