A multiscale framework for the elasto-plastic constitutive equations of crosslinked epoxy polymers considering the effects of temperature, strain rate, hydrostatic pressure, and crosslinking density
Introduction
Elucidating the plastic behavior of amorphous polymeric materials requires a thorough investigation of their structure-property relationships under various conditions including temperature, strain rate, and hydrostatic pressure. Influences of these parameters on the constitutive response of polymers have been extensively studied both experimentally (Bauwens-Crowet et al., 1969; Bauwens-Crowet et al., 1972; Gómez-del Río and Rodríguez, 2012; Richeton et al., 2005; Richeton et al., 2006; Richeton et al., 2007a; Richeton et al., 2007b) and simulationally (Odegard et al., 2014; Park et al., 2018a; Sundararaghavan and Kumar, 2013; Vu–Bac et al., 2015) to understand the non-linear variations of the yield and post-yield stresses due to the relaxation of polymer segments. In other words, the dissipation of applied stress during deformation is determined by intrinsic chain relaxations, which are largely dependent on time, temperature, and pressure. Focusing on the results of relaxation behaviors of glassy polymers, quantification of the structure-property relations in both the elastic and plastic regimes has been conducted considering the influence of the crosslinking ratio (Wan et al., 2011), molecular structures of resins and curing agents (Jeyranpour et al., 2015; Park et al. 2018b), molecular weight (Hossain et al., 2010), chain interactions (Jatin et al., 2014), and reinforcements (Kim et al., 2017).
In order to examine polymer plasticity characteristics, classical theories describing the yielding of amorphous polymers have been developed and experimentally validated (Argon, 1973a; Argon, 1973b; Bauwens-Crowet et al., 1969; Bauwens-Crowet et al., 1972; Eyring, 1936; Gómez-del Río and Rodríguez, 2012; Richeton et al., 2005; Richeton et al., 2006; Richeton et al., 2007a; Richeton et al., 2007b; Robertson, 1966). First, Eyring proposed a theory for the yielding of glassy polymers by modeling the amount of energy required for the initiation of plastic flow based on the transition state theory (Eyring, 1936). Eyring's theory describes the yielding of polymers as a single activation process that exhibits a linear dependence of the yield stress on the logarithm of the strain rate. Afterward, Robertson (1966) established a correlation between the yield stress, strain rate, and temperature by assuming cis–trans molecular conformation transition mechanisms. Robertson's theory quantified the yield stress with the glass transition temperature of a material by employing the Williams–Landel–Ferry equation (Williams et al. 1955). Another widely known model that characterizes polymer's plasticity is Argon's theory, which focuses on the derivation of an activation-free enthalpy for the thermally activated formation of molecular kinks (Argon, 1973a; Argon, 1973b). That theory postulated the irreversible conformational change of polymer chains with double-kinked elastic cylinders due to the introduction of a wedge disclination loop that was adopted from the plastic deformation mechanism of crystalline materials. The Argon theory describes the kinking activation energy for polymer yielding with the mechanical properties of materials such as the elastic modulus and Poisson's ratio. The subsequent yielding models for polymer's plasticity such as the Ree–Eyring (Ree and Eyring, 1958) and cooperative (Fotheringham et al., 1976; Fotheringham and Cherry, 1978; Richeton et al., 2005; Richeton et al., 2006; Richeton et al., 2007a; Richeton et al., 2007b) ones were developed by focusing on the accurate prediction of the yield stress in broad ranges of the strain rate and temperature. They are based on Eyring's equation assuming the involvement of multiple relaxations in the plastic deformation of glassy polymers and describe the nonlinear yielding at either extremely high strain rates or low temperatures that significantly limit local molecular movements. Additional activation processes contribute to the accurate estimation of abrupt changes in the yield stress at high strain rates of approximately 103/s or temperatures around −50°C (Bauwens-Crowet et al., 1969; Bauwens-Crowet et al., 1972; Gómez-del Río and Rodríguez, 2012; Richeton et al., 2005; Richeton et al., 2006; Richeton et al., 2007a; Richeton et al., 2007b).
Recently, these classical yielding theories have been validated not only by the systematic estimations of yielding characteristics, but also by the direct observations of atomic properties during deformations coupled with atomistic simulations (Park et al., 2018a; Sundararaghavan and Kumar, 2013; Vu-Bac et al., 2015). Several atomistic studies reported significant deviations of the yield stresses determined by molecular dynamics (MD) simulations from experimental values due to their nonlinear dependences on the strain rate following the Ree–Eyring and cooperative models (Nazarychev et al., 2016; Odegard et al., 2014; Park et al., 2018a). In particular, a systematic derivation of the yield stress of amorphous polymer (Nazarychev et al., 2016) revealed that Eyring's first equation, which exhibited a linear relation between the reduced yield stress and the logarithm of the strain rate, failed to predict a quasi-static yield stress in a broad strain rate range spanning from the quasi-static to MD values of about 109/s. Direct observations of polymer molecules also helped to achieve a better understanding of possible deformation mechanisms. Atomistic studies on the deformation mechanisms of amorphous polymers revealed that interchain non-bonding interactions mainly accommodated the applied deformation by consuming most of the deformation energy (Hossain et al., 2010; Jatin et al., 2014; Park et al., 2018b), leading to vigorous local molecular movements during plastic deformation. In thermoplastic polymers, elastic deformation proceeds by the dominant non-bonding interactions that increase the free volume. Afterward, dihedral angle transitions become the dominant factor affecting plastic deformations; the transition population of the dihedral angle from the trans to gauche state is maximized in the vicinity of the yield point, owing to the increased space between various polymer segments (Hossain et al., 2010; Jatin et al., 2014). A similar trend was observed for thermosetting polymers. Because their monomers are crosslinked with curing agents, irreversible deformations mainly occur in localized areas (unlike those in thermoplastics) (Park et al., 2018b). It was found that the dihedral angles of epoxy polymers in the vicinity of benzene rings were strongly involved in the irreversible folding of the epoxy network, which was the main origin of plastic strain accumulation (Park et al., 2018b). In addition, MD simulations of epoxy polymers (Sundararaghavan and Kumar, 2013) reproduced the molecular kinks characterizing the correlation between the sharp stress drop and the irreversible folding of the epoxy network described by the Argon theory.
