A new visco–elasto-plastic model via time–space fractional derivative

To characterize the visco–elasto-plastic behavior of metals and alloys we propose a new constitutive equation based on a time–space fractional derivative. The rheological representative of the model can be analogous to that of the Bingham–Maxwell model, while the dashpot element and sliding friction element are replaced by the corresponding fractional elements. The model is applied to describe the constant strain rate, stress relaxation and creep tests of different metals and alloys. The results suggest that the proposed simple model can describe the main characteristics of the experimental observations. More importantly, the model can also provide more accurate predictions than the classic Bingham–Maxwell model and the Bingham–Norton model.     

一种新的聚合物基复合材料应力松弛经验模型

视黏弹性元件弹簧和粘壶的特性参数为时间的函数,建立双变参Maxwell模型,求解其本构方程及松弛函数,探讨了其松弛函数与经验KWW函数的关系,提出了一种新的应力松弛经验模型。运用最小二乘原理,建立该模型参数的确定方法。开展了玻璃纤维增强聚合物(GFRP)复合材料的长期应力松弛实验,并通过实验数据的分析验证了该模型的正确性。研究表明,新应力松弛经验模型拟合曲线与实测曲线吻合较好,相关系数达到96%以上,说明了该模型适合描述GFRP复合材料的黏弹性松弛特性。     

Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as relaxation test is well fitted by power law decay then the fractional constitutive law involving Caputo’s derivative directly appears. It will be shown that fractional model is proper for studying visco-elastic behavior, since it may capture both relaxation and creep tests, requiring the identification of two parameters only. This consideration is assessed by the good agreement between experimental tests on creep and relaxation and the fractional model proposed. Experimental tests, here reported are performed on two polymers having different chemical physical properties such that the fractional model may cover a wide range of visco-elastic behavior.     

Multi-scale modeling of time-dependent response of smart sandwich constructions

Polymer and polymer based composite structures exhibit time-dependent response, leading to their being described as viscoelastic bodies. The rate of creep (or stress relaxation) in viscoelastic bodies increases with increasing the temperature of the bodies. In this study, we are interested in analyzing the time-dependent response of smart sandwich composites comprising of glass fiber reinforced polymer (GFRP) skins, polyurethane foam core, and lead zirconate titanate (PZT) wafers embedded in the GFRP skins. The PZT is used to monitor lifetime performance of sandwich composites. A multi-scale model is developed to integrate different constitutive models of the constituents in the sandwich structures. Quasi-static and creep tests are conducted for bulk epoxy, GFRP, polyurethane foam, and sandwich specimens under uniaxial tension and bending. The tests were done at room temperature and at 80 °C. The experimental data are used for material characterization and model verification. The multi-scale model that is developed can be used to understand the effect of different responses of the constituents on the overall time-dependent behavior of sandwich structures and examine the feasibility of using PZT wafers for monitoring lifetime performance of sandwich structures.     

Nonlinear constitutive model for time-dependent behavior of FRP-concrete interface

The constitutive behavior of the FRP-concrete interface is a prime issue when evaluating the strengthening effects of concrete structures strengthened with externally bonded fiber-reinforced polymer (FRP) sheets or plates. This paper is devoted to developing a new nonlinear viscoelastic model for the study of the long-term behavior of the FRP-concrete interface. The model has the ability to describe the creep of the FRP-concrete adhesive layer and the creep fracture propagation along the FRP-concrete interface. The linear viscoelastic behavior of the FRP-concrete interface is taken into account by using Maxwell’s generalized rheological model through a step-by-step time increment procedure. The nonlinear time-dependent behavior of the adhesive layer is considered through the micro slip criterion model (MSCM), which depends on a displacement failure criterion and the interfacial fracture energy. The proposed model is implemented with a finite element model and it has been calibrated by using the results of time-dependent double-lap shear test specimens. The results demonstrate the reliability of the proposed model and its capability to predict the time-dependent debonding propagation along the FRP-concrete interface. It is also shown that the model can be used to predict a wide range of creep fractures not only under low sustained loading but also under high sustained loading.     

分数算子描述的粘弹性材料的本构关系研究

从分数导数的定义出发,提出了在经典粘弹性模型理论中采用Abel粘壶取代传统牛顿粘壶的新观点.将有机硅高分子材料在MTS831.10材料试验机上进行动态力学行为试验,对试验结果分别用经典粘弹性模型和分数导数模型进行拟合.结果表明,分数导数Kelvin模型可以同时精确地拟合高分子材料的存储模量和损耗模量随频率变化的曲线,而且其形式简单、统一,在计算过程中需要调整的参数很少.     

