Numerical simulations of annular extrudate swell of polymer melts

Numerical viscoelastic simulations were carried out using a K-BKZ type of separable integral constitutive equation. Both reversible and irreversible models were tried for several types of damping functions to calculate the annular extrudate behavior of high-density polyethylene (HDPE). There are two aims in this study; first, to clarify the properties of these dumping functions, and second, to investigate the influence of rheological characteristics on annular extrudate swell. In these numerical simulations, relaxation spectrum and shear viscosity were fixed, and the other characteristics were varied. The reversional response of the damping function mainly has an effect on the magnitude of the area swell even if the die is straight. The irreversible model expresses the experimental results of annular extrudate swell better than the reversible model. The accurate fitting of N1 by the damping model is important for predicting it. The magnitude of N1 predicted from the Wagner exponential model is lower than that of the PSM model, and the area swell shows the same tendency as N1. A modified PSM model that allows the N1 curve to shift can fit the magnitude of area swell. The relationship between the diameter and thickness of the extrudate depends on N2/N1, and it was estimated by simple linear elasticity of solids. The time dependent viscosity varies with the type of damping function, and it influences the time-dependent swell.     

Overall numerical simulation of extrusion blow molding process

This paper focuses on the overall numerical simulation of the parison formation and inflation process of extrusion blow molding. The competing effects due to swell and drawdown in the parison formation process were analyzed by a Lagrangian Eulerian (LE) finite element method (FEM) using an automatic remeshing technique. The parison extruded through an annular die was modeled as an axisymmetric unsteady nonisothermal flow with free surfaces and its viscoelastic properties were described by a K‐BKZ integral constitutive equation. An unsteady die‐swell simulation was performed to predict the time course of the extrudate parison shape under the influence of gravity and the parison controller. In addition, an unsteady large deformation analysis of the parison inflation process was also carried out using a three‐dimensional membrane FEM for viscoelastic material. The inflation sequence for the parison molded into a complex‐shaped mold cavity was analyzed. The numerical results were verified using experimental data from each of the sub‐processes. The greatest advantage of the overall simulation is that the variation in the parison dimension caused by the swell and drawdown effect can be incorporated into the inflation analysis, and consequently, the accuracy of the numerical prediction can be enhanced. The overall simulation technique provides a rational means to assist the mold design and the determination of the optimal process conditions.     

Visco‐hyperelastic constitutive model for modeling the quasi‐static behavior of polyurethane foam in large deformation

Flexible polyurethane foam is widely used in numerous applications such as seats and mattresses, due to its low stiffness and its ability to absorb deformation energy. The main objective of this article is to model the quasi‐static mechanical behavior of three types of polyurethane foam in large deformation and to compare these three foams with three proposed models. The uniaxial compression/decompression tests at three different strain rates were performed. The test results show that the three foams present different plateau stresses, maximum stresses, and abilities to absorb energy. Moreover, polyurethane foam also presents a nonlinear hyperelastic behavior and a viscoelastic behavior in large deformation. Three visco‐hyperelastic models which include a hyperelastic component and a memory component are proposed to model these behaviors. Model parameters were identified using the experimental data and a proper identification method. These models were validated on these three types of foam with the aim to present comparison results. The comparison results show that Ogden's viscoelastic model best agrees with the experimental results. POLYM. ENG. SCI., 55:1795–1804, 2015. © 2014 Society of Plastics Engineers     

L形异型材共挤胀大及变形的三维非等温数值模拟

采用PTT本构方程,应用Arrhenius方程来描述温度对黏度的影响,建立了L形双层共挤模型,通过有限元方法分析了聚丙烯（PP）和聚苯乙烯（PS）熔体的三维非等温黏弹流动过程,对比分析了2种材料在不同组合情况下口模出口面的速度场、剪切速率场以及共挤出胀大和变形情况。结果表明,L形分层共挤的胀大和变形不仅与2种熔体黏度差异有关,还与口模截面形状有关;黏度较低的PP会向黏度较高的PS一侧偏转,且PP有包覆PS的趋势;黏度较低的PP位于L形内侧时共挤出胀大和变形程度大于其位于外侧时,且两熔体黏度差异越大,两方案的共挤出胀大和变形程度的差异越大。     

Effect of characteristic scale on the extrudate swelling behavior of polypropylene melt in a micro-extrusion process

The swelling behavior in micro-extrusion has a significant effect on the dimensional and shape accuracy of microproducts. In this study, the effect of characteristic scale, defined as the gap of die land in an annular micro-extrusion die, on the extrudate swelling behavior of viscoelastic melt is analyzed through numerical simulations and micro-extrusion experiments. The results show that the swelling behavior displays an obvious dependence on the characteristic scale. An increase in the characteristic scale reduces the swell ratio and retards the process to reach the equilibrium state. In contrast, a decrease in the characteristic scale results in a larger magnitude of change in velocity field and faster relaxation development of stress field. The location of the maximum velocity layer for the laminar flow gradually deviates from the geometric center of channel toward the wall of mandrel with the increase in the characteristic scale. Moreover, an increase in the flow rate results in a larger swell ratio for a constant characteristic scale. The elastic effect plays a more dominant role than the viscous effect in determining the viscoelastic swell behavior. It is imperative to consider the complicated swelling behavior and remarkable viscoelastic effect simultaneously in micro-extrusion process.     

