Measurement of microscopic plastic-strain distributions in the region of a crack tip

Two independent experimental techniques are used to measure the strain distribution within the plastically deformed region around a crack tip. Moiré grid interference is used to measure the in-plane strain with the specimen grid engraved directly on the specimen surface. Optical interference is used to measure the through-the-thickness strain over the same engraved area. The testing arrangement allows measurement of at-load strain as well as residual strain. The measured strain distribution is compared with recent work by Swedlow using a finite-element numerical technique and with results of the etch-pit technique used by Hahn and Rosenfield.     

Generation of sub-micron-sized droplets,by continuous extensional flow

A method is devised to generate sub-micron-sized droplets without surfactants by using a continuous extensional flow. By using the extensional flow judiciously the physical limits on the viscosity ratio of the dispersed and dispersing media could be overcome. The strain hardening behavior of the Boger fluid under the extensional flow gave a synergic effect to generate the smaller droplets. It was shown that the drop size could be decreased to a sub-micron scale and the droplet size distribution was much narrower compared to the case of the conventional shear (stirring) method. This method is also applicable when the two phases are compatible.     

The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts

The extensional viscosity of some flexible chain polymers and a thermotropic liquid crystalline polymer was measured in uniaxial extensional flow at constant extension rate. Power law functions were found for the dependence of the extensional viscosity at constant accumulated strain on strain rate. The stress growth curves were compared with measurements in axisymmetric entry flow, where both elongation and shear occur. The comparison showed that the values of the extensional viscosity calculated from the measurements in the entry flow correspond to the ones calculated from the viscosity growth measured in uniaxial elongation and averaged over extensional strain equal to what is accumulated on the fluid as it flows from the barrel into the capillary.     

Assessment of nonlinear strain measures for extensional and shearing flows of polymer melts

From stress-strain experiments in extensional and shearing flows, nonlinear strain measures and effective damping functions are derived for a polyisobutylene melt. The strain measures determined in planar extensional flow and in simple shear flow coincide. Experimental results are compared with predictions of two molecular theories, the Doi-Edwards model and the molecular stress function approach of Wagner and Schaeffer. Discrepancies between theories and experiment lead to a reconsideration of the classification of extensional flows. The symmetry of the flow field is identified and quantified as an important parameter influencing the strain measure, and a unifying strain measure for general extensional and shearing flows of polymer melts is presented.     

A finite element formulation for analysis of functionally graded plates and shells

Summary A finite element formulation is derived for the thermoelastic analysis of functionally graded (FG) plates and shells. The power-law distribution model is assumed for the composition of the constitutent materials in the thickness direction. The procedure adopted to derive the finite element formulation contains the analytical through-the-thickness integration inherently. Such formulation accounts for the large gradient of the material properties of FG plates and shells through the thickness without using the Gauss points in the thickness direction. The explicit through-the-thickness integration becomes possible due to the proper decomposition of the material properties into the product of a scalar variable and a constant matrix through the thickness. The nonlinear heat-transfer equation is solved for thermal distribution through the thickness by the Rayleigh-Ritz method. According to the results, the formulation accounts for the nonlinear variation in the stress components through the thickness especially for regions with a variation in martial propperties near the free surfaces.     

Extensional rheology of a shear-thickening cornstarch and water suspension

A filament-stretching rheometer is used to measure the extensional viscosity of a shear-thickening suspension of cornstarch in water. The experiments are performed at a concentration of 55 wt.%. The shear rheology of these suspensions demonstrates a strong shear-thickening behavior. The extensional rheology of the suspensions demonstrates a Newtonian response at low extension rates. At moderate strain rates, the fluid strain hardens. The speed of the strain hardening and the extensional viscosity achieved increase quickly with increasing extension rate. Above a critical extension rate, the extensional viscosity goes through a maximum and the fluid filaments fail through a brittle fracture at a constant tensile stress. The glassy response of the suspension is likely the result of jamming of particles or clusters of particles at these high extension rates. This same mechanism is responsible for the shear thickening of these suspensions. In capillary breakup extensional rheometry, measurement of these suspensions demonstrates a divergence in the extensional viscosity as the fluid stops draining after a modest strain is accumulated.     

