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TECHNICAL PAPERS

Size Effects on Cavitation Instabilities

[+] Author and Article Information
Christian F. Niordson, Viggo Tvergaard

Department of Mechanical Engineering, Solids Mechanics,  Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

J. Appl. Mech 73(2), 246-253 (May 10, 2005) (8 pages) doi:10.1115/1.2074747 History: Received April 26, 2005; Revised May 10, 2005

In metal-ceramic systems the constraint on plastic flow leads to so high stress triaxialities that cavitation instabilities may occur. If the void radius is on the order of magnitude of a characteristic length for the metal, the rate of void growth is reduced, and the possibility of unstable cavity growth is here analyzed for such cases. A finite strain generalization of a higher order strain gradient plasticity theory is applied for a power-law hardening material, and the numerical analyses are carried out for an axisymmetric unit cell containing a spherical void. In the range of high stress triaxiality, where cavitation instabilities are predicted by conventional plasticity theory, such instabilities are also found for the nonlocal theory, but the effects of gradient hardening delay the onset of the instability. Furthermore, in some cases the cavitation stress reaches a maximum and then decays as the void grows to a size well above the characteristic material length.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Cell model for a material with an array of voids. (a) Hexagonal distribution of voids with a cylindrical cell indicated by the circular dashed line. (b) A part of a layer of voids. (c) Using the symmetry of the problem half a void can be modeled in an axisymmetric cell. The cell radius and length are denoted Rc and Lc, respectively, and the void radius is denoted Rv.

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Figure 2

Typical finite element meshes used for the analyses for (a) a large void and (b) a small void. In (a) the two sides and the node used as control degrees of freedom for the Rayleigh-Ritz procedure are highlighted.

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Figure 3

Conventional results for a material without voids and for two materials containing voids where the in-plane spacing equals the out-of-plane spacing (Lc∕Rc=1). For one of the materials Rv∕Rc=0.2 and for the other material Rv∕Rc=0.4. The analyses are carried out for three different values of the ratio of transverse stress to axial stress ρ. The material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the overall response in terms of the true stress as a function of logarithmic strain, and (b) shows the relative void growth.

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Figure 4

Size dependent results for a material with large voids (Rv∕Rc=0.2) with equal in-plane and out-of-plane spacing (Lc∕Rc=1). Both conventional and gradient dependent results with l*∕Rv=1.0 and l*∕Rv=2.0 are shown for two different stress ratios. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the overall response in terms of the true stress as a function of strain, and (b) shows the relative void growth.

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Figure 5

Contours of effective plastic strain for ρ=0.7 at an overall strain of ϵ1=0.15. The conventional material parameters are given by σy∕E=0.004,ν=1∕3,n=10. The contours of effective plastic strain differs by ΔϵP=0.1.

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Figure 6

Size dependent results for sparsely distributed voids (Rv∕Rc=0.001). Both conventional and gradient dependent results with l*∕Rv=1.0 and l*∕Rv=2.0 are shown for two different stress ratios. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the overall response in terms of the true stress as a function of strain, and (b) shows the relative void growth.

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Figure 7

Size dependent results for a material with Rv∕Rc=0.05 and Lc∕Rc=1. Both conventional and gradient dependent results with l*∕Rv=1.0, 2.0, 3.0, and 4.0 are shown for two different stress ratios. The curves correspond to a material with a fixed length parameter and a fixed initial void volume fraction, but with different void size and void spacing. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the overall response in terms of the true stress as a function of strain, and (b) shows the relative void growth.

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Figure 8

Size dependent results for sparsely distributed voids (Rv∕Rc=0.001) for different values of ρ. Both conventional and gradient dependent results with l*∕Rv=2.0 are shown. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the overall response in terms of the true stress as a function of strain, and (b) shows the relative void growth.

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Figure 9

Size dependent results for sparsely distributed voids (Rv∕Rc=0.001). Both conventional and gradient dependent results with l*∕Rv=2.0 and 3.0 are shown for two different stress ratios. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10. (a) Shows the relative void growth as a function of strain, and (b) shows the ratio of the void size in the x1 direction to the void size in the x2 direction for ρ=0.9.

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Figure 10

Relative void growth for sparsely distributed voids (Rv∕Rc=0.001) using the general theory with each of the three length parameter activated one by one keeping the other two equal to zero. For comparison results for the single parameter version is included. The conventional material parameters are given by σy∕E=0.004, ν=1∕3, and n=10.

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