Creep constitutive models for viscoelastic materials based on fractional derivatives

Xu, Huanying; Jiang, Xiaoyun

doi:10.1016/j.camwa.2016.05.002

Cited by 112 publications

(54 citation statements)

References 22 publications

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“…At this moment, we cease the discussion on the Kohlrausch function since it is not so popular in the polymer rheology even though some successful applications reported [152].…”

Section: Kohlrausch Function As a Memory Kernel In A Fractional Operatormentioning

confidence: 99%

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Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

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“…At this moment, we cease the discussion on the Kohlrausch function since it is not so popular in the polymer rheology even though some successful applications reported [152].…”

Section: Kohlrausch Function As a Memory Kernel In A Fractional Operatormentioning

confidence: 99%

“…Let us consider a fractional model consisting two fractional elements arranged in series. Then, the stress-strain constitutive equation coming from the Scott-Blair models [130][131][132] should be [152]…”

Section: An Example With Spring-dashpot Elements: Two Asymptotic Casesmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

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“…Celauro et al [11] proposed a fractional derivative model to describe the viscoelastic behaviour of asphalt mixtures. In addition, some scholars [12][13][14][15][16][17] have developed relevant creep models using fractional calculus. Although fractional calculus is widely adopted in developing creep models, a key limitation is its inability to describe the accelerated creep phase of geotechnical materials; in terms of the mathematical description of the obtained curves, fractional calculus cannot suitably describe the accelerated creep stage of rocks.…”

Section: Introductionmentioning

confidence: 99%

New Maxwell Creep Model Based on Fractional and Elastic‐Plastic Elements

Tang

Wang

Duan

2020

Advances in Civil Engineering

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Creep models are mainly used to describe the rheological behaviour of geotechnical materials. An important research focus for studying creep in geotechnical materials is the development of a model with few parameters and good simulation performance. Hence, in this study, by replacing the Newtonian dashpot and spring in the classical Maxwell model with fractional and elasticplastic elements, a new Maxwell creep model based on fractional derivatives and continuum damage mechanics was developed. One-and three-dimensional (1D/3D) creep equations of the new Maxwell creep model were derived. e 1D creep equation of the new model was used to fit existing creep data of rock salt, and the 3D creep equation was used to fit the creep data of remolded loess. e model curves matched the creep data very well, showing considerably higher accuracy than other models. Furthermore, a sensitivity study was carried out, showing the effects of the fractional derivative order β and exponent α on the creep strain of rock salt.is new model is simple with few parameters and can effectively simulate the complete creep behaviour of geotechnical materials.

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“…Although Riemann-Liouville, Caputo, and Grunwald-Letnikov fractional derivatives [21][22][23][24][25][26][27][28] are widely used in physics, mathematics, medicine, economics, and engineering as shown above, these derivative definitions lack some of the agreed properties for classical differential operator, such as the chain rule. The conformable derivative can be regarded as a natural extension of the classical differential operator, which satisfies most important properties, such as the chain rule [29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk

et al. 2019

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A new chaotic financial system is proposed by considering ethics involvement in a fourdimensional financial system with market confidence. A five-dimensional conformable derivative financial system is presented by introducing conformable fractional calculus to the integerorder system. A discretization scheme is proposed to calculate numerical solutions of conformable derivative systems. The scheme is illustrated by testing hyperchaos for the system. among interest rates, investments, prices, and savings. Chen [12] presented a fractional form of nonlinear financial system. Wang, Huang and Shen [13] established an uncertain fractional-order form of the financial system. Mircea et al. [14] set up a delayed form of the financial system. Xin, Chen, and Ma [15] proposed a discrete form of financial system. Yu, Cai, and Li [16] extended the

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Creep constitutive models for viscoelastic materials based on fractional derivatives

Cited by 112 publications

References 22 publications

Response functions in linear viscoelastic constitutive equations and related fractional operators

Response functions in linear viscoelastic constitutive equations and related fractional operators

New Maxwell Creep Model Based on Fractional and Elastic‐Plastic Elements

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk

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