异形医用双腔导管挤出离模膨胀变形机理与调控

如何准确预测和精密调控异形医用双腔导管的离模膨胀变形是实现其精密成型的关键技术。通过传统和气辅精密成型的对比分析，研究表明离模膨胀变形是由熔体的第二法向应力差驱动的径向二次流动所诱发，传统挤出成型会产生较大的第二法向应力差，第二法向应力差驱动诱发的径向二次流动是离模膨胀变形的直接驱动力，从而导致传统挤出成型的异形医用双腔导管不仅产生离模膨胀，而且还产生椭圆度误差，其最大离模膨胀比和椭圆度误差分别为1.86和6.3 %。异形医用双腔导管的气辅精密挤出成型基本可以消除熔体的第二法向应力差，必然消除了挤出成型过程的径向二次流动，从而实现了异形医用双腔导管的精密控形。     

聚合物多层共挤成型离模膨胀过程的三维黏弹性数值模拟

共挤成型中,聚合物黏弹特性与过程参数波动的耦合作用会产生波动的离模膨胀,使得根据共挤制品的形状设计相应的共挤定型口模在工程上仍是一项技术挑战。基于这一技术问题,通过建立的稳态有限元数值算法,系统研究了过程参数和黏弹性流变性能参数对共挤成型离模膨胀的影响规律和机理。研究结果表明,多层共挤口模芯壳层熔体离模膨胀是由熔体的二次流动引起,主要取决于芯壳层熔体二次流动的方向与强度。熔体二次流动的方向与第二法向应力差的正负号有关,而熔体二次流动的强度则与第二法向应力差大小呈正比。芯层熔体的离模膨胀与口模出口和混合区进口处芯层熔体向外的二次流动强度呈正比,而壳层熔体的离模膨胀取决于壳层熔体内外界面向外的二次流动的相对强度。研究还表明芯、壳层熔体及口模整体的离模膨胀随着壳层熔体黏度的增大而增加,而随着壳层熔体进口流量的增大而减小。     

High weissenberg number simulation of an annular extrudate swell using the differential type constitutive equation

Annular extrudate swell simulations at high Weissenberg numbers were made using a differential type constitutive equation. The streamline-upwinding method with a sub-element for extra stress components, which is called SU4 × 4, is one of the best mixed finite element methods for computation of viscoelastic flows. Planar and capillary extrudate swell calculations at high Weissenberg numbers (We > 1000) were accomplished by SU4 × 4. However, annular extrudate swell simulations at high We by SU4 × 4 were not successful. The calculated We was less than about 4. A new calculation technique using a Newton-Raphson discretization of the equation of motion was developed. This technique is called a “new under-relaxation method.” The calculated We of annular extrudate swell simulation by the new under-relaxation method with SU4 × 4 was about 6～250 times larger than those by SU4 × 4. Reasonable calculation results were obtained in an annular flow and a capillary extrudate swell by this method, and the reliability and the utility of the new under-relaxation method are shown. It is now possible to consider the swell shapes of annular extrudate under industrially useful conditions. The calculated swelling ratios were also compared with experimental ones.     

A macromolecular deformation model to estimate viscoelastic flow effects in polymer melts

The elastic deformation of polymer macromolecules in a shear field is used as the basis for quantitative predictions of viscoelastic flow effects in a polymer melt. Non-Newtonian viscosity, capillary end correction factor, maximum die swell, and die swell profile of a polymer melt are predicted by the model. All these effects can be reduced to generic master curves, which are independent of polymer type. Macromolecular deformation also influences the brittle failure strength of a processed polymer glass. The model gives simple and accurate estimates of practically important processing effects, and uses fitting parameters with the clear physical identity of viscoelastic constants, which follow well established trends with respect to changes in polymer composition or processing conditions.     

