Constitutive modeling for the elastic-viscoplastic behavior of high density polyethylene under cyclic loading
Introduction
High density polyethylene (HDPE) is widely used for industrial applications due to its excellent shape flexibility, lightweight, cost effectiveness, and corrosion resistance. For instance, plastic fuel tanks and cooling bottles made of HDPE are more and more popular in automobiles manufactured by Ford Motor Company and others. In general, the loads of these HDPE applications include cyclic changes of stress and temperature which may result in different failure modes, such as necking, buckling and fatigue cracking. Better comprehension of constitutive behavior of HDPE will help us to predict failure mode and lifetime of HDPE under cyclic loading, which is significant to improve the robustness of our designs while delivering lighter and cheaper products, such as automobile fuel tanks. Therefore, the ultimate target of our present work is to obtain an accurate and time-efficient constitutive model to predict stress-strain response of HDPE under cyclic loading for the long-term durability analysis including failure mode prediction and lifetime evaluation of its applications using CAE tools.
As a typical semi-crystalline polymer, the deformation behavior of HDPE is much more complicated than metals and amorphous polymers. The constitutive behavior of HDPE shows strong strain rate and temperature dependence even around room temperature, while the constitutive behavior of metals at elevated temperatures exhibits strain rate and temperature dependence (Abdel-Karim and Khan, 2010; Cyr et al., 2015; Farrokh and Khan, 2009; le Graverend et al., 2014; Khan and Baig, 2011; Salvati and Korsunsky, 2017; Shutov and Ihlemann, 2014; Voyiadjis and Samadi-Dooki, 2016). The constitutive model of HDPE have to consider the relationship between the crystalline phase and the amorphous phase, while there is no necessity to consider that for amorphous polymers (Ames et al., 2009; Anand et al., 2009; Baghani et al., 2012; Bouvard et al., 2013; Horstemeyer and Bammann, 2010; Khan et al., 2006; Srivastava et al., 2010). Besides constitutive models, there are abundant valuable researches on the deformation mechanism of metals and amorphous polymers using various methods (Malekmotiei et al., 2015; Oleinik, 1991; Samadi-Dooki et al., 2016; Voyiadjis et al., 2018; Zheng et al., 2018). Due to the difference from metals and amorphous polymers, it is necessary to study deformation mechanism and constitutive behavior of semi-crystalline polymers such as HDPE. Depending on the loading rate, the relevant works can be sorted into three categories: models for static-loading, quasi-static loading and dynamic cyclic loading.
For HDPE under static loading, the constitutive behavior typically refers to creep or stress relaxation. Current studies focus on constitutive models to predict strain increase with time in creep or stress decay with time in relaxation. These models are helpful for the understanding of the viscous properties of HDPE, but only suitable for those under constant or monotonic load conditions. Initially, Lai and Bakker (1995) studied the nonlinear tensile creep of HDPE influenced by aging and employed an equation to forecast the creep behavior at different stress levels. Later, Elleuch and Taktak (2006a, 2006b) studied the viscoelastic behavior of HDPE under both tensile and compressive loads, and computed the compressive creep and stress relaxation using a mathematical equation. In recent years, investigators show great interest in modeling the creep behavior of polymers including HDPE using different theories and methodologies (Bozorg-Haddad et al., 2010; Huang et al., 2011, Khan and Yeakle, 2011; Kühl et al., 2017; Xu and Jiang, 2017). In addition, metal at elevated temperature performs evident creep or stress relaxation behavior, which is similar to polymer (Wen et al., 2018).
The majority of the studies on the constitutive behavior of HDPE under quasi-static loading are devoted to developing constitutive models based on the motion of molecular chains to simulate the macro mechanical behavior. These contributions reveal that the deformation mechanism is physically related with the microstructures, so they account for the temperature, strain rate and crystallinity influences on the mechanical properties of HDPE. However, it is necessary to modify these models to achieve high numerical accuracy and predict the cyclic-loading behavior. Boyce et al. (1988a, 1988b, 1989a, 1989b) proposed a constitutive model based on the macromolecular structure and micro mechanism of plastic flow, which can be used to predict the large inelastic deformation of glassy polymers and explain the temperature and strain rate effects. Arruda and Boyce. (1993a, 1993b), Arruda et al. (1993) extended Boyce's model to amorphous polymers as well as rubber elastic materials, and discussed the initial anisotropy impact on the deformation behavior. Subsequently, many constitutive models based on their work have been developed for semi-crystalline thermoplastics, including HDPE (Ayoub et al., 2010, 2011, 2014; van Dommelen et al., 2003; Popa et al., 2014; Uchida and Tada, 2011, 2013). Bergstrom (2015) made great progress in numerical implementations of these constitutive models. Furthermore, other independent constitutive models have been developed based on the fact that semi-crystalline thermoplastic consists of the crystalline phase and the amorphous phase (Balieu et al., 2013; Gonzalez et al., 2017; Maurel-Pantel et al., 2015; Ponçot et al., 2013; Regrain et al., 2009; Rozanski and Galeski, 2013; Shojaei and Li, 2013).
