Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates

A combined experimental and analytical investigation has been performed to understand the mechanical behavior of two amorphous polymers—polycarbonate and poly(methyl methacrylate)—at strain rates ranging from 10⁻⁴ to 10⁴ s⁻¹. This range in strain rates was achieved in uniaxial tension and compression tests using a dynamic mechanical analyzer (DMA), a servo-hydraulic testing machine, and an aluminum split-Hopkinson pressure bar. DMA tension tests were used to characterize the viscoelastic behavior of these materials, with focus on the rate-dependent shift of material transition temperatures. Uniaxial compression tests on the servo-hydraulic machine (10⁻⁴ to 1 s⁻¹) and the split-Hopkinson pressure bar (10³ to 10⁴ s⁻¹) were used to characterize the rate-dependent yield and post-yield behavior. Both materials were observed to exhibit increased rate sensitivity of yield under the same strain rate/temperature conditions as the β-transition of the viscoelastic behavior. A physically based constitutive model for large strain deformation of thermoplastics was then extended to encompass high-rate conditions. The model accounts for the contributions of different molecular motions which become operational and important in different frequency regimes. The new features enable the model to not only capture the transition in the yield behavior, but also accurately predict the post-yield, large strain behavior over a wide range of temperatures and strain rates.     

Half-fringe photoelasticity: A new approach to whole-field stress analysis

This paper presents a new method for whole-field stress analysis based on a symbiosis of two techniques—classical photoelasticity and modern digital image analysis. The resulting method is called ‘half-fringe photoelasticity (HFP)’. Classical photoelasticity demands materials with high birefringence, which leads to extensive use of plastics as model materials. Since the behavior of these materials is often different from that of the prototype materials, their use distorts the similitude relationships. In many contemporary problems this distortion is untenable. HFP offers a way out of this dilemma. It permits materials and loads to be chosen so that no more than one half of a fringe order appears in the area of interest. Thus, for example, glass, which behaves linearly up to high stress levels and over a wide range of temperatures, could be used as model material. Alternatively, models from polymeric materials could be used under very low load in order to stay within the linear part of the stress-strain diagram and to prevent large deformations. The half-fringe-photoelasticity system, which is described here, utilizes the resulting low levels of birefringence for effective stress analysis. This paper describes the system. It outlines a calibration routine and illustrates its application to two simple problems using glass models.     

Characteristic relations of flow birefringence

This is the first of two closely related papers on the flow-birefringence response to the velocity vector field of a particular liquid representing a certain class of birefringent bodies.^* The flowing material under study was the aqueous solution of the compound known under the name NGS 1828 and commonly known as “milling yellow” or “acid yellow”. This solution appears to exhibit all three major mechanisms of birefringence. The physical parameters characterizing this material depend strongly on temperature, concentration and age and, therefore, it can be considered as representing a typical class of liquids used in flow-model experiments. The paper presents the experimental evidence that the flow birefringence cannot be explained and described by the simple mathematical model of birefringence in solid continuum which relates the changes of the components of dielectric tensor to the components of stress and strain tensors, or their derivatives, and which neglects the influence of the spectral frequency (wavelength of radiation). Results are presented for transmission birefringence (and for scattered-light birefringence in the second paper) in the visible and the infrared bands of radiation. It is shown that:

- the amount of birefringence depends strongly and non-monotonically on wavelength of radiation;

- the linear range of optical response to shear-strain rate depends on wavelengths of radiation;

- the directions of optic axes strongly depend both on the shear-strain rate and on the wavelength of radiation, even in the linear range of mechanical response.

It is further shown that there exists a relation between the absorption bands, the maximum transmittance, the dispersion of birefringence, the spectral dependence of optic-axes direction, and the linear range of optical response. Within the maximum transmittance band and the linear range of mechanical response the linear range of birefringence is maximum and the dispersion of birefringence is minimum with respect to the shear-strain rate; the corresponding dispersion of optic axis is also minimum. Samples of typical recordings are given in the visible and the infrared radiation for typical flow patterns. One of the practical conclusions is that to optimize the flow-birefringence studies of engineering problems it is advisable to choose the radiation in the near-infrared range. The evidence presented shows that the common trend in engineering research toward simplification of the model of the flow-birefringence response is not necessary.     

