Fractional Derivative Consideration on Nonlinear Viscoelastic Statical and Dynamical Behavior under Large Pre-Displacement

2007 ◽  
pp. 363-376 ◽  
Author(s):  
Hiroshi Nasuno ◽  
Nobuyuki Shimizu ◽  
Masataka Fukunaga
Keyword(s):  
Fractional Derivative ◽  
Dynamical Behavior ◽  
Nonlinear Viscoelastic

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

2005 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu ◽  
Hiroshi Nasuno
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Dynamical Behavior ◽  
Variable Coefficients ◽  
Rate Of Change ◽  
Order Derivative ◽  
Nonlinear Viscoelastic ◽  
Higher Order Derivative ◽  
The Difference ◽  
Different Order

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.

A nonlinear viscoelastic fractional derivative model of infant hydrocephalus

2011 ◽  
Vol 217 (21) ◽  
pp. 8693-8704 ◽  
Author(s):  
K.P. Wilkie ◽  
C.S. Drapaca ◽  
S. Sivaloganathan
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivative Model ◽  
Nonlinear Viscoelastic

Dynamic analysis of a fractional order system

2015 ◽  
Vol 08 (06) ◽  
pp. 1550079
Author(s):  
M. Javidi ◽  
N. Nyamoradi
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Dynamic Analysis ◽  
Fractional Order ◽  
Local Stability ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Order System ◽  
Stability Conditions ◽  
Hurwitz Stability

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Basic Assumption ◽  
Dynamical Behavior ◽  
Impact Dynamics ◽  
Viscoelastic Materials ◽  
Free Response ◽  
Scale Free ◽  
Collective Response ◽  
Elementary Constituent ◽  
Entropy Elasticity

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Finite Deformation ◽  
Dynamical Behavior ◽  
Strain Tensor ◽  
Constitutive Models ◽  
Cauchy Stress ◽  
Viscoelastic Materials ◽  
Kirchhoff Stress Tensor ◽  
Biot Stress

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

scholarly journals Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Salih Djilali ◽  
Behzad Ghanbari
Keyword(s):  
Fractional Derivative ◽  
Dynamical Behavior ◽  
Time Derivative ◽  
Interaction Model ◽  
Integration Rule ◽  
Existence And Stability ◽  
Natural Result ◽  
Biological Environment ◽  
Fractional Time Derivative ◽  
The Impact

AbstractThe behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

2011 ◽  
Vol 189-193 ◽  
pp. 3391-3394 ◽  
Author(s):  
Qing Zhao Yao ◽  
Lin Chao Liu ◽  
Qi Fang Yan
Keyword(s):  
Fractional Derivative ◽  
Mechanical Behavior ◽  
Constitutive Relations ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Bernoulli Beam ◽  
The Laplace Transform ◽  
Static Mechanical ◽  
Euler Bernoulli Beam

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

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