A new method for formulating linear viscoelastic models

Su, Xianglong; Yao, Donggang; Xu, Wenxiang

doi:10.1016/j.ijengsci.2020.103375

Cited by 23 publications

(12 citation statements)

References 20 publications

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“…In the limits a  0 and a  1, a spring-pot realizes the constitutive relations of a spring and a dashpot. Interpretations of the order a in the constitutive relation (5) have been developed by representing spring-pots as infinite arrangements of springs and dashpots. Examples include representations of spring-pots as ladder models [17,19], GM models [22,23,25], GKV models [22], and their respective variants [18,20,21,24].…”

Section: Fractional Modelsmentioning

confidence: 99%

“…Mechanical circuits consisting of springs, dashpots, and fractional viscoelastic elements, so-called spring-pots [1], provide an intuitive and mathematically tractable framework to analyze the rheological properties of materials. Such models, which we hereafter refer to as viscoelastic models, well capture the deformation behavior of many materials and thus find a wide range of applications from food [2,3] and engineering materials [4,5] to rocks [6][7][8]. Despite merely linear combinations of passive mechanical elements, viscoelastic models have a wellestablished physical basis.…”

Section: Introductionmentioning

confidence: 99%

“…Transfer function-based formulation [5] is a promising approach to avoid the difficulties caused by the nonuniqueness. The transfer function T s ( ) of a viscoelastic model is defined as the Laplace transform of the ratio of the stress s t ( ) to the strain e t , ( ) i.e., transfer functions correspond to the models' constitutive relation [hereafter, the Laplace transform of a function f t ( ) of time t is denoted as f s ( )  where s is the complex frequency]:…”

Section: Introductionmentioning

confidence: 99%

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Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

Suda,

Nagahama

2023

Phys. Scr.

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Viscoelastic models aim to characterize material properties by extracting the parameters that best describe observed deformations. However, due to the equivalence of various model structures, the conventional model parameters as constants assigned to each model element cannot be uniquely determined for a given deformation and, therefore, are unfit to represent material properties. To resolve this problem, we formulate the framework of viscoelastic models based on their transfer functions. We explicitly derive the functional form of the transfer functions of viscoelastic models and show that a set of parameters is uniquely determined for each equivalent set of viscoelastic models. The newly identified parameters correspond to the instantaneous modulus and the distribution of the characteristic time scales of the system, and based on this perspective, we show that the differentiation order of a fractional viscoelastic element, spring-pot, is determined solely by the recursive nature of the characteristic time scales of the system. Furthermore, we show that all viscoelastic models are equivalent to four canonical model structures.

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Section: Fractional Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

Suda,

Nagahama

2023

Phys. Scr.

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“…Our approach of incorporating non-linearities into a fractional description of viscoelastic behavior using an FMM linear kernel in the Wagner constitutive equation can be further extended to an even wider class of materials such as non-soft materials or viscoelastic solids by replacing the FMM kernel with a Fractional Zener kernel, Fractional Kelvin-Voigt [26], or the more generalized Modified Fractional Maxwell (MFM) introduced by Xu et al [69,70]. The MFM has been shown to accurately describe the linear viscoelastic properties of complex liquids, rubber-like materials, and glassy viscoelastic solids across a wide range of frequencies.…”

Section: Generalized Damping Functionmentioning

confidence: 99%

Incorporating Rheological Nonlinearity into Fractional Calculus Descriptions of Fractal Matter and Multi-Scale Complex Fluids

2021

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In this paper, we use ideas from fractional calculus to study the rheological response of soft materials under steady-shearing flow conditions. The linear viscoelastic properties of many multi-scale complex fluids exhibit a power-law behavior that spans over many orders of magnitude in time or frequency, and we can accurately describe this linear viscoelastic rheology using fractional constitutive models. By measuring the non-linear response during large step strain deformations, we also demonstrate that this class of soft materials often follows a time-strain separability principle, which enables us to characterize their nonlinear response through an experimentally determined damping function. To model the nonlinear response of these materials, we incorporate the damping function with the integral formulation of a fractional viscoelastic constitutive model and develop an analytical framework that enables the calculation of material properties such as the rate-dependent shear viscosity measured in steady-state shearing flows. We focus on a general subclass of fractional constitutive equations, known as the Fractional Maxwell Model, and consider several different analytical forms for the damping function. Through analytical and computational evaluations of the shear viscosity, we show that for sufficiently strong damping functions, for example, an exponential decay of fluid memory with strain, the observed shear-thinning behavior follows a power-law response with exponents that are set by the power-law indices of the linear fractional model. For weak damping functions, however, the power-law index of the high shear rate viscosity is set by the terminal behavior of the damping function itself at large strains. In the limit of a very weak damping function, the theoretical formulation predicts an unbounded growth of the shear stress with time and a continuously growing transient viscosity function that does not converge to a meaningful steady-state value. By determining the leading terms in our analytical solution for the viscosity at both low and high shear rates, we construct an approximate analytic expression for the rate-dependent viscosity. An error analysis shows that, for each of the damping functions considered, this closed-form expression is accurate over a wide range of shear rates.

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“…In the case of linear viscoelastic materials, there is a linear connection between the stress and strain values at any given time, while nonlinear viscoelastic materials present a nonlinear connection. These materials are often characterized by their stress relaxation and creep behaviors [1,[6][7][8][9][10][11]. PA6 is widely used as a structural element; thus, the time dependence of the material was neglected in our investigations because such elements are generally subjected to quasi-static loads.…”

Section: Introductionmentioning

confidence: 99%

Optical Investigation of the Limits of Modeling the Nonlinear Elastic Behavior of PA6 Using Linear Elastic Material Models

2022

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One of the most critical issues during polymer finite element simulations is the selection of the proper material models. The widely used and accepted multilinear material models require load case-specific material tests, which are time and cost demanding. Data for these characteristics must be acquired by standardized measurements. On the other hand, the parameters required to create a linear elastic material model in most cases are easy to obtain, and the establishment of the model is a shorter process. This research is aimed to provide information to engineers about the possibility of modeling the nonlinear elastic materials by using linear elastic material models and about the limits of such models. To create the most accurate material models, laboratory measurements were performed on polyamide (PA6) material, which is a widely used raw material in the industry. Test specimens were manufactured to obtain material constants according to the ISO 527-2 standard, and for validating the effectiveness of the applied material models, three different tensile specimens were created, which were tested under quasi-static loading in the elastic region. A comprehensive finite element investigation was performed, and the numerical results were then compared to laboratory measurements using the GOM Aramis digital image correlation (DIC) system. By comparing the optically measured strain data to the numerical results, it was determined that the nonlinear elastic materials can be modeled using linear elastic models in a well identifiable strain range with sufficient accuracy.

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A new method for formulating linear viscoelastic models

Cited by 23 publications

References 20 publications

Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

Incorporating Rheological Nonlinearity into Fractional Calculus Descriptions of Fractal Matter and Multi-Scale Complex Fluids

Optical Investigation of the Limits of Modeling the Nonlinear Elastic Behavior of PA6 Using Linear Elastic Material Models

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