## Abstract

Ultra-high molecular weight polyethylene (UHMWPE) used in biomedical applications, e.g., as a bearing surface in total joint arthroplasty, has to possess superior tribological properties, high mechanical strength, and toughness. Recently, equal channel angular extrusion (ECAE) was proposed as a processing method to introduce large shear strains to achieve higher molecular entanglement and superior mechanical properties of this material. Finite element analysis (FEA) can be utilized to evaluate the influence of important manufacturing parameters such as the extrusion rate, temperature, geometry of the die, back pressure, and friction effects. In this paper, we present efficient FEA models of ECAE for UHMWPE. Our studies demonstrate that the choice of the constitutive model is extremely important for the accuracy of numerical modeling predictions. Three considered material models (J2-Plasticity, Bergstrom-Boyce, and the three-network model) predict different extrusion loads, deformed shapes, and accumulated shear strain distributions. The work has also shown that the friction coefficient significantly influences the punch force and that the two-dimensional (2D) plane strain assumption can become inaccurate in the presence of friction between the billet and the extrusion channel. Additionally, a sharp corner in the die can lead to the formation of the so-called “dead zone” due to a portion of the material lodging into the corner and separating from the billet. Our study shows that the presence of this material in the corner substantially affects the extrusion force and the resulting distribution of accumulated shear strain within the billet.

## 1 Introduction

Equal channel angular extrusion (ECAE) was initially proposed as a method to deliver large amounts of uniform shear deformation to a metal specimen in order to influence the material microstructure and improve its mechanical and physical properties, see for example, Ref. [1]. This technique is widely used for metal alloy processing and is being actively developed and improved, see Refs. [2–4].

Originally developed for metals, ECAE is also being considered for the processing of polymer materials to form oriented structures and improve properties, see Ref. [5]. One of the earliest numerical modeling studies on extrusion of the polymer material (polycarbonate) was published in Ref. [6]. The authors assumed two-dimensional (2D) plane strain and used a J2-Plasticity material model. More advanced 2D plane strain numerical simulations of the angular extrusion were performed in Refs. [7,8] for the high-density polyethylene (HDPE) material. These publications implemented a hypoelastic viscoplastic constitutive model to account for the strain rate sensitivity of the polymer. The authors investigated how the die geometry and number of passes affect stress and strain fields within the polymer billet. The experimental study on how angular extrusion affects mechanical properties of polypropylene (PP), supported by numerical modeling of the extrusion process, was reported in Ref. [9]. Similarly, numerical and experimental studies on the applicability of the ECAE process and its influence on the properties and behavior of polypropylene and HDPE were presented in Ref. [10]. The authors performed a set of parametric numerical studies on the extrusion rate, billet-die friction coefficient, and back pressure sensitivity for PP and HDPE showing that those parameters are crucial.

Recently, ECAE was proposed as a means of achieving higher molecular entanglement and/or superior mechanical properties of ultra-high molecular weight polyethylene (UHMWPE) [11] and UHMWPE-based composites [12]. The numerical modeling of the process could be used to better understand its mechanics and how it influences physical and mechanical properties of the resulting material. In particular, it can be used to evaluate the importance of the processing parameters such as friction between the billet and the die, extrusion rate, extrusion angle, back pressure, and processing temperature. However, to the best of the authors’ knowledge, there are no published results on the numerical simulations of the ECAE for the UHMWPE material. This paper is based on the research presented at the IMECE 2020 conference [13]. It extends beyond the results published in the conference proceeding by including temperature-dependent simulations and providing a comparison with available experimental data.

This paper is organized as follows: Sec. 2 describes the experimental setup for the ECAE of UHMWPE and its implementation in finite element analysis (FEA). Three constitutive material models utilized in the extrusion simulations are presented in Sec. 3. Section 4 provides a comparison of our modeling approach to the published results for the extrusion of HDPE. Section 5 presents several numerical studies including a comparison between 2D plane strain and three-dimensional (3D) models, a friction sensitivity study, and a comparative analysis of the constitutive material models used for large deformations of polymers during extrusion. The issues related to the extrusion channel shape and the process temperature are also addressed in the section. Conclusions are provided in Sec. 6.

