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.

Viscoelasticity of Fractional Order: New Restrictions on Constitutive Equations with Applications

2020 ◽  
Vol 20 (13) ◽  
pp. 2041011
Author(s):  
Teodor M. Atanacković ◽  
Marko B. Janev ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Fractional Order ◽  
Constitutive Equations ◽  
Linear Viscoelasticity ◽  
Fractional Derivatives ◽  
Second Law Of Thermodynamics ◽  
Weak Form ◽  
Second Law ◽  
Isothermal Conditions ◽  
Complex Order ◽  
Derivatives Of

In this paper, we analyze the restrictions on the coefficients in the constitutive equations of linear Viscoelasticity that follow from the Second Law of Thermodynamics under isothermal conditions. Especially, we analyze the constitutive equations in which fractional derivatives of real and complex order appear. We present the conditions that follow after application of the Bochner–Schwartz theorem. Conditions derived here, representing in certain cases a weak form of the Second law of Thermodynamics, are more general (weaker) than the classical Bagley–Torvik conditions widely used in Viscoelasticity Theory. Several examples that illustrate the theory are presented.

scholarly journals Certain Subclasses of Analytic Multivalent Functions Associated with Petal-Shape Domain

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 291
Author(s):  
Lei Shi ◽  
Hari M. Srivastava ◽  
Muhammad Ghaffar Khan ◽  
Nazar Khan ◽  
Bakhtiar Ahmad ◽  
...  
Keyword(s):  
Analytic Functions ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
New Class ◽  
Special Cases ◽  
Petal Shape

In this article, we introduce a new class of multivalent analytic functions associated with petal-shape region. Furthermore, some useful properties, such as the Fekete–Szegö inequality, and their consequences for some special cases are discussed. For some specific value of function f, we obtain sufficient conditions for multivalent starlike functions connected with petal-shape domain. Finally, in the concluding section, we draw the attention of the interested readers toward the prospect of studying the basic or quantum (or q-) generalizations of the results, which are presented in this paper. However, the (p,q)-variations of the suggested q-results will provide a relatively minor and inconsequential development because the additional (rather forced-in) parameter p is obviously redundant.

The Effect of Elastic Modulus Rate in Stress Controlled Tests on the Modified Jeffrey’s Model for Structured Fluids

2015 ◽  
Vol 751 ◽  
pp. 102-108
Author(s):  
Tainan Gabardo Miranda dos Santos ◽  
Hilbeth Parente Azikri de Deus ◽  
Cezar Otaviano Ribeiro Negrão
Keyword(s):  
Numerical Simulation ◽  
Elastic Modulus ◽  
Constitutive Equation ◽  
Constitutive Equations ◽  
Coupled System ◽  
Structural Level ◽  
Kinetics Equation ◽  
Viscoelastic Models ◽  
Structured Fluids ◽  
Linear Viscoelastic

Structured fluids are known to be dependable of their structural level. Examples of such fluids may be found in different industries as chemical, biomedical, manufacturing, food and oil. The mathematical models to describe structured fluids are normally composed by a coupled system: one constitutive equation (based on viscoelastic models) and one kinetics equation (an equation which describes the structural level evolution in time of the material). The works found in the literature use linear viscoelastic constitutive equations which do not account the dependence of the elastic modulus with the microstructural level in their fundamental hypothesis. In this sense, the present work aims to evaluate, through numerical simulation, the effect of a new constitutive equation in rheological tests and compare its results to those of the model developed by Souza Mendes and Thompson (2013), in which those considerations are not made.

scholarly journals On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 264
Author(s):  
Mohamed I. Abbas ◽  
Maria Alessandra Ragusa
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Sufficient Conditions ◽  
New Class ◽  
Hybrid Fixed Point Theorem ◽  
Fractional Differential ◽  
The Given ◽  
Derivatives Of ◽  
Increasing Function

This paper deals with a new class of hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain continuously differentiable and increasing function ϑ. By means of a hybrid fixed point theorem for a product of two operators, an existence result is proved. Furthermore, the sufficient conditions of the continuous dependence on the given parameters are investigated. Finally, a simulative example is given to highlight the acquired outcomes.

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. .

Wave propagation dynamics in a fractional Zener model with stochastic excitation

2020 ◽  
Vol 23 (6) ◽  
pp. 1570-1604
Author(s):  
Teodor Atanacković ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Wave Propagation ◽  
Constitutive Equation ◽  
Equations Of Motion ◽  
Weak Form ◽  
Stochastic Excitation ◽  
Expansion Theorem ◽  
Zener Model ◽  
Dissipation Inequality ◽  
Regularity Properties ◽  
Fractional Zener Model

Abstract Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

scholarly journals Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

scholarly journals A New Class of Estimators Based on a General Relative Loss Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1138
Author(s):  
Tao Hu ◽  
Baosheng Liang
Keyword(s):  
Loss Function ◽  
Asymptotically Normal ◽  
New Class ◽  
Statistical Inferences ◽  
Relative Loss ◽  
Special Cases ◽  
Class Of Estimators ◽  
Box Cox Transformation ◽  
Loss Estimator ◽  
Prostate Cancer Study

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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