Fractional Order Viscoelasticity Research Articles

A Hybrid Stress Finite Element Method for Integro-Differential Equations Modelling Dynamic Fractional Order Viscoelasticity

A Hybrid Stress Finite Element Method for Integro-Differential Equations Modelling Dynamic Fractional Order Viscoelasticity

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type

AbstractLinear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Dynamic behavior of nanoplate on viscoelastic foundation based on spatial-temporal fractional order viscoelasticity and thermoelasticity

The nanodevices often work in a thermal environment and lead to the viscoelastic and thermoelastic coupling which becomes more complicated when further coupled with the size effects. A spatial-temporal fractional order model is proposed to study the dynamic behavior of thermoelastic nanoplate in the present work. The integer order differential is extended to the fractional order differential with the integer order differential as a special case. This extension makes the modeling of small-scale effects of mechanical behavior and generalized heat conduction more flexible. Non-local strain gradient elasticity and non-local heat conduction are used to describe the spatial non-local effects while the fractional order differential to describe the complicated history-dependent effects or the temporal nonlocal effects. The standard solid viscoelastic model and generalized heat conduction model with the fractional order differentials lead to the fractional-order differential governing equations. The fractional order differential equation is solved by the Laplace transform method, and the analytical solution of the response is expressed in terms of the Mittag-Leffler function. In order to verify the reliability of the analytical solution, the numerical solution is also provided for comparison. Based on the numerical results of dynamic response under the step load, the effects of thermoelastic fractional order parameters, viscoelastic fractional order parameters, and small-scale parameters are discussed. The new spatial-temporal fractional order model is a natural extension of the traditional integer order model which can be recovered from the present model.

One-dimensional haemodynamic model of a vascular network with fractional-order viscoelasticity

Abstract We propose a computational framework for a one-dimensional haemodynamic model with the arterial walls described by the fractional-order viscoelastic material constitutive law. This framework is used to compare blood flow characteristics for simulations with elastic and fractional-order viscoelastic walls. We use three well-established benchmark tests: a single pulse wave in a long vessel, flow in a 37-segment network of elastic tubes, and flow in anatomically detailed arterial network consisting of 61 arterial segments. All results for elastic model are in a good agreement with analytical solutions, in vitro data and other well-established approaches. Fractional-order model demonstrates noticeable differences in pulse wave propagation speed and minor differences in pressure and flow profiles. Differences in profiles are negligible in major vessels, but more profound in vessels beyond the third or fourth generation.

Numerous analytical wave solutions to the time-fractional unstable nonlinear Schrödinger equation with beta derivative

Fractional nonlinear evolution equations are mathematical representations used to explain a wide range of complex phenomena occurring in nature. By incorporating fractional order viscoelasticity, these equations can accurately depict the intricate behaviour of materials or mediums, requiring fewer parameters compared to classical models. Furthermore, fractional viscoelastic models align with molecular theories and thermodynamics, making them highly compatible. Consequently, the scientific community has shown significant interest in fractional nonlinear evolution equations and their soliton solutions. This study employs the extended Kudryashov method to derive soliton solutions for the time-fractional unstable nonlinear Schrödinger equation, utilizing the Atangana–Baleanu fractional derivative known as the beta derivative. The obtained solitons exhibit various shapes, including V-shaped, periodic, singular periodic, flat kink, and singular bell, under specific conditions. To better understand their physical characteristics, 3D and contour plots are presented by assigning parameter values to certain solutions. Additionally, 2D graphs are generated to observe how the fractional parameter affects the solutions. The Hamiltonian function is determined to further analyse the dynamics of the phase plane. The simulations were conducted using Mathematica and MATLAB software tools. The outcomes of this research contribute to a deeper understanding of the behaviour of fractional nonlinear evolution equations and their soliton solutions, offering insights into the complex dynamics of viscoelastic systems.

