On the Decay Rates of Porous Elastic Systems

In this paper we analyze the porous elastic system. We show that viscoelasticity is not strong enough to make the solutions decay in an exponential way, independently of any relationship between the coefficients of wave propagation speed. However, it decays polynomially with optimal rate. When the porous damping is coupled with microtemperatures, we give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of wave propagation speed. Numerical experiments using finite differences are given to confirm our analytical results. It is worth noting that the result obtained here is different from all existing in the literature for porous elastic materials, where the sum of the two slow decay processes determine a process that decay exponentially.     

Exponential decay in one-dimensional porous-thermo-elasticity

This paper concerns the one-dimensional problem of the porous-thermo-elasticity. Two kinds of dissipation process are considered: the viscosity type in the porous structure and the thermal dissipation. It is known that when only thermal damping is considered or when only porous damping is considered we have the slow decay of the solutions. Here we prove that when both kinds of dissipation terms are taken into account in the evolution equations the solutions are exponentially stable.     

L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

We study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t ?? ??, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy??s law. In this paper, we prove that any L ^?? weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural L ¹ topology with decay rates to the Barenblatt profile of the porous medium equation. The density function tends to the Barenblatt solution of the porous medium equation while the momentum is described by Darcy??s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.     

关于一类非线性阻尼模型控制问题的一个注记

利用模态方法进行柔性体控制，就必须作模态截断。在线性粘性阻尼模型情形，阻尼正比于频率，高阶模态会很快衰减。但对于非线性阻尼模型，情况就复杂了。结果表明，高阶模态响应不总能很快衰减，并依赖于初始条件、因为对于非线性阻尼模型的控制，如何进行模态截断，是一个需要谨慎处理的问题。     

Thermoelasticity of bodies with microstructure and microtemperatures

This paper is concerned with a linear theory of thermodynamics for elastic materials with microstructure, whose microelements possess microtemperatures. It is shown that there exists the coupling of microrotation vector field with the microtemperatures even for isotropic bodies. Uniqueness and continuous dependence results are presented. The theory is used to establish the solution corresponding to a concentrated heat source acting in an unbounded continuum.     

Wave propagation analysis of porous functionally graded piezoelectric nanoplates with a visco-Pasternak foundation

This study investigates the size-dependent wave propagation behaviors under the thermoelectric loads of porous functionally graded piezoelectric(FGP) nanoplates deposited in a viscoelastic foundation. It is assumed that(i) the material parameters of the nanoplates obey a power-law variation in thickness and(ii) the uniform porosity exists in the nanoplates. The combined effects of viscoelasticity and shear deformation are considered by using the Kelvin-Voigt viscoelastic model and the refined hi...     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...     

Decay of Benjamin–Ono solitons under the influence of dissipation

The adiabatic decay of Benjamin–Ono algebraic solitons is studied when the influence of various types of small dissipation and radiative losses due to large scale Coriolis dispersion are taken into consideration. The physically most important dissipations are studied, Rayleigh and Reynolds dissipation, Landau damping, dissipation in a laminar boundary layer and Chezy friction on a rough bottom. The decay laws for the soliton parameters, that is amplitude, velocity and width, are found in analytical form and are compared with the results of direct numerical modelling.     

Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers

This paper aims to study the nonlinear-forced vibrations of a viscoelastic cantilever with a piecewise piezoelectric actuator layer on its top surface using the method of Multiple Scales. The governing equation of motion is a second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities which appear in stiffness, inertia, and damping terms. The nonlinear terms are due to the piezoelectricity, viscoelasticity, and geometry of the system. Forced vibrations of the system are investigated in the cases of primary resonance and non-resonance hard excitation including subharmonic and superharmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including damping coefficient, thickness to width ratio of the beam, length and position of the piezoelectric layer, density of the beam, and the piezoelectric coefficient on the frequency-response curves are discussed for each case. It is shown that in all these cases, the response of the system follows a softening behavior due to the existence of the piezoelectric layer. The piezoelectric layer provides an effective tool for active control of vibration. In addition, the effect of the viscoelasticity of the beam on passive control of amplitude of vibration is illustrated.     

Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus

Linear stability analysis of thermal convection is studied for a viscoelastic fluid in a rotating porous cylindrical annulus. The modified Darcy–Jeffrey model with the addition of the Coriolis term in a rotating frame of reference is applied to characterize the non-Newtonian rheology in porous media. We investigate how the interaction among the Coriolis force, viscoelasticity, and bounded sidewalls affects the preferred mode at the onset of convection. The results show that for a slowly rotating case, the oscillatory mode is always preferred for any considered cylindrical radii. However, for a moderately rotating case, the oscillatory preferred mode only arises intermittently as the outer cylindrical radius gradually increases. This result is quite different from the case for viscoelastic fluids in a rotating porous layer or in a porous cylinder without rotation. Further, we discover that for a pair of given cylindrical radii when the Taylor number exceeds a critical value depending on the viscoelastic parameters, the oscillatory convection does not occur. We also examine how the variations of the Taylor number and the viscoelastic parameters affect the patterns of temperature disturbance at the onset of convection.     

Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion

The nonisothermal finite strain dynamics of a porous solid containing a viscous fluid is developed on the basis of a new thermodynamics of open systems and irreversible processes. The same theory is applicable to the mechanics of a nonporous solid with thermomolecular diffusion of a substance in solution. New fundamental concepts of “thermobaric” and “convective” potentials are presented in the context of porous solids. Field equations and Lagrangian equations with generalized coordinates are derived directly from a variational principle of “virtual dissipation”. Inclusion of nonlinear viscoelasticity and plastic behavior is indicated. Partial saturation of pore fluid is discussed. The theory is applicable to the mechanics of a non porous solid with thermolecular diffusion of several molecular species in solution, and under certain conditions to the analogous case of a porous solid containg a fluid mixture. It is shown how the Lagrangian equations provide the foundation of finite element methods.     

