Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

On the three-dimensional undulation instability of a rectangular viscoelastic composite plate in biaxial compression

Within the frame work of the second version of small precritical deformation in the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB), the undulation instability problem for a simply supported rectangular plate made of a viscoelastic composite material is investigated in biaxial compression in the plate plane. The corresponding boundary-value problem is solved by employing the Laplace transformation and the principle of correspondence. For comparison and estimation of the accuracy of results given by the TDLTSDB, the same problem is also solved by using various approximate plate theories. The viscoelasticity properties of the plate material are described by the Rabotnov fractional-exponential operator. The numerical results and their discussion are presented for the case where the plate is made of a multilayered viscoelastic composite material. In particular, the variation range of problem parameters is established for which it is necessary to investigate the undulation instability of the viscoelastic composite plate by using the TDLTSDB. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 93–108, January–February, 2009.     

On the plane deformation of fibre-reinforced viscoelastic plates

The realization method of elastic solutions is used to solve the problem of bending of a viscoelastic plate reinforced unidirectionally by elastic fibres. Numerical computations are carried out for three kinds of external load. The plane deformation of this plate is discussed.     

Well‐posedness of initial boundary value problems on longitudinal impact on a composite linear viscoelastic bar

We investigate the correctness of the initial boundary value problem of longitudinal impact on a piecewise‐homogeneous semi‐infinite bar consisting of a semi‐infinite elastic part and finite length visco‐elastic part whose hereditary properties are described by linear integral relations with an arbitrary difference kernel. Introducing nonstationary regularization in boundary conditions and in the contact conditions, the well‐posedness of the considered problem is proved. Copyright © 2017 John Wiley & Sons, Ltd.     

Long-term failure of a layered viscoelastic composite material with a crack under a time-dependent load

The long-term failure of a layered viscoelastic composite caused by precritical propagation of a coin-shaped crack is studied. It is assumed that the crack is located inside a viscoelastic layer (the layer of binder) parallel to the layer orientation. The crack development due to stretching of the composite massive by uniformly distributed external forces increasing with time is described. It is assumed that these forces act perpendicularly to the plane of crack propagation. The investigation is carried out within the framework of Boltzmann-Volterra linear theory for resolving integral operators with difference kernels describing the deformation of a material with time-dependent rheological properties. An irrational function of the viscoelastic integral operator is presented in the form of a proper continued fraction and transformed using the method of operator continued fractions. Numerical solutions are obtained for resolving integral operators with the kernel in the form of Rabotnov exponential-fractional function. The kinetics of crack growth with a prefailure zone commensurable with the crack length is described. A comparison with the results obtained in terms of the concept of thin structure of the crak tip is given.Timoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 545–558, July–August, 2000.     

复合材料大层数层压矩形截面杆的扭转问题分析

本文根据有效弹性模量理论［１］，采用三维八节点等参数有限元和整体—局部方法，对复合材料大层数矩形厚截面层压杆的扭转问题及其自由边缘效应进行了分析研究，通过算例计算给出了剪切应力在横截面内的分布规律、杆的扭转变形及其在自由边缘区域层间应力的分布情况·由于本文的分析方法可根据需要仅在应力梯度较大的局部区域，按单层逐层划分单元或在单层内再细化单元，以求得单层内精确的应力场和位移场，因此能显著节约计算量与机时，为具有大层数层压杆的扭转强度计算提供了一种有效的方法·     

On a dynamical hierarchical model for prismatic shells in the theory of elastic mixtures

The present paper is devoted to the design of a hierarchy of two‐dimensional models for dynamical problems within the theory of multicomponent linearly elastic mixtures in the case of prismatic shells with thickness which may vanish on some part of its boundary. The hierarchical model is obtained by a semidiscretization of the three‐dimensional problem in the transverse direction. In suitable weighted Sobolev spaces we investigate the well‐posedness of the two‐dimensional problems, prove pointwise convergence of the sequence of approximate solutions restored from the solutions of the reduced problems to the exact solution of the original problem and estimate the rate of convergence. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamic vibrations of a damageable viscoelastic beam in contact with two stops

A model for the material damage, due to dynamic vibrations of a Kelvin‐Voigt viscoelastic beam whose tip is constrained to move between two stops, is presented and numerically analyzed. The contact of the free tip with the stops is described by the normal compliance condition. The evolution of damage of the beam's material, which measures the reduction of its load carrying capacity, is modeled with a parabolic inclusion. The existence of the unique local solution is stated. A numerical algorithm is presented, in which spatially it is approximated by finite elements, and the time derivatives are discretized with the Euler scheme. Error estimates are derived for sufficiently regular solutions, and four numerical simulations are shown. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

On the asymptotic behavior of a linear viscoelastic fluid

We study the asymptotic behavior of an incompressible viscoelastic fluid and prove that the temporal decay of the energy is similar to one of the memory kernel. The innovative aspect of this research lies in considering the evolutive problem with non‐zero external sources and/or initial histories. Copyright © 2012 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].     

Blow-up of solutions for a viscoelastic equation with nonlinear damping

The initial boundary value problem for a viscoelastic equation with nonlinear damping in a bounded domain is considered. By modifying the method, which is put forward by Li, Tasi and Vitillaro, we sententiously proved that, under certain conditions, any solution blows up in finite time. The estimates of the life-span of solutions are also given. We generalize some earlier results concerning this equation.      

A new general decay for a transmission problem of viscoelastic Timoshenko systems

In this paper, we investigate the long-time behavior for a transmission problem of viscoelastic Timoshenko systems with different speeds of wave propagation. By constructing a new Lyapunov functional and combining the technique of perturbation energy with some precise estimates for multipliers, we establish a general uniform decay estimates for the energy.     

General decay of energy for a viscoelastic equation with nonlinear damping

In this paper we are concerned with a nonlinear viscoelastic equation with nonlinear damping. The general uniform decay of the energy is obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

A note on the polynomial decay of a weakly dissipative viscoelastic system

In this note we study an abstract class of weakly dissipative second-order systems with finite memory. We establish the polynomial decay of Rivera, Naso and Vegni for the solution of the system under a very weak condition on the relaxation function.     

Decay rates for the energy of a singular nonlocal viscoelastic system

This work deals with decay rates for the energy of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. We prove the decay rates for the energy of a singular one‐dimensional viscoelastic system with a nonlinear source term and nonlocal boundary condition of relaxation kernels described by the inequality $g_i^{\prime }\left(t\right)\le ?H\left(g_i\left(t\right)\right),\left(i=1,2\right)$ for all t ≥ 0, with H convex.     

Global existence and decay of solutions of a singular nonlocal viscoelastic system with a nonlinear source term,nonlocal boundary condition,and localized damping term

The paper deals with the existence of a global solution of a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition, and localized frictional damping a(x)u_t using the potential well theory. Furthermore, the general decay result is proved. We construct a suitable Lyapunov functional and make use of the perturbed energy method.