## Abstract

A new phenomenological deformation theory with strain gradient effects is proposed. This theory, which belongs to nonlinear elasticity, fits within the framework of general couple stress theory and involves a single material length scale l. In the present theory three rotational degrees of freedom ω_{i} are introduced in addition to the conventional three translational degrees of freedom u_{i}. ω_{i} has no direct dependence upon u_{i} and is called the micro-rotation, i.e. the material rotation θ_{i} plus the particle relative rotation. The strain energy density is assumed to only be a function of the strain tensor and the overall curvature tensor, which results in symmetric Cauchy stresses. Minimum potential principle is developed for the strain gradient deformation theory version. In the limit of vanishing l, it reduces to the conventional counterparts: J_{2} deformation theory. Equilibrium equations, constitutive relations and boundary conditions are given in details. Comparisons between the present theory and the theory proposed by Shizawa and Zbib (Shizawa, K., Zbib, H.M., 1999. A thermodynamical theory gradient elastoplasticity with dislocation density Censor: fundamentals. Int. J. Plast. 15, 899) are given. With the same hardening law as Fleck et al. (Fleck, N.A., Muller, G.H., Ashby, M.F., Hutchinson, J.W., 1994 Strain gradient plasticity: theory and experiment. Acta Metall. Mater 42, 475), the new strain gradient deformation theory is used to investigate two typical examples, i.e. thin metallic wire torsion and ultra-thin metallic beam bend. The results are compared with those given by Fleck et al, 1994 and Stolken and Evans (Stolken, J.S., Evans, A.G., 1998. A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109). In addition, it is explained for a unit cell that the overall curvature tensor produced by the overall rotation vector is the work conjugate of the overall couple stress tensor.

Original language | English |
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Pages (from-to) | 971-995 |

Number of pages | 25 |

Journal | International Journal of Plasticity |

Volume | 18 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 2002 |

Externally published | Yes |

## Keywords

- Constitutive behavior
- Couple stress theory
- Microstructure
- Nonlinear elasticity
- Strain gradient effects