Fractional Dashpot Research Articles

Hidden jerk in universal creep and aftershocks.

Most materials exhibit creep memory under the action of a constant load. The memory behavior is governed by Andrade's creep law, which also has an inherent connection with the Omori-Utsu law of earthquake aftershocks. Both empirical laws lack a deterministic interpretation. Coincidentally, the Andrade law is similar to the time-varying part of the creep compliance of the fractional dashpot in anomalous viscoelastic modeling. Consequently, fractional derivatives are invoked, but since they lack a physical interpretation, the physical parameters of the two laws extracted from curve fit lack confidence. In this Letter, we establish an analogous linear physical mechanism that underlies both laws and relates its parameters with the material's macroscopic properties. Surprisingly, the explanation does not require the property of viscosity. Instead, it necessitates the existence of a rheological property that relates strain with the first order time derivative of stress, which involves jerk. Further, we justify the constant quality factor model of acoustic attenuation in complex media. The obtained results are validated in light of the established observations.

A three-dimensional fractional visco-hyperelastic model for soft materials

Soft materials have attracted widespread attention and brought a wave of novel potential applications. Accurate mechanical characterization is crucial for improving the ration design of the desired soft elements and understanding the influence of deformation properties on themselves. However, the intrinsic complexity of mixed elasticity and viscosity in terms of their material property, posing new challenges for constitutive modeling. In this work, a fractional visco-hyperelastic constitutive model is developed by introducing the fractional viscoelasticity into finite strain: two branches are incorporated where a hyperelastic spring is used to describe the equilibrium response, and a non-linear fractional dashpot is introduced to characterize the time-dependent material response by employing the 3D fractional viscoelasticity. It provides a method from the perspective of fractional mechanics to avoid the increased number of governing equations and the associated complex calibration process caused by introducing the internal variables. The results show that the proposed model can successfully describe the rate dependent mechanical behavior of soft materials as well as predict the material behavior in different deformation modes with reasonable accuracy.

A Nonlinear Triaxial Damage Creep Model for Granite Based on Atangana–Baleanu Fractional Derivative

Finding a fractional derivative that can more accurately describe the characteristics of a dynamic process is one of the objectives for fractional derivative modeling. In this paper, Atangana–Baleanu (AB) fractional order dashpots with constant and variable coefficients are constructed by using AB fractional derivatives with nonlocal and nonsingular kernel characteristics. Compared with Riemann–Liouville (RL) and conformable (KA) fractional order dashpot, AB fractional order dashpot can describe the viscoelastic properties of materials in full time. A new damage AB fractional derivative (DABFD) creep model is proposed by improving the Nishihara model with the help of the variable coefficient AB fractional dashpot. The parameters for the DABFD creep model are determined by fitting creep experimental data of Beishan granite. Compared with the Nishihara, damage RL fractional derivative (DRLFD) and damage KA fractional derivative (DKAFD) creep models, DABFD creep model has a better fitting and prediction accuracy. The parameter sensitivity for the DABFD creep model shows that different creep curves can be demonstrated with different DABFD creep model parameters. The model can better represent the viscoelastic properties and three stages of the creep process for rock.

Elastic waves propagation through a liquid-saturated poroelastic interlayer based on the fractional viscoelastic BISQ model

The reflection and transmission problems of elastic waves, through a liquid-saturated poroelastic interlayer sandwiched between two elastic solid half-spaces, are investigated. The reflection and transmission coefficients are derived based on the Biot/squirt (BISQ) model, which incorporates both the Biot and squirt-flow mechanisms. Moreover, the viscoelasticity of the solid frame and the pore fluid are also considered. The fractional Zener model and the fractional dashpot model (Abel dashpot) are adopted for the solid frame and the pore fluid, respectively. The numerical results are provided and shown graphically. The dependences of the attenuation, the reflection and transmission coefficients upon the characteristic squirt flow length are numerically discussed. Comparing with that predicted by Biot theory without considering the squirt flow effects, it is observed that the reflection and transmission coefficients and the attenuation due to the sandwiched interlayer are affected noticeably by the effect of the squirt flow.

