Application of fractional calculus methods to viscoelastic behaviours of solid propellants

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190291 ◽  
Author(s):  
Changqing Fang ◽  
Xiaoyin Shen ◽  
Kuai He ◽  
Chao Yin ◽  
Shasha Li ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Complex Modulus ◽  
Viscoelastic Model ◽  
Relaxation Modulus ◽  
Advanced Materials ◽  
Solid Propellants ◽  
Static Modulus ◽  
Materials Modelling ◽  
Viscoelastic Constitutive Model

A three-branch viscoelastic model based on fractional derivatives is proposed for the viscoelastic behaviours of solid propellants. The simulation results show a satisfactory agreement with the stress relaxation modulus and complex modulus of solid propellants. As a comparison, the static modulus is also characterized by traditional viscoelastic model with integer-order derivatives. Results show that the application of the fractional derivatives to the viscoelastic constitutive model can effectively reduce the number of the required parameters while giving an accurate prediction of viscoelastic behaviours of solid propellants. Moreover, a simple and effective direct search method based on simulated annealing and Powell's mothed is proposed for the data fitting. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives'.

Estimating the ‘In-Service’ Modulus of Elasticity and Length of Polyester Mooring Lines via a Non Linear Viscoelastic Model Governed by Fractional Derivatives

2012 ◽  
Author(s):  
Georgios I. Evangelatos ◽  
Pol D. Spanos
Keyword(s):  
Modulus Of Elasticity ◽  
Field Data ◽  
Fractional Derivatives ◽  
Viscoelastic Model ◽  
Mooring Lines ◽  
Static Modulus ◽  
Proposed Model ◽  
Non Linear ◽  
Linear Viscoelastic ◽  
Linear Viscoelastic Model

In this paper a non linear viscoelastic model governed by fractional derivatives is presented for modeling the in-service behavior of polyester mooring lines. In the formulation an iterative approach utilizing the Gauss-Newton minimization algorithm in conjunction with the catenary equations used to determine the static modulus of elasticity and the effective length of polyester mooring lines corresponding to calm sea conditions. Upon establishing the accuracy of the static modulus via comparison with field data, the catenary equations and the offshore platform’s position versus time are used to identify the polyester strain under developed-sea conditions. In this manner, time histories of stress and strain for polyester ropes in service conditions are obtained. Then, a non linear viscoelastic model involving fractional derivative terms is used to capture the in service polyester line behavior. For this, the tension of the proposed model corresponding to the actual polyester strain is compared at each time step to the tension obtained from the field data. Finally, the parameters of the proposed model are derived by minimizing the error in the least-squares sense over a large number of data points using the Levenberg-Marquardt algorithm. The numerically derived force-strain relationship is found to be in reasonable agreement with supplementary field and laboratory experimental data, the field data pertain to an offshore structure moored in position using polyester mooring lines operated in the Gulf of Mexico during Hurricane Katrina (August of 2005).

A Theory Relating Creep and Relaxation for Linear Materials With Memory

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
R. C. Koeller
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Creep Compliance ◽  
Complex Modulus ◽  
Relaxation Times ◽  
Relaxation Modulus ◽  
Experimental Methods ◽  
Generalized Function ◽  
Materials With Memory ◽  
With Memory

The purpose of this paper is to suggest a linear theory of materials with memory, which gives a description for the similarities resulting when the various analytical and experimental methods used to reduce the creep and relaxation data are imposed on the observational changes in curvature that take place in both the creep compliance and relaxation modulus graphs. On a Log-Log graph both have one, two, or at most three pairs of changes in curvature depending on whether the material is a fluid or solid. These changes in curvature have been observed in many experiments and various regions have been discussed and classified. Section 1 gives a few of the many applications of fractional calculus to physical problems. In Sec. 2 an equation that contains both integration and differentiation is presented using geometrical observations about the relationship between the changes in curvature in the relaxation modulus and creep compliance based on published experiments. In Sec. 3 the generalized function approach to fractional calculus is given. In Sec. 4 a mechanical model is discussed. This model is able to share experimental data between the creep and relaxation functions, as well as the real and imaginary parts of the complex compliance or the complex modulus. This theory shares information among these three experimental methods into a unifying theory for solid materials when the loads are within the linear range. Under a limiting case, this theory can account for flow so that the material need not return to its original shape after the load is removed. The theory contains one physical parameter, which is related to the speed of sound and a group of phenomenological parameters that are functions of temperature and the composition of the material. These phenomenological parameters are relaxation times and creep times. This theory differs from the classical polynomial constitutive equations for linear viscoelasticity. It is a special case of Rabotnov’s equations and Torvik and Bagley’s fractional calculus polynomial equations, but it imposes symmetry conditions on the stress and strain when the material is a solid. Sections 56 are comments and conclusions, respectively. No experimental results are given at this time since this paper presents the foundations of materials with memory as related to experimental data. The introduction of experimental data to fit this theory will result in the breakdown of an important part of this research.