Determination of the elongational viscosity of polymer melts by melt spinning experiments. A comparison with different experimental techniques

In this work, melt spinning experiments were tentatively used for the determination of the elongational viscosity of polymer melts at different levels of tensile strain and strain rate. The materials examined were two high-density polyethylene grades for blow moulding with similar number-average molecular mass but different polydispersity index. The data from melt spinning tests were compared with transient extensional viscosity data obtained by uniform isothermal tensile tests, performed by means of an extensional rheometer, as well as with those produced by converging flow tests (Cogswell model). The results showed that for high strain and strain rate levels, the melt spinning experiments provide elongational viscosity data quite close to the transient extensional viscosity values obtained from the tensile tests.     

Surface tension driven jet break up of strain-hardening polymer solutions

Experimental studies attempting to ascertain the influence of viscoelasticity on the atomization of polymer solution are often hindered by the inability to decouple the effect of shear thinning from the effect of extensional hardening. Here, the influence of viscoelasticity on the jet break up of a series of non-shear-thinning viscoelastic fluids is quantified. Previous characterization using an opposed-nozzle rheometer identified the critical extensional rates for strain hardening of these model fluids. The strain hardening fluids exhibit a beads-on-string structure with reduction or elimination of satellite drops. Capillary instabilities grow on the filaments connecting the spheres and eventually break the filaments up into a string of very small drops about one order of magnitude smaller than the satellite drops formed by a Newtonian fluid with the same shear viscosity, surface tension, and density. These results confirm that strain hardening is the key rheological property in jet break up and that the critical extensional rate of a fluid is pertinent in determining the final characteristics of break up. Results suggest that the opposed-nozzle rheometer does probe extensional behavior in the range of extensional rates that are relevant to jet break up, providing a tool to roughly predict jet break up.     

A new look at extensional rheology of low-density polyethylene

The nonlinear rheology of three selected commercial low-density polyethylenes (LDPE) is measured in uniaxial extensional flow. The measurements are performed using three different devices including an extensional viscosity fixture (EVF), a homemade filament stretching rheometer (DTU-FSR) and a commercial filament stretching rheometer (VADER-1000). We show that the measurements from the EVF are limited by a maximum Hencky strain of 4, while the two filament stretching rheometers are able to probe the nonlinear behavior at larger Hencky strain values where the steady state is reached. With the capability of the filament stretching rheometers, we show that LDPEs with quite different linear viscoelastic properties can have very similar steady extensional viscosity. This points to the potential for independently controlling shear and extensional rheology in certain rate ranges.     

The effect of step-stretch parameters on capillary breakup extensional rheology (CaBER) measurements

Extensional rheometry has only recently been developed into a commercially available tool with the introduction of the capillary breakup extensional rheometer (CaBER). CaBER is currently being used to measure the transient extensional viscosity evolution of mid to low-viscosity viscoelastic fluids. The elegance of capillary breakup extensional experiments lies in the simplicity of the procedure. An initial step-stretch is applied to generate a fluid filament. What follows is a self-driven uniaxial extensional flow in which surface tension is balanced by the extensional stresses resulting from the capillary thinning of the liquid bridge. In this paper, we describe the results from a series of experiments in which the step-stretch parameters of final length, and the extension rate of the stretch were varied and their effects on the measured extensional viscosity and extensional relaxation time were recorded. To focus on the parameter effects, well-characterized surfactant wormlike micelle solutions, polymer solutions, and immiscible polymer blends were used to include a range of characteristic relaxation times and morphologies. Our experimental results demonstrate a strong dependence of extensional rheology on step-stretch conditions for both wormlike micelle solutions and immiscible polymer blends. Both the extensional viscosity and extensional relaxation time of the wormlike micelle solutions were found to decrease with increasing extension rate and strain of the step-stretch. For the case of the immiscible polymer blends, fast step-stretches were found to result in droplet deformation and an overshoot in the extensional viscosity which increased with increasing strain rates. Conversely, the polymer solutions tested were found to be insensitive to step-stretch parameters. In addition, numerical simulations were performed using the appropriate constitutive models to assist in both the interpretation of the CaBER results and the optimization of the experimental protocol. From our results, it is clear that any rheological results obtained using the CaBER technique must be properly considered in the context of the stretch parameters and the effects that preconditioning has on viscoelastic fluids.     

