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.     

Molecular approach in linear polymer dynamics: Theory and numerical experiment

A constitutive equation for polymer solutions and melts is obtained on the basis of the dynamics of noninteracting dumbbells moving in a nonlinear anisotropic fluid. The equation obtained is used to describe nonlinear effects under conditions of simple shear and steady-state flow in a circular tube and for the numerical investigation of a flow in a finite cylinder with a rotating end face. Barnaul. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–13, January–February, 2000.     

Viscoplastic flow around a cylinder kept between parallel plates

Numerical simulations have been undertaken for the creeping pressure-driven flow of a Bingham plastic past a cylinder kept between parallel plates. Different gap/cylinder diameter ratios have been studied ranging from 2:1 to 50:1. The Bingham constitutive equation is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both the yielded and practically unyielded regions. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient for a wide range of Bingham numbers. The present results extend previous simulations for creeping flow of a cylinder in an infinite medium and provide calculations of the drag coefficient around a cylinder in the case of wall effects.     

Interchain tube pressure effect in extensional flows of oligomer diluted nearly monodisperse polystyrene melts

We have derived a constitutive equation to explain the extensional dynamics of oligomer-diluted monodisperse polymers, if the length of the diluent has at least two Kuhn steps. These polymer systems have a flow dynamics which distinguish from pure monodisperse melts and solutions thereof, if the solvent has less than two Kuhn steps, e.g. is not a chain. The constitutive equation is based on a phenomenological tube-based model within the methodology of the molecular stress function approach. The nonlinear dynamics have been explained as a consequence of a constant thermal interchain pressure originating from the short polymer chains (e.g. the oligomers) on the wall of the tube containing the long chains. The nonlinear dynamics are uniquely defined by the Rouse time and the maximal extensibility of the long polymer chains. Both are linked to the entanglement length. The relation between the Rouse times and entanglements have been established based on published extensional experiments on nearly monodisperse polystyrene melts. The constitutive equation has shown agreement with the experimental startup of and steady extension data from Huang et al. (Macromolecules 46:5026–5035, 2013a) based on 285 and 545 kg/mol polystyrenes diluted in styrene oligomers containing 3.3 (1.92 kg/mol) and 7.3 (4.29 kg/mol) Kuhn steps.     

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.     

Hybrid analysis of plate problems

For the problem of plates on yielding subgrades, with deflections larger than allowed according to the classical theory of thin plates, a geometric nonlinear theory has been introduced, which led to two simultaneous differential equations depending on one another. These constitutive equations cannot be solved without further information about the response of the subgrade. Therefore, and hybrid technique has been applied. By means of moiré and photoelastic methods, experimental data have been obtained. These are introduced into the constitutive equations, which then can be solved without any mathematical difficulties.     

Uniform flow of viscoelastic fluids past a confined falling cylinder

Uniform steady flow of viscoelastic fluids past a cylinder placed between two moving parallel plates is investigated numerically with a finite-volume method. This configuration is equivalent to the steady settling of a cylinder in a viscoelastic fluid, and here, a 50% blockage ratio is considered. Five constitutive models are employed (UCM, Oldroyd-B, FENE-CR, PTT and Giesekus) to assess the effect of rheological properties on the flow kinematics and wake patterns. Simulations were carried out under creeping flow conditions, using very fine meshes, especially in the wake of the cylinder where large normal stresses are observed at high Deborah numbers. Some of the results are compared with numerical data from the literature, mainly in terms of a drag coefficient, and significant discrepancies are found, especially for the constant-viscosity constitutive models. Accurate solutions could be obtained up to maximum Deborah numbers clearly in excess of those reported in the literature, especially with the PTT and FENE-CR models. The existence or not of a negative wake is identified for each set of model parameters.     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

高导无氧铜圆柱-平板冲击实验及不同本构模型效果比较

利用一级气炮对高导无氧铜(OFHC)进行了圆柱以205 m/s速度冲击平板实验,并进行了数值模拟。用锰铜应力计测试了靶中应力随时间的变化,并进行了回收观测。采用Johnson-Cook(J-C)、Zerilli-Armstrong(Z-A)、Steinberg-Cochran-Guinan(S-C-G)3种本构模型对实验进行了数值模拟。实验结果与数值模拟结果比较表明:就峰值应力而言,采用J-C、Z-A及S-C-G本构模型的计算结果都比较接近实验;就圆柱变形而言,Z-A及S-C-G模型的计算较J-C模型结果更符合实验。然而,速度为500 m/s冲击实验的数值模拟结果表明:3种本构模型的计算结果差异明显。     

填药柱壳在内部爆炸载荷下的变形和破坏

采用一种计及三轴因子的损伤模型,以本构关系的内变量理论为基础得到了热塑性本构关系的普适显武表达式;得到了改进的Johnson-Cook本构模型的增量形式; 考虑温度和损伤对材料参数的影响,计入温度和损伤对材料塑性变形发展的耦合作用,给出完备的计算方程组;用Lagrange显示差分的方法对填药柱壳在内部爆炸载荷下的变形和破坏进行了数值模拟,并对结果进行分析,与实验结果进行了比较.     