热空气老化下黏弹性阻尼硅橡胶寿命预测模型

以压缩永久变形为寿命分析研究对象,研究了常用工况下老化时间对黏弹性阻尼硅橡胶的储能模量、损耗模量及损耗因子的影响,同时采用热空气加速老化试验方法探究了黏弹性阻尼硅橡胶在不同温度（348,363,378,393 K）下压缩永久变形性能保持系数随老化时间的变化规律,并通过对两种老化动力学模型进行研究分析和修正获得了黏弹性阻尼硅橡胶的老化反应速率,此外,还对不同老化动力学模型下老化反应速率对Arrhenius模型的非线性行为进行了分段研究,推测了黏弹性阻尼硅橡胶的储存寿命。结果表明,在室温（298 K）下两种老化动力学模型在储存寿命为10.0 a时的压缩永久变形性能保持系数均为0.60;当选择压缩永久变形性能保持系数为0.50作为失效判据时,黏弹性阻尼硅橡胶的储存寿命分别为20.0 a和19.2 a,使用两种模型所得黏弹性阻尼硅橡胶的储存寿命误差仅为4%。     

Mechanical Behavior of Butyl Adhesives

Butyl adhesives are widely used to bond elastomeric membranes. The results of uniaxial tension tests and simple shear tests conducted on bulk samples of butyl adhesives are presented in this paper. Butyl adhesives were found to have a rate-sensitive mechanical response with very low tensile and shear strengths. The stress-strain curves of the adhesive are characterized by an initial elastic response followed by a region of large plastic flow. A three-element viscoelastic model was used to model the stress-strain behavior of the adhesive. The model effectively combines a viscous dashpot and a network spring to capture the plastic flow in the material after the initial yield. The parameters of the model are calibrated to simulate a wide range of mechanical response of the adhesive.     

Assessment of hyperelastic material models for the application of adhesive point-fixings between glass and metal

For the investigation of adhesive point-fixings a computationally demanding finite element model is required. The accuracy of the numerical results depends highly on the validity of the used material models, which describe the deformation behaviour of the adhesive. The material models are derived from curve-fitting the mathematical expressions to experimental data mostly derived from uniaxial and equibiaxial experiments. In literature the suitability of the used material models is determined by comparing the numerical results from the same uniaxial and equibiaxial experiments to the experimental results. In contrast, in this contribution, the material models are validated by two additional validation experiments, i.e. an adhesive point-fixing loaded in uniaxial tension and an adhesive point-fixing loaded in a combination of tension and shear.After comparison of the numerical and experimental displacements, it appears that the material models that are calibrated by shear tests or by a combination of shear tests yield the best results. In addition, most numerical load-displacement curves have an almost linear gradient at small strains. Such behaviour is also demonstrated in the experimental measurements of the deformation.     

AC-13级配钢渣沥青混合料粘弹塑本构模型研究

为了研究钢渣沥青混合料非线性粘弹塑性变形特性,提出Schapery模型与改进Swchartz模型组合的积分型粘弹塑本构模型。采用钢渣替换AC-13级配中粒径2.36 mm以上的石灰石粗骨料,制作得到钢渣沥青混合料试件。设计并开展一系列的单轴压缩蠕变实验,通过应力递增蠕变回复实验,获得不同应力条件下材料的弹性、粘弹性应变和粘塑性应变,进而拟合确定本构模型参数。利用0.4 MPa、1.0 MPa下的蠕变回复实验验证模型有效性。结果表明,模型不仅能准确刻画钢渣沥青混合料蠕变过程中的弹性、粘弹性与粘塑性变形,还可用于预测不同应力水平下钢渣沥青混合料蠕变变形规律。     

数值模拟黏弹流体通过椭圆环模具的挤出胀大

The numerical simulation of extrudate swell is significant in extrusion processing.Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products.A mathematical model of three-dimensional(3D)viscoelastic flow through elliptical ring die for polymer extrusion was investigated.The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem.The discrete elastic viscous split stress(DEVSS)and streamline-upwind PetrovGalerkin(SUPG)technology were used to obtain stable simulation results.Free surface was updated by updating the streamlines which needs less memory space.According to numerical simulation results,the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight,but with the increase of volumetric flow rate and relax time the extrudate swell ratio increased markedly.Finally,the numerical simulation of extrudate swell flow for low-density polyethylene(LDPE)melts was investigated and the results agreed well with others’work.These conclusions provided quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape.     

Investigation of the effect of chamfer size on the behaviour of hybrid joints made by adhesive bonding and riveting

The paper reports the results of a numerical study performed on: (a) purely adhesive joints and (b) new hybrid single lap joints with a variable adherend thickness in the lap region. The variable thickness creates chamfer defined by a geometric parameter ch which has a very positive influence on the mechanical response of the joint. The novelty in this paper is the investigation of the effect of chamfer size on the behaviour of hybrid joints made by 2 simple techniques: adhesive bonding and riveting. In particular, 10 types of chamfer geometries are considered, each causing a different stiffness of the adherends being joined. As a result, the strength of the connection is increased and its weight reduced, which is of vital importance in aircraft constructions.The adherends and rivets are assumed to be made of aluminium, i.e., an elastic-plastic material, and subjected to gradual degradation due to tension. The adhesive layer is modelled as a semi-brittle material with progressive degradation using cohesive elements. Following the creation of 3D finite element models, the samples are subjected to quasi-static uniaxial deformation (nonlinear analysis with ABAQUS/Explicit).The numerical results lead to the conclusion that the variable geometry, i.e., chamfering, has a very positive effect. At the maximum chamfer length equal to 10 mm, the increase in the maximum force was about 32.8% compared to the model without chamfer.     

