DECAY RATES AND CONVERGENCE OF SOLUTIONS TO SYSTEM OP ONE-DIMENSIONAL VISCOELASTIC MODEL WITH DAMPING

This paper considers the asymptotic behavior of solutions to the system of one-dimensional viscoelastic model with damping and prove that the corresponding solutions time-asymptotically behave like nonlinear diffusion wave as in [4,11]. In addition, It is also shown that the system of one-dimensional viscoelastic model with damping is a viscosity approximation of a hyperbolic conservation laws with damping.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

General decay for a quasilinear system of viscoelastic equations with nonlinear damping

In this paper, we consider a system of coupled quasilinear viscoelastic equations with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.     

Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.     

Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with strong damping is considered. We prove that, under suitable assumptions on relaxation functions and certain initial data, the decay rate of the solutions energy is exponential.     

一个带有阻尼项和源项的非线性粘弹性方程组解的整体衰减

本文考虑了一个带有非线性阻尼项的粘弹性方程组.通过使用扰动能量的方法,我们得到了整体解的能量泛函依据松弛函数的衰减速率按指数衰减或者多项式衰减.     

A novel analysis method for damping characteristic of a type of double-beam systems with viscoelastic layer

The double-beam system with a viscoelastic layer is a classical mechanical model for many beam-type composite structures. However, few studies have been able to optimize the structure from the perspective of structural damping characteristics. To fully understand the damping characteristics of the viscoelastic double-beam system, an analysis method based on dynamic stiffness method and Wittrick-Williams algorithm is presented in this paper. Through numerical case studies, five typical parameters of the viscoelastic double-beam system are discussed to investigate their influence on the damping characteristic of the system. Finally, the conclusions are used to parametric analysis for a kind of double-sheathing cable systems. Results show that the damping coefficient of the connection layer have a significant effect on the damping characteristic of the double-sheathing cable system compared with other design parameters. The proposed methods and conclusions obtained in this paper are helpful to design and optimize the structural parameters of engineering structures, thus having certain application and promotional value.     

On a system of nonlinear wave equations with Balakrishnan–Taylor damping

In this paper, we study the initial-boundary value problem for a coupled system of nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan–Taylor damping terms. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Also, we show that nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of stronger damping.     

Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism

In this paper, we prove the global existence of small smooth solutions to the three-dimensional incompressible Oldroyd-B model without damping on the stress tensor. The main difficulty is the lack of full dissipation in stress tensor. To overcome it, we construct some time-weighted energies based on the special coupled structure of system. Such type energies show the partial dissipation of stress tensor and the strongly full dissipation of velocity. In the view of treating “nonlinear term” as a “linear term”, we also apply this result to 3D incompressible viscoelastic system with Hookean elasticity and then prove the global existence of small solutions without the physical assumption (div–curl structure) as previous works.     

On decay and blow-up of solutions for a system of nonlinear wave equations

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

考虑细胞外基质黏弹性行为的细胞黏附力学模型

细胞外基质由大量胶原蛋白和纤维蛋白组成,这些基质蛋白形成复杂的交联网络状结构,具有黏弹性力学特性.研究表明,黏弹性基质能显著影响细胞迁移、增殖和分化等生理行为,还能影响癌症转移和组织纤维化等疾病的发生与发展.然而,细胞感知细胞外基质黏弹性力学特性的分子机制仍不清楚.该文通过建立细胞黏附力学模型,从分子层次揭示细胞黏附在细胞响应外界黏弹性力学微环境中的作用.结果表明,细胞能通过调控细胞黏附动力学（包括黏附周期和黏附形成时间）响应细胞外基质的黏弹性力学特性.通过将模型计算结果与实验现象相比较,验证了模型的正确性.细胞黏附力学模型将为组织工程中细胞力学微环境的构建奠定理论基础.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

On hydrodynamics of viscoelastic fluids

In this paper, we study a hydrodynamic system describing fluids with viscoelastic properties. After a brief examination of the relations between several models, we shall concentrate on a few analytical issues concerning them. In particular, we establish local existence and global existence (with small initial data) of classical solutions for an Oldroyd system without an artificially postulated damping mechanism. © 2005 Wiley Periodicals, Inc.     

Global existence and uniform decay for a nonlinear viscoelastic equation with damping

In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established.     

On the distribution of real eigenvalues in linear viscoelastic oscillators

In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.     

Optimal constrained layer damping with partial coverage

This paper deals with the optimal damping of beams constrained by viscoelastic layers when only one or several portions of the beam are covered. An efficient finite element model for dynamic analysis of such beams is used. The design variables are the dimensions and prescribed locations of the viscoelastic layers and the objective is the maximum viscoelastic damping factor. The method for non-linear programming in structural optimization is the so-called method of moving asymptotes.     

Parameter Identification and FE Implementation of a Viscoelastic Constitutive Equation Using Fractional Derivatives

Damping in viscoelastic materials can be described in several ways. In FE codes for transient calculations with direct integration usually Rayleigh‐damping is provided. However, it is known that this model is not qualified to represent the damping properties of viscoelastic material over a broad range of time or frequency. Another approach uses fractional time derivatives of stresses and strains in the constitutive equations. This model requires few parameters, provides good curve fitting properties and is physically proved. In this paper a parameter identification for the fractional 3‐parameter model will be carried out and its implementation into an FE code will be demonstrated.     

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].