Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method

Mould filling processes, in which a material flow front advances through a mould, are typical examples of moving boundary problems. The moving boundary is accompanied by a moving contact line at the mould walls causing, from a macroscopic modelling viewpoint, a stress singularity. In order to be able to simulate such processes, the moving boundary and moving contact line problem must be overcome. A numerical model for both two- and three-dimensional mould filling simulations has been developed. It employs a pseudo-concentration method in order to avoid elaborate three-dimensional remeshing, and has been implemented in a finite element program. The moving contact line problem has been overcome by employing a Robin boundary condition at the mould walls, which can be turned into a Dirichlet (no-slip) or a Neumann (free-slip) boundary condition depending on the local pseudo-concentration. Simulation results for two-dimensional test cases demonstrate the model's ability to deal with flow phenomena such as fountain flow and flow in bifurcations. The method is by no means limited to two-dimensional flows, as is shown by a pilot simulation for a simple three-dimensional mould. The reverse problem of mould filling is the displacement of a viscous fluid in a tube by a less viscous fluid, which has had considerable attention since the 1960's. Simulation results for this problem are in good agreement with results from the literature. © 1998 John Wiley & Sons, Ltd.     

Stability analysis of radial inflation of incompressible composite rubber tubes

The inflation mechanism is examined for a composite cylindrical tube composed of two incompressible rubber materials, and the inner surface of the tube is subjected to a suddenly applied radial pressure. The mathematical model of the problem is formulated, and the corresponding governing equation is reduced to a second-order ordinary differential equation by means of the incompressible condition of the material, the boundary conditions, and the continuity conditions of the radial displacement and the radial stress of the cylindrical tube. Moreover, the first integral of the equation is obtained. The qualitative analyses of static inflation and dynamic inflation of the tube are presented. Particularly, the effects of material parameters, structure parameters, and the radial pressure on radial inflation and nonlinearly periodic oscillation of the tube are discussed by combining numerical examples.     

Some exact solutions of the thin-sheet stamping problem

Stamping, which reduces to the flattening of a viscous blank or preform between two rigid surfaces, is one of the basic methods of obtaining thin-walled metal and especially plastic parts. In a number of cases during stamping the material is able to spread freely in the gap between mould and dye (free-edge stamping) and the preform is considerably deformed. Below, the problem of the evolution of the shape of the preform is considered and it is shown that given reasonable simplifying assumptions the problem admits a complete set of integrals of motion which make it possible to obtain a series of nontrivial exact solutions. An approach to the stamping problem similar to that adopted here was proposed in [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 24–33, March–April, 1992.     

Biaxial deformation of dough using the bubble inflation technique. II. Numerical modelling

In Part I the bubble inflation test was used to measure the stress-strain relationship of dough. A large disagreement was found between the stress-strain curve based on experimental data and the curve derived from Bloksma's analytical model. In Part II, a numerical simulation of the bubble inflation test is performed using Finite Element Analysis, in order to obtain further information regarding the accuracy of the analytical predictions. A hyperelastic model is assumed for the dough, with a strain energy potential described by the compressible form of the Mooney-Rivlin model. Four cases were investigated, corresponding to various combinations of material parameters of the Mooney-Rivlin model. The numerical results reinforce the conclusions drawn in Part I of the study, specifically that Bloksma's analysis of the bubble inflation could lead to large errors in the stress-strain curve. It was further concluded that the accuracy of the analysis was dependent on the material properties. For a neo-Hookeian material, the analysis leads to accurate results. This is because, for this material, all the assumptions made in the analysis regarding the bubble shape, the material's incompressibility and the bubble wall thickness distribution are accurate.     

A numerical approach to model non-isothermal viscous flow through fibrous media with free surfaces

A numerical simulation is presented to predict the free surface and its interactions with heat transfer and cure for flow of a shear-thinning resin through the fibre preform the flow part of the simulation is based on the finite element/control volume method. Since the traditional control volume approach produces an error associated with a mass balance inconsistency, a new method which overcomes this issue is proposed, the element control volume method. The heat transfer and cure analysis in the simulation are based on the finite difference/control volume method. Since heat conduction is dominant in the through-thickness direction and most of the heat convection is in-plane, heat transfer and cure are solved in fully three-dimensional form. A simple concept of the boundary condition constant is introduced which models a realistic mould configuration with a heating element located at a distance behind the mould wall. The varying viscosity throughout the mould associated with the strain rate, temperature and degree of cure distribution may be accounted for in calculating the mould-filling pattern. This introduces a two-way coupling between momentum and energy transport in fibrous media during mould filling.     

Finite deformation of elastic membranes with application to the stability of an inflated and extended tube

A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.     

