An analytical approach for the design optimization of bearing land in the metal extrusion of shaped sections

Bearing region plays an important role in controlling material flow and its optimal design could lead to high quality extruded products. On the other hand, too much of bearing causes the process load to increase. Thus, there must be an optimum point where the bearing lands and the extrusion pressure are just the right values. Determining the proper bearing length is often performed using trial and error methods in the extrusion industry and numerical analysis. The aim of this study is to optimize the bearing length in forward extrusion dies using upper bound method for non-axisymmetric sections. A generalized kinematically admissible velocity field is employed to obtain uniform velocity at the exit surface of the die. Dead metal zone and bearing region define the geometry of the deformation zone. The multi-objective optimization using response surface methodology was applied to optimize the relative extrusion pressure and the deviation of the mean value for the velocity at die exit. Using this method, the proper bearing length is determined. Optimization of bearing land is performed for extrusion of rectangular and L-shaped profiles. The proposed analytical method was verified by physical modelling experiments and numerical simulations. A unique answer for the bearing design could be obtained using the suggested method in a few seconds opposing to numerical method which required many timely and costly trials. This method would be useful for die designers to get the appropriate bearing land and at the same time not to increase the process load excessively.

     

Multi-scale phenomena induced by fluoropolymer processing aids during the extrusion of linear-low density polyethylene

Using polymer processing aids (PPA) based on fluoropolymers not only reduces die pressure, but also postpones extrusion defects for linear polyolefins. The purpose of this work is to investigate the mechanisms explaining PPA efficiency. An experimental set-up was developed to study the PPA actions at different scales, based on a slit die with transparent walls and removable steel inserts, in line with a laboratory extruder. Original results are presented, correlating the evolution of die pressure, surface defects, velocity profiles measured by Laser Doppler Velocimetry and PPA die coating analyzed by Scanning Electronic Microscopy. During extrusion, PPA droplets and stretched aggregates appear first randomly on the whole die surface. Then, coating morphology progressively changes, leading to a network of interconnected and heterogeneous PPA streaks. Quantification of surface deposition shows that the PPA concentration grows towards the die exit, in concordance with spatially evolutionary wall slip. Surface defects elimination is correlated to the presence of discontinuous deposits at the die exit.     

The use of the capillary rheometer for the rheological evaluation of extrudable cement-based materials

In this paper, the rheological properties of an extrudable cement-based paste are investigated by means of an original ram extrusion apparatus (capillary rheometer). The experimental results indicate that a careful measurement of the die pressure is necessary to obtain a realistic viscosity vs shear rate curve, as required in extrusion technology. In particular, it is shown that the optimal test configuration is when the pressure measurement is made directly inside the rheometer die. By applying this rheological methodology in steady-state conditions, it has been observed that the extrudable cement-based material here evaluated obeys to a simple power–law equation, in the range of shear rates investigated, which are suitable for an industrial extrusion process. This paper was presented at the third Annual European Rheology Conference (AERC) held in Hersonissos, Crete, Greece, April 24–27, 2006.     

Simulation of coating flows with slip effects

This work is concerned with the numerical prediction of wire coating flows. Both annular tube‐tooling and pressure‐tooling type extrusion–drag flows are investigated for viscous fluids. The effects of slip at die walls are analysed and free surfaces are computed. Flow conditions around the die exit are considered, contrasting imposition of no‐slip and various instances of slip models for die wall conditions. Numerical solutions are computed by means of a time marching Taylor–Galerkin/pressure–correction finite element scheme, that demonstrates how slip conditions on die walls mitigate stress singularities at the die exit. For pressure‐tooling and with appropriate handling of slip, reduction in shear rate at the die exit may be achieved. Maximum shear rates for tube‐tooling are about one quarter of those encountered in pressure‐tooling. Equivalently, extension rates peak at land entry, and tube‐tooling values are one third of those observed for pressure‐tooling. With slip and tube‐tooling, peak shear values at die exit may be almost completely eliminated. Nevertheless, in contrast to the pressure‐tooling scenario, this produces larger peak shear rates upstream within the land region than would otherwise be the case for no‐slip. Copyright © 2000 John Wiley & Sons, Ltd.     

Solution for plane strain forward and backward extrusions with a fractional reductionR=0.5 by the integartion depending on a parameter

A parameter t is introduced to boundary slip line of rigid regions for plane strain and indirect extrusions with a fractional reduction R=0.5. Integration by substitution has been used along the boundary slip line in order to obtain the extrusion pressure. By the integration depending on a parameter, the following results are obtained, and die pressure is 5.14k for backward extrusion; and pad average pressure is 2.57k for forward extrusion. All the results from this method are the same as those of the conventional solution.     

