Index of content:
Volume 10, Issue 1, March 1966

Elasticity in Steady Flow
View Description Hide DescriptionIn rheology the use of the concepts of elasticity—or mechanical stored energy—or recoverable shear in steady flow has been used for some time, but by no means universally accepted. In this paper, the foundations of the statistical mechanics of polymer solutions, as developed by Kuhn, Rouse, and Zimm, together with numerous others, are reviewed for the purpose of developing the basic ideas of elasticity and birefringence in steady flow and investigating what assumptions and limitations the present theory has. The quantitative relations between a number of mathematical theories are tabulated and a survey of the experimental results is given. The questions still unsolved are formulated. Especially the correlation among viscosity, flowbirefringence, dynamic properties, and normal stress measurements is emphasized.

Mechanical and Optical Characterization of Plasticized Polyvinyl Chloride
View Description Hide DescriptionMechanical and birefringent properties of plasticized polyvinyl ehloride in the glass transition region were determined by means of quasistatic tests at different temperatures and sinusoidal oscillation tests at room temperature. In the former method, mechanical creep and photocreep tests were conducted at temperatures ranging from −80 to 115°F. The method of reduced variables (temperature‐time equivalence principle) was used to obtain continuous curves for relaxation modulus and stress fringe value corresponding to room temperature and extending up to 24 decades of time. The resulting curves show a broad transition region for the material. The shift function was plotted versus temperature and was found to be about the same for both mechanical and optical properties. Short‐time (dynamic) properties obtained by the temperature‐time equivalence principle were compared with those obtained at room temperature by direct sinusoidal oscillation tests at different frequencies. The complex modulus and complex stress fringe value were determined as functions of frequency and converted to a relaxation modulus and stress fringe value as functions of time by means of an approximate interrelationship. The validity of the temperature‐time equivalence principle in this case was proven by the good agreement between these results and those obtained from creep tests at different temperatures.

Comments on the Experiments of Bodner and of Fitzgerald and Woodward on Laterally Vibrating Beams under Axial Tension
View Description Hide DescriptionBodner and Fitzgerald and Woodward have described dynamic tests on beams of aluminum alloy and polytetrafluoroethylene, respectively, in which the laterally vibrating bars were subjected to tension by coaxial wires. In both cases, the direct effect of the wires was disregarded. This paper presents an analysis of the wire‐bar systems, based on the theories of vibrating bars and strings. The calculations show that resonance patterns generally resembling those found experimentally should be expected.

Thermistor Analog Study of Dynamic Shear in an Ideal Viscous Material
View Description Hide DescriptionAs part of a continuing study of the mechanical behavior of materials with temperature‐dependent properties, dynamic responses of an ideal viscousmaterial are considered. An electric analog circuit containing thermistors and ideal inductors is presented. The characteristics of this circuit simulate the highly nonlinear characteristics of the model of the material. The experiments show how the conversion of mechanical work to heat can lead to stress and time‐dependent apparent viscosities and to thermal instability in velocity‐driven Couette flow. When the boundary velocity is sinusoidal, the heating can lead to the appearance of a pseudo elasticity as well as to thermal instability. The thermal stability of the system is shown to depend on the frequency as well as the amplitude of the excitation. A group of similarity parameters are described which are ratios of characteristic times for heating, velocity diffusion,thermal diffusion, and loading. These provide a description of the model mechanical experiment and make the connection with the electrical analog.

Time‐Dependent Concentrated Surface Load Moving with Diminishing Velocity on a Viscoelastic Half Space
View Description Hide DescriptionThe vertical and horizontal displacements are obtained for a time‐dependent moving load on a viscoelastic half space. The load is assumed to increase exponentially with time and its velocity to decrease exponentially with time. The problem is solved as a quasistatic problem since the velocity of the load is small compared to stress wave velocities in the half space. The half space is characterized in shear by the four‐parameter model and is assumed to be elastic in its volumetric behavior. The elasticviscoelastic analogy is used in the solution of the problem.

Theoretical and Experimental Studies on Temperature Dependence of the Mechanical Behavior of Polymer Solids by a Creep Recovery Method. Part 2. Penetration of a Steel Ball
View Description Hide DescriptionA general theory of linear viscoelasticity under complex stress is analysed by utilizing the Laplace transformation and is applied to Hertz's contact problem for measuring the creep recovery in the penetration of a steel ball. As in the similar method of uniaxial compression, the temperature dependence of characteristic relaxation times can be computed from the experimental results on (residual depth of penetration creep recovery) ∼θ (temperature) curves by using the analytical results. The temperature dependence is in good agreement with that obtained from the results in uniaxial compression. The method using the RN (Rockwell Number) ∼θ curves obtained at a constant rate of temperature rise by a conventional Rockwell testing machine, is recommended as the simplest method and may be successfully used for various polymer solids. The effect of differences in molecular structure on the temperature dependence of mechanical behavior can be demonstrated clearly by this method. The results on thermosetting laminates are also presented. Further, the merits of this creep recovery are discussed.

