Nonlinear constitutive equations for transient creep of viscoelastic polymers including the effect of hydrostatic pressure

The behaviour of polymers is known to be significantly influenced by the hydrostatic pressure in creep deformation or elastic-plastic deformation. The effect of the third stress invariant on the nonlinear viscoelastic deformation is much smaller than that of the hydrostatic pressure. In this paper, a constitutive equation for transient creep is proposed, which includes the effect of the hydrostatic pressure on the yield function. The creep and plastic strains or the creep strain rate converge to zero with increasing hydrostatic pressure. The proposed constitutive equation is in good agreement with the actual creep data of cellulose nitrate and cellulose acetate, under various combinations of superimposed tensile and hydrostatic loadings.     

Creep behaviour in unloading process of cellulose nitrate under superimposed tensile and hydrostatic loading

The creep behaviour in unloading process under various combinations of superimposed tensile and hydrostatic loading are quantitatively investigated using non-linear-viscoelastic cellulose nitrate heated at 65° C. The creep strain and the creep strain-rate in the unloading process are quite influenced by the effect of hydrostatic pressure, but are not so influenced as those in the loading process mentioned in a previous paper. The deformation properties in the unloading process are also discussed with experiments for proportional loading (namely, uniform rate of stress increases or decreases with time). The stress-strain relation in the unloading process of the creep behaviour under superimposed loading is deduced by using the invariant theory. The deduced relation gives good agreement with the actual observations under the superimposed loading.     

Mechanical behaviour and formulation of stress-strain relation for non-linear viscoelastic cellulose nitrate under cyclic loadings

The cyclic deformations under various repeated stresses are quantitatively investigated using non-linear viscoelastic cellulose nitrate heated to 60° C. The non-elastic strain or creep-plastic strain is remarkably influenced by the repeated stress and the stress rate. The cyclic deformations corresponding to the repeated stress less than a certain stress level attain the saturated state called the shake down after some cycles. The stress-strain relations of the non-linear viscoelastic media in the loading and unloading processes are deduced from the invariant theory using an hypothesis of creep potential. The non-linear viscoelastic observations obtained on the cellulose nitrate at 60° C under cyclic loadings are found to fit the deduced relations for the loading and unloading processes independent of the repeated stress and the stress rate.     

A Generalized Yield Criterion

A generalized yield criterion is proposed based on the metai plastic deformation mechanics and the fundamental formula in theory of plasticity. Using the generalized yield criterion, the reason is explained that Mises yield criterion and Tresca yield criterion do not completely match with experimental data. It has been shown that the yield criteria of ductile metals depend not only on the quadratic invariant of the deviatoric stress tensor J2, but also on the cubic invariant of the deviatoric stress tensor J3 and the ratio of the yield stress in pure shear to the yield stress in uniaxial tension κ/σs. The reason that Mises yield criterion and Tresca yield criterion are not in good agreement with the experimental data is that the effect of J3 and κ/σs is neglected.     

Cyclic tensile creep behaviour of cellulose nitrate and evaluation of constitutive equations

The cyclic creep behaviour under axial tension was quantitatively investigated using non-linear viscoelastic cellulose nitrate heated at 65°C. The instantaneous strains at the instants of loading and unloading for three creep cycles and the creep strain rates during the three creep cycles were found to be influenced by the cycle numbers. However, the effect of the cycle number on the loading process was quite different from that on the unloading process in the cyclic creep deformation. The evaluation of the creep constitutive equations for the loading and unloading processes at one loading cycle deduced in the previous papers is discussed for the cyclic creep deformation. The deduced creep equations give good agreement with the actual observations for the three creep cycles independent of the cycle numbers and of the creep stress levels.     

Constitutive equation for transient creep of cellulose nitrate under hydrostatic pressure

The behaviour of polymers is influenced by the hydrostatic pressure during the unloading creep period as well as during loading creep. Moreover, the behaviour during the unloading creep is also influenced by the value of the unloading strain at the final point of the loading process. In this paper, a constitutive equation for transient creep in the unloading process is proposed by using the same postulates as in the loading process of the previous paper, which includes the effect of the hydrostatic pressure and the effect of the unloading strain. The proposed unloading creep equation is in good agreement with the actual creep data on cellulose nitrate.     

Multiple-factor dependence of the yielding behavior to isotropic ductile materials

Extensively experimental evidences have shown that the yielding behaviors for many isotropic materials exhibit both the pressure and stress-state dependence. The strength-differential in tension and compression results not only from the hydrostatic stress, but also from the loading type. Different materials often demonstrate different yielding features in stress space even though they have the same ratio of compression yield-stress to tension yield-stress. Bases on a physical hypothesis introduced herein, a yield criterion is proposed to fulfill the experimental observations. The yield criterion can be represented finally with three independent invariants of the stress tensor and the deviatoric stress tensor. The yield criterion can well predict the yielding behavior arising from the yielding mechanism of the multiple-factor dependence. The yielding analysis with the effect of the hydrostatic stress does not have a limit when regardless of the strength-differential in tension and compression. The yield criterion assures the convexity in a wider range of material properties so that it can be used as the plastic potential in the implementation of predicting the subsequent yield surface. The yield criterion reveals a clearly transforming procedure with respect to its simplified form when applied to different materials from a general compressive isotropic material to an ideal incompressive isotropic material, back to the Mises’s criterion. Divers experiments on metallic and polymeric materials are compared. The results show that the new yield criterion is in fairly good agreement with all referred experiments.     

