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Strain Gradient Plasticity: A Variety of Treatments and Related Fundamental Issues

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From Creep Damage Mechanics to Homogenization Methods

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 64))

Abstract

Different theoretical interpretations and possible mathematical expressions for the higher-order strain gradient plasticity theory initiated by Aifantis are investigated. These different interpretations of the theory result in different computational procedures. The effects of the orders of finite-element shape functions and the number of Gaussian quadrature points on the qualities of numerical solutions are examined for different formulations.

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Notes

  1. 1.

    In Kuroda and Tvergaard (2010), somewhat different definitions for \(\pmb g^\mathrm {p}\) were used; i.e., \(\pmb g^\mathrm {p}~=~-~\nabla \varepsilon ^\mathrm {p}\) without the inclusion of \(\beta \) for the simplest theory, and \(\pmb g^\mathrm {p}=-l_*^2h\nabla \varepsilon ^\mathrm {p}\) to discuss an alternative formulation (corresponding to Treatment 4 in the present study) of Fleck-Hutchinson theory (Fleck and Hutchinson 2001), where \(l_*\) is a length-scale parameter. In the present paper, to enable a broader and extended discussion, the more general definition given by Eq. (9.7) is employed.

  2. 2.

    This type of derivation of constitutive relations for the higher-order stresses was presented in the context of gradient crystal plasticity theory (Gurtin 2002). The crystal plasticity version of Eq. (9.14) also appeared in Gurtin (2002). Thermodynamical formulations of crystal plasticity were also discussed in Forest et al. (2002).

  3. 3.

    Gurtin and Anand (2005a) used the quantity \(\dot{E}_\mathrm {p} = \sqrt{(\dot{\varepsilon }^\mathrm {p})^2 + l_\mathrm {dis}^2|\nabla \dot{\varepsilon }^\mathrm {p}|^2}\) for the denominator of Eq. (9.42).

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Mechanical Systems Engineering, Yamagata University, Jonan 4-3-16, Yonezawa, Yamagata, 992-8510, Japan

    Mitsutoshi Kuroda

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  1. Mitsutoshi Kuroda
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Correspondence to Mitsutoshi Kuroda .

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Editors and Affiliations

  1. Otto-von-Guericke-Universität Magdeburg, Fakultät für Maschinenbau, Institut für Mechanik, Lehrstuhl für Technische Mechanik, Universitätsplatz 2, 39106 Magdeburg, Germany

    Holm Altenbach

  2. Department of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan

    Tetsuya Matsuda

  3. Department of Mechanical Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

    Dai Okumura

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Kuroda, M. (2015). Strain Gradient Plasticity: A Variety of Treatments and Related Fundamental Issues. In: Altenbach, H., Matsuda, T., Okumura, D. (eds) From Creep Damage Mechanics to Homogenization Methods. Advanced Structured Materials, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-319-19440-0_9

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  • DOI: https://doi.org/10.1007/978-3-319-19440-0_9

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  • Online ISBN: 978-3-319-19440-0

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