A permeameter for unsaturated soil

A permeameter for unsaturated soil was developed by observing the way in which pore water recovers hydrostatic equilibrium. It works like an hour glass that is turned upside-down everytime the state of reference (or hydrostatic equilibrium) is reached. The hydraulic conductivity is deduced from the curves of evolution of pore-water pressure and from the distribution of partial density of water at hydrostatic equilibrium. Roman Letters a is defined by (10), kg m^–3 - A _n coefficients of the analytic solution, kgm^–3 - C ₁, C ₂, C ₃, C ₄ constants and constants of integration - D diffusivity, m² s^-1 - g gravity constant, m s^-2 - g gravity vector field - K hydraulic conductivity defined by (2), m⁵ s^-1 J^-1 - K _w hydraulic conductivity defined by (5), m ^-1 - k permeability - L length of soil sample, m - n integer in (22) - n porosity - p absolute pore water pressure, Pa - p ₀ absolute pore water pressure, Pa - p _a absolute air pressure, Pa - q volume flux or Darcy's velocity, m s^-1 - r exponent defined by (13) - S _w degree of saturation, % - t time variable, sec - u _n , v _n are defined by (22b), (22c) - x(x, y, z) space variable Greek Letters , are defined by (11), (13) - _w dynamic viscosity - water partial density, kg m^–3. It is the ratio of the mass of water to total volume of a representative elementary volume - ₀, _l water partial densities, kgm^–3 - _w density of water, kgm^–3 - _s density of solid particles, kgm^–3 - differences of partial density, kgm^–3 - p differences of water pressure, Pa - pi - , · gradient operator, divergence operator - Laplacian operator - volumetric water content, % - piezometric head, m     

Direct derivation of the two-shaft system balance conditions for the second order reciprocal inertia forces on the V-type eight cylinders internal combustion engines with a plane crankshaft

By employing the four shafts balance concept paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a two-shafts balance regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the two-shafts balance concept, successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance regime.     

On various variational principles in nonlinear theory of elasticity

In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu’s monograph [1].     

Variational principle of thermoelectroelasticity and its application to the problem of vibrations of a thin‐wall member

The dynamic behavior of thinwall members manufactured from materials with the pyroelectric effect was studied. A variational formulation of the problem is used, and a variational principle is formulated that differs from the wellknown one. Correct boundaryvalue problems describing the tension, compression, and bending of a thinwall pyroelectric member are constructed using the variational principle and a number of hypotheses on the distribution of the components of physical fields along the width of the member.     

Die Querschnittsverwölbung eines schubdeformierten elastischen Hohlzylinders — Theorie und Experiment

Übersicht Für die durch Tangentialschub hervorgerufene Deformation eines rotationssymmetrischen, inkompressiblen elastischen Körpers wird einerseits eine Elastizitätstheorie zweiter Ordnung entwickelt. Sie führt auf lineare Randwertprobleme zweiter bzw. vierter Ordnung für die Primärverschiebung in Umfangsrichtung und die den Sekundärverschiebungen in der Meridianebene zugeordnete Verschiebungsfunktion. Im Fall eines endlich langen Hohlzylinders kann die Primärdeformation analytisch bestimmt und für die Sekundärdeformation ein äquivalentes Variationsproblem angegeben werden, das einer numerischen Lösung unmittelbar zugänglich ist. Andererseits wurde die Verwölbung des spannungsfreien Endquerschnitts eines schubbeanspruchten Hohlzylinders experimentell ermittelt. Durch Vergleich mit der theoretischen Oberflächenform läßt sich das Verhältnis der beiden maßgeblichen elastischen Stoffkonstanten bestimmen.

