In-line measurement of rheological properties of polymer melts

Shear viscosity (), first normal stress difference (N ₁), and extensional viscosity ( _E ) of polymer melts measured under processing conditions are important in process modeling, quality control, and process control. A slit rheometer that could simultaneously measure , N ₁, and the planar extensional viscosity ( _p ) was designed and tested by attaching it in-line to a laboratory model single-screw extruder. A tube (circular cross-section) rheometer to measure and the uniaxial extensional viscosity ( _u ) simultaneously was also designed and tested. Two commercial grades of LDPE (low density polyethylene) with melt index values of 6 and 12 were used as test materials for the study. Exit and hole pressure methods were used to estimate N ₁, and the entrance pressure drop method using the analyses of Cogswell, Binding, and Gibson (the last analysis used with the axisymmetric case only) was used to estimate _E .The hole pressure method was considered better than the exit pressure method to estimate N ₁ (due to the greater susceptibility of the latter to experimental errors). From the hole pressure method N ₁ was obtained from 100 kPa to 500 kPa over a range of shear rates from 40 s^–1 to 700 s^–1. Among the analyses used to estimate the extensional viscosity, Cogswell's is recommended due to its simpler equations without loss of much information compared to the other analyses. The range of extension rates achieved was 1 to 30 s^–1. The combination of the hole pressure and entrance pressure drop methods in a slit rheometer is a feasible design for a process rheometer, allowing the simultaneous measurement of the shear viscosity, first normal stress difference and planar extensional viscosity under processing conditions. Similarly, combining the entrance pressure drop measurements with a tube rheometer is also feasible and convenient.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u _ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.     

Effect of transverse stress on switch-toughening of ferroelectrics

Testing data indicated that T stress significantly altersR-curves of ferroelectrics. The model of stress-induced polarization switching is adopted to evaluate the fracture toughness of ferroelectrics under K field and T stress. Analytical solution is obtained to estimate the steady state fracture resistance of mono-domain ferroelectrics. The result is generalized to multi-domain ferroelectric ceramics via Reuss approximation, which enables us to explain quantitatively R-curves under two testing configurations.     

Effect of different size and orientation of rectangular rotary blades on quality of puddling

The effect of different combinations of size and orientation of rectangular blades of a wet land puddler, on puddling index, force requirement and performance index was investigated under controlled conditions in a soil-bin from the study. It was observed with the blade size of 15-cm width × 8.5-cm depth and a blade angle of 30° with respect to shaft gave better performance than other combinations in terms of minimum puddling index, minimum horizontal force, vertical force, specific energy and maximum performance index. The values are, respectively, 67.95%, 105.12 N, 33.85 N, and 9.71 kJ/m³ and 6.69. The above-mentioned parameters of the blade were found to be very close to the predicted values obtain through MREG computer programme for optimization of size and orientation of puddler blades.     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

Direct simulation of isolated elliptic vortices and of their radiated noise

The aerodynamic evolution and the acoustic radiation of elliptic vortices with various aspect ratios and moderate Mach numbers are investigated by solving numerically the full compressible Navier–Stokes equations. Three behaviours are observed according to the aspect ratio σ = a/b where a and b are the major and minor semi-axes of the vortices. At the small aspect ratio σ = 1.2, the vortex rotates at a constant angular velocity and radiates like a rotating quadrupole. At the moderate aspect ratio σ = 5, the vortex is initially unstable. However the growth of instability waves is inhibited by the return to axisymmetry which decreases its aspect ratio. The noise level becomes lower with time and the radiation frequency increases. For vortices with larger aspect ratios σ ≥ 6, the return to axisymmetry does not occur quickly enough to stop the growth of instabilities, which splits the vortices. Various mergers are then found to occur. For instance in the case σ = 6, several successive switches between an elliptic state and a configuration of two co-rotating vortices are observed. The present results show that the initial value of the aspect ratio yields the relative weight between the return to axisymmetry which stabilizes the vortex and the growth of instabilities which tends to split it. Moreover the noise generated by the vortices is also calculated using the analytical solution derived by Howe (J. Fluid Mech. 71:625–673, 1975) and is compared with the reference solution provided by the direct computation. This solution is found to be valid for σ = 1.2. An extended solution is proposed for higher aspect ratios. Finally, the pressure field appears weakly affected by the switches between the two unstable configurations in the case σ = 6, which underlines the difficulty to detect the split or the merger of vortices from the radiated pressure. This study also shows that elliptic vortices can be used as a basic configuration of aerodynamic noise generation.      

