Thermistor analog study of dynamic shear in model viscoelastic materials

An electric transmission line analog for studies of the dynamic plane shear of an incompressible viscoelastic material which has a temperature dependent viscosity is described. Thermistors are used to simulate the temperature dependent viscosity. Criteria are established for the similarity of the line and the material. Experiments analogous to constant rate of deformation studies of an elastic material and a material with a single viscoelastic relaxation time are described. More detailed experiments analogous to the deformation of a viscous material at a constant rate of deformation and at constant stress are also described. These show phenomena analogous to necking and fracture.     

Rheotens-mastercurves and elongational viscosity of polymer melts

In a Rheotens experiment, the tensile force needed for elongation of an extruded filament is measured as a function of the draw ratio. For thermo-rheologically simple polymer melts, the existence of Rheotens-mastercurves was proved by Wagner, Schulze, and Göttfert (1995). Rheotens-mastercurves are invariant with respect to changes in melt temperature and changes in the average molar mass. By use of purely viscous models, we convert Rheotens-mastercurves of a branched and a linear polyethylene melt to elongational viscosity as a function of strain rate. The resulting elongational viscosity from constant force extension experiments is found to be in general agreement with what is expected as steady-state viscosity of polyethylene melts measured in either constant strain-rate or constant stress mode.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Material aspects of dynamic neck retardation

Neck retardation in stretching of ductile materials is promoted by strain hardening, strain-rate hardening and inertia. Retardation is usually beneficial because necking is often the precursor to ductile failure. The interaction of material behavior and inertia in necking retardation is complicated, in part, because necking is highly nonlinear but also because the mathematical character of the response changes in a fundamental way from rate-independent necking to rate-dependent necking, whether due to material constitutive behavior or to inertia. For rate-dependent behavior, neck development requires the introduction of an imperfection, and the rate of neck growth in the early stages is closely tied to the imperfection amplitude. When inertia is important, multiple necks form. In contrast, for rate-independent materials deformed quasi-statically, single necks are preferred and they can emerge in an imperfection-free specimen as a bifurcation at a critical strain. In this paper, the interaction of material properties and inertia in determining neck retardation is unraveled using a variety of analysis methods for thin sheets and plates undergoing plane strain extension. Dimensionless parameters are identified, as are the regimes in which they play an important role.     

Deformation and the strip necking zone in a cracked steel sheet

When a tensile stress is applied to a thin cracked plate, a strip necking region results ahead of a crack tip. The relative opening displacement between the crack surfaces and between the upper and lower boundaries of the strip necking region were measured by the moiré method. The strains ahead of the strip necking region and the thickness reduction (therein) were also measured. The measured relative opening displacements were compared with the calculated values using the Dugdale strip necking model. The thickness reduction in the strip necking region is equal to the relative opening displacement.     

Drop formation dynamics of constant low-viscosity,elastic fluids

The dynamics of drop formation under gravity has been investigated as a function of elasticity using a set of low-viscosity, ideal elastic fluids and an equivalent Newtonian glycerol–water solution. All solutions had the same shear viscosity, equilibrium surface tension, and density, but differed greatly in elasticity. The minimum drop radius in the early stages of drop formation (necking) was found to scale as expected from potential flow theory, independent of the elasticity of the solutions. Thus, during this stage of drop formation when viscous force is still weak, the dynamics are controlled by a balance between inertial and capillary forces, and there is no contribution of elastic stresses of the polymer. However, upon formation of the pinch regions, there is a large variation in the drop development to break-off observed between the various solutions. The elastic solutions formed secondary fluid threads either side of a secondary drop from the necked region of fluid between the upper and lower pinches, which were sustained for increasing amounts of time. The break-off lengths and times increase with increasing elasticity of the solutions. Evolution of the filament length is, however, identical in shape and form for all of the polymer solutions tested, regardless of differing elasticity. This de-coupling between filament growth rate and break-up time (or equivalently, final filament length at break-up) is rationalised. A modified force balance to that of Jones and Rees [48] is capable of correctly predicting the filament growth of these low-viscosity, elastic fluids in the absence of any elastic contributions due to polymer extension within the elongating filament. The elongation of the necked region of fluid (which becomes the filament) is dominated by the inertia of the drop, and is independent of the elasticity of the solution. However, elasticity does strongly influence the resistance of the pinch regions to break-off, with rapid necking resulting in extremely high rates of surface contraction on approach to the pinch point, initiating extension of the polymer chains within the pinch regions. This de-coupling phenomenon is peculiar to low-viscosity, elastic fluids as extension does not occur prior to the formation of the pinch points (i.e. just prior to break-up), as opposed to the high viscosity counterparts in which extension of polymers in solution may occur even during necking. Once steady-state extension of the polymers is achieved within the pinch at high extension rates, the thread undergoes elasto-capillary break-up as the capillarity again overcomes the viscoelastic forces. The final length at detachment and time-to-break-off (relative to the equivalent Newtonian fluid) is shown to be linearly proportional to the longest relaxation time of the fluid.     

