Dilute solution rheology: experimental results and finitely extensible nonlinear elastic dumbbell theory

Experimental results for intrinsic viscosity and for intrinsic complex viscosity of polymer solutions were compared with the rheological predictions of the finitely extensible, nonlinear elastic (FENE) dumbbell theory of a dilute suspension. The FENE dumbbell adequately models the intrinsic viscosity of flexible polymers, but less successfully portrays the behavior in small amplitude oscillatory motion. Expressions for the high frequency asymptotic limit of the intrinsic complex viscosity of a FENE dumbbell suspension, and the mean-square end-to-end distance of FENE dumbbells in steady shear flow are given.     

The langevin approach to the dumbbell kinetic problem in rheology

An alternative formalism to the dumbbell kinetic problem is proposed which is believed to be more fundamental than the classical Liouville one. The new formulation provides a logical approach to non-conservative systems and systems with varying frictional coefficients. A non-linear dumbbell with internal viscosity and varying frictional factor for the beads is examined. It is proved that the centre of gravity of the dumbbell must move affinely with the solvent continuum. A useful class of approximation is suggested to reduce the constitutive equation to an explicit form. The response of the model is computed for a number of flow fields. For shear flows, the introduction of the internal viscosity results in a shear-thinning phenomenon. The onset of non-Newtonian behaviour occurs at the correct order of magnitude of the dimensionless shear rate. Also, a negative second normal stress difference is found which varies with shear rate. In an oscillatory shear flow, the internal viscosity leads to a finite limiting value of the dynamic viscosity at high frequencies. In elongational flow the effects of the varying frictional coefficient dominate that of the internal viscosity. Interesting phenomena include the presence of a hysteresis loop and the ability of the dumbbell to maintain an extended configuration at moderate elongational rates. Clearly, these are relevant in turbulent drag reduction applications. The model has sufficient merits to deserve more investigation.     

Reciprocity in elastic media with rheology

The classic reciprocity theorem of Betti (Nuovo Cimento 2(VII–VIII):69–97, 1872) for perfectly elasto-static media has been extended to the dynamic fields by Graffi (Ann. Mat. Pura Appl. IV(18):173–200, 1939) for isotropic media with variable density. In this note the theorem is extended to a wide class of media with rheology.     

The Schwinger-Dyson approximation in elastic dumbbell models for flexible polymers

We present a set of nonlinear integral equations as an approximation scheme for the correlation and response functions in an elastic dumbbell model for flexible macromolecules suspended in a Newtonian fluid. Material functions such as the viscosity, the normal stress coefficients and the end-to-end distance as well as two-time-correlation functions are calculated. The results are in excellent agreement with exact results where they are available.     

A study of conformation-dependent friction in a dumbbell model for dilute solutions

We consider predictions of rheological behavior in a variety of shear and extensional flows for an elastic dumbbell model with a nonlinear spring, and conformation-dependent hydrodynamic properties. The latter include a conformation-dependent anisotropic bead friction coefficient, and a related conformation-dependent degree of inefficiency for rotation in straining flows. With these features, the dumbbell exhibits hydrodynamic behavior consistent with a particle of finite axis ratio over the complete set of possible polymer conformations, from random-coil to a fully extended thread-like configuration. The predicted rheological behavior in shear flow is improved, relative to data, by the inclusion of anisotropy and strain-inefficiency in the frictional properties of the model, while other desirable features such as the sudden onset of fully extended states at a critical value of the velocity gradient, the presence of a hysteresis-loop in end-to-end dimension as a function of the velocity gradient, and the correlation of end-to-end distance (or birefringence) with the eigenvalue of the velocity gradient tensor for a wide variety of two-dimensional flows, are maintained.     

Unilateral transmission joints with friction elastic rings

Summary Frictional couplings, transmitting torque in only one direction, are studied in order to find the maximum value of the transmissible torque. For this purpose, the elastic equilibrium configuration of the helical spring interposed between driver and driven shafts is analyzed, and relations between the geometrical and dynamical parameters are found. The results are obtained through numerical calculations and tabulated for different values of the influencing physical factors.

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Sommario Sono studiati accoppiamenti per attrito atti a trasmettere coppie torcenti in una sola direzione; per tali accoppiamenti si determina il valor massimo della coppia trasmissibile.A tale scopo, viene analizzata la configurazione di equilibrio elastico della molla elicoidale interposta tra gli alberi motore e condotto, e sono ottenute le relazioni che collegano parametri geometrici e dinamici.I risultati sono ottenuti attraverso calcoli numerici, e sono tabulati per differenti valori dei fattori fisici influenti sul funzionamento del giunto.

