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.     

Bending test for capturing the vivid behavior of giant reeds,returned through a proper fractional visco-elastic model

This paper presents results of experimental investigations made to evaluate the vivid behavior of giant reed Arundo donax. In particular, attention was paid to the relationship between visco-elastic properties and moisture content, which is widely recognized as one of the key factor that influences the mechanical properties of all wood-based materials. To this aim, after a controlled drying treatment on samples of reed, stress relaxation tests in three point bending configuration were performed to evaluate the effects of moisture content on visco-elastic behavior of the giant reed. Further, the novel aspect of this paper is that of using an Euler–Bernoulli model embedded with an advanced visco-elastic constitutive law to fit experimental data of bending tests. Such a model of continuum beam takes into account different constitutive laws of viscoelasticity, being natural materials visco-elastic.     

A computational comparison of the inelastic constitutive models of hart and miller

Summary The uniaxial response behavior of Hart's and Miller's nonelastic constitutive equations is compared. These models have been selected because they are fully developed and have been applied on the uniaxial nonelastic response behavior of different materials. Among these for stainless steel AISI 316 the complete set of material parameters for both models has been published. Based on these parameter sets a comparison of both models is performed including monotonic strain controlled tensile tests, creep tests, load relaxation tests and cyclic tests. The predictions of both models are compared with available experimental data.Both models can not describe the whole range of experimental data. For Hart's model one essential flow parameter had to be adjusted to obtain a reasonable simulation of creep experiments. Further it gives unrealistic predictions for strain cycling. The incorporation of a so-called negative strain rate sensitivity severly restricts the practical applicability of Miller's model. Additionally in the high temperature regime the response curves for load relaxation tests deviate considerably from the experimentally observed ones at low strain rates. Both models have to be improved for practical applications.With 11 Figures     

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.     

分数指数模型的热力学分析及其应用

本文论证了两种经典粘弹性固体模型的等价性并指出了其存在的问题。给出了热力学对分数指数模型 ^[1]参数的限制条件。计算与实验结果比较表明:因为该模型具有适当多的参数,采用同一组参数可以做到同时与同一材料的蠕变和松弛试验结果很好吻合;并能做到松弛模量和蠕变柔量的Stieltjes卷积近似等于单位阶跃函数;在很宽广的频率范围内能同时很好地模拟真实材料的存储模量和损耗模量。由于其计算速度快,能与大多数真实材料的性能实验结果相拟合,可以广泛应用于工程实际中的粘弹性静力和动力问题的计算。     

Simple method for obtaining viscoelastic parameters of polymeric materials by incorporating physical-aging effects

A simple method for obtaining viscoelastic parameters from the results of static tensile tests is presented herein. Viscoelastic parameters were obtained by fitting experimental results and calculated results based on the power law model and linear viscoelasticity. The static tensile tests were carried out at various pre-aging times and the effect of physical aging was determined. The data confirmed that the physical aging process has a significant effect on the viscoelastic behavior. A creep test was conducted in order to discuss the validity of the prediction using the results of the static tensile test. It was confirmed that the predictions based on the viscoelastic parameters obtained from static tensile tests cannot adequately model actual viscoelastic behavior. The effective time theory was incorporated into the prediction in order to account for the progress of physical aging. It was verified that incorporating effective time theory into the prediction allows for the precise prediction of the long-term viscoelastic behavior.     

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.     

A cyclic viscoplastic and creep damage model for lead free solder alloys

The reliability of microelectronic components under cyclic thermomechanical loading is an important problem especially for new leadfree solder alloys. To investigate the low cycle fatigue strength of solder joints, material models are required, that can describe the constitutive inelastic deformation and damage behavior of solder materials. Such models form the basis for advanced numerical analyses by the finite element method. In the present contribution an appropriate material model that combines the viscoplastic constitutive model of Chaboche-type with the damage law of A.C.F. Cocks for porous creep will be introduced. The algorithm is reported for an implementation as a user defined material subroutine into the FEM-code ABAQUS®. The necessary parameters of the material model are identified using results of miniaturized double lap-shear experiments and tensile tests for a Sn96Ag3Cu1 solder alloy at various temperatures. The comparison of experimental and numerical results shows a good agreement with respect to strain rate sensitivity, relaxation and damage behavior of the investigated solder material. Finally, some numerical applications to surface mounted microelectronic devices are presented.     

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.     

Mathematical description of the mechanical behaviour of metallic materials under creep conditions

The metallic materials creep behaviour has been described and a complete model is presented. The basic constitutive equation, as well as the structure parameters, have been derived from a mathematical analysis that represents the dominant physical procedures and mechanisms. The model is very general because it is referred to all stages of creep and describes the creep behaviour of all metallic materials, including those strengthened by a dispersion of second-phase particles. A creep function has been derived from the constitutive equation describing all three stages of creep under constant loading. The function has the minimum possible number of fitting, parameters. The dependence of the fitting parameters on the loading conditions has been described using very simple mathematical relations. Applications and predictions have been carried out in a wide range of metallic materials. Good agreement has been shown by a comparison made also between the creep curves determined experimentally, and those obtained from creep function and determined fitting parameters.     