Configurational mechanics of necking phenomena in engineering thermoplastics

Necking is a significant part of the yielding process in many thermoplastics. It starts as strain localization associated with microshear banding and/or cavitations and appears as a domain of oriented (drawn) material, i.e., a “neck”, separated from the domain of original (isotropic) material by a narrow transition zone, which appears as a distinct boundary of the neck region. On further increase of displacement, the neck propagates through the test specimen under constant draw stress. Strain localization such as crazing and shear bending is associated with necking on micro- and sub-microscales. As a result material toughness, i.e., resistance to cracking, as well as durability, i.e., service lifetime under various service conditions, are related to the material ability to necking and specific characteristics of necking process. Necking is manifested in significant changes in a characteristic length scale, e.g., the distance between equally spaced marks in the reference state may increases by factor of 2 in amorphous polymers and up to a factor of 10 in some semicrystalline thermoplastics. There is also a characteristic relaxation time change during the necking. Thus from continuum mechanics viewpoint, the changes of intrinsic material space-time metric are the most fundamental manifestation of necking. Therefore we model necking phenomena as space-time scales transformation and introduce a four-dimensional (4D) Riemannian metric tensor of a material space-time imbedded into 4D Newtonian (laboratory) space-time with a Euclidean metric. Kinetic equation of necking, i.e., evolution equation for material metric tensor is derived using extremal action principle. An example of traveling wave solution for neck propagation in a tensile bar is presented. Analysis of the solution and comparison with experimental observations are discussed.     

Crack tip field in a linear elastic–plastic strain-hardening material

The strip necking model for strain-hardening materials is studied in this paper, in which the stress distributed over the strip necking zone is assumed to be ultimate stress. The bi-linear stress–strain relation which can model certain features of plastic flow is adopted in this model. The stress and strain fields are calculated based on this model in this paper. The size of the strip necking region is determined by balancing the stress intensity factor due to remote loading with that due to assumed closing forces equal to the ultimate tensile strength of the material distributed over the strip necking zone. It is interesting that the strip necking region size and the crack tip opening displacement depend not only on the remote load, but also the material hardening parameters, which is different from the results of strip yield model. The results agree with experiments well, and the model has wider application.     

Effects of spatial grain orientation distribution and initial surface topography on sheet metal necking

The finite element method is used to numerically simulate localized necking in AA6111-T4 under stretching. The measured EBSD data (grain orientations and their spatial distributions) are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. We assume that localized necking is associated with surface instability, the onset of unstable growth in surface roughening. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and the artificial initial imperfection necessitated by the macroscopic M–K approach [Marciniak and Kuczynski (1967). Int. J. Mech. Sci. 9, 609–620] is not relevant in the present analysis. The effects of spatial orientation distribution, material strain rate sensitivity, texture evolution, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution. It is also demonstrated that the initial surface topography has only a small influence on necking.     

Numerical modelling of step–strain for stretched filaments

This article addresses the modelling of filament-stretching/step–strain deformation under viscoelastic capillary break-up configurations of the CaBER-type. Start-up, prior to step–strain, is conducted under constant stretch-rate synchronous plate retraction with impulsive sessation of plate motion. The study encompasses variation in material rheology, appealing to Oldroyd, Geisekus and Phan-Thien/Tanner-type models, which display differences in shear and extensional viscosity properties (shear thinning/extension hardening). Two different viscosity ratio settings are considered to reflect high- and low-solvent viscosity constituent components; the former representing typical Boger fluids, the latter high-polymer concentration fluids. We compare and contrast results for three alternative filament aspect ratios at the onset of step–strain. Throughout the step–strain period, we have been able to successfully capture such physical features as drainage to the filament feet, necking at the filament centre, and periods with travelling waves through the axial filament length. In addition, we have identified the suppressive influence that larger capillary forces have upon radial fluctuations, and the minor impact that gravitational forces have upon the ensuing deformation. From this study, estimates for rheometrical data have been derived in terms of characteristic material relaxation time and apparent extensional viscosity. The computational techniques employed include a compressed-mesh (CM) procedure, an Arbitrary Lagrangian–Eulerian scheme (ALE) and a free-surface particle tracking technique. Spatial discretisation of the problem is accomplished through a hybrid finite element/finite volume algorithm implemented in the form of a time-stepping incremental pressure-correction formulation.     