The State of Stress and Strain in an Orthotropic Heterogeneous Viscoelastic Cylinder with Burning Inner Boundary
View Description Hide DescriptionThe analysis of stress and strain in an orthotropic heterogeneous linear viscoelastic cylinder which has a burning inner boundary and which is subjected to a time‐dependent internal pressure, has been worked out. The assumptions of isothermal conditions and infinitesimal strains are consistently made and the conditions of a quasistatic plane strain are adopted. Numerical solution for an orthotropic heterogeneous viscoelastic cylinder is established and which was anticipated to be quite stable. The established solution can be reduced to that of a homogeneous orthotropic cylinder. As special cases from the general problem, solutions for an isotropic compressible cylinder and an isotropic incompressible cylinder can also be derived. Numerical examples have been worked out to illustrate the influence of anisotropy, heterogeneity, and viscosity on the deformation of a burning cylindrical grain in a solid propellant rocket. The results for both short‐time and long‐time behavior are presented for a relaxation function of a form similar to that obtainable from a seven‐parameter Maxwell model which has been shown to represent the rheological behavior of a filled elastomer fairly well.

Correlation of Shear Behavior of Solutions of Polyisobutylene
View Description Hide DescriptionNew data are presented for the shear behavior of solutions of polyisobutylene in toluene at 25°C. Five samples (viscosity‐average molecular weight range of to ) were studied at shear rates from 1.0 to Both Couette and capillary viscometers were used in the investigation. The results are correlated by a standard reduced variable plot and compared with earlier data. A method for predicting the viscosity of straight chain polymer solutions from minimal data is discussed.

Moderately Concentrated Polymer Solutions: Correlation of Flow Data by Simple Models
View Description Hide DescriptionA mathematical model was proposed recently which appears to correlate the flow curves for several polymers in aqueous solution over a wide range of shear rates and concentrations. New data for several nonaqueous systems as well as data from literature sources support the idea that the same equation can be applied to nonaqueous solutions. The equation relates relative viscosity at shear rate and shear stress τ to two parameters, the zero‐shear viscosity and the power input at the inflection point B as follows: where is the error function of q. The parameter B appears to be a function of intrinsic viscosity but is not sensitive to polymer type or concentration. A comparison is made which indicates that the proposed model has advantages over other two‐parameter flowequations for real polymer systems where concentration is varied.

Hypothesis on a Certain Flow Instability in Polymer Melts
View Description Hide DescriptionApplication is made of Pao's theory for viscoelasticflow together with measurements of the relaxation spectrum to analyze the flow behavior of polymer melts. One result for linear polyethylene is that over a limited range of shear rate, the shear stress becomes a many‐valued function of shear rate. This behavior arises as a natural consequence of the viscoelastic nature of polymers and appears to unify many previous experimental observations of the behavior of polymer melts at high shear rates. The theory predicts an instability in the flow curve of linear polyethylene. It predicts that this discontinuity occurs at different values depending on whether the shear stress is increasing or decreasing. The recoverable strain is predicted to pass through a maximum and the velocity of a material particle in the unstable region can have a number of values at different instants in time. One interpretation of this would be that the flow can occur as a “stick‐slip” process. The effect of hydrostatic pressure on the flow curve is also discussed. All of the results of this analysis have been observed experimentally by others.

An Experimental Appraisal of Viscoelastic Models
View Description Hide DescriptionEleven differential and nine integral rheological models for viscoelastic fluids are tabulated in a uniform notation. Included among these are several very recently published models and three previously unpublished models. Representative experimental data are presented and used in qualitative and quantitative evaluations of the models. Comparisons between the models and the experimental data are summarized in a table where a six‐point rating scheme is used to assess the ability of several models to quantitatively describe seven material functions. The table indicates which models show promise for future work.

Elementary Flows of Anisotropic Fluids
View Description Hide DescriptionThe Ericksen theory of anisotropic fluids is examined with respect to its usefulness for describing the constitutive properties of real fluids. It is found that except in the limit of low shear rates, anisotropic effects may not be observed in simple laminar shearing flows or in helical flows, in both of which the behavior of these materials is indistinguishable from the predictions of the isotropic theory of simple fluids, and such flows do not suffice to determine uniquely all the material functions needed to describe other configurations. In unsteady flows, the theory as presently developed is limited to applications in which the fluid time constant is much smaller than the characteristic time of the flow field, i.e., to flows for which the Deborah number is sufficiently small. In these, the theory of simple fluids again appears to suffice with but the same exception.