Uniaxial and biaxial extensional viscosity measurements of dilute and semi-dilute solutions of rigid rod polymers

Effective uniaxial extensional and biaxial extensional viscosities of dilute and semi-dilute solutions of collagen, a rigid rod molecule, have been measured with an opposing jet apparatus. The concentration of collagen in the glycerin/water solvent ranged from 50 to 2300 ppm. The data agree quantitatively with a theory developed by Batchelor describing the extensional viscosity of perfectly aligned rigid rods. The viscosity measured for the dilute rigid rod solutions is independent of the rate of strain as predicted by Batchelor's theory. Data taken on the semi-dilute, strain-thinning solutions at strain rates sufficiently high to align the rods in the extension direction also agree with the predictions of Batchelor's theory. The measured viscosity of semi-dilute solutions at low strain rates agree qualitatively with a theory developed by Doi and Edwards describing the strain-thinning behavior of semi-dilute rigid rod solutions.     

Strain hardening behavior of polymer blends with fibril morphology

We investigate the rheological behavior of the polymer blends with fibril morphology, with special focus on the effect of fibril morphology on the extensional properties under uniaxial extension. We add a small amount of the dispersed phase to the matrix, and control the blend morphology by changing the viscosity ratio. When the fibril morphology is maintained, the blend shows not only a significant increase of the extensional viscosity but the strain hardening behavior. The extensional viscosity increases depending on the aspect ratio of the fibers, while the strain hardening behavior originates from the restricted stretching of deformable fibers, which has been confirmed theoretically by introducing the concept of rigidity of the fiber. It suggests a way to induce the strain hardening behavior by introducing deformable fibrils into the matrix, that is, by the design of polymer blends with a small amount of dispersed phase such that the fibril structure is maintained.     

An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.     

Dynamics of worm-like micelles: the Cox–Merz rule

The viscoelastic behaviour of worm-like micelles in small-amplitude oscillatory, steady simple shear and uniaxial extensional flows are analyzed with a model that couples the Oldroyd-B constitutive equation with a kinetic equation that accounts for the structural changes induced by the flow. In some cases, the constitutive equation predicts a viscoelastic behaviour that is consistent with the Cox–Merz rule. Departures from this rule are also predicted. Experimental data obtained for two worm-like micellar systems indicate that in these solutions, the Cox–Merz rule is not usually followed, in agreement with the predictions of our model. In uniaxial extensional flow, the model predicts a strain hardening in the extensional viscosity at low extensional rates and a strain-thinning at high extensional rates.     

Effect of a controlled pre-deformation history on extensional viscosity of dilute polymer solutions

We use a modified filament stretching rheometer to quantify the influence of a known controlled pre-shear history on the transient extensional viscosity of a dilute polymer solution. Two different types of pre-deformation are explored; both influence the subsequent stretching significantly, albeit in opposite ways. Small-amplitude oscillatory straining parallel to the direction of stretching enhances strain hardening and accelerates the tensile stress growth toward the steady-state value. Conversely, steady torsional shearing orthogonal to the direction of stretching retards strain hardening and results in a delayed approach to steady-state elongational flow. In both cases, the final steady-state extensional viscosity is the same as that observed with no pre-shearing. Calculations using a finitely extensible nonlinear elastic Peterlin dumbbell model qualitatively capture the trends observed in experiments, enabling interpretation of these observations in terms of the degree of polymer chain stretching imposed by the flow before extensional stretching.     

Design of new HDPE/silica nanocomposite and its enhanced melt strength

Linear polymers are restricted to use in processes that involve severe extensional deformation, such as fiber spinning, film blowing, and thermoforming. To extend their applicability, the extensional properties of polymer melts should be enhanced such that strain hardening, which is defined as an increase in extensional viscosity under a large strain that deviates from the linear viscoelastic curve, is pronounced. In this study, a novel preparation method of linear polymer/inorganic nanocomposites was proposed with a main focus on enhanced melt strength. The design of molecular structure consists of three components—linear polymer, compatibilizer, and surface-modified particles. High-density polyethylene was used as a linear polymer while polyethylene grafted with maleic anhydride was used as a compatibilizer. Silica particles were synthesized and modified on their surfaces by 3-aminopropyltriethoxysilane. The strain hardening behavior of the surface-modified silica composites was pronounced. However, such a result was not observed for the composites of the same composition with pure-silica. In addition, the dispersion of the modified silica was much better than that of pure-silica.     