Polypropylene-polyethylene blends

Summary Die swell behaviour and morphology of melt blends of isotactic polypropylene (PP) and high density polyethylene for pure polymers and blends with 25, 50 and 75 weight % PP are described in the present study. A light interference contrast microscopy technique was used for the morphological characterization of melt blends and extrudate samples of the blends obtained with an Instron capillary rheometer. The results indicate that the domains from blends where the dispersed phase has higher viscosity than the continuous phase remain as continuous domains in the extrudate whereas domain destruction takes place when blends where the continuous phase has the higher viscosity are extruded.The die swell behaviour as well as the fiber forming properties of extrudates of melts having unstable domains extruded at high shear stresses resemble the behaviour of homopolymers, whereas samples with stable domains are significantly different, die swell increases with temperature at constant shear stress and stable fibers cannot be obtained after necking.With 10 figures and 1 table     

低碳钢单轴拉伸实验颈缩段的应力分析

为了深入研究塑性材料在单轴拉伸过程中的颈缩应力分布,结合Aramis三维应变测量系统对Q235钢和Q345钢进行了单轴拉伸实验。基于已有文献的颈缩外形理论,结合实验数据提出了颈缩阶段几何尺寸的变化规律公式,并与传统经验公式进行了对比。采用本文给出的颈缩阶段几何尺寸的变化规律公式计算,Q235钢误差率为27.73%,Q345钢误差率为20.49%;采用传统经验公式计算,Q235钢误差率为64.33%,Q345钢误差率为70.78%。结果表明,本文提出的变化规律公式精确度远高于传统公式精确度。基于此,在考虑材料系数的基础上推导出了包含材料系数的半解析半经验应力分布方程。     

Influence of molecular structure on the extensional behaviour of polyethylene melts

The elongational behaviour of polyethylene samples having different molecular structure has been tested. Elongational viscosity measurements have been carried out using the isothermal melt spinning technique. The extensional behaviour of the different samples is analysed as a function of total strain. The effect of long-chain branching on elongational viscosities is described. A comparison is presented between elongational viscosity and melt strength data.Some of the results reported here were presented at the VIIIth International Congress on Rheology, Naples, September 1–5, 1980, cf. [16].     

Extensional rheology of concentrated poly(ethylene oxide) solutions

The shear and extensional rheology of three concentrated poly(ethylene oxide) solutions is examined. Shear theology including steady shear viscosity, normal stress difference and linear viscoelastic material functions all collapse onto master curves independent of concentration and temperature. Extensional flow experiments are performed in fiber spinning and opposed nozzles geometries. The concentration dependence of extensional behavior measured using both techniques is presented. The zero-shear viscosity and apparent extensional viscosities measured with both extensional rheometers exhibit a power law dependence with polymer concentration. Strain hardening in the fiber spinning device is found to be of similar magnitude for all test fluids, irrespective of strain rate. The opposed nozzle device measures an apparent extensional viscosity which is one order of magnitude smaller than the value determined with the fiber spinline device. This could be attributed to errors caused by shear, dynamic pressure, and the relatively small strains developed in the opposed nozzle device. This instrument cannot measure local kinematics or stresses, but averages these values over the non-homogenous flow field. These results show that it is not possible to measure the extensional viscosity of non-Newtonian and shear thinning fluids with this device. Fiber spin-line experiments are coupled with a momentum balance and constitutive model to predict stress growth and diameter profiles. A one-mode Giesekus model accurately captures the plateau values of steady and dynamic shear properties, but fails to capture the gradual shear thinning of viscosity. Giesekus model parameters determined from shear rheology are not capable of quantitatively predicting fiber spinline kinematics. However, model parameters fit to a single spinline experiment accurately predict stress growth behavior for different applied spinline tensions.     

On neck propagation in amorphous glassy polymers under plane strain tension

Unlike metals, necking in polymers under tension does not lead to further localization of deformation, but to propagation of the neck along the specimen. Finite element analysis is used to numerically study necking and neck propagation in amorphous glassy polymers under plane strain tension during large strain plastic flow. The constitutive model used in the analyses features strain-rate, pressure, and temperature dependent yield, softening immediately after yield and subsequent orientational hardening with further plastic deformation. The latter is associated with distortion of the underlying molecular network structure of the material, and is modelled here by adopting a recently proposed network theory developed for rubber elasticity. Previous studies of necking instabilities have almost invariably employed idealized prismatic specimens; here, we explicitly account for the unavoidable grip sections of test specimens. The effects of initial imperfections, strain softening, orientation hardening, strain-rate as well as of specimen geometry and boundary conditions are discussed. The physical mechanisms for necking and neck propagation, in terms of our constitutive model, are discussed on the basis of a detailed parameter study.     

