Phenomenological isotropic visco-hyperelasticity: a differential model based on fractional derivatives

Bouzidi, Safia; Bechir, Hocine; Brémand, Fabrice

doi:10.1007/s10665-015-9818-6

Cited by 8 publications

(8 citation statements)

References 70 publications

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“…The fractional derivatives theory imposes the values of the real number 0 < α < 1, 0 < β < 1 and we have 0 < ξ < 1 [63]. We note that the relaxation behavior of filled rubbers is characterized by a very rapid decrease of the stress at the beginning of a hold time, followed by an extremely slow decay during several hours [71].…”

Section: Which Can Be Neglected In Comparison Tomentioning

confidence: 91%

“…The right and left Cauchy-Green strain tensors are denoted by C = F T F and B = FF T , respectively. The principal invariants of C (or B) are given by Combining both the one-dimensional fractional linear constitutive equation of standard linear solid (SLS) and the concept of dual variables [64], Bouzidi et al [63] obtained the following constitutive equation…”

Section: Presentation Of the Modelmentioning

confidence: 99%

“…In the present work, we have developed the integral version of the constitutive equation, i.e. equation ( 24) of Bouzidi et al [63], in which the thixotropy was not accounted for. To model the Payne effect, the so-called intrinsic time or shift time concept was used, which arises from a change of the material microstructure.…”

Section: Introductionmentioning

confidence: 99%

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Modeling Payne effect on basis of linearization of a visco-hyperelastic model

Bouzidi¹,

Bechir²

2022

Modelling Simul. Mater. Sci. Eng.

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The present work concerns the modeling of the Payne effect in nonlinear viscoelasticity. This effect is a characteristic property of filled elastomers. Indeed, under cyclic loading of increasing amplitude, a decrease is shown in the storage modulus and a peak in the loss modulus. In this study, the Payne effect is assumed to arise from a change of the material microstructure, i.e., the thixotropy. The so-called intrinsic time or shift time was inferred from solving a differential equation that represents the evolution of a material's microstructure. Then, the physical time is replaced by the shift time in the framework of a recent fractional visco-hyperelastic model, which was linearized in the neighborhood of a static pre-deformation. As a result, we have investigated the effects of static pre-deformation, frequency, and magnitude of dynamic strain on storage and loss moduli in the steady state. Thereafter, the same set of parameters identified from the complex Young's modulus was used to predict the stress in the pre-deformed configuration. Finally, it is demonstrated that the proposed model is reasonably accurate in predicting Payne effect.

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Section: Which Can Be Neglected In Comparison Tomentioning

confidence: 91%

Section: Presentation Of the Modelmentioning

confidence: 99%

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Modeling Payne effect on basis of linearization of a visco-hyperelastic model

Bouzidi¹,

Bechir²

2022

Modelling Simul. Mater. Sci. Eng.

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“…A fractional-order model of anomalous cosmic ray diffusion with a finite velocity of free particle motion is considered in [34]. An efficient fractionalderivative based model for the prediction of multiaxial visco-hyperelastic behavior of elastomers is constructed in [8]. Fractional dynamical systems with types of attractors that are distinct from attractors of integer-order systems are considered in [10].…”

Section: Introductionmentioning

confidence: 99%

An Operator-Based Approach for the Construction of Closed-Form Solutions to Fractional Differential Equations

Navickas

Telksnys

Timofejeva

et al. 2018

Mathematical Modelling and Analysis

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An operator-based approach for the construction of closed-form solutions to fractional differential equations is presented in this paper. The technique is based on the analysis of Caputo and Riemann-Liouville algebras of fractional power series. Explicit solutions to a class of linear fractional differential equations are obtained in terms of Mittag-Leffler and fractionally-integrated exponential functions in order to demonstrate the viability of the proposed technique.

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“…The stress‐strain relation of a visco‐hyperelastic model is formulated using a convolution integral as in the case of small‐strain viscoelasticity . The visco‐hyperelastic material model is widely used to characterize the viscoelastic behavior of materials in the finite‐strain regime .…”

Section: Introductionmentioning

confidence: 99%

Closed‐form stress solutions for incompressible visco‐hyperelastic solids in uniaxial extension

Kossa

2017

Z Angew Math Mech

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This paper is concerned with the theory of incompressible visco-hyperelastic solids. Closed-form stress solutions are derived for five different visco-hyperelastic material models in uniaxial extension. Two loading scenarios are considered: uniaxial extension with constant true strain-rate; uniaxial extension with constant engineering strain-rate. The obtained stress solutions are expressed using only elementary functions, when the applied stretch is linear in the true strain. However, for the constant engineering strain-rate case, all the solutions are expressed using special functions, even for the very simple Hencky's hyperelastic material model. The novel solutions can be used to improve the performance and accuracy of the parameter-fitting procedure of a particular visco-hyperelastic solid. In addition, the closed-form expressions for the stresses may serve reference solution for benchmark problems.

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Phenomenological isotropic visco-hyperelasticity: a differential model based on fractional derivatives

Cited by 8 publications

References 70 publications

Modeling Payne effect on basis of linearization of a visco-hyperelastic model

Modeling Payne effect on basis of linearization of a visco-hyperelastic model

An Operator-Based Approach for the Construction of Closed-Form Solutions to Fractional Differential Equations

Closed‐form stress solutions for incompressible visco‐hyperelastic solids in uniaxial extension

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