Influence of long-chain branching on time-pressure and time-temperature shift factors for polystyrene and polyethylene

Rheological characterizations were carried out for two polystyrenes. One was a linear polymer with M _w =222,000 g/mol and M _w /M _n =2, while the other was a randomly branched polystyrene with M _w =678,000 g/mol and a broad molecular weight distribution. Experiments performed included oscillatory shear to determine the storage and loss moduli as functions of frequency and temperature, viscosity as a function of shear rate and pressure, and multi-angle light scattering to determine the radius of gyration as a function of molecular weight. The presence of branching in one sample was clearly revealed by the radius of gyration and the low-frequency portion of the complex viscosity curve. Data are also shown for three polyethylene copolymers, one (LLDPE) made using a Ziegler catalyst and two made using metallocene catalysts, one (BmPE) with and one (LmPE) without long-chain branching (LCB). While the distribution of comonomer is known to be much more uniform in LmPE than in LLDPE, the pressure shift factors were the same for these two polymers. The pressure and temperature shift factors of the two polystyrenes were identical, but, in the case of polyethylene, the presence of a small amount of LCB in the BmPE had a definite effect on the shift factors. These observations are discussed in terms of the relative roles of free volume and thermal activation in the effects of temperature and pressure.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Evaluation of calculated entanglement spacings for undiluted linear amorphous polymers

Conclusions In reality, entanglements based on rheological measurements have been essentially defined operationally as a kinetic or time delay effect present in concentrated polymer systems. This is where intuitively molecular loops or overlap and entanglements of polymer chains are most likely to occur. The evaluation in fig. 2 suggests that the theory of rubber elasticity, even discounting the inherent imprecision of measurements, gives anM _e which is not relatable to polymer composition 1and chain structure. Conversely reliable and internally consistent values ofM _c can come from the molecular weight dependence of: 1.Newtonian viscosities, 2.NMR relaxation times, and 3 the shear dependence of viscosities, see fig. 1. However, entanglement spacings calculated in this way lack adequate theoretical substantiation particularly as to the mechanism of flow. Nonetheless, the correlation ofM _c with composition and structure is convincing and provides a sound empirical correlation and the basis for a renaissance in theoretical interpretation of viscoelastic behavior of linear amorphous polymers. Further theory must rationalize the fact that the shear compliance, which is proportional toM _e , depends on a combination of molecular weight averages involvingZ andZ+1. Yet the break in viscosity-molecular weight curves yieldingM _c has been generally confirmed to depend on only weight average molecular weight. The values ofM _c appear worthy of further experimental and theoretical evaluation. The values ofM _e appear to be based on experimental techniques that at present do not provide sufficient precision to warrant citation of such values molekular weight distribution is thought to have an important yet ill-defined influence on calculated values of M _e .     

Squeezing flow of elastic liquids

The theoretical analysis is made of the relation between applied force and plate separation for squeezing flows of viscoelastic liquids between closely-spaced parallel disks. The lubrication approximation and the quasi-steady-state assumption are employed in the development. Elastic effects are incorporated through inclusion of normal stresses. Solutions are presented for liquids with power-law viscometric functions, and a numerical procedure is used for fluids having viscometric functions of arbitrary form. For fast and slow squeezing, calculated values of t_{$\text{1}\text{2}$}, the time required to squeeze out half the fluid, are found to agree with the constant force data of Leider [1,2].     

Fatigue crack growth through residual stress fields—theoretical and experimental studies on thick-walled cylinders

Experiments are described which determined the effects of various residual stress distributions on the growth rate of fatigue cracks. For each stress distribution, the contribution (K_RS) to the net stress intensity at the crack tip is determined, and a comparison is then made with the behaviour predicted using a fracture mechanics approach based on a weight function analysis and a simple crack closure model. The example studied is a thickwalled pressure vessel containing a longitudinal crack which grows radially from the inner surface; fatigue cracks were grown under laboratory conditions in ring test specimens. sectioned from vessels which had been cold-expanded by different amounts to increase their pressure limits, and so contained various complex residual stress distributions. The experiments provide direct evidence that the effects of residual stress (and by extension, thermal stress) on the crack tip stress intensity may be modelled conveniently using weight function techniques, and can be incorporated satisfactorily in fatigue crack growth analyses.     

Three-dimensional elastic behavior of a nonhomogeneous layer

Summary  This paper deals with the theoretical treatment of a three-dimensional elastic problem governed by a cylindrical coordinate system (r,θ,z) for a medium with nonhomogeneous material property. This property is defined by the relation G(z)=G ₀(1+z/a) ^m where G ₀,a and m are constants, i.e., shear modulus of elasticity G varies arbitrarily with the axial coordinate z by the power product form. We propose a fundamental equation system for such nonhomogeneous medium by using three kinds of displacement functions and, as an illustrative example, we apply them to an nonhomogeneous thick plate (layer) subjected to an arbitrarily distributed load (not necessarily axisymmetric) on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacement and stress components are shown graphically. Received 10 May 1999; accepted for publication 15 August 1999     

