A flexible multiscale algorithm based on an improved smoothed particle hydrodynamics method for complex viscoelastic flows

Viscoelastic flows play an important role in numerous engineering fields,and the multiscale algorithms for simulating viscoelastic flows have received significant attention in order to deepen our understanding of the nonlinear dynamic behaviors of viscoelastic fluids.However,traditional grid-based multiscale methods are confined to simple viscoelastic flows with short relaxation time,and there is a lack of uniform multiscale scheme available for coupling different solvers in the simulations of viscoelastic fluids.In this paper,a universal multiscale method coupling an improved smoothed particle hydrodynamics(SPH)and multiscale universal interface(MUI)library is presented for viscoelastic flows.The proposed multiscale method builds on an improved SPH method and leverages the MUI library to facilitate the exchange of information among different solvers in the overlapping domain.We test the capability and flexibility of the presented multiscale method to deal with complex viscoelastic flows by solving different multiscale problems of viscoelastic flows.In the first example,the simulation of a viscoelastic Poiseuille flow is carried out by two coupled improved SPH methods with different spatial resolutions.The effects of exchanging different physical quantities on the numerical results in both the upper and lower domains are also investigated as well as the absolute errors in the overlapping domain.In the second example,the complex Wannier flow with different Weissenberg numbers is further simulated by two improved SPH methods and coupling the improved SPH method and the dissipative particle dynamics(DPD)method.The numerical results show that the physical quantities for viscoelastic flows obtained by the presented multiscale method are in consistence with those obtained by a single solver in the overlapping domain.Moreover,transferring different physical quantities has an important effect on the numerical results.

BOUNDARY ELEMENT METHOD FOR SOLVING DYNAMICAL RESPONSE OF VISCOELASTIC THIN PLATE(Ⅱ)──THEORETICAL ANALYSIS

In this paper, the necessary theoretical analysis for the approximation boundary element method to solve dynamical response of viscoelastic thin plate presented in [1] is.discussed. The theorem of existence and uniqueness of the approximate solution andthe error estimation are also obtained. Based on these conclusions , the principle forchoosing the mesh size and the number of truncated terms in the fundamental solution are given. It isshown that the theoretical ana analysis in this paper are consistent with thenumerical results in [1].

Dynamic Viscoelasticity of Polyester/Rubber Composites under Cyclic Loading

This paper focuses on the influence of dynamic viscoelasticity and surface temperature on the fatigue mechanism and fatigue lifetime of polyester/rubber composites. Rubber composites show significant viscoelasticity during fatigue process. The variations of dynamic elastic modulus, mechanical loss angle, loss energy per cycle exhibit different trend in fatigue initial stage and final stage. Due to high viscoelasticity high heat generation occurs under cyclic loading, which leads to a high surface temperature. It is found that the variation of specimen surface temperature depends strongly on cycling frequency and stress amplitude. SEM (scanning electron microscopy) observation and static residual stiffness studies reveal that the surface temperature affects fracture morphology and fatigue lifetime of rubber composites strongly because of heat aging.

QUASI-STATIC AND DYNAMICAL ANALYSIS FOR VISCOELASTICTIMOSHENKO BEAM WITH FRACTIONAL DERIVATIVECONSTITUTIVE RELATION

The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. ne quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed.

Controller design based on viscoelasticity dynamics model and experiment for flexible drive unit

In order to ensure that the system has the advantage of light weight and vibration absorption, the steel rope is used as a flexible transmission part. A flexible drive unit(FDU) is developed, whose features are guided by steel rope, increasing force by the movable pulley group, modular, convenient and flexible. Dynamics model for controller is deduced based on the constitutive equation of viscoelasticity. Controller is designed for position control and is based on the viscoelasticity dynamics model compensation control strategy proposed. The control system is based on the TURBO PMAC multi-axis motion control card.Prototype loading experiments and velocity experiments results show that the FDU can reach 2 Hz with no load and the max speed of 30(°)/s. The FDU has the capability of the load torque 11.2 N·m and the speed of 24(°)/s simultaneously, and the frequency response is 1.3 Hz. The FDU can be used to be the pitch joint of hip for biped robot whose walking speed is 0.144 km/h theoretically.

