Statistical origins of fractional derivatives in viscoelasticity (original) (raw)
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Short Communication Statistical origins of fractional derivatives in viscoelasticity
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 apparently- non-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.
Entropy
We calculate the transverse velocity fluctuations correlation function of a linear and homogeneous viscoelastic liquid by using a generalized Langevin equation (GLE) approach. We consider a long-ranged (power-law) viscoelastic memory and a noise with a long-range (power-law) auto-correlation. We first evaluate the transverse velocity fluctuations correlation function for conventional time derivativesĈ NF − → k , t and then introduce time fractional derivatives in their equations of motion and calculate the corresponding fractional correlation function. We find that the magnitude of the fractional correlationĈ F − → k , t is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus G F (ω) andĈ F − → k , t is also calculated analytically. The difference between the values of G (ω) for two specific viscoelastic fluids is quantified. Our model calculation shows that the fractional effects on this measurable quantity may be three times as large as compared with its non-fractional value. The fact that the dynamic shear modulus is related to the light scattering spectrum suggests that the measurement of this property might be used as a suitable test to assess the effects of temporal fractional derivatives on a measurable property. Finally, we summarize the main results of our approach and emphasize that the eventual validity of our model calculations can only come from experimentation.
Notes on computational aspects of the fractional-order viscoelastic model
Journal of Engineering Mathematics
This paper deals with the computational aspect of the investigation of the relaxation properties of viscoelastic materials. The constitutive fractional Zener model is considered under continuous deformation with a jump at the origin. The analytical solution of this equation is obtained by the Laplace transform method. It is derived in a closed form in the terms of the Mittag-Leffler function. The method of numerical evaluation of the Mittag-Leffler function for arbitrary negative arguments which corresponds to physically meaningful interpretation is demonstrated. A numerical example is given to illustrate the effectiveness of this result.
Fractional-order relaxation laws in non-linear viscoelasticity
Continuum Mechanics and Thermodynamics, 2007
Viscoelastic constitutive equations are constructed by assuming that the stress is a nonlinear function of the current strain and of a set of internal variables satisfying relaxation equations of fractional order. The dependence of the relaxation equations on the strain can also be nonlinear. The resulting constitutive equations are examined as mapping between appropriate Sobolev spaces. The proposed formulation is easier to implement numerically than history-based formulations.
International Journal of Plasticity, 2003
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. #
Fractional Calculus and Waves in Linear Viscoelasticity
2010
Readership: Graduate and PhD students in applied mathematics, classical physics, mechanical engineering and chemical physics; academic institutions; research centers. Key Features • Contains accessible mathematical language for easy understanding • Features ample examples to reiterate concepts in the book • Makes extensive use of graphical images • Includes a large and informative general bibliography for further research
Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity
Physical Review E, 2016
Many of the most interesting complex media are non-Newtonian and exhibit time-dependent behavior of thixotropy and rheopecty. They may also have temporal responses described by power laws. The material behavior is represented by the relaxation modulus and the creep compliance. On the one hand, it is shown that in the special case of a Maxwell model characterized by a linearly time-varying viscosity, the medium's relaxation modulus is a power law which is similar to that of a fractional derivative element often called a springpot. On the other hand, the creep compliance of the time-varying Maxwell model is identified as Lomnitz's logarithmic creep law, making this possibly its first direct derivation. In this way both fractional derivatives and Lomnitz's creep law are linked to time-varying viscosity. A mechanism which yields fractional viscoelasticity and logarithmic creep behavior has therefore been found. Further, as a result of this linking, the curve-fitting parameters involved in the fractional viscoelastic modeling, and the Lomnitz law gain physical interpretation.
Response functions in linear viscoelastic constitutive equations and related fractional operators
Mathematical Modelling of Natural Phenomena
This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mi...