On the use of fractional derivative operators to describe
viscoelastic damping in structural dynamics
Finite element formulation of sandwich beams and approximation
of fractional derivatives by using the Gα scheme

EADS Corporate Research Center France, Suresnes
Jean-François DEÜ
LMSSC, Cnam, Paris
François DUBOIS
Chaire de calcul scientifique, Cnam, Paris
Laboratoire de Mathématiques, Analyse Numérique et Équations aux Dérivées Partielles,
Université Paris Sud, Orsay

Many investigations have demonstrated the potential of viscoelastic materials to improve the dynamics of lightly damped structures. There are numerous techniques to incorporate these materials into structures. The constrained layer passive damping treatment is already largely used to reduce structural vibrations, especially in conjunction with active control [2, 10, 16]. One of the crucial questions is how to quantify such a material damping if the viscoelastic solid has a weak frequency dependence on its dynamic properties over a broad frequency range. Classical linear viscoelastic models, using integer derivative operators, convolution integral or internal variables, become cumbersome due to the high quantity of parameters needed to describe the material behavior. In order to overcome these difficulties, fractional derivative operators acting on both, strain and stress can be employed.
Until the beginning of the 80s, the concept of fractional derivatives associated to viscoelasticity was regarded as a sort of curve-fitting method. Later, Bagley and Torvik [1] gave a physical justification of this concept in a thermodynamic framework. Their fractional model has become a reference in literature. Special interest is today dedicated to the implementation of fractional constitutive equations into FE formulations. In this context, the numerical methods in the time domain are generally associated with the Grünwald formalism for the fractional order derivative of the stress-strain relation in conjunction with a time discretization scheme. Most of the approaches found in literature are restricted to single-degree-of-freedom systems and bar-type structures although the numerical investigation can be sophisticated (see, for example, [15, 6, 5]). On the other side, the numerical community is interested in the approximation of fractional derivatives. One refers to the pioneering theoretical work of Lubich [13] and the state of the art proposed by Diethelm et al. [3]. Most applications use the discrete convolution formula proposed by Grünwald-Letnikov [11, 12]. Another direction could be autonomous systems in the context of diffusive representations [14, 18, 17].
This work is split up in two parts. The first one concerns a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on Euler-Bernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects. This investigation was carried out in [9] and thoroughly described in the PhD thesis [7].
The second part of this work focus on the development of a numerical method based on the Gear scheme – which is a three-level step algorithm, backward in time and second order accurate – for the approximation of fractional derivatives (see [8]). After a first tentative [4] for preliminary tests using semi-derivatives, we focus on the analytic determination of the coefficients of the numerical scheme and a set of preliminary tests in order to derive orders of convergence. Herein, the formal power of the Gear scheme is proposed to approximate fractional derivative operators in the context of finite difference methods. Some numerical examples are presented and analyzed in order to show the effectiveness of the present Gear scheme at the power α (Gα-scheme) when compared to the classical Grünwald-Letnikov approximation. In particular, for a fractional damped oscillator problem, the combined Gα-Newmark scheme is shown to be second-order accurate.


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Laboratoire de Mécanique des Structures et des Systèmes Couplés - LMSSC