A viscoelastic–plastic constitutive model for uniaxial ratcheting behaviors of polycarbonate

In this paper, a modified viscoelastic–plastic constitutive model has been proposed on the framework of Anand's work to describe the uniaxial ratcheting behavior of polycarbonate (PC) under tension–tension cyclic loading. The experimental observation illustrates that the previously accumulated deformation has an assignable influence on the subsequent material response during the ratcheting process of PC. Thus, the deformation resistance in the viscoelastic micromechanism is assumed to be evolving with the local accumulated inelastic strain rather than keeping unchanged in the original Anand's model. The proposed model is validated firstly by the monotonic tension and creep experiment results of PC. Then, its capability to describe the uniaxial ratcheting behaviors is compared with Anand model. Finally, the modified model is adopted to study the effect of mean stress, stress amplitude, loading rate, and peak holding time on the ratcheting behaviors of PC. It is shown that the proposed model can predict reasonably the uniaxial tension–tension ratcheting behavior of polymer. POLYM. ENG. SCI., 55:2559–2565, 2015. © 2015 Society of Plastics Engineers     

Viscoelastic constitutive model for uniaxial time‐dependent ratcheting of polyetherimide polymer

Based on the experimental observations, a cyclic nonlinear viscoelastic constitutive model was proposed to describe the uniaxial time‐dependent ratcheting of polyetherimide (PEI) polymer under tension–compression and tension–tension cyclic loading. The model was constructed by extending the nonlinear viscoelastic Schapery model (Schapery, Polym. Eng. Sci., 9, 295 (1969)). The extension emphasized the changes of parameter functions used in the original model, which enabled the model to describe the ratcheting of polymer material. Comparing the simulations with corresponding experimental results, the capability of the extended model to predict the uniaxial time‐dependent ratcheting of PEI was verified. It is shown that the extended model can reasonably describe the uniaxial time‐dependent ratcheting of the polymer under the tension–compression and tension–tension cyclic loading with different peak‐holdings, stress rates, and stress levels. POLYM. ENG. SCI., 52:1874–1881, 2012. © 2012 Society of Plastics Engineers     

A simple constitutive model for cyclic compressive ratchetting deformation of polyteterafluoroethylene (PTFE) with stress rate effects

Pure polyteterafluoroethylene (PTFE) were prepared by using the process of cold pressure and sintering. A series of unaxial cyclic compression tests were carried out on solid cylindrical specimens of pure PTFE. The focus of study was to investigate time‐dependent ratchetting behavior of PTFE under cyclic loading. It is shown that the stress–strain hysteresis loops exhibit pronounced nonlinearity with high stress range. The cyclic hardening is rate and stress range dependent. However, mean stresses have less effect on cyclic hardening than stress ranges. It is shown that viscoelasticity is obvious in the beginning cycles. The same stress range produces almost the same viscoelastic delay at the beginning of the compression cycles. A modified universal ratchetting model (URM), which is rate dependent is employed to predict the ratchetting strain under compressive cyclic loading condition at room temperature. POLYM. ENG. SCI., 48:29–36, 2008. © 2007 Society of Plastics Engineers     

A Mathematical Model for Amorphous Polymers Based on Concepts of Reptation Theory

Mechanical behaviors of amorphous polymers have been investigated in all aspects from macroscopic thermodynamics to molecular dynamics in past five decades. Most models either have too complex mathematics or can only explain mechanical behaviors of specific materials under certain defined conditions. In this article, a mathematical model is proposed to understand mechanical behaviors of amorphous polymers with aid of the concepts of reptation theory. This new model is capable to match most experimental results of different amorphous polymers for a wide range of time and temperature effect from rubber zone to glassy zone. Above glass transitional temperature, the model shows hyperelastic behavior. Below glass transitional temperature, elastic–viscoplastic properties can be obtained. In the proposed model, no yielding surface is assumed. Hyperelasticity and Mullin's effect are illustrated in a different way without assuming strain energy function in advance. Yielding stress is controlled by Young's moduli, defect density, and defect velocity of molecular chains. Anisotropic plasticity is simply controlled by anisotropic Young's moduli. Therefore, no additional anisotropic parameters are needed to define anisotropic yielding surface. Strain rate, temperature, and hydrostatic pressure effects on yielding stress are through their effect on Young's moduli. Linear elastic, hyperelastic, viscoelastic, and viscoplastic models are put into one single equation, which makes the mathematical structure very easy to understand and easy to use. This model is validated by comparing with five existed experimental data. Proposed model also shares some features similar to the old well‐known large deformation models for amorphous polymers. POLYM. ENG. SCI., 59:2335–2346, 2019. © 2019 Society of Plastics Engineers     

