Laplace变换法及其在粘弹性波中传播的应用

针对波传播分析理论的发展历程进行了简要的综述,详细介绍了几种处理粘弹性波传播问题的分析方法,重点讲解Laplace变换法以及Laplace变换在粘弹性波中的应用,对比分析几种方法在各自应用上的优劣,由于Laplace变换法能准确地描述应力波在任意时刻、任意点的波动情况,在处理大尺寸混凝土类构件中应力波传播问题时具有其独特的优势。     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

Crack propagation in the power-law nonlinear viscoelastic material

I.IntroductionCrazingdamageisacommonphenon1enonoffractureofpolymericmaterials.Theformationofcrazezoneisamid-stateinthefractureprocessofthematerialsfromperfectstatetofaiIurc.Microscopically,inthisregionthereexistssomefibrilslinkingthetwocracksurfacesandres…     

The extended Padé-Laplace method for efficient discretization of linear viscoelastic spectra

The discretization of linear viscoelastic spectra is valuable as a starting point for non-linear viscoelastic modeling. However, obtaining the parameters of the generalized Maxwell model from linear viscoelastic experiments with naive least squares procedures is known to be an ill-posed problem. A novel technique, the Padé-Laplace method was recently elucidated (Fulchiron et al., 1993) for robustly extracting the parameters of the generalized Maxwell model from stress relaxation experiments, without any a priori assumption about the number of Maxwellian modes. We extend this method for obtaining the Maxwellian modes from dynamic data and discuss the relationship between continuous viscoelastic spectra and the Maxwellian modes obtained by this procedure. Furthermore, the applicability of this method with experimental data in limited time/frequency windows is clarified. Finally, a procedure for assembling the discretized spectrum with the Padé-Laplace method applied to both stress relaxation and dynamic data with typical experimental time/frequency cutoffs is developed.Dedicated to Professor H. Janeschitz-Kriegl on the occasion of his 70th birthday.     

Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear

In linear viscoelastic investigations the frequency dependence of the phase shift between stress and strain appears to be very characteristic of the molecular structure of the material. This function is also a good approximation of the slope of the double logarithmic plot of the absolute value of the shear modulusG _d vs. the angular frequency. The product (G _d /) sin 2 comes very close to the relaxation spectrumH(), with = 1/, in all physical states of the material.The experimentally observed separability of time and strain effects in nonlinear viscoelasticity of highly viscous isotropic polymer fluids imposes restraints to the form of the constitutive equation. A single integral superposition equation of the Boltzmann type containing the product of a time function and a nonlinear strain function gives good results in describing experimental data in shear as well as in elongation. The molecular structure affects both functions in a different way. A universal definition of the nonlinear tensorial strain measure has not yet been developed. There are some indications that a definition on the basis of the principal stretch ratios may be fruitful.Invited paper, presented at the First Conference of European Rheologists at Graz (Austria), April 14–16, 1982.     

平面粘弹性问题的辛求解方法

将弹性力学辛对偶求解方法与Laplace变换相结合,提出了一个求解粘弹性平面问题的新方法。首先利用Laplace变换,将粘弹性平面问题转化为一个准弹性问题,在辛弹性力学的框架下,利用分离变量和辛本征展开法对其进行求解,然后由逆变换得到原问题的解。为证明方法的有效性,求解分析了矩形域平面粘弹性圣维南问题,得到了令人满意的结果。     

Linear viscoelastic behavior of asphalt and asphalt based mastic

The rheological properties of an asphalt mastic and its matrix are investigated. For the purpose of comparison a sample of thermal aged asphalt matrix is also considered. Dynamic and creep shear measurements are reported. The reduced shear rate concept proposed by Ohl and Gleissle is used to correlate mechanical properties of the three materials at the same temperature. We found that the concept gives only qualitative trends. A similar conclusion is found concerning the applicability of the time-temperature superposition principle for each sample. Our experimental results show also that the increase in viscosity due to thermal treatment or to the inclusion of solid particles is not uniform with temperature. The differences in the increase of the Vogel temperature from the asphalt to the mastic, or to the thermally aged asphalt, relate to the different mechanisms involved. Sedimentation of steel spheres in asphalt and mastic, is studied next. The Newtonian wall correction factor for the Stokes drag law holds for the three samples. Despite the similar behavior observed in conventional shear tests, Stokes' law gives the correct trend for the two asphalts although it overestimates the experimental settling velocity by a factor of approximately two in the case of the mastic. Received: 8 June 1999/Accepted: 19 October 1999     