Besides, many studies on the constitutive behavior of HDPE under quasi-static loading pay more attention to the numerical simulation accuracy other than the microstructures. The most popular approach on this subject is the viscoplasticity theory based on overstress (VBO), which ignores both the yield surface and the difference between loading and unloading (Chaboche, 2008; Krempl, 2000; Ho and Krempl, 2002; Krempl and Khan, 2003; Khan and Krempl, 2005). Although VBO is quite different from classical plasticity theory, it is adopted to calculate the constitutive response of HDPE under quasi-static loading and unloading due to the precise prediction (Colak, 2005; Colak and Dusunceli, 2006; Dusunceli and Colak, 2006, 2008; Dusunceli, 2010). Scalet et al. (2015) put forward a one-dimensional numerical model based on the phase transition approach and Fischer-Burmeister complementary function to describe the loading and unloading behavior of HDPE.
Compared with static and quasi-static loading, the mechanical behavior of HDPE under dynamic cyclic loading exhibits a couple of features which are challenging for numerical modeling. At first, the viscous behavior becomes much more complicated under cyclic loading. For instance, the mean strain increase due to the non-zero mean stress in the cyclic-loading test is different from the strain increase under the same constant stress in the creep test. The mean strain increase with cycle is known as the ratcheting or cyclic creep which has attracted a lot of interests for both polymers and metals (Abdel-Karim, 2009, 2010; Dafalias and Feigenbaum, 2011; Guo et al., 2011, 2013; Lee et al., 2014;Zhu et al., 2014a,b). Additionally, the mechanical properties such as yield stress and hardening can decay cyclically with the increase of the specimen temperature as a result of the energy dissipation of the inelastic deformation in a cyclic-loading test (Henann and Anand, 2009; Khan and Farrokh, 2006). The yield behavior that can happen in the quasi-static or dynamic cyclic-loading test also draws attentions a lot in recent years (Hachour et al., 2014; Li and Buckley, 2010; Pandey et al., 2013; Sundararaghavan and Kumar, 2013). Moreover, damage accumulation due to fatigue crack growth can also weaken the mechanical properties such as elastic modulus in the cyclic-loading test. Damage which can occur in static, quasi-static or dynamic cyclic-loading tests for different materials has been always a stimulating topic (Kruch and Chaboche, 2011; Mozaffari and Voyiadjis, 2016; Shojaei et al., 2013; Tekog˜lu and Pardoen, 2010; Voyiadjis et al., 2012; Xu et al., 2014; Zaïri et al., 2008, 2011).