Optimization of the deflagration to detonation transition: reduction of length and time of transition

The aim of this experimental investigation is the study of Deflagration to Detonation Transition (DDT) in tubes in order to (i) reduce both run-up distance and time of transition (L _DDT and t _DDT) in connection with Pulsed Detonation Engine applications and to (ii) attempt to scale L _DDT with λ_CJ (the detonation cellular structure width). In DDT, the production of turbulence during the long flame run-up can lead to L _DDT values of several meters. To shorten L _DDT, an experimental set-up is designed to quickly induce highly turbulent initial flow. It consists of a double chamber terminated with a perforated plate of high Blockage Ratio (BR) positioned at the beginning of a 26 mm inner diameter tube containing a “Shchelkin spiral” of BR ≈ 0.5. The study involves stoichiometric reactive mixtures of H₂, CH₄, C₃H₈, and C₂H₄ with oxygen and diluted with N₂ in order to obtain the same cell width λ_CJ≈10 mm at standard conditions. The results show that a shock-flame system propagating with nearly the isobaric speed of sound of combustion products, called the choking regime, is rapidly obtained. This experimental set-up allows a L _DDT below 40 cm for the mixtures used and a ratio L _DDT/λ_CJ ranging from 23 to 37. The transition distance seems to depend on the reduced activation energy (E _a/RT _c) and on the normalized heat of reaction (Q/a ₀ ²). The higher these quantities are, the shorter the ratio L _DDT/λ_CJ is. PACS 47.40.Rs · 47.60.+i · 47.70.Pq · 47.80.CbThis paper was based on the work that was presented at the 19th International Colloquium on the Dynamics of Explosions and Reactive Systems, Hakone, Japan, July 27–August 1, 2003.     

Predictions of first-ply failure load of laminated composite shafts

The failure strengths of laminated composite shafts subjected to static bending load and/or torque are studied using both theoretical and experimental approaches. We investigate optimal angle-ply orientations of laminated composite shafts with the design of symmetric main fiber lay-ups [θ/−θ/...] _S and anti-symmetric reinforcing fiber lay-ups [α/−α/...]_−S for maximum stiffness. Different analytical methods, together with various failure criteria, are used to predict the first-ply failure strengths of the composite shafts. A hand lay-up and pressure bag technique was used to manufacture the composite shafts, which includes the stacking of pieces of prepreg in the desired orientations and the curing process. The first-ply failure loads of laminated composite shafts were determined using acoustic emission. The failure modes of the shafts are studied and experimental results are used to verify the theoretical predictions. The experimental results have been proven to be efficient and effective in the theoretical prediction of first-ply failure loads of laminated composite shafts.     

Spectral Analysis of the Balance Equation of Ground Water Hydrology

The spectral analysis of the balance equation of ground water flow, associated to an asymptotic expansion of the conductivity (K) and head (h) fields, permits to show that the high wave number components of the source terms, F, and of the conductivity, K, are attenuated when h is computed as solution to the balance equation. This has important consequences on the inverse mapping (h, F) → K: in fact it is not possible to recover in a reliable way the high wave number components of K, because small errors on the corresponding components of h are amplified so that they can hidden the true signal.     

Forced oscillations of a layer of a viscoelastic material under the action of a convective pressure wave

The longitudinal and transverse components of deformation of the surface of a flat layer of a viscoelastic material glued onto a solid base under the action of a traveling pressure wave are determined. The coating compliance is described by two components corresponding to two components of surface displacement. The dimensionless compliance components depend only on the viscoelastic properties of the material, the ratio of the wave length to the layer thickness λ/H, and the ratio of the wave velocity to the velocity of propagation of shear oscillations V/C _t ⁰ . Data on the dynamic compliance are presented for 0.3 < λ/H < 30 and 0.1 < V/C _t ⁰ < 10. The compliance is demonstrated to be determined by its absolute value and by the phase lag of strain from pressure. The effect of viscous losses in the material and compressibility of the latter on the dynamic compliance is analyzed. An anomalous behavior of the compliance with the wave velocity being greater than a certain critical value is explained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 90–97, March–April, 2007.     