Influence of die surface on the efficiency of fluoropolymer processing aids during the extrusion of linear-low density polyethylene

Nowadays, fluoropolymer based Polymer Processing Aids (PPA) are currently used to eliminate surface defects and to reduce die pressure in linear polyolefins extrusion. For 10 years, the main origins of the PPA actions are known: PPA discontinuously coat the die surface and create wall slip that delays the sharkskin defect. However, the microscopic parameters that govern PPA efficiency are not really understood. Based on several experimental measurements (Laser Doppler Velocimetry, Scanning Electron Microscopy, surface analysis), we propose an analysis of the mechanisms of PPA actions in various extrusion conditions. Different kinds of die surfaces have been studied and the results show that PPA efficiency can be greatly improved by controlling die topology and chemistry.     

挤出平缝口模优化设计

挤出平缝口模通常用于加工膜和片材，对产品厚度的一致性有很高要求。本文给出了结合聚合物成型模拟技术、设计灵敏度分析和数值优化技术的口模形状优化设计方法。以最小压力降为目标函数，口模出口处任意点的速率与出口已知速率相对误差的平方和小于容许误差为约束函数，口模形状参数为优化设计变量，给出了目标函数的表达式，推导了日标函数、约束函数对优化设计变量的灵敏度公式。利用灵敏度分析和基于梯度的优化算法即序列二次规划算法(SQP法)求解最优设计参数。通过算例表明，应用该法进行口模优化设计可以减小压力降和口模出口速率变化率。     

Isolating curvature effects in computing wall-bounded turbulent flows

An adjoint optimization method is utilized to design an inviscid outer wall shape required for a turbulent flow field solution of the So–Mellor convex curved wall experiment using the Navier–Stokes equations. The associated cost function is the desired pressure distribution on the inner wall. Using this optimized wall shape with a Navier–Stokes method, the abilities of various turbulence models to simulate the effects of curvature without the complicating factor of streamwise pressure gradient are evaluated. The one-equation Spalart–Allmaras (SA) turbulence model overpredicts eddy viscosity, and its boundary layer profiles are too full. A curvature-corrected version of this model improves results, which are sensitive to the choice of a particular constant. An explicit algebraic stress model does a reasonable job predicting this flow field. However, results can be slightly improved by modifying the assumption on anisotropy equilibrium in the model's derivation. The resulting curvature-corrected explicit algebraic stress model (EASM) possesses no heuristic functions or additional constants. It slightly lowers the computed skin friction coefficient and the turbulent stress levels for this case, in better agreement with experiment. The effect on computed velocity profiles is minimal.     

Upper Bound Analysis for Extrusion of T-section Bar from Square Billet Through Square Dies

The present study is devoted to the upper bound analysis of extrusion of T-section bars from square billets through square dies using the modified SERR (Spatial Elementary Rigid Region) technique. Optimized values of the non-dimensional average extrusion pressure at various area reductions have been computed and compared with experimental results available in literature. The optimization process in this study consisted of three stages: (i) optimization of the extrusion pressure with respect to appropriate system variables, (ii) determination of the optimum scheme of discretization of the deformation zone for each of the three formulations (single-point, double-point and triple-point), and (iii) identifying the formulation that gives the lowest upper bound.     

An experimental study of planar entry flow of rubber compound

An automated rheometer based on an injection molding machine is developed for the evaluation of entry flow problems. Several entry flow geometries having different contraction and expansion angles and different channel lengths are tested. Two pressure transducers are flushmounted along the die length and a displacement transducer is installed to measure the screw motion. Signals generated by the pressure transducers and displacement transducer are supplied to an A/D converter and an IBM PC/AT computer. The pressure losses for a rubber compound are measured between two cross-sections along the flow direction. The time evolution of pressure with overshoot during flow before and after the entry region is observed. At low flow rates the pressure drops of the expansion flow are larger than those of the contraction flow. At high flow rates the pressure drops of the contraction flow become higher than those of the expansion flow. A ratio of the pressure drop to absolute pressure before the entry is found to be almost independent of flow rate.     

挤压凹模强度的光弹性与有限元优化综合分析

本文介绍用光弹性试验测定挤压凹模工作时承受的内压力分布规律,在此基础上作了凹模的有限元优化设计,最后进行了光弹性实验验证.这种综合分析方法为提高模具的承载能力、节省模具材料和改进模具结构设计提供了有益的经验.     