Viscoelasticity, viscoplasticity, and creep failure of polypropylene/clay nanocomposites

Observations are reported on polypropylene/clay nanocomposites in tensile tests with various strain rates, relaxation tests at various strains, and creep tests with various stresses at room temperature. New constitutive equations are derived in viscoelasticity and viscoplasticity of nanocomposites. Adjustable parameters are found by fitting the experimental data. The stress–strain relations are applied to the analysis of creep rupture. It is demonstrated that reinforcement of polypropylene with 1 wt.% of nanoclay induces an increase in time to failure by an order of magnitude.     

Viscoplastic analysis of structural polymer composites using stress relaxation and creep data

A structural carbon based composite material has been investigated for its high temperature viscoplastic properties using a model based on an overbearing stress concept and using the data obtained from load relaxation and creep. The time dependent viscoplastic properties were obtained at several load and temperature levels. An elastic–viscoplastic constitutive model (proposed by Gates) was used for the modeling efforts. The model is based on an overstress concept appropriate to inelastic properties of composites. The materials parameters for the model are obtained from a set of load relaxation experiments. The model predictions have been compared to the results of creep tests. The results show that the model is capable of predicting the creep behavior at shorter time periods and lower temperatures. As the temperature is increased or as the creep is prolonged the model predictions deviate from the experimental results.     

考虑温度效应的沥青混合料参数模型

选取修正Burgers模型作为沥青混合料的粘弹性本构模型,基于最小二乘法原理对单轴压缩蠕变试验数据进行拟合,获得本构模型的粘弹性参数。考虑温度变化对沥青混合料粘弹性的影响,建立修正Burgers模型的粘弹性参数模型,运用实验数据拟合粘弹性参数模型的分项系数,理论计算考虑温度效应的蠕变柔量,与实验结果对比分析表明,参数模型有较好的适用性,拟合精度高。粘弹性参数模型能将有限的实验数据推广到其他温度工况,节约大量实验,具有很好适用性。     

Application of fractional derivative models in linear viscoelastic problems

Appropriate knowledge of viscoelastic properties of polymers and elastomers is of fundamental importance for a correct modelization and analysis of structures where such materials are present, especially when dealing with dynamic and vibration problems. In this paper experimental results of a series of compression and tension tests on specimens of styrene-butadiene rubber and polypropylene plastic are presented; tests consist of creep and relaxation tests, as well as cyclic loading at different frequencies. Experimental data are then used to calibrate some linear viscoelastic models; besides the classical approach based on a combination in series or parallel of standard mechanical elements as springs and dashpots, particular emphasis is given to the application of models whose constitutive equations are based on differential equations of fractional order (Fractional Derivative Model). The two approaches are compared analyzing their capability to reproduce all the experimental data for given materials; also, the main computational issues related with these models are addressed, and the advantage of using a limited number of parameters is demonstrated.     

膨胀土卸荷蠕变特性及其非线性蠕变模型

针对黏土蠕变的非线性性质,以成都黏土为研究对象展开蠕变试验,发现黏土变形包括瞬时弹性变形、衰减蠕变变形、稳态蠕变变形和加速蠕变变形;黏土长期弹性模量随时间和应力的增加非线性软化;黏滞系数随应力的增加非线性软化,随时间的增加非线性硬化。基于流变学理论、分数阶微积分理论和Harris衰减函数,分别构建了分数阶导数元件、非线性弹性元件和非线性黏滞元件,从而建立了形式简单、参数较少和概念清晰的非线性分数阶导数蠕变模型。将非线性分数阶导数蠕变模型和Burgers蠕变模型进行对比拟合分析,发现非线性分数阶导数蠕变模型各阶段的拟合结果更好,对黏土非线性蠕变的描述更合理,可准确地反映黏土蠕变全过程,表明了所建立非线性分数阶导数蠕变模型的科学合理性。     

Constitutive equations that describe creep stress relaxation for 316H stainless steel at 550°C

Abstract

The prediction of the stress relaxation behaviour of welding induced residual stresses in thick section 316H austenitic stainless steel welded component provides an input for quantifying reheat crack initiation observed in the heat affected zone. The cracks occur after service at a temperature range from 490 to 520°C. The present work reviews some of the widely applied stress relaxation models. The relative strengths and weaknesses of these existing models are discussed. An improved constitutive equation derived from a forward uniaxial creep deformation law is proposed. The relative importance of the parameters selected in the new constitutive model, when compared with experimental data, is discussed. The importance of a better understanding of the role of the internal stress and its measurement is highlighted.     

A formulation to model the nonlinear viscoelastic properties of the vascular tissue

Nearly all soft tissues, including the vascular tissue, present a certain degree of viscoelastic material response, which becomes apparent performing multiple relaxation tests over a wide range of strain levels and plotting the resulting stress relaxation curves, nonlinear viscoelasticity of the tissue. Changes in relaxation rate at each strain may occur at multiple strain levels. A constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these nonlinear viscoelastic phenomena is here presented and used to fit stress?Cstretch curves from experimental relaxation tests. This constitutive model was used to fit several data set of in vitro experimental stress relaxation tests performed on ovine and porcine aorta. The good fitting of the experimental data shows the capability of the model to reproduce the viscoelastic response of the vascular tissue.     

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.