Identification by a non-integer order model of the mechanical behaviour of an elastomer

Damping in a dynamic system results from the transformation of mechanical energy (the sum of kinetic and potential energies) into another type of energy (heat, noise, etc.). Such dissipation is partly due to the viscoelastic properties of the different components of the system under consideration. To allow for these viscoelastic properties, behaviour is generally assumed to be either viscous or hysteretic, although these hypotheses are no longer experimentally confirmed for strongly dissipative materials. The study we present here is concerned with modelling viscoelastic behaviour by operators which are non-integer order time differential. This model is considered for small deformations, a hypothesis which makes it possible to postulate the linearity of the behaviour operator. We were especially concerned with the case of solids for which there is a natural state (non-deformed and non-stressed) which is deemed to be the initial state. After a review of the general form of the constitutive law of a viscoelastic, homogeneous isotropic non-ageing medium in small deformations, we introduce fractional models as specific cases of these so-called hereditary continuous media. Each fractional model, defined by a generalised differential equation, can be associated with a memory; this allows us to obtain a second possible classification. The study of the four-parameter model which we develop afterwards allows us to classify the viscoelastic behaviour of the material in time and frequency domains. This study also makes it possible to define two characteristic parameters of the model, f(phi) and eta(phi), which are better suited to an identification method. The final part is an attempt to determine the coefficients of the constitutive law for elastomers. The identification is carried out by limiting a deviation constructed from experimental data and an analytical expression of displacement. Copyright (C) 1996 Elsevier Science Ltd.