The macroscopic yielding behaviour of polymers in multiaxial stress fields

Experimental data on yielding of polyoxymethylene (POM) and polypropylene (PP) have been obtained in a wide range of complex triaxial stress conditions. These complex triaxial stress states have been produced by superimposition of simple stresses such as uniaxial compression, uniaxial tension and pure shear on hydrostatic pressure of various intensities. The actual yield surfaces of both polymers were constructed using the data. The yield surface of POM is a cone-shaped with a pointed apex and straight edges, while the yield surface of PP is cone-lime and non-linear, also with a pointed apex. A yield criterion is shown to very closely predict the observed behaviour of both polymers.     

Creep behaviour in cellulose nitrate under superimposed tensile and hydrostatic loading

The creep behaviour under various combinations of superimposed tensile and hydrostatic loading are quantitatively investigated using nonlinear-viscoelastic cellulose nitrate heated at 65° C. The creep strain and the creep strain-rate are seriously affected by the effect of hydrostatic pressure. The trends of behaviour under superimposed tensile and hydrostatic loading cannot be predicted by the concepts which apply to uniaxial loading. The stress—strain relation of the creep behaviour under superimposed loading is deduced from the invariant theory using an hypothesis of creep potential. The creep data obtained on the cellulose nitrate at 65° C under superimposed loading, as well as uniaxial loading, are found to fit to the deduced relation well.     

基于拉伸、压缩、剪切单轴强度试验值的材料三参数广义强度准则

基于材料的第一应力不变量和应力偏张量第二应力不变量,给出了同时考虑拉伸、压缩、剪切强度参数影响下的三参数广义强度准则.该强度准则公式严格满足纯拉伸、纯压缩、纯剪切情况下的强度值,在物理意义上体现了静水压力和静水拉力对材料强度的不同影响和剪切强度对应力偏张量的影响.基于此三参数广义强度准则,讨论了材料在各种典型应力状态下的强度包线,并同传统的强度准则进行了比较分析.最后,以ROHACELL闭孔PMI泡沫为例,给出了三种典型泡沫材料的强度破坏包线曲线和试验验证数据,比较结果证明,采用该强度准则可以较好地预测ROHA-CELL刚性泡沫材料的破坏强度.     

基于能量的弹塑性损伤实用本构模型

建立一个实用的弹塑性损伤本构模型。在有效应力空间采用经验公式计算塑性变形,能将模型塑性部分与损伤部分解耦,降低模型的数值处理复杂性,同时大大简化模型塑性应变的计算。结合不可逆热力学理论,基于损伤能量释放率建立损伤准则,损伤能量释放率由修正后的弹性Helmholtz自由能导出,模型中将弹性Helmholtz自由能分解为应力球量部分和应力偏量部分,将其应力球量部分产生的损伤取为零,同时根据应力状态引入折减系数对其应力偏量部分进行修正,使得模型能较为准确的模拟混凝土材料在双轴加载下的本构行为。将应力张量谱分解为正、负两部分以分别定义材料受拉、受压损伤演化,并采用受拉损伤变量、受压损伤变量分别模拟混凝土材料在拉、压加载下的本构特性。引入一个加权损伤变量使得模型能较准确的反映混凝土材料的“拉-压软化效应”。最后该文给出初步试验验证,证明了该文模型的有效性。     

Bulk and shear characterization of bituminous mixtures in the linear viscoelastic domain

Traffic loads and temperature variations produce three-dimensional stress–strain fields inside road pavements, and therefore the characterization of bituminous mixtures in different deformation modes is important for prediction of the performance of pavement structures. This paper presents a methodology for the bulk and shear characterization of bituminous mixtures in the linear viscoelastic domain, under the hypothesis of material isotropy, by means of uniaxial harmonic tests with the measurement of axial and transverse strains. The theoretical approach was based on the application of the elastic–viscoelastic correspondence principle and was validated by performing tension–compression tests at selected frequencies and temperatures on asphalt concrete specimens characterized by different volumetric properties. The results showed that since uniaxial tests induced both volume and shape variations, the simultaneous measurement of the complex bulk and shear moduli was possible. The validity of the time–temperature superposition principle was also verified for both deformation modes, allowing the construction of master curves for the bulk and shear moduli. The results also showed that the total dissipated energy could be decomposed into its volumetric and deviatoric fractions with excellent accuracy.     

