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.     

Calculation of discrete relaxation modulus and creep compliance

Müntz theorems from classical approximation theory are used to lay the mathematical foundations for discrete spectral representations of the relaxation modulus and creep conpliance. Algorithms that can automatically generate nonnegative discrete relaxation and retardation strengths are formulated by using the nonnegative least square method. Received: 30 September 1998 Accepted: 20 May 1999     

Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance

We describe a recursive computer algorithm which generates line spectra from relaxation modulus or creep compliance data without producing negative spectrum lines. We apply the algorithm here to data read from mathematical models for the relaxation modulus. Since these data were thus free of the usual experimental error, we could use a relatively simple form of the basic algorithm that is applicable also to smoothed data. The spectra faithfully reproduced the input functions and may serve for data storage as well as for predicting other experimental responses.     

Investigation of relaxation properties of polymer melts by comparison of relaxation time spectra calculated with different algorithms

In this paper a recently introduced algorithm (Brabec and Schausberger, 1995) for the calculation of relaxation time spectra is compared with two standard methods, i.e., Weese's regularization, and Baumgaertel's and Winter's regression algorithm. A reasonable agreement between those three algorithms is found for the relaxation properties of mono-, polydisperse, bi-, and multimodal polystyrene samples. All three numerical methods reproduce the relaxation properties for long and medium times correctly, but they show some disagreement at short times because of sparse experimental data. The high numerical accuracy opens the possibility to test and improve the physical models which underlie the calculations. The good agreement of the different algorithms suggests that small inconsistencies to physical models are not due to a failure of the numerical methods, but due to an insufficiency of the generalized Maxwell model.     

Calculation of discrete retardation spectra from creep data — II. Analysis of measured creep curves

An algorithm for the calculation of discrete, logarithmic equidistant retardation spectra from creep or recovery is applied to experimental data in different regions of consistency. Spectra in the glass-rubber transition region are given for technical poly(styrene), poly(methylmethacrylate), and poly(carbonate) as well as the course of all characteristic compliance type functions. The spectrum in the terminal region of a poly(styrene) of narrow molecular weight distribution is calculated both from creep and recovery data. The course of the dynamic moduli calculated from the spectra and by direct conversion is found in excellent agreement to measurements by means of a dynamic viscometer.Dedicated to Prof. Dr. H. Janeschitz-Kriegl at the occassion of his 70th birthday.     

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.     

大尺度网格下DSMC仿真碰撞概率的修正方法

从DSMC碰撞概率的实现途径出发,通过引入最小二乘法及亚松弛的思想给出一种大尺度网格下分子仿真碰撞概率的修正方法,将该方法应用于非结构四面体网格下的DSMC模拟中,计算分析了文献中钝锥模型的绕流结果,数值结果初步验证了本文方法的合理性和有效性。     

基于变形场测量数据主元压缩的模型参量反求方法

利用主元分析法对数字图像相关技术所测结构变形信息进行数据压缩,并进一步用于力学模型未知参量的反求计算。首先,为降低数字图像相关技术所测庞大数据的应用成本,提出利用主元分析法对结构表面变形场数据进行压缩,实现在保留结构表面变形信息主要特征的前提下显著降低数据量的目的;其次,针对压缩后的数据建立了基于最小二乘法的力学模型参量反求模型,并利用高斯牛顿法进行求解;最后,以具体算例从计算精度、收敛速度和抗噪性等方面验证了数据压缩对模型参量反求的效果。研究结果表明,所提方法在显著降低使用数据量的前提下,能够有效提高力学模型参量反求计算的收敛速度,特别是对于包含多个模型参数的反求问题,具有较高的精度和较好的稳定性。     

A delta-function method for the n-th approximation of relaxation or retardation time spectrum from dynamic data

A function series g(x; n, m) is presented that converges in the limiting case n and m = constant to the delta-function located at x = = 1. For every finite n, there exists 2n+1(–nmn) approximations of the delta-function _(n)(x–x _n,m ). x _n,m is the argument where the function reaches its maximum. A formula for the calculation is given.The delta-function approximation is the starting point for the approximative determination of the logarithmic density function of the relaxation or retardation time spectrum. The n-th approximation of density functions based on components of the complex modulus (G*) or the complex compliance (J*) is given. It represents an easy differential operator of order n.This approach generalizes the results obtained by Schwarzl and Staverman, and Tschoegl. The symmetry properties of the approximations are explained by the symmetry properties of the function g(x; n, m). Therefore, the separate equations for each approximation given by Tschoegl can be subsumed in a single equation for G and G, and in another for J and J.     

An erosion model for the discrete element method

A shear impact energy model (SIEM) of erosion suitable for both dilute and dense particle flows is proposed based on the shear impact energy of particles in discrete element method (DEM) simulations. A number of DEM simulations are performed to determine the relationship between the shear impact energy predicted by the DEM model and the theoretical erosion energy. Simulation results show that nearly one-quarter of the shear impact energy will be converted to erosion during an impingement. According to the ratio of the shear impact energy to the erosion energy, it is feasible to predict erosion from the shear impact energy, which can be accumulated at each time step for each impingement during the DEM simulation. The total erosion of the target surface can be obtained by summing the volume of material removed from each impingement. The proposed erosion model is validated against experiment and results show that the SIEM combined with DEM accurately predicts abrasive erosions.     