## 2 Finite Element Model of Equal Channel Angular Extrusion Experiment

The finite element model has been developed for the ECAE process illustrated in Fig. 1. In this particular setup, a steel angular channel with a square cross section of 50 mm × 50 mm and a sharp right-angle connection (Φ = 90 deg, Ψ = 0) is used. A set of heating cartridges is embedded into the channel walls to maintain constant temperature during the extrusion. An aluminum pressure punch, which extrudes the material billet is velocity controlled and a back pressure plate, which delivers resistance to the billet motion in the horizontal part of the channel is force controlled in order to be able to prescribe desired extrusion rate and back pressure. The virgin polymer powder is placed to the vertical portion of the channel and then compressed at an elevated temperature of 162.5 °C for 2.5 h for the purpose of polymer consolidation. A colored pigment can be added to layers of the powder to make the polymer billet striped and allow tracking its deformations. After that, the specimen is extruded at an elevated temperature of 162.5 °C and with the pressure punch velocity of 15 mm/min. Once the extrusion is complete, the billet is extracted from the horizontal part of the channel using a displacement-controlled extraction punch. More detailed description of the experimental setup is given in Refs. [11,12].

The schematics provided in Fig. 1 illustrates the case when angle Ψ = 0 (sharp corner). For extrusion of UHMWPE, this design of the die can lead to the formation of the so-called “dead zone”—a portion of the material billet which gets stuck in the lower corner of the channel and eventually separates during the extrusion, see Fig. 2. To deal with this problem, the channel geometry can be improved by either increasing angle Φ (Φ > 90 deg), or introducing a smooth fillet in the lower corner by increasing angle Ψ (Ψ > 0), or adding a mitre-fillet-like insert to the lower left corner of the die which can potentially prevent the material from jamming and separating during the extrusion.

In the numerical models, the extrusion channel, pressure punch, back pressure plate, and the extraction punch are modeled as rigid surface contact bodies. The extrusion channel is fixed, the pressure punch has its vertical displacement and velocity prescribed, the back pressure plate is force controlled, and the extraction punch is displacement controlled. The extrusion billet is modeled using ∼15,000 2D plane strain Herrmann quadrilateral finite elements in the case of 2D plane strain assumption and ∼120,000 3D tetrahedral linear finite elements in the case of 3D simulations (Fig. 3). The mesh size for each model was chosen based on the tradeoff between successful convergence of the simulations and their computational efficiency.

The contact interaction between deformable finite elements and rigid contact bodies is modeled as bilinear shear friction sliding (see Ref. [14]). Extrusion and extraction processes are simulated by two consecutive load cases. The extrusion takes 600 s which corresponds to the experimental extrusion time. All simulations are performed using MSC.MARC Mentat software.^{2}

## 3 Material Models for Extrusion of Ultra-High Molecular Weight Polyethylene

One of the important steps in numerical modeling of such a complex process as ECAE of a polymeric material is to pick an adequate constitutive material model. We begin with the assumptions that the billet material will be non-porous before the beginning of the extrusion process due to compaction at elevated temperature, and its behavior can be described by continuum solid mechanics equations. The material extrusion is performed at elevated temperatures, so the thermal sensitivity of a chosen material model is especially important. There are two approaches to include temperature dependence of the material behavior. The first approach is to perform characterization tests for the material at each level of temperature which the material is processed at. Such an approach is needed when the material model does not directly account for thermal sensitivity, for example, J_{2}-Plasticity or Bergstrom-Boyce (BB) [15]. The second approach is to perform characterization experiments at certain levels of temperature and use them to calibrate the material's temperature-dependent response. This approach is used when the constitutive model explicitly accounts for the thermal sensitivity of the material properties. Additionally, the material model has to be applicable to large compressive and shear deformations of the polymer. It has to allow for accounting of the strain rate sensitivity of the material. It also needs to be easily calibrated based on limited experimental data (usually simple tension and compression tests).