Circular arc rules of complex plane plot for model parameters determination of viscoelastic material

We present a new approach to determine the rheological parameters of a mechanical model of viscoelastic materials. The fractional derivative solid model composed of a spring in series with a fractional derivative Kelvin–Voigt element has been employed to characterize the dynamic mechanical response of a real viscoelastic material. In dynamic mechanical analysis (DMA) measurements, the frequency-dependent loss modulus is plotted against the storage modulus in the form of complex plane plot. We find that the complex plane plot of the fractional derivative solid model is a depressed or distorted semicircle with its center below the real axis. The model parameters could be identified graphically via its complex plane curve, from which the spring constants can be obtained through the two intercepts of the extrapolated circular arc with the real axis, whereas the order of fractional dashpot can be estimated by the displacement of the semicircle center. The graphical method allows us to easily find rheological parameters, without using complicated calculation and special algorithms.

A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters

Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

A fractional dashpot for nonlinear viscoelastic fluids

In this paper, we present a fractional dashpot that is capable of continuously adjusting the degree of dissipation between the whole dashpot and the Maxwell model (spring and dashpot in serial connection). It is related to the springpot by a frequency shift of the relaxation modulus in the complex domain. In contrast to the springpot, which is a viscoelastic solid-like model, the fractional dashpot is an intrinsically viscoelastic fluid model. We demonstrate that the fractional dashpot allows one to better model the polydispersity effect of typical viscoelastic fluids, overcoming undesired stationary predictions and reducing or even eliminating multiple modes in data fitting. The linear version of the fractional dashpot model is also scaled up to large deformation by incorporation of corotational tensor derivatives and a projected strain rate tensor for constructing an objective fractional constitutive equation. The resulting model having five model parameters (one for fractionality, two for linear viscoelasticity, one for straining, and the last one for rotation) is able to fit startup shear viscosity of a high molecular weight polystyrene solution in high accuracy, and yet using only a single mode.

Oscillations and damping in the fractional Maxwell materials

This paper examines the oscillatory behaviour of complex viscoelastic systems with power law-like relaxation behaviour. Specifically, we use the fractional Maxwell model, consisting of a spring and fractional dashpot in series, which produces a power-law creep behaviour and a relaxation law following the Mittag-Leffler function. The fractional dashpot is characterised by a parameter beta, continuously moving from the pure viscous behaviour when beta=1 to the purely elastic response when beta=0. In this work, we study the general response function and focus on the oscillatory behaviour of a fractional Maxwell system in four regimes: stress impulse, strain impulse, step stress, and driven oscillations. The solutions are presented in a format analogous to the classical oscillator, showing how the fractional nature of relaxation changes the long-time equilibrium behaviour and the short-time transient solutions. We specifically test the critical damping conditions in the fractional regime, since these have a particular relevance in biomechanics.

Research on viscoelastic behavior and rheological constitutive parameters of metallic glasses based on fractional-differential rheological model

