Measurement of the first and second normal stress differences: Correlation of four experiments on a polyisobutylene/decalin solution “D1”

Pressure distribution measurements for a polyisobutylene/decalin solution D1 in the Truncated Cone-and-Plate (TCP) apparatus are combined with elastic hole pressures obtained for the same solution on the Lodge Stressmeter^® in order to provide two independent estimates of the second normal-stress difference (N ₂). The values ofN ₂ from the TCP apparatus, obtained by numerical differentiation of a function of the center-hole pressure and the pressure gradient, are in good agreement with measurements made on the same sample by Tanner et al. with a direct method, namely the Tilted Trough Experiment, and by Christiansen et al. with a method that requires an extrapolation to the pressure at the free surface of coneand-plate rheogoniometer data obtained with flush-mounted pressure transducers. The viscosities from the modified Stressmeter for low shear rates extend over five decades of shear rate, including a zero-shear-rate region, and agree with the data of Christiansen on a torque-driven flow. The Higashitani-Pritchard-Baird-Lodge (HPBL) equation relatingN ₁–N ₂ to the hole pressure gives good agreement with the data over a certain range of shear stress. The Newtonian hole pressures for several liquids at 20 and 46 °C compare well with a finite-element calculation for a two-dimensional Poiseuille flow. When the elastic hole pressures from the Stressmeter are combined with the extrapolated rim pressures from the TCP Apparatus in order to extract the value ofN ₂, an agreement betweenN ₂ from the center-hole pressure andN ₂ from the rim pressure can only be obtained up to a shear rate of about 40 s^–1, beyond which the value of –N ₂ from the rim pressure diverges abruptly to negative values. It is possible that this constitutes the first quantitative estimate of an edge effect in cone-and-plate rheometry. Alternatively, the elastic hole pressure in cone-and-plate flow is not equivalent to the elastic hole pressure in Poiseuille flow, at least at high shear rates. The data of Christiansen et al. with flush-mounted pressure transducers appear to confirm this second possibility. Finally, a single set of shift factors obeying the Williams, Landel and Ferry equation superposes the viscosity, the first and the second normal-stress difference within experimental scatter, which can be less than 1% for a certain combination of normal-stress differences. The data were recorded at 3, 20, 30, and 46 °C in the shear rate range 1–260 s^–1.     

An experimental study of the first normal stress difference — shear stress relationship in simple shear flow for concentrated shear thickening suspensions

For many polymeric solutions in a simple shear flow a plot of the logarithm of the first normal stress difference N ₁ against the logarithm of the shear stress , for a range of temperatures results in a linear relationship. For such polymer solutions these plots yield a straight line of slope very close to 2 when measured at low shear rates. This relationship is tested against a polymer solution (polyacrylamide in a 50/50 mixture of glycerol/water), a polymer melt (polyethylene), and three concentrated suspensions. These are Superclay (supplied by English China Clays, Cornwall, U.K.) in water, polyvinylchloride in dioctylphthalate and polystyrene latex in water, at volume concentrations of 40% 48%, and 60%, respectively. It was found that the log N ₁ — log relationship is applicable to the polymer solution and melt over a significant range of shear rates and temperatures. In the cases of concentrated suspensions the relationship holds to the point of onset of the shear thickening behaviour. Beyond this point a different relationship exists, however, flow instabilities are apparent. A comment on the contribution of N ₁ and N ₂ to the flow instabilities is made.     

Qualitative relation between reflected light intensity by Kalliroscope flakes and velocity field in the Couette–Taylor flow system

We have used particle image velocimetry adapted to the Couette–Taylor system in order to measure the flow characteristics (axial and radial velocity components, vorticity fields, kinetic energy, etc.) and their spatio-temporal dependence. By looking for similarity between spatio-temporal diagrams of reflected light intensity and those of velocity fields, we have established that the intensity of light reflected by Kalliroscope flakes is related to the radial velocity component when the outer cylinder is fixed.     

