Critical stress and thermal activation of crystal plasticity in polyethylene: Influence of crystal microstructure and chain topology
Graphical abstract
Introduction
The plastic yielding behavior of solid polymers has been intensively investigated in the last fifty years from the standpoint of predictive modelling in consideration that it determines the strain and stress thresholds of large deformations accompanied with irreversible damage processes of these materials in use conditions [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Both crazing and shear banding can contribute to the yielding of glassy and semi-crystalline polymers for which inclusion of rubber particles proved to be an efficient toughening issue [5].
Particular attention has been focused on the mechanisms of yielding of semi-crystalline polymers, noticeably polyethylene, owing to massive production and relatively complex microstructure in comparison to glassy polymers [6], [7], [8], [9], [10].
The present paper especially addresses the modeling of shear yielding of semi-crystalline polymers. This phenomenon has been previously studied using three kinds of theoretical approaches. The most common one is the dynamic approach based on the Eyring's concept of thermally activated rate processes that is often used for modeling the yielding of glassy polymers [11], [12], [13], [14], [15], [16], [17], [18], [19]. This phenomenological modelling enables to account for the strain rate and temperature dependences of the yield stress. The advantages is that it enables establishing fairly good correlations with the viscoelastic relaxations of the material that are associated with the thermally activated motions of the polymer chains, for both amorphous and semi-crystalline polymers. However it cannot account for the structural characteristics of the semi-crystalline polymers that are well known to influence their mechanical properties. Secondly, plastic yielding of semi-crystalline polymers has been approached from a metallurgical standpoint introducing the concept of dislocation in the crystalline phase. This approach is based on the critical phenomenon of nucleation of dislocations that initiate plastic deformation via crystal slip [20], [21], [22], [23], [24], [25], [26], [27], [28]. It does account for the structural characteristics of the material through the crystal thickness that determines the level of elastic energy to be overcome for the dislocation nucleation. Moreover, in this approach, nucleation is considered as a thermally activated rate process since thin crystal platelets such as polymers crystalline lamellae do not afford natural dislocation sources [20]. This aspect is implicitly taken into account is the activation energy of the nucleation. However, only few authors have explicitly considered the thermally activated aspect of plastic flow in parallel to the dislocation-driven modelling of semi-crystalline polymers [14], [16], [18], [29]. Thirdly, the so-called constitutive law approach is an attempt to describe in an empirical manner the stress-strain behaviour including the yielding phenomenon [30], [31]. Recent developments in this kind of approach have been achieved via micromechanical modelling that considers the semi-crystalline polymer as a micro-composite in which the two components exhibit thermally activated deformation processes [32], [33], [34], [35], [36], [37], [38]. However, this third approach does not actually takes into account the dimensional aspects of the semi-crystalline microstructure.
A common point to all these approaches of yielding of semi-crystalline polymers is that they do not consider the occurrence of cavitation that may bias the activation of shear yielding [39]. Moreover, although some of these approaches integrate structural factors, the role of the intercrystalline tie chains and chain entanglements has been rarely considered in the modelling of shear yielding [40], [41]. These characteristic features of the molecular topology yet should be taken into account in the mechanical modelling of semi-crystalline polymers since they contribute to transmit the stress between the two phases. Indeed, it is largely recognized that these stress transmitters, ST, are determining factors for the long-term mechanical behavior of polyethylene [42], [43], [44], short-term phenomena such as yielding [40], [45], [46] and strain-induced cavitation [47], [48], and on the elastic modulus of the amorphous phase as well as [49], [50]. It is to be noticed that the density of these molecular stress transmitters cannot be directly measured, they can only be estimated via experimental data such as the natural draw ratio or the strain hardening coefficient [44], [51] or by theoretical computation based polymer chain statistics at equilibrium [43], [52].
Due to the aforementioned complex structural features, it is not an easy task to analyze both the microscopic and the dynamic aspects of the mechanisms involved at the initiation of crystal plastic yielding of semi-crystalline polymers. In a previous study [53], we reported that low draw temperature and thick crystalline lamella are favorable for the occurrence local crystal shear prior to macroscopic yielding. Shear of crystalline lamella is believed to starts at the anchoring points of the stress transmitters at the surface of lamella owing local stress concentration [40], [41], [54], [55]. This suggests that chain topology should be taken onto account in any kind of theoretical approach of yielding. Cavitation which depends on temperature and microstructure [47] also depends on the chain topology as tie chains and chain entanglements are likely to prevent void opening [47], [48]. This local damaging phenomenon could strongly promote the occurrence of plastic events [39], [47], [56], [57], [58] prior to yielding by causing local stress concentration around the induced cavities or defects.
It seems therefore necessary to define a yield strain criterion close to the appearance of early local plasticity events in contrast to the yield point of the stress-strain curve which corresponds to the threshold of macroscopic plastic flow due to accumulation of local plasticity events and damaging phenomena. It also appears that the stress for the initiation of local plasticity results from a combination of structural and topological factors that deserve to be taken into consideration in the modelling of the yielding phenomenon together with the thermal activation aspect.
Thus, the aim of this paper is first to propose a new way to assess the strain and stress for initiation of local crystal plasticity in polyethylene from in situ WAXS experiments and to compare these data with the conventional yield strain and stress data for a collection of samples covering a large crystallinity range. Secondly, an endeavor in made to combine both microstructural and topological factors with the approach of the plastic flow as a thermally activated rate process. Finally, comparison is made between the present experimental stress data with data computed from theoretical values of elastic constant borrowed from literature.
Section snippets
Materials and sample preparation
Four polyethylenes with close molecular weights and different hexane co-unit concentration were provided by Total Petrochemical (Feluy, Belgium). These materials are the same ones as those used in previous publications [47], [50], [53]. Details on the molecular characteristics of these materials are given in Table 1.
The four polyethylenes were compression molded into 500 μm sheets between aluminum foils in a heating press and then subjected to three different thermal treatments, namely
Results
A yielding criterion associated to the initiation of crystal plasticity has been defined in previous papers benefiting from in situ WAXS experiments during tensile testing [50], [53]. The (200) reflection in polar sector of the WAXS patterns arise from lamellae lying in equatorial region of the spherulites. This (200) plane is normal to the tensile stress so that its spacing changes directly in relation to the local stress in tensile direction, i.e. it can be taken as a local stress gauge. As
Modified thermal activation approach
Lets us first remind the screw dislocation nucleation model which predicts the dependence of crystal yield stress, σyi, on crystal thickness, Lc. Brooks et al. [27] proposed the following revised relation that has been successfully used by the present authors [28].where K is the shear modulus in the (200) slip plane of the dislocation, b the Burgers vector, ro and Eo the core size and the core energy of the dislocation and ΔG the activation energy barrier
Conclusion
A method for estimation of the stress and strain at initiation of crystal plastic deformation in equatorial region of polyethylene spherulites, i.e. the crystal yield stress, σyi, and strain, εyi, has been proposed by using in-situ WAXS experiment during uniaxial tensile tests. Influences of temperature and microstructure have been analyzed thanks to a series of samples covering a wide range of microstructural characteristics in terms of crystallinity and crystal thickness. The local crystal
Acknowledgements
The authors are indebted to Total-Petrochemicals (Feluy, Belgium) for supplying the polymers and their molecular characteristics. The authors are grateful to the European Synchrotron Radiation Facility (Grenoble, France) for time allocation on the BM02 beamline and to Dr. C. Rochas for assistance in the SAXS experiments. The China Scholarship Council (2011008081) is acknowledged for the grant of a doctoral fellowship to B. Xiong.
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