Quantitative relation between the relaxation time and the strain rate for polymeric solids under quasi‐static conditions

Relaxation time is an essential physical quantity reflecting the hysteresis of the microstructure of materials. To associate the relaxation time with the strain rate, the stress–strain curves of six types of polymers at low strain rate were normalized, and a nondimensional generalized Maxwell model incorporating strain‐rate‐dependent relaxation times was obtained by the internal variable theory of irreversible thermodynamics. The results indicate that the constitutive equation may capture well the normalized stress–strain behaviors that are not related to the strain rate. The ratio of the initial modulus to the secant modulus at the maximum stress was also found to not rely on the strain rate anymore. Furthermore, strain‐rate independence occurred only when the relaxation time was proportional to the time interval for stress from zero to the maximum stress. The relaxation time varied in a power law with the strain rate. The explicit relation is helpful for providing a concise and promising solution for predicting the quasi‐static mechanical response of viscoelastic solids. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 133, 44114.     

The dynamic response and failure of Polycarbonate plate by soft body impact

This article is mainly to experimentally and numerically investigate the dynamic response and failure of Polycarbonate (PC) plate against strike by soft body. The experimental results show that high speed soft body impact leads to a large global displacement for PC sheet, though, the obtained strain data shows deformation of PC material are still small. This evidence allows us to employ a thermo‐viscoelastic constitutive model we proposed in our previous work, where the model parameters are determined based on the uniaxial tension test data of PC materials, to describe the PC plate. Then, the simulation is made in finite element (FE) software LS‐DYNA and computational results get a fair agreement with experiments including displacement, strain, and the crack propagations at high velocity impact. The temperature effect on mechanical behavior of PC sheet under impact is numerically studied as well. It is found that the effect gets more significant with the increase of impact velocity, and the higher temperature of PC sheet would lead to its larger deflection but smaller maximum resistance force and principal stress. POLYM. ENG. SCI., 56:1160–1168, 2016. © 2016 Society of Plastics Engineers     

Tensile creep of a long‐fibre glass mat thermoplastic (GMT) composite. II. Viscoelastic‐viscoplastic constitutive modeling

In Part I of this article, the short‐term tensile creep of a 3‐mm‐thick continuous long‐fibre glass mat thermoplastic composite was characterized and found to be linear viscoelastic up to 20 MPa. Subsequently, a nonlinear viscoelastic model has been developed for stresses up to 60 MPa for relatively short creep durations. The creep response was also compared with the same composite material having twice the thickness for a lower stress range. Here in Part II, the work has been extended to characterize and model longer term creep and recovery in the 3‐mm composite for stresses up to near failure. Long‐term creep tests consisting of 1‐day loading followed by recovery were carried out in the nonlinear viscoelastic stress range of the material, i.e., 20–80 MPa in increments of 10 MPa. The material exhibited tertiary creep at 80 MPa and hence data up‐to 70 MPa has been used for model development. It was found that viscoplastic strains of about 10% of the instantaneous strains were developed under load. Hence, a non‐linear viscoelastic–viscoplastic constitutive model has been developed to represent the considerable plastic strains for the long‐term tests. Findley's model which is the reduced form of the Schapery non‐linear viscoelastic model was found to be sufficient to model the viscoelastic behavior. The viscoplastic strains were modeled using the Zapas and Crissman viscoplastic model. A parameter estimation method which isolates the viscoelastic component from the viscoplastic part of the nonlinear model has been developed. The model predictions were found to be in good agreement with the average experimental curves. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers     

