A new and efficient constitutive model based on fractional time derivatives for transient analyses of viscoelastic systems
Introduction
Among the fundamental early studies regarding the use of fractional calculus to describe the stress–strain relationships for certain viscoelastic media [1], the works by Bagley and Torvik [2], [3], [4], [5], [6] are of great relevance. The authors [2], [3], [4], [5], [6] suggest the existence of some inadequacies related to the classical complex modulus approach, such as the non-causal response to impulsive forces and the problem of expressing the dynamic equations in the time-domain by applying the Fast Fourier Transforms [7]. To overcome it, Bagley and Torvik have proposed a very attractive generalized fractional derivative model (FDM) to be used in conjunction with the finite element (FE) method for transient analyses of various polymers. They have constructed a rubbery, transition and glassy (RTG) model composed only by four parameters to be identified by curve-fitting from experimental data [5], [6]. However, it was limited to one-dimensional constitutive law. Later, Makris [8] attempted to extend it to three dimensions to deal with isotropic and incompressible viscoelastic materials by using a fractional order solid model. In the same way, Schmidt and Gaul (S&G) [9] have provided a new perspective for the three-dimensional constitutive equations of viscoelastic materials. In their developments, they have employed the concept of hydrostatic and deviatoric parts of the total stress tensor proposed initially by Makris [8] and Flügge [13]. The interest was to construct a four-parameter FDM with the aim of aiding the curve-fitting process and its integration with the FE method. However, it results in costly computations, sometimes unfeasible, due to the fact that, all the non-local stress history of the viscoelastic material is self-dependent.
Hence, it has been observed that few works have attempted to reformulate the existent three-dimensional FDM formulation proposed up to now in order to describe the dynamic behavior of general systems with viscoelastic materials. The main idea is to remove the integral of the non-local stress field in order to alleviate computational costs and storage memory frequently required in transient analyses. This is the motivation behind this contribution. Galucio et al. [10] have used the concept of internal variables to represent the anelastic displacements of a three-layer viscoelastic sandwich beam. The advantage of using such approach is that, it represents a straightforward transient FDM formulation in which modifications by the viscoelastic substructure are introduced on the stiffness of the elastic host structure. Those modifications are, in fact, time independent, and by adding the time-dependent viscoelastic forces to the original applied external excitations only in the right-rand side of the dynamic equations. However, if it is used in a viscoelastic sandwich plate modelled according to the First Shear-Order Deformation Theory (FSDT), for instance, a poor representation of the damped responses is achieved, as detailed in [11]. Sales et al. [24] proposed a typical section model presenting non-smooth, free-play type nonlinearities in their control surface, which is equipped with a viscoelastic torsional damper to mitigate its oscillating motion, reducing limit cycle oscillation amplitude and completely eliminating of subcritical behavior. The fractional derivative constitutive model used by the authors is the same as proposed by Galúcio [10]. In the same way, Cortés and Elejabarrieta [12] have used a five-parameter FD model derived from the local equation of the linear momentum for a cantilever beam fully treated by a free soft viscoelastic layer only. By this approach, only the displacements and external applied loadings must be stored. Consequently, it reduces the computational cost involved in the analysis, but the kinematics assumptions assumed for the base-beam and the viscoelastic layer are particular to this application and it is not general. Gong et al. [25] presented a mathematical model of a Vibrating Flip-Flow Screens (VFFS), a device that provides an effective solution for screening highly moist and fine-grained minerals. An important correlation between experimental results and a unidimensional viscoelastic fractional derivative constitutive model have been made, supplying the literature with useful experimental data. The Grunwald-Letnikov approximation of the Riemann-Liouville definition of fractional derivative is used. Their mathematical model is fashionably conceived to consider only the viscoelastic dissipative effort generated by viscoelastic shear springs, disregarding its stiffness. As a result, the viscoelastic stiffness self-dependency is avoided. Nguyen et al. [26] present the study of a nonlinear Magnetorheological elastomers (MRE) consisted among others of a fractional viscoelastic model. In the same line as Gong et al. [25], they have created an element combining four different springs into one single element: magnetic, frictional, elastic and fractional Maxwell viscoelastic spring. A device has been constructed and experimental evaluations performed. All analyses were carried in the frequency domain. Lin and Ng [27] dedicated their effort in creating a definition of a general form of a fractional vibration system to determine its eigenvalue problem and methods of solution and frequency response functions. Also, they attempted to develop a conventional modal analysis, and a novel method of equivalent eigensystem (MEE) and its solution, in a more efficient manner. However, only uniaxial viscoelastic elements were used. Freed and Diethelm [28] have developed a model to represent human biological tissue assuming it be isotropic and considering its viscoelastic properties. However, the model was conceived after the evaluation of a candidate sets of functions to represent the elastic and viscoelastic dynamic behavior of the human tissue. Concerning the viscoelastic functions used, two of them were originated from the fractional calculus. From their research a new class of viscoelastic materials arose. Fukanaga and Shimizu [29] proposed a three-dimensional fractional derivative viscoelastic constitutive model based on the continuum mechanics treatment using the generalized Maxwell model. Thus, a constitutive law is obtained in terms of the second Piola-Kirchhoff stress tensor and the right Cauchy-Green strain tensor. A comparison with the models proposed by the authors and others existent in the literature [30] have shown the applicability and effectiveness of their approach.