scholarly journals Advanced materials modelling via fractional calculus: challenges and perspectives

2020 ◽  
Vol 378 (2172) ◽  
pp. 20200050
Author(s):  
Giuseppe Failla ◽  
Massimiliano Zingales
Keyword(s):  
Fractional Calculus ◽  
Long Memory ◽  
Experimental Studies ◽  
Advanced Materials ◽  
Fractional Operators ◽  
Theme Issue ◽  
Materials Modelling ◽  
Local Media ◽  
Non Local ◽  
Complex Phenomena

Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Fractional-order modelling of epoxy resin

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190292 ◽  
Author(s):  
J. A. Tenreiro Machado ◽  
António M. Lopes ◽  
Rui de Camposinhos
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Epoxy Resins ◽  
Clustering Algorithm ◽  
Electrical Impedance Spectroscopy ◽  
Advanced Materials ◽  
Mathematical Tool ◽  
Materials Modelling ◽  
Hierarchical Clustering Algorithm ◽  
Two Stages

This paper describes epoxy resins by means of electrical impedance spectroscopy (EIS) and the mathematical tool of fractional calculus (FC). Two stages are considered: first, the EIS is used for testing the samples and, second, the measured data are approximated using integer and fractional order models. The FC-based modelling describes the epoxy resins using a small number of parameters that reflect their main characteristics. The EIS data gathered for the epoxies samples are compared with those of different adhesives and sealants by means of a hierarchical clustering algorithm that unravels the relationships between the distinct materials. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Study on Transforming Dynamic Modulus Master Curve into Static Relaxation Modulus Master Curve Basing on the Relaxation Time Spectrum of Asphalt Mixture

2013 ◽  
Vol 321-324 ◽  
pp. 268-272
Author(s):  
Feng Xia Chi ◽  
Li Jian Wang ◽  
Xiao Ning Zhang
Keyword(s):  
Relaxation Time ◽  
Dynamic Modulus ◽  
Master Curve ◽  
Complex Modulus ◽  
Asphalt Mixture ◽  
Relaxation Modulus ◽  
Time Spectrum ◽  
Relaxation Time Spectrum ◽  
Static Modulus ◽  
Static Relaxation

In order to transform dynamic complex modulus into static relaxation modulus, an advanced Shear Rheometer is used to get complex modulus master curve of asphalt mixture. Dynamic modulus master curve is transformed into static modulus master curve basing on the relaxation time spectrum. Results show it is feasible that the dynamic modulus master curve can be transformed into relaxation modulus master curve in wide frequency domain basing on the relaxation time spectrum.

scholarly journals On the thermodynamical restrictions in isothermal deformations of fractional Burgers model

2020 ◽  
Vol 378 (2172) ◽  
pp. 20190278 ◽  
Author(s):  
Teodor M. Atanacković ◽  
Marko Janev ◽  
Stevan Pilipović
Keyword(s):  
Constitutive Equations ◽  
Fractional Derivatives ◽  
Qualitative Difference ◽  
Weak Form ◽  
Entropy Inequality ◽  
Advanced Materials ◽  
Creep Function ◽  
Theme Issue ◽  
Burgers Model ◽  
Materials Modelling