Modeling the uniaxial rate and temperature dependent behavior of amorphous and semicrystalline polymers

Strain rate and temperature dependent constitutive equations are proposed for polymer materials based on existing isotropic formulations of viscoplasticity. The proposed formulations are capable of simulating some of the important features of deformation behavior of amorphous and semicrystalline polymers. The materials model is based on the assumption that the evolution of flow stress is dependent on the rate of deformation, temperature, and an appropriate set of internal variables. The proposed theory is capable of modeling yielding, strain softening, and the orientation hardening exhibited by amorphous polymers. It is also possible to model the initial viscoplastic and subsequent nonlinear hardening behavior shown by semicrystalline polymers at large strains. Uniaxial tensile tests with uniform and hourglass specimens are made at temperatures ranging from 23 to 100°C and under various crosshead speeds. Both amorphous polycarbonate and semicrystalline polypropylene sheet materials are tested to characterize the stress and strain behavior of these materials and to determine their appropriate material constants. Load relaxation experiments are also conducted to obtain the necessary material constants describing the rate and temperature dependent flow stress behavior of polypropylene. Simulation results compare favorably against experimental data for these polymer materials.     

Nonlinear viscoelastic modeling of soft polymers

A one‐dimensional phenomenological constitutive model, representing the nonlinear viscoelastic behavior of polymers is developed in this study. The proposed model is based on a modification of the well‐known three element standard solid model. The linear dashpot is replaced by an Eyring type one, while the nonlinearity is enhanced by a nonlinear, strain dependent spring constant. The new constitutive model was proved to be capable of capturing the main aspects of nonlinear viscoelastic response, namely, monotonic and cyclic loading, creep and stress relaxation, with the same parameter values. Model validation was tested on the experimental results at various modes of deformation for two elastomeric type materials, performed elsewhere. A very good agreement between model simulations and experimental data was obtained in all cases. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015 , 132, 42141.     

Viscoplastic deformation of an epoxy resin at elevated temperatures

The tensile behavior under monotonic loading and stress‐relaxation testing of an epoxy resin has been studied. Experimental data at various strain rates and three temperatures from ambient up to just below T_g were performed, to study the transition from the brittle behavior to a ductile and therefore viscoplastic one. Dynamic mechanical analysis was applied to study the glass transition region of the material. Furthermore, a three‐dimensional viscoplastic model was used to simulate the experimental results. This model incorporates all features of yield, strain softening, strain hardening, and rate/temperature dependence. The multiplicative decomposition of the deformation tensor into an elastic and viscoplastic part has also been applied, following the element arrangement in the mechanical model. A stress‐dependent viscosity was controlling the stress–strain material behavior, involving model parameters, calculated from the Eyring plots. A new equation for the evolution of the activation volume with deformation was proposed, based on a probability density function. The model capability was further verified by applying the same set of parameters to predict with a good accuracy the stress‐relaxation data as well. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 101: 2027–2033, 2006     

A procedure for modeling the nonlinear viscoelastoplastic creep of HDPE at small strains

HDPE pipes are frequently laid in buried or submerged conditions and are often subjected to considerable internal pressure. This context requires the consideration of HDPE as a structural material and demands constitutive models to predict failure possibilities in short and long terms. This article presents an approximate procedure to simulate the viscoelastoplastic nature of HDPE's material behavior under creep conditions. While a generalized Kelvin‐Voigt model based on Prony series is used to model viscoelasticity, the power law of Zapas‐Crissman is adopted to account for viscoplastic effects. The associated material parameters are obtained from experimental creep‐recovery tests evaluated at different stress levels and constant temperature. As this type of test allows an uncoupled procedure for identifying the viscoelastic and viscoplastic material parameters, this task is divided into two stages: (i) a constrained nonsmooth optimization problem is defined and solved for the viscoelastic parameters, and (ii) the viscoplastic parameters are determined by linear regression. Thereafter, the viscoelastic and viscoplastic parameters obtained for each experimental stress level are interpolated linearly for intermediate stress conditions. Finally, a numerical‐experimental example is presented, showing that the proposed procedure is able to reproduce adequately more complex loading conditions. POLYM. ENG. SCI., 57:144–152, 2017. © 2016 Society of Plastics Engineers     

