Response functions in linear viscoelastic constitutive equations and related fractional operators (original) (raw)
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Following the modelling of Zener, we establish a connection between the fractional Fokker-Planck equation and the anomalous relaxation dynamics of a class of viscoelastic materials which exhibit scale-free memory. On the basis of fractional relaxation, generalisations of the classical rheological model analogues are introduced, and applications to stress-strain relaxation in filled and unfilled polymeric materials are discussed. A possible generalisation of Reiner's Deborah number is proposed for systems which exhibit a diverging characteristic relaxation time. #
A variable order fractional constitutive model of the viscoelastic behavior of polymers
International Journal of Non-linear Mechanics, 2019
The multiple timescale evolution of polymers' microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. In the presented work, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.
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The study addresses the physical background and modeling of linear viscoelastic response functions and their reasonable relationships to the Caputo-Fabrizio fractional operator via the Prony (Dirichlet series) series decomposition. The problem of interconversion with power-law and exponential (single and multi-term functions) has been discussed. Special attentions have been paid on the Prony series decomposition approach, the related interconversion problems and the expression of the viscoelastic constitutive equations in terms of Caputo-Fabrizio fractional operator.
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Three-dimensional constitutive viscoelastic laws with fractional order time derivatives
Synopsis In this article the three-dimensional behavior of constitutive models containing fractional order time derivatives in their strain and stress operators is investigated. Assuming isotropic viscoelastic behavior, it is shown that when the material is incompressible, then the one-dimensional constitutive law calibrated either from shear or elongation tests can be directly extended in three dimensions, and the order of fractional differentiation is the same in all deformation patterns. When the material is viscoelastically compressible, the constitutive law in elongation involves additional orders of fractional differentiation that do not appear in the constitutive law in shear. In the special case where the material is elastically compressible, the constitutive laws during elongation and shear are different; however the order of fractional differentiation remains the same. It is shown that for an elastically compressible material, the four-parameter fractional solid—the rubbery, transition, and glassy model, which has been used extensively to approximate the elongation behavior of various polymers, can be constructed from the three-parameter fractional Kelvin—the rubbery transition model in shear and the elastic bulk modulus of the material. Some of the analytical results obtained herein with operational calculus are in agreement with experimental observations reported in the literature. Results on the viscoelastic Poisson behavior of materials described with the fractional solid model are presented and it is shown that at early times the Poisson function reaches negative values.
37th Structure, Structural Dynamics and Materials Conference, 1996
Numerical procedures for the time integration of the spatially discretized finite element equations for viscoelastic structures governed by a constitutive equation involving fractional derivative operators are presented. To avoid difficulties concerning fractional order initial conditions, a form of the fractional calculus model of viscoelasticity involving a convolution integral with a singular memory kernel of Mittag-Leffler type is used. The constitutive equation is generalized to three-dimensional states for isotropic materials. A certain property of the memory kernel is used, in connection with Griinwald's definition of fractional differentiation and a Backward Euler rule, for the time evolution of the convolution term. A desirable feature of this process is that no actual evaluation of the memory kernel is needed. This together with the Newmark's method for time integration enables the direct calculation of the time evolution of the nodal degrees of freedom. To illustrate the ability of the numerical procedure a few numerical examples are presented. In one example the numerically obtained solution is compared with a time series expansion of the analytical solution.
Statistical origins of fractional derivatives in viscoelasticity
Journal of Sound and Vibration, 2005
Many linear viscoelastic materials show constitutive behavior involving fractional order derivatives. Linear, time invariant systems without memory have exponential decay in time but, contradictorily, not the power law decay associated with fractional derivatives. The physics literature has noted that apparentlynon-exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates. Many engineer-researchers interested in viscoelastic damping, however, seem unaware of these observations. In this letter I give an unoriginal explanation, but with a fresh engineering flavor, for the appearance of these fractional order derivatives. By this explanation, fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms. r
On fractional modelling of viscoelastic mechanical systems
Mechanics Research Communications, 2016
Since Leibniz's fractional derivative, introduced by Lazopoulos [1], has physical meaning contrary to other fractional derivatives, the viscoelastic mechanical systems are modelled with the help of Leibniz fractional derivative. The compliance and relaxation behaviour of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems is presented.
International Journal of Polymeric Materials, 2004
Polymeric materials are known to be more or less dispersive and absorptive. Dispersion has a consequence that the dynamic modulus is frequency dependent, and absorption is exhibited by the fact that these materials have the ability to absorb energy under vibratory motion. The phenomenon of dispersion in conjunction with the powerful notion of complex Modulus of Elasticity (MOE), permits to establish the relation between the real and the imaginary components of the MOE, that is, respectively the Storage and loss moduli. The loss factor is simply determined through taking the Ratio of these two MOE components. The theoretical background for the interrelations between the Storage modulus and the loss modulus is found in the Kramers-Kronig relations. However, due to the mathematical difficulties encountered in using the exact expressions of these relations, approximations are necessary for applications in practical situations. On the other hand, several simple models have been proposed to explain the viscoelastic behavior of materials, but all fail in giving a full account of the phenomenon. Among these models, the standard viscoelastic model, better known as the Zener model, is perhaps the most attractive. To improve the performance of this model, the concept of fractional derivates has been incorporated into it, which results in a four-parameter model. Applications have also shown the superiority of this model when theoretical predictions are compared to experimental data of different polymeric materials. The aim of this article is to present the results of applying this model to rubber, both natural and filled, and to some other selected more general polymer.