An intermediate model method for obtaining a discrete relaxation spectrum from creep data

A convenient method is described for obtaining a discrete stress relaxation spectrum from linear viscoelastic creep data by means of a three-stage process. In stage one, a discrete retardation spectrum is fitted to the creep data using a least squares procedure, subject to the constraint that the discrete spectrum must be a specified order of polynomial function of the retardation time. In stage two, the resulting generalised Voigt model is solved numerically for an imposed step in strain, to determine the stress relaxation modulus function of time. In stage three, a discrete relaxation spectrum is fitted to the calculated stress relaxation modulus function. Although three stages are involved instead of the usual two, the procedure has been found to have certain practical advantages. These advantages make it suitable for the generation of relaxation spectra needed in viscoelastic stress analyses of solids, for example by the finite element method. In order to illustrate the proposed procedure it is applied to both artificial data and experimental creep data for poly(methyl methacrylate) at 70°C and at the glass transition.     

A fundamental relationship between the relaxation spectra and the tangent distribution function in the theory of linear viscoelasticity

The tangent distribution function (TDF) is analyzed within the theory of linear viscoelasticity on mechanical properties. A proof is given that both the relaxation and retardation spectra can be derived from the TDF, through a Fredholm integral equation. Furthermore, the relaxation strength can be calculated as a consequence of this relationship. Finally, as an example, the relationship is applied to discrete spectra.     

Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G() andG(), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersg _i , _i ,i = 1,2, N, of the discrete spectrum to obtain a best fit ofG, G, and it was found to be essential that bothg _i and _i are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesg _i,0, _i,0,N ₀. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ ₀,, J _d,i and _i of the creep complianceJ(t) can then readily be calculated using the Laplace transform.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method

For the numerical determination of a continuous relaxation time spectrum from rheological data so-called regularization methods are necessary. The regularization method based on Tikhonov regularization and commonly used in rheology assumes the relaxation time spectrum to be smooth. Accordingly, this method is not able to properly resolve spectra containing edges or at least large curvatures as proposed in the literature. In order to reconstruct such spectra a novel edge preserving regularization (EPR) method was developed which extends the properties of the commonly used method. In order to demonstrate the efficiency of EPR it is compared with the commonly used regularization method by means of simulated and experimental data. This reveals that EPR is able to prove whether or not the theoretically predicted edge containing spectra for high-molecular-weight, monodisperse polymers are compatible with experimental data. Received: 11 June 1999/Accepted: 1 September 1999     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

On the damping function of shear relaxation modulus for entangled polymers

Published data of the damping function of the shear relaxation modulus, h(), are reviewed. This is the ratio of the relaxation modulus measured at a finite magnitude of shear, , to that at the limit of = 0. Majority of the data are in accord with the universal function of the Doi-Edwards tube model theory, in which the damping or the decrease of h() is attributed to the contraction along the tube of extended polymer chains. The weaker damping seems to be attributed to 1) comb-branching such as in LDPE; 2) lack of entanglement in too short chains; 3) bimodal molecular weight distribution. However, a star-branching does not cause a deviation from the tube model theory and a broadness of molecular weight distribution is not a major origin of a weaker damping. A star-branched polystyrene with 15 arms exhibits no strain dependence: h() = 1. For highly entangled systems with more than 50 entanglement points per molecule, the strain dependence is stronger than that of the Doi-Edwards theory. This could be due to a slip or an instability of deformation in the material.     

A direct method for obtaining discrete relaxation spectra from creep data

A direct method for obtaining discrete relaxation spectra from creep data is proposed. The conversion of creep data to relaxation data is avoided and the discrete relaxation times are freely adjustable. The nonnegative least square method is used to generate nonnegative discrete relaxation strength. Received: 10 November 1999 Accepted: 22 September 2000     

The accuracy of some formulae for the interconversion of creep and stress-relaxation data

The accuracy of the approximation formulaeJ (t) ~ 1/G (t) andd lnJ (t)/d lnt ~ —d lnG (t)/d lnt, which interconnect stress-relaxation modulusG (t) and creep complianceJ (t) and their double logarithmic rates are investigated. For glassy polymers, the errors in the first formula are less than 1–2%, and in the second, they are generally in the order of a few percent, too. Similar estimates can also be found for the real parts of the analogous complex functionsJ ^* () andG ^* ().     

A new formulation for fictitious mass of viscous dynamic relaxation method

Abstract

In this article, a new relationship is proposed for the fictitious mass of viscous dynamic relaxation (DR) method. First, incremental equations are derived for DR steps. Using transformed Gershgörin theory, a new relationship is achieved for fictitious mass of viscous DR by formulating modified time step ratio. This procedure presents a new algorithm for the viscous DR method. To evaluate the numerical efficiency of the proposed method, some 2D and 3D truss and frame structures are analyzed with elastic linear and geometrically nonlinear behaviors. Results show that by using the proposed algorithm for fictitious mass, the convergence rate of the viscous DR method is improved so that the proposed algorithm presents the structural response with lower iterations in comparison with other common DR techniques.