Prediction of three-dimensional general shape extrudates by an implicit iterative scheme

This paper presents a numerical technique for solving three-dimensional free surface problems in extrusion applications. The method is fully implicit in the sense that a Newton-Raphson scheme is applied on all variables, and geometrically general. In particular, the die section shape may be complex and contains multiple corners: very few restrictions apply on the mesh generation because the method does not require the nodes to be located on straight lines (spines). A clear distinction is introduced between the directions associated with the kinematic condition and the remeshing rules. As a difference with respect to earlier publications, these concepts are handled separately. Only Stokes problems are solved in this paper and we have not introduced surface tension. Therefore corners in the die section propagate discontinuities in the extrudate shape, an a method for relocating corners without losing the quadratic convergence of the scheme is presented. Data structures used for the implementation are briefly discussed. We present results on the extrusion of various profiles, including a rectangular die (a benchmark problem) and various complex sections containing multiple corners.     

Upper bound analysis of thick wall tubes extrusion process through rotating curved dies

In this study, extrusion process of thick wall tubes through rotating curved dies is investigated by the method of upper bound. Total deformation region is divided into four deformation zones and a velocity field is developed for each deformation zone. The twist moments generated on container and mandrel surfaces are calculated and by equating them with the twist moment exerted by rotating die, the twisting length of tube inside the container is determined. Then, the internal powers, the powers dissipated on frictional and velocity discontinuity surfaces for a rigid-perfectly plastic material are evaluated and they are used in upper bound model. By optimizing the total power with respect to the slippage parameter between die and the tube material, the required relative extrusion pressure for a given process conditions and die angular velocity is determined. The results of finite element simulations are also presented and satisfactory agreement between the calculated and FEM results are demonstrated.     

Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method

Accurate prediction of extrudate (die) swell in polymer melt extrusion is important as this helps in appropriate die design for profile extrusion applications. Extrudate swell prediction has shown significant difficulties due to two key reasons. The first is the appropriate representation of the constitutive behavior of the polymer melt. The second is regarding the simulation of the free surface, which requires special techniques in the traditionally used Eulerian framework. In this paper we propose a method for simulation of extrudate swell using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional-step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step, the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation besides allowing for natural evolution of the free surface with the flow. In the second step, mesh is moved in ALE mode and the associated convection of the variables due to relative motion of the mesh is performed using a Godunov type scheme. While the mesh is fixed in space in the die region, the nodal points of the mesh on the extrudate free surface are allowed to move normal to flow direction with special rules to facilitate the simulation of swell. A differential exponential Phan Thien Tanner (PTT) model is used to represent the constitutive behavior of the melt. Using this method we simulate extrudate swell in planar and axisymmetric extrusion with abrupt contraction ahead of the die exit. This geometry allows the extrudate to have significant memory for shorter die lengths and acts as a good test for swell predictions. We demonstrate that our predictions of extrudate swell match well with reported experimental and numerical simulations.     

Shear band propagation from a crack tip subjected to Mode II shear wave loading

Results are reported for pressure–shear plate impact experiments in which pre-cracked 4340 steel plates are subjected to Mode II loading. Experiments show the propagation of a shear band ahead of the initial crack. Finite element simulations are used to interpret the results. Normal and transverse velocity–time profiles measured at the rear surface of the target can be simulated reasonably well using even an elastic model for the material response. A propagating shear band is obtained when the material is modeled as having reduced shearing resistance described by a thermo-viscoplastic power law, and complete loss of shearing resistance when the shear strain reaches a critical value. However, the predicted speed of propagation of the tip of the shear band is substantially less than required to explain the lengths of the bands observed in the experiments. Adjustments of parameters of the power-law model have little effect on the overall length of the band. Possible reasons for differences between predicted and measured shear band speeds are examined. Further reduction in the shearing resistance in the shear band appears to be essential for the simulated bands to be as long as those observed in the experiments.     

Effect of secondary zone dimensions on melt fracture manifestations of high density polyethylene

An experimental study was performed on melt fracture phenomena in extrusion of high density polyethylene. The purpose of the work was to study the sensitivity of melt fracture driven roughness to the size of recirculation zones, viz. secondary zones. The experimental apparatus consists of a right angle die and a hypodermic needle used as a capillary. The position of the needle relative to the die was adjusted using a special fixture. The roughness of the extrudate was studied as a function of penetration depth. A developed procedure provides a comparison between profile lengths of extruded strands. The computed mean, median, and mode values for roughness were presented as a function of capillary position. A qualitative analysis was conducted for the force oscillations during extrusion with a separate set of dies, equipped with the fixed capillaries of identical lengths and different depths of penetration. It was observed that the oscillatory pattern is sensitive to the sizes of the secondary zones. This qualitative observation supports the conclusions from the quantitative analysis that the roughness of the extrudate can be controlled through an adjustment of the secondary zone sizes.Partly presented at the 58th Annual Meeting of the Society of Rheology, Tulsa, Oklahoma, October 20–23, 1986     

Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry

The oscillating flow instability of a molten linear high-density polyethylene is carefully studied using a single screw extruder equipped with a transparent slit die. Experiments are performed using laser Doppler velocimetry in order to obtain the local velocities field across the entire die width. At low flow rate, the extrusion is stable and steady state velocity profiles are obtained. During the instability, the velocity oscillates between two steady state limits, suggesting a periodic stick-slip transition mechanism. At high flow rate, the flow is mainly characterized by a pronounced wall slip. We show that wall slip occurs all along the die land. An investigation of the slip flow conditions shows that wall slip is not homogeneous in a cross section of the slit die, and that pure plug flow occurs only for very high flow rates. A numerical computation of the profile assuming wall slip boundary conditions is done to obtain the true local wall slip velocity. It confirms that slip velocities are of the same order of magnitude as those measured with a capillary rheometer.     

Laminar and melt fracture flow of polymer melts through successive capillaries

Summary The flow behavior of a polystyrene and a polyethylene melt has been studied during extrusion through successive capillaries arranged to produce converging and diverging flow patterns through the two orifices. Laminar flow and flow instability regions were included in the experiments.The pressure drop at fixed mass flow rate in converging or diverging flow through the combined capillaries equalled the sum of the pressure drops in each die if the ratio of die diameters was 1.43 in this case. The combined pressure drop exceeded the sum of the individual pressure drops, however, during converging melt fracture flow through die combinations with higher contraction ratios. Die swell measurements through successive orifices indicate that extrudate diameter may be influenced by the contraction between the reservoir and capillary. This effect is particularly noticeable at lower shear rates.

---

Zusammenfassung Das Fließverhalten einer Polystyrol- und einer Polyäthylen-Schmelze wurde bei der Extrusion durch zwei hintereinandergeschaltete Kapillaren untersucht. Diese ließen sich so anordnen, daß darin entweder konvergente oder aber divergente Strömungsformen erzeugt werden konnten.Der Druckabfall bei konstantgehaltenem Massenstrom war in den kombinierten Kapillaren sowohl bei konvergenter als auch bei divergenter Strömung gleich der Summe der Druckabfälle in den Einzelkapillaren, solange das Durchmesserverhältnis 1,43 blieb. Dagegen überstieg der Druckabfall in der kombinierten Anordnung die Summe der einzelnen Druckabfälle, wenn bei höheren Kontraktionsverhältnissen die konvergente Strömung Schmelzenbruch zeigte. Die Messung der Strangaufweitung (die swell) nach hintereinandergeschalteten Düsen zeigt, daß der Extrudatdurchmesser durch die Verengung zwischen Reservoir und Kapillare beeinflußt werden kann. Dieser Effekt tritt besonders ausgeprägt bei kleineren Schergeschwindigkeiten auf.

---

---

With 4 figures and 1 table     

Self-reinforcement of polymers as a consequence of elongational flow

In this study, attention is directed towards the abilities of elongational flow to induce the chain orientation and, as a consequence, to improve the properties of extruded profiles. A novel-designed extrusion die with semihyperbolic convergency of the channel, by which a high percentage of elongational flow in the melt is generated, was connected with a conventional single-screw extruder. Commercial grades of high-density polyethylene and isotactic polypropylene were used as model commodity polymers. Round extrudates prepared by this processing line were assessed as self-reinforced products. Detailed structural observation carried out with the aid of scanning electron microscopy and wide-angle X-ray scattering evidenced that the structure of extruded rods is uniquely fibrous and highly oriented throughout the profile. As a result, the self-reinforced extrudates possess outstanding storage modulus in a wide range of temperatures and considerably higher melting point and fusion heat than rods produced by conventional extrusion.This paper presented at the Annual European Rheology Conference (AERC) 2005, April 21–23, 2005 in Grenoble, France     

Crack profiles in mortar measured by holographic interferometry

In order to test the accuracy of theoretical fracture models for mortar and concrete, it is necessary to have accurate measurements of the crack profiles. In this study, sandwich holographic interferometry has been used to find the crack profiles in a center-notched plate specimen loaded at the center of the notch. The results have shown that at low load levels with corresponding short crack lengths, there is little difference between the measured crack profiles and elastic crack profiles computed by finite-element analysis. At high loads with long crack lengths there is a large difference between measured and computed elastic crack profiles. The data suggest the presence of a closing pressure at the crack tip and that there may be a limit to crack-tip-opening displacement (CTOD) before the crack propagates.