The J_{2}-Plasticity (J_{2}) model is often the simplest choice as in Refs. [6,16] where it is used for the modeling of ECAE of polycarbonate and HDPE. According to the model, the yielding in the material occurs when the second invariant of deviatoric stress (J_{2}) reaches a critical value. Expressing J_{2} in terms of principal stresses *σ*_{1}, *σ*_{2}, and *σ*_{3}, the yield criterion can be written as follows: $\sigma \xaf=\sigma y$, where *σ*_{y} is the tensile yield strength and $\sigma \xaf=(1/2)\u22c5((\sigma 1\u2212\sigma 2)2+(\sigma 2\u2212\sigma 3)2+(\sigma 3\u2212\sigma 1)2)$ is the effective von Mises stress. In the numerical implementations of the model, the evolution of the yield stress is usually specified as a piecewise linear function of the equivalent plastic strain $\epsilon \xafp$.^{3} However, the J_{2} material model was developed for metallic materials, and hence, it is usually not suitable for highly rate and temperature-dependent polymers [17]. In addition, the plastic behavior of polymeric materials is dependent on the first invariant of stress tensor (see for example, Refs. [18–20]) which is not reflected by the J_{2} model.

A more advanced option is the BB model [15]. This model was developed for elastomers; however, it can be applied to UHMWPE as its high molecular weight causes the polymer chains to entangle and hence act like crosslinks producing a pseudo-elastomeric response of the amorphous region. The model includes strain rate sensitivity of the polymer and can be calibrated with only two loading-unloading tests performed at different strain rates. Obviously, having more experimental data will improve its accuracy. This model is usually included in commercial FE software packages which makes it easy to implement. In addition, the Bergstrom-Boyce model was validated for UHMWPE [17] in several simple mechanical tests (tension and compression). According to BB, the mechanical response of a material is governed by two networks *A* and *B* acting in parallel (see Fig. 4). The first network *A* controls the equilibrium behavior of the material, and the second network *B* governs the time-dependent deviation from equilibrium. The total Cauchy stress is a sum of stresses acting on networks *A* and *B*. Appendix A provides explicit expressions for the stresses acting on networks *A* and *B* along with the governing equations for the viscous response of network *B*. The major disadvantage of the model is its inability to directly take the temperature sensitivity into account requiring additional sets of experiments at each temperature of interest.

The three-network model (TNM) presented in Ref. [21] explicitly includes dependence on both strain rate and temperature sensitivity. It has been developed specifically for UHMWPE and has shown good agreement with the experimental data. The initial viscoplastic response of the material is governed by two energy activation mechanisms which correspond to amorphous and semicrystalline domains, and the large strain behavior is governed by entropic resistance. Rheologically, the model consists of three networks *A*, *B*, and *C* which act in parallel (Fig. 5). The network *A* is a temperature-dependent version of an eight-chain model similar to Refs. [22,23]. The *B* network is similar to network *A* except that the effective bulk modulus of the network evolves with plastic strain which allows the transition from initial yielding to large strains flow to be captured. The network *C* is governed by an eight-chain model including dependence on stretch tensor invariants similarly to the Mooney–Rivlin model with non-Gaussian chain statistics [24,25]. The total stress acting on the model is a sum of the stresses acting on each network *A*, *B*, and *C*. Appendix B provides explicit expressions for the stresses acting on networks *A*, *B*, and *C* along with the governing equations for the viscous response of networks *A* and *B*. The challenge for this model is that it requires a significant number of characterization tests to calibrate and implement due to a large number of material parameters.

Three constitutive models (J_{2}, BB, and TNM) were implemented in the MSC.MARC simulations utilizing add-on software POLYUMOD.^{4} To calibrate material models, we used MCALIBRATION software^{5} based on a set of characterization uniaxial experiments published for HDPE in Ref. [7] and for UHMWPE GUR 1050 in Ref. [26].

## 4 Validation for Extrusion of High-Density Polyethylene

As part of the validation effort, we compared predictions by our models with the published data for ECAE of HDPE, see Ref. [7]. We constructed a 2D plane strain finite element model based on the description provided in Ref. [7]. A billet of 10 mm × 10 mm × 100 mm is extruded through a rigid right-angle channel with Φ = 90 deg and Ψ = 0 and zero friction at the contact surface between the billet and the die. The extrusion is performed at a constant rate of 6 mm/min and at two values of temperature (25 °C and 60 °C). In our comparative simulations, we utilized the J_{2}-Plasticity model and calibrated it using the material characterization data published in the same paper [7]. The obtained model parameters are summarized in Table 1, where *E* is Young's modulus, *v* is Poisson's ratio, *ɛ*_{p} is the equivalent plastic strain, and $\sigma \xafY$ is the equivalent material yield stress.