服从分数导数Kelvin本构模型的粘弹性阻尼器的阻尼性能分析及试验研究

从分数导数的定义出发,提出了在经典粘弹性模型理论中采用Abel粘壶取代传统牛顿粘壶的新观点.将高分子粘弹性阻尼器在MTS831.10材料试验机上进行动态力学行为试验,对试验结果用分数导数模型进行拟合.结果表明,分数导数Kelvin模型可以同时精确地拟合高分子材料的存储模量和损耗模量随频率变化的曲线,而且其形式简单、统一.在计算过程中需要调整的参数很少.最后将分数导数模型引入静态特性公式,得出了圆筒状粘弹性阻尼器动态刚度与阻尼的表达式.     

Polynomial operators,stieltjes convolution,and fractional calculus in hereditary mechanics

Summary Fractional calculus is used to describe the general behavior of materials with memory. An expression for the fractional derivative or the fractional integral is developed in terms of the Stieltjes convolution and the Riesz distribution. The general fractional calculus polynomial operator constitutive equation is reduced to a Stieltjes convolution. A constitutive equation which depends on a memory parameter for an isotorpic viscoelastic material is presented. The proposed creep compliance has an initial response, a primary creep region, a secondary creep region and a tertiary creep region. The corresponding relaxation modulus has a glassy region, a leathery region, a rubbery region and a liquid region.     

分数阶导数线性流变固体模型及其应用

采用Koeller 弹壶元件替代标准线性固体模型中的Newton黏壶, 得到分数阶导数线性流变固体模型, 给出了表征模型动态黏弹特性的存储模量、损耗模量和损耗因子以及表征模型静态黏弹特性的蠕变柔量和松弛模量。采用分数阶导数线性流变固体模型、标准线性固体模型和五参量固体模型对聚丙烯材料应力松弛特性进行分析。讨论了Mittag-Leffler函数的求和截断误差。结果表明分数阶导数线性流变固体模型能更准确描述聚丙烯材料的应力松弛行为。     

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.     

A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers

We formulate a constitutive framework for biodegradable polymers that accounts for nonlinear viscous behavior under regimes with large deformation. The generalized Maxwell model is used to represent the degraded viscoelastic response of a polymer. The large-deformation, time-dependent behavior of viscoelastic solids is described using an Ogden-type hyperviscoelastic model. A deformation-induced degradation mechanism is assumed in which a scalar field depicts the local state of the degradation, which is responsible for the changes in the material’s properties. The degradation process introduces another timescale (the intrinsic material clock) and an entropy production mechanism. Examples of the degradation of a polymer under various loading conditions, including creep, relaxation and cyclic loading, are presented. Results from parametric studies to determine the effects of various parameters on the process of degradation are reported. Finally, degradation of an annular cylinder subjected to pressure is also presented to mimic the effects of viscoelastic arterial walls (the outer cylinder) on the degradation response of a biodegradable stent (the inner cylinder). A general contact analysis is performed. As the stiffness of the biodegradable stent decreases, stress reduction in the stented viscoelastic arterial wall is observed. The integration of the proposed constitutive model with finite element software could help a designer to predict the time-dependent response of a biodegradable stent exhibiting finite deformation and under complex mechanical loading conditions.     

竖向集中力作用下分数导数型半无限体粘弹性地基变形分析

研究成果表明地基土体具有粘弹性性质,在长期荷载作用下会产生蠕变变形。为了准确预测地基变形,必须建立精确的土体本构模型。分数导数粘弹性模型具有精确度高,确定模型所需的实验参数少,应用范围广的特点。针对半无限体粘弹性地基,采用分数导数粘弹性本构模型模拟土的力学行为,运用弹性-粘弹性对应原理分析了粘弹性地基的变形。研究发现,采用分数导数粘弹性本构模型得到的地基沉降量较经典粘弹性模型要小,经典粘弹性模型不能很好地反映集中力对周围地基沉降的影响以及地基的蠕变性质。     

聚合物基复合材料有效蠕变响应与单轴拉伸行为的细观力学模拟

基于变分渐近均匀化理论框架建立表征线性黏弹性聚合物基复合材料有效蠕变响应和宏观应力-应变行为的细观力学模型。从线性黏弹性聚合物基复合材料本构方程中构建能量泛函变分表达式出发,采用变分渐近法求解线性黏弹性聚合物基复合材料的有效蠕变柔度系数,并以此为基础计算聚合物基复合材料的时变和单轴拉伸行为。通过算例验证了构建模型的适用性和准确性。由于所有计算均在时间域内完成,不再需要传统线黏弹性复合材料使用的Laplace转换和反演,计算效率大为提高。      