BOUNDARY ELEMENT METHOD FOR SOLVING DYNAMICAL RESPONSE OF VISCOELASTIC THIN PLATE(I)

In this paper, a boundary element method for solving dynamical response of viscoelastic thin plate is given. In Laplace domain, we propose two methods to approximate the fundamental solution and develop the corresponding boundary element method. Then using the improved Bellman's numerical inversion of the Laplace transform, the solution of the original problem is obtained. The numerical results show that this method has higher accuracy and faster convergence.

DYNAMICAL STABILITY OF VISCOELASTIC COLUMN WITH FRACTIONAL DERIVATIVE CONSTITUTIVE RELATION

The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.

The asymptotic field of a dynamically growing crack in a viscoelastic materia

A mechanical model of a fracturing viscoelastic material was developed to investigate viscous effects in a dynamically growing crack-tip field. It was shown that in the stable creep-growing phase, elastic deformation and viscous deformation are equally dominant in the near-tip field, and stress and strain have the same singularity, namely, （σ-e）∝r^-1/（n-1）. The asymptotic solution of separating variables of stress, stain and displacement in the crack-tip field was obtained by asymptotic analysis, and the resulting numerical value of stress and strain in the crack-tip field was obtained by the shooting method and the boundary condition of a mode I crack. Through numerical calculation, it was shown that the near-tip fields are mainly governed by the creep exponent n and Mach number M. When n →∞, the asymptotic solution of a viscoelastic material can be degenerated into that of Freund＇s elastic-ideally plastic material by analyzing basic equations.

GENERAL DYNAMIC EQUATION AND DYNAMICAL CHARACTERISTICS OF VISCOELASTIC TIMOSHENKO BEAMS

In this paper, a governing differential equation of viscoelastic Timoshenko beam including both extension and shear viscosity is developed in the time domain by direct method. To measure the complex moduli and three parameters of standard linear solid, the forced vibration technique of beam is successfully used for PCL and PMMA specimens. The dynamical characteristics of viscoelastic Timoshenko beams, especially the damping properties, are derived from a considerable number of numerical computations. The analyses show that the viscosity of materials has great influence on dynamical characteristics of structures, especially on damping, and the standard linear solid model is the better one for describing the dynamic behavior of high viscous materials.

Reflective cracking viscoelastic response of asphalt concrete under dynamic vehicle loading

In order to investigate the mechanical response of reflective cracking in asphalt concrete pavement under dynamic vehicle loading, a finite element model is established in ABAQUS. The viscoelastic behavior is described by a prony series which is calculated through nonlinear fitting to the creep test data obtained in the laboratory. Based on the viscoelastic theory, the time-temperature equivalence principle, fracture mechanics and the dynamic finite element method, both the Jintegral and the mix-mode stress intensity factor are utilized as fracture evaluation parameters, and a half-sine dynamic loading is used to simulate the vehicle loading. Finally, the mechanical response of the pavement reflective cracking is analyzed under different vehicle speeds, different environmental conditions and various damping factors. The results indicate that increasing either the vehicle speed or the structure damping factor decreases the maximum values of fracture parameters, while the structure temperature has little effect on the fracture parameters. Due to the fact that the vehicle speed can be enhanced by improving the road traffic conditions, and the pavement damping factor can become greater by modifying the components of materials, the development of reflective cracking can be delayed and the asphalt pavement service life can be effectively extended through both of these ways.

Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations

This investigation focuses on the nonlinear dynamic behaviors in the trans- verse vibration of an axiMly accelerating viscoelastic Timoshenko beam with the external harmonic excitation. The parametric excitation is caused by the harmonic fluctuations of the axial moving speed. An integro-partial-differential equation governing the transverse vibration of the Timoshenko beam is established. Many factors are considered, such as viscoelasticity, the finite axial support rigidity, and the longitudinally varying tension due to the axial acceleration. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the governing equation. Based on the numerical solutions, the bifurcation diagrams are presented to study the effect of the external transverse excitation. Moreover, the frequencies of the two excitations are assumed to be multiple. Further, five different tools, including the time history, the Poincaré map, and the sensitivity to initial conditions, are used to identify the motion form of the nonlinear vibration. Numerical results also show the characteristics of the quasiperiodic motion of the translating Timoshenko beam under an incommensurable re- lationship between the dual-frequency excitations.

THE DYNAMIC BEHAVIORS OF VISCOELASTIC PIPE CONVEYING FLUID WITH THE KELVIN MODEL

Based on the differential constitutive relationship of linearviscoelastic material, a solid-liquid coupling vibration equation forviscoelastic pipe conveying fluid is derived by the D'Alembert'sprinciple. The critical flow velocities and natural frequencies ofthe cantilever pipe conveying fluid with the Kelvin model (flutterinstability) are calculated with the modified finite differencemethod in the form of the recurrence for- mula. The curves betweenthe complex frequencies of the first, second and third mode and flowvelocity of the pipe are plotted. On the basis of the numericalcalculation results, the dynamic behaviors and stability of the pipeare discussed. It should be pointed out that the delay time ofviscoelastic material with the Kelvin model has a remarkable effecton the dynamic characteristics and stability behaviors of thecantilevered pipe conveying fluid, which is a gyroscopicnon-conservative system.

ANALYSIS OF NONLINEAR DYNAMIC RESPONSE FOR VISCOELASTIC COMPOSITE PLATE WITH TRANSVERSE MATRIX CRACKS

Based on the Schapery three-dimensional viscoelastic constitutive relationship with growing damage, a damage model with transverse matrix cracks for the unidirectional ?bre rein- forced viscoelastic composite plates is developed. By using Karman theory, the nonlinear dynamic governing equations of the viscoelastic composite plates under transverse periodic loading are es- tablished. By applying the ?nite di?erence method in spatial domain and the Newton-Newmark method in time domain, and using the iterative procedure, the integral-partial di?erential gov- erning equations are solved. Some examples are given and the results are compared with available data.

DYNAMIC RESPONSES OF VISCOELASTIC AXIALLY MOVING BELT

Based on the Kelvin viscoelastic differential constitutive law and the motion equation of the axially moving belt, the nonlinear dynamic model of the viscoelastic axial moving belt was established. And then it was reduced to be a linear differential system which the analytical solutions with a constant transport velocity and with a harmonically varying transport velocity were obtained by applying Lie group transformations. According to the nonlinear dynamic model, the effects of material parameters and the steady-state velocity and the perturbed axial velocity of the belt on the dynamic responses of the belts were investigated by the research of digital simulation . The result shows:1) The nonlinear vibration frequency of the belt will become small when the relocity of the belt increases . 2) Increasing the value of viscosity or decreasing the value of elasticity leads to a deceasing in vibration frequencies. 3) The most effects of the transverse amplitudes come from the frequency of the perturbed velocity when the belt moves with harmonic velocity .

TWO-MODE GALERKIN APPROACH IN DYNAMIC STABILITY ANALYSIS OF VISCOELASTIC PLATES

The dynamic stability of viscoelastic thin plates with large deflections was investigated by using the largest Liapunov exponent analysis and other numerical and analytical dynamic methods. The material behavior was described in terms of the Boltzmann superposition principle. The Galerkin method was used to simplify the original integro-partial-differential model into a two-mode approximate integral model,which further reduced to an ordinary differential model by introducing new variables. The dynamic properties of one-mode and two-mode truncated systems were numerically compared.The influence of viscoelastic properties of the material,the loading amplitude and the initial values on the dynamic behavior of the plate under in-plane periodic excitations was discussed.

