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A strain-gradient plasticity theory with a corner-like effect: a thermodynamics-based extension

  • IUTAM Paris 2015
  • Published:
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Abstract

A higher-order strain gradient plasticity theory with a corner-like effect, which was recently proposed by the author, is updated to a form suitable for its thermodynamics-based extensions. It is assumed that the free energy is augmented by a defect energy that has a non-quadratic dependence on the plastic strain gradients. Predictive feature of the extended theory is examined via finite element analyses of a constrained simple shear problem and a plane strain tension problem involving plastic flow localization.

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Notes

  1. One can construct constitutive models directly into \((\bar{\mathbf{L}}^{\mathrm{p}})_{\mathrm{sym}}\), as well as into \((\bar{\mathbf{L}}^{\mathrm{p}})_{\mathrm{anti{\text {-}}sym}}\), in the conceptual intermediate configuration. In the case of crystal plasticity, construction of constitutive relations in the current configuration and that in the intermediate configuration are visibly equivalent. For a single slip system, one can assume \(\bar{\mathbf{L}}^{\mathrm{p}}=\dot{\gamma }{} \mathbf{s}\otimes \mathbf{m}\), where \(\dot{\gamma }\) is the crystallographic slip, and s and m are unit vectors of slip direction and slip plane normal that reside in the intermediate configuration (i.e., the lattice space). According to Eq. (2), we can see \(\mathbf{L}^{\mathrm{p}}=\mathbf{F}^{\mathrm{e}}\cdot \bar{\mathbf{L}}^{\mathrm{p}}\cdot \mathbf{F}^{\mathrm{e}-1}=\mathbf{F}^{\mathrm{e}}\cdot (\dot{\gamma }{} \mathbf{s}\otimes \mathbf{m})\cdot \mathbf{F}^{\mathrm{e}-1}=\dot{\gamma }{} \mathbf{s}^{*}\otimes \mathbf{m}^{*}\). The vectors \(\mathbf{s}^{*}=\mathbf{F}^{\mathrm{e}}\cdot \mathbf{s}\) and \(\mathbf{m}^{*}=\mathbf{F}^{\mathrm{e}-\mathrm{T}}\cdot \mathbf{m}\) are simply viewed as the slip direction and slip plane normal in the current configuration.

  2. The \(\tau _\mathrm{e} \) and \(\dot{\varepsilon }^{\mathrm{p}}\) in Eqs. (8) and (9) also satisfy the relations \(\mathbf{M}^{\mathrm{e}}{:}\,\bar{\mathbf{L}}^{\mathrm{p}}=\tau _\mathrm{e} \dot{\varepsilon }^{\mathrm{p}}\) and \(\tau _\mathrm{e} =\sqrt{\frac{3}{2}}{} \mathbf{M}^{\mathrm{e}}{:}\,\bar{\mathbf{N}}^{\mathrm{p}}\) with \(\bar{\mathbf{L}}^{\mathrm{p}}=\dot{\phi }{} \mathbf{F}^{\mathrm{e}-1}\cdot \mathbf{N}^{\mathrm{p}}\cdot \mathbf{F}^{\mathrm{e}}=\dot{\phi } \bar{\mathbf{N}}^{\mathrm{p}}\) and \(\mathbf{M}^{\mathrm{e}}=\mathbf{F}^{\mathrm{eT}}\cdot {\varvec{\uptau }} \cdot \mathbf{F}^{\mathrm{e}-\mathrm{T}}\), where \(\mathbf{M}^{\mathrm{e}}\) is called the Mandel stress.

  3. Here, we use the relation \({\varvec{\uptau }} {:}\,\delta \mathbf{D}^{\mathrm{e}}=(\mathbf{F}^{\mathrm{e}-1}\cdot {\varvec{\uptau }} \cdot \mathbf{F}^{\mathrm{e}-\mathrm{T}}){:}\,(\delta \dot{\mathbf{F}}^{\mathrm{eT}}\cdot \mathbf{F}^{\mathrm{e}})_{\mathrm{sym}} =\mathbf{T}^{\mathrm{e}}{:}\,\delta \dot{\mathbf{E}}^{\mathrm{e}}\) with \(\delta \dot{\mathbf{F}}^{\mathrm{e}}=\delta \mathbf{L}^{\mathrm{e}}\cdot \mathbf{F}^{\mathrm{e}}\).

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Acknowledgments

This work was partly supported by JSPS Grant-in-Aid for Scientific Research (B), Grant Number 25289001.

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Mechanical Engineering, Yamagata University, Yonezawa, Yamagata, 992-8510, Japan

    Mitsutoshi Kuroda

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  1. Mitsutoshi Kuroda
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Kuroda, M. A strain-gradient plasticity theory with a corner-like effect: a thermodynamics-based extension. Int J Fract 200, 115–125 (2016). https://doi.org/10.1007/s10704-015-0055-9

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