Fractional Viscoelasticity Research Articles

Finite element simulations of quartz crystal microbalances (QCM): from Sauerbrey to fractional viscoelasticity under water

2D finite element simulations are performed on QCM working in the thickness-shear mode and loaded with different homogeneous films. They include a purely elastic film, a viscoelastic Maxwellian liquid, viscoelastic-Voigt solid, and the fractional viscoelastic (power-law) version of each case. Unlike single-relaxation kind models, fractional viscoelasticity considers the relaxation-time spectrum often found in polymeric materials. The films are tested in air or covered with liquids of different viscosities. Two substrate thicknesses are tested: 100 nm and 500 nm, the latter being close to the condition that promotes the resonance of the adsorbed film. In all cases the simulations are compared with small-load approximation theory (SLA). The 100 nm films follow the theory closely, although significant deviations of the SLA are observed as the overtone number n increases, even in purely elastic films. We also show that it is possible to identify the viscoelastic ‘fingerprint’ of the 100 nm films in air using raw data and Sauerbrey’s equivalent thickness obtained with the QCM in the 3 < n < 13 range. These numerical data are validated by experimental measurements of crosslinked polydimethylsiloxane films with thicknesses ∼150 nm. In contrast, the 500 nm films deviate notoriously from the SLA, for all viscoelastic models and overtones, with the largest deviation observed in the elastic film. When a liquid layer covers the QCM without an adsorbed film, the only overtone that numerically reproduces the theoretical value is the fundamental, n = 1. For n > 1, strong coupling between the solid and liquid is detected, and the original vibration modes of the crystal are altered by the presence of the liquid. Finally, the numerical simulations suggest that it is possible to detect whether a viscoelastic film is formed under a liquid layer using only the information from n = 1. In these film/liquid systems we also observe the so-called missing-mass effect, although the theory and simulations exhibit different levels of impact of such effect when the liquid viscosity is high.

Thermoviscoelastic response of skin tissue via fractional Pennes bioheat and fractional viscoelasticity

In this paper, the thermoviscoelastic response of skin tissue, as a typical class of viscoelastic biological tissue, subject to thermal shock is studied based on the fractional Kelvin-Voigt model and fractional Pennes bioheat conduction. A time-fractional biothermoviscoelastic governing equation is derived and initial-boundary value conditions are given for monolayer skin tissue of finite thickness. The temperature change and thermal stress of viscoelastic skin tissue are obtained by the Laplace transform and its inversion using matlab software. The influences of viscosity coefficient, relaxation times, fractional-order viscosity/heat parameter, and blood perfusion rate of skin tissue on the temperature distribution and the elastic displacement are analyzed. The numerical results show that fractional-order viscosity does not affect temperature, whereas fractional-order heat flow does. Moreover, fractional-order heat flow affects the sensitivity of blood perfusion rate to the thermodynamic response.

Electro-mechanical analogy for Prabhakar-like fractional viscoelasticity

The aim of this paper is to set up a formal equivalence between a mechanical system and an electrical one. Specifically, we consider the Maxwell-Prabhakar linear viscoelastic model, based on Prabhakar fractional operators. Therefore, we find the analogous expression for the electric current due to a step potential. The expression for the resulting electric current depends on the variable characterizing the viscoelastic model and its behaviour is then discussed with the support of some interesting plots.

Effects of fractional viscoelasticity material of electrostatic micro-resonators on performances of delayed proportional-derivative control

Effects of fractional viscoelasticity material of electrostatic micro-resonators on performances of delayed proportional-derivative control

Dynamic Modeling and Response Analysis of Dielectric Elastomer Incorporating Fractional Viscoelasticity and Gent Function

Dielectric Elastomer (DE) has been recognized for its remarkable potential in actuation and sensing applications. However, the functionality of most DE materials is restricted by their high viscoelastic effects. Currently, there is a lack of dynamic models that consider both viscoelasticity and stiffening effects. To address this research gap, we propose a fractional-order model in this study. Specifically, the model comprehensively integrates both viscoelastic and stiffening effects under electromechanical coupling, utilizing the principle of virtual work. Further, the effects of the system parameters are analyzed. The results indicate that the fractional-order derivative influences the hysteresis behaviors during the transient state and affects the duration of the transient process. Furthermore, when the system’s energy surpasses a certain threshold, the steady-state response can transition between two distinct potential wells. Through the manipulation of electromechanical coupling parameters, bifurcation can be induced, and the occurrence of snap-through and snap-back behaviors can be controlled. These findings have significant implications for the design and optimization of DE materials in various applications.

