Continuum-based modeling large-strain plastic deformation of semi-crystalline polyethylene systems: Implication of texturing and amorphicity
Introduction
Semi-crystalline polyethylene systems have a large range of applications, in sectors like oil industry, automobile, aeronautic, robotic, biomechanics and civil engineering, in which they may be subjected to large strains under in-service or manufacturing process. These materials present an extremely large variety of molecular architectures with a crystallinity index ranged from about seventy percent down to a few percent only, passing the morphology from well-organized crystalline lamellae to discrete crystalline lamellae randomly dispersed within an amorphous matrix (Peacock, 2000). The broad morphology range leads basically to an extremely large variety of responses from the plastic response characteristics of thermoplastics to the nonlinear elastic response characteristics of elastomers (Ayoub et al., 2011; Abdul-Hameed et al., 2014) with a profound influence of the amorphous and crystalline features such as density of stress transmitters in the amorphous layers and crystal dimensions (Hillmansen et al., 2000; Argon et al., 2005; Kazmierczak et al., 2005; Bartczak and Galeski, 2010). The development of continuum-based models is a prerequisite for designing and performance predicting polyethylene-based products whether in the form of fibers, films or massive parts.
Modeling response of semi-crystalline polyethylene systems is to relate to their complex organization hierarchy from the nano-sized lamellar structure to the macroscopic scale. Purely phenomenological approaches using theories such as viscoelasticity (Zhang and Moore, 1997) or viscoplasticity (Drozdov and Gupta, 2003; Colak and Dusunceli, 2006; Zaïri et al., 2006; Khan and Krempl, 2006; Ben Hadj Hamouda et al., 2007; Dusunceli and Colak, 2008; Drozdov et al., 2013) have the advantage of simplicity. Nonetheless, these models provide only a mathematical description of the different aspects of the material response, generally under low-strain and without obvious connection to the actual microstructure and its deformation-induced evolution. Over the years, continuum-based models have been developed with the concern to introduce microstructural specificities as precisely as possible in the aim to provide a deep understanding of the separate and synergic effects of key microstructural parameters governing the macroscopic response. The link with the microstructure strongly depends on the approach used for the constitutive representation and the observation scale from which the model starts. In the most physically consistent models, the microstructure approximation of semi-crystalline polymer systems is generally based on composite-type representations.
Following the pioneering work of Haward and Thackray (1968), continuum-based models have been proposed to constitutively describe the large-strain material behavior of semi-crystalline polymers by the combination of resistances representing intermolecular and molecular network micro-mechanisms (Ayoub et al., 2010, 2011; Abdul-Hameed et al., 2014; Makki et al., 2017; Sepiani et al., 2018; Chen et al., 2019; Deplancke et al., 2019; Qi et al., 2019; Bernard et al., 2020). The latter micro-mechanism describes the resistance to deformation of the amorphous molecular structure to simulate the molecular orientation/relaxation process during the strain-hardening stage of the stress-strain response (Boyce et al., 2000). In the above-cited models, a simple averaging homogenization is used and only for the intermolecular resistance which does not allow to represent the crystallinity effect on the deformation-induced orientation. The active interaction between crystalline and amorphous domains is a first-order factor which requires to be taken into account using more sophisticated micromechanics-based approaches. For instance, micromechanics-based models using the Eshelby inclusion theory were proposed through the matrix-inclusion constitutive representation of the semi-crystalline polymer system (Bédoui et al., 2006; Gueguen et al., 2010; Anoukou et al., 2014; Hachour et al., 2014). These models were restricted to the elastic stiffness and yield strength predictions, and more recently extended to the small-strain post-yielding behavior (Mesbah et al., 2021). Multi-scale homogenization-based constitutive models have been also developed by considering at the mesoscopic scale an aggregate of two-phase layered composite inclusions consisting in parallel crystalline lamellae and amorphous layers. Using this concept, the elastic-viscoplastic deformation behavior of high-density polyethylene was predicted at small strain levels (Nikolov and Doghri, 2000; Nikolov et al., 2002) and moderate strain levels by simulating the texture evolution (Lee et al., 1993a, 1993b; van Dommelen et al., 2003; Agoras and Ponte Castaneda, 2012; Uchida and Tada, 2013; Mirkhalaf et al., 2019). The models are generally identified using loading modes in which the crystal plasticity is a first-order phenomenon, such as compression, channel die compression and shear. While the local events involved in yielding due to tensile straining are not connected with clear crystal shearing but with the phenomena occurring in the amorphous phase and the relation to crystal thickness is not expected, rather to crystallinity degree, read amorphicity (Seguela et al., 1998a, 1998b; Bartczak and Kozanecki, 2005; Bartczak and Galeski, 2010; Rozanski and Galeski, 2013). The latter is an influential and decisive parameter in tensile yielding, the strength and consistency of the amorphous phase being thus first-order factors (Mesbah et al., 2021).
