On the extremal stress and displacement in Hencky plasticity.

*(English)*Zbl 0548.73022Some earlier studies have established the existence of an extremal displacement field \(\bar u\) for an elasto-perfectly plastic material employing Hencky’s law. The construction of the extremal stress field \({\bar\sigma }\) from \(\bar u\) and duality between the spaces of stresses and strains have also been investigated. In the present paper, considering a pair of extremal fields \(\bar u\), \({\bar\sigma }\), it is proved that the singular density of the deformation \(\epsilon^ D(u)\) is determined by the absolutely continuous tensor \({\bar\sigma }\)(D) via a relation stated in the paper. The proof has been presented with the help of several theorems and lemmas and requires a good knowledge of mathematical analysis.

Reviewer: V.K.Arya

##### MSC:

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

49J40 | Variational inequalities |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

##### Keywords:

extremal displacement; Hencky’s law; extremal stress; pair of extremal fields; singular density of the deformation; determined by the absolutely continuous tensor
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##### References:

[1] | G. Anzellotti, Pairings between measures and bounded functions and compensated compactness , to appear in Ann. Mat. Pura e Appl. · Zbl 0572.46023 |

[2] | G. Anzellotti, On the minima of functionals with linear growth , · Zbl 0589.49027 |

[3] | G. Anzellotti, On the existence of the rates of stress and displacement for Prandtl-Reuss plasticity , Quart. Appl. Math. 41 (1983/84), no. 2, 181-208. · Zbl 0521.73030 |

[4] | G. Anzellotti and M. Giaquinta, Existence of the displacement field for an elastoplastic body subject to Hencky’s law and von Mises yield condition , Manuscripta Math. 32 (1980), no. 1-2, 101-136. · Zbl 0465.73022 |

[5] | G. Anzellotti and M. Giaquinta, On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium , J. Math. Pures Appl. (9) 61 (1982), no. 3, 219-244 (1983). · Zbl 0467.73044 |

[6] | E. Giusti, Minimal surfaces and functions of bounded variation , Department of Pure Mathematics, Australian National University, Canberra, 1977. · Zbl 0402.49033 |

[7] | R. Kohn Ph.D. thesis, Princeton, 1979. |

[8] | R. Kohn and R. Temam, Dual spaces of stresses and strains, with application to Hencky plasticity , · Zbl 0532.73039 |

[9] | R. Temam and G. Strang, Functions of bounded deformation , Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 7-21. · Zbl 0472.73031 |

[10] | R. Temam, Existence theorem for a variational problem of plasticity , Nonlinear Problems of Analysis in Geometry and Mechanics (Proc. Sympos., Univ. Paul Sabatier, Toulouse, 1979) eds. Attia, Bancel, and Gumowski, Res. Notes in Math., vol. 46, Pitman, London, 1981, pp. 57-70. · Zbl 0455.73036 |

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