Decoupled second-order transient schemes for the flow of viscoelastic fluids without a viscous solvent contribution

The simulation of transient flows is relevant in several applications involving viscoelastic fluids. In the last decades, much effort has been spent on deriving time-marching schemes able to efficiently solve the governing equations at low computational cost. In this direction, decoupling schemes, where the global system is split into smaller subsystems, have been particularly successful. However, most of these techniques only work if inertia and/or a large Newtonian solvent contribution is included in the modeling. This is not the case for polymer melts or concentrated polymer solutions.In this work, we propose two second-order time-integration schemes for discretizing the momentum balance as well as the constitutive equation, based on a Gear and a Crank–Nicolson scheme. The solution of the momentum and continuity equations is decoupled from the constitutive one. The stress tensor term in the momentum balance is replaced by its space-continuous but time-discretized form of the constitutive equation through an Euler scheme implicit in the velocity. This adds velocity unknowns in the momentum equation thus an updating of the velocity field is possible even if inertia and solvent viscosity are not included in the model. To further reduce computational costs, the non-linear relaxation term in the constitutive equation is taken explicitly leading to a linear system of equations for each stress component.Four benchmark problems are considered to test the numerical schemes. The results show that a Crank–Nicolson based discretization for the momentum equation produces oscillations when combined with a Crank–Nicolson based scheme for the constitutive equation whereas, if a Gear based scheme is implemented for the constitutive equation, the stability is found to be dependent on the specific problem. However, the Gear based scheme applied to the momentum balance combined with both second-order methods used for the constitutive equation is stable and accurate and performs much better than a first-order Euler scheme. Finally, a numerical proof of the second-order convergence is also carried out.     

Experimental investigation on resistance characteristics in micro/mini cylinder group

Friction factors associated with forced flow of de-ionized water over staggered and in-line micro/mini cylinder group plates with 3.5 mm width and 40 mm length, which are made of micro/mini cylinders with hydraulic diameter of 0.5 mm and the heights of 1.0 mm, 0.75 mm, 0.5 mm and 0.25 mm, respectively, have been experimentally investigated over Reynolds number ranging from 25 to 800. The flux and the pressure drop between the inlet and the outlet of micro/mini cylinder group are measured and the experimental results are compared with those of convectional correlations. The investigation shows the value of fRe is approximately the constant in micro/mini cylinder group plates when the flow is purely laminar state. Except test sections with 0.25 mm cylinder height, the values of fRe for other test sections increase when Re > 100, as the results of the appearance of vortex resistance, the enhancement of stream pulse and the acceleration of stream frequency. For test section with 0.25 mm cylinder height, the values of fRe rapidly and oscillatingly increase at Re > 150 due to the influence of the scale effect, tip clearance effect and the roughness effect on the cylinder surface and bottom of micro/mini cylinder group plates. The friction factor in a staggered array is much larger than that at in-line array for micro/mini cylinder group plates and the higher the cylinder height is, the lower the friction factor becomes.     

Error Bounds for Approximate Solutions for Bending of Unsymmetrically Laminated Plates

Abstract

Certain types of heterogeneous plates exhibit coupling between membrane and flexural effects in their constitutive relations. Such a situation commonly occurs in unsymmetrically laminated plates and in reinforced concrete slabs after cracking. Approximate solutions, principally the “reduced bending stiffness” approximation, have been proposed in the past. The accuracy of this approximation has been examined for several specific cases, but no general investigations have been reported. This paper presents a method for determining bounds on the relative mean square error of approximate solutions to general coupled plate bending problems.     

An empirical constitutive equation of integral type for viscoelastic liquids

A 5-constant constitutive equation is proposed. The analytical form for the relaxation modulus as a function of flow conditions was chosen based on experimental data for stress-relaxation in solid polymers. The resulting formulae for the material functions in simple and oscillatory shear flow fulfil the empirical Cox-Merz rule as well as other phenomenological relations formulated by Coleman and Markowitz. The theoretical results are compared with experimental data obtained by Han for various polymer melts. Good agreement between theory and experiment is found.     