Influence of Molecular Conformation on the Constitutive Response of Polyethylene: A Comparison of HDPE, UHMWPE, and PEX

The current work presents the characterization and comparison of the mechanical response of three different industrial forms of polyethylene. Specifically, high-density polyethylene (HDPE), ultra high molecular weight polyethylene (UHMWPE), and cross-linked polyethylene (PEX) were tested in compression as a function of temperature (−75 to 100°C) and strain-rate (10⁻⁴ to 2,600 s⁻¹). The responses of UHMWPE and PEX are very similar, whereas HDPE exhibits some differences. The HDPE samples display a significantly higher yield stress followed by a flat flow behavior. Conversely UHMWPE and PEX both exhibit significant strain hardening after yield. The temperature and strain-rate dependence are captured by simple linear and logarithmic fits over the full range of conditions investigated. The yield behavior is presented in terms of an empirical mapping function that is extended to analytically solve for the mapping constant. The power-law dependence on strain-rate observed in some polymers is explained using this mapping function.     

Comparison of viscous and elastic properties of polyolefin melts in shear and elongation

Viscous and elastic properties of a linear polypropylene (PP) and a long-chain branched low-density polyethylene (LDPE) have been investigated by creep and creep–recovery experiments in shear and elongation. The data obtained verify the ratios between the linear values of the viscosities and the steady-state elastic compliances in shear and elongation predicted by the theory of linear viscoelasticity. In the nonlinear range, no simple correlation between the viscous behaviour in shear and elongation exists. The elongational viscosity of the PP decreases with increasing stress analogously to the shear thinning observed; the linear range extends to higher stresses in elongation than in shear, however. The LDPE shows thinning in shear and strain hardening in elongational flow. For the LDPE, a linear steady-state elastic tensile compliance corresponding to one third of the linear steady-state elastic compliance in shear was determined. For the PP, this theoretically predicted value is approximately reached. Analogous to the viscous behaviour, the linear range extends to higher stresses in elongation than in shear. For both materials, the steady-state elastic compliances in the nonlinear range decrease with increasing stress in shear as well as in elongation. However, the decrease in elongation is more pronounced.     

A model long-discontinuous-fiber filled thermoplastic melt in extensional flow

The shear cell model works for dilute fiber filled systems in extensional flow. This research investigates the suitability of the idea for highly aligned fibers in a concentrated suspension. A model fiber-filled polymer system made from nylon fibers in low-density polyethylene provided a means of controlling the material parameters. Two systems, with fiber aspect ratios of 20 and 100, containing 50% 0.5 mm fibers by volume are investigated. The thickness of the polymer layer, i.e. with fibers this size, allows bulk viscosity data to be compared with the data from the filled fluid. A weaving process created the discontinuous fiber/polyethylene preforms with high alignment of the fibers and with control of the fiber to fiber overlap. Testing the polyethylene in simple shear and extending the nylon/polyethylene provided the data needed to check the micro mechanics. A cone and plate rheometer and a capillary instrument produced the viscosity/strain rate data that characterized the specific polyethylene used in the composite. A furnace inset placed in an Instron hydraulic test machine allowed extension of the filled system at strain rates from 0.002 to 0.4 s⁻¹. The shear experiments show that the low-density polyethylene is a simple shear-thinning melt that provides a good model fluid. The extension of the filled systems shows an increase of the apparent extensional viscosity from that of neat polyethylene. Apparent viscosity rises two to three orders of magnitude for the systems investigated. The micromechanics allowed the conversion of the extensional data from the two filled systems to the shear viscosity of the polymer surrounding the fibers. The calculated polyethylene viscosity compares well with the data from the standard rheometers. The shear cell approach may be applied to highly aligned, high fiber-volume-fraction suspensions when the viscosity of the polymer is known at the scale of the film surrounding each fiber.     

低应变率下硅铝质泡沫陶瓷复合材料的力学性能研究

以泡沫陶瓷复合材料在防护工程中的应用为背景,利用MTS(Material Test System,材料试验机)对该型材料进行了准静态压缩实验。得到了应变率在10-5~10-3s-1范围内的应力应变曲线,并对实验结果进行了理论分析和数值模拟。研究表明,泡沫陶瓷复合材料的力学性能在准静态一维应力压缩条件下显示出明显的应变率效应,同时其应力应变曲线可用一种经验的脆性材料本构模型进行较好地拟合。而在一维应变压缩条件下,材料的应力应变曲线则显示出明显的三段式特征:弹性段、平台段和密实段,同时材料的吸能幅值随着应变率的增大而增加。     

考虑破碎影响的颗粒材料亚塑性模型及应变局部化模拟

在基于2ndP-K应力率的亚塑性模型基础上,通过引入一个能够考虑颗粒破碎影响的孔隙比-平均压力临界状态方程,形成了一个能够模拟颗粒破碎影响的颗粒材料亚塑性模型,数值算例考查了颗粒破碎对应变局部化模式及位移-承载曲线的影响,结果表明,所建议模型具有模拟破碎对颗粒材料应变局部影响的良好性能。     

Linear limit stresses of some photoelastic and mechanical model materials

Certain laws of similarity must be observed in structural-model analyses. In this paper, one aspect of model similarity—that of linearity—is examined quite extensively. Most model analyses assume that both prototype and model materials obey Hooke's law. But the plastics often used for structural or photoelastic models are viscoelastic or photoviscoelastic. The stress-strain and stress-birefringence relations are time dependent and may be nonlinear. Through careful calibration of model materials and proper design of model tests, potential errors due to the time dependence of material properties can usually be avoided. If the results of the test are to be interpreted conveniently and accurately, the stresses in the model material must be within the linear range. This range is limited and time dependent for most plastics. The linear range may extend only to stresses considerably below the ultimate or fracture strength of the material. Hence, analyses based don linearity may be in error if the initial stresses are too high and/or if given stresses are sustained too long before desired information is collected. The stresses which limit the linear range, called linear limit stresses, were determined for both stress-strain and stress-birefringence relations for four commonly used plastics: CR-39 (Cast Optics Co.), PS-1 and PS-2 (Photolastic, Inc.) and P6-K (B.A.S.F., Germany). A graphical presentation of the time-dependent photoelastic and mechanical properties is employed. It was concluded that linear limit stresses for birefringence are approximately equal to those based on strain and can therefore be used to establish, within reasonable bounds of accuracy, the linear range of behavior of the material.     

Techniques for measuring stress-strain relations at high strain rates

The qualitative dependence of the mechanical behavior of some materials on strain rate is now well known. But the quantitative relation between stress, strain and strain rate has been established for only a few materials and for only a limited range. This relation, the so-called constitutive equation, must be known before plasticity or plastic-wave-propagation theory can be used to predict the stress or strain distribution in parts subjected to impact stresses above the yield strength. In this paper, a brief review of some of the experimental techniques for measuring the stress, strain, strain-rate relationship is given, and some of the difficulties and shortcomings pointed out. Ordinary creep or tensile tests can be used at plastic-strain rates from 10^?8 to about 10^?1/sec. Special quasi-static tests, in which the stress- and strain-measuring devices as well as the specimen geometry and support have been optimized, are capable of giving accurate results to strain rates of about 10²/sec. At higher strain rates, it is shown that wave-propagation effects must be included in the design and analysis of the experiments. Special testing machines for measuring stress, strain and strain-rate relationships in compression, tension and shear at strain rates up to 10⁵/sec are described, and some of the results presented. With this type of testing machine, the analysis of the data requires certain assumptions whose validity depends upon proper design of the equipment. A critical evaluation of the accuracy of these types of tests is presented.     

A comparison of quasi-static uniaxial-strain and hugoniot tests for quartz-phenolic composite

The purpose of this paper is to show that, within experimental uncertainty, the change in volume with stress obtained by quasi-static uniaxial-strain tests matches that obtained by hugoniot experiments over the same pressure range for quartz phenolic. The result of these tests shows that comparing the data by both techniques is meaningful. In addition, the use of the relatively simple and inexpensive quasi-static uniaxial-strain test (strain rates of 10^?4/sec) may provide designers and materials engineers a method for rapid surveying of materials for their hugoniot properties.     

Hopkinson拉杆的入射波波形整形实验研究

目前,分离式Hopkinson杆实验技术已经被广泛用于测试材料在10~2~10~4s~(-1)应变率范围内的动态力学特性。为了抑制入射波的高频振荡,实现恒定应变率加载,本文利用分离式Hopkinson拉杆(SHTB)实验装置,研究了加载金属短杆(2A12T4铝合金)及整形垫片(纸板、PVC软塑料及带磁性胶皮)对入射波波形的影响。实验结果表明,整形垫片降低了入射应力脉冲的高频振荡,获得了比较平滑的入射应力脉冲,延长了上升时间。同时,利用所得的波形整形结果,对2A12T4铝合金进行了拉伸应力波脉冲加载的拉伸和断裂实验测试。     

Rheo-optical measurements of the elongational flow of aqueous polymer solutions

An opposed jets rheometer was used to investigate the elongational viscosity as a function of the strain rate for dilute aqueous solutions of polyvinylformamide and polyacrylamide. Critical strain rates at which the enhancement in elongational viscosity occurs were determined for both systems. The influence of the polymer concentration on the elongational viscosity was investigated. The measurements were performed with polymer concentrations less than the critical polymer concentration c_p^* c_p^* . In order to assess the deformation and orientation of the macromolecules, flow-induced birefringence was measured simultaneously.     

Effects of crystal content on the mechanical behaviour of polyethylene under finite strains: Experiments and constitutive modelling

The mechanical stress-strain behaviour of polyethylene (PE) materials under finite strains is studied both experimentally and theoretically. In order to gain insight into the structure and physical properties of investigated PE materials, a series of thermal (DSC and DMTA) and microstructural (small-angle X-ray scattering and AFM) characterizations have been undertaken. The influence of crystallinity on the various features of the tensile stress-strain response is considered over a large strain range, implying thermoplastic-like to elastomer-like mechanical behaviour. A physically-based hyperelastic-viscoplastic approach was adopted to develop a pertinent model for describing the mechanical behaviour of PE materials under finite strains. The semicrystalline polymer is being treated as a heterogeneous medium, and the model is based on a two-phase representation of the microstructure. The effective contribution of the crystalline and amorphous phases to the overall intermolecular resistance to deformation is treated in a composite framework, and coupled to a molecular network resistance to stretching and chain orientation capturing the overall strain hardening response. In order to extract the individual constitutive response of crystalline and amorphous phases, a proper identification scheme based on a deterministic approach was elaborated using the tensile test data of PE materials under different strain rates. Comparisons between the constitutive model and experiments show fair agreement over a wide range of crystallinities (from 15% to 72%) and strain rates. The constitutive model is found to successfully capture the important features of the observed monotonic stress-strain response: the thermoplastic-like behaviour for high crystallinity includes a stiff initial response, a yield-like event followed by a gradual increase of strain hardening at very large strains; for the elastomer-like behaviour observed in the low crystallinity material, the strain hardening response is largely predominant. Strain recovery upon unloading increases with decreasing crystallinity: this is quantitatively well reproduced for high crystallinity materials, whereas predictions significantly deviate from experiments at low crystallinity. Model refinements are finally proposed in order to improve the ability of the constitutive equations to predict the nonlinear unloading response whatever the crystal content.     

A linear viscoelastic model for solid polyethylene

Stress relaxation tests have been carried out on a blue, pipe grade PE 80 medium density polyethylene (BP Chemicals), to provide thermo-viscoelastic rheology for use in calculating thermal stresses in pipe production. Stresses up to 4 MPa were used, with strains up to about 2%, in tests at temperatures from 23° to 90°C. Within this range a linear viscoelastic model was applicable, provided the initial ramp strain rate was less than 7×10^–5 s^–1. The stress relaxation data was fitted directly by a model incorporating an elastic response to volumetric strains, and a generalised linear solid model, consisting of two Maxwell elements and a purely elastic element in parallel, for deviatoric strains. Arrhenius type temperature dependence of relaxation times and shear moduli is found, and within experimental accuracy the temperature dependence of all these model parameters is the same. As a consequence, and provided that the duration of the strain ramp is sufficiently short relative to relaxation times, the model leads to time-temperature superposition of the relaxation moduli, using the same shift factor on both the response magnitude and time axes.     

Flow of branched polymer melts in a lubricated cross-slot channel: a combined computational and experimental study

Numerical simulations have been performed to evaluate the accuracy of the multimode Giesekus model in predicting the flow behavior of a rheologically well characterized low-density polyethylene melt in a lubricated cross-slot channel. Specifically, the fidelity of the numerical results is established by detailed comparison with flow-induced birefringence measurements in a new optical rheometer with lubricated side walls that allows the creation of ideal two-dimensional flow kinematics that lead to the elimination of end effects commonly encountered in flow birefringence measurements. Based on these comparisons, the ability of the multimode Giesekus model to capture the flow characteristics with reasonable accuracy in the experimentally available Wi range of 21 to 29 has been established. However, it should be noted that the model predictions are, at best, qualitative in the vicinity of the stagnation point. The discrepancy between numerically predicted and experimentally observed stresses in this region is mainly attributed to the inaccuracy of the experimental data that stem from the occurrence of multiple orders of retardation within the measurement volume. Overall, these studies have paved the way for the development of a hi-fidelity lubricated cross-slot channel rheometer.     

Effect of the rate of deformation on the crystallization behavior of polymers

Deformation has a significant influence on the crystallization process in a number of polymers. In this paper, the response of a recently developed model for crystallizing polymers is investigated when subject to uni-, bi-axial and constant width extensions for a range of strain rates. Both the loading and unloading behavior are examined for these deformations. The particular model studied here was developed to capture the effect of strain induced crystallization in polymers and has been applied to model crystallization in polyethylene terephthalate at temperatures just above its glass transition temperature. The model has been formulated using the notion of multiple natural configurations within a full thermodynamic framework. The connection between micro-structural changes taking place in the polymer and the form of the model are elucidated. The interplay between the relaxation processes, the rate of deformation and their combined effect on crystallization is illustrated. The results show an earlier onset of crystallization for high strain rates due to stretching of the polymer network. At low strain rates however, crystallization is not observed as the polymer network is able to relax during the deformation. A sharp upturn in the stress is observed after the onset of crystallization due to the formation of a rigid crystalline phase. The unloading curves clearly show a hysteric behavior with the amount of dissipation increasing for increasing values of strain rate. These results compare favorably with experimental observations available in literature.     

Localised flow-induced crystallisation of a polyethylene melt

The sensitivity of flow-induced crystallisation (FIC) to the nature of flow type is demonstrated using a high-density polyethylene (HDPE) for two different flow geometries. A contraction–expansion slit geometry was used to create a mixed, but primarily simple shear flow, while a cross-slot geometry provided a region within the flow of high extension. Flow-induced birefringence was captured at a melt processing temperature of 155 °C to identify the principal stress difference within the two flows and determine regions of higher stress within the HDPE. The experiments were then repeated at 125 °C, and FIC was identified using bright-field observation. Crystallisation was observed within the regions that previously exhibited high stress levels. It was found that lower deformation rates in pure shear were required when compared with simple shear to create the crystal filaments.     

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.     

A shear-compression specimen for large strain testing

A new specimen geometry, the shear-compression specimen (SCS), has been developed for large strain testing of materials. The specimen consists of a cylinder in which two diametrically opposed slots are machined at 45° with respect to the longitudinal axis, thus forming the test gage section. The specimen was analyzed numerically for two representative material models, and various gage geometries. This study shows that the stress (strain) state in the gage, is three-dimensional rather than simple shear as would be commonly assumed. Yet, the dominant deformation mode in the gage section is shear, and the stresses and strains are rather uniform. Simple relations were developed and assessed to relate the equivalent true stress and equivalent true plastic strain to the applied loads and displacements. The specimen was further validated through experiments carried out on OFHC copper, by comparing results obtained with the SCS to those obtained with compression cylinders. The SCS allows to investigate a large range of strain rates, from the quasi-static regime, through intermediate strain rates (1–100 s⁻¹), up to very high strain rates (2×10⁴s⁻¹ in the present case).