A viscoelastic constitutive model for human femoropopliteal arteries

High failure rates present challenges for surgical and interventional therapies for peripheral artery disease of the femoropopliteal artery (FPA). The FPA’s demanding biomechanical environment necessitates complex interactions with repair devices and materials. While a comprehensive understanding of the FPA’s mechanical characteristics could improve medical treatments, the viscoelastic properties of these muscular arteries remain poorly understood, and the constitutive model describing their time-dependent behavior is absent. We introduce a new viscoelastic constitutive model for the human FPA grounded in its microstructural composition. The model is capable of detailing the contributions of each intramural component to the overall viscoelastic response. Our model was developed utilizing fractional viscoelasticity and tested using biaxial experimental data with hysteresis and relaxation collected from 10 healthy human subjects aged 57 to 65 and further optimized for high throughput and automation. The model accurately described the experimental data, capturing significant nonlinearity and hysteresis that were particularly pronounced circumferentially, and tracked the contribution of passive smooth muscle cells to viscoelasticity that was twice that of the collagen fibers. The high-throughput parameter estimation procedure we developed included a specialized objective function and modifications to enhance convergence for the common exponential-type fiber laws, facilitating computational implementation. Our new model delineates the time-dependent behavior of human FPAs, which will improve the fidelity of computational simulations investigating device-artery interactions and contribute to their greater physical accuracy. Moreover, it serves as a useful tool to investigate the contribution of arterial constituents to overall tissue viscoelasticity, thereby expanding our knowledge of arterial mechanophysiology. Statement of significanceThe demanding biomechanical environment of the femoropopliteal artery (FPA) necessitates complex interactions with repair devices and materials, but the viscoelastic properties of these muscular arteries remain poorly understood with the constitutive model describing their time-dependent behavior being absent. We hereby introduce the first viscoelastic constitutive model for the human FPA grounded in its microstructures. This model was tested using biaxial mechanical data collected from 10 healthy human subjects between the ages of 57 to 65. It can detail the contributions of each intramural component to the overall viscoelastic response, showing that the contribution of passive smooth muscle cells to viscoelasticity is twice that of collagen fibers. The usefulness of this model as tool to better understand arterial mechanophysiology was demonstrated.

Modelling of laminated glass interlayer by fractional viscoelasticity

Fractional calculus, i.e. the theory of derivatives and integrals of non-integer order, can be efficiently used for the theoretical modelling of viscoelastic materials. Our research is focused on the polyvinyl butyral which is used as an interlayer for the laminated glass. Polyvinyl butyral can be classified as a viscoelastic material and the introduction of the fractional viscoelasticity seems to be appropriate tool for its description. This paper briefly introduces the springpot element and its connection into more complex theoretical models. We mainly consider the generalized Maxwell model in its standard and fractional form and show their application by fitting the data obtained by experimental analysis.

Study on the performance of variable-order fractional viscoelastic models to the order function parameters

In variable-order fractional calculus, the order is allowed to be in a functional form, which provides a novel way to describe the time-dependent physical processes. In this work, the performance of variable-order fractional viscoelastic models to the order function parameters is comprehensively investigated. Firstly, the variable-order fractional constitutive model is deduced from the theory of constant-order fractional viscoelasticity. The effect of order function is studied which shows that the linear order function can effectively describe the viscoelastic properties. Then the variable-order fractional viscoelastic models are developed considering three kinds of viscoelastic behaviors, including stress-strain relationship, creep, and stress relaxation. The sensitivity of the viscoelastic behaviors to the parameters in the linear order function is analyzed in detail. Finally, the effectiveness of the proposed models is validated by comparing them to the existing experimental data of typical viscoelastic behaviors, and the laws of the order function for each case are discussed based on the fractional viscoelastic theories. It is proved that the variable-order fractional models can give an accurate description of the viscoelastic behaviors under various loading conditions, and the change of the fractional order can visualize the evolution of the mechanical properties of the material.