The effect of pre-shear on the extensional rheology of wormlike micelle solutions

The effect of initial microstructural deformation, alignment, and morphology on the response of wormlike micelle solutions in transient uniaxial extensional flows is investigated using a pre-shear device attached to a filament stretching rheometer. In filament stretching experiments, increasing the strength and the duration of the pre-shear just before stretch is found to delay the onset of strain hardening. In these experiments, the wormlike micelle solution filaments fail through a rupture near the axial midplane. The value of the elastic tensile stress at rupture is found to decrease with increasing pre-shear rate and duration. The most dramatic effects are observed at shear rates for which shear banding has been independently observed. The reduction in the strain hardening suggests that pre-shear before filament stretching might break down the wormlike micelles reducing their size before stretch. Strain hardening is also observed in capillary breakup rheometry experiments; however, the pre-sheared wormlike micelle solutions strain harden faster, achieve larger steady-state extensional viscosities and an increase in the extensional relaxation time with increasing shear rate and duration. The difference between the response of the wormlike micelles in filament stretching and capillary breakup experiments demonstrates the sensitivity of these self-assembling micelle networks to pre-conditioning.     

The triple jet: influence of shear history on the stretching of polymer solutions

A kerosene-based aircraft safety fuel and aqueous solutions of poly (ethylene oxide) and polyacrylamide are examined using the “triple jet” system. This device allows the solution to be stretched as it flows from a capillary tube and the axial stress, strain and strain rate in the liquid are measured.The shear history of the solution is altered by placing cylindrical inserts in the capillary tube. This is shown to have a large effect on the extensional behaviour of aircraft safety fuel, a moderate effect on the extensional behaviour of poly (ethylene oxide) solution and little effect on the behaviour of polyacrylamide solution. The extensional viscosity of the aircraft fuel is raised by an order of magnitude when a long period of high shear is used; the effects last for periods of up to one second, though traditional methods suggest a relaxation time of the order of 10^?3 seconds. A liquid of shear viscosity 4 centipoise may have an extensional viscosity of over 100 poise.Plots of the extensional modulus of the jet as a function of distance along the jet emphasize the importance of shear history for the first two types of solution and suggest that the latter stages of the stretching process are elastic in character. Typical extensional moduli for the solutions tested are in the range 1.3–5.0 × 10⁴ dyn.cm^?2.The relevance of the interplay between shearing and stretching flow to the phenomena of lubrication and turbulence suppression is mentioned.     

Comparison of the melt fracture behavior of metallocene and conventional polyethylenes

The influence of sparse long-chain branching and molecular weight distribution on the melt fracture behavior of polyethylene melts was investigated. Four commercial polyethylene resins were employed for this study: a conventional low-density polyethylene, a conventional linear low-density polyethylene, a linear metallocene polyethylene, and a sparsely branched metallocene polyethylene. Rheological measurements were obtained for both shear and extensional deformations, and melt fracture experiments were carried out using a controlled rate capillary rheometer. A single capillary geometry was used to focus on the effects of material properties rather than geometric factors. For the linear polyethylenes, surface melt fracture, slip-stick fracture, and gross melt fracture were all observed. Conversely, the branched PE resins did not exhibit a slip-stick regime and the degree of gross fracture was observed to be much more severe than the linear resins. These variations can be explained by the effects that long-chain branching has on the onset of shear-thinning behavior (slip-stick fracture) and the degree of extensional strain hardening (gross melt fracture). Although there is some indication that the breadth of molecular weight distribution indirectly influences surface melt fracture, the results remain inconclusive.     

Suspensions of monodisperse spheres in polymer melts: particle size effects in extensional flow

Suspensions in polymeric, viscoelastic liquids have been studied in uniaxial extensional flow. The fibre wind-up technique has been used for this purpose. The effects of particle size and particle volume fraction have been investigated, using monodisperse, spherical particles. The results have been compared with shear flow data on the same materials. The values of the relative extensional viscosities at low stretching rates are in agreement with the relative shear viscosities and relative moduli. This indicates that hydrodynamic forces are stronger than the particle interaction forces. At larger strain rates strain hardening occurs; it is suppressed when particles are added. Small aggregating particles reduce the strain hardening more strongly than larger particles; strain hardening can even be totally eliminated. When further increasing the stretching rate, hydrodynamic effects dominate again and the effect of particle size effect on strain hardening disappears.