It is of primary importance to quantitatively predict polymer yielding by atomistic simulations as well as to elucidate the plastic deformation mechanisms at the atomic scale. The quantitative estimation of constitutive response provides alternative ways for the construction of finite element (FE) models considering various physical and chemical variables that have significant effects on deformation parameters (Park et al., 2018a; Vu-Bac et al., 2015). Owing to the existence of such multiple microscopic factors that influence the deformation properties of polymers, estimates of effects of the corresponding parameters and applications to the FE models become increasingly important in material design. Furthermore, such multiscale models constructed by MD and FE simulations could reduce time and cost consumption associated with the traditional trial–and–error experimental methods. Hence, the importance of constructing a well-established simulation framework for evaluating the elasto–plastic constitutive response of these materials and predicting their macroscopic structural characteristics has been acknowledged by many researchers (Arash et al., 2019; Baek et al., 2019; Danielsson et al., 2007; Kim et al., 2015; Kim et al., 2017; Miehe and Schänzel, 2014; Shin et al., 2019; Venkatesan and Basu, 2015; Xie et al., 2016).
The quantitative estimations of the constitutive response of polymers would require considering time-scale extensions because the time step of atomistic simulations is around one femto-second, which is extremely short. It means that the MD simulations are inevitably conducted for approximately several nanoseconds owing to the limited computational resources. Due to the short-used time for deformation simulations, the strain rate of MD simulations is generally ranging from 107/s to 1010/s, which is higher than laboratory tests. Thus, the stress determined via atomistic simulations significantly deviates from the response in the quasi-static range due to the strain rate dependence of the stress (Park et al., 2018a; Sundararaghavan and Kumar, 2013; Vu-Bac et al., 2015). To overcome these timescale limitations, two different schemes for determining the quasi-static yield stresses of amorphous polymers via atomic simulations were developed. The first approach involved the 0 K solution of the Argon theory that is described by the elastic properties of materials (Sundararaghavan and Kumar, 2013; Vu-Bac et al., 2015). Although this method has been successfully applied to several polymer materials, its assumption on temperature dependence of the yield stress deviated from experimental observations. This method used a linear dependence to represent the yield stress-temperature relation that remained unaffected by the strain rate, whereas both the experimental observations and classical yield models (Bauwens-Crowet et al., 1969; Bauwens-Crowet et al., 1972; Gómez-del Río and Rodríguez, 2012; Richeton et al., 2005; Richeton et al., 2006; Richeton et al., 2007a; Richeton et al., 2007b) revealed that the yield stress nonlinearly varied with temperature, especially in a low-temperature region. The second approach involved the derivation of a quasi-static yield stress using the equivalence of time and temperature during polymer relaxations (Park et al., 2018a). The slope of the plot of the reduced yield stress versus the logarithm of the strain rate at elevated temperatures was used for the prediction of the quasi-static yield stress considering prominent relaxation characteristics under both conditions. Although the nonlinear properties of polymer yielding can be accurately predicted by the last approach, it would require conducting time-consuming computations to obtain reliable slope values at various temperatures.
In this study, a method using the 0 K solution of Argon's theory to derive the quasi-static yield stress was further modified to include nonlinear polymer plasticity characteristics by taking into account the influence of the hydrostatic pressure and crosslinking density. The cooperative model was adopted to represent nonlinear dependences of the yield stress at various strain rates and predict quasi-static solutions at different temperatures using the glass transition temperature as a criterion for the extinction of the internal yield stress. The derived quasi-static constitutive laws were utilized for the construction of a numerical FE plasticity model based on the paraboloidal yield surface (Tschoegl, 1971) to evaluate the multi-axial loading behaviors of an open-hall specimen at different crosslinking densities.
Section snippets
Microscopic model
In this section, quasi-static constitutive laws were derived from the results of MD simulations. Systematic evaluations of the yielding characteristics of epoxy polymers were conducted in the MD environment, and the obtained data were calibrated at different temperatures, strain rates, hydrostatic pressures, and crosslinking ratios.
Macroscopic constitutive model
In this section, the macroscopic constitutive model previously developed by Melro et al. (2013a) is described and utilized for the FE analysis of the macroscopic structure of epoxy polymer based on the quasi-static constitutive laws derived in the previous section. The FE analysis procedure included 1-element patch test and open-hall structure test whose objective was not only to validate the macroscopic constitutive response with the quasi-static constitutive laws, but also to examine the
Conclusion
In this work, multiscale simulations for the elasto-plastic deformation of epoxy polymer were conducted by considering the influence of temperature, strain rate, hydrostatic pressure, and crosslinking density. To construct a framework without experimental data, a method for predicting quasi-static yield behavior was developed based on the results of a previous work (Sundararaghavan and Kumar, 2013) that utilized the 0 K solution of the Argon theory. Furthermore, the nonlinear dependence of the
CRediT authorship contribution statement
Hyungbum Park: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Maenghyo Cho: Conceptualization, Funding acquisition, Project administration, Supervision, Validation, Visualization, Writing - review & editing.
Declaration of interests
The authors declare no competing interests.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) Republic of Korea grant funded by the Korea government (MSIP) (No. 2012R1A3A2048841).
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