Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations

The fractional viscoelastic equation (FVE), which is a second-order differential equation with fractional derivatives describing the dynamical behavior of a single-degree-of-freedom viscoelastic oscillator, is considered. Some viscoelastic damped mechanical systems may be described by FVEs. However, FVEs with conventional nonzero initial values cannot generally be solved. In this paper, the prehistories of the unknown functions before the initial times, referred to as the initial functions, are taken into account to solve FVEs. Mathematically, appropriate initial functions are essential for unique solutions of FVEs. Physically, the initial functions reflect the processes of giving the initial values. FVEs are solved for some initial functions both by analytical and numerical methods. The initial functions affect the solutions of FVEs. It is discussed how the solutions depend on the initial functions. Implication of the solutions to viscoelastic materials will be discussed.     

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Fractional differential equations are more and more used in modeling memory(history-dependent,nonlocal,or hereditary) phenomena.Conventional initial values of fractional differential equations are define at a point,while recent works defin initial conditions over histories.We prove that the conventional initialization of fractional differential equations with a Riemann–Liouville derivative is wrong with a simple counter-example.The initial values were assumed to be arbitrarily given for a typical fractional differential equation,but we fin one of these values can only be zero.We show that fractional differential equations are of infinit dimensions,and the initial conditions,initial histories,are define as functions over intervals.We obtain the equivalent integral equation for Caputo case.With a simple fractional model of materials,we illustrate that the recovery behavior is correct with the initial creep history,but is wrong with initial values at the starting point of the recovery.We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry

The paper is concerned with analysis of time-fractional diffusion-wave equation with Caputo fractional derivative in a half-space. Several examples of problems with Dirichlet and Neumann conditions at the boundary of a half-space are solved using integral transforms technique. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Numerical results are presented graphically for various values of order of fractional derivative.     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.     

Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders

Analytical solutions to rotating functionally graded hollow and solid long cylinders are developed. Young＇s modulus and material density of the cylinder are assumed to vary exponentially in the radial direction, and Poisson＇s ratio is assumed to be constant. A unified governing equation is derived from the equilibrium equations, compatibility equation, deformation theory of elasticity and the stress-strain relationship. The governing second-order differential equation is solved in terms of a hypergeometric function for the elastic deformation of rotating functionally graded cylinders. Dependence of stresses in the cylinder on the inhomogeneous parameters, geometry and boundary conditions is examined and discussed. The proposed solution is validated by comparing the results for rotating functionally graded hollow and solid cylinders with the results for rotating homogeneous isotropic cylinders. In addition, a viscoelastic solution to the rotating viscoelastic cylinder is presented, and dependence of stresses in hollow and solid cylinders on the time parameter is examined.     

Exact Averaging of Stochastic Equations for Transport in Random Velocity Field

We present new examples of exactly averaged multi-dimensional equation of transport of a conservative solute in a time-dependent random flow velocity field. The functional approach and a technique for decoupling the correlations are used. In general, the averaged equation is non-local. We study the special cases where the averaged equation can be localized and reduced to a differential equation of finite-order, where the problem of evolution of the initial plume (Cauchy problem) can be solved exactly. We present in detail the results of the analyses of two cases of exactly averaged problems for Gaussian and telegraph random velocity with an identical exponential correlation function, which are informative and convenient models for continuous and discontinuous random functions. The problems in which the field has sources of solute and boundaries are also examined. We study the behavior of different initial plumes for all times (evolutions and convergence) and show the manner in which they approach the same asymptotic limit for two stochastic distributions of flow-velocity. A comparison between exact solutions and solutions derived by the method of perturbation is also discussed.     

Stability analysis of a fractional viscoelastic plate strip in supersonic flow under axial loading

The stability of a viscoelastic plate strip, subjected to an axial load with the Kelvin–Voigt fractional order constitutive relationship is studied. Based on the classical plate theory, the structural formulation of the plate is obtained by using the Newton’s second law and the aerodynamic force due to the fluid flow is evaluated by piston theory. The Galerkin method is employed to discretize the equation of motion into a set of ordinary differential equations. To determine the stability margin of plate the obtained set of ordinary differential equations are solved using the Laplace transform method. The effects of variation of the governing parameters such as axial force, retardation time, fractional order and boundary conditions on the stability margin of fractional viscoelastic panel are investigated and finally some conclusions are outlined.     

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

黏弹性准饱和土中球空腔动力特性

在频率域内研究了内水压力作用下分数导数型黏弹性准饱和土中球空腔的稳态动力响应. 通 过引入与孔隙率有关的应力系数合理地确定了介质和孔隙水共同承担的内水压力值. 将土骨 架和衬砌分别视为具有分数导数本构模型的黏弹性体和多孔柔性材料, 基于Biot两相介质模 型, 通过引入位移势函数解耦得到了内水压力作用下分数导数型黏弹性准饱和土中半封闭球 形空腔的位移、应力和孔隙水压力解析表达式. 考察了物性和几何各参数对球形空腔动力响 应的影响, 结果表明: 分数导数本构模型更合理地描述了土体的动力学行为; 饱和度对应力 和孔隙水压力影响较大, 而对位移影响较小.     