Determination of the molecular weight distribution of linear polymers by inversion of a blending law on complex viscosities

The method presented in this paper allows to calculate the molecular weight distribution (MWD) of linear homopolymer melts from the complex shear modulus data measured in a wide frequency domain. An empirical blending law on complex viscosities is first developed; as a consequence, the variations of the storage and loss modulus as a function of MWD are presented. This simulation demonstrates also the role of the shape of the MWD itself, and shows that one should not postulate a priori the shape of the MWD. An efficient numerical approach based on a Tikhonov regularization method with constraint is used to solve this ill-posed problem; the MWD is hence derived without any assumption on its shape. This method is first applied on simulated data to prove its numerical efficiency. Then the inversion method is applied on complex moduli data of various commercial polymers (polypropylene, polyethylene and polystyrene) and on an artificial mixture of polystyrene that have been presented in the literature. For amorphous polymers, the coupling of the terminal relaxation domains with the transition region at higher frequency leads to errors in the low molecular weight tail: one way to solve this problem is to cut off the experimental data at the high frequencies. This general method needs only a few physical parameters, namely the scaling law for the Newtonian viscosity η₀=f(M _w ) and the plateau modulus G _N ⁰, and leads to reasonable results with respect to the simplicity of the viscoelastic model used. Received: 27 October 1997 Accepted: 24 February 1998     

Effect of flow diverters on free convection heat transfer from a pair of vertical arrays of isothermal cylinders

Laminar free convection heat transfer from two vertical arrays of five isothermal cylinders separated by flow diverters is studied experimentally using a Mach-Zehnder interferometer. The width of flow diverters is kept constant to two-cylinder diameters and the cylinders vertical center-to-center spacing is equal to three-cylinder diameter. Effect of the ratio of the horizontal spacing between two cylinder arrays to their diameter (S_h/D) on heat transfer from the cylinders is investigated for various Rayleigh numbers. The experiments are performed for S_h/D = 2-4, and the Rayleigh number based on the cylinder diameter ranging from 10³ to 3 × 10³. It is observed that for small S_h/D ratios, the flow diverters have a negative effect on the total rate of heat transfer from the arrays; while by increasing the horizontal center to center spacing, they tend to enhance the overall cooling rate of the array. Moreover, increasing Ra and S_h/D generally results in a higher average Nusselt number for each cylinder in the array.     

Influence of M w of LDPE and vinyl acetate content of EVA on the rheology of polymer modified asphalt

Asphalt binder was modified by low-density polyethylene (LDPE) and ethyl vinyl acetate (EVA) polymers to investigate the structure–property relationships of polymer-modified asphalt (PMA). The PMA was prepared in a high-shear blender at 160 °C. The optimum blending time (OBT) for each polymer was determined following a separate investigation. OBT was influenced by M_w, MWD, and polymer structure. The influence of M_w of LDPE and vinyl acetate (VA) content of EVA on PMAs was studied by rheological tools. Polymer modification improved the rheological properties of base asphalt. EVA–PMAs were found to be less temperature sensitive than LDPE-modified asphalts. LDPE modification increased flow activation energy (E_a) but EVA modification decreased E_a. Both VA content and M_w of LDPE have influenced the storage stability of PMAs. The low-temperature properties of PMAs and short ageing tests were not influenced by polymer type. On the other hand, the high-temperature properties of PMAs were strongly influenced by M_w of LDPE and VA content of EVA. Overall, EVA with low VA content showed the best temperature resistance to high- temperature deformations, the highest upper service temperature as well as the best storage stability.     

A Critique of Atomistic Definitions of the Stress Tensor

Current interest in nanoscale systems and molecular dynamical simulations has focussed attention on the extent to which continuum concepts and relations may be utilised meaningfully at small length scales. In particular, the notion of the Cauchy stress tensor has been examined from a number of perspectives. These include motivation from a virial-based argument, and from scale-dependent localisation procedures involving the use of weighting functions. Here different definitions and derivations of the stress tensor in terms of atoms/molecules, modelled as interacting point masses, are compared. The aim is to elucidate assumptions inherent in different approaches, and to clarify associated physical interpretations of stress. Following a critical analysis and extension of the virial approach, a method of spatial atomistic averaging (at any prescribed length scale) is presented and a balance of linear momentum is derived. The contribution of corpuscular interactions is represented by a force density field f. The balance relation reduces to standard form when f is expressed as the divergence of an interaction stress tensor field, T ⁻. The manner in which T ⁻ can be defined is studied, since T ⁻ is unique only to within a divergence-free field. Three distinct possibilities are discussed and critically compared. An approach to nanoscale systems is suggested in which f is employed directly, so obviating separate modelling of interfacial and edge effects.      

Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

A Near-Asymptotics analysis of the turbulence energy spectrum is presented that accounts for the effects of finite Reynolds number recently reported by Mydlarski and Warhaft [21]. From dimensional and physical considerations (following Kolmogorov and von Karman), proper scalings are defined for both low and high wavenumbers, but with functions describing the entire range of the spectrum. The scaling for low wavenumbers uses the kinetic energy and the integral scale, L, based on the integral of the correlation function. The fact that the two scaled profiles describe the entire spectrum for finite values of Reynolds number, but reduce to different profiles in the limit, is used to determine their functional forms in the “overlap” region that both retain in the limit. The spectra in the overlap follow a power law, E(k) =Ck _{−5/3 + μ}, where μ and C are Reynolds number dependent. In the limit of infinite Reynolds number, μ → 0 and C → constant, so the Kolmogorov/Obukhov theory is recovered in the limit. Explicit expressions for μ and the other parameters are obtained, and these are compared to the Mydlarski/Warhaft data. To get a better estimate of the exponent from the experimental data, existing models for low and high wavenumbers are modified to account for the Reynolds number dependence. They are then used to build a spectral model covering all the range of wavenumbers at every Reynolds number. Experimental data from grid-generated turbulence are examined and found to be in good agreement with the theory and the model. Finally, from the theory and data, an explicit form for the Reynolds number dependence of φ = ɛL/u ₃ is obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

Measurement of extensional viscosity of polymer solutions

A filament stretching technique for measuring the extensional viscosity of polymer solutions at constant stretch rate is presented. The liquid sample is held between two coaxial discs and stretched by moving the bottom disc downwards with a speed that increases exponentially with time. This is illustrated using a constant viscosity, elastic fluid consisting of 0.185% polyisobutylene in a solvent of kerosene and polybutene. For the case of this particular fluid, two distinct stretch rate regions are found to arise. The stretch rate in the first region is much higher than in the second, which is, in most cases, close to the overall stretch rate imposed on the sample. Nonetheless, all the results of any given run can be represented using an average extensional rate. The extensional stress growth data, plotted as the Trouton ratio against time, show an initial linear viscoelastic region where T_R rises to a value of 3, independent of extensional rate. Beyond this region, T_R depends on the stretch rate and rises dramatically to values in excess of 10³; the higher the extensional rate, the faster is the increase in T_R. These data do not seem to reach a steady state and appear to be similar to polymer melt data obtained by others in the past. The reproducibility of the results is very good and all this suggests that it is now possible to obtain unambiguous constant-stretch-rate stress-growth data for polymer solutions stretched from a state of rest.     

Study of high Deborah number flow of polyisobutylene in square die

A large capacity RAM extruder was designed which provides the opportunity to study high Deborah number (D) flows, with D < 1,000. A modified version of particle image velocimetry was developed to enable the measurement of the velocity field in dies of arbitrary cross section. During the measurement process, tracer particles were simultaneously illuminated by both a focused laser beam locally and a lamp globally. Only those particles that passed through the laser beam were taken into account. The beam was scanned to achieve full field measurements. This method of measurement allowed us to find the location of a particle in the direction of the optical axis of the lens, i.e. that which makes the particle image on the CCD detector of the video camera. A device employing this method was designed and used to measure velocity profiles. The results of these measurements in two cross sections of the square die, at three values of pressure, are presented. The velocity was defined as the ratio of displacement to the elapsed time during which this displacement occurred. Errors in measurements of the coordinates and the observation time of particles were estimated as ±20 μm and less than 0.1%, respectively. A large plateau in the axial velocity profile was found at relatively small Deborah numbers, e.g. D ≈ 28. In flows with higher Deborah numbers, e.g. D ≈ 766, an almost flat velocity profile was detected. Two components of velocity, one longitudinal and one transversal, were measured simultaneously. However, the transversal component appeared to be less than the error of measurements and less than 1% of the axial velocity. Received: 4 August 1998 Accepted: 5 April 1999     