Damping Characterization and Viscoelastic Behavior of Laminated Polymer Matrix Composites Using a Modified Classical Lamination Theory

The damping property of materials can be defined as the ratio of dissipated energy over the total strain energy during the loading–unloading process, called the specific damping capacity (SDC). In this study, in order to characterize the damping properties of materials, a test plan in designed to extract the SDC of a single layer composite from hysteresis data. The theory of linear viscoelasticity incorporates a varying Young’s Modulus by using a complex stiffness modulus. Considering different lay-ups, the modified classical lamination plate theory is modified to represent both stiffness and SDC of laminates. The results are compared with experimental results for symmetric laminated specimen. This evaluation shows a very good agreement between theoretical and experimental results in the range of low frequency loading from 0.2 to 4 Hz. The complex compliance matrix changes the governing equation in to a complex form which contains both stiffness and damping properties.     

An Internal Damping Model for the Absolute Nodal Coordinate Formulation

Introducing internal damping in multibody system simulations is important as real-life systems usually exhibit this type of energy dissipation mechanism. When using an inertial coordinate method such as the absolute nodal coordinate formulation, damping forces must be carefully formulated in order not to damp rigid body motion, as both this and deformation are described by the same set of absolute nodal coordinates. This paper presents an internal damping model based on linear viscoelasticity for the absolute nodal coordinate formulation. A practical procedure for estimating the parameters that govern the dissipation of energy is proposed. The absence of energy dissipation under rigid body motion is demonstrated both analytically and numerically. Geometric nonlinearity is accounted for as deformations and deformation rates are evaluated by using the Green–Lagrange strain–displacement relationship. In addition, the resulting damping forces are functions of some constant matrices that can be calculated in advance, thereby avoiding the integration over the element volume each time the damping force vector is evaluated.     

Influence of rotation on the damping of surface waves in a cylindrical layer of viscous fluid

A study is made of the oscillations of the free surface of a cylindrical layer of viscous incompressible fluid attracted by gravity to a solid, uniformly rotating cylinder. Logarithmic decay rates are found for the damping of the surface gravitational waves when large Reynolds numbers are assumed. It is shown that rotation introduces an asymmetry into the damping of the waves traveling in and against the direction of rotation of the fluid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 171–173, November–December, 1984.     

Elongational rheology of polyethylene melts — temporary network constitutive laws

The uniaxial elongational properties of various polyethylenes have been evaluated using an elongational rheometer and a melt-strength apparatus. It is possible to derive the data obtained in elongation from the distribution of relaxation times obtained from oscillatory shearing measurements (linear viscoelasticity), using a Wagner constitutive equation. The effects of the molecular parameters of the samples have been studied, in particular the effect of polydispersity on the shape of the damping function.     

Difference between the longitudinal Frenkel-Biot waves in water-and gas-saturated porous media

The problem of the propagation of longitudinal Biot waves in a porous medium saturated with a weakly compressible liquid (water) or a gas is considered theoretically. The frequency dependence of the phase velocities and damping coefficients is investigated numerically. It is shown that for a certain relationship between the parameters of the porous medium and the saturating fluid there is a “critical” frequency at which the properties of longitudinal waves of both kinds are identical. An analytical expression for this “critical” frequency is obtained. It is shown that for a gas-saturated porous medium, at a certain frequency, in both longitudinal waves the relative gas-matrix motion changes type. Assuming that the saturating-gas behavior corresponds to an adiabatic equation of state, an estimate is obtained for the threshold pore pressure necessary for the restructuring of the relative motion. The wave associated with matrix deformation is shown to have a high damping coefficient in a porous medium saturated with a weakly compressible liquid (water in the case considered) but to be only weakly damped in a gas-saturated porous medium.     

The wave attenuation on the mud bed

In the paper the wave attenuation in a two layer fluid system is studied. The fluid in the top layer is ideal and that in the lower layer is the Voigt model of the viscoelastic medium. A dispersion relation is derived and the rate of the wave decay is computed. The approximate explicit expressions of the decay rate for different water depth are given, where the viscoelasticity is either very large or very small. Compared with the numerical results, our results are very accurate, which can be used by an engineer. The project supported by the National Natural Science Foundation of China and by the Lianyungang Port Office, China     

Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours

The Laplace–Carson transform classically used for homogenization of linear viscoelastic heterogeneous media yields integral formulations of effective behaviours. These are far less convenient than internal variables formulations with respect to computational aspects as well as to theoretical extensions to closely related problems such as ageing viscoelasticity. Noticing that the collocation method is usually adopted to invert the Laplace–Carson transforms, we first remark that this approximation is equivalent to an internal variables formulation which is exact in some specific situations. This result is illustrated for a two-phase composite with phases obeying a compressible maxwellian behaviour. Next, an incremental formulation allows to extend at each time step the previous general framework to ageing viscoelasticity. Finally, with the help of a creep test of a porous viscoelastic matrix reinforced with elastic inclusions, it is shown that the method yields accurate predictions (comparing to reference results provided by periodic cell finite element computations).     

The effect of molecular weight distribution on the elongational properties of linear polymers

The elongational properties of a series of six polypropylene and two polystyrene samples have been studied at constant rate of strain. A Wagner-type constitutive equation has been used to fit the experimental data, and the shape of the damping function has been correlated with the polydispersity index of the samples. As the memory function or relaxation function of linear viscoelasticity may be derived from the molecular-weight distribution using either molecular or phenomenological models, it is therefore possible to calculate the stress growth function of a linear polymer in elongation from its molecular-weight distribution.