T = 25 °C | T = 60 °C | ||
---|---|---|---|

E, MPa | 500 | E, MPa | 150 |

ν | 0.38 | ν | 0.38 |

ɛ_{p} | $\sigma \xafY$, MPa | ɛ_{p} | $\sigma \xafY$, MPa |

0 | 10.0 | 0 | 3.6 |

0.0027 | 16.4 | 0.0184 | 9.1 |

0.0807 | 25.1 | 0.0771 | 15.9 |

0.1614 | 31.5 | 0.1124 | 19.1 |

0.2595 | 36.0 | 0.1537 | 22.1 |

0.2133 | 39.4 | 0.1905 | 24.3 |

0.3306 | 44.8 | 0.2299 | 26.6 |

T = 25 °C | T = 60 °C | ||
---|---|---|---|

E, MPa | 500 | E, MPa | 150 |

ν | 0.38 | ν | 0.38 |

ɛ_{p} | $\sigma \xafY$, MPa | ɛ_{p} | $\sigma \xafY$, MPa |

0 | 10.0 | 0 | 3.6 |

0.0027 | 16.4 | 0.0184 | 9.1 |

0.0807 | 25.1 | 0.0771 | 15.9 |

0.1614 | 31.5 | 0.1124 | 19.1 |

0.2595 | 36.0 | 0.1537 | 22.1 |

0.2133 | 39.4 | 0.1905 | 24.3 |

0.3306 | 44.8 | 0.2299 | 26.6 |

Figure 6 shows a comparison of the predictions for the time history of the extrusion punch force in the experiment conducted at 25 °C and 60 °C.

The relative percentage difference as compared with the peak load of the “published model” and the steady extrusion force was calculated to evaluate the level of agreement. The difference between the “present model” and the “published model” peak loads is 26% and 3.2%, whereas the difference at the steady portion of the curves (at 400 s) is 6% and 4.7% for 25 °C and 60 °C, respectively. Thus, it can be seen that predictions practically coincide for 60 °C. For the 25 °C extrusion, a noticeable difference is observed in the peak load. Note that in Ref. [7], the authors used a different material model (hypoelastic viscoplastic) which directly accounts for rate sensitivity requiring sets of characterization experiments at all considered temperatures. Apparently, utilizing J_{2} instead of their model leads to a later transition from compression to the shearing phase in the simulated extrusion process at 25 °C.

## 5 Results and Discussion

We performed a set of parametric studies for numerical models simulating the ECAE process of the UHMWPE GUR 1050 material. This material has a crystallinity level of $50.4\xb13.3%$, density 0.933 g/cm^{3}, and 5.5–6 × 10^{6} g/mole molar mass as described in Ref. [26]. Section 5.1 presents results on the sensitivity of the predictions to the billet-die friction parameters, namely, the friction coefficient between the billet and the die. In Sec. 5.2, we investigate the accuracy of the 2D plane strain assumption by comparing two-dimensional and three-dimensional modeling results. Section 5.3 provides an insight on the importance of the proper choice of the material constitutive model. Section 5.4 considers how the introduction of a triangular prismatic insert to the lower corner of the channel changes the force applied to the pressure punch by partially replicating the formation and separation of the dead zone and its subsequent impact on channel geometry.

The simulations were performed for the extrusion punch velocity of 15 mm/min which corresponds to the shear strain rate on the order of 0.05 s^{−1}, and a back pressure of 36 MPa. Five different values of the friction coefficient and four different values of the extrusion temperature were considered.

