Fractional Kelvin Research Articles

Investigation on the creep mechanism of PA6 films based on quasi point defect theory

Polyamide 6 (PA6) films with significant α relaxation process was selected as the model system. The creep behavior and rheological mechanism during deformation in the amorphous regions of semi-crystalline polymers are systematically investigated by carrying out creep experiments. Based on the quasi point defect (QPD) theory, the complete physical process of PA6 film creep behavior from elasticity to viscoelasticity and viscoplasticity was analyzed and modeled from the perspective of structural heterogeneity. The results demonstrate that the creep deformation of PA6 film is a typical thermo-mechanical coupling and nonlinear mechanics process, and potential creep mechanisms corresponds to stress-induced local shear deformation enhancement and thermal activation-induced particle flow diffusion. The elastic-plastic transition involved in the creep deformation process of semi-crystalline polymer originates from the activation of quasi-point defective sites in the amorphous region, the expansion of sheared micro-domains and irreversible fusion. The generalized fractional Kelvin (GFK) model is proposed, and the physical meaning of parameters is explained by combining the quasi point defect theory and creep delay spectrum(L(τ)). Finally, the effectiveness of the GFK model and the QPD theory in studying the deformation behavior of PA6 films was validated by comparing experimental data with theoretical results, which theoretically reveals the structural evolution of PA6 film during creep process.

Magnetohydrodynamics flow and heat transfer of novel generalized Kelvin–Voigt viscoelastic nanofluids over a moving plate

In this work, the unsteady magnetohydrodynamics boundary layer flow and heat transfer of novel generalized Kelvin–Voigt viscoelastic nanofluids over a moving plate are investigated. The classical Kelvin–Voigt constitutive relation is generalized to incorporate a time-fractional derivative to characterize the fluid behavior, which is proved to be of significance and physically justified. The newly developed fractional Kelvin–Voigt constitutive correlation and a dual-phase-lagging constitutive equation are applied to the momentum and energy equations, respectively, for a nanofluid model over a moving plate. The formulated integrodifferential velocity and thermal boundary layer equations are solved using the finite difference method together with a fast algorithm, which reduces the consumed central processing unit time significantly. Several numerical examples are presented to illustrate the influence of the critical parameters on the nanofluid motion and thermal characteristics. Compared to the fractional Maxwell nanofluid model, the velocity boundary layer for the fractional Kelvin–Voigt nanofluid model is thinner. Although the fractional indexes show similar effects on the velocity boundary layer, the impacts of the relaxation parameters are in contrast. This work provides valuable insights into the feasibility of using the fractional Kelvin–Voigt viscoelastic model to depict the fluid flow and heat transfer characteristics of nanofluids.

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

Unveiling the role of hydroxyapatite and hydroxyapatite/silver composite in osteoblast-like cell mineralization: An exploration through their viscoelastic properties

Mechanical properties are becoming fundamental for advancing the comprehension of cellular processes. This study addresses the relationship between viscoelastic properties and the cellular mineralization process. Osteoblast-like cells treated with an osteogenic medium were employed for this purpose. Additionally, the study explores the impact of hydroxyapatite (HA) and hydroxyapatite/silver (HA/Ag) composite on this process. AFM relaxation experiments were conducted to extract viscoelastic parameters using the Fractional Zener (FZ) and Fractional Kelvin (FK) models. Our findings revealed that the main phases of mineralization are associated with alterations in the viscoelastic properties of osteoblast-like cells. Furthermore, HA and HA/Ag treatments significantly influenced changes in the viscoelastic properties of these cells. In particular, the HA/Ag treatment demonstrated a marked enhancement in cell fluidity, suggesting a possible role of silver in accelerating the mineralization process. Moreover, the study underscores the independence observed between fluidity and stiffness, indicating that modifications in one parameter may not necessarily correspond to changes in the other. These findings shed light on the factors involved in the cellular mineralization process and emphasize the importance of using viscoelastic properties to discern the impact of treatments on cells.

An analytical approach for shock response of fractional viscoelastic beams with tip mass

This paper specifically analyzes the behavior of a beam with fractional viscoelastic properties and a tip mass under shock. The shock is applied longitudinally to the beam, and a fractional Kelvin–Voigt model is employed to describe the viscoelastic properties. Additionally, the paper examines the fractional viscoelastic power as a function of time during the shock. The equation of motion for the fractional viscoelastic beam with a tip mass is derived using Hamilton’s principle. In order to ensure accurate analysis of the results, the graphs are carefully analyzed in both the time and frequency domains. To solve the equations, the special technique provided for differential equations of variable order (VO) is used. Two distinct methods, namely, absolute method and square root of the sum of squares method, are thoroughly examined to calculate the absolute acceleration. Subsequently, the most suitable method is selected based on the evaluation results. The study’s results demonstrate that the damping parameters of the viscoelastic beam have a significant impact on shock transmissibility. Additionally, the findings demonstrate that the ability to transfer shocks can be manipulated by modifying the system’s geometry and mass.

Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Abstract The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.

Creep characteristics and damage model of coal–rock combinations with different height ratios

Numerous coal pillars are left after the coal mining process. The composite structure comprising a roof and coal pillar has prominent creep characteristics, which threaten safe underground mining. Therefore, the creep characteristics of coal-rock combinations should be studied to ensure the safety of quarry and surface. Uniaxial creep tests under static load axial pressure and different height ratios were performed using a self-designed rock creep disturbance test device to determine the effect of height ratio and axial pressure on the creep characteristics of coal–rock combinations. From the test results, a creep damage model for coal–rock combinations was established by combining the elastomer, fractional Kelvin body, plastic body, Abel dashpot, and modified nonlinear viscoplastic body; introducing damage variables D related to stress, height ratio, and time; and deriving a one-dimensional creep equation. An improved nonlinear least squares method based on pattern search was utilized to invert the creep parameters. The results of the creep equation calculation were fitted with the experimental results with good results. The creep curve with a height ratio of 2:1 was predicted with good results. The research results provide theoretical references for long-term stability analysis of rock engineering.

Dynamic analysis of viscoelastic foundation plate with fractional Kelvin–Voigt model using shifted Bernstein polynomials

The fractional Kelvin–Voigt (FKV) viscoelastic constitutive model has been employed to build the fractional differential governing equation of viscoelastic foundation plate. Shifted Bernstein polynomial algorithm is implemented to solve the governing equations on a direct time domain numerically. Furthermore, the validity of the method is demonstrated by convergent analysis and error estimation of a math example. Finally, the dynamic analysis of viscoelastic foundation plate under different uniform loads. The movement of plate with different aspect ratios in viscoelastic foundation under the same load is studied. The stress of viscoelastic foundation plates of two materials under identical loading situations is investigated.

Nonlinear dynamics of fractional viscoelastic PET membranes with linearly varying density

In this paper, the nonlinear dynamics of fractional viscoelastic polyethylene terephthalate (PET) membranes with linearly varying density are elaborated. The viscoelasticities of the PET membranes are characterized with the fractional Kelvin–Voigt model, and the density distribution is considered a linear fluctuation in the lateral direction. The geometrically nonlinear formulation is established with the von-Karman theory and the Hamilton principle. The Bubnov–Galerkin method is used to solve the resulting fractional differential equations, and a refined numerical scheme combining the finite difference with a discrete Caputo-type fractional derivative approximation is developed. Comparison examples are provided to substantiate the accuracy and effectiveness of the present method. A detailed parametric analysis is performed to illuminate the effects of the fractional order, density coefficient and various system parameters on the time response of the viscoelastic PET membranes. This study paves the way for fractional modelling and vibrational analysis of viscoelastic substrates in flexible electronics manufacturing.

Dynamic response analysis of vehicle–pavement-coupled system based on RIOHTrack full-size accelerated loading test

The interaction between vehicles and asphalt pavements can cause different degrees of damage to the pavement structure. Due to this interaction, deformation of pavement is commonly observed, potentially affecting ride quality and driving safety. Investigating the responses of the vehicle–pavement interaction to different pavement structures is therefore a curial task. To this end, we proposed a vehicle–pavement-coupled dynamic model, based on the accelerated loading test of the Research Institute of Highway MOT Track (RIOHTrack). This model is derived by coupling a heavy-duty truck vibration model and the well-known huge multi-layer rectangular plate model. The heavy-duty truck vibration model which describes a 3D vehicle–pavement dynamics system is constructed according to test conditions. We formulate the coupling based on the variable-order fractional Kelvin foundation and the random dynamic load. In particular, to obtain the random dynamic load, we first analyze the time domain equation of road roughness from the field data set of the RIOHTrack test track and then use the Galerkin method for numerical calculation. Using the proposed model, we simulated 19 types of pavement structures of the RIOHTrack test track, which characterize the primary features of the empirical data set. Furthermore, we discuss the bifurcation consensus of the dynamic response of the front and rear wheels under different loads and analyzes comprehensively the dynamic responses of different pavement structures. The results show that the vehicle–pavement-coupled dynamic response model proposed in this paper has better applicability to simulate deformation for different asphalt pavement structures and better reflects the history-dependent process and memory effect of pavement deformation.

