Stability Analysis of Maxwell Viscoelastic Pipes Conveying Fluid with Both Ends Simply Supported

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热环境中黏弹性功能梯度材料及其结构的蠕变

基于经典的对应原理, 将 Mori-Tanaka 方法等细观力学结果推广于定常温度环境下的黏弹性情形. 根据泊松比与时间呈弱相关的特点, 给出 Laplace 象空间中功能梯度材料的松弛模量和热膨胀系数, 并直接建立耦合热应变的多维黏弹性本构关系. 在此基础上, 求解黏弹性功能梯度圆柱薄壳在热环境中的轴对称弯曲蠕变变形问题. 考虑材料热物参数的温度相关性, 首先确定稳态温度场, 导出相空间中轴对称弯曲变形的解析解, 采用数值反演得到蠕变变形. 算例表明, 蠕变初期, 热环境的影响明显, 随着时间增加, 热应力松弛, 影响逐渐消失. 当圆柱薄壳受轴压时, 相比于两端固支, 两端简支的端部变形更加明显. 通过圆柱薄壳的轴对称弯曲求解, 给出体积含量呈任意分布的黏弹性功能梯度结构在热机载荷下的蠕变分析途径.      

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发,推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程,采用改进的有限差分法,分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

纤维沥青混凝土FRA黏弹性预测模型

纤维沥青混凝土FRA(Fiber reinforced asphalt concrete)的力学行为取决于沥青混合料和纤维的物理属性及几何特征。通过三参数固体模型,考虑纤维的几何特性,构造了一种预测FRA黏弹性行为模型,推导出该模型的本构方程、蠕变柔量和松弛模量表达式。采用该模型研究了纤维体积分、比半径和弹性模量等对FRA松弛模量和蠕变柔量的影响。结果表明,纤维掺入量和比半径对FRA的松弛模量和蠕变柔量影响较大,而弹性模量影响较小。通过试验,验证了FRA预测模型具有较高的精度。在研究试验范围内,SMA-13沥青混凝土的聚丙烯腈纤维(PAN)最佳掺入量和长径比分别为0.3%～0.4%和600～900。     

A two parameter elasto-plastic formulation for hardening pressure-dependent materials

A new pressure-dependent yield function is proposed by introducing a plastic Poisson's ratio within the theoretical formulation of the plastic potential. In analogy with other classical models, an equivalent stress and an equivalent plastic strain increment are defined. Then, according to these definitions, the equivalent stress–strain curve is derived and an exponential hardening law is introduced. The advantage of the proposed formulation over alternative approaches relies in explicit closed-form expressions of the flow rules and of the plastic multiplier.     

From dynamic modulus via different relaxation spectra to relaxation and creep functions

The main goal of the paper is to compare predictive power of relaxation spectra found by different methods of calculations. The experimental data were obtained for a new family of propylene random copolymers with 1-pentene as a comonomer. The results of measurements include flow curves, viscoelastic properties, creep curves and rubbery elasticity of copolymer melts. Different relaxation spectra were calculated using independent methods based on different ideas. It lead to various distributions of relaxation times and their “weights”. However, all of them correctly describe the frequency dependencies of dynamic modulus. Besides, calculated spectra were used for finding integral characteristics of viscoelastic behaviour of a material (Newtonian viscosity, the normal stress coefficient, steady-state compliance). In this sense all approaches are equivalent, though it appears impossible to estimate instantaneous modulus. The most crucial arguments in estimating the results of different approaches is calculating the other viscoelastic function and predicting behaviour of a material in various deformation modes. It is the relaxation and creep functions. The results of relaxation curve calculations show that all methods used give rather similar results in the central part of the curves, but the relaxation curves begin to diverge when approaching the high-time (low-frequency) boundary of the relaxation curves. The distributions of retardation times calculated through different approaches also appear very different. Meanwhile, predictions of the creep curves based on these different retardation spectra are rather close to each other and coincide with the experimental points in the wide time range. Relatively slight divergences are observed close to the upper boundary of the experimental window. All these results support the conclusion about a rather free choice of the relaxation time spectrum in fitting experimental data and predicting viscoelastic behaviour of a material in different deformation modes. Received: 15 March 2000 Accepted: 18 September 2000     

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.     