### 5.1 Friction Study.

Friction between the die and the billet plays a significant role in the extrusion process, see Ref. [27]. A parametric study was conducted in order to investigate how the friction coefficient *µ* influences the predictions for the extrusion forces during ECAE. A set of 3D FE simulations using the 3D J_{2}-Plasticity model was performed with various values of friction coefficient: *µ* = 0.01, 0.05, 0.1, 0.15, and 0.2. Note that in the simulations, *µ* = 0.01 corresponds to the frictionless sliding of the billet within the channel. The small number is chosen instead of *µ* = 0 to improve numerical convergence. The use of frictionless contact is justified by the experimental data published in Ref. [28] where the authors show that friction can be neglected if mean contact pressure is larger than 30 MPa. The value *µ* = 0.2 was chosen as a maximum value based on the data reported in Ref. [29].

Figure 7 confirms that the friction coefficient is an important parameter and must be accurately evaluated for the realistic simulation of the extrusion process. As can be seen by comparing with no friction results (*µ* = 0.01), the amount of force needed to overcome friction during the extrusion phase of the process reaches 30% of the total force for *µ* = 0.1 and 80% for *µ* = 0.2.

### 5.2 Two Dimension Plane Strain versus Three Dimension Modeling.

Three-dimensional finite element models contain a significant number of finite elements which can result in long simulation times and accumulation of a round-off error when modeling such highly nonlinear process as ECAE for UHMWPE. A 2D plane strain approximation can be a good option to reduce computational effort when the distribution of stresses and strains does not significantly vary in the direction normal to the lateral sides of the extrusion channel. Thus, a numerical study was performed to evaluate the applicability of the 2D plane strain assumption for ECAE of UHMWPE. In the presented simulations, the J_{2}-Plasticity model was used because it is the simplest to implement. Since the plane strain assumption is geometrical and does not involve any constraints on the physical behavior of the material, the obtained results will also be applicable to more complex material models.

As seen in Fig. 8, the predicted deformed shapes of the extracted billets are similar for 2D and 3D simulations. However, as shown in Fig. 9, the punch force versus displacement curves for 3D and 2D are not similar when the friction coefficient *µ* is not negligible. The 3D model predicts a larger punch force which can be explained by the contribution of friction between the billet and the side walls of the extrusion channel. If the friction is small (*µ* = 0.01), the difference between 2D and 3D predictions decreases significantly. This means that in the case of significant friction, the 2D assumption might become inaccurate and 3D modeling has to be considered.

### 5.3 Sensitivity to the Choice of the Material Model.

A set of 2D plane strain finite element simulations was conducted to evaluate the performance of three different constitutive models, J_{2}, BB, and TNM. All three material models were calibrated using experimental data from Ref. [26] assuming that the process is performed at room temperature (20 °C). Note that this temperature regime is not typical for the processing of polymer materials and was used only to investigate general trends in the models’ predicting abilities as applied to UHMWPE. The simulations were performed with friction coefficient *µ* = 0.01 representing frictionless sliding of the billet within the channel.

Figure 10 compares the deformed shapes predicted by all considered material models with the actual shape of the UHMWPE billet extracted from the angular channel after the extrusion. Based on the mutual orientation of the stripes within the billet and the outline of the deformed shape, the predictions by BB and TNM models appear to be closer to the experiment than the J_{2}-Plasticity model. However, there is still a noticeable discrepancy between the actual billet and the predictions by BB and TNM exhibited by lower curvature and higher inclination angle of the stripes in the numerical results. We attribute this discrepancy to the formation of the “dead zone” during the actual extrusion process that was not included in the numerical models. Another potential reason is that the constitutive models were not calibrated for the process temperature of 162.5 °C.

The punch force versus displacement plot shown in Fig. 11 supports the observations made based on the deformed shapes comparison. Namely, the J_{2}-Plasticity model performs differently from BB and TNM. It predicts that a significantly larger load is needed to perform the extrusion which can be explained by linear hardening of the material in the model but is not the case for the considered polymer. At the same time, BB and TNM models have similar predictions for both the deformed shape and the punch force.

Figure 12 shows the distribution of the maximum absolute value of shear strain experienced by the material during the ECAE process as predicted by the J_{2}, BB, and TNM material models. This parameter is important because it could be correlated with changes in molecular entanglement densities and/or material properties of UHMWPE. It can be seen that, similarly to the force–displacement curves, the BB and TNM models predict comparable levels of shear strain, while the strains predicted by J_{2} are lower and more uniform.