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The deformation of a nonlinear-elastic hollow cylinder under tangential shear — theory and experiment

Summary A second-order elasticity theory is developed for the axially symmetric deformations of an incompressible isotropic elastic material due to tangential shear. It leads to second-order and fourth-order linear boundary value problems for the primary displacements in azimuthal direction and for the displacement function of the secondary deformations, resp., which take place within a plane through the axis of symmetry. In the case of a hollow cylinder of finite length, an analytic solution for the primary deformations exists, and for the secondary deformations an equivalent variational problem can be formulated and solved numerically. On the other hand, the shape of the free surface of the hollow cylinder was found out by experiments. Comparing the data with the theoretical results, the ratio of the two relevant elastic constants can be determined.

     

Balance relations for classical mixtures containing a moving non-material surface with application to phase transitions

This work is concerned with an extension of classical mixture theory to the case in which the mixture contains an evolving non-material surface on which the constituents may interact, as well as be created and/or annihilated. The formulation of constituent and mixture jump balance relations on/across such a non-material surface proceed by analogy with the standard volume or bulk constituent and mixture balance relations. On this basis, we derive various forms of the constituent mass, momentum, energy and entropy balances assuming (1), that the constituent in question is present on both sides of the moving, non-material surface, and (2), that it is created or annihilated on this surface, as would be the case in a phase transition. In particular, we apply the latter model to the transition between cold and temperate ice found in polythermal ice masses, obtaining in the process the conditions under which melting or freezing takes place at this boundary. On a more general level, one of the most interesting aspects of this formulation is that it gives rise to certain combinations of the limits of constituent and mixture volume fields on the moving mixture interface which can be interpreted as the corresponding surface form of these fields, leading to the possibility of exploiting the surface entropy inequality to obtain restrictions on surface constitutive relations.     

Limitation and improvement of PIV

In this second part of the paper, the Particle Image Distortion (PID) technique is described. It is proposed to overcome the limitations of conventional PIV due to the local deformations u/x, u/y, v/x and v/y in two-dimensional flows. Both simulation and experiment demonstrate that high accuracy and high spatial resolution are possible with this technique. The large time required to compute the cross-correlations, however, limits its wide applications at present.     

Analysis of slip and temperature jump coefficients in a binary gas mixture

The possibility of simplifying the formulas obtained by the Maxwell-Loyalka method for the velocity _u, temperature _T and diffusion _d slip coefficients and the temperature jump coefficient in a binary gas mixture with frozen internal degrees of freedom of the molecules is considered. Special attention is paid to gases not having sharply different physicochemical properties. The formulas are written in a form convenient for use without linearization in the thermal diffusion coefficient. They are systematically analyzed for mixtures of inert gases, N₂, O₂, CO₂, and H₂ at temperatures extending from room temperature to 2500°K. It is shown that for the molecular weight ratios m^* = m₂/m₁ considered the expressions for _u and can be radically simplified. With an error acceptable for practical purposes (up to 10%) it is possible to employ expressions of the same structural form as for a single-component gas: for _u if 1 m^* 6, and for if 1 m^* 3. When 1 m^* 2 the expression for _T can be simplified with a maximum error of 5%. Within the limits of accuracy of the method the expression for _t can be linearized in the thermal diffusion coefficient. An approximate expression convenient for practical calculations is proposed for _d Finally, the , _u, and _T for a single-component polyatomic gas with easy excitation of the internal degrees of freedom of the molecules are similarly analyzed; it is shown that these expressions can be considerably simplified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 152–159, November–December, 1990.     

Various reciprocal theorems and variational principles in the theories of nonlocal micropolar linear elastic mediums

In the first part of our paper, we have extended the concepts of the classical convolution and the convolution scalar product given by I. Hlaváck and presented the concepts of the convolution vector and the convolution vector scalar product, which enable us to extend the initial value as well as the initial-boundary value problems for the equation with the operator coefficients to those for the system of equations with the operator coefficients.In the second part of this paper, based on the concepts of the convolution vector and the convolution vector scalar product, two fundamental types of reciprocal theorems of the non-local micropolar linear elastodynamics for inhomogeneous and anisotropic solids are derived.In the third part of this paper, based on the concepts and results in the first and second parts as well as the Lagrange multiplies method which is presented by W. Z. Chien, four main types of variational principles are given for the nonlocal micropolar linear elastodynamics for inhomogeneous and anisotropic solids. These are the counterparts of the variational principles of Hu-Washizu type, Hellinger-Reissner type and Gurtin type in classical elasticity as well as Hlaváck type and Iesan type in local micropolar and nonlocal elasticity. Finally, we have proved the equivalence of the last two main variational principles which are given in this paper.     