A viscoelastic–plastic constitutive model for uniaxial ratcheting behaviors of polycarbonate

In this paper, a modified viscoelastic–plastic constitutive model has been proposed on the framework of Anand's work to describe the uniaxial ratcheting behavior of polycarbonate (PC) under tension–tension cyclic loading. The experimental observation illustrates that the previously accumulated deformation has an assignable influence on the subsequent material response during the ratcheting process of PC. Thus, the deformation resistance in the viscoelastic micromechanism is assumed to be evolving with the local accumulated inelastic strain rather than keeping unchanged in the original Anand's model. The proposed model is validated firstly by the monotonic tension and creep experiment results of PC. Then, its capability to describe the uniaxial ratcheting behaviors is compared with Anand model. Finally, the modified model is adopted to study the effect of mean stress, stress amplitude, loading rate, and peak holding time on the ratcheting behaviors of PC. It is shown that the proposed model can predict reasonably the uniaxial tension–tension ratcheting behavior of polymer. POLYM. ENG. SCI., 55:2559–2565, 2015. © 2015 Society of Plastics Engineers     

Finite difference solution for flow from a slit die

A finite difference solution for an isothermal viscoelastic liquid flowing through a film forming die is investigated. The fluid is described by a Maxwell model in which the time derivative has been replaced by an Oldroyd's convective derivative and the numerical technique used combines features of the Solution Algorithm (SOLA), Simplified Marker and Cell method (SMAC), and SOLASMAC. The numerical scheme was tested with a Newtonian fluid and high density polyethylene (HDPE). Two slit dies with gap-to-length ratios 4 and 16 were used. In the two cases, die swell has occurred: A maximum of 4.5% swell was observed for the Newtonian fluid, while up to 77% swell was attained with HDPE. The simulated flow behavior of HDPE exhibited high amplitude oscillations at dimensionless time greater than 0.18. These oscillations are thought to be related to the nature of sheet flow, which can be unstable. This study shows that with some improvement the finite difference method can be used for studying the extrusion of polymer through slit and capillary dies.     

Thermodynamic approach of inflation process of K‐BKZ polymer sheet with respect to thermoforming

The problem of modeling and the dynamic finite element simulation of thermoforming process for viscoelastic sheet are considered. The pressure load used in modeling is thus deduced from the thermodynamic law of ideal gases. The viscoelastic behavior of the K‐BKZ model is considered. The Lagrangian formulation together with the assumption of the membrane theory is used in the finite element implementation. The numerical validation is performed by comparing the theoretical solution for the uniaxial and equibiaxial hencky deformation with numerical results. Moreover, the influence of the K‐BKZ constitutive model, for three linear time distribution of airflow rate loading, on the thickness and on the stress distribution in thermoforming of containers made of HDPE are analyzed. POLYM. ENG. SCI., 45:1319–1335, 2005. © 2005 Society of Plastics Engineers     

熔体挤出速度对共挤吹塑型坯离模膨胀影响的数值模拟

基于三维非等温黏弹性熔体多相分层流动有限元数值模拟技术,模拟研究了熔体挤出速度对多层共挤吹塑成型环坯离模膨胀和初始温度场的影响规律,揭示了型坯离模膨胀的产生机理。结果表明,多层共挤吹塑成型环坯离模膨胀是由熔体的二次流动诱发而产生,与熔体流出机头进入自由膨胀段的二次流动强度成正比,而其二次流动强度随着熔体挤出速度的增大而增强,因而导致环坯离模膨胀随着熔体挤出速度的增加而增大;多层共挤吹塑成型熔体的二次流动强度与其第二法向应力差成正比关联关系,这与Debbaut的试验研究结论完全吻合,表明二次流动是由第二法向应力差驱动而产生。     

The effect of extrudate swell on modeling the film blowing process

The extrudate swell effect has not received sufficient attention in modeling the film blowing process. This effect is addressed in this paper, and as an ab initio study, only viscous fluids were considered. The problem region was separated into two zones; the extrudate swell zone and the film blowing zone. The annular extrudate swell problem was solved using a finite element method. The film blowing process was modeled following Pearson and Petrie's (4) work. Although only viscous fluids were considered, the simulation results show a remarkable difference when swelling was included in the modeling. Viscoelastic fluids, which are more realistic for polymer melts, were not investigated here because of the so called high Weisenberg number problem. This is an open area still under investigation.