Distributed-order fractional wave equation on a finite domain: creep and forced oscillations of a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of distributed fractional order type for the special choice of weight functions. Prescribing boundary conditions on displacement and stress, we obtain, as special solutions, cases corresponding to creep and forced oscillations. In solving system of differential and integro-differential equations, we use the Laplace transformation in the time domain.     

粘弹性固体的精细积分有限元算法

粘弹性固体本构方程的数学表达式分为微分型和积分型两种，其数值求解主要是时域上离散计算。文中从微分型表达式出发导出其状态空间方程的数学表达式，通过严格推导论证了它与微、积分型表达式的等价性；引入状态空间方程，从而利用精细积分格式来求解粘弹性固体本构方程；给出了粘弹性固体本构方程的精细积分有限元算法，为求解粘弹性固体本构方程的数值解提供了一个新的途径，具有计算简便，求解精度高等优点。     

层状分数阶黏弹性饱和地基与梁共同作用的时效研究

饱和地基与梁共同作用问题的研究在力学领域及工程界都具有重要意义.采用分数阶Merchant模型研究饱和地基的流变固结,该模型比常用整数阶黏弹性模型更能精确反映地基的时变特征.基于层状正交各向异性黏弹性饱和地基的固结解答,采用有限元法与边界元法耦合的方法,研究梁与分数阶黏弹性饱和地基的共同作用问题.依据Timoshenk...     

Dynamic analysis of beams on viscoelastic foundation

This study is intended to analyze dynamic behavior of beams on Pasternak-type viscoelastic foundation subjected to time-dependent loads. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate exactly the dynamic stiffness matrix of the problem. The solutions obtained are transformed to the real space using the Durbin's numerical inverse Laplace transform method. The dynamic response of beams on viscoelastic foundation is analyzed through various examples.     

分数导数型本构关系描述粘弹性梁的振动分析

本文研究粘弹性梁在周期激励作用下的受迫振动问题。梁的材料满足Kelvin-Volgt分数导数型本构关系。基于动力学方程、本构关系和应变－位移关系建立了小变形粘弹性梁的振动方程。采用分离变量法分析粘弹性梁的自由振动,导出模态坐标满足的常微分－积分方程和模态函数满足的常微分方程,对于两端简支的截面梁给出了固有频率和模态函数。对于简谐激励作用下粘弹性梁的受迫振动,利用模态叠加得到了稳态响应。最后给出数值算例说明本文方法的应用。     

Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.The project supported by the National Natural Science Foundation of China (10272067, 10426024), the Doctoral Program Foundation of the Education Ministry of China (20030422046) and the Natural Science Foundation of Shandong University at Weihai. The English text was polished by Keren Wang.     

The analyticity of solutions of the Stefan problem

The Stefan problem of a semi-infinite body with arbitrarily prescribed initial and boundary conditions is studied. One of the objectives of the paper is to investigate the analyticity of the solutions. For this purpose, the prescribed initial and boundary conditions are considered to be series of fractional powers of their arguments. It is found that the exact solutions of the problem for various forms of the initial and boundary conditions can be established in series of parabolic cylinder functions and time t. Existence and convergence of the series solutions are studied and proved. The present solutions include the known exact solutions as special cases. On the basis of the present solutions, the question of the analyticity of solutions of the Stefan problem, raised by Rubinstein in his book, can be answered. Conditions for analyticity of the solutions with various initial and boundary conditions are fully discussed.     

Numerical-analytic investigation of the dynamics of viscoelastic and porous elastic bodies

This paper presents the results of mathematical and discrete modeling of linear dynamics problems for three-dimensional viscoelastic and porous elastic bodies. The employed methods and approaches are based on formulating boundary integral equations solved using boundary elements. The model of a standard viscoelastic body is employed as the viscoelastic model. The properties of porous elastic materials are described using the full Biot model with four basic functions. Examples of numerical solutions of the problems are compared with known results of solutions.     

A Method for Determining the Viscoelastic Characteristics of Composites

An efficient method for determining the deformation function of a composite is discussed. The method is based on a fractional exponential representation of the deformation functions of the composite components. The viscoelastic solution is obtained using the Volterra principle. The deformation function is represented as a function of a base operator. Thus, the problem is solved by approximating the deformation function by a continued fraction and applying the method of operator continued fractions. A computational procedure is detailed and illustrated using data on longitudinal relaxation of polymethylmethacrylate. As an example, the deformation of a polymethylmethacrylate-based fibrous composite with viscoelastic properties is analyzed__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 9–21, May 2005.