On motions which preserve ellipsoidal holes

Let be an ellipsoid in ³ contained in a region . Suppose one body occupies the region – in a certain stress-free reference configuration while a second body, the inclusion, occupies the region in a stress-free reference configuration. Assume the inclusion is free to slip at . Now suppose that by changing some variable such as the temperature, pressure, humidity, etc., we cause the trivial deformation y(x)=x of the inclusion to become unstable relative to some other deformation. For example, the inclusion may be made out of such a material that if it were removed from the body, it would suddenly change shape to another stress-free configuration specified by a deformation y=Fx, F F=C, C being a fixed tensor characteristic of the material, at a certain temperature. However, with an appropriate material model for the surrounding body, we expect it will resist the transformation, and both body and inclusion will end up stressed.In a recent paper, Mura and Furuhashi [1] find the following unexpected result within the context of infinitesimal deformations: certain homogeneous deformations of the ellipsoid which take it to a stress-free configuration also leave the surrounding body stress-free. These are essentially homogeneous, infinitesimal deformations which preserve ellipsoidal holes. In this paper, we find all finite homogeneous deformations and motions which preserve ellipsoidal holes.     

Second-Order Estimates for the Macroscopic Response and Loss of Ellipticity in Porous Rubbers at Large Deformations

This work presents the application of a recently proposed second-order homogenization method (Ponte Castañeda, 2002) to generate estimates for effective behavior and loss of ellipticity in hyperelastic porous materials with random microstructures that are subjected to finite deformations. The main concept behind the method is the introduction of an optimally selected linear thermoelastic comparison composite, which can then be used to convert available linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. In this paper, explicit results are provided for the case where the matrix is taken to be isotropic and strongly elliptic. In spite of the strong ellipticity of the matrix phase, the homogenized second-order estimates for the overall behavior are found to lose ellipticity at sufficiently large compressive deformations corresponding to the possible development of shear band-type instabilities (Abeyaratne and Triantafyllidis, 1984). The reasons for this result have been linked to the evolution of the microstructure, which, under appropriate loading conditions, can induce geometric softening leading to overall loss of ellipticity. Furthermore, the second-order homogenization method has the merit that it recovers the exact evolution of the porosity under a finite-deformation history in the limit of incompressible behavior for the matrix. Mathematics Subject Classifications (2000) 49S05, 74B20, 74Q15, 74Q05.     

The effect of a singular perturbation on nonconvex variational problems

We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter 0. The technique utilizes the notion of -convergence of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

General viscosity-composition relationship for dispersions,solutions and binary liquid systems