热超弹性圆筒的不稳定性

应用有限变形弹性理论分析了受内压和轴向拉伸作用的不可压热超弹性圆筒发生非均 匀变形的不稳定性问题. 受内压和轴向拉力作用的薄壁圆筒,当内压较小时,圆筒发生稳定 的均匀膨胀变形;当内压大于某一临界值时,圆筒产生复杂的非均匀变形,其一部分膨胀变 形很大,形如``灯泡'状,而另一部分仅仅是轻微膨胀,且此时的变形是不稳定的. 但对厚 壁圆筒而言,不论压力如何,总是发生稳定的均匀膨胀变形. 根据圆筒的变形曲线,给出了 圆筒可以发生不稳定变形的临界厚度. 同时,讨论了轴向拉伸和温度场对圆筒变形的影响.     

Non-steady plane-strain ideal plastic flow

Ever since the ideal forming theory has been developed for process design purposes, application has been limited to sheet forming and, for bulk forming, to two-dimensional steady flow. Here, application for the non-steady case was performed under the plane-strain condition based on the theory previously developed. In the ideal flow, material elements deform following the minimum plastic work path (or mostly proportional true strain path) so that the ideal plane-strain flow can be effectively described using the two-dimensional orthogonal convective coordinate system. Besides kinematics, for a prescribed final part shape, schemes to optimize a preform shape out of a class of initial configurations and also to define the evolution of shapes and boundary tractions were developed. Discussions include the two problematic issues on internal tractions and the non-monotonous straining. For demonstration purposes, numerical calculations were made for a bulk part under forging.     

On the response and stability of an inflated toroidal membrane under radial loading

Two flat annular hyperelastic membranes, stacked and bonded together at both the boundaries (equators), form a closed inflatable structure of toroidal topology. The response and stability of the inflated toroidal structure subjected to a radial line force distribution at the inner boundary are studied. The forcing is considered under constant pressure and constant amount of gas inflation conditions. Two hyperelastic models described by the corresponding relaxed strain energy density functions are considered for the membrane material. The influence of geometry, material and level of inflation on the response and stability of the structure under load has been brought out. The structure exhibits pressure limit points with increasing levels of inflation. The force–deflection (stiffness) behaviour is found to be qualitatively different below and above the pressure limit points. Below the pressure limit point, wrinkling and pull-in under loading are revealed for different inflation conditions, and the stability boundaries are determined. Under certain conditions, a counter-intuitive stretch-softening behaviour is also observed.     

Nonlinear Electroelastic Deformations

Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.     

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为了研究爆炸及燃烧产生的“蘑菇状”烟云,采用中尺度气象模式RAMS(Regional Atmospheric Modeling System)模拟计算了冲击波过后的爆炸烟云运动过程。对于稳定层结大气条件,模拟结果显示爆炸烟云造成的流动是一个多层次涡-波运动的过程。爆炸烟云在浮力作用下会形成浮力涡环结构,在到达最大高度以后,其烟云外侧边缘部分涡度变号,形成了环套环的多涡结构。随时间增加,涡能量的逐步衰减,涡运动转变为重力内波运动。通过模拟不同爆炸当量和不同稳定度烟云的上升最大高度,得到了与目前常用的爆炸烟云上升公式一致的形式。     

Inflation of spherical rubber balloons

When a spherical rubber balloon of the sort used in meteorological applications is inflated, the onset of aspherical deformation is observed after the pressure maximum has been attained. Upon further inflation the balloon regains its spherical shape. Here, the rubber balloon is idealized as an elastic membrane and inflation is taken to be accomplished by a prescribed increase in enclosed volume. The axisymmetric equilibrium states of slightly imperfect membranes are determined numerically by means of the Ritz-Galerkin method. Several particular material models representative of the behavior of rubberlike solids are employed in order to illustrate a number of feautres associated with the aspherical deformation.     

The behavior of elastic tubes in a cohesion-free continuum

A model is made of a plain cross section of a tube (tunnel lining) in a cohesion-free continuum (rolling material, sand) which is represented by steel rollers of different diameters. The compactibility of sand in this model is represented by rubber inserts around the tunnel lining. The stress of the tube is measured by photoelasticity. These experiments are the start of a large program of investigation to calculate the stress deformation and buckling of elastic tubes under different loading conditions in rolling material.     

Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

On deformations with constant modified stretches

Two plane, non-homogeneous, non-conformal deformations with constant modified stretches (the inflation of a circular cylindrical tube and the bending of a rectangular block into a sector of a circular tube) are shown to be possible in the absence of body forces in isotropic elastic solids that satisfy both the classical pressure-compression inequality and the Baker-Ericksen inequality.     