### 5.4 Dead Zone Study: Process Temperature Influence.

The results presented in Secs. 5.1–5.3 assume that the material stays continuous even at high levels of stresses with no separation or damage. Because of this, the punch force–displacement curves (Figs. 6 and 11) exhibit a well pronounced hill when the material is pulled from the sharp lower corner of the die. However, our experiments show that the formation of the “dead zone” in the corner results in the separation of the portion of the billet. It is presently challenging to model separation of material (propagation of crack or damage) directly as the proper fracture (or damage) initiation parameters for UHMWPE subjected to high compressive and shear stresses at elevated temperature are not available.

To evaluate the extrusion in the absence of material fracture, we considered the modification of the extrusion channel model by a mitre-fillet-like insert preventing the formation of a dead zone by mimicking its shape. The modified extrusion channel and the insert are shown in Fig. 13. The dimensions of the insert are based on the size of the separated material portion. Note that the introduction of the insert in our models was not aimed to provide justification for change of the actual die shape but rather to observe the effect of the dead zone formation on the punch force and the accumulated maximum shear strain in the material. A comprehensive numerical study on how the geometric shape of the extrusion channel influences the billet damage level during ECAE is shown for example in Ref. [30].

A set of numerical simulations was performed with the models incorporating the triangular insert to investigate its influence on the deformed shape of the billet, extrusion force, and maximum shear strains. Figure 14(a) shows a comparison of the deformed shape of the billet after the extrusion with and without the insert. It can be seen that the introduction of the insert results in the outline of the billet becoming more similar to what is observed in the experiment, see the lower surface of the billet. Also, the orientation of the stripes corresponds better to the experimental image. However, there is still a noticeable difference between the experimental and predicted billets, especially at the upper surface of the deformed shape. The curvature of the stripes at the top surface of actual billet is likely due to severe local plastic deformations caused by the sharp upper corner of the die not captured by the FE models. The prediction of the extrusion load gets affected as well, see Fig. 14(b). The bump in the force around 40 mm of the punch displacement, corresponding to the material getting stuck at the lower corner of the billet, disappears and the load drops at the stable portion of the curve by 30 kN.

Figure 15 shows the prediction of the maximum accumulated shear strain in the UHMWPE billet with and without triangular insert. According to the prediction, the introduction of the triangular insert significantly lowers the maximum shear stain throughout the billet (roughly two times) and results in a more uniform strain distribution. Accurately predicting, maximizing, and ensuring homogeneity of this effect is crucial as shear deformation under these conditions is hypothesized to fundamentally alter the microstructure and therefore the mechanical properties.

Another important parameter for ECAE of UHMWPE is the processing temperature. As stated above, all of the material models were calibrated for UHMWPE GUR 1050 using the experimental data from Ref. [26]. The J_{2} and BB models do not include the temperature dependence of material properties (even though it can be incorporated by conducting characterization experiments for temperatures of interest). Unlike the J_{2} and BB models, TNM allows explicit incorporation of temperature dependence. However, the experiments reported in Ref. [26] were performed at two values of temperature only (20 °C and 37 °C). In contrast, the actual extrusion process was conducted at a temperature of 162.5 °C for which the material characterization data are not presently available. Since the material model was calibrated for much lower temperatures, the numerical process loses its stability for temperatures above 140 °C. Thus, to investigate the thermal sensitivity of the extrusion process, we limited the range of considered temperatures to 20–140 °C. We are aiming not to exactly match experimental data but rather observe the trends in the predicted response. For this purpose, we report force–displacement curves for the extrusion process as simulated using TNM at 20 °C, 90 °C, and 140 °C.

Figure 16 compares the predictions for punch force as a function of the punch displacement to the actual experimental data. As seen in this image, the extrusion force drops by almost a third of what is predicted by the model at 20 °C when the temperature is increased to 140 °C. Since the constitutive model was calibrated using data for much lower temperatures, the predicted force at 140 °C is lower than measured even though the experimental measurements were taken at 162.5 °C. This proves that the proper reflection of the thermal sensitivity of the material properties is crucial for accurate modeling of the ECAE process at elevated temperatures.