An Entropy Dynamics Approach for Deriving and Applying Fractal and Fractional Order Viscoelasticity to Elastomers

Abstract Entropy dynamics is a Bayesian inference methodology that can be used to quantify time-dependent posterior probability densities that guide the development of complex material models using information theory. Here, we expand its application to non-Gaussian processes to evaluate how fractal structure can influence fractional hyperelasticity and viscoelasticity in elastomers. We investigate how kinematic constraints on fractal polymer network deformation influences the form of hyperelastic constitutive behavior and viscoelasticity in soft materials such as dielectric elastomers, which have applications in the development of adaptive structures. The modeling framework is validated on two dielectric elastomers, VHB 4910 and 4949, over a broad range of stretch rates. It is shown that local fractal time derivatives are equally effective at predicting viscoelasticity in these materials in comparison to nonlocal fractional time derivatives under constant stretch rates. We describe the origin of this accuracy that has implications for simulating large-scale problems such as finite element analysis given the differences in computational efficiency of nonlocal fractional derivatives versus local fractal derivatives.

Stochastic analysis of small-scale beams with internal and external damping

Random flexural vibrations of small-scale Bernoulli–Euler beams with internal and external damping are investigated in the paper. Such a kind of problem occurs in design and optimization of structural components of smart miniaturized electromechanical systems. Internal damping is related to the viscoelastic stress–strain law describing the beam time-dependent behavior while the external damping represents the interaction between the structure and the surrounding viscous environment. In the proposed formulation, Boltzmann superposition integral and fractional-order viscoelasticity are exploited to capture the above mentioned non-conventional phenomena. Size effects caused by long range interactions arising at small-scale are also taken into account and modeled by means of an integral nonlocal formulation based on the stress-driven approach. Moreover, in order to reproduce stochastic vibrations of small-size devices, such as resonators and sensors, the external loading are modeled by taking into account their random nature. The relevant dynamic problem is thus described by a fractional-order stochastic partial differential equation in space and time equipped with standard and non-classical boundary conditions. A proper differential eigenanalysis for the nonlocal viscoelastic bending problem is performed to find the modal shapes and closed-form expressions of power spectral density and stationary variance of displacements are provided. The proposed methodology, accounting for size-effects, hereditariness behaviors, viscous properties of surrounding environment and randomness of external loadings, is suitable to model and capture the effective behavior of miniaturized devices. The presented theoretical findings and numerical outcomes show the influence of size-effects and viscoelastic parameters on the structural response.

Fractional and fractal order effects in soft elastomers: Strain rate and temperature dependent nonlinear mechanics

Soft dielectric elastomers are commonly used in applications where large deformation is prevalent such as energy harvesters, sensors, and soft actuators. The thermo-mechanical response of these elastomers is important in order to accurately describe the viscoelastic behavior over a broad range of operating conditions. Fractal characteristics of the dielectric elastomer VHB 4905 and their connection to fractional order viscoelasticity are analyzed to better understand viscoelasticity over a range of elevated temperatures starting at room temperature. We extend prior work in viscoelasticity to include excluded volume effects as a function of temperature. A fractal hyperelastic model is combined with a fractional order viscoelastic model and validated experimentally. Bayesian uncertainty methods are used to quantify the material parameters and their influence on temperature dependent viscoelasticity measurements. The model fits are compared to previously collected temperature dependent viscoelastic measurements on VHB 4905 for temperatures ranging from 23 °C to 60 °C. Based on fractional order viscoelasticity, we infer that the excluded volume parameter is negative and initially decreases before reaching a constant value near 50 °C.

Finite deformation and fractional order viscoelasticity of an auxetic foam

Auxetic foams exhibit novel mechanical properties due to their unique microstructure for improved energy-absorption and cavity expansion applications that have fascinated the scientific community since their inception. Given the advancements in material processing and performance of polymer open cell auxetic foams, there is a strong desire to fully understand the nonlinear rate-dependent deformation of these materials. The influence of nonlinear compressibility is introduced here along with relaxation effects to improve model predictions for different stretch rates and finite deformation regimes. The viscoelastic behavior of the material is analyzed by comparing fractional order and integer order calculus models. All results are statistically validated using maximum entropy methods to obtain Bayesian posterior densities for the hyperelastic, auxetic, and viscoelastic parameters. It is shown that fractional order viscoelasticity provides [Formula: see text]–[Formula: see text] improvement in prediction over integer order viscoelastic models when the model is calibrated at higher stretch rates where viscoelasticity is more significant.