Fracture of austenitic steel subject to a wide range of stress triaxiality ratios and crack deformation modes

The stress triaxiality ratio (hydrostatic pressure divided by von Mises equivalent stress) strongly affects the fracture behaviour of materials. Various fracture criteria take this effect into consideration in their effort to predict failure. The dependency of the fracture locus on the stress triaxiality ratio has to be investigated in order to evaluate these criteria and improve the understanding of ductile fracture.This was done by comparing the experimental results of austenitic steel specimens with a strong variation in their stress triaxiality ratios. The specimens had cracks with varying depths and crack tip deformation modes; tension, in-plane shear, and out-of-plane shear. The crack growth in fracture mechanics specimens was compared with the failure of standard testing specimens for tension, upsetting and torsion. By associating the experimental results with finite element simulations it was possible to compare the critical plastic equivalent strain and stress triaxiality ratio values at fracture. In the investigated triaxiality regime an exponential dependency of the fracture locus on the stress triaxiality ratio was found.     

An activated rate theory approach to the hydrostatic extrusion of polymers

An analysis of the mechanics of the hydrostatic extrusion process for polymers is presented, in which the predicted extrusion pressure is considered to be influenced by the effects of strain, strain rate and pressure on the material flow stress, as well as by the billet—die friction. The extrusion behaviour of both crystalline and amorphous polymers is discussed with reference to experimental results for linear polyethylene, polyoxymethylene and polymethylmethacrylate. Particular attention is paid to the method of incorporating the flow behaviour of the polymer into the analysis. A modified form of the Eyring equation for an activated rate process is proposed, in which the effects of strain rate and pressure on the flow stress are assumed to be separable, but related to strain by the large strain dependence of the stress activation volume. Moreover, a direct equivalence between the pressure effect and the friction between the polymer and the die is proposed for hydrostatic extrusion, following previous work on the adhesive mechanism for friction in polymers. This results in a formally identical analysis for both crystalline and amorphous polymers, in which the strain rate sensitivity, pressure sensitivity and friction coefficients all increase markedly with material strain during the process.     

塑性体积成形过程的变形分区及可视化研究

针对塑性体积成形过程复杂且金属流动行为难于分析的现状,从塑性加工力学角度,提出了利用J2不变量和罗德系数μσ等应力场特征量的变形分区方法,可对塑性区的范围及材料所处应变类型进行定量分析.并通过二次开发实现了塑性体积成形过程的可视化,能够直观地给出塑性区内任意部位材料所处应力应变的定量分布规律.     

Limits of linear viscoelastic behavior of polymers

In the present study different approaches for determination of the limits of linear viscoelastic (LVE) behavior are considered on examples of some thermoplastic and thermosetting polymers. Stress or strain level, commonly considered as a limit of LVE behavior, are interrelated time-dependent functions strongly influenced by action of external factors. The concept of energy threshold has an advantage of combining into one physical function the effects of both stress and strain in initiating nonlinear behavior. The value of the stored deviatoric energy is considered as a limit of LVE behavior and is a material characteristic. The experimental data on tension at various constant strain rates and tensile creep at various stresses, temperatures, and moisture conditions are considered. It is proved for some polymers that LVE limits in stress-strain representation fall on a common curve that is an energy curve independent of time. Decrease of the test rate or growth of temperature or moisture content appears only in a shift down along the energy curve to the lower limit stresses and higher limit strains.     

The mechanical behaviour of polystyrene under pressure

Tensile deformation of polystyrene carried out under pressure up to 4 kbar has shown that the pressure-transmitting fluid (silicon oil) acts as a stress crazing and cracking agent. Unsealed specimens showed a brittle-to-ductile transition at 2.95 kbar, while specimens sealed with Teflon tape and rubber showed the same transition at only 0.35 kbar. Analysis of the stress-strain curves for the sealed specimens indicated that the pressure dependency of the craze initiation stress differs from that of shear band initiation stress. The brittle-to-ductile transition occurs when the initiation stresses of both processes become equal. The principal stress for craze initiation showed almost no pressure dependency, suggesting that crazes initiate when the principal stress level of the tensile specimen reaches a critical value irrespective of the applied hydrostatic pressure. Similarly, no pressure dependency was observed for the principal ductile fracture stress. The pressure dependency of yield stress agreed well with a non-linear pressure dependent von Mises yield criterion.     

Improvement of the flow formulation by recovery of the hydrostatic pressure using the finite point method

Most of the existing numerical methods based on penalty technique, lead to pressure fields, which are not satisfactory especially when crude meshes are used. The present work for the first time, using a simple and effective finite point approach approximates the total hydrostatic stresses. By this new technique, the nodal hydrostatic pressure values are calculated so that the gradient of hydrostatic pressure satisfies the equilibrium equation, with regard to the current distribution of deviatoric stresses. This approach is verified using a cantilever beam with known analytical stress field. It is also applied to a typical 2D axisymmetric bulk metal forming process. The calculated pressure field is then compared with available solutions. The results of the present approach are very encouraging. Copyright © 2005 John Wiley & Sons, Ltd.