Photomechanical behavior of cellulose acetate under viscoelastoplastic deformation

To confirm the possibilities of cellulose acetate as a material for a model analysis during viscoelastoplastic deformation, the time-dependent photomechanical properties of the material were examined by means of creep tests under constant stress and recovery tests after removal of stress. Consequently, though the strain and the fringe order of cellulose acetate during creep and recovery are greatly influenced by stress and room temperature, both of them can be described simply by a power function of time, and the coefficient of each of these formulas can be represented by a function of the ratio of active stress to yield stress only. The effect of temperature is included in the formulation of the yield stress. In addition, the strain and the fringe order can be represented by the viscous-viscoelastic model proposed by Findleyet al.,^1,2 in which both of them are divided into four components: elastic, plastic, time-dependent irrecoverable viscous and time-dependent recoverable viscoelastic. The relation between viscoelastic strain and viscoelastic fringe order, and the relation between viscous strain and viscous fringe order were verified to be equivalent to that between plastic strain and plastic fringe order, all of which do not depend on stress, temperature or time. Therefore, the strain distribution of cellulose acetate under viscoelastoplastic deformation can be determined directly from the value of the fringe order measured.     

Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model

Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melts, using a fractional model. In this paper the authors propose an advanced model for capturing the non-linear trend of the shear viscosity of polymer melts as function of the shear rate. Results obtained with the fractional model are compared with those obtained using a classical model which involves classical Maxwell elements. The comparison between experimental data and the theoretical model shows a good agreement, emphasizing that fractional model is proper for studying viscoelasticity, even if the material exhibits a non-linear behavior.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

Subcritical crack growth in an aging plate with variable elastic modulus

The paper addresses subcritical growth of a crack in a thin isotropic plate made of an aging viscoelastic material with time-dependent elastic modulus. The behavior of the material is described by Arutyunyan’s creep theory. To simulate fracture, a modified Leonov–Panasyuk–Dugdale model and a critical crack opening displacement criterion are used. An equation describing the subcritical growth of the crack is derived assuming that Poisson’s ratio is constant. As an example, the critical loads are determined, and curves of subcritical crack growth are plotted for a specific material. The results are compared with the case of constant elastic modulus     

Nonlinear viscoelastic-degradation model for polymeric based materials

This study presents a phenomenological constitutive model for describing response of solid-like viscoelastic polymers undergoing degradation. The model is expressed in terms of recoverable and irrecoverable time-dependent parts. We use a time-integral function with a nonlinear integrand for the recoverable part and another time-integral function is used for the irrecoverable part, which is associated with the degradation evolution in the materials. Here, the degradation is attributed to the secondary and tertiary creep stages. An ‘internal clock’ concept in viscoelastic materials is used to incorporate the accelerated failure in the materials at high stress levels. We ignore the effect of heat generation due to the dissipation of energy and possible healing in predicting the long-term and failure response of the polymeric materials. Experimental data on polymer composites reported by Drozdov (2011) were used to characterize the material parameters and validate the constitutive model. The model is shown capable of predicting response of the polymer composites under various loading histories: creep, relaxation, ramp loading with a constant rate, and cyclic loadings. We observed that the failure time and number of cycles to failure during cyclic loadings are correlated to the duration of loading and magnitude of the prescribed mechanical loads. A scalar degradation variable is also introduced in order to determine the severity of the degradation in the materials, which is useful to predict the lifetime of the structures subject to various loading histories during the structural design stage.     