A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

Fractional viscoelasticity in fractal and non-fractal media: Theory, experimental validation, and uncertainty analysis

In this paper, fractional and non-fractional viscoelastic models for elastomeric materials are derived and analyzed in comparison to experimental results. The viscoelastic models are derived by expanding thermodynamic balance equations for both fractal and non-fractal media. The order of the fractional time derivative is shown to strongly affect the accuracy of the viscoelastic constitutive predictions. Model validation uses experimental data describing viscoelasticity of the dielectric elastomer Very High Bond (VHB) 4910. Since these materials are known for their broad applications in smart structures, it is important to characterize and accurately predict their behavior across a large range of time scales. Whereas integer order viscoelastic models can yield reasonable agreement with data, the model parameters often lack robustness in prediction at different deformation rates. Alternatively, fractional order models of viscoelasticity provide an alternative framework to more accurately quantify complex rate-dependent behavior. Prior research that has considered fractional order viscoelasticity lacks experimental validation and contains limited links between viscoelastic theory and fractional order derivatives. To address these issues, we use fractional order operators to experimentally validate fractional and non-fractional viscoelastic models in elastomeric solids using Bayesian uncertainty quantification. The fractional order model is found to be advantageous as predictions are significantly more accurate than integer order viscoelastic models for deformation rates spanning four orders of magnitude.

Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters

In this paper, we investigate the free damped vibration of a nanobeam resting on viscoelastic foundation. Nanobeam and viscoelastic foundation are modeled using nonlocal elasticity and fractional order viscoelasticity theories. Motion equation is derived using D’Alambert’s principle and involves two retardation times and fractional order derivative parameters regarding to a nanobeam and viscoelastic foundation. The analytical solution is obtained using the Laplace transform method and it is given as a sum of two terms. First term denoting the drift of the system’s equilibrium position is given as an improper integral taken along two sides of the cut of complex plane. Two complex conjugate roots located in the left half-plane of the complex plane determine the second term describing the damped vibration around equilibrium position. Results for complex roots of characteristic equation obtained for a single nanobeam without viscoelastic foundation, where imaginary parts represent damped frequencies, are validated with the results found in the literature for natural frequencies of a single-walled carbon nanotube obtained from molecular dynamics simulations. In order to examine the effects of nonlocal parameter, fractional order parameters and retardation times on the behavior of characteristic equation roots in the complex plane and the time-response of nanobeam, several numerical examples are given.

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

We study a second order hyperbolic initial‐boundary value partial differential equation (PDE) with memory that results in an integro‐differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016

Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials, is formulated for temporal semidiscretization of the problem. Stability estimates of the discrete problem are proved, that are used to prove optimal order a priori error estimates. The theory is illustrated by a numerical example.

Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity

A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions . Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity , under the appropriate assumptions on data. • Well-posedness of a hyperbolic integro-differential equation is studied. • The model arises in fractional viscoelasticity with two Mittag-Leffler type kernels. • Homogeneous Dirichlet and nonhomogeneous Neumann boundary conditions are considered. • Existence, uniqueness and regularity of the solution are proved by Galerkin's method. • We extend the method to prove regularity of any order for models with smooth kernels.

Fractional order viscoelasticity in characterization for atrial tissue

Atrial tissue due to its solid-like and fluid-like constituents shows highly viscoelastic properties. Up to now, the distribution pattern of muscle fiber in heart is not well established, and it is hard to establish the constitutive model for atrial tissue completely based on the microstructure level. Consider the equivalence between the fractional viscoelasticity and the fractal spring-dashpot model, a generalized fractional order Maxwell model is proposed to model the porcine atrial tissue in the phenomenological sense. This model has a simple expression and intuitively physical meanings. The constitutive parameters in the model are estimated in the complex domain by a genetic algorithm. Final results illustrate the proposed model gets a well agreement with the experimental data.

Spatially fractional-order viscoelasticity, non-locality, and a new kind of anisotropy

A class of non-local viscoelastic equations of motion including equations of fractional order with respect to the spatial variables is studied. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind of anisotropy, associated with azimuthal dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Relaxation effects are additionally accounted for by replacing second-order time derivatives by lower-order fractional derivatives. Explicit fundamental solutions of the Cauchy problem for scalar equations with isotropic and anisotropic non-locality are constructed. For some particular choices of the parameters, numerical solutions are constructed.

The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneous mixed Dirichlet and Neumann boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then used to prove a priori error estimates. The theory is illustrated by a numerical example.