On shear bands in ductile materials.

*(English)*Zbl 0625.73041(Authors’ summary.) Spatially non-uniform solutions are found here for the field relations that govern the behaviour in shear of a class of rigid-plastic materials that have rate-independent responses and, after a certain amount of plastic flow, exhibit strain softening, i.e., a decline of yield stress with further flow. As strain softening can destabilize homogeneous configurations and result in the concentration of strain in narrow bands, the constitutive relations have been chosen to account for a possibly important influence upon the stress of the spatial variation of the accumulated strain. It is shown that for simple non-steady shearing flows, the shear strain, as a function of time and position, can be expressed in terms of easily evaluated elliptic functions. The theory yields strain fields showing shear bands that are remarkably similar to those observed in ballistic tests of metals under conditions in which inertial forces are negligile but the combined effects of adiabatic heating and thermal softening result in apparent strain softening. Although quantitative comparison with observation is not yet possible, it is expected that the theory will be applicable to slow deformations of geological materials that are ductile at elevated pressures and temperatures and are often softer after flowing as the result of an accumulation of internal damage with deformation.

Reviewer: M.N.L.Narasimhan

##### MSC:

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

74R99 | Fracture and damage |

##### Keywords:

Spatially non-uniform solutions; rigid-plastic materials; rate- independent responses; strain softening; concentration of strain; shear bands
PDF
BibTeX
XML
Cite

\textit{B. D. Coleman} and \textit{M. L. Hodgdon}, Arch. Ration. Mech. Anal. 90, 219--247 (1985; Zbl 0625.73041)

Full Text:
DOI

##### References:

[1] | 1911 [1] Kármán, Th. von, Festigkeitsversuche unter allseitigem Druck, Z. Vereines Deut. Ing, 55, 1749–1757. |

[2] | 1944 [1] Zener, C., & J. H. Holloman, Effect of strain rate upon plastic flow · doi:10.1063/1.1707363 |

[3] | 1950 [1] Milne-Thomson, L. M., Jacobian Elliptic Function Tables, Dover (New York). · Zbl 0041.44702 |

[4] | 1957 [1] Handin, J., & R. V. Hager, Jr., Experimental deformation of sedimentary rocks under confining pressure: tests at room temperature on dry samples, Bull. |

[5] | 1958 [1] Noll, W., A mathematical theory of the mechanical behavior of continuous media, Ar · Zbl 0083.39303 · doi:10.1007/BF00277929 |

[6] | 1960 [1] Griggs, D. T., F. J. Turner, & H. C. Heard, Deformation of rocks at 500\(\deg\) to 800\(\deg\)C in Rock Deformation, eds.: D. T. Griggs & J. Handin, Geological Society of America Memoir 79, Waverly Press (Baltimore) pp. 39–105; particularly 62–65. |

[7] | 1964 [1] Milne-Thomson, L. M., § 16, Jacobian elliptic functions and theta functions; § 17, Elliptic integrals, in Handbook of Mathematical Functions, eds.: M. Abra-Mowitz & L. A. Stegun, National Bureau of Standards (Washington), Appl. Math. Series, Vol. 55, pp. 567–626. |

[8] | 1965 [1] Murrell, S. A. F., The effect of triaxial stress systems on the strength of rocks at atmospheric · doi:10.1111/j.1365-246X.1965.tb03155.x |

[9] | 1965 [2] Truesdell, C., & W. Noll, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, Vol. III/3, Springer (Berlin, Heidelberg, New York). · Zbl 0779.73004 |

[10] | 1966 [1] Coleman, B. D., H. Markovitz, & W. Noll, Viscometric Flows of Non-Newtonian Fluids, Springer Tracts in Natural Philosophy, Vol. 5, Springer (Berlin, Heidelberg, New York). · Zbl 0137.21903 |

[11] | 1979 [1] Costin, L. S., E. E. Crisman, R. H. Hawley, & J. Duffy, On the Localization of Plastic Flow in Mild Steel Tubes under Dynamic Torsional Loading, Technical Report (January 1979), Division of Engineering, Brown University (Providence). |

[12] | 1979 [2] Jaeger, J. C., & N. G. W. Cook, Fundamentals of Rock Mechanics, 3rd Edition, Chapman & Hall (London). |

[13] | 1979 [3]Rogers, H. C., Adiabatic plastic deformatio · doi:10.1146/annurev.ms.09.080179.001435 |

[14] | 1981 [1] ELeiche, A.-S. M., Strain-rate history and temperature effects on the torsional-shear behavior of a mild · doi:10.1007/BF02325768 |

[15] | 1983 [1] Coleman, B. D., Necking and drawing in polymeric fibers under tension, Ar · Zbl 0535.73016 · doi:10.1007/BF00282158 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.