We investigate, in the distributional setting, the restrictions on the constitutive equation for a fractional Burgers model of viscoelastic fluid that follow from the weak form of the entropy inequality under isothermal conditions. The results are generalized, from the Burgers model, to an arbitrary class of linear constitutive equations with fractional derivatives. Our results show that the restrictions obtained here on the coefficients of constitutive equations are weaker when compared with the restrictions obtained by Bagley–Torvik method. We show the precise relation between restrictions derived here and those derived by Bagley–Torvik. We deal with the creep test, for the case when Bagley–Torvik conditions are violated, and new conditions obtained in this work are satisfied. The results show a qualitative difference in the form of creep function. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

STRESS RELAXATION OF MAGNETORHEOLOGICAL FLUIDS

2002 ◽  
Vol 16 (17n18) ◽  
pp. 2655-2661
Author(s):  
W. H. LI ◽  
G. CHEN ◽  
S. H. YEO ◽  
H. DU
Keyword(s):  
Stress Relaxation ◽  
Viscoelastic Model ◽  
Relaxation Modulus ◽  
Damping Function ◽  
Mr Fluids ◽  
Large Step ◽  
Linear Viscoelastic ◽  
Predicted Values ◽  
Two Parameters ◽  
Step Strain

In this paper, the experimental and modeling study and analysis of the stress relaxation characteristics of magnetorheological (MR) fluids under step shear are presented. The experiments are carried out using a rheometer with parallel-plate geometry. The applied strain varies from 0.01% to 100%, covering both the pre-yield and post-yield regimes. The effects of step strain, field strength, and temperature on the stress modulus are addressed. For small step strain ranges, the stress relaxation modulus G(t,γ) is independent of step strain, where MR fluids behave as linear viscoelastic solids. For large step strain ranges, the stress relaxation modulus decreases gradually with increasing step strain. Morever, the stress relaxation modulus G(t,γ) was found to obey time-strain factorability. That is, G(t,γ) can be represented as the product of a linear stress relaxation G(t) and a strain-dependent damping function h(γ). The linear stress relaxation modulus is represented as a three-parameter solid viscoelastic model, and the damping function h(γ) has a sigmoidal form with two parameters. The comparison between the experimental results and the model-predicted values indicates that this model can accurately describe the relaxation behavior of MR fluids under step strains.

On Fractional Hamilton Formulation Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Eqab M. Rabei
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Fractional Integration ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Caputo Fractional Derivatives ◽  
Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

scholarly journals Study on Relaxation Damage Properties of High Viscosity Asphalt Sand under Uniaxial Compression

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Yazhen Sun ◽  
Zhangyi Gu ◽  
Jinchang Wang ◽  
Chenze Fang ◽  
Xuezhong Yuan
Keyword(s):  
Uniaxial Compression ◽  
Viscoelastic Model ◽  
Relaxation Modulus ◽  
Maxwell Model ◽  
High Viscosity ◽  
Secondary Development ◽  
Compression Tests ◽  
Crack Control ◽  
Different Temperatures ◽  
Damage Process

Laboratory investigations of relaxation damage properties of high viscosity asphalt sand (HVAS) by uniaxial compression tests and modified generalized Maxwell model (GMM) to simulate viscoelastic characteristics coupling damage were carried out. A series of uniaxial compression relaxation tests were performed on HVAS specimens at different temperatures, loading rates, and constant levels of input strain. The results of the tests show that the peak point of relaxation modulus is highly influenced by the loading rate in the first half of an L-shaped curve, while the relaxation modulus is almost constant in the second half of the curve. It is suggested that for the HVAS relaxation tests, the temperature should be no less than −15°C. The GMM is used to determine the viscoelastic responses, the Weibull distribution function is used to characterize the damage of the HVAS and its evolution, and the modified GMM is a coupling of the two models. In this paper, the modified GMM is implemented through a secondary development with the USDFLD subroutine to analyze the relaxation damage process and improve the linear viscoelastic model in ABAQUS. Results show that the numerical method of coupling damage provides a better approximation of the test curve over almost the whole range. The results also show that the USDFLD subroutine can effectively predict the relaxation damage process of HVAS and can provide a theoretical support for crack control of asphalt pavements.

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