Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body

2014 ◽  
pp. 49-82
Author(s):  
Teodor M. Atanacković ◽  
Stevan Pilipović ◽  
Bogoljub Stanković ◽  
Dušan Zorica
Keyword(s):  
Fractional Derivative ◽  
Viscoelastic Body

Nonlinear Statical and Dynamical Models of Fractional Derivative Viscoelastic Body

2005 ◽  
Author(s):  
H. Nasuno ◽  
N. Shimizu
Keyword(s):  
Fractional Derivative ◽  
Power Function ◽  
Exponential Function ◽  
Theoretical Consideration ◽  
Viscoelastic Body ◽  
Nonlinear Behavior ◽  
Dynamical Models ◽  
Dynamical Behaviors ◽  
Viscoelastic Coefficient ◽  
Sinusoidal Excitation

The authors have been conducting experiments on the investigation of nonlinear quasi-statical and dynamical behaviors of a viscoelastic body described by the fractional derivative law. Pre-stress due to pre-displacement induces high damping performance during sinusoidal excitation. To understand this behavior, nonlinear statical and dynamical models are investigated by theoretical consideration. The authors establish and propose appropriate models to describe the nonlinear behavior of the fractional derivative viscoelastic body. The nonlinearity of the viscoelastic coefficient for quasi-statical compressive displacement may be described by the power function with respect to pre-displacement and the nonlinearity of the viscoelastic coefficient for sinusoidal excitation may be described by the exponential function with respect to pre-displacement. Some discussions on the values of the viscoelastic coefficients.

Dynamic Modeling of Viscoelastic Body in Multibody System

2011 ◽  
Vol 105-107 ◽  
pp. 587-594
Author(s):  
Da Zhi Cao ◽  
Zhi Hua Zhao ◽  
Ge Xue Ren
Keyword(s):  
Fractional Derivative ◽  
Multibody System ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Time Integration ◽  
Three Dimensional ◽  
Viscoelastic Body ◽  
Integration Scheme ◽  
Large Rotation ◽  
Time Integration Scheme

Dynamic equations of viscoelastic bodies with fractional constitutive are derived base on the principle of virtual work and the theory of continuum mechanics. The three-dimensional fractional derivative viscoelastic constitutive model is implemented into the flexible multibody system (FMBS), using the 3D solid element based on the absolute nodal coordinate formulation (ANCF), which can exactly describe the geometric nonlinearities due to large rotation and large deformation. The BDF time integration scheme in conjunction with the Grünwald approximation of fractional derivative and the Newton-Raphson algorithm are used to solve the equations of motion. Several numerical examples are presented to demonstrate the use of the modeling procedure presented in this investigation and the effects of parameters in the fractional derivative model.

scholarly journals On a fractional derivative type of a viscoelastic body

2002 ◽  
pp. 27-38 ◽  
Author(s):  
Teodor Atanackovic ◽  
Branislava Novakovic
Keyword(s):  
Stress State ◽  
Fractional Derivative ◽  
Constitutive Equations ◽  
Viscoelastic Body ◽  
Special Cases ◽  
Derivative Type

We study a viscoelastic body, in a linear stress state with fractional derivative type of dissipation. The model was formulated in [1]. Here we derive restrictions on the model that follow from Clausius-Duhem inequality. Several known constitutive equations are derived as special cases of our model. Two examples are discussed. .

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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