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.     

Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range

The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William–Landel–Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations.     

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.     

Fractional dynamical system and its linearization theorem

Nowadays, it is known that the solution to a fractional differential equation can’t generally define a dynamical system in the sense of semigroup property due to the history memory induced by the weakly singular kernel. But we can still establish the similar relationship between a fractional differential equation and the corresponding fractional flow under a reasonable condition. In this paper, we firstly present some results on fractional dynamical system defined by the fractional differential equation with Caputo derivative. Furthermore, the linearization and stability theorems of the nonlinear fractional system are also shown. As a byproduct, we prove Audounet–Matignon–Montseny conjecture. Several illustrative examples are given as well to support the theoretical analysis.     

Numerical solutions of differential equations with fractional L-derivative

Fractional differential equations are solved with L-fractional derivatives, using numerical procedures. Two characteristic fractional differential equations are numerically solved. The first equation describes the motion of a thin rigid plate immersed in a Newtonian fluid connected by a massless spring to a fixed point, and the other one the diffusion of gas in a fluid.     

A stability study of Navier-Stokes equations (III)

In this paper, the necessary conditions of the existence of C² solutions in some initial problems of Navier-Stokes equations are given, and examples of instability of initial value (at t=0) problems are also given. The initial value problem of Navier-Stokes equation is one of the most fundamental problem for this equation various authors studies this problem and contributed a number of results. J. Lerav, a French professor, proved the existence of Navier-Stokes equation under certain defined initial and boundary value conditions. In this paper, with certain rigorously defined key concepts, based upon the basic theory of J. Hadamard partial differential equations¹, gives a fundamental theory of instability of Navier-Stokes equations. Finally, many examples are given, proofs referring to Ref. [4].     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

Integration of third order ordinary differential equations possessing two-parameter symmetry group by Lie’s method

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: (1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). (2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.     

A note on the definition of fractional derivatives applied in rheology

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives—Riemann–Liouville (R–L) definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R–L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model (Scott–Blair model). We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R–L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R–L derivatives with lower terminal a →∞ (or, equivalently, the Caputo derivatives with lower terminal a →∞) not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott–Blair model and give an exact solution of this equation under given initial conditions.     

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.     

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.     

Nonlocal-in-time kinetic energy in nonconservative fractional systems,disordered dynamics,jerk and snap and oscillatory motions in the rotating fluid tube

In this article we generalize the basic theoretical properties of nonlocal-in-time kinetic energy approach introduced in the framework of nonlocal classical Newtonian mechanics for the case of fractional dynamical systems explored in the context of the fractional actionlike variational approach. Two independent fractionally Lagrangians weights are considered independently: the Riemann-Liouville fractional weight and the extended exponentially fractional weight. For each weight, the corresponding nonlocal fractional Newton's law of motion is derived. Three main physical applications were discussed in details: free particles, oscillators and dynamics of particles in a rotating tube with earth frame. A number of differential equations depending on fractional and nonlocal-in-time parameters were obtained and their solutions are discussed accordingly. For specific parameters and particular initial conditions, it was observed that the dynamics exhibit a kind of strange phase plot trajectories that indicate the presence of disordered motions. However one of the main results concerns the physics of particles in the rotating tube which display, for specific values of fractional and nonlocal-in-time parameters, oscillatory motions controlled by the nonlocal-in-time parameter.     

强非线性动力系统的频率增量法

提出一类强非线性动力系统的暧时频率增量法,将描述动力系统的二阶常微分方程,化为以相位为自变量、瞬廛频率为未知函数的积分方程;用谐波平衡原理,将求解瞬时频率的积分问题,归结为求解以频率增量的Fourier系数为独立变量的线性代数方程组;给出了若干例子。     

Exact solutions for some non-conservative hyperbolic systems

In this paper, we study initial value problem for some non-conservative hyperbolic systems of partial differential equations of first order. The first one is the Riemann problem for a model in elastodynamics and the second one the initial value problem for a system which is a generalization of the Hopf equation. The non-conservative products which appear in the equations do not make sense in the classical theory of distributions and are understood in the sense of Volpert (Math. USSR Sb. 2 (1967) 225). Following Lax (Comm. Pure Appl. Math. 10 (1957) 537) and Dal Maso et al. (J. Math. Pures Appl. 74 (1995) 483), we give an explicit solution for the Riemann problem for the elastodynamics equation. The coupled Hopf equation is studied using a generalization of the method of Hopf (Comm. Pure Appl. Math. 3 (1950) 201).     

Stability analysis of gradient elastic beams by the method of initial value

The buckling of a bar is studied analytically on the basis of a simple linear theory of gradient elasticity in the frame of the method of initial values. The method of initial values provides the values of the displacements and stress resultants throughout the bar once the initial displacements and initial stress resultants are known. We use probably for the first time the method of initial values to get critical loads of a strain gradient beam under completely different boundary conditions at the two end faces of the beam. Exact carryover matrix is presented for the classical beam and gradient beam analytically. The first mode shapes of classical beam and gradient beam are plotted. The method of initial values is also applied to the beams with variable cross-section. The priorities of the method of initial values are depicted. The variational approach gives a sixth-order ordinary differential equation for a beam in buckling. The additional boundary conditions are used to obtain critical loads. It is observed that critical loads increase dramatically for increasing values of the gradient coefficient.     

The contraction ratios of perfect elastoplasticity under biaxial controls

We studied contraction ratios, one rate form and one total form, of the Prandtl–Reuss model under combined axial and torsional controls. In the transition point of elasticity and plasticity, the rate form contraction ratio may undergo a discontinuous jump, which, depending on the control paths and initial stresses, may be positive, zero, or negative. For the total form contraction ratio, no similar jump phenomenon is observed in the elasticity–plasticity transition point. Depending on initial stresses both ratios may be greater than 1/2 . In the simulations of the axial–torsional strain control tests, the hoop and radial strains are not known a priori and hence can not be viewed as inputs. This greatly complicates the constitutive model analyses because the resulting differential equations become highly non-linear. To tackle this problem, we devise a new parametrization of the axial and shear stresses, deriving a first order differential equation to solve for the parameter variable, with which the consistency condition and initial conditions are fulfilled automatically. For mixed controls, the responses can be expressed directly in terms of the parameter without solving the first order differential equation. In particular, when control paths are rectilinear exact solutions can be obtained.     

Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.     

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.