Determination of the full-field stress and displacement using photoelasticity and sampling moiré method in a 3D-printed model

The quantitative characterization of the full-field stress and displacement is significant for analyzing the failure and instability of engineering materials. Various optical measurement techniques such as photoelasticity, moiré and digital image correlation methods have been developed to achieve this goal. However, these methods are difficult to incorporate to determine the stress and displacement fields simultaneously because the tested models must contain particles and grating for displacement measurement; however, these elements will disturb the light passing through the tested models using photoelasticity. In this study, by combining photoelasticity and the sampling moiré method, we developed a method to determine the stress and displacement fields simultaneously in a three-dimensional (3D)-printed photoelastic model with orthogonal grating. Then, the full-field stress was determined by analyzing 10 photoelastic patterns, and the displacement fields were calculated using the sampling moiré method. The results indicate that the developed method can simultaneously determine the stress and displacement fields.     

On the interplay between fast reaction and slow diffusion in the concrete carbonation process: a matched-asymptotics approach

A matched-asymptotics approach is proposed to show the occurrence of two distinct characteristic length scales in the carbonation process. The separation of these scales arises due to the strong competition between reaction and diffusion effects. We show that for sufficiently large times τ the width of the carbonated region is proportional to \(\sqrt{\tau}\), while the width of the reaction front is proportional to \(\tau^{\frac{p-1}{2(p+1)}}\) for carbonation-reaction rates with a power law structure like k[CO₂] ^p [Ca(OH)₂] ^q , where k>0 and p,q>1 and identify the proportionality coefficient asymptotically. We emphasize the occurrence of a water barrier in the reaction zone which may hinder the penetration of CO₂ by locally filling with water air parts of the pores. This non-linear effect may be one of the causes why a purely linear extrapolation of accelerated carbonation test results to natural carbonation settings is (even theoretically) not reasonable. Finally, we compare our asymptotic penetration law against measured penetration depths from Bune (Zum Karbonatisierungsbedingten Verlust der Dauerhaftigkeit von Außenbauteilen aus Stahlbeton, 1994). The novelty consists in the fact that the factor multiplying \(\sqrt{\tau}\) is now identified asymptotically by solving a non-linear system of ordinary differential equations, and hence, fitting arguments are not necessary to estimate its size. We offer an alternative to the (asymptotic) \(\sqrt{\tau}\) expression of the carbonation-front position obtained in Papadakis et al. (AIChE J. 35:1639, 1989).     

Elastic interactions between multi-valued foldons and anti-foldons for the (2+1)-dimensional variable coefficient Broer–Kaup system in water waves

With the help of the improved tanh-function method, some exact variable separation solutions for a (2+1)-dimensional variable coefficient Broer–Kaup system in water waves are found. The detailed investigation indicates that these seemly independent variable separation solutions actually depend on each other. Based on the exact variable separation solution, completely and noncompletely elastic interactions between multi-valued foldons and anti-foldons are studied analytically and graphically.     

Bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings

Auxiliary bearings are utilized in practical installations of magnetically suspended rotating machines with the main functions to provide support to the rotating machines during their non-operational period, and to protect the magnetic bearings and the rotating assembly from being damaged due to power loss during operation. The relatively small clearances of these bearings, which are typically half of that of the magnetic bearings, may at time cause contact between the rotor and these bearings to occur even during normal operation of the rotating machines. The work presented herein examines the bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between the rotor and the auxiliary bearings. Numerical results revealed the occurrence of period-doubling bifurcation resulting in vibrations of period-2, -4, -8, -16 and -32, as well as quasi-periodic and chaotic vibrations. The results further showed that for a relatively small rotor imbalance magnitude, which is within the prescribed level of certain classes of practical rotating machinery, such nonlinear dynamical phenomena would not have been discovered had the auxiliary bearings forces been omitted in the model of the rotor-bearing system. As these bearings are essential elements in practical installations of magnetically supported rotating machines, failure to include them in the rotor-bearing model may result in incorrect prediction of the rotor’s vibration response.     