Temperature effects on the dynamic response of viscoelastic dielectric elastomer

Viscoelasticity and temperature can significantly affect the performance of a dielectric elastomer. In the current study, we use a thermodynamic model to describe the effect of temperature and viscoelasticity on the electromechanical response undergoing a cyclic electric load by taking into account of the temperature dependent dielectric constant. Because of the significant viscoelasticity in the dielectric elastomer, the deformation and the nominal electric displacement can not keep in phase with the electric field at low frequencies. The results show that the magnitude of the cyclic electromechanical actuation strain increases with the decrease of the temperature and decreases with the increasing frequency, and viscoelasticity can result in significant hysteresis for dielectric elastomers under a relative low temperature and a low frequency.

A viscoelastic-plastic constitutive model with Mohr-Coulomb yielding criterion for sea ice dynamics

A new viscoelastic-plastic （VEP） constitutive model for sea ice dynamics was developed based on continuum mechanics. This model consists of four components： Kelvin-Vogit viscoelastic model, Mohr-Coulomb yielding criterion, associated normality flow rule for plastic rehololgy, and hydrostatic pressure. The numerical simulations for ice motion in an idealized rectangular basin were made using smoothed particle hydrodynamics （SPH） method, and compared with the analytical solution as well as those based on the modified viscous plastic（VP） model and static ice jam theory. These simulations show that the new VEP model can simulate ice dynamics accurately. The new constitutive model was further applied to simulate ice dynamics of the Bohai Sea and compared with the traditional VP, and modified VP models. The results of the VEP model are compared better with the satellite remote images, and the simulated ice conditions in the JZ20-2 oil platform area were more reasonable.

Dynamic stability of viscoelastic circular cylindrical shells taking into account shear deformation and rotatory inertia

The present work discusses the problem of dynamic stability of a viscoelastic circular cylindrical shell, according to revised Timoshenko theory, with an account of shear deformation and rotatory inertia in the geometrically nonlinear statement. Pro- ceeding by Bubnov-Galerkin method in combination with a numerical method based on the quadrature formula the problem is reduced to a solution of a system of nonlinear integro-differential equations with singular kernel of relaxation. For a wide range of variation of physical mechanical and geometrical parameters, the dynamic behavior of the shell is studied. The influence of viscoelastic properties of the material on the dynamical stability of the circular cylindrical shell is shown. Results obtained using different theories are comDared.

A NEW ALGORITHM OF TIME STEPPING IN DYNAMIC VISCOELASTIC PROBLEMS

A new scheme of time stepping for solving the dynamic viscoelastic problems are presented. By expanding variables at a discrete time interval, FEM based recurrent formulae are derived. A self-adaptive algorithm for different sizes of time steps can be carried out to improve computing accuracy. Numerical validation shows satisfactory performance.

NONLINEAR DYNAMIC RESPONSE AND ACTIVE VIBRATION CONTROL OF THE VISCOELASTIC CABLE WITH SMALL SAG

The problem considered is an initially stressed viscoelastic cable with small sag. The cable material is assumed to be constituted by the hereditary differential type. The partial differential equations of motion is derived first. Then by applying Galerkin's method, the governing equations are reduced to a set of third order nonlinear ordinary differential equations which are solved by Runge-Kutta numerical integration procedures. Only after the transverse vibration of the plane is considered and the nonlinear terms are neglected, can the nonlinear ordinary differential equations be expressed as a continuous state equation in the state space. The matrix of state transition is approximated stepwise by the matrix exponential; in addition, the state equation is discretized to a difference equation to improve the computing efficiency. Furthermore, an optimal control of procedure system based on the minimization of a quadratic performance index for state vector and control forces is developed. Finally, the effect of dynamic response of the cable, which is produced by viscoelastic parameters, is testified by the research of digital simulation.