Kinetics of microcrack blunting ahead of macrocrack approaching shear wave speed

The fracturing of glass and tearing of rubber both involve the separation of material but their crack growth behavior can be quite different, particularly with reference to the distance of separation of the adjacent planes of material and the speed at which they separate. Relatively speaking, the former and the latter are recognized, respectively, to be fast and slow under normal conditions. Moreover, the crack tip radius of curvature in glass can be very sharp while that in the rubber can be very blunt. These changes in the geometric features of the crack or defect, however, have not been incorporated into the modeling of running cracks because the mathematical treatment makes use of the Galilean transformation where the crack opening distance or the change in the radius of curvature of the crack does not enter into the solution. Change in crack speed is accounted for only via the modulus of elasticity and mass density. For this simple reason, many of the dynamic features of the running crack have remained unexplained although speculations are not lacking. To begin with, the process of energy dissipation due to separation is affected by the microstructure of the material that distinguishes polycrystalline from amorphous form. Energy extracted from macroscopic reaches of a solid will travel to the atomic or smaller regions at different speeds at a given instance. It is not clear how many of the succeeding size scales should be included within a given time interval for an accurate prediction of the macroscopic dynamic crack characteristics. The minimum requirement would therefore necessitate the simultaneous treatment of two scales at the same time. This means that the analysis should capture the change in the macroscopic and microscopic features of a defect as it propagates. The discussion for a dual scale model has been invoked only very recently for a stationary crack. The objective of this work is to extend this effort to a crack running at constant speed beyond that of Rayleigh wave. Developed is a dual scale moving crack model containing microscopic damage ahead of a macroscopic crack with a gradual transition. This transitory region is referred to as the mesoscopic zone where the tractions prevail on the damaged portion of the material ahead of the original crack known as the restraining stresses, the magnitude of which depends on the geometry, material and loading. This damaged or restraining zone is not assumed arbitrarily nor assumed to be intrinsically a constant in the cohesive stress approach; it is determined for each step of crack advancement. For the range of micronotch bluntness with 0 < β < 30° and 0.2 σ_∞/σ₀ 0.5, there prevails a nearly constant restraining zone size as the crack approaches the shear wave speed. Note that β is the half micronotch angle and the applied stress ratio is σ_∞/σ₀ with σ₀ being the maximum of the restraining stress. For σ_∞/σ₀ equal to or less than 0.5, the macrocrack opening displacement COD is nearly constant and starts to decrease more quickly as the crack approaches the shear wave speed. For the present dual scale model where the normalized crack speed v/c_s increases with decreasing with the one-half microcrack tip angle β. There prevails a limit of crack tip bluntness that corresponds to β 36° and v/c_s 0.15. That is a crack cannot be maintained at a constant speed if the bluntness is increased beyond this limiting value. Such a feature is manifestation of the dependency of the restraining stress on crack velocity and the applied stress or the energy pumped into the system to maintain the crack at a constant velocity. More specifically, the transitory character from macro to micro is being determined as part of the unknown solution. Using the energy density function dW/dV as the indicator, plots are made in terms of the macrodistance ahead of the original crack while the microdefect bluntness can vary depending on the tip geometry. Such a generality has not been considered previously. The macro-dW/dV behavior with distance remains as the inverse r relation yielding a perfect hyperbola for the homogeneous material. This behavior is the same as the stationary crack. The micro-dW/dV relations are expressed in terms of a single undetermined parameter. Its evaluation is beyond the scope of this investigation although the qualitative behavior is expected to be similar to that for the stationary crack. To reiterate, what has been achieved as an objective is a model that accounts for the thickness of a running crack since the surface of separation representing damage at the macroscopic and microscopic scale is different. The transitory behavior from micro to macro is described by the state of affairs in the mesoscopic zone.     

Effect of melt viscosity of polypropylene on fibrillation of thermotropic liquid crystalline polymer in in situ composite film

 Various grades of polypropylene were melt blended with a thermotropic liquid crystalline polymer, a block copolymer of p-hydroxy benzoic acid and ethylene terephthalate (60/40 mole ratio). The blends were extruded as cast films at different values of draw ratio (slit width/film thickness). Fibrillation of TLCP dispersed phase with high fiber aspect ratio (length/width) was obtained with the matrix of low melt flow rate, i.e., high viscosity and with increasing film drawing. Melt viscosities of pure components and blends measured using capillary rheometer were found to decrease with increasing shear rate and temperature. Viscosity ratios (dispersed phase to matrix phase) of the systems being investigated at 255 °C at the shear rate ranged from 10² to 10⁴ s⁻¹, were found to lie between 0.04 and 0.15. The addition of a few percent of elastomeric compatibilizers; a tri-block copolymer SEBS, EPDM rubber and maleated-EPDM, was found to affect the melt viscosity of the blend and hence the morphology. Among these three compatibilizers, SEBS was found to provide the best fibrillation. Received: 10 January 2000/Accepted: 24 January 2000