Nonlinear viscoelastic and viscoplastic response of glassy polymers

The main features of inelastic mechanical behavior of glassy state were studied theoretically and experimentally in terms of tensile stress‐strain and tensile creep experiments. A theoretical treatment introduced in earlier work, which takes into account the viscoelastic path at small strains and the viscoplastic one at higher stresses, proved to be capable of describing the main aspects of mechanical response of glassy polymers, i.e. nonlinear viscoelasticity during creep procedure, and yield stress, yield strain, strain softening and rate effect in a constant crosshead speed test.     

Natural rubber protein as interfacial enhancement for bio‐based nano‐fillers

Natural rubber was enhanced with soy protein nano‐aggregates and carbon black using a hybrid process. The rubber composites reinforced with an optimum amount of soy protein or soy protein/carbon black showed useful tensile properties. The stress‐strain behaviors were analyzed with a micro‐mechanical model that describes the stress–strain measurements well. The model analysis provides insight into filler network characteristics and entanglement modulus. The composites were also analyzed with both linear and nonlinear viscoelastic properties. Temperature and frequency dependent modulus as well as the model analysis of stress softening effect describe the ability of soy protein to constraint polymer chains in the highly filled composites. For the composites reinforced with soy protein, the good tensile properties are attributed to good filler‐polymer adhesion through the compatibilization effect of natural rubber protein. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 130: 2188–2197, 2013     

Modeling of the transient viscosity for polymer melts after startup of shearing and elongational deformations

In order to describe the transient stress growth for polymer melts, the empirical model proposed by Seo for the viscosity of steady‐state flow is combined with a phenomenological viscoelastic model of a differential type (the White–Metzner model) along the lines proposed by Souvaliotis and Beris. The relaxation time is taken as a function of the invariant of the stress tensor (hence that of the configuration tensor) rather than that of the rate of the deformation tensor. Numerical results show a good correlation with experimental data. The model predictions approach steady‐state values at long times after the startup. The nonlinear form of the model correlates very well with the experimental data over many decades of the deformation rate, both in shearing and elongational deformations. The proposed model is a simple one that can also describe the overshoot in the transient stress growth. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 88: 510–515, 2003     

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.     

Anisotropic mechanical behavior of semi‐crystalline polymers: Characterization and modeling of non‐monotonic loading including damage

In this work, the mechanical response of high density polyethylene (HDPE) to complex uniaxial tensile loadings is firstly characterized experimentally, taking into account the damage occurring in large deformation and the initial anisotropy induced by the forming process. Anisotropic effects are characterized through tensile tests using several complex loading paths involving large deformation, and for different orientation with respect to the extrusion direction. A mechanical model is then developed, based on a non‐equilibrium thermodynamic approach of irreversible processes, resulting in a new thermodynamic potential describing both the elasto‐viscoelastic–viscoplastic behavior and the volume variation due to damage. Results show that transverse strains and volume strain of HDPE highly depend on specimen orientation, whereas the apparent Young's modulus is not affected by this orientation. The developed model is validated for HDPE, and satisfyingly predicts the complex response of HDPE to complex loadings paths. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017 , 134, 44468.     

