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A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading

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Abstract

In this paper, we have developed a linear stability analysis to predict the formation of necking instabilities in porous ductile plates subjected to dynamic biaxial stretching. The mechanical behavior of the material is described with the Gurson–Tvergaard–Needleman constitutive relation for progressively cavitating solids (Gurson 1977; Tvergaard 1981, 1982; Tvergaard and Needleman 1984) which considers the voids to be spherical and the matrix material isotropic with yielding defined by the von Mises (1928) criterion. The analytical model is formulated in a two-dimensional framework in which the multiaxial stress state that develops inside the necked region is approximated with the Bridgman (1952) correction factor, superimposing a hydrostatic stress state to the uniform stress field that develops in the plate before localization. As opposed to the linear stability models published so far to model dynamic necking in ductile plates, which consider the material to be fully dense and incompressible, the approach developed in this paper provides new insights into the interplay between porosity and inertia on plastic localization. In addition, the predictions of the theoretical model for the critical strain leading to necking formation have been compared with unit-cell finite element calculations performed in ABAQUS/Explicit (2019). Satisfactory quantitative and qualitative agreement has been found between the theoretical and the computational approach for loading paths ranging from plane strain tension to nearly equibiaxial tension, loading rates varying from 100 to \(10{,}000~\text {s}^{-1}\), and different values of the initial void volume fraction ranging from 0.01 to 0.1. Both analytical and finite element results suggest that the influence of porosity on necking localization increases, due to voids coalescence, as the loading rate increases and the loading path approaches equibiaxial tension. The original formulation developed in this paper serves as a basis for analytically modeling the dynamic formability of porous ductile plates, and it can be readily extended to consider more complex porous plasticity theories, e.g. constitutive models which consider the anisotropy of the material (Benzerga and Besson 2001) and/or voids with different shapes (Gologanu et al. 1993; Monchiet et al. 2008).

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Funding

The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme. Project PURPOSE, grant agreement 758056.

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Authors and Affiliations

  1. Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, Avda. de la Universidad, 30, 28911, Leganés, Madrid, Spain

    M. Anil Kumar, K. E. N’souglo & J. A. Rodríguez-Martínez

Authors
  1. M. Anil Kumar
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  2. K. E. N’souglo
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  3. J. A. Rodríguez-Martínez
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Contributions

MAK: Conceptualization; Data curation; Formal analysis; Investigation; Software; Validation; Writing—original draft. KEN: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Supervision; Validation; Writing—original draft; Writing—review and editing. JAR-M: Conceptualization; Formal analysis; Funding acquisition; Investigation; Methodology; Project administration; Resources; Supervision; Validation; Visualization; Writing—original draft; Writing—review and editing.

Corresponding author

Correspondence to K. E. N’souglo.

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The authors declare that they have no conflict of interest.

Supplementary material

In the following link we provide access the codes developed in Wolfram Mathematica to compute the stability analysis results presented in Sect. 3: https://doi.org/10.21950/2XVATI.

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Kumar, M.A., N’souglo, K.E. & Rodríguez-Martínez, J.A. A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading. Int J Fract 232, 181–198 (2021). https://doi.org/10.1007/s10704-021-00599-8

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