## 6 Conclusions

An efficient numerical procedure was developed to simulate ECAE of the UHMWPE material in order to better understand the mechanics of the process and improve the processing parameters with the goal of achieving superior mechanical properties of UHMWPE. Three constitutive material models (J_{2}, BB, and TNM), calibrated using published characterization experiments for GUR 1050 UHMWPE [26], were utilized in the simulations. A set of parametric studies was performed to investigate how numerical predictions are affected by friction parameters, geometrical assumptions (2D versus 3D), constitutive model choice, “dead zone” formation, and process temperature.

We demonstrated that the value of the friction coefficient significantly affects the predicted extrusion punch force. For example, if the friction coefficient is increased from *µ* = 0.01 to *µ* = 0.2, the force applied to the extrusion punch increases by 80%.

The geometrical assumption study has shown that the 2D plane strain assumption can be a good option for the ECAE process simulation if the friction between the billet and the die is small or can be completely neglected. However, if the friction is not small, then the 2D assumption becomes inaccurate.

The applicability of three constitutive models (J_{2}, BB, and TNM) to the prediction of the UHMWPE behavior during the ECAE process was investigated. It has been shown that the prediction based on J_{2}-Plasticity is not in good agreement with the experimental results, both by deformed shape and punch force comparison. In addition, the J_{2} model does not account for the material strain rate and temperature sensitivity and hence cannot accurately predict the polymer behavior. In contrast, two other material models, the BB and TNM, showed similar performance and their utilization resulted in a more accurate prediction of the billet shapes after extrusion and of the punch forces. Even though both models account for the strain rate sensitivity of the UHMWPE, only TNM explicitly includes temperature in its constitutive equations. So, it comes to a tradeoff between running a set of characterization tests at each value of temperature to calibrate the BB model and calibrating the more advanced TNM which requires more extensive characterization testing and includes more material parameters. Another factor to consider is that the BB model is included in most commercial FE software packages.

Note that our simulations are performed at the macroscale; they do not explicitly include microstructure of the material (crystallinity, molecular entanglement, possible porosity, etc.) and its evolution. Incorporation of microstructure evolution (which is usually implemented by assigning the material properties at each integration point of the FE mesh based on changes in the associated representative volume element during each time increment) would improve both accuracy and spatial resolution of the models. At the same time, such multiscale simulations would require more information on macromolecular dynamics and significant computational resources.

During extrusion, it is possible that a portion of the material gets stuck in the sharp corner (“dead zone”) of the die leading to an increased punch force and possible material separation. Modeling of the material separation is challenging, as it is unclear which damage initiation and/or fracture propagation criteria would be appropriate. We proposed to avoid modeling the separation of the “dead zone” by introducing a mitre-fillet-like insert in the lower corner of the extrusion channel. This resulted in the improved prediction of the deformed shape of the billet. The prediction of the punch force as compared with the experimental data improved both qualitatively and quantitatively (by 24%). However, this measure is artificial and can be used only if the simulation of the actual “dead zone” formation and material separation is not feasible.

The influence of the processing temperature on the numerical predictions for the ECAE of UHMWPE was investigated. As the TNM constitutive model was calibrated using characterization experimental data obtained at much lower temperatures than the actual level during the extrusion, the prediction was not aimed to obtain an exact match with the experimental data but observe the trend. It has been shown that with the increase in temperature, the extrusion force drops significantly.

One of the important simulation results is the distribution of maximum accumulated shear strains in extruded UHMWPE billet. It can be used to better understand how the polymer chain entanglement level may vary within the billet and how the processing parameters (such as temperature, extrusion rate, back pressure, and friction) affect the local microstructure. It has been shown that the most influential factor is the presence of the “dead zone” as it significantly affects the magnitude and distribution of the maximum shear strain. Namely, when the triangular insert is introduced, the predicted magnitude of the shear strain is roughly two times lower as compared with the model without the insert, but its distribution is much more uniform. The information on the distribution of accumulated shear strain could be a powerful means for predicting localized alterations to polymer chain conformation as well as mechanical properties within experimentally produced billets.

## Footnotes

## Acknowledgment

The authors would like to thank Dr. Jorgen Bergström for his help with accessing the constitutive material modeling software.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

## Funding Data

National Science Foundation EPSCoR award (#1757371).