碳纤维增强树脂复合材料细观蠕变性能

碳纤维增强树脂复合材料以其优异的性能,在各领域得到广泛应用。由于树脂基体具有黏弹性,使其合成的复合材料也表现出黏弹性行为。蠕变是材料黏弹性行为中最典型的一类现象,因此对碳纤维增强树脂复合材料细观蠕变性能的研究具有重要意义。室温下利用纳米压痕技术对碳纤维增强树脂复合材料中的基体、界面及纤维相在不同峰值载荷下的细观蠕变行为进行分析。结果表明:在相同的蠕变时间下,最大载荷为2 mN和10 mN的纤维蠕变位移约为基体蠕变位移的1/3和1/2,界面的蠕变位移介于两者之间;稳态蠕变阶段的蠕变速率小于0.1%;基体、界面、纤维的蠕变应力指数分别为3.6、2.9和2.1。同时根据Kelvin-Voigt模型得到了基体、界面及纤维的第一、第二复数模量、黏度系数及蠕变柔量。      

Three-dimensional, finite deformation, viscoplastic constitutive models for polymeric materials

A general methodology for developing three-dimensional. finite deformation, viscoplastic constitutive models for polymeric materials is presented. The development begins with the presentation of a one-dimensional spring and dashpot construction which exhibits behavior typical of polymeric materials, namely strain-rate dependence, stress relaxation, and creep. The one-dimensional construction serves as a starting point for the development of a three-dimensional, finite deformation, viscoplastic constitutive model which also exhibits typical polymeric behavior. Furthermore, the three-dimensional constitutive model may be easily generalized to incorporate an arbitrary number of inelastic processes, representing (inelastic) microstructural deformation mechanisms operating on different time scales. Strain-rate dependence, stress relaxation, and creep phenomena are discussed in detail for a simple version of the constitutive model. Test data for a particular polymer is used to validate the simple model. It is concluded that the methodology provides a flexible approach to modeling polymeric materials over a wide range of loading conditions.     

Modeling short- and long-term time-dependent nonlinear behavior of polyethylene

A new methodology for developing macromechanical constitutive formulations for time-dependent materials is presented in this article. In particular, two phenomenological constitutive models for polymer materials are illustrated, describing time-dependent and nonlinear mechanical behavior. In this new approach, short-term creep test data are used for modeling both short-term and long-term responses. The differential form of a model is used to simulate typical nonlinear viscoelastic polymeric behavior using a combination of springs and dashpots. Unified plasticity theory is then used to develop the second model, which is a nonlinear viscoplastic one. Least squares fitting is applied for the determination of material parameters for both models, based on experimental results. Due to practical constraints, experimental data are usually available for short-term time-frames. In the presented proposed formulation, the material parameters determined from short-term testing are used to obtain material parameter relationships for predicting the long-term material response. This is done by extending short-term information for longer time frames. Finally, theoretical and experimental results of tensile tests on polyethylene subjected to various load levels and test times are compared and discussed. Very good agreement of the modeling results with experimental data shows that the developed formulation provides a flexible and reliable framework for predicting load responses of polymers.     

Circular Functions Based Comprehensive Analysis of Plastic Creep Deformations in the Fiber Reinforced Composites

Analytically based model is presented for behavioral analysis of the plastic deformations in the reinforced materials using the circular (trigonometric) functions. The analytical method is proposed to predict creep behavior of the fibrous composites based on basic and constitutive equations under a tensile axial stress. New insight of the work is to predict some important behaviors of the creeping matrix. In the present model, the prediction of the behaviors is simpler than the available methods. Principal creep strain rate behaviors are very noteworthy for designing the fibrous composites in the creeping composites. Analysis of the mentioned parameter behavior in the reinforced materials is necessary to analyze failure, fracture, and fatigue studies in the creep of the short fiber composites. Shuttles, spaceships, turbine blades and discs, and nozzle guide vanes are commonly subjected to the creep effects. Also, predicting the creep behavior is significant to design the optoelectronic and photonic advanced composites with optical fibers. As a result, the uniform behavior with constant gradient is seen in the principal creep strain rate behavior, and also creep rupture may happen at the fiber end. Finally, good agreements are found through comparing the obtained analytical and FEM results.     

Time-dependent and residual behavior of pultruded GFRP beams subjected to sustained intensities and cold temperature

This paper presents the time-dependent response and residual behavior of pultruded glass fiber reinforced polymer (GFRP) beams. A total of nine beams are tested in four-point bending. One beam is served as control and eight beams are loaded to failure after exposing to three levels of sustained intensities (20%, 40%, and 60% of the static capacity) at room (25 °C) and cold (− 30 °C) temperatures for 2000 h. Time-dependent material parameters are obtained from the test. Analytical approaches are used to predict the behavior of test beams, based on mechanics-based failure criteria and Findley's creep theory. Three-dimensional finite element models are also developed, based on the experimentally obtained material parameters. The GFRP beams demonstrate time-dependent material degradation due to the sustained load. Cold temperature alters the load-carrying capacity and creep response of the beams. Brittleness of the GFRP is accelerated when the beams are exposed to sustained intensities and cold temperature. The contribution of shear deformation to the deflection of the beams increases with sustained load. Although the proposed modeling approaches agree with the experiment, further development is recommended to account for micro-level material deterioration characteristics.