In order to capture the above features, researchers have conducted a few elementary investigations for HDPE under cyclic loading. Kang and his group devoted great efforts in constitutive models of polymers under cyclic loading, such as ultra-high molecular weight polyethylene and polyamide 6 as well as metals (Yu et al., 2013, 2018; Zhu et al., 2014a,b, 2016, 2017). Their model is extremely complicated due to glass transition and moisture diffusion, though this physical model can be used to predict the cyclic-loading deformation trend (Yu et al., 2017a, 2017b). Drozdov. (2007, 2010a, 2010b) established a numerical approach with different strain-based parameters for loading and unloading to simulate the mechanical behavior of thermoplastics including HDPE under cyclic loading by distinguishing the first cycle from the subsequent cycles. In Drozdov's work, calibration can be obtained with the complicated simulation process and numerous parameters strongly dependent on the tests. Ghorbel (2008) improved the classical thermodynamic model originally for metal by modifying its parabolic Drucker-Prager criterion to calculate the cyclic-loading behavior of some polymers excluding HDPE. Krairi and Doghri (2014) constructed a model by incorporating viscoelasticity governed by Prony series, viscoplasticity characterized by isotropic and kinematic hardening, and ductile damage related with plastic strain, which could accurately predict the quasi-static behavior of HDPE but could only moderately predict the cyclic-loading behavior. Haouala and Doghri (2015) developed a numerical method by extending the two-scale time homogenization theory to viscoelastic-viscoplastic model to compute stress-strain response of UHMWPE under cyclic loading, which provides an interesting way to predict the cyclic-loading behavior of polymers, but the simulation procedure is complex and the prediction precision is improvable. Actually, the cyclic loading brings about difficulties in modelling deformation behavior for various materials including metals, heterogeneous composites, nano-composites, rock-like materials, and so on (Barai and Weng, 2011; Brassart et al., 2012; Brepols et al., 2014; Cao et al., 2018; Gao et al., 2011; Lahellec and Pierre Suquet, 2013; Launay et al., 2011; Mercier and Molinari, 2009; Yang et al., 2013). The publications on metals under cyclic loading are much more developed than polymers and other materials (Castelluccio and McDowell, 2017; Ghorbanpour et al., 2017; Guan et al., 2017; Hazeli et al., 2015; Lee et al., 2015; Muhammad et al., 2017; Qiao et al., 2015; Smith et al., 2018; Wang et al., 2013; Yoshida et al., 2015; Zecevic et al., 2017).
All in all, it is still imperative to devote more efforts to develop an accurate and time-efficient constitutive model for HDPE under cyclic loading. Based on the contributions mentioned above, the following information can be summarized for simulating the constitutive behavior of HDPE accurately and time-efficiently. First, an advanced constitutive model should be able to predict the mechanical response for static, quasi-static and dynamic cyclic loadings. Second, the model should contain an elastic-plastic network for the crystalline phase and the paralleled viscoelastic networks for the amorphous phase. Third, model calibration should consider sufficient test data in order to get an accurate prediction.
Thus, the deformation mechanism is explored by performing relaxation-unloading tests to extract the time-independent elastic-plastic behavior of HDPE from the overall elastic-viscoplastic deformation. The experimental result designates nonlinear elasticity and parallel framework consisting of elastic-plastic and viscoelastic networks. Consequently, the original PRF is developed based on the relaxation-unloading tests and calibrated by considering creep, relaxation, tension and cyclic-loading test data. The basic PRF in Abaqus was proposed by Lapczyki et al. (2012) and Hurtado et al. (2013) to capture the nonlinear time-dependent mechanical response of polymers subjected to large deformation. In addition, the developed constitutive model is verified by comparing the predicted data with the experimental data of cyclic-loading tests.
Section snippets
Deformation mechanism of HDPE
In this section, the deformation mechanism of HDPE was examined by conducting uniaxial tension tests and relaxation-unloading tests. In the uniaxial tension tests, HDPE had the totally coupled nonlinear elastic-viscoplastic deformation behavior, while in the relaxation-unloading tests, the time-independent elastic-plastic deformation could be extracted from the overall time-dependent deformation by experimentally removing the viscous effects. More importantly, the relaxation-unloading tests
Development and calibration of the constitutive model
In this section, a constitutive model for HDPE based on the experimental results of relaxation-unloading tests is obtained by developing the nonlinear springs and generalizing the Norton-Baily law in parallel rheological framework. Besides, a methodology based on parameter optimization is proposed to calibrate the material model by embedding Abaqus FEM model into ModeFrontier main program.
Model validation and discussion
In order to validate the developed model, cyclic-loading tests with a stress ratio equal to zero were performed and predicted at different stress levels and frequencies. The stress-strain response of HDPE under cyclic loading could be represented by the ellipse hysteresis loop evolution. For the convenience of subsequent investigations on failure and fatigue of HDPE under cyclic loading, three parameters were used to characterize the position and shape change of the hysteresis loop. Therefore,
Conclusions
The study in this paper provided a useful way of exposing the deformation mechanism, developing a constitutive model, and determining parameters for the model. It is important to emphasize the following points.
- (1)
The constitutive behavior of HDPE was totally elastic-viscoplastic coupled when the load was applied. The special designed relaxation-unloading test could decouple the time-independent elastic-plastic behavior from the overall time-dependent behavior by removing the viscous effects via
Acknowledge
The authors would like to express their sincere appreciations to Dr. Carlos Engler-Pinto, Dr. Katherine Avery and Prof. Hong-Tae Kang for the helpful discussions on constitutive modeling and Mr. Frisch Robert for his supportive work on the testing setups.
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