Concentration Redistribution Phenomena in the Shear Flow of Monolayers of Suspended Particles
View Description Hide DescriptionThe full range of phenomena occurring in the flow of suspensions remains still unexplored. This paper describes phenomena observed in a neutrally buoyant suspension consisting of a single layer (a monolayer) of rigid spherical particles, flowing through a 220‐cm long rectangular channel of high‐aspect ratio (1:16) with its minor dimension 1.6 times larger than the particle diameter. The flow was investigated over a range of particle Reynolds numbers between 101 and 407, and of concentrations between 1.7 and 5%. Concentration distributions obtained from a detailed analysis of direct photographic observations show the existence of four consecutive regions as the suspension moves down the channel: (a) an initial region, close to the channel entrance, characterized by essentially uniform particle distribution. (b) A region in which the particles move away from the wall forming a particle‐free layer next to it. The core containing the particles does not have a uniform concentration distribution but exhibits marked concentration peaks at its outer margin. (c) A region in which the core has narrowed, the peaks have merged, and the concentration has become uniform. (d) A region of instability characterized by the onset of waves in the configuration of the core, followed, as the flow progresses downstream, by a breaking up of the core and by the formation of groups of particles separated by axial gaps of suspending liquid. The transitions between these regions are not sharply bounded, each region evolving gradually into the next. The process is accelerated by increases in flow velocity, and decelerated by increases in concentration. The phenomena point to the existence of a hydrodynamic force leading to the formation of the core, and possibly of concentration‐dependent particle interactions opposing such a formation and contributing to the onset of instabilities.

Dynamic Compression of Clay under an Explosive Pulse
View Description Hide DescriptionThe dynamic stress‐strain‐time response of a clay in compression is studied using a high‐speed explosive pulse type of apparatus. The rise times of the applied stress pulses are the order of magnitude of 0.2 msec and the entire test durations are approximately 3.0 msec. Compressive loading is obtained with a piston using the high pressure developed by firing a Springfield 30–06 cartridge into a combustion chamber. Measurements include forces at the top and bottom of the test specimen, deformation of the specimen, and acceleration as well as velocity of the upper platen. Dynamic stress‐strain response exhibits an under‐linear type of nonlinearity for a wide range of consistencies (moisture contents) generally encountered in field applications. Effects of moisture content are conveniently handled in terms of a nondimensional dynamic stress‐strength parameter using the static ultimate unconfined compressive strength as a consistency index. The nonlinear effects are represented in terms of a two‐constant hyperbolic stress‐strain relation. The two hyperbolic constants are measures of the initial tangent modulus and the ultimate strength of the clay. These constants are easily obtained as the intercept and slope of the straight line approximation of the experimental data when presented in transformed hyperbolic coordinates. Because of the multiphase nature of clay, the results may be of value in studying the response characteristics of other composite materials.

Molecular Fractionation in the Flow of Polymeric Fluids
View Description Hide DescriptionThe kinetic theory of liquids has been used to derive two crude models which rationalize the establishment of a molecular weight gradient during the capillary extrusion of polydisperse thermoplastic melts. A molecular weight decrease near the polymer/die wall interface is predicted along with a decay in the degree of extrudate swelling with increasing die length, each model giving equations relating decay rate with die dimensions and shear rate. Polyethylene data indicate some shortcomings in the theory, but both models fit experimental extrudate swelling data well, and appreciably better than an empirical equation of exponential decay. Further in keeping with theory, surface sections of broad molecular weight distribution polyethylene extrudates have lower molecular weights than whole extrudates, the magnitude of the effect increasing with increasing die length. No significant molecular effect was found for a fractionated linear polyethylene. The decay in extrudate swelling was also found to be much greater for broad molecular weight polyethylene samples than for the fractionated polymer. It is suggested that molecular fractionation in capillary flow is a significant contributing cause to the difference in die length variation of melt viscosity and melt elasticity.

Viscoelastic Behavior of Polyethylene in Capillary Flow Expressed with Three Material Functions
View Description Hide DescriptionIt was shown that three material functions are required to relate various phenomena in the capillary flow of polyethylene. The phenomena examined are the entrance pressure loss, laminar flow through a capillary, memory decay in the flow, and extrudate swelling. A unified picture was constructed for the capillary flow and the energy balance was calculated.

Vibrational Measurements with Large Amplitudes
View Description Hide DescriptionUsing a new instrument, the influence of the shear amplitude on the dynamic behavior of polymer solutions up to 700% shear was investigated. At amplitudes <∼100% shear, the dynamic viscosity remained constant, as required by linear viscoelasticity; however, the shear storage modulus decreased considerably. The steady‐state flow curve forms the envelope of curves measured with different amplitudes and frequencies, predominantly in the range of higher rates of shear. The more complete analysis of the results is forthcoming.

Study of Turbulent Flows of Dilute Polymer Solutions in a Couette Viscometer
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Steady Flow of Non‐Newtonian Fluids in a Square Duct
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Some Experimental Observations on the Stored Energy Function for Small Values of Strain Invariants
View Description Hide DescriptionIn order to evaluate the stored energy function W for elastomeric materials, at least biaxial experiments must be made. Two of the biaxial testers developed at the Jet Propulsion Laboratory are an inflated cylinder device and a second biaxial sheet tester. Some preliminary experimental results obtained with these two devices at small values of strain invariants are presented for a polyurethane elastomer. These results are analyzed without making the assumption that the elastomer is incompressible, a fact which modifies the analysis of multiaxial or large‐strain behavior. The results from the biaxial sheet tester are consistent with those obtained from an inflated cylinder of the same material, though experimental difficulties prevent a detailed comparison. Nevertheless, both series of experiments show that all three terms, where are the three strain invariants, depend strongly on the invariants. Moreover, is negative, in agreement with theoretical expectations. Thus, the form of W in the region where is very complex.