Rheological models and hysteresis effects.

*(English)*Zbl 0633.73001This 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##### References:

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