Concentration dependence of the linear viscoelastic properties of particle suspensions

The response under small amplitude oscillatory deformations of a suspension of non-Brownian spheres dispersed in a viscoelastic fluid is investigated. The correspondence principle of linear viscoelasticity is used to derive a simple constitutive model from a model for a suspension in a Newtonian liquid. The theory predicts that for a specific particulate system the concentration dependence of the viscoelastic properties should collapse to a single master curve when the values are normalized with those of the carrier fluid alone. Measurements with the micro-Fourier rheometer using oscillatory squeeze flow are carried out on two suspensions of 60 and 80 μm sized particles dispersed in polymeric fluid and in silicon oil, and the master curve is verified. Received: 27 April 1999/Accepted: 15 October 1999     

The effect of temperature on the viscoelastic behavior of linear low-density polyethylene

Summary Observations are reported on linear low-density polyethylene in isothermal torsional oscillation and relaxation tests at various temperatures ranging from room temperature to 110 C. Constitutive equations are derived for the viscoelastic response of a semicrystalline polymer at small strains. The polymer is treated as an equivalent network of strands bridged by junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is thought of as an ensemble of meso-regions with various potential energies for rearrangement of strands. Two types of meso-domains are introduced: active, where strands separate from temporary junctions as they are excited by thermal fluctuations, and passive, where detachment of strands is prevented by the surrounding macromolecules. The time-dependent behavior of the ensemble reflects separation of active strands from their junctions and merging of dangling strands with the network. Stress–strain relations are developed by using the laws of thermodynamics. The governing equations involve six material constants that are found by fitting the experimental data. The study focuses on the effects of (i) temperature, (ii) the deformation mode (torsion versus bending), and (iii) the loading program (oscillations versus relaxation) on the adjustable parameters.This work was partially supported by the West Virginia Research Challenge Grant Program     

A series solution for the effective properties of incompressible viscoelastic media

This paper presents a series solution for the homogenization problem of a linear viscoelastic periodic incompressible composite. The method uses the Laplace transform and the correspondence principle which are combined with the classical expansion along Neumann series of the solution of the periodic elasticity problem in Fourier space. The terms of the Neumann series appear as decoupled, containing geometry dependent terms and viscoelastic properties dependent terms which are polynomial fractions whose inverse Laplace transforms are provided explicitly.     

Optimization of the collocation inversion method for the linear viscoelastic homogenization

The effective behaviour of linear viscoelastic heterogeneous material can be derived from the correspondence principle and the inversion of the obtained symbolic homogenized behavior. Various numerical methods were proposed to carry out this inversion. The collocation method, widely used, within this framework rests on a discretization of the characteristic spectrum in a sum of discrete lines for which it is necessary to determine the intensities and the positions by the minimization of the difference between the exact temporal function and its approximation. The classical method is based on a priori choice of the lines positions and on the optimization of their intensities. It is shown here that the combined optimization of the positions and the (positive) intensities lead to a minimization problem under constraints. In the simple case of an incompressible isotropic two-phase material, the assessment of the effective relaxation function with a continuum spectra or made up of discrete lines proves that the proposed method improves the predictions of the classical approach.     

Comparison between the dynamic moduli of fully non-linear and quasilinear viscoelastic materials

This paper deals with the non-linear viscoelastic model with multiple hereditary integrals (MHI) in the frequency domain and the conditions that it reduces to single hereditary integral or the quasilinear viscoelastic (QLV) model. It is shown that when the higher order complex moduli are related to the first-order modulus as the MHI model reduces to the QLV model. The coefficients of quasilinearity should be real and independent of amplitude and frequency.     

Theoretical and experimental analysis of longitudinal wave propagation in cylindrical viscoelastic rods

Wave propagation in viscoelastic rods is encountered in many applications including studies of impact and fracture under high strain rates and characterization of the dynamic behavior of viscoelastic materials. For viscoelastic materials, both material and geometric dispersion are possible when the diameter of the rod is of the same order as the wavelength. In this work, we simplify the Pochhammer frequency equation for low and intermediate loss viscoelastic materials and formulate corrections for geometric dispersion for both the phase velocity and attenuation. The formulation is then experimentally verified with measurements of the phase velocity and attenuation in commercial polymethylmethacrylate rods that are 12 and in diameter. Without correcting for geometric dispersion, the usable frequency range for determining the phase velocity and attenuation for the rod is about , and about for the rod. Using the correction procedure developed here, it was possible to accurately determine the phase velocity and attenuation up to frequencies exceeding for the rod and for the rod. These corrections are applicable to many polymers and other viscoelastic materials. From thereon, the viscoelastic properties of the material can be determined over a wide range of frequencies.     

