Onset of shear localization in viscoplastic solids.

*(English)*Zbl 0612.73046An outstanding problem in mechanics is the modeling of the phenomenon of initiation and development of localized shear bands, in materials whose inelastic deformation behavior is inherently rate-dependent. R. J. Clifton [Materials response to ultra-high loading rates, NRC Rpt. NMAB- 356, p. 129] and Y. L. Bai [e.g.: J. Mech. Phys. Solids 30, 195-207 (1982; Zbl 0491.73037)] have presented a linear perturbation stability analysis for the initiation of shear bands in viscoplastic solids deforming in simple shear. This localization analysis is essentially one- dimensional in nature.

In this paper, we present a three-dimensional generalization of this linear perturbation stability analysis for the onset of shear localization. We neglect elastic effects and consider isotropic, incompressible, viscoplastic materials which exhibit strain hardening (or softening), strain-rate hardening, thermal softening and pressure hardening. For this class of materials, we derive the general characteristic stability equation. Next, we focus our attention on the special version of this equation for plane motions and consider the two physically important limiting cases of (1) quasi-static, isothermal deformations, and (2) dynamic, adiabatic deformations. For these cases, we derive (a) the critical conditions for the formation of shear bands, (b) the most probable directions along which the bands can form, and (c) information regarding the incipient rate of growth of the emergent shear bands.

The results, which are presented in detail in the body of the paper, provide new insight into the important phenomenon of shear localization under both quasi-static and rapid deformation of many materials including both metals and polymers.

In this paper, we present a three-dimensional generalization of this linear perturbation stability analysis for the onset of shear localization. We neglect elastic effects and consider isotropic, incompressible, viscoplastic materials which exhibit strain hardening (or softening), strain-rate hardening, thermal softening and pressure hardening. For this class of materials, we derive the general characteristic stability equation. Next, we focus our attention on the special version of this equation for plane motions and consider the two physically important limiting cases of (1) quasi-static, isothermal deformations, and (2) dynamic, adiabatic deformations. For these cases, we derive (a) the critical conditions for the formation of shear bands, (b) the most probable directions along which the bands can form, and (c) information regarding the incipient rate of growth of the emergent shear bands.

The results, which are presented in detail in the body of the paper, provide new insight into the important phenomenon of shear localization under both quasi-static and rapid deformation of many materials including both metals and polymers.

##### MSC:

74B99 | Elastic materials |

74C99 | Plastic materials, materials of stress-rate and internal-variable type |

74D99 | Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) |

74H55 | Stability of dynamical problems in solid mechanics |

74A15 | Thermodynamics in solid mechanics |

##### Keywords:

initiation; development; localized shear bands; inelastic deformation; rate-dependent; three-dimensional generalization; linear perturbation stability analysis; onset of shear localization; isotropic, incompressible, viscoplastic materials; strain hardening; softening; strain-rate hardening; thermal softening; pressure hardening; general characteristic stability equation; quasi-static, isothermal deformations; dynamic, adiabatic deformations
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\textit{L. Anand} et al., J. Mech. Phys. Solids 35, 407--429 (1987; Zbl 0612.73046)

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##### References:

[1] | Bai, Y.L., (), 277 |

[2] | Bai, Y.L., J. mech. phys. solids, 30, 1982, (1982) |

[3] | Clifton, R.J., Material response to ultra-high loading rates, NRC rpt. NMAB-356, 129, (1980) |

[4] | Clifton, R.J.; Duffy, J.; Hartley, K.A.; Shawki, T.G., Scripta metall., 18, 443, (1984) |

[5] | Fressengeas, C.; Molinari, A., Acta metall., 33, 387, (1985) |

[6] | Hoff, N.J., (), 29 |

[7] | Hutchinson, J.W.; Obrecht, H., (), 101 |

[8] | Rabotnov, G.N.; Shesterikov, S.A., J. mech. phys. solids, 6, 27, (1957) |

[9] | Rice, J.R., Theoretical and applied mechanics, (), 207 |

[10] | Rudnicki, J.W.; Rice, J.R., J. mech. phys. solids, 23, 371, (1975) |

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