Simulating hyperelasticity and fractional viscoelasticity in the human heart

Biomechanics plays an important role in the diagnosis and treatment of pathological conditions of the heart. Computational models are paving the way for personalized therapeutic treatment but they rely on accurate constitutive equations for predicting their biomechanical behavior. Even so, viscoelasticity remains under-explored in computational modeling despite experimental observations. To facilitate the viscoelastic modeling of cardiovascular soft tissues, we previously developed a fractional viscoelastic modeling approach, which extends existing hyperelastic models. This has comparable computational costs to the conventional hyperelastic model and only requires two additional material parameters for the viscoelastic response. This approach was demonstrated to be able to accurately capture the viscoelastic response of the human myocardium. However, the numerical properties of this fractional viscoelastic approach have not yet been examined. In this work, we present its implementation in Finite Element Analysis, examine its numerical properties in uniaxial extension and 2D inflation test examples, and examine its physiological implication in a computational model of an idealized left ventricle in a fully idealized circulatory system. Optimal convergence properties were observed and the importance of viscoelasticity during passive filling, ventricular motion, and regional fiber strain and stresses were explained.

Physics-motivated fractional viscoelasticity model for dynamic relaxation in amorphous solids

Dynamic mechanical relaxation is an important metric to understand the mechanical/physical properties of amorphous solids which are of viscoelastic nature. Due to the heterogenous microstructure, the relaxation behavior of amorphous solids usually shows strong deviation from the Debye relaxation. The distribution of relaxation time derived from either the stretched exponential function (KWW function) or the power law form is probably the most adopted paradigm to describe the non-Debye relaxation. They are essentially the continuous spectrums given in analytical forms. However, whether a real amorphous material conforms to such distribution law remains to be discussed. Here we test the assumption in typical metallic glasses (MGs) as representatives of the general amorphous solids. The mechanical spectrum of a Cu46Zr47Al7 MG in wide frequency domain is probed by the dynamic mechanical analysis technique. It is found that both the KWW function and the modified fractional (MF) model based on power law can well describe the experimental data. As a step forward, we combine the quasi-point defect theory with the MF model to theoretically reveal the feature of temperature-dependent structural evolution in MG. Finally, the distribution of relaxation time corresponding to the experimental data is discretized to argue the theoretically predicted microstructural heterogeneity in the MGs.

A GENERAL RETURN-MAPPING FRAMEWORK FOR FRACTIONAL VISCO-ELASTO-PLASTICITY.

We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin-Voigt, Maxwell, Kelvin-Zener and Poynting-Thomson. We also consider a fractional quasi-linear version of Fung's model to account for stress/strain nonlinearity. The fractional viscoelastic models are combined with a fractional visco-plastic device, coupled with fractional viscoelastic models involving serial combinations of Scott-Blair elements. We then develop a general return-mapping procedure, which is fully implicit for linear viscoelastic models, and semi-implicit for the quasi-linear case. We find that, in the correction phase, the discrete stress projection and plastic slip have the same form for all the considered models, although with different property and time-step dependent projection terms. A series of numerical experiments is carried out with analytical and reference solutions to demonstrate the convergence and computational cost of the proposed framework, which is shown to be at least first-order accurate for general loading conditions. Our numerical results demonstrate that the developed framework is more flexible, preserves the numerical accuracy of existing approaches while being more computationally tractable in the visco-plastic range due to a reduction of 50% in CPU time. Our formulation is especially suited for emerging applications of fractional calculus in bio-tissues that present the hallmark of multiple viscoelastic power-laws coupled with visco-plasticity.

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.

Experimental and numerical study of rate-dependent mode-I failure of a structural adhesive

ABSTRACT We present an experimental and numerical study of the rate dependence of the mode-I failure of adhesive joints, focussing on aluminium plates bonded with Araldite® 2015. For the experimental part, we tested 24 double-cantilever beams (DCB) at six different prescribed speeds, from 0.1 to 5000 mm/min. The numerical simulations use a previously proposed cohesive-zone model (CZM) based on fractional viscoelasticity and a novel finite element combining a Timoshenko beam and an interface element. The CZM had previously been validated for a rubber interface, so here we present a procedure to identify its input parameters and validate its capability to predict the failure of joints made with an epoxy adhesive. An effective procedure is also developed to evaluate the dependence of the fracture energy on the crack speed without experimentally measuring the crack speed. The adhesive response was found to be markedly rate dependent. Within the range of tested speeds, the fracture energy of the adhesive more than doubles its value and the shape of the ‘fracture energy-crack speed’ curve resembles a sigmoidal shape, but more tests are needed at higher speeds to better determine the maximum value of the fracture energy and the actual shape of the complete curve.