Investigation of the strain‐rate‐dependent mechanical behavior of a photopolymer matrix composite with fumed nano‐silica filler

With the evolution of additive manufacturing, there is an increasing demand to produce high strength and stiffness polymers. Photopolymers are very commonly used in stereolithography and fused deposition modeling processes, but their application is limited due to their low strength and stiffness values. Nano‐sized fibers or particles are generally embedded in the polymer matrix to enhance their properties. In this study, we have studied the effect of fumed nano‐sized silica filler on the elastic and viscoelastic properties of the photopolymer. The uniaxial testing coupons with different concentrations of silica filler have been fabricated via casting. We observed improvement in mechanical properties by the addition of the nano‐sized filler. As polymers exhibit time‐dependent mechanical response, we have conducted tensile tests at different strain rates as it is one of the most common modes of deformation, and is commonly used to characterize the parameters of the rate‐dependent material. We noticed significant dependence of the mechanical properties on the strain rate. quasi‐linear viscoelastic (QLV) model, which combines hyperelastic and viscoelastic phenomena, has been employed to capture the response of the material at different strain rates. We found out that the QLV model with Yeoh strain energy density function adequately describes the rate‐dependent behavior of the material and has reasonable agreement with the experimental results. POLYM. ENG. SCI., 59:1695–1700 2019. © 2019 Society of Plastics Engineers     

AC-13级配钢渣沥青混合料粘弹塑本构模型研究

为了研究钢渣沥青混合料非线性粘弹塑性变形特性,提出Schapery模型与改进Swchartz模型组合的积分型粘弹塑本构模型。采用钢渣替换AC-13级配中粒径2.36 mm以上的石灰石粗骨料,制作得到钢渣沥青混合料试件。设计并开展一系列的单轴压缩蠕变实验,通过应力递增蠕变回复实验,获得不同应力条件下材料的弹性、粘弹性应变和粘塑性应变,进而拟合确定本构模型参数。利用0.4 MPa、1.0 MPa下的蠕变回复实验验证模型有效性。结果表明,模型不仅能准确刻画钢渣沥青混合料蠕变过程中的弹性、粘弹性与粘塑性变形,还可用于预测不同应力水平下钢渣沥青混合料蠕变变形规律。     

Nonlinear stress relaxation of silica filled solution‐polymerized styrene–butadiene rubber compounds

The stress relaxation of silica (SiO₂) filled solution‐polymerized styrene–butadiene rubber (SSBR) has been investigated at shear strains located in the nonlinear viscoelastic regions. When the characteristic separability times are exceeded, the nonlinear shear relaxation modulus can be factorized into separate strain‐ and time‐dependent functions. Moreover, the shear strain dependence of the damping function becomes strong with an increase in the SiO₂ volume fraction. On the other hand, a strain amplification factor related to nondeformable SiO₂ particles can be applied to account for the local strain of the rubbery matrix. Furthermore, it is believed that the damping function is a function of the localized deformation of the rubbery matrix independent of the SiO₂ content. The fact that the time–strain separability holds for both the unfilled SSBR and the filled compound indicates that the nonlinear relaxation is dominated by the rubbery matrix, and this implies that the presence of the particles can hardly qualitatively modify the dynamics of the polymer. It is thought that the filler–rubber interaction induces a coexistence of the filler network with the entanglement network of the rubbery phase, both being responsible for the nonlinear relaxation. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Modeling of visco‐hyperelastic behavior of transversely isotropic functionally graded rubbers

In this article, visco‐hyperelastic constitutive model is developed to describe the rate‐dependent behavior of transversely isotropic functionally graded rubber‐like materials at finite deformations. Zener model that consists of Maxwell element parallel to a hyperelastic equilibrium spring is used in this article. Steady state response is described by equilibrium hyperelastic spring and rate‐dependence behavior is modeled by Maxwell element that consists of a hyperelastic intermediate spring and a nonlinear viscous damper. Modified and reinforced neo‐Hookean strain energy function is proposed for the two hyperelastic springs. The mechanical properties and material constants of strain energy function are graded along the axial direction based on exponential function. A history‐integral method has been used to develop a constitutive equation for modeling the behavior of the model. The applied history integral method is based on the Kaye‐BKZ theory. The material constant parameters appeared in the formulation have been determined with the aid of available uniaxial tensile experimental tests for a specific material and the results are compared to experimental results. It is then concluded that, the proposed constitutive equation is quite proficient in forecasting the behavior of rubber‐like materials in different deformation and wide ranges of strain rate. POLYM. ENG. SCI., 56:342–347, 2016. © 2016 Society of Plastics Engineers