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

2018 ◽  
Vol 10 (09) ◽  
pp. 1850099 ◽  
Author(s):  
Hesam Khajehsaeid
Keyword(s):  
Fractional Order ◽  
Differential Operators ◽  
Filler Content ◽  
Viscoelastic Model ◽  
Constitutive Models ◽  
Viscoelastic Behavior ◽  
Fractional Model ◽  
Material Parameters ◽  
Integer Order ◽  
Viscoelastic Models

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.

Identification of Nonlinear Viscoelastic Models of Flexible Polyurethane Foam From Uniaxial Compression Data

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
Yousof Azizi ◽  
Patricia Davies ◽  
Anil K. Bajaj
Keyword(s):  
Uniaxial Compression ◽  
Viscoelastic Model ◽  
Viscoelastic Behavior ◽  
Model Parameters ◽  
Viscoelastic Models ◽  
Nonlinear Viscoelastic ◽  
Compression Data ◽  
Computationally Intensive ◽  
Nonlinear Viscoelastic Model ◽  
Flexible Polyurethane

Flexible polyethylene foam is used in many engineering applications. It exhibits nonlinear and viscoelastic behavior which makes it difficult to model. To date, several models have been developed to characterize the complex behavior of foams. These attempts include the computationally intensive microstructural models to continuum models that capture the macroscale behavior of the foam materials. In this research, a nonlinear viscoelastic model, which is an extension to previously developed models, is proposed and its ability to capture foam response in uniaxial compression is investigated. It is hypothesized that total stress can be decomposed into the sum of a nonlinear elastic component, modeled by a higher-order polynomial, and a nonlinear hereditary type viscoelastic component. System identification procedures were developed to estimate the model parameters using uniaxial cyclic compression data from experiments conducted at six different rates. The estimated model parameters for individual tests were used to develop a model with parameters that are a function of strain rates. The parameter estimation technique was modified to also develop a comprehensive model which captures the uniaxial behavior of all six tests. The performance of this model was compared to that of other nonlinear viscoelastic models.

Finite deformation and fractional order viscoelasticity of an auxetic foam

2022 ◽  
pp. 1045389X2110643
Author(s):  
Eugenia Stanisauskis ◽  
Paul Miles ◽  
William Oates
Keyword(s):  
Fractional Order ◽  
Finite Deformation ◽  
Viscoelastic Behavior ◽  
Integer Order ◽  
Relaxation Effects ◽  
Viscoelastic Parameters ◽  
Rate Dependent ◽  
Improve Model ◽  
Model Predictions ◽  
And Performance

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.

Identification of Nonlinear Viscoelastic Models of Flexible Polyurethane Foam From Uniaxial Compression Data

2012 ◽  
Author(s):  
Yousof Azizi ◽  
Patricia Davies ◽  
Anil K. Bajaj
Keyword(s):  
Uniaxial Compression ◽  
Viscoelastic Model ◽  
Viscoelastic Behavior ◽  
Model Parameters ◽  
Viscoelastic Models ◽  
Nonlinear Viscoelastic ◽  
Compression Data ◽  
Computationally Intensive ◽  
Nonlinear Viscoelastic Model ◽  
Flexible Polyurethane

Flexible polyethylene foam, which is used in many engineering applications, exhibits nonlinear and viscoelastic behavior. To date, several models have been proposed to characterize the complex behavior of foams from the computationally intensive microstructural models to continuum models that capture the macroscale behavior of the foam materials. A nonlinear viscoelastic model, which is an extension of previously developed models, is proposed and its ability to capture foam response in uniaxial compression is investigated. It is assumed in the model that total stress is decomposed into the sum of a nonlinear elastic component, which is modeled by a higher order polynomial, and a nonlinear hereditary type viscoelastic component. System identification procedures are developed to estimate the model parameters using uniaxial compression data from experiments conducted at different rates. The performance of this model is compared to that of other nonlinear viscoelastic models.

