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. .

Dynamics of a Fractional Derivative Type of a Viscoelastic Rod With Random Excitation

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Teodor Atanacković ◽  
Marko Nedeljkov ◽  
Stevan Pilipović ◽  
Danijela Rajter-Ćiri
Keyword(s):  
Fractional Derivative ◽  
Weak Solutions ◽  
Constitutive Equations ◽  
Fractional Derivatives ◽  
Random Excitation ◽  
Bounded Noise ◽  
White Noise Process ◽  
Derivative Type ◽  
Axial Vibrations ◽  
Viscoelastic Rod

AbstractThe axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations as well as their stochastic moments are determined.

On a new class of constitutive equations for linear viscoelastic body

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Diana Dolićanin-Đekić
Keyword(s):  
Constitutive Equation ◽  
Constitutive Equations ◽  
Viscoelastic Body ◽  
Sufficient Conditions ◽  
Second Law Of Thermodynamics ◽  
Weak Form ◽  
New Class ◽  
Special Cases ◽  
Linear Viscoelastic ◽  
Derivatives Of

AbstractWe study a viscoelastic body involving a constitutive equation with distributed order fractional derivatives of complex order. Using a dissipation inequality in a weak form, we derive a sufficient conditions on coefficients of a model that guarantee that the Second law of thermodynamics under isothermal conditions is satisfied. Several known constitutive equations follow from our model as special cases. As an application, a new constitutive equation is related to an equation of motion of a generalized linear oscillator.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals Determination of the Stress State of a Piecewise Homogeneous Elastic Body with a Row of Cracks on an Interface Surface Subject to Antiplane Strains with Inclusions at the Tips

2015 ◽  
Vol 2015 ◽  
pp. 1-22 ◽  
Author(s):  
Ali Golsoorat Pahlaviani ◽  
Suren Mkhitaryan
Keyword(s):  
Stress State ◽  
Elastic Body ◽  
Linear Equations ◽  
The Body ◽  
Shear Stresses ◽  
Interface Surface ◽  
Special Cases ◽  
Crack Problems ◽  
Fourier Sine Series

The stress state of a bimaterial elastic body that has a row of cracks on an interface surface is considered. It is subjected to antiplane deformations by uniformly distributed shear forces acting on the horizontal sides of the body. The governing equations of the problem, the stress intensity factors, the deformation of the crack edges, and the shear stresses are derived. The solution of the problem via the Fourier sine series is reduced to the determination of a singular integral equation (SIE) and consequently to a system of linear equations. In the end, the problem is solved in special cases with inclusions. The results of this paper and the previously published results show that the used approach based on the Gauss-Chebyshev quadrature method can be considered as a generalized procedure to solve the collinear crack problems in mode I, II, or III loadings.

Deriving Plastic Constitutive Equations of the Material in Complex Stress State by Means of Simple Test Results

2011 ◽  
Vol 109 ◽  
pp. 100-104 ◽  
Author(s):  
Xiao Jiu Feng ◽  
Li Fu Liang
Keyword(s):  
Stress State ◽  
Constitutive Equations ◽  
Complex Stress ◽  
Complex Stress State ◽  
One Dimension ◽  
Isotropic Hardening ◽  
Flow Rule ◽  
Test Results ◽  
Loading Function ◽  
Simple Tension

By conducting simple tension and torsion tests to material, constitutive equations of one dimension are obtained. Plastic theories of continuum mechanics are used for analyzing deformation behavior of the material after yielding. Here, material is presumed to have isotropic hardening characteristic. By using Mises loading function and the associative flow rule, the derivations are made to extend the constitutive equations of one dimension in the simple tension and torsion tests to that of multi-dimension and obtain the plastic constitutive equations of the material in complex stress state , respectively.

Dynamic Interaction of Viscoelastic Soil and Fractional Derivative Type Lining with a Circular Tunnel

2012 ◽  
Vol 170-173 ◽  
pp. 1542-1545
Author(s):  
Min Jie Wen ◽  
Zi Ping Su ◽  
Hui Tuan He
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Dynamic Interaction ◽  
Constitutive Behavior ◽  
Steady State Response ◽  
Circular Tunnel ◽  
Resonance Effects ◽  
Material Parameters ◽  
State Response ◽  
Derivative Type

Coupled harmonic vibration of viscoelastic soil and fractional derivative type lining system with a deeply buried circular tunnel is investigated in the frequency domain. Based on theory of elastic and fractional derivative, steady state response of the viscoelastic soil and lining system is studied. Regarding the lining as a medium with fractional derivative constitutive behavior, and the analytical expressins of the displacement and stress of the soil and lining are respectively obtained by the continuity conditions on the inner boundary of lining and the interface between the soil and the lining. The order of fractional derivative model has a greater influence on system dynamic response, and it dependent on the material parameters of lining. With the frequency increasing, the resonance effects of system decrease.

scholarly journals Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions

2021 ◽  
Vol 7 (2) ◽  
pp. 3006-3026
Author(s):  
Imran Abbas Baloch ◽  
◽  
Thabet Abdeljawad ◽  
Sidra Bibi ◽  
Aiman Mukheimer ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Convex Functions ◽  
Integral Identity ◽  
Fractional Derivatives ◽  
Order Derivative ◽  
Special Cases

<abstract><p>Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.</p></abstract>

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