Cracking Resistance of Recycled Rubber Asphalt Binder Composed of Warm-Mix Additives.

Warm-mix asphalt technology has been applied to recycled rubber asphalt binder (RAB), which forms warm-mixed crumb rubber-modified asphalt binder (W-RAB) as a “green” material for environmental conservation and to enhance road performance. Furthermore, low-temperature cracking is one of the major distresses for asphalt pavement, which drastically restricts ride quality and service level. Therefore, the main objective of this study is to comparatively analyze the low-temperature properties of W-RABs based on thermal stress and the simple fractional model. W-RABs were obtained by mixing 60 mesh recycled rubber (CR) and two different types of warm-mix additives, namely viscosity reducer (1, 2, and 3%) and surfactant (0.4, 0.6, and 0.8%). First, Hopkins and Hamming’s numerical algorithm and the Boltzmann superposition principle were used for obtaining thermal stress . Subsequently, critical cracking temperature was derived using the single asymptote procedure (SAP) theory. Second, the simple fractional viscoelasticity model was used to calculate the creep compliance, damping ratio, and dissipation energy ratio, and the results were compared with the Superpave protocol results obtained with bending beam rheometer (BBR) tests. The results showed that a combination of CR and warm-mix additives could slightly improve the thermal crack resistance of the asphalt binder. The addition of 0.6% surfactant yielded the optimum performance, while only a high dosage (3%) of viscosity reducer provided a marked improvement in efficiency, which decreased with a decrease in temperature. This study recommends the use of RAB composited with 0.6% surfactant for areas with extremely low temperature.

A bridge between the fractional viscoelasticity and time-varying viscosity model: physical interpretation and constitutive modeling

A bridge between the fractional viscoelasticity and time-varying viscosity model: physical interpretation and constitutive modeling

A macroscopic viscoelastic model of magnetorheological elastomer with different initial particle chain orientation angles based on fractional viscoelasticity

In this paper, a macroscopic viscoelastic modeling method for magnetorheological elastomer (MRE) based on fractional derivative model is presented to describe the dynamic viscoelastic properties of MRE with different initial particle chain orientation angles. The angle between the particle chain and the applied magnetic field is used as an indicator to describe the directionality of the particle chain. MRE samples with different initial inclination angles have been designed and fabricated. The dynamic viscoelastic properties of different MRE samples under shear working mode were measured using a parallel plate rheometer. The dynamic viscoelastic properties of MRE with different initial inclination angles are analyzed under the test conditions of different strain amplitude, frequency and magnetic flux density. The test results show that the initial inclination angle of the particle chain in the MRE has a significant effect on the dynamic viscoelastic properties of the MRE. A polynomial function is used to describe the relationship between the initial particle chain orientation angle and the magneto-induced modulus of MRE. A phenomenological model of magneto-induced modulus is established based on the fractional derivative model. The model parameters are identified using the nonlinear least square method. The predicted values of the model are in good agreement with the experimental results, indicating that the model can well describe the dynamic viscoelastic properties of MRE.