轴对称压电-压磁圆柱夹层结构的圣维南端部效应

研究了半无限长轴对称压电-压磁夹层结构的圆柱体圣维南端部效应的衰减问题。圆柱的端部承受自平衡磁电弹载荷;圆柱的内外表面为机械自由表面,但承受不同的电磁边界条件,即电学短路或电学开路及磁学短路或磁学开路边界条件。基于横贯各向同性压电或压磁材料在轴对称圆柱坐标系下的本构方程,推导了关于衰减率的特征方程并求得问题的数值解。结果表明,边界条件、内外径之比、材料厚度比对结构的衰减率都有显著的影响。     

Viscoelastic properties in plane squeeze-film flows

In this paper, we consider the problem of plane squeeze-film flows in a kinematics based on the formalism of convected (moving and deforming) coordinates. The flows discussed are treated as instantaneous motions with proportional stretch histories (cf. [16]). Certain simplifications in the constitutive equations of an incompressible simple fluid (cf. [10]) have been achieved for moderately low Deborah numbers.Approximate solutions of plane flows are obtained either for slightly viscoelastic fluids or in a form valid in the vicinity of any arbitrarily chosen instant of time. The conditions of improved lubrication, leading to inequalities imposed on material constants or kinematic quantities, are discussed in detail. Also, the necessary conditions are discussed under which the time-dependent distance between the plates may decrease non-monotonically, showing some “bounces”.     

On equations of the linear theory of shells with surface stresses taken into account

We construct equations of equilibrium and constitutive relations of linear theory of plates and shells with transverse shear strain taken into account, which are based on reducing the spatial elasticity relations with surface stresses taken into account to two-dimensional equations given on the shell median surface. We analyze the influence of surface elasticity moduli on the effective stiffness of plates and shells.     

The rational mechanics and thermodynamics of polymeric fluids based upon the concept of a variable relaxed state

The conceptual framework of polymer continuum mechanics based upon Eckart's idea of a variable relaxed state is developed. No constitutive models are explicitly used. The theory admits four constitutive functions only, the scalar specific internal energy, the vectorial heat flux, and two tensorial fluxes representing non-elastic stress and flow (slippage). The non-linearity of the constitutive relations includes self-induced anisotropy (Leonov) with Reiner-Rivlin's equation representing a special example for this. — The effectiveness of this non-linear theory is demonstrated by treating elongational flows of polymer melts.     

A theory for swaging of discs and lugs

A practical theory for swaging bored holes within plates and cylinders is proposed which can take into account work-hardening in the presence of small plastic strains based upon equivalent stress-strain data. With the appropriate choice of yield function, this theory applies to the swaging of both thin and thick plates under respective plane stress and plane strain conditions. The theory can be adapted further to the autofrettage of open and closed-ended, thick-walled cylinders where similar plane deformations conditions apply. Here swaging refers to the practice in which an oversized plug or sphere is forced into the bore thereby expanding it permanently to leave a residual circumferential compression in the bore material upon removal of the expanding tool. A similar effect results from applying an initial over-pressure to a long thick-walled cylinder in an autofrettage process. Both treatments are employed to enhance the fatigue resistance when the service loading upon the disc or cylinder amounts to a cyclic, circumferential tension within its bore. Strain gauges bonded to the entry face of the plate are used to monitor the circumferential and radial strain distributions both during and after the swaging process. Experimental results presented for swaging of thin and thin annular discs in aluminium alloy show that the measured residual strain distributions concord with the theory for large discs with a 10/1 diameter ratio. The agreement is less satisfactory with the loss in axial symmetry for parallel-sided lugs with a width to hole diameter ratio of 4/1.     

Determination of rheological constants of an elastico-viscous fluid by kinematical analysis of oscillatory motions

This paper contains experimental and theoretical work on a rotatory oscillatory flow of an elastico-viscous fluid. Assuming a linear constitutive equation, the motion of the fluid contained in a spherical or circular cylindrical vessel is computed. In particular, the limited height of the cylinder is taken into account for the boundary conditions. The solutions yield the velocities of the fluid particles as functions of the constitutive equation parameters, everywhere in the flow field.The experimental study is essentially based upon quantitative vizualisation techniques with solid particles: the amplitude and phase of the displacements of the tracers suspended in the fluid are obtained from an analysis of the photographs.Using the above-mentioned theoretical results we deduce the measurement of characteristic coefficients and functions of the materials tested. An experimental investigation of moderately concentrated aqueous solutions of polyethylene oxide is carried out.