One-dimensional rheological consolidation study on two-sided semi permeable boundary and fractional viscoelastic saturated soil under cyclic loading

Considering the viscoelastic of soft clay, the fractional kelvin model is used to describe the constitutive relationship of soft clay. At the same time, considering that both sides of the soft clay layer have certain permeability, the two-sided semi-permeable boundary is used as its boundary condition. Combined with the cyclic load of different waveforms, the one-dimensional consolidation settlement of saturated soft clay is analyzed. Firstly, the consolidation equation is obtained through Laplace transformation, and the semi-analytic result is obtained by Laplace inverse transformation. The accuracy of the solve method and the model’s applicability are checked by the degradation of boundary conditions and the model. Finally, one-dimensional consolidation settlement under cyclic load with different waveforms is compared.

Effective properties of fractional viscoelastic composites via two‐scale asymptotic homogenization

Driven by the growing interest in fractional constitutive modeling, we adopt a two‐scale asymptotic homogenization approach to study the effective properties of fractional viscoelastic composites. We focus on a purely mechanical setting and derive the cell and homogenized problems corresponding to the balance of linear momentum equation in the absence of body forces and inertial terms. In doing this, we reformulate the original framework in the Laplace–Carson domain and discuss how to obtain the effective coefficients in the time domain. We particularize the general setting of our work by considering memory functions that describe special types of fractional linear viscoelastic behaviors, and after presenting the limit cases of our selections, we framed the homogenization results to account for benchmark problems with different combinations of constitutive models. Specifically, these latter involve elastic, fractional Kelvin–Voigt, fractional Zener and fractional Maxwell constituents. Our results permit us to reinforce the interpretation of the theoretical findings and to elucidate the role of the fractional constitutive models on the effective properties of the composites under investigation.

Investigating Fractional Damping Effect on Euler–Bernoulli Beam Subjected to a Moving Load

In this work, the dynamic response of Euler–Bernoulli beams of four different boundary conditions with fractional order internal damping under a traversing moving load is investigated. The load is assumed to be moving with different values of constant velocity. A proposed approach to obtain the closed-form solution of the problem based on Green’s functions combined with a decomposition technique in the Laplace transform domain is introduced. Several cases are studied and compared to the literature; for instance, if simply supported beam is considered, the following three cases are to be explored: the case of elastic (or undamped) beam, the damped (or viscously damped) beam, and finally the fractionally damped beam modeled by the fractional Kelvin–Voigt model. The effects to the beam dynamic response induced by magnitude of moving load velocity, damping ratio, and fractional damping order are explored. The results expressed sufficient agreement with similar problems found in literature and evidenced that the dynamic response of beams is significantly affected by varying the fractional order of beam damping as well as the moving load velocity. Accordingly, using fractionally damped materials exhibits better realistic behavior of beams and intermediate between elastic and viscous beam behaviors.

Fractional model for simulating Long-Term fracture conductivity decay of shale gas and its influences on the well production

Due to the high-content of clay and kerogen, shale will always show viscoelastic mechanical behaviors during the development of shale gas, which will lead to the increase of proppant embedment and do harm to gas recovery from shale reservoirs. In the previous studies, viscoelastic mechanical behaviors of reservoir rock had not been coupled with productivity model of gas well and the long-term fracture conductivity was overestimated. In order to figure out the creep damage to long-term fracture conductivity and predict the estimated ultimate recovery of shale gas wells accurately, a novel time-varying fracture conductivity model based on fractional calculus was developed for simulating the long-term decay of fracture conductivity. Proppant deformation, proppant embedment, and shale creep described by fractional Kelvin model were taken into consideration in the time-varying fracture conductivity model and it was coupled with embedded discrete fracture model (EDFM) to simulate the well performance. Simulation results show that: Daily gas production curve simulated by the EDFM with creep damage under field conditions of a shale gas well in Sichuan Province, China was aligned with its production data. The damage rate of cumulative production considering shale creep is more than twice that only caused by proppant deformation and proppant embedment. And the formation viscous modulus determines the strength of the shale viscoelastic mechanical behaviors. According to formation viscous modulus, the shale reservoirs can be divided into three types: Type I with strong creep damage (E2v < 5GPa), Type Ⅱ with moderate creep damage (5GPa < E2v < 30GPa), and Type III with few creep damage (E2v > 30GPa).

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

Identification of the extended standard linear solid material model by means of experimental dynamical measurements

A novel algebraic procedure for the non-parametric identification of the material model by means of dynamical test measurements is proposed. An extended Standard Linear Solid (SLS) material model is taken into account to model the material linear visco-elastic behavior. It consists of the series arrangement of fractional Kelvin model elements adopting real parameters and integer and non-integer order time differential operators, and of hysteretic Kelvin model elements adopting complex parameters and integer order time differential operators. Hysteretic Kelvin model elements are introduced to take account of the material hysteretic behavior. The material E(j⋅ω) complex modulus, is analytically modeled as the ratio of pseudo polynomials (non-integer power terms) in the j⋅ω Fourier variable. A multi-step, iterative, material model identification technique is here proposed to identify the unknown polynomial coefficients and the non-integer exponent values starting from E(j⋅ω) material discrete estimates from input-output dynamical measurements made on a beam specimen at different ω frequency values. Computational, nonphysical SLS elements resulting from the application of the identification procedure can be found and eliminated, so that a low order optimal model result. Some results obtained by applying the proposed identification technique with real experimental measurements are shown and discussed.

Constitutive modeling of human cornea through fractional calculus approach

In this work, the fractional calculus approach is considered for modeling the viscoelastic behavior of human cornea. It is observed that the degree of both elasticity and viscosity is easy to describe in terms of the fractional order parameters in such an approach. Modeling of the human cornea when subjected to simple stress up to the level of 250 MPa by fractional order Maxwell model along with the Fractional Kelvin Voigt Viscoelastic Model is reported. For the Maxwell governing fractional equation, two fractional parameters α and β have been considered to model the stress–strain relationship of the human cornea. The analytical solution of the fractional equation has been obtained for different values of α and β using Laplace transform methods. The effect of the fractional parameter values on the stress-deformation nature has been studied. A comparison between experimental values and calculated values for different fractional order of the Maxwell model equation defines the parameters which depict the real-time stress–strain relationship of the human cornea. It has been observed that the fractional model converges to the classical Maxwell model as a special case for α = β = 1.

Methodologies and models for measuring viscoelastic properties of cancer cells: Towards a universal classification

Different methods and several physical models exist to study cell viscoelasticity with the atomic force microscope (AFM). In search of a robust mechanical classification of cells through AFM, in this work, viscoelastic parameters of the cancer cell lines MDA-MB-231, DU-145, and MG-63 are obtained using two methodologies; through force–distance and force–relaxation curves. Four mechanical models were applied to fit the curves. The results show that both methodologies agree qualitatively on the parameters that quantify elasticity but disagree on the parameters that account for energy dissipation. The Fractional Zener (FZ) model represents well the information given by the Solid Linear Standard and Generalized Maxwell models. The Fractional Kelvin (FK) model concentrates the viscoelastic information mainly in two parameters, which could be an advantage over the other models. Therefore, the FZ and FK models are proposed as the basis for the classification of cancer cells. However, more research using these models is needed to obtain a broader view of the meaning of each parameter and to be able to establish a relationship between the parameters and the cellular components.

Polynomial stability of a transmission problem involving Timoshenko systems with fractional Kelvin–Voigt damping

In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we investigate three cases: The first one when the damping is localized in the bending moment, the second case when the damping is localized in the shear stress, we prove that the energy of the system decays polynomially with rate in both cases. In the third case, the fractional Kelvin–Voigt is acting on the shear stress and the bending moment simultaneously. We show that the system is polynomially stable with energy decay rate of type , provided that the two dampings are acting in the same subinterval. The method is based on the frequency domain approach combined with multiplier technique.

Study on viscoelastic materials at micro scale pondering supramolecular interaction impacts with DMA tests and fractional derivative modeling

AbstractViscoelastic materials are commonly utilized in seismic vibration suppression of building structures. Supramolecular modification can extend the effective damping temperature range and improve the damping performance of viscoelastic materials. In present work, the mechanical behaviors of the nitrile rubber viscoelastic material are investigated by dynamic thermomechanical analyzer (DMA) with different strain amplitudes, excitation frequencies and environmental temperatures. The supramolecular interaction and molecular structures modified fractional Kelvin (SMF Kelvin) model taking into account the temperature and frequency impacts is proposed and verified with the DMA test results of the nitrile rubber viscoelastic material. The test results of the viscoelastic damper with different temperatures are employed to further prove the effectiveness of the new proposed SMF Kelvin model.