A meshless method of a thin plate

A meshless approach to analysis of arbitrary Kirchhoff plates by the local boundary integral equation(LBIE) method is presented. The method combines the advantageous features of, all the three methods: the Galerkin finite element method (GFEM), the boundary element method (BEM) and the element-free Galerkin method (EFGM). It is a truly meshless method, which means that the discretization is independent of geometric subdivision into elements or cells, but is only based on a set of nodes (ordered or scattered) over a domain in question. It involves only boundary integration, however, over a local boundary centered at the node in question; It poses no difficulties in satisfying the essential boundary conditions while leading to banded and sparse system matrices using the moving least square (MLS) approximations. It is shown that high accuracy can be achieved for arbitrary geometries for clamped and simply-supported edge conditions. The method is found to be simple, efficient, and attractive. Project supported by the National Science Foundation of China (No. 19972019).     

An improved algorithm for calculating relaxation time spectra from material functions of polymers with monodisperse and bimodal molar mass distributions

Using the basic concept of Emri and Tschoegl, the algorithm for calculating relaxation time spectra has been improved such that excellent results are provided in the difficult case of polymers with narrow molar mass distributions. These spectra can be compared with those calculated by nonlinear regularization (Weese 1992), which we regard as a very exact method, and show equally good results with even less mathematical effort. Examples of dense relaxation time spectra (up to eight points per decade) are given for nearly monodisperse polystyrene melts and for mixtures of these up to four components. The relaxation time spectra describe the dynamic mechanical experimental data in each case with an overall error of less than 3%. The filtering method used to avoid physically senseless oscillations has been proven to resolve the characteristic peaks contributed by monodisperse polymers accurately.     

A discrete element model for simulating saturated granular soil

A numerical model is developed to simulate saturated granular soil, based on the discrete element method. Soil particles are represented by Lagrangian discrete elements, and pore fluid, by appropriate discrete elements which represent alternately Lagrangian mass of water and Eulerian volume of space. Macro-scale behavior of the model is verified by simulating undrained biaxial compression tests. Micro-scale behavior is compared to previous literature through pore pressure pattern visualization during shear tests. It is demonstrated that dynamic pore pressure patterns are generated by superposed stress waves. These pore-pressure patterns travel much faster than average drainage rate of the pore fluid and may initiate soil fabric change, ultimately leading to liquefaction in loose sands. Thus, this work demonstrates a tool to roughly link dynamic stress wave patterns to initiation of liquefaction phenomena.     

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.     

An improved sample-and-hold reconstruction procedure for estimation of power spectra from LDA data

Techniques for deriving the auto or power spectrum (PSD) of turbulence from laser Doppler anemometry (LDA) measurements are reviewed briefly. The low pass filter and step noise errors associated with the sample-and-hold process are considered and a discrete version of the low pass filter for the resampled signal is derived. This is then used to develop a procedure by which the PSD estimates obtained from sample and hold measurements can be corrected. The application of the procedures is examined using simulated data and the results show that the frequency range of the analysis can be extended beyond the Nyquist frequency based on the mean sample rate. The results are shown to be comparable to those obtained using the method of Nobach et al. (1998) but the new procedures are more straightforward to implement. The technique is then used to determine the PSD of real LDA data and the results are compared with those from a hot wire anemometer.     

A method for the evaluation of design limits for structural materials in a cyclic state of creep

The purpose of the present paper is to demonstrate how the minimum theorems proposed in an accompanying paper (Ponter and Boulbibane, 2002) can be utilised in the prediction of the deformation and life assessment of structures subjected to cyclic mechanical and thermal loadings. The developed method, which is based upon bounding theorems and an associate programming method, the Linear Matching method, takes into account the changes in residual stress field occurring within a cycle. Although the solution provided a bound on the inelastic work, it also appears that generally the displacements predicted by this solution are smaller than those that would be predicted by the rapid cycle solution. By way of illustration a simple non-linear viscous model is adopted and a number of solutions are presented involving a Bree plate problem subjected to cyclic histories of load and temperature. An elastic follow-up factor is identified as a key design parameter for high temperature dwell periods.     

松弛方法在计算凝聚炸药爆轰问题中的应用

利用松弛近似,将非线性的凝聚炸药爆轰控制方程转化为线性的松弛方程组,并采用五阶WENO格式和五阶线性多步显隐格式对线性松弛方程组进行空间方向和时间方向的离散,由此建立具有高精度和高分辨率性质的计算凝聚炸药爆轰的松弛方法。建立的松弛方法可以避免求解Riemann问题及计算非线性通量的Jacobi矩阵,同时无需分裂处理反应源项。通过对凝聚炸药的平面一维定常爆轰波结构及球面一维聚心、散心爆轰起爆和传播过程的数值模拟,验证了所建立的松弛方法能够很好地计算凝聚炸药爆轰问题。