Crystal plasticity based analysis of localized necking in aluminum tube under internal pressure

The finite element method is used to numerically simulate localized necking in aluminum alloy tube under internal pressure. The measured electron backscatter diffraction (EBSD) data are directly incorporated into the finite element model and the constitutive response at an integration point is described by the single crystal plasticity theory. The tube is assumed sufficiently long, so that length changes as well as the end effects can be ignored and a plane strain analysis can be performed. Localized necking is assumed to be associated with surface instability, the onset of unstable thinning. It is demonstrated that such a surface instability/necking is the natural outcome of the present approach, and an artificial initial imperfection required by other approaches is not necessary in the present analysis. The effects of spatial grain orientation distribution, material strain rate sensitivity, work hardening, and initial surface topography on necking are discussed. It is found that localized necking depends strongly on both the initial texture and its spatial orientation distribution, while the initial surface topography has a negligible effect on necking.     

Rheological behaviour of nano-composites based on polyamide 6 under shear and elongational flow at high strain rates

In this work, the rheological behaviour of high molecular mass polyamide 6 (PA6)/organo-montmorillonite nano-composites, obtained via melt blending, was investigated under shear and extensional flow. Capillary rheometry was used for the measurement of high shear rate steady state shear viscosity and die entrance pressure losses; further, by the application of a converging flow method (Cogswell model) to these experimental results, elongational viscosity data were indirectly calculated. The extensional behaviour was directly investigated by means of melt spinning experiments, and data of apparent elongational viscosity were determined. The results evidenced that the presence of the organo-clay in filled PA6 melts modifies the rheological behaviour of the material, with respect to the unfilled polymer, in dependence on the type of flow experienced by the fluid. In shear flow, the nano-composites showed a slightly lower viscosity than neat PA6, whereas in elongation, they appeared much more viscous, in dependence on the organo-clay content.     

Strain Localization in Mild (Low Carbon) Steel Observed by Acoustic Emission - ESPI Coupling during Tensile Test

Two strain localization modes: the Piobert-Lüders band propagation and the development of necking, were investigated in uniaxial tensile tests for a low alloyed and low carbon steel. These two macroscopic localization phenomena were simultaneously monitored by speckle interferometry (ESPI) and acoustic emission (AE). The coupling of these two experimental techniques gives complementary information about strain localization features and mechanisms. For Lüders bands, it was found that the acoustic activity heard during the travel of the Piobert-Lüders band varies in closely correlated to the tensile force fluctuations, the relations between strain rate, band velocity, band width and plastic strain were investigated. Although the strain rate in the wake of the wave front is not always zero, the acoustic activity remains concentrated in the wave front itself. For necking, the acoustic activity is found to decrease regularly through the homogeneous plasticity stage and the diffuse necking stage and then increases when the localized necking starts, while ESPI patterns show a gradual strain concentration.     

Analysis of necking based on a one-dimensional model

Dimensional reduction is applied to derive a one-dimensional energy functional governing tensile necking localization in a family of initially uniform prismatic solids, including as particular cases rectilinear blocks in plane strain and cylindrical bars undergoing axisymmetric deformations. The energy functional depends on both the axial stretch and its gradient. The coefficient of the gradient term is derived in an exact and general form. The one-dimensional model is used to analyze necking localization for nonlinear elastic materials that experience a maximum load under tensile loading, and for a class of nonlinear materials that mimic elastic-plastic materials by displaying a linear incremental response when stretch switches from increasing to decreasing. Bifurcation predictions for the onset of necking from the simplified theory compared with exact results suggest the approach is highly accurate at least when the departures from uniformity are not too large. Post-bifurcation behavior is analyzed to the point where the neck is fully developed and localized to a region on the order of the thickness of the block or bar. Applications to the nonlinear elastic and elastic-plastic materials reveal the highly unstable nature of necking for the former and the stable behavior for the latter, except for geometries where the length of the block or bar is very large compared to its thickness. A formula for the effective stress reduction at the center of a neck is established based on the one-dimensional model, which is similar to that suggested by Bridgman (1952).     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.