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This research was presented at the Euromech 22 Colloquium on Mechanisms, held at Newcastle (U.K.) in September, 1970.     

The rheology of polymer solution elastic strands in extensional flow

We apply micro-oscillatory cross-slot extensional flow to a semi-dilute poly(ethylene oxide) solution. Micro-particle image velocimetry (μPIV) is used to probe the real local flow field. Extreme flow perturbation is observed, where birefringent strands of extended polymer originate from the stagnation point. This coincides with a large increase in the extensional viscosity. The combination of stagnation point flow and μPIV enables us to investigate directly the stress and strain rates in the strand and so determine the true extensional viscosity of the localised strand alone. The Trouton ratio in the strand is found to be ~4000, amongst the highest values of Trouton ratio ever reported. Consideration of the flow in the exit channels surrounding the highly elastic strand suggests a maximum limit for the pressure drop across the device and the apparent extensional viscosity. This has implications for the understanding of high Deborah number extensional thinning reported in other stagnation point flow situations.     

Shakedown in elastic contact problems with Coulomb friction

Elastic systems with frictional interfaces subjected to periodic loading are sometimes predicted to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan’s theorem might apply to them – i.e., that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.In this paper, we prove this result for ‘complete’ contact problems in the discrete formulation (i) for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions and (ii) for certain two-dimensional problems in which the friction coefficient at each node is less than a certain critical value. We also present counter-examples for all systems that do not fall into these categories, thus giving a definitive statement of the conditions under which Melan’s theorem can be used to predict whether such a system will shake down.     

Effect of friction in wedging of elastic solids

In this paper the contact problem for an elastic wedge of arbitrary angle is considered. It is assumed that the external load is applied to the medium through a rigid wedge and the coefficient of friction between the loading wedge and the elastic solid is constant. The problem is reduced to a singular integral equation of the second kind with the contact pressure as the unknown function. An effective numerical solution of the integral equation is described and the results of three examples are presented. The comparison of these results with those obtained from the frictionless wedge problem indicates that generally friction has the tendency of reducing the peak values of the stress intensity factors calculated at the wedge apex and at the end points of the contact area.This work was supported by NASA-Laugley under the Grant NGR 39-007-011 and by NSF under the Grant GK-42771X.     

The non-isothermal elastic dumbbell: A model for the thermal conductivity of a polymer solution

In a flowing polymeric liquid, molecular orientation will give rise to anisotropic conduction of heat. In this paper, a theory is presented relating the thermal conductivity tensor to the deformation history of the fluid. The basis of this theory is formed by the Hookean dumbbell. It is shown that the anisotropy of the thermal conductivity is proportional to the polymer contribution to the extra-stress tensor. This stress-thermal law makes it relatively simple to incorporate anisotropic heat conduction into the numerical simulation of a flowing polymeric liquid.     

Micromechanical analysis of friction anisotropy in rough elastic contacts

Computational contact homogenization approach is applied to study friction anisotropy resulting from asperity interaction in elastic contacts. Contact of rough surfaces with anisotropic roughness is considered with asperity contact at the micro scale being governed by the isotropic Coulomb friction model. Application of a micro-to-macro scale transition scheme yields a macroscopic friction model with orientation- and pressure-dependent macroscopic friction coefficient. The macroscopic slip rule is found to exhibit a weak non-associativity in the tangential plane, although the slip rule at the microscale is associated in the tangential plane. Counterintuitive effects are observed for compressible materials, in particular, for auxetic materials.     

Determination of rough-surface skin friction coefficients from wake profile measurements

A technique for determining the skin friction coefficients from wake profile measurements is described, and is applied to symmetric turbine airfoils with rough surfaces, which operate in a compressible, high-speed flow environment. The procedure involves the measurement of profiles of streamwise momentum in the wakes which form downstream of different airfoils with different levels of surface roughness. Different physical phenomena which affect such wake profiles are discussed and related to different effects, such as surface roughness, form drag, flow separation zones, and laminar-to-turbulent transition. With the same inlet experimental condition for each case, overall skin friction coefficients for the rough airfoils are determined. Resulting values increase considerably as the magnitude of equivalent sandgrain roughness size increases.List of symbols A test airfoil surface area - A _i test section inlet area - A _e test section exit area - c chord length of airfoil - C _f/2 skin friction coefficient - (C _f/2)_smooth skin friction coefficient for smooth airfoil - (C _f/2)_rough skin friction coefficient for rough airfoil - F _s force from surface shear stress - F _p force from form drag due to airfoil blockage and separated flow - F _w force imposed by the top and bottom test section walls - F _s,smooth force from surface shear stress for smooth airfoil - F _s,rough force from surface shear stress for rough airfoil - h height of test section - k roughness height - k _s equivalent sand grain roughness - L total length of airfoil surface from leading edge to trailing edge - p airfoil passage effective pitch - P _o stagnation pressure - P _oe exit local stagnation pressure - P _oe, exit freestream stagnation pressure - P _oi inlet stagnation pressure - P _s static pressure - P _se exit static pressure - s distance along airfoil surface from leading edge - u local streamwise velocity - u _i local streamwise velocity at test section inlet - u _e local streamwise velocity at test section exit - u local freestream streamwise velocity at test section exit - w width of test section - x linear distance along airfoil centerline from airfoil leading edge - y normal coordinate measured from airfoil centerline Greek symbols ratio of specific heats - _s roughness parameter - _i local static air density at test section inlet - _e local static air density at test section exit - local static air density in freestream at test section exit     