Experimental Validation of Non-Conventional Viscoelastic Models via Equivalent Damping Estimates

2008 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Damping Ratio ◽  
Viscoelastic Model ◽  
Measurement Data ◽  
Vibration Measurement ◽  
Modal Parameter ◽  
Structural Dynamic ◽  
General Validity ◽  
Integer Order ◽  
Equivalent Damping ◽  
Viscoelastic Models

Non-conventional rheological models based on non-integer order differential operators can be used to describe the viscoelastic behavior of materials, especially of polymers. These models are usually selected and then validated by means of creep and relaxation tests. However, engineers dealing with structural dynamic problems may need to obtain model identification from vibration measurement data. In this case, however, the direct identification of an optimal set of parameters of a viscoelastic model from time or frequency domain measurements is a difficult task, especially if the structural dissipative contributions are slight. In this paper, an indirect approach is adopted, based on the concept of damping ratio. When dealing with standard linear viscous dissipative models, a damping ratio modal parameter ζn can be analytically defined and experimentally estimated. But this theoretical parameter shows a dependency from the modal frequency that may dramatically fail in fitting the experimental data. On the contrary, it is known that a better agreement between theory and experiments can be achieved by means of non-integer order differential models, even though in this case analytical expressions for ζn are difficult to find. To overcome this difficulty, a method of general validity for viscoelastic models is developed, based on the concept of equivalent damping ratio and on the circle-fit technique. The proposed method is applied to experimental damping estimates from plane flexural vibrations of clamped-free beams, obtained from specimens of different size made of materials such as Polyethylene, Polyvinyl-chloride and Delrin.

A Nonlinear Viscoelastic Model for the Relaxation Behavior of Tendon

2012 ◽  
Author(s):  
Frances M. Davis ◽  
Raffaella De Vita
Keyword(s):  
Stress Relaxation ◽  
Relaxation Function ◽  
Viscoelastic Model ◽  
Viscoelastic Behavior ◽  
Relaxation Behavior ◽  
Successful Model ◽  
Viscoelastic Models ◽  
Nonlinear Viscoelastic ◽  
Linear Viscoelastic ◽  
Nonlinear Viscoelastic Model

Tendons are viscoelastic materials which undergo stress relaxation when held at a constant strain. The most successful model used to describe the viscoelastic behavior of tendons is the quasi-linear viscoelastic (QLV) model [1]. In the QLV model, the relaxation function is assumed to be a separable function of time and strain. Recently, this assumption has been shown to be invalid for tendons [2] thus suggesting the need for new nonlinear viscoelastic models.

Fractional-order viscoelastic model of musculoskeletal tissues: correlation with fractals

2021 ◽  
Vol 477 (2249) ◽  
pp. 20200990
Author(s):  
Jianqiao Guo ◽  
Yajun Yin ◽  
Gang Peng
Keyword(s):  
Fractional Order ◽  
Cross Sections ◽  
Viscoelastic Model ◽  
Temporal Correlation ◽  
Human Cartilage ◽  
Functional Changes ◽  
Viscoelastic Models ◽  
Musculoskeletal Tissues ◽  
Relaxation Experiments ◽  
Spatio Temporal

Self-similar fractals are widely obtained from biomaterials within the human musculoskeletal system, and their viscoelastic behaviours can be described by fractional-order derivatives. However, existing viscoelastic models neglect the internal correlation between the fractal structure of biomaterials and their fractional-order temporal responses. We further expanded the fractal hyper-cell (FHC) viscoelasticity theory to investigate this spatio-temporal correlation. The FHC element was first compared with other material elements and spring–dashpot viscoelastic models, thereby highlighting its discrete and fractal nature. To demonstrate the utility of an FHC, tree-like, ladder-like and triangle-like FHCs were abstracted from human cartilage, tendons and muscle cross-sections, respectively. The duality and symmetry of the FHC element were further discussed, where operating the duality transformation generated new types of FHC elements, and the symmetry breaking of an FHC altered its fractional-order viscoelastic responses. Thus, the correlations between the staggering patterns of FHCs and their rheological power-law orders were established, and the viscoelastic behaviour of the multi-level FHC elements fitted well in stress relaxation experiments at both the macro- and nano-levels of the tendon hierarchy. The FHC element provides a theoretical basis for understanding the connections between structural degeneration of bio-tissues during ageing or disease and their functional changes.