Metallic glasses offer novel physical, chemical and mechanical properties and have promising potential applications. Recently, exploring the structure and deformation mechanism of metallic glasses according to the rheological mechanical behavior in the nominal elastic region has been the object of intensive research. Physically the mechanical analogues of fractional elements can be represented by self-similarity spring-dashpot fractal networks. In light of the fractal distribution features of the structural heterogeneities, a fractional differential rheological model is introduced to explore the viscoelastic a behavior of metallic glasses in this paper. To investigate the viscoelastic deformation mechanism, carefully designed nanoindentation tests at ambient temperature are proposed in this study. Three kinds of metallic glasses with different Poisson's ratio and glass transition temperature, which have the chemical compositions of Pd40Cu30Ni10P20, Zr48Cu34Pd2Al8Ag8, and (Fe0.432Co0.288B0.192Si0.048Nb0.04) 96Cr4 are selected as the model materials. Experimental and theoretical results clearly indicate that in the nominal elastic region, these metallic glasses exhibit linear viscoelasticity, implying a loading rate-dependent character. Based on the fractional calculus and Riemann-Liouville definition, experimental results are analyzed by the fractional-differential and integer order rheology models respectively. According to the stability of the fitting parameters, here we show that the fractional-differential Kelvin model, which consists of a spring and a fractional dashpot element, can fit the experimental viscoelastic deformation data of the investigated metallic glasses better than that with integer order rheological model. The extracted elastic modulis E1 of the spring in the fractional-differential Kelvin model are comparable to those of samples measured by traditional methods. Such a similarity can be well explained by the mechanical analogue of fractal model proposed for describing the distribution features of the structural heterogeneities in metallic glasses. The rheological parameters obtained here including viscosity index A and fractional order are capable of reflecting the rheological features and the flowing tendency of the above-mentioned metallic glasses. It is found that there exists a clear relationship between the rheological parameters and the reduced glass transition temperature as well as Poisson's ratio, which is helpful for understanding the correlation between plasticity and Poisson's ratio from the micro-structural point of view. The current work provides a rheological model-structure-property relation that may be applicable to a wide range of glassy materials.

Rheological representation of fractional order viscoelastic material models

We develop rheological representations, i.e., discrete spectrum models, for the fractional derivative viscoelastic element (fractional dashpot or springpot). Our representations are generalized Maxwell models or series of Kelvin-Voigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Accordingly, the number of parameters of the rheological representation is independent of the number of rheological units. We prove that the representations converge to the corresponding fractional model in the limit as the number of units tends to infinity. The representations extend to compound fractional derivative models such as the fractional Maxwell model, fractional Kelvin-Voigt model, and fractional standard linear solid. Computational experiments show that the rheological representations are accurate approximations of the fractional order models even for a small number of units.

Identification of viscoelastic model parameters by means of cyclic nanoindentation testing

Abstract A method for the identification of model parameters describing the time-dependent material behavior by means of cyclic nanoindentation is presented. The complex shear modulus of the material sample is determined from the prescribed amplitude in the load history, the measured amplitude in the penetration history, and the phase shift between the peak values for the load and the penetration. The parameters for a specific viscoelastic model are obtained by comparing the experimentally-obtained storage and loss moduli – both depending on the frequency used during cyclic testing – with the analytical expressions for the respective viscoelastic model. The presented method is applied to low-density polyethylene, giving access to the parameters of the fractional dash-pot which is used to describe the viscoelastic behavior. The results are compared with results from nanoindentation (static) creep tests, considering different maximum loads. Finally, the performance of the presented method is assessed by comparing the creep-compliance functions corresponding to the model parameters identified by nanoindentation with the macroscopic creep-compliance function obtained from bending-beam-rheometer tests.

On the thermodynamics of fractional damping elements

Constitutive models of viscoelasticity in combination with fractional differential operators are successfully used by many authors to describe the mechanical properties of polymers. The topic of the present paper is the investigation of rheological models incorporating fractional damping elements from the point of view of thermodynamics. We take the Clausius Duhem inequality as admissibility criterion and investigate uniaxial and three-dimensional deformation processes at constant temperature. We specify sufficient conditions and show that rheological models, which consist of springs in combination with fractional dashpots, are compatible with the dissipation principle. As a new aspect of the subject, we present a systematic method for deriving the free energy functionals. With the help of two examples we demonstrate that the free energy of fractional systems can be derived as a generalisation of related discrete systems. To illustrate this method, we study in detail an isolated fractional dashpot (also known as a power-law model) and a fractional standard solid (Zener model). Finally, we propose a three-dimensional formulation of the fractional Zener model and specify the corresponding free energy functional.