The sliding interface crack with friction between elastic and rigid bodies

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair (b,a) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G_tip, defined at each crack tip by the J_ε-integral along a semi-circular path, centered at the crack tip, with vanishing radius ε, vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J-integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.     

Elastic perfectly plastic structures with temperature dependent elastic coefficients

We study the evolution of elastic perfectly plastic structures where the elastic coefficients depend on temperature, as they are subjected to classical loading and given variation of the temperature field. We prove variational theorems for the instantaneous fields of velocities and stress rates, and establish the generalized differential equation for the evolution of the stress field. To cite this article: B. Halphen, C. R. Mecanique 333 (2005).     

Hertz problem for elastic solids with variable poisson coefficients

International Applied Mechanics -     

Reflection and transmission coefficients for three-dimensional plane waves in elastic media

Problems for transient line and point load sources in a multilayered elastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The inverse Laplace transform of the ray integral can be found in closed form by applying the Cagniard method. For problems in the Cartesian coordinates for line load sources emitting cylindrical waves consistent with either the plane strain conditions or the antiplane strain conditions and for problems in the cylindrical coordinates for axisymmetric and asymmetric point load sources emanating spherical waves, it is well known that: (1) the system of incident, reflected, and transmitted cylindrical/spherical waves at an interface separating two dissimilar media can be divided into two independent of each other, if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spherical waves can be constructed by linear superposition of the Laplace-transformed plane P and SV waves, or the plane SH waves, and (3) the potential reflection and transmission coefficients for the plane P, SV, and S H waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordinates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the displacement potentials are coupled in the boundary conditions at the interface, the total wave motion at the interface is composed of the coupled spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superposition of the three-dimensional coupled plane P and S waves. Since such a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systematically these coefficient formulas.     

Shear rheology of polymer solutions near the critical condition for elastic instability

The use of constant viscosity, highly elastic polymer solutions, so called Boger fluids, has been remarkably successful in elucidating the behavior of polymeric materials under flowing conditions. However, the behavior of these fluids is still complicated by many different physical processes occurring within a narrow window of observation time and applied shear rate. In this study, we investigate the long-time shear behavior of an ideal Boger fluid: a well characterized, athermal, dilute, binary solution of high molecular weight polystyrene in oligomeric polystyrene. Rheological measurements show that under an applied steady shear flow, this family of polymer solutions undergoes a transient decay of normal stresses on a timescale much longer than the polymer molecule's relaxation time. Rheological and flow visualization results demonstrate that the observed phenomenon is not caused by polymer degradation, phase separation, viscous heating, or secondary flows from elastic instabilities. Although the timescale is much shorter than that associated with polymer migration in the same solutions (MacDonald and Muller, 1996), the appearance of this phenomenon only at the rates where migration has been observed suggests that it may be a prerequisite for observing migration. In addition, we note that through sufficient preshearing of the sample, the normal stress decrease suppresses the elastic instability. These results show that there is considerable uncertainty in choosing the appropriate measure of the fluid relaxation time for consistently modeling the critical condition for the elastic instability, the decay of normal stresses, and the migration of polymer species.     

Self-excited vibration caused by dry friction between two elastic structures

Based on our previous work^[1], self-excited vibration of a multi-degree-of-freedom system caused by dry friction between two elastic structures is investigated using the Chinese cultural relic dragon washbasin as an example. Some new characteristics of the self-excited vibration in this kind of system are found. The conditions under which self-excited vibration occurs at low-order or high-order modes are discussed. Effects of changes in parameters of the system on the self-excited vibration are analyzed. The vibration mechanism of the water droplets spurting phenomenon of the Chinese cultural relic dragon washbasin is further explained. This investigation presents a new idea for modeling the self-excited vibration caused by dry friction interaction between two elastic structures. The project supported by the National Natural Science Foundation of China