Bayesian inference and uncertainty propagation using efficient fractional-order viscoelastic models for dielectric elastomers

2020 ◽  
pp. 1045389X2096984
Author(s):  
Paul R Miles ◽  
Graham T Pash ◽  
Ralph C Smith ◽  
William S Oates
Keyword(s):  
Bayesian Inference ◽  
Fractional Order ◽  
Viscoelastic Model ◽  
Viscoelastic Behavior ◽  
Efficient Estimation ◽  
Dielectric Elastomers ◽  
Mcmc Methods ◽  
Accurate Estimation ◽  
Parameter Uncertainties ◽  
Modeling Framework

Dielectric elastomers are employed for a wide variety of adaptive structures. Many of these soft elastomers exhibit significant rate-dependencies in their response. Accurately quantifying this viscoelastic behavior is non-trivial and in many cases a nonlinear modeling framework is required. Fractional-order operators have been applied to modeling viscoelastic behavior for many years, and recent research has shown fractional-order methods to be effective for nonlinear frameworks. This implementation can become computationally expensive to achieve an accurate approximation of the fractional-order derivative. Accurate estimation of the elastomer’s viscoelastic behavior to quantify parameter uncertainty motivates the use of Markov Chain Monte Carlo (MCMC) methods. Since MCMC is a sampling based method, requiring many model evaluations, efficient estimation of the fractional derivative operator is crucial. In this paper, we demonstrate the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of estimating the uncertainty of a nonlinear viscoelastic model. We also demonstrate the use of parameter subset selection techniques to isolate parameters that are identifiable in the sense that they are uniquely determined by measured data. For those identifiable parameters, we employ Bayesian inference to compute posterior distributions for parameters. Finally, we propagate parameter uncertainties through the models to compute prediction intervals for quantities of interest.

scholarly journals On infinite order differential operators in fractional viscoelasticity

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Andrea Giusti
Keyword(s):  
Fractional Order ◽  
Constitutive Equations ◽  
Linear Viscoelasticity ◽  
Infinite Order ◽  
Differential Operators ◽  
Fractional Viscoelasticity ◽  
Viscoelastic Models ◽  
General Properties ◽  
Derivatives Of ◽  
Short Times

AbstractIn this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2.

Fractional Model for Malaria Disease

2013 ◽  
Author(s):  
Carla M. A. Pinto ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dynamical Evolution ◽  
Order Model ◽  
Fractional Model ◽  
Integer Order ◽  
A Value ◽  
Parameter Values ◽  
Future Work ◽  
Disease Free

In this paper we study a fractional order model for malaria transmission. It is considered the integer order model proposed by Chitnis et al [1] and we generalize it up to become a fractional model. The new model is simulated for distinct values of the fractional order. Are considered two initial conditions and a set of parameter values satisfying a value of the reproduction number, R0, less than one, for the integer model. In this case, there is co-existence of a stable disease free equilibrium and an endemic equilibrium. The results are in agreement with the integer order model and reveal that we can extend the dynamical evolution up to new types of transients. Future work will focus on analytically prove some of the results obtained.

Experimental Identification of a Fractional Derivative Linear Model for Viscoelastic Materials

2005 ◽  
Author(s):  
Giuseppe Catania ◽  
Silvio Sorrentino
Keyword(s):  
Frequency Response ◽  
Differential Operators ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Frequency Domain Method ◽  
Stress Strain ◽  
General Technique ◽  
Integer Order ◽  
Experimental Identification ◽  
Linear Viscoelastic

Non integer, fractional order derivative rheological models are known to be very effective in describing the linear viscoelastic dynamic behaviour of mechanical structures made of polymers [1]. The application of fractional calculus to viscoelasticity can be physically consistent [2][3][4] and the resulting non integer order differential stress-strain constitutive relation provides good curve fitting properties, requires only a few parameters and leads to causal behaviour [5]. When using such models the solution of direct problems, i.e. the evaluation of time or frequency response from a known excitation can still be obtained from the equations of motion using standard tools such as modal analysis [6]. But regarding the inverse problem, i.e. the identification from measured input-output vibrations, no general technique has so far been established, since the current methods do not seem to easily work with differential operators of non integer order. In this paper a frequency domain method is proposed for the experimental identification of a linear viscoelastic model, namely the Fractional Zener also known as Fractional Standard Linear Solid [5], to compute the frequency dependent complex stress-strain relationship parameters related to the material. The procedure is first checked with respect to numerically generated frequency response functions for testing its accuracy, and then to experimental inertance data from a free-free homogeneous beam made of High Density Polyethylene (HDPE) in plane flexural and axial vibration.

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