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**Rheological models and hysteresis effects.**
*(English)*
Zbl 0633.73001

This paper proposes several types of constitutive relations in an attempt to describe mathematically the mechanical behaviour of a variety of materials. Such laws are derived from models consisting of serial or parallel combinations of two basic rheological elements in parallel or series, respectively. The emphasis is on rate-independent models consisting of elastic, plastic and memory elements to describe elasto- plastic behaviour with hysteresis effects. In particular the classical Preisach model for ferromagnetism is deduced as a special case. In an attempt to describe local fracture the author proposes a new type of rheological element. This feature of absolute irreversibility renders such a model useful in many applications outside continuum mechanics.

Assuming that deformations are small, several related dynamical problems are formulated in terms of systems of variational equations for two types of model. The first is a parallel arrangement of several rheological models each consisting of a linear elastic element and a viscous element in series. Replacing the viscous elements by perfectly plastic elements results in the classical model for elasto-plasticity with strain hardening. The second is a serial arrangement of several components each consisting of an elastic and a viscous element in parallel. Two special cases are considered. In one the elastic elements are linear and in the other the viscous elements are linear. The latter case is also generalised by replacing the elastic elements by memory elements. Existence and uniqueness theorems are proved for each of these problems.

This is an interesting paper for those researchers concerned with rheological modelling. However it is only likely to be understood by those with some ability in modern mathematical analysis.

Assuming that deformations are small, several related dynamical problems are formulated in terms of systems of variational equations for two types of model. The first is a parallel arrangement of several rheological models each consisting of a linear elastic element and a viscous element in series. Replacing the viscous elements by perfectly plastic elements results in the classical model for elasto-plasticity with strain hardening. The second is a serial arrangement of several components each consisting of an elastic and a viscous element in parallel. Two special cases are considered. In one the elastic elements are linear and in the other the viscous elements are linear. The latter case is also generalised by replacing the elastic elements by memory elements. Existence and uniqueness theorems are proved for each of these problems.

This is an interesting paper for those researchers concerned with rheological modelling. However it is only likely to be understood by those with some ability in modern mathematical analysis.

Reviewer: P.J.Barratt

### MSC:

74A20 | Theory of constitutive functions in solid mechanics |

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

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

76A99 | Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena |

### Keywords:

rate-independent models; memory elements; elasto-plastic behaviour; hysteresis effects; classical Preisach model for ferromagnetism; local fracture; absolute irreversibility; dynamical problems; parallel arrangement of several rheological models; linear elastic element; viscous element; perfectly plastic elements; strain hardening; serial arrangement of several components; Existence; uniqueness theorems; rheological modelling
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\textit{A. Visintin}, Rend. Semin. Mat. Univ. Padova 77, 213--241 (1987; Zbl 0633.73001)

### References:

[1] | H. Brezis , Problèmes unilateraux , J. Math. Pure Appl. , 51 ( 1972 ), pp. 1 - 68 . MR 428137 | Zbl 0237.35001 · Zbl 0237.35001 |

[2] | A. Damlamian - A. Visintin , Une généralisation vectorielle du modèle de Preisach pour l’hystéresis , C. R. Acad. Sc. Paris , t. 297 (24 oct. 1983 ), serie I , pp. 437 - 440 . MR 732853 | Zbl 0546.35068 · Zbl 0546.35068 |

[3] | G. Duvaut - J. L. LIONS, Inequalities in mechanics and physics , Springer , Berlin , 1976 . MR 521262 | Zbl 0331.35002 · Zbl 0331.35002 |

[4] | W.N. Findley - J.S. Lai - K. Onaran , Creep and relaxation of nonlinear viscoelastic materials , North-Holland , Amsterdam , 1976 . MR 431847 | Zbl 0345.73034 · Zbl 0345.73034 |

[5] | S. FLÜGGE ed., Elasticity and Plasticity, volume VI of Encyclopedia of Physics , Springer-Verlag , Berlin , 1958 . · Zbl 0103.16403 |

[6] | M.A. Krasnosel’ski , Equations with nonlinearities of hysteresis type (Russian) , Abh. Akad. Wiss. DDR , 3 ( 1977 ), pp. 437 - 458 . (English abstract in Zbt. 406 - 93032 ). MR 528438 |

[7] | M.A. Krasnosel’ski et. al., Hysterant operator , Soviet Math. Dokl. , 11 ( 1970 ), pp. 29 - 33 . · Zbl 0212.58002 |

[8] | M.A. Krasnosel’skii - A. V. POKROVSKIĭ, Modeling transducers with hysteresis by means of continuous systems of relays , Soviet Math. Dokl. , 17 ( 1976 ), pp. 447 - 451 . Zbl 0344.68033 · Zbl 0344.68033 |

[9] | J Nečas - J. Hlavaček , Mathematical theory of elastic and elasto-plastic bodies: an introduction , Elsevier , Amsterdam , 1981 . Zbl 0448.73009 · Zbl 0448.73009 |

[10] | G.S. Nguyen , Sur les matériaux standards généralisés , J. de Mécanique , 14 ( 1975 ), pp. 39 - 63 . MR 416177 | Zbl 0308.73017 · Zbl 0308.73017 |

[11] | P. Suquet , Plasticité et homogeneisation, Thèse , Paris , 1882 . |

[12] | A. Visintin , A model for hysteresis of distributed systems, Ann , Mat. Pura Appl. , 131 ( 1982 ), pp. 203 - 231 . MR 681564 | Zbl 0494.35052 · Zbl 0494.35052 · doi:10.1007/BF01765153 |

[13] | A. Visintin , On the Preisach model for hysteresis , Nonlinear Analysis T.M.A. , 9 ( 1984 ), pp. 977 - 996 . MR 760191 | Zbl 0563.35007 · Zbl 0563.35007 · doi:10.1016/0362-546X(84)90094-4 |

[14] | A. Visintin , Evolution equations with hysteresis in the source , SIAM J. Math. Anal. , 17 ( 1986 ), 1113 - 1138 . MR 853520 | Zbl 0618.35053 · Zbl 0618.35053 · doi:10.1137/0517079 |

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