On the sensitivity of interconversion between relaxation and creep

The interconversion equation of linear viscoelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J(t). It is widely utilised in rheology to estimate J(t) from measurements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J(t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first measure G(t) and then determine J(t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J(t) from G(t) compared with that of G(t) from J(t). Although algorithms are available that work adequately in both directions, numerical experimentation strongly suggests that the recovery of J(t) from G(t) measurements is the more stable. An elementary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J(t). It explicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspective, about the implications and implementations of the results.     

Influence of partial constrained layer damping on the bending wave propagation in an impacted viscoelastic sandwich

This paper presents a parametric model to study the transient bending wave propagation in a viscoelastic sandwich plate due to impact loading. The effect of partial constrained layer damping (PCLD) geometry on wave propagation is investigated by comparing with propagation in single layer elastic plate. Several boundary conditions are also considered, and their effect on wave propagation is highlighted.The equation of motion is obtained from Lagrange’s equations. For the single layer plate, the governing equation is solved in time domain using Newman and Wilson method. For the plate with PCLD, the frequency dependant viscoelastic behavior of the core is represented by Prony series; the equation of motion is converted into frequency domain using Fourier transform the displacement is obtained in the frequency domain and is converted into time domain with the Inverse Fast Fourier Transform.The model was validated in our previous paper (Khalfi and Ross (2013)) with experimental results, additional validation is carried in this paper with literature, and good agreement is recorded. The results show that the plate covered with PCLD remains a dispersive medium. The shape of the wave is mainly related to the sandwich stiffness while the viscoelastic layer contributes in reducing the amplitude and speed of propagation. The particularity of this transient model lies in its ability to follow the shape of the bending wave at all times to observe formation, propagation and disappearance. With this model, the influence of any structural input parameters on the bending wave can be studied. The findings presented will also serve as a research base for more advanced horizons.     

MICROMECHANICS ANALYSIS ON EVOLUTION OF CRACK IN VISCOELASTIC MATERIALS

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Wave propagation in a fractional viscoelastic Andrade medium: Diffusive approximation and numerical modeling

This study focuses on the numerical modeling of wave propagation in fractionally-dissipative media. These viscoelastic models are such that the attenuation is frequency-dependent and follows a power law with non-integer exponent within certain frequency regimes. As a prototypical example, the Andrade model is chosen for its simplicity and its satisfactory fits of experimental flow laws in rocks and metals. The corresponding constitutive equation features a fractional derivative in time, a non-local-in-time term that can be expressed as a convolution product whose direct implementation bears substantial memory cost. To circumvent this limitation, a diffusive representation approach is deployed, replacing the convolution product by an integral of a function satisfying a local time-domain ordinary differential equation. An associated quadrature formula yields a local-in-time system of partial differential equations, which is then proven to be well-posed. The properties of the resulting model are also compared to those of the Andrade model. The quadrature scheme associated with the diffusive approximation, and constructed either from a classical polynomial approach or from a constrained optimization method, is investigated. Finally, the benefits of using the latter approach are highlighted as it allows to minimize the discrepancy with the original model. Wave propagation simulations in homogeneous domains are performed within a split formulation framework that yields an optimal stability condition and which features a joint fourth-order time-marching scheme coupled with an exact integration step. A set of numerical experiments is presented to assess the overall approach. Therefore, in this study, the diffusive approximation is demonstrated to provide an efficient framework for the theoretical and numerical investigations of the wave propagation problem associated with the fractional viscoelastic medium considered.     

Memory response on thermal wave propagation in an elastic solid with voids

Abstract

Fractional derivative is a widely accepted theory to describe the physical phenomena and the processes with memory responses which is defined in the form of convolution having kernels as power functions. Due to the shortcomings of power law distributions, some other forms of derivatives with few other kernel functions are proposed. This present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for an infinite porous material subjected to the presence of distributed time-dependent heat source acting over the plane area. The heat transport equation for this problem is involving the memory dependent derivative on a slipping interval in the context of three-phase-lag (3PL) model of generalized thermoelasticity. Employing the Laplace transform as a tool, the analytical results for the distributions of the change in volume fraction field, temperature, stress, and displacement are obtained on solving the vector-matrix differential equation using eigenvalue approach. The numerical inversion of the Laplace transform is performed using the Zakian method. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative and delay time also.

Communicated by Nickolay Banichuk.     

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.