Pascal Cosson

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

Abstract 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, fϑ and ηϑ, 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.

Methodology of combined aplpication of directional derivatives and the extended finite element method (X-FEM) for solving vibrations eigenvalue problems

This paper presents a new methodology for solving the eigenvalue problem for time dependent structures. The time dependent structures of interest are structures with a moving discontinuity such as crack or structures with moving free (external/internal) surfaces. For the last case, they can result from a removal of material during a machining process or from a deterioration of the structure’s geometry. The methodology that we developed, is based on a combination of the eXtended Finite Element Method (X-FEM) and the Directional Derivatives method. X-FEM enables to overcome the drawbacks of conformity and remeshing: indeed, using standard FEM, a moving discontinuity in time within a structure requires not only that the mesh must conform to the discontinuity geometry but also to fully remesh the structure as much as necessary to follow the discontinuity in time. In order to alleviate this last point, the directional derivatives are a powerful tool because they allow to estimate the evolution of quantities from on reference domain to another one. In our case, they will allow to estimate the solutions of the eigenvalue problem. We suggest on the first sections to remind the main keys of both methods and we present then the combined methods in order to solve an eigenvalue problem. The application will be done on a one-dimensional eigenvalue problem and the numerical results will be presented to demonstrate the accuracy and the advantages of selected approaches. We conclude on the future prospects of the current work that mainly consist of to develop the methodology at the second order in order to increase the accuracy and to find a criteria in order to automatize the combined methods.

Numerical Simulation of mechanical behavior of woven composite at different strain rate by a collaborative elasto-plastio-damage model with fractionnal derivatives

This paper deals with a collaborative model to represent hysteresis behavior at different strain rates. The model consists of two sub-models. The first one treats the behavior during loading path. The elastic and in-elastic strains are computed as well as the in-ply damages. The strain-rate sensitivity is also taken into account. The second sub-model involves a fractional derivate approach to describe viscoelastic material response during unloading path. The hysteresis loops and strain rate sensitivity are taken into account by fractional con-stitutive law. Fractional model involves a few parameters which are easily identified through an optimization procedure from the experimental data. The model is validated for thermoset and thermoplastic composite materials at different strain rates.