Concerning time and length scale estimates made from burst-mode LDA autocorrelation measurements

 An investigation was performed in grid-generated turbulence to highlight the difficulties in estimating integral and microscales from discrete autocorrelation measurements made using a burst-mode laser Doppler anemometer (LDA). Comparison was made to hot-wire results and the theoretical growth laws determined by the decay of turbulent energy. The chief concern was the presence of a spike in the measured autocorrelation coefficient functions at t=0. It was found that renormalizing the autocorrelation to a value at t=0 determined by “backfitting” a quadratic equation to the slots near t=0 led to microscale estimates that showed the same trend as the decay law and were in agreement with those determined using a hot-wire. Additionally, practical guidelines are presented for making accurate autocorrelation measurements by LDA. Received: 12 January 1995/Accepted: 9 June 1997     

Experimental investigation on the propagation of fatigue cracks emanating from sharp notches

The effect of notch geometry on the propagation of fatigue cracks emanating from sharp V-shaped notches is investigated by means of an experimental campaign performed on Al-7075-T651 specimens carrying notches with opening angles of 45°, 90°, and 135°. The samples were tested using a servohydraulic machine under different loading directions and at several loading levels. The crack deflection induced by the variation in loading direction was determined my measuring the kinking angle and by studying the crack propagation plane through fractographic analysis. A linear elastic fracture mechanics approach was adopted for the analysis of experimental results. Stress intensity factors were calculated using an appropriate weight function set up for studying inclined edge cracks emanating from sharp V-notches. The influence of K _II on the crack propagation was discussed on the basis of theoretical and semi empirical models.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

Normal stress difference in liquid lubricants sheared under high pressure

Although normal stress differences in liquids have conventionally been associated with polymers, aspects of rheological behavior in lubricated concentrated contacts suggest that normal stress difference may be significant in even low molecular weight liquids sheared under high pressure and high shear stress. A torsional flow rheogoniometer was constructed for use at high (300 MPa) pressure. Four typical liquid lubricants were investigated, including one polymer/mineral oil solution. Shear stress and N ₂-N ₂ are reported as functions of shear rate. The effect of pressure variation is reported for two liquids. Results are compared with predictive techniques and a molecular dynamics simulation. Simple low molecular weight lubricant base oils can generate measurable and significant normal stress differences when sheared at high shear stress.     

An improved experimental method for the determination of the critical stress intensity factor KIc of brittle materials

The critical stress intensity factor K_Ic is determined by a simple and accurate method, using small test specimens and a simple procedure in this paper.Single edge V-notched tension specimens made of PMMA are subjected to a load which is slowly increased until the crack begins to move from the notch tip. During the crack propagation event shadow patterns at the tip of the crack are recorded in a video recorder. Under these loading conditions, the creating real crack propagate slowly until the crack propagation velocity take an abrupt increase and the entire fracture of the specimen takes place. The stress intensity factor which correspond to the transition from the slow to fast crack speed, is the critical stress intensity factor K_Ic and it can be the fracture toughness of the material.The results are accurate and in good agreement with those values of K_Ic which are calculated by approximate theoretical expressions.The purpose of this paper is to introduce an improved, simple and accurate experimental method for the determination of fracture toughness of brittle materials.     

Nonlinear cellular convection and heat transport in a porous medium

Nonlinear steady cellular convection in a fluid-saturated porous medium is investigated using the technique of spectral analysis. The effect of permeability is shown to contract the cell and to damp the convection process. The influence of Prandtl number, though small, is seen only in the fourth order term. The cross-interactions of the higher modes caused by nonlinear effects are considered through the modal Rayleigh number R . The possibility of the existence of a steady solution with two self-excited modes in certain regions is predicted. A detailed discussion of the heat transport is made. The theoretical value of the Nusselt number is found to be in good agreement with the experimental results. The similarities and qualitative differences between the present analysis and that of the power integral technique are brought out.     

Modelling of rotational factor in notched bend specimen under general and local yield situation

The location of the plastic hinge axis in a three point SEN bend specimen is a highly controversial issue. An unambiguous and reliable estimation of rotational factor (r_p) is very essential for the accurate determination of CTOD data. In contrast to the numerous studies reported on the r_p determination in a cracked situation, limited information is available for a blunt notch situation, although many engineering structures do contain notchlike defects with finite root radius. An attempt is made to determine r_p for two situations, namely well below the general yield and around the general yield. The work is based on a theoretical estimation of the plastic zone size using the stress concentration factor and the elastic as well as the elastic-plastic stress distribution. A theoretical estimation of r_p in both the pseudo-elastic and the elastic-plastic situation is estimated through analytical modelling involving factors like plastic zone size, bend angle and notch opening displacement. The values of the rotational factor are found to increase from a small value to around 0.29 in a well below general yield situation to 0.53 to 0.54 in a general yield situation with continued loading. A wide discrepancy in the P/P_GY ratio separating the two situations, i.e. well below general yield and around general yield, is observed. Consideration of the elastic and the elasto-plastic stress distribution indicates a much smaller value of P/P_GY as compared to the ratio obtained from experimental load-displacement plots.     