The effect of boundary curvature on the stress response of linear and branched polyethylenes in a contraction�Cexpansion flow

The effect of flow-boundary curvature on the principal stress difference (PSD) profiles observed through a contraction?Cexpansion (CE) slit flow is evaluated for three different polyethylenes exhibiting increasing levels of branching. Studies were performed using both experimental optical techniques and computational simulations, in the latter case to evaluate the ability of constitutive models to predict these complex flows. The materials were characterised using linear and extensional rheology, which were fitted to the multi-mode ROLIE-POLY and POM-POM models depending upon material branching. Three CE-slit geometries were used; with sharp corners, and with rounding equal to one quarter and one half of the slit length. These created a mixed, but primarily simple shear flow, with different levels of extension and shear depending upon the level of curvature. The PSD developed from an initial Newtonian profile to increasing levels of asymmetry between the inlet and the outlet flow as the level of material branching increased. The rounding was found to lead to the delocalisation of PSD within the flow and removal of the stress singularity from the corner of the CE-slit. It also led to a decrease in the pressure drop across the geometry and an ??opening out?? of features such as downstream stress fangs to create downstream ??crab-claws??. Matching between experiments and simulations for the time evolution of flow from start up for each material in the various geometries illustrated good agreement for both models.     

On fallacies in the decision between the Caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator

Recently Dal [Dal, F., 2011. Multiple time scale solution of an equation with quadratic and cubic nonlinearities having fractional-order derivative. Mathematical and Computational Applications 16 (1), 301–308] presented ‘a new analytical scheme’ to calculate the dynamic response of a fractionally damped nonlinear oscillator possessing both quadratic and cubic nonlinearities via the method of multiple time scales. It has been claimed that damping features are modeled via the Caputo fractional derivative. In the present paper, it is shown that both the scheme and the object of investigation are not new, and moreover, the above mentioned author's statement is inconsistent, since under the assumptions made in the paper under consideration these two fractional-order derivatives coincide. Besides, the utilized procedure was inconsequential. It has been proved that the investigation of the dynamic response of a nonlinear viscoelastic oscillator presents the case that, with some minimal restrictions, the Riemann–Liouville and Caputo definitions produce completely equivalent mathematical models of the nonlinear viscoelastic phenomenon.     

The Neumann boundary problem for axisymmetric fractional heat conduction equation in a solid with?cylindrical hole and associated thermal stress

The theory of thermoelasticity based on the heat conduction equation with the Caputo time-fractional derivative of order α is used to study thermal stress in an infinite medium with a cylindrical hole. Two types of Neumann boundary conditions are considered: the constant value of the normal derivative of the temperature and constant heat flux at the surface of a cavity. The solution is obtained applying Laplace and Weber integral transforms. Numerical results are illustrated graphically.     

Non-uniform, axisymmetric misfit strain： in thin films bonded on plate substrates/substrate systems： the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system’s curvatures and the misfit strain. These relations feature a ‘‘local’’ part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a ‘‘non-local’’ part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a ‘‘non-local’’ dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.     

The plane elasticity problem of an isotropic wedge under normal and shear distributed loading––application in the case of a multi-material problem

The problem of a multi-material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution. The stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r. The proposed solution is applied to the elastostatic problem of a composite isotropic k-materials infinite wedge under distributed loading along its external faces. Applications are made in the case of the two-materials composite wedge under linearly distributed loading along its external faces and in the case of a three-materials composite wedge under a parabolically distributed loading along its external faces.     

Numerical simulation of initiation and cessation of flow between coaxial cylinders in the case of a viscoplastic Bingham-Il’yushin medium

A flow of a viscoplastic Bingham-Il’yushin medium in the annulus between two rotating cylinders (the circular Couette flow) is considered. The processes of flow initiation and cessation are numerically simulated.     

Rayleigh–Lamb Waves in a Compound Wave Guide. Reflection from and Transmission Through the Vertical Interface between Media with Different Impedances

The method of superposition is used to study the first normal wave reflecting from and transmitting through the interface in a compound waveguide consisting of two rigidly joined elastic half-strips with equal width and different mechanical properties. We study how the impedances of the contacting media influence the transformation of the energy of the incident wave to those of the reflected and transmitted waves. Two cases are considered — propagating waves of higher orders appear in the reflected wavefield earlier than in the transmitted wavefield and propagating waves of higher orders appear in the transmitted wavefield earlier than in the reflected wave field. For both cases, the impedances vary so that the incident wave can propagate in both more rigid and softer media. It is shown that by increasing the impedances of the contacting media, the interface can be made more transparent