The aim of the present article is to examine the ability of a multi-scale homogenization-based constitutive model to capture the polyethylene large-strain tensile response variation with crystal concentration. The model is applied on a high-density polyethylene with 0.72 crystal content, a low-density polyethylene with 0.3 crystal content and an ultra-low-density polyethylene with 0.15 crystal content. The results of our simulations are criticized by analyzing the model ability to capture a series of experimental observations under monotonic, loading-unloading and oligo-cyclic stretching over a large strain range. The effect of the amorphous phase fraction on the microstructure evolution is finally discussed thanks to the model.
The present paper is organized as follows. Section 2 presents the fully three-dimensional continuum-based constitutive theory. The model-experiments comparisons are presented and discussed in Section 3. Concluding remarks are finally given in Section 4.
The following notation is used throughout the text. Tensors and vectors are denoted by normal boldfaced letters and italicized boldfaced letters, respectively, while scalars and individual components of vectors and tensors are denoted by normal italicized letters. The superposed dot designates the time derivative. The superscript T indicates the transpose quantity.
Section snippets
Constitutive representation
The organization hierarchy of the semi-crystalline polyethylene structure depends on the amorphicity. As illustrated in Fig. 1, decreasing the crystallinity leads to a morphology transition from micro-sized interconnecting spherulites to individually crystalline lamellae randomly dispersed in the rubbery amorphous matrix. According to the microstructure approximation based on a composite-type description, the morphology may be regarded as a percolated crystalline matrix at high crystallinity
Materials
Three commercial polyethylene systems, differing by their molecular topology and molar masses, were ordered from Total Petrochemicals and DOW Chemicals: a Ziegler-Natta high-density ethylene-hexene copolymer (HD) from Total Petrochemicals, a linear low-density ethylene-octene copolymer (LL) and an ultra-low density ethylene-octene copolymer (UL) from DOW Chemicals, both issued from metallocene catalysis. The molecular characteristics of the polyethylene pellets determined by Gel Permeation
Concluding remarks
In this paper, the highly nonlinear and rate-dependent mechanical response of semi-crystalline polyethylene systems covering a wide spectrum of the crystallinity was investigated within a multi-scale homogenization-based approach. The model ability to capture the large-strain response variation with the crystallinity was criticized. The model exhibited weaknesses in small-strain region but it was able to reproduce, within large-strain region, the rate-dependent monotonic response variation with
Credit statement
Zhu Yan: Software, Formal analysis, Visualization, Methodology, Writing – original draft preparation, Qiang Guo: Software, Formal analysis, Methodology, Fahmi Zaïri: Supervision, Methodology, Conceptualization, Writing – original draft preparation, Project administration, Ali Zaoui: Supervision, Conceptualization, Writing – original draft preparation, Qifeng Jiang: Methodology, Conceptualization, Project administration, Xiaobing Liu: Supervision, Project administration.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Sichuan Science and Technology Program (2020YFH0152), National Natural Science Foundation of China (52079118), National Key R&D Program of China (2018YFB0905200), Open Fund of Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University (szjj2019-012).
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2022, International Journal of PlasticityCitation Excerpt :Hachour et al. (2014) conducted experimental investigation of the biaxial yielding of HDPE using butterfly-shaped and notched round bar specimens and further proposed a micromechanics-based two-phase criterion for the yield onset of SCPs considering the effects of different crystallinities. Recently, Yan et al. (2021) proposed a viscoplastic–viscohyperelastic model based on a multi-scale homogenization approach to consider the amorphous–crystalline interfacial interaction. The model was applied to semi-crystalline PE systems with various crystallinities under both monotonic and oligo-cyclic tensile loadings.