Viscoelasticity and diffusion in miscible blends of saturated hydrocarbon polymers

Linear viscoelasticity and tracer diffusion were investigated as functions of temperature, component molecular weight and blend composition for entangled, single-phase blends of nearly monodisperse poly(ethylene-alt-propylene) (PEP) and head-to-head polypropylene (HHPP). Both components are non-polar and, despite evidence for slight differences of component glass temperatures in their blends, the viscoelastic data obey time-temperature superposition rather well. The properties of the blends were compared at constant T-T _g (blend) with predictions of the tube-model theories. The composition dependence of viscosity agrees best with the double-reptation prediction, as had been found earlier for molecular weight blends. The variation in plateau modulus with composition is consistent with reptation, but the changes are too small to provide a definitive test. The tracer diffusion coefficients, D ^* _PEP and D ^* _HHPP are nearly independent of composition, consistent with the reptation prediction and in sharp contrast with tracer diffusion for blends with specific associations. Results for the recoverable compliance depart from this pattern, varying differently and much less strongly with composition than the predictions of either single or double reptation. It thus seems that microstructural blends may behave in significantly more complex ways than molecular weight blends even for components with only weak dispersive interactions and rather modest differences in glass temperature and plateau modulus.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Dispersion of elastic guided waves in piezoelectric infinite plates with inversion layers

In this work, we study the dispersion of elastic waves in piezoelectric infinite plates with ferroelectric inversion layers. The motivation is to analyze the effect of ferroelectric inversion layers on wave dispersion and resonant behavior under impulsive line loads. A semi-analytical finite-element (SAFE) method has been adopted to analyze the problem. Two model problems are considered for analysis. In one, the plate is composed of a layer of 36° rotated y-cut LiNbO₃ with a ferroelectric inversion layer. In the other, material is PZT-4 with a ferroelectric inversion layer. Comparison with experimental results, reported in the literature for isotropic materials, shows a very good agreement with theoretical predictions obtained using SAFE method. Furthermore, comparison of the resonance frequencies of the S₁ modes, calculated using KLM approximation (f₀ = C_d/2h) and SAFE method, are illustrated for each problem. The frequency spectra of the surface displacements show that resonant peaks occur at frequencies where the group velocity vanishes and the phase velocity remains finite, i.e., a minimum in the dispersion curve below the cut-off frequency. The effect of the ratio of the thicknesses of the inversion layer (IL) and the plate on the frequencies and strength of the resonant peaks is examined. It is observed that for PZT-4 with 50% IL to plate thickness ratio the frequency for the second resonant peak is about twice that for the first one. Results are presented showing the dependence of resonant frequencies on the material properties and anisotropy. Materials selection for single-element harmonic ultrasound transducers is a very important factor for optimum design of transducers with multiple thickness-mode resonant frequencies. The theoretical analysis presented in this study should provide a means for optimum ultrasound transducer design for harmonic imaging in medical applications.     

Normal stress measurements on drag-reducing polymer solutions

Summary The elastic properties of very dilute solutions of a number of drag-reducing polymers differing either in chemical nature or molecular weight were investigated over a range of values of shear stress using the jet thrust method. Parallel drag reduction measurements were also made with the solutions. The results indicate a general relationship between the value of the first normal stress difference at the wall, (p ₁₁-p ₂₂, and the dragreducing ability.The data tends to confirm the generality of the correlation between the value of theWeissenberg number and the drag reduction reported byMetzner for a single polymer sample.     

Local weight functions for semi-elliptical surface cracks in finite thickness plates

Weight functions for any local point, 0 < Φ < π/2 along a semi-elliptical surface crack in finite thickness plates were derived from an assumed approximate general weight function and two reference stress intensity factors. The resulting weight functions were verified using available finite element results for two nonlinear stress fields and good agreement was achieved. When used together with weight functions for Φ = 0 and Φ = π/2 the weight functions are suitable for the calculation of stress intensity factors anywhere along the crack front for semi-elliptical surface cracks in complex stress fields with aspect ratios in the range 0 ≤ a/c ≤ 1 and relative depths 0 ≤ a/t ≤ 0.8.     

Computation of conjugate heat transfer in the turbulent mixed convection regime in a vertical channel with multiple heat sources

This paper presents the results of a comprehensive numerical study to analyze conjugate, turbulent mixed convection heat transfer from a vertical channel with four heat sources, uniformly flush-mounted to one of the channel walls. The results are presented to study the effect of various parameters like thermal conductivity of wall material (k _s), thermal conductivity of flush-mounted discrete heat source (k _c), Reynolds number of fluid flow (Re _s), modified Richardson number (Ri ⁺) and aspect ratio (AR) of the channel. The standard k-ε turbulence model, modified by including buoyancy effects with physical boundary conditions, i.e. without wall functions, has been used for the analysis. Semi-staggered, non-uniform grids are used to discretise the two dimensional governing equations, using finite volume method. A correlation, encompassing a wide range of parameters, is developed for the non-dimensional maximum temperature (T ^*) using the asymptotic computational fluid dynamics (ACFD) technique.