Hence, in this paper, a new, simpler and general FDM formulation is suggested based on the development of a recurrence term, starting from the three-dimensional FDM formulation proposed initially by Schmidt and Gaul [9]. Based on the proposed method, no simplification must be made on the three-dimensional constitutive laws, making this approach suitable for general dynamic analyses of structures with viscoelastic materials. As an example, a three-layered viscoelastic sandwich plate is addressed herein and the main features and capabilities of the proposed method are highlighted.
Section snippets
Background on the three-dimensional viscoelastic constitutive law
In this section, the non-local three-dimensional viscoelastic constitutive equation is summarized, based on the developments made by Makris [8] and Flügge [13]. By using the hydrostatic (index ) and deviatoric (index ) concepts, the deviatoric and hydrostatic constitutive relations are established as [9]:where and are the shear and bulk modulus at low frequency range, respectively, and are the fractional order of
New constitutive equation based on a recurrence formula
At this time, it must be highlighted that, the use of Eq. (3), as proposed by Schmidt and Gaul [9], to perform transient analyses of structures containing viscoelastic materials can lead to costly computations, sometimes unfeasible. Since, it is necessary to perform the integration of the stress history of each viscoelastic element in a FE mesh and store it at each time step. Clearly, it depends also on the complexity of the FE model of the structure treated with viscoelastic materials, but, as
Introducing the new constitutive equation into FE models
In this section, Eq. (9) is combined with the FE model of a moderately thin three-layer sandwich plate proposed by Khatua and Cheung [18] formed by a base plate (1), a viscoelastic core (2) and a constraining layer (3), as shown in Fig. 2. The kinematic assumptions assumed to integrate the elementary FE mass and stiffnesses matrices of each layer can be found in details in de Lima et al. [19], [20], [21] and Guedri et al. [22]. However, for the purposes of this study, it is important to define
Numerical examples
To demonstrate the accuracy and efficiency of the new constitutive equation (9) based on the recurrence formula proposed herein, numerical simulations have been performed with a simply supported three-layer viscoelastic sandwich plate. The physical properties of the elastic layers made of aluminum are: Young’s modulus, , Poisson ratio, , mass density, . For the 3 M ISD112 material (, ) it is used the FDM model, whose parameters are defined in
Concluding remarks
The fractional calculus is a powerful tool to derive the constitutive equations of viscoelastic media, especially when the interest is related to its use in conjunction with the FE method to perform transient analyses of engineering structures with viscoelastic materials. Starting from the classical three-dimensional constitutive equations for a viscoelastic material, it has been demonstrated that, by performing a differentiation of the deviatoric and hydrostatic parts of the total stress and
CRediT authorship contribution statement
A.G. Cunha-Filho: Methodology. Y. Briend: Methodology. A.M.G. de Lima: Supervision, Visualization, Investigation. M.V. Donadon: Supervision, Visualization, Investigation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are grateful to CNPq for the continued support to their research activities through the research grant 302026/2016-9 (A.M.G. de Lima) and 301053/2016-2 (M.V. Donadon). It is also important to express the acknowledgements to the FAPEMIG, state agency, especially to the research projects APQ-01865 and PPM-0058-18 (A.M.G. de Lima) and APQ-01865-18 (Manzanares-Filho).
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