Modeling and measuring visco-elastic properties: From collagen molecules to collagen fibrils

Collagen is the main structural protein in vertebrate biology, determining the mechanical behavior of connective tissues such as tendon, bone and skin. Although extensive efforts in the study of the origin of collagen exceptional mechanical properties, a deep knowledge of the relationship between molecular structure and mechanical properties remains elusive, hindered by the complex hierarchical structure of collagen-based tissues. Understanding the viscoelastic behavior of collagenous tissues requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar and molecular level. Hence, we investigate the viscoelastic properties at the molecular level by using an atomistic modeling approach, performing in silico creep tests of a collagen-like peptide. The results are compared with creep and relaxation tests at the level of isolated collagen fibrils performed previously using a micro-electro-mechanical systems platform. Individual collagen molecules present a non-linear viscoelastic behavior, with a Young's modulus increasing from 6 to 16 GPa (for strains up to 20%), a viscosity of 3.84±0.38 Pa s, and a relaxation time in the range of 0.24–0.64 ns. At the fibrils level, stress–strain–time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell–Weichert model. The fibrils showed an elastic modulus of 123±46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell–Weichert model with a fast time response of 7±2 s and a slow time response of 102±5 s.     

Effect of pressure-dependent slip on flow curve multiplicity

Various microstructural pictures for slip at polymer/solid interfaces lead to relations which have a region where multiple values of slip velocity are predicted for the same shear stress. This leads to the expectation of multivalued flow curves, which has been verified in specific cases by numerous researchers. We study the effect of pressure dependence on flow curve multiplicity using a simple multivalued slip relation to model the phenomena of hysteresis and spurt flow in polymer extrusion. A continuation technique is used to trace out the boundaries of the region of flow curve multiplicity as pressure drop and die length to diameter (L/D) ratio are changed. Results for Newtonian, shear thinning and viscoelastic constitutive equations show that, despite the multivalued nature of the slip model, multiplicity (and thus hysteresis) is absent at high L/D.  For the sake of completeness, we also carry out time-dependent simulations at constant piston speed taking fluid compressibility into account. These simulations show that oscillations in the pressure drop and exit volumetric flow rate result only if the system is operated in the multiplicity region of the steady state flow curve, in agreement with the results of similar simulations by researchers using various multivalued slip models without pressure dependence. The results demonstrate that a multivalued slip model does not guarantee multiplicity in the flow curve for the constant pressure drop operation, nor oscillations for constant piston speed operation. Received: 18 August 1997 Accepted: 30 March 1998     

Analytical parametric analysis of the contact problem of human buttocks and negative Poisson's ratio foam cushions

Analytical investigations on the contact problems between two homogeneous and isotropic soft bodies were performed to simulate the contact of human buttocks and seat cushions. The cushion materials' Poisson's ratio were allowed to be negative. The human buttocks were modeled as an ideal sphere with radius 15 cm, and assumed to have a low Young's modulus and a Poisson's ratio close to 0.5. These parameters were held constant during our analysis. Peak contact pressure was reduced by adjusting the contour curvature of cushions according to Hertz theory, as expected. Moreover, analysis by both the Hertz model and a finite thickness 3D elasticity model showed that using negative Poisson's ratio cushions could further reduce the pressure. Negative Poisson's ratio cushions may be beneficial in the prevention of pressure sores or ulcers in the sick and in reduction of pressure-induced discomfort in seated people.     

Remarks on properties of photoviscoelastic model materials

One of the basic problems of structural-model analysis, model photoelasticity and photoelastic coatings in the problem of mechanical and optical creep, relaxation and related phenomena. It is pointed out that, in spite of creep or relaxation, it is possible to achieve physical similarity between model and object if the model material behaves in a certain range as a linear viscoelastic material. Such a material is called a “momentarily linear material.” Several model materials behave in this way in a certain range of stress and time. Because of creep and relaxation, the common tensile tests are, in general, not quite adequate for evaluation of physical properties of plastics used for models. Also the bending test is not always adequate. It is shown how to obtain sufficiently accurate relations between stress, strain, birefringence and time, using tapered specimens. The problem of biaxial creep of model materials is discussed, and a simple method of evaluating the suitability of a given plastic as a model material is shown. Some conclusions concerning time-dependent factors are formulated, and some possible areas of investigation are shown.     

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

     

An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains