Nonlinear mechanical response of high density polyethylene. Part II: Uniaxial constitutive modeling

Based on the experimental data presented in Part I, two uniaxial constitutive models are constructed. The first, a nonlinear viscoelastic (NVE) model, is formulated using the mechanical analogy consisting of one independent spring and six Kelvin elements in series. Creep data are used to determine the model parameters. The second model, a viscoplastic (VP) formulation, is developed using the viscoplastic theory proposed by Bodner to characterize the uniaxial viscoplastic behavior of metals. Inelastic strain rate is introduced into the state variable in addition to inelastic work to depict the strong rate dependent behavior of HDPE. Experimental data from constant strain rate tests are employed to construct the material functions of the model. Limitations in the application of each model are discussed in conjunction with possibilities for future work.     

PE80燃气管道的应力松弛模型与实验验证

采用实验方法研究聚乙烯(PE)管道在机械载荷条件下的应力松弛行为,通过卷积分解析法和基于Maxwell模型的有限单元数值模拟法对其进行验证。结果表明,在小应变下采用Boltzmann叠加原理推导出的PE黏弹性本构模型可用于预测PE管道的应力松弛行为;基于实验参数,利用有限单元法能有效地对PE的黏弹性行为进行预测。     

Viscoplastic analysis of fiber reinforced polymer matrix composites under various loading conditions

The present study aims modeling the viscoplastic behavior of polymer matrix composite laminates under different loading conditions. The adopted model is based on the one‐parameter plasticity model employed to predict the plastic part of the recognized nonlinear behavior of fiber composites [C.T. Sun and J.L. Chen, J. Compos. Mater., 23, 1009 (1989)]. This model evolved to a three‐parameter constitutive viscoplastic model used to describe successfully the strain‐rate dependent mechanical behavior. Based on this model, designated as 3PV, a numerical implementation was made, based on the Classical Laminate Theory (CTL), to simulate non‐linear behavior of general laminates. The validation of this numerical implementation was performed using experimental data reported in literature. The first step was to assess the model predictions under high strain rates. The model was able to model high strain rate mechanical response, of an epoxy system reinforced with glass fibers, between 10 and 2,500 s⁻¹. Furthermore the model proved to be reasonable accurate to simulate creep, stress relaxation and constant strain rate loading of the same material system under high temperatures. Following the experimental observations of vsicoplasticity strain evaluation during load and unloading made by Kim and Tsai [J. Compos. Mater. 36(6), 745 (2002)], a simple model modification is suggested. This modified model proved accurate enough to simulate relaxation successfully of a composite during loading and unloading. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers     

Some aspects of the mechanical response of BMI 5250‐4 neat resin at 191°C: Experiment and modeling

The inelastic deformation behavior of BMI‐5250‐4 neat resin, a high‐temperature polymer, was investigated at 191°C. The effects of loading rate on monotonic stress–strain behavior as well as the effect of prior stress rate on creep behavior were explored. Positive nonlinear rate sensitivity was observed in monotonic loading. Creep response was found to be significantly influenced by prior stress rate. Effect of loading history on creep was studied in stepwise creep tests, where specimens were subjected to a constant stress rate loading followed by unloading to zero stress with intermittent creep periods during both loading and unloading. The strain‐time behavior was strongly influenced by prior deformation history. Negative creep was observed on the unloading path. In addition, the behavior of the material was characterized in terms of a nonlinear viscoelastic model by means of creep and recovery tests at 191°C. The model was employed to predict the response of the material under monotonic loading/unloading and multi‐step load histories. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2008     

Creep-based characterization of nonlinear relaxation behavior of plastics

It is a matter of fact that creep experiments can be conducted more easily and accurately than stress relaxation experiments, since it is easier to maintain a stress constant (for instance by a “dead weight”) than a strain constant. Nevertheless, in practice, structural parts made of plastics (which are nonlinear viscoelastic materials) are very often loaded under stress-relaxation conditions. The present paper presents an approach to predict the behavior of a nonlinear viscoelastic material under stress-relaxation-type loading, based on data obtained from creep-type experiments. The nonlinear creep compliance is described mathematically by an exponential series with a limited number of terms and a single nonlinearity function depicting the transient behavior. The nonlinear behavior of the material under constant strain (i.e., stress relaxation) is then obtained by dividing the considered time range into very short time intervals in which constant stresses are acting, while the different values of the applied stresses are chosen in a manner that guarantees the same stain at the end of each interval. In this way, one performs a numerical nonlinear superposition of the effects of the loadings in the various intervals, leading to the desired results under stress relaxation. A comparison of theoretical results with experiments conducted on some thermoplastic materials shows good agreement.     

An integral constitutive equation for nonlinear plasto-viscoelastic behavior of high-density polyethylene

In the present paper an effort is made to model the time-dependent behavior of high-density polyethylene (HDPE) with a one-dimensional integral representation. Owing to the plasto-viscoelastic behavior of the material, we assume that the total strain can be decomposed into a recoverable viscoelastic strain and an irrecoverable plastic strain. The viscoelastic deformation is represented by the Schapery thermodynamic theory. The plastic deformation is assumed to be accumulated during the loading history. An effective time concept is introduced for the plastic deformation, so that the response due to complex loading can be accounted for. The present representation gives a very good prediction of the responses of creep and recovery, two-step creep, and constant stress rate loading and unloading. It is also applied successfully to describe the process of preconditioning of semicrystalline polymers.     

Viscoelastic material characterization and modeling for polyethylene

An extensive set of stress relaxation and constant strain rate tests for characterizing the mechanical responses of a medium density polyethylene and a high density polyethylene that are commonly used in natural gas distribution piping is described and analyzed. The development of coherent master curves for the relaxation modulus, maximum stress, and the time-to-failure for pressurized pipes through a combination of both horizontal and vertical shifting is presented. The relaxation data are used to develop a nonlinear Viscoelastic material model. The model is assessed by making comparisons of the predicted stress-strain response with the measured response in the constant strain rate tests.     

Experimental approach to mechanical property variability through a high‐density polyethylene gas pipe wall

An experimental investigation was designed to establish the distribution of mechanical properties throughout a high‐density polyethylene (HDPE) gas pipe wall. The proposed approach used a continuous and uniform filament that was automatically machined from the pipe on a precision lathe at a very low cutting speed and an optimal depth of cut to minimize heating and structural disturbances. Typical engineering stress–strain curves, in every layer, were obtained on a testing machine especially designed for polymers, and they were statistically analyzed. The stress–strain behavior of HDPE pipe material could basically be divided into three distinctive zones, the second of which remained important. The average stress level illustrating cold drawing for a given layer was almost constant throughout the pipe wall. The measured stresses and moduli correlated very well with the pipe thickness, and they increased from the outer layers toward the inner layers. This was explained by the crystallinity evolution because the pipe production process was based on a convective water‐cooling system with a temperature gradient, which generated residual stresses. Computed statistical stress–strain correlations at yielding, the onset of cold drawing, and fracture points revealed acceptable linear relations for an error level of p ≤ 0.05. On the other hand, an increasing linear correlation characterized the relationship of the yield stress and elastic modulus. This result was confirmed by literature for standard specimens, prepared by compression molding, that did not represent an actual pipe structure with respect to an extrusion thermomechanical history. Such an approach to mechanical property variability within an HDPE pipe wall highlighted the complexity of the hierarchical structure behavior in terms of stress–strain and long‐term brittle failure. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 97: 272–281, 2005     

Effects of temperature,strain rate,and cyclic loading on mechanical behavior of silane-modified polyurethane sealant

To evaluate the mechanical properties of modified polyurethane sealants in engineering applications, the influences of temperature, strain rate, and cyclic loading on the mechanical properties of silane-modified polyurethane sealant were experimentally investigated. The monotonic tensile experiments with various strain rates and temperatures were conducted, and strain rate and temperature dependent nonlinear stress–strain curves were obtained. The results showed that the silane-modified polyurethane sealant exhibited temperature dependence at constant strain rate and rate dependence at room temperature. However, it is shown no obvious rate dependence at temperature of 150°C. In addition, the multi-step cyclic loading experiments with mean strain decrease and increase at each step were carried out to analyze the influence of cyclic loading and cyclic loading history at different temperatures. The results demonstrated that the viscous behavior of the materials was evidently observed in the first step and disappeared in other steps for the four-step cyclic loading with mean strain decrease case. Moreover, the cyclic stress relaxation of the materials was not obvious due to the prior cyclic loading with higher mean strain history, while the cyclic stress relaxation of the material continued to occur for the prior cyclic loading with lower mean strain history, and the cyclic strength of the materials decreased with the increase of temperature.     

The response of a glassy polymer in a loading/unloading deformation: The stress memory experiment

A ‘stress memory’ experiment was designed to expose the nonlinear viscoelastic relaxation processes in a glassy epoxy polymer. The stress memory experiment consists of (i) constant strain rate uniaxial loading to a pre-yield, yield or post-yield condition, (ii) unloading at the same strain rate to zero stress, (iii) holding the strain constant and (iv) monitoring the subsequent stress memory response, where the stress first increases to a maximum and then relaxes to an equilibrium value for that strain. This is an analog to the classic volume memory experiment by Kovacs (Fortschr Hochpolym Forsch, 3, 394, 1964). The stress memory response showed a strong dependence on the loading/unloading strain rate which cannot be predicted by linear viscoelasticity and also provides a significant challenge to a current nonlinear constitutive models. A recently developed Stochastic Constitutive Model (J Rheol, 57(3), 949, 2013) qualitatively predicts the effect of strain rate on the stress memory response.     

A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers

A thermodynamically consistent nonlinear viscoelastic constitutive theory is derived to capture the wide range of behavior observed in glassy polymers, including such phenomena as yield, stress/volume/enthalpy relaxation, nonlinear stress-strain behavior in complex loading histories, and physical aging. The Helmholtz free energy for an isotropic, thermorheologically simple, viscoelastic material is constructed, and quantities such as the stress and entropy are determined from the Helmholtz potential using Rational Mechanics. The constitutive theory employs a generalized strain measure and a material clock, where the rate of relaxation is controlled by the internal energy that is likewise determined consistently from the viscoelastic Helmholtz potential. This is perhaps the simplest model consistent with the basic requirements of continuum physics, where the rate of relaxation depends upon the thermodynamic state of the polymer. The predictions of the model are compared with extensive experimental data in the following companion paper.     

Prediction of the relationship between the rate of deformation and the rate of stress relaxation in glassy polymers

Kim, et al. (Polymer, 54(15), 3949, 2013) recently reported on the unexpected relaxation behavior of an amorphous polymer in the T_g-region, where the rate of stress relaxation increased with deformation at a strain rate of 1.5 × 10⁻⁴ s⁻¹ but decreased at a strain rate of 1.2 × 10⁻⁵ s⁻¹. This inversion in the ordering with strain rate challenges the underlying structure of the existing nonlinear viscoelastic and viscoplastic constitutive models, where the key nonlinearity is a deformation dependent material clock. The nonlinear stress relaxation predictions of a recently developed stochastic constitutive model, SCM, (Medvedev, et al., J. Rheology, 57(3), 949, 2013) that acknowledge dynamic heterogeneity of the glass have been investigated. The SCM predicts the inversion in the ordering of the mobility with the loading strain rate as reported by the stress relaxation response. The change in perspective on the nonlinear viscoelastic behavior of glassy polymers engendered by the SCM is discussed.     

On the nonlinear characterization of the long term behavior of polymeric materials

In the linear viscoelastic range the long term behavior of viscoelastic materials—such as polymers—can be described by using exponential series with a limited number of terms for the approximation of the relaxation modulus or of the creep compliance. This procedure can be extended to the nonlinear viscoelastic range by multiplying the linear parameters of the material by certain nonlinearity factors, which depend upon the level of the applied loading. Application of this method to stress relaxation data of several polymers has shown that nonlinearity factors can be approximated as linear functions of the applied constant strain. From creep tests, on the other hand, one can observe that the immediate strain response to the suddenly applied stress is linear elastic even in the nonlinear viscoelastic range of the investigated polymer. The computation of the linear viscoelastic material parameters as well as of the nonlinearity factors is conducted numerically by using least squares techniques. Good agreement between computed results and experimental data can be observed in the presented examples.     

Numerical determination of internal stress and viscoelastic parameters for xerographic papers

A computerized method has been developed for determining, from one experimentally obtained loading curve for a paper sample: (a) the viscoelastic parameters for predicting paper deformation, and (b) the internal or residual stress level in the paper. The loading curve comprises a stress-time curve obtained at a constant straining rate, followed by a stress relaxation curve at a constant strain level. The paper deformation is modeled by the Halsey, White, and Eyring model and a nonlinear viscoelastic model.     

Experimental study of elongational flow and failure of polymer melts

A new and simple instrument for measurement of elongational flow response of polymer melts in constant uniaxial extension rate experiments is described. Quantitative stress development data are presented for a series of low-density polyethylene (LDPE), high-density polyethylene (HDPE), polystyrene (PS), polypropylene (PP), and poly(methyl methacrylate) (PMMA) melts. For small elongation rate E, linear viscoelastic behavior was observed; while for large E, LDPE and PS showed exponential stress growth, while HDPE and PP showed only linear stress growth. Stress relaxation experiments were carried out for several of the same melts in the instrument. Elongation to break and mechanisms of filament failure were studied. HDPE and PP have a tendency to neck and exhibit ductile failure, while at high E, LDPE and PS seem to show cohesive fracture. The elongational flow stress response data were compared to predictions of nonlinear viscoelastic fluid theory, specifically the Bogue-White formulation. The qualitative differences in responses of the melts studied were explained in terms of different dependences of the effective relaxation times on deformation rate and, more specifically, on values of the a parameter in the theory.     

Cyclic deformation behavior of an epoxy polymer. Part I: Experimental investigation

A series of uniaxial cyclic tests were carried out on solid cylindrical specimens of an epoxy resin, Epon 826/Epi‐Cure Curing Agent 9551. The focus of the study was to investigate time‐dependent viscoelastic behavior of this thermosetting polymer material under cyclic loading and to develop a constitutive model with the capabilities to simulate the observed deformation response. The tests include stress‐controlled or strain‐controlled cyclic loading with/without mean stress or mean strain at various amplitudes and loading rates. It was found that the cyclic stress‐strain response of this material is amplitude‐dependent and rate‐dependent, and the response to axial tension is different from that in compression. The stress‐strain loops exhibit more pronounced nonlinearity with high amplitudes or low loading rates. For stress‐controlled cyclic loading with mean stress, ratcheting strain is accumulated, which is of viscoelastic nature, and this is confirmed by its full recovery after load removal. For strain‐controlled cyclic loading with mean strain, the mean stress relaxation occurs, which contributes to the observed longer life in comparison to the stress‐controlled cyclic loading with mean stress. Polym. Eng. Sci. 44:2240–2246, 2004. © 2004 Society of Plastics Engineers.     

Influence of talc fillers on bimodal polyethylene composites for ground heat exchangers

In this study, a commercial grade of talc is used as filler in a bimodal high-density polyethylene (HDPE) used for the pressure pipe application. The composites are characterized by thermogravimetric analysis (TGA), differential scanning calorimetry, dynamic mechanical thermal analysis, and tensile testing. The results illustrate that the presence of talc has a considerable effect on the material properties and the pipe life-length. It is presented that the thermal stability measured by TGA is enhanced, while the oxidation induction time decreases in cooperation of the talc. The nucleation behavior of talc particles during crystallization has no obvious effect on melting temperature; however, an increase in crystallization temperature is evidenced. Storage modulus as recorded from the dynamic mechanical analysis is also increased in all composites, furthermore, the temperature of the α relaxation is shifted toward higher temperature and finally the strain hardening modulus for the HDPE/talc composites is assessed and compared to the neat HDPE as a measure of environmental stress crack resistance.     

Pre‐necking and post‐necking relaxation and creep behavior of polycarbonate: A phenomenological study

Experiments have been performed to investigate the mechanical response of unfilled polycarbonate vis‐à‐vis the influence of prior deformation on stress relaxation and creep. Piecewise linear deformation histories, which involve strain‐controlled tensile loading of a specimen to a maximum load and partial unloading to a target strain/stress point as prologue to a relaxation test, have been shown to qualitatively influence the recorded stress‐time behavior. In particular, the stress magnitude during relaxation first increases and is then followed by a decrease. Analogously, in creep tests during unloading, the strain might decrease and then increase. Time characteristics for this U‐turn in the deformation response are influenced by the placement of the test. The influence of prior specimen conditioning on this phenomenon is investigated by comparing test data from virgin samples to that of specimens having high (～85%) inelastic strain from prior tensile elongation. Findings suggest that the observed persistence in the occurrence of this reversal effect for both types of specimens is evidence of the need to incorporate this behavior into the fold of material modeling. Additionally, this novel relaxation and creep behavior has been observed in other amorphous (poly(phenylene oxide)) and crystalline (high‐density polyethylene) polymers. Polym. Eng. Sci. 44:1783–1791, 2004. © 2004 Society of Plastics Engineers.     

Temperature-dependent mechanical properties and constitutive equation of cellulose nitrate

The strain-time response under tensile loading (creep tests) and the stress strain response under constant tensile stress rate (proportional loading tests) have been evaluated at 4 temperatures 20, 45, 55, and 65°C, for samples of cellulose nitrate. A time-dependent constitutive equation (or stress-strain relation) for the nonlinear visco-elastic material is deduced from invariant theory with a hypothesis of a creep potential. The procedure for determining the seven material constants involved in the deduced constitutive equation is described for the creep and proportional loading tests and the variation of these constants with temperature is presented. The deduced constitutive equation gives good agreement with the actual observations for the creep and proportional loading tests, independent of the values of temperature, creep stress, or stress rate.     

The effect of barium titanate ceramic loading on the stress relaxation behavior of barium titanate-silicone elastomer composites

The stress relaxation behavior of barium titanate (BTO)-elastomer (Ecoflex) composites, as used in large strain sensors, is studied using the generalized Maxwell-Wiechert model. In this article, we examine the stress relaxation behavior of ceramic polymer composites by conducting stress relaxation tests on samples prepared with varying the particle loading by 0, 10, 20, 30, and 40 wt% of 100 and 200 nm BTO ceramic particles embedded in a Ecoflex silicone-based hyperelastic elastomer. The influence of BTO on the Maxwell-Wiechert model parameters was studied through the stress relaxation results. While a pristine Ecoflex silicone elastomer is predominantly a hyperelastic material, the addition of BTO made the composite behave as a visco-hyperelastic material. However, this behavior was shown to have a negligible effect on the electrical sensing performance of the large strain sensor.