轴向拉伸作用下超弹性管充气失稳的内压和形貌研究

超弹性管充气局部失稳现象与生物工程中动脉瘤形成机理相似。本文以两端密封的超弹性管为研究对象,从实验和理论两方面研究其在轴向拉伸作用下充气时的失稳现象。实验采用高频压力传感器测量充气过程中超弹性管内部压力的变化,高速摄像系统同步记录形貌变化的图像,分析超弹性管的几何尺寸和轴向拉力的改变对失稳临界压力的影响。基于连续介质力学理论,建立超弹性管充气时的控制方程并运用打靶法求解,理论分析轴向拉力与几何尺寸对超弹性管充气失稳的影响,并与实验结果比较;同时,预测超弹性管在任意时刻的形貌变化,分析特征点的变化趋势。     

An explicit method to determine response coefficients in finite elasticity

Some classical solutions to problems involving controllable deformations of isotropic incompressible elastic solids are recapitulated. It is shown that, when only one deformation mechanism is inhomogeneous, a parametric differentiation of the load-deformation relations yields a possibility to determine explicitly the material response coefficients from data conveniently measurable at tests. The method is illustrated for two cases; extension, inflation and torsion of a tube and inflation of a spherical shell.

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Résumé Quelques solutions classiques de problèmes en déformation pour des solides élastiques isotropes incompressibles sont rappelies. On montre que, quand un seul mécanisme est nonhomogène, une dérivation parametrique des relations force-déformation rend possible la détermination explicite des coefficients characteristiques du solide à partir de données expérimentales facilement accessibles.Deux examples illustrent la méthode: cas d'un tube soumis à une extension, une expansion et une torsion et cas d'une coque sphérique en expansion.

     

On the bursting of linear polymer melts in inflation processes

Molten LLDPE and HDPE plates (thickness 2 mm) have been inflated into a circular cylinder (inner radius 31 mm) under isothermal conditions. Low deformation rates allow the plates to be inflated considerably into the cylinder, and at high inflation rates an early burst is observed.Axis-symmetric numerical simulation of the inflations have been performed, using a constitutive equation in the form of a separable memory integral where the strain dependence is described by the Linear Molecular Stress Function (L-MSF) model with dissipative convective constraint release. The material parameters in the constitutive model are obtained using liner viscoelastic (oscillatory shear) and uni-axial elongational measurements.The numerical simulations were performed for inflation of a flat plate and a perturbed plate, where a small circular cone was removed from the centre of the surface of the plate. This was done in order to investigate the stability of the inflations. It is shown that all of the inflations are hydrodynamically unstable, though the effect on the occurrence of the burst is limited. One exception is at slow inflation, where an unexpected burst may appear as a consequence of minute deviations from an ideal flat plate. All of the numerical calculations show quantitative agreement with the experiments for a wide range of experimental conditions. This strongly suggests that the initiation of the burst is a hydrodynamic phenomenon.The critical parameters in the inflation of molten linear polymers have been investigated using the Gel equation as a memory function (M(s)=Ans ^–(1+n)) and inflating the plate with a constant velocity for the top of the plate. The hydrodynamic burst in a linear polymer is mainly associated with the linear viscoelastic properties and only slightly with the non-linear strain dependence. Increased (linear) elasticity reduces the inflated volume, at the same inflation velocity, before the burst occurs. Furthermore, the critical parameter for the occurrence of the burst (whether or not the burst occurs) is related to the crossover point (G=G) in linear viscoelasticity.     

New approach to the autofrettage of high-strength cylinders

The usual method of autofrettage (cold working) for gun tubes utilizes hydraulic pressure applied directly to the bore in order to plastically deform the wall of the tube so that favorable residual-stress patterns are produced. The strength of the tube is effectively increased, providing many associated benefits; however, ultra-high hydraulic pressures are required for high-strength steels since plastic-flow pressure is directly proportional to the yield strength of the material. A new method for the autofrettage of high-strength steel cylinders requiring greatly reduced pressures is developed and described herein. An oversize mandrel is forced through the tube to plastically deform the walls. Three methods of forcing the mandrel are investigated. Mechanical-push swaging is used in the autofrettage of short 5-in. long specimens with pull swaging and hydraulic-push swaging being used on specimens 40 in. long. All specimens are made from 4340 steel heat treated to various strengths. Cylinders with wall ratios ranging from 1.5 to 2.8, yield strengths ranging from 90,000 to 180,000 psi, and percent enlargements at the bore ranging from 1.0 to 5.0 are utilized. An engineering analysis is made investigating such factors as percent enlargement and elastic recovery at the bore, the ratio of pressure required for pushing the mandrel to the yield strength of the material, the effects of various lubricants on the frictional forces involved, and the induced three-dimensional stresses in the cylinder walls. Sach's boring-out technique is used to evaluate induced residual-stress patterns. Strains are recorded with electric-resistance strain gages and the associated dynamic and static instrumentation is described. Results are presented in graph form.