### Appendix A: Bergstrom-Boyce Material Model

This model has been originally proposed in Ref. [15] and widely used for the modeling of polymeric materials’ behavior, see for example, Refs. [17,31]. It can directly account for strain rate sensitivity of the material. However, for temperature-dependent response, this model requires characterization experiments at each value of temperature of interest.

**is a sum of the Cauchy stresses acting on networks**

*σ**A*and

*B*

*A*is as

*μ*

_{A}is the initial shear modulus, $\lambda Alock$ is the limiting chain stretch,

*κ*is the bulk modulus, $BA*=(J)\u22122/3FAFAT$ is the distortional portion of the left Cauchy–Green tensor, $FA$ is the deformation gradient, $\lambda *\xaf=tr(BA*)/3$ is the effective chain stretch, $J=det(FA)$ is the volume change, and

*L*

^{−1}is the inverse Langevin function.

*B*is expressed as

*B*, $\lambda Be*\xaf=tr(BBe*)/3$ is the chain stretch in the elastic part of network

*B*, $\lambda Block=\lambda Alock$ is the limiting chain stretch, $BBe*=(JBe)\u22122/3FBeFBeT$ is the finger (left Cauchy–Green) tensor of the elastic part of network

*B*, and $FBe$ is the elastic part of the deformation gradient tensor.

*B*is governed by the effective strain rate

*B*$\sigma B\u2032$ is expressed as $\sigma B\u2032=\mu Bdev(BBe)$. In Eq. (A4),

*μ*

_{B}is the shear modulus of the network

*B*, and

*C*,

*m*, and

*τ*

_{base}are the material parameters.

### Appendix B: Three-Network Material Model

This model has been originally proposed in Ref. [21] and widely used for the modeling of polymeric materials’ behavior including temperature sensitivity of the material response, see, for example, Refs. [32,33]. It directly takes into account both strain rate and temperature dependence.

**is a sum of the Cauchy stresses acting on networks**

*σ**A*,

*B*, and

*C*:

*A*is given by

*A*,

*μ*

_{A}is the initial shear modulus, $\lambda Alock$ is the limiting chain stretch,

*θ*is the current temperature,

*θ*

_{0}is the reference temperature, $\theta ^$ is the material parameter specifying the temperature response of the stiffness, $BAe=(JAe)\u22122/3FAeFAeT$ is the finger (left Cauchy–Green) tensor of the elastic part of network

*A*, $\lambda Ae*\xaf=tr(BAe*)/3$ is the effective chain stretch based on the eight-chain topology assumption,

*κ*is the bulk modulus,

*L*

^{−1}is the inverse Langevin function, and $I$ is the unit tensor.

*A*is governed by the effective strain rate

*p*

_{A}= −((

*σ*_{A})

_{11}+ (

*σ*_{A})

_{22}+ (

*σ*_{A})

_{33})/3 is the hydrostatic pressure,

*R*(

*x*) = (

*x*+ |

*x*|)/2 is the ramp function, and $\tau ^A$,

*α*,

*m*

_{A}, and

*n*are the material parameters.

*B*is governed by the following equation:

*B*, $BBe=(JBe)\u22122/3FBeFBeT$ is the finger (left Cauchy–Green) tensor of the elastic part of network

*B*, and $\lambda Be*\xaf=tr(BBe*)/3$ is the effective chain stretch based on the eight-chain topology assumption.

*μ*

_{B}evolves from

*μ*

_{Bi}to

*μ*

_{Bf}with plastic strain according to the following rule:

*A*, the viscous flowrate for network

*B*is given by

*p*

_{B}= −((

*σ*_{B})

_{11}+ (

*σ*_{B})

_{22}+ (

*σ*_{B})

_{33})/3, and $\tau ^B$ and

*m*

_{B}are the material parameters.

*C*is

*μ*

_{C}is the initial shear modulus, $B*=J\u22122/3FFT$ is the Cauchy–Green deformation tensor, $\lambda *\xaf=tr(B*)/3$ is the effective chain stretch,

*q*is the material parameter, and $I1*$ and $I2*$ are the stretch tensor first and second invariants.

## References

^{®}2019 Volume A: Theory and User Information