Communicated by Joerg Fehr.     

A non-linear method for the calculation of the loss tangent distribution function

A procedure to obtain the tangent distribution function from internal friction peaks in solids is described. The method was applied to simulated data in order to show that the distribution can reproduce accurately the internal friction data in transition region as well as in most of the terminal region. Finally, the procedure is applied to experimental data in glass transition peaks of polystyrene.     

Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint

The relaxation spectrum is estimated from dynamic experiments using Bayesian analysis and a new regularization constraint. In the Bayesian framework, a probability can be calculated for each estimate of the spectrum. This offers several advantages; (1) an optimal estimate of the relaxation spectrum may be calculated as the mean of a large number of estimates, and (2) reliable errors for the optimal estimate can be provided using the deviation of all estimates from the mean. Furthermore, the Bayesian approach (3) gives an estimate of the overall noise level of the experiment, which is usually an important but unknown parameter for the calculation of relaxation spectra from dynamic experiments by indirect methods (determining the regularization parameter), and finally, (4) the information content in a given set of experimental data can be quantified. The validity of the Bayesian approach is demonstrated using simulated data.     

A nonlinear regularization method for the calculation of relaxation spectra

It is well known that the relaxation spectrum characterizing the linear viscoelastic properties of a polymer melt or solution is not directly accessible by an experiment. Therefore, it must be calculated from data for a material function. With Tikhonov regularization the relaxation spectrum in the terminal and plateau region can be calculated from data for a material function in the corresponding region. Serious difficulties arise however, if the spectrum should be determined in a larger range. These difficulties are caused by the considerably different contributions at short and long relaxation times. We show that these difficulties can be avoided by a nonlinear regularization method.     

A regularization-free method for the calculation of molecular weight distributions from dynamic moduli data

There are several models for the determination of molecular weight distributions (MWDs) of linear, entangled, polymer melts via rheometry. Typically, however, models require a priori knowledge of the critical molecular weight, the plateau modulus, and parameters relating relaxation time and molecular weight (e.g., k and in =kM). Also, in an effort to obtain the most general MWD or to describe certain polymer relaxation mechanisms, models often rely on the inversion of integral equations via regularization. Here, the inversion of integral equations is avoided by using a simple double-reptation model and assuming that the MWD can be described by an analytic function. Moreover, by taking advantage of dimensionless variables and explicit analytic relations, we have developed an unambiguous and virtually parameter-free methodology for the determination of MWDs via rheometry. Unimodal MWDs have been determined using only a priori knowledge of the exponent and dynamic moduli data. In addition, the uncertainty in rheological MWD determinations has been quantified, and it is shown that the reliability of the predictions is greater for the high-molecular-weight portion of the distribution.     

基于实测数据的扁平钢箱温度梯度疲劳应力谱分析方法

为研究扁平钢箱梁温度疲劳应力谱,以南溪长江大桥悬索桥主梁为研究对象,基于温度传感器长期实测数据,筛选实测数据的日温度极值,运用广义极值模型描述季节极值概率分布并采用极值外推方法得到设计基准期极值模型。引入拉丁超立方抽样(LHS)法对极值模型进行抽样,得到日温度极值样本。结合日温度极值样本和正弦函数模型,构建服役期内关注点的温度时程曲线。基于有限元ANSYS软件平台,分析不同温度梯度下关注点的应力效应,回归温度梯度与疲劳应力的线性关系式,依据温度梯度时程曲线与线性关系式模型,采用雨流计数法得到钢箱梁温度梯度疲劳应力谱。研究表明,模拟抽样生成的温度样本数据符合温度场的季节变化特征,样本概率模型与实测数据概率模型相对吻合。关注点温度梯度疲劳应力谱能够为扁平钢箱梁疲劳寿命设计提供参考。     

A new simple method of implicit time integration for dynamic problems of engineering structures

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai’s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems. The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China (G2000048702, 2003CB716707), the National Science Fund for Distinguished Young Scholars (10025208), the National Natural Science Foundation of China (Key Program) (10532040), the Research Fund for Oversea Chinese (10228028). The English text was polished by Keren Wang.     

A method for calculating the lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system

The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimensionD _L ^(A) of the autonomous system, the definition of the Lyapunov dimensionD _L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely,D _L ^(A) −D_L=1. For a quasi-periodically excited dynamical system, similar conclusions are formed. Project supported by the National Natural Science Foundation of China (No. 19772027), the Science Foundation of Shanghai Municipal Commission of Education (99A01) and the Science Foundation of Shanghai Municipal Commission of Science and Technology (No. 98JC14032).