پیش بینی رشد ناپیوسته ترک خستگی در پلی اتیلن با تراکم بالا بر اساس تئوری لایه ترک همراه با پارامتر های متغیر لایه ترک Prediction of discontinuous fatigue crack growth in high density polyethylene based on the crack layer theory with variable crack layer parameters

Recently, high density polyethylene (HDPE) is widely used in water and natural gas distribution pipes. The expected lifespan of HDPE pipes is several decades in field conditions [1]. Owing to the long-term failure process of thermoplastic structural elements, accelerated experiments have been widely used in the laboratory setting [2,3]. Here, environmental and loading conditions may be manipulated to execute short-term testing through increasing temperature and load beyond their normally encountered values [4–۷]. Specifically, cyclic loading is widely employed for accelerated testing of creep [8]. In these experiments, it is critical to identify the relationship between short-term experiments and long-term field failure processes, such as the connection between fatigue and creep failure. For this purpose, a comprehensive understanding of fracture mechanisms under various loading conditions is essential. The fracture mechanism of thermoplastics under fatigue including HDPE [1,9] varies as a function of applied stress, stress ratio, and frequency, as well as temperature and environment. The chain and molecular structures of HDPE undergo substantial breakage and recrystallization in the case of relatively high load conditions, which is known as ductile failure. Under lower load conditions, however, final failure may occur in the form of brittle fracture without large amounts of plastic deformation, which is often observed in field failures of HDPE pipes. To obtain meaningful results, the variation in loading conditions of the accelerated testing must reflect the same fracture mechanisms of actual field failures. Brittle fracture of HDPE can be conventionally divided into three stages during creep and fatigue loading conditions. In the first stage, micro-crazes accumulate in the vicinity of pre-existing defects. When the accumulated damage exceeds a specified critical level, a macroscopic crack starts to grow. In the second stage, the crack grows slowly in a brittle manner through a quasi-static crack growth process, also known as slow crack growth (SCG). Finally, in the third stage, global instability leads to catastrophic failure with rapid crack propagation (RCP) [10]. In many filed failure cases, the time for SCG is considered the major contribution of the lifespan of HDPE. Therefore, accurate modeling of the SCG process is critical to predicting the lifespan of HDPE. Hence, many studies have been conducted to investigate the SCG process of HDPE [2,3,10–۱۳]. Evidently, SCG of HDPE proceeds in a continuous or discontinuous manner depending on the applied load, temperature, and crack size. Differences in SCG patterns are accompanied by significantly different SCG kinetics. The point at which the mechanism and kinetics of the SCG distinctly change is referred to as the ductilebrittle transition of the second kind (DBT2), since it corresponds to the transition from ductile (creep) to brittle fracture of the microfibers within the process zone (PZ) [2,9,12,14,15]. There are several empirical models of SCG rate based on the conventional Paris law [16–۱۸]. However, these modified models still do not competently capture the discontinuous SCG mode of HDPE. This can be plainly seen from the fact that the empirical models use the crack driving force exclusively, i.e., the stress intensity factor, without consideration for the interaction between the crack and PZ [10]. Morphological and microstructural studies on the crack as well as the PZ in front of the crack tip are necessary to address these shortcomings. The crack layer (CL) theory is the first model that allows the PZ to evolve independently of the crack and explicitly considers the interaction between the main crack and PZ (namely, damages ahead of crack tip). The system including the main crack and PZ, which surrounds the crack tip, comprises the CL [19]. In the case of engineering thermoplastics, localized strain areas (damage) surrounding the main crack tip may vary, depending on the molecular architecture and morphology of the polymer in addition to the stress state [20]. Polyethylene (PE), especially HDPE, displays a simple wedge shape PZ, consisting of cold drawn fibers and membranes, with a relatively sharp boundary between the PZ and the surrounding original material. Therefore, in this case a superposition method is applicable. It enables simplification of the CL parameters [13,21,22]. That is, the morphological characteristics of the PZ and crack can be utilized to construct CL theory for simulating SCG behavior of various thermoplastics, including HDPE. During the CL simulation, several parameters reflecting the material properties, microstructure, and test conditions were employed, e.g., the closing stress acting on PZ boundary, the initial surface fracture energy, the enthalpy transition for material transformation, and the natural drawing ratio, amongst others. Although each CL parameter has a clear physical meaning, the experimental evaluation of such parameters are commonly required. So, the application of CL theory for prediction of thermoplastic lifespan in brittle fracture is somewhat limited. One effective way of improving the applicability of CL theory is to conduct a systematic analysis of the correlations between SCG behavior and various CL parameters. An investigation on the effect of temperature, which is one of the accelerating factors, on the enthalpy of cold drawing, i.e. material transformation from its original isotropic state to a highly oriented drawn status, was performed [9]. However, studies on the other CL parameters are also necessary. In this study, a parametric analysis was performed to estimate the lifespan and SCG mode of field conditions through a combination of short-term tests and numerical simulations of SCG and PE lifespan. SCG behavior of HDPE under fatigue loading conditions was numerically simulated by applying CL theory. A variation of CL input parameters was considered for the parametric analysis. The experimental results reported by Parsons et al. [2,12] were compared with our simulation results, and the CL parameter dependency on fatigue loading conditions, such as maximum load level, R-ratio, and loading frequency, were constructed. Since that the CL theory suggests a system of two coupled differential equations describing the crack and PZ evolution, the phenomenological observations used in the present study can be extended to other conditions, as long as the crack-PZ interaction mechanism is the same. Therefore, one can accurately estimate the SCG behavior and the lifespan of HDPE at fatigue and creep loading conditions through application of the CL algorithm presented in this study.