The effects of viscoelasticity on residual strain in aortic soft tissues

Residual stress is thought to play a critical role in modulating stress distributions in soft biological tissues and in maintaining the mechanobiological stress environment of cells. Residual stresses in arteries and other tissues are classically assessed through opening angle experiments, which demonstrate the continuous release of residual stresses over hours. These results are then assessed through nonlinear biomechanical models to provide estimates of the residual stresses in the intact state. Although well studied, these analyses typically focus on hyperelastic material models despite significant evidence of viscoelastic phenomena over both short and long timescales. In this work, we extended the state-of-the-art structural tensor model for arterial tissues from Holzapfel and Ogden for fractional viscoelasticity. Models were tuned to capture consistent levels of hysteresis observed in biaxial experiments, while also minimizing the fractional viscoelastic weighting and opening angles to correctly capture opening angle dynamics. Results suggest that a substantial portion of the human abdominal aorta is viscoelastic, but exhibits a low fractional order (i.e. more elastically). Additionally, a significantly larger opening angle in the fully unloaded state is necessary to produce comparable hysteresis in biaxial testing. As a consequence, conventional estimates of residual stress using hyperelastic approaches over-estimate their viscoelastic counterparts by a factor of 2. Thus, a viscoelastic approach, such as the one illustrated in this study, in combination with an additional source of rate-controlled viscoelastic data is necessary to accurately analyze the residual stress distribution in soft biological tissues. Statement of significanceResidual stress plays a crucial role in achieving a homeostatic stress environment in soft biological tissues. However, the analysis of residual stress typically focuses on hyperelastic material models despite evidence of viscoelastic behavior. This work is the first attempt at analyzing the effects of viscoelasticity on residual stress. The application of viscoelasticity was crucial for producing realistic opening dynamics in arteries. The overall residual stresses were estimated to be 50% less than those from using hyperelastic material models, while the opening angles were projected to be 25% more than those measured after 16 hours, suggesting underestimated residual strain. This study highlights the importance viscoelasticity in the analysis of residual stress even in weakly dissipative materials like the human aorta.

A viscoelastic model for human myocardium

Understanding the biomechanics of the heart in health and disease plays an important role in the diagnosis and treatment of heart failure. The use of computational biomechanical models for therapy assessment is paving the way for personalized treatment, and relies on accurate constitutive equations mapping strain to stress. Current state-of-the art constitutive equations account for the nonlinear anisotropic stress-strain response of cardiac muscle using hyperelasticity theory. While providing a solid foundation for understanding the biomechanics of heart tissue, most current laws neglect viscoelastic phenomena observed experimentally. Utilizing experimental data from human myocardium and knowledge of the hierarchical structure of heart muscle, we present a fractional nonlinear anisotropic viscoelastic constitutive model. The model is shown to replicate biaxial stretch, triaxial cyclic shear and triaxial stress relaxation experiments (mean error ∼7.68%), showing improvements compared to its hyperelastic (mean error ∼24%) counterparts. Model sensitivity, fidelity and parameter uniqueness are demonstrated. The model is also compared to rate-dependent biaxial stretch as well as different modes of biaxial stretch, illustrating extensibility of the model to a range of loading phenomena. Statement of SignificanceThe viscoelastic response of human heart tissues has yet to be integrated into common constitutive models describing cardiac mechanics. In this work, a fractional viscoelastic modeling approach is introduced based on the hierarchical structure of heart tissue. From these foundations, the current state-of-the-art biomechanical models of the heart muscle are transformed using fractional viscoelasticity, replicating passive muscle function across multiple experimental tests. Comparisons are drawn with current models to highlight the improvements of this approach and predictive responses show strong qualitative agreement with experimental data. The proposed model presents the first constitutive model aimed at capturing viscoelastic nonlinear response across multiple testing regimes, providing a platform for better understanding the biomechanics of myocardial tissue in health and disease.

A fractional finite strain viscoelastic model of dielectric elastomer

Dielectric elastomer (DE) is an intelligent soft material and it is regarded as one significant artificial muscle. The nonlinear large deformation, electromechanical coupling and nonlinear viscoelasticity of DE material post great challenges for constitutive modeling and the associated dynamics. Fractional viscoelasticity has encountered some successes in the dynamics of complex materials. In the current study, a 3D constitutive relation of DE based on the fractional Kelvin-Voigt viscoelastic model by considering finite strain is established. The Cauchy stress under the condition of uniaxial and biaxial stretches is derived. The fractional dynamics equations of DE membrane are also formulated based on the virtual work principle. The composite integral rule for fractional derivative and the Runge-Kutta method are combined to numerically solve the dynamics system. The constitutive model is fitted to the uniaxial and biaxial stretch experimental data, and the material parameters are obtained. Subject to a cyclic deformation, the stress-softening effect of the DE membrane is simulated. Furthermore, the dynamics responses of DE membrane are also calculated. A convenient approach is given to evaluate the equilibrium point of the system. It reveals that the model could be used to predict the creep behaviors for both free vibration and parametric excitation vibration.

Excluded volume effects and fractional viscoelasticity in polymers

Excluded volume effects and fractional viscoelasticity in polymers