An equation for the viscosity of a mixture of two imaginary Newtonian liquids is derived. In the derivation the mathematical assumption is used that the effective activation energy for viscous flow of a binary liquid mixture is a linear combination of the reciprocals of the activation energy of the components. It contains two dependent fitting constants and has the same structure as the Mooney equation for dispersions of spherical solid particles, the Huggins equation for polymer solutions and is identical to an equation by Hoffmann and Rother, when written in the variables that the last authors used.As a consequence it can be shown that the viscosity of binary liquid mixtures, liquid resion solutions, dispersions of solid spherical particles and polymer solutions can be described very well by one and the same equation, up to the highest concentrations.It has further been found that the viscosity of dispersions of non-spherical particles, solutions of solids in organic solvents and solutions of electrolytes and non-electrolytes in water can also be described by this formula. The equation permits the construction of a straight line on which all liquids can be plotted.An algebraic analysis of the equation shows that each series of viscosity composition data can be placed in one of three rheological groups independent of the type of fraction that is used to characterize the composition.Seventy-four binary systems, covering a wide range of liquids have been used to show the applicability of the developed equation.It has been found that in most cases the data are best described by splitting them into two regions, each with its own set of dependent constants. General symbol for the fraction or concentration of the component with the higher viscosity determining the composition of a binary mixture [—] - _v Volume fraction of the component with the higher viscosity [—] - _w Weight fraction of the component with the higher viscosity [—] - _mw Molecular weight fraction of the component with the higher viscosity [—] - c Concentration of the component with the higher viscosity [g/cm³] - E ₂,E ₁,E Activation energy for viscous flow referring to the component with the higher viscosity, the lower viscosity and the viscosity of the binary mixtures, respectively [J] - ₂, ₁, Experimental parameter (with the dimension of energy) referring to the component with the higher viscosity, the component with the lower viscosity and to the binary mixtures, respectively [J] - ₁, ₂ Viscosity of the component with the lower and the higher viscosity, respectively [Pa · s] - Viscosity of a binary mixture [Pa · s] - [] The usual intrinsic viscosity of the component with the highest viscosity [cm³/g] - _r / ₁ [—] - _{sp r} – 1 [—] - [] -intrinsic viscosity [—] - [] _v Volume intrinsic viscosity [—] - [] _w Weight intrinsic viscosity [—] - [] _c Concentration intrinsic viscosity, identical to [] [cm³/g] - T _e Temperature at which the two liquids have the same viscosity [K] - _e Viscosity at temperatureT _e [Pa · s] - P ₁,P ₂ Density of the component with the lower and the higher viscosity, respectively - R Gas constant [J · Mol^–1 · K^–1]     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Die theorie der nichtmetrischen Spannungen in Kristallen

Summary Inhomogeneous quasiplastic deformations give rise to internal stresses in a solid body. These non-metric stresses are investigated in the stationary case by means of non-linear continuum theory based on a non-metric, non-Euclidean geometry. The theory is developed for crystals. It is also applicable to polycrystalline and amorphous bodies.The quasiplastically deformed crystal is geometrically described by an elastic metric tensor and by a lattice connexion, which is considered to be non-metric with respect to the elastic metric. The covariant derivative of the elastic metric tensor, based on the lattice connexion, is regarded as source-function of non-metric stresses. Source functions for thermal and magnetostrictive stresses are explicitly established.It is a well known fact that non-metric stresses may be included into the linear continuum theory of dislocations if the concept of quasidislocation-density is used. Generalizing these ideas, it is shown that, within non-linear continuum theory, in addition to quasidislocation-density the concept of quasidisclination-density is absolutely necessary. With the help of these quantities a complete analogy between non-metric stresses and stresses caused by crystal dislocations and crystal disclinations may be established. A unified non-linear theory for quasiplastic deformations, crystal dislocations and crystal disclinations is developed. It turns out that in general non-metric stresses cannot be compensated completely by dislocation movements.Zusammenfassung Im Festkörper führen inhomogene, quasiplastische Deformationen zu inneren Spannungen. Diese nichtmetrischen Spannungen werden für den stationären Fall im Rahmen einer nichtlinearen Kontinuumstheorie untersucht. Der Theorie liegt eine nichtmetrische, nichteuklidische Geometrie zugrunde. Die Theorie wird für Einkristalle entwickelt. Sie gilt aber auch für vielkristalline und für amorphe Körper.Der quasiplastisch verzerrte Kristall wird geometrisch mit Hilfe einer Gitterkonnexion und eines elastischen Metriktensors beschrieben, wobei die Gitterkonnexion nichtmetrisch ist bezüglich der elastischen Metrik. Die kovariante Ableitung des elastischen Metriktensors bezüglich der Gitterkonnexion ist die Quellenfunktion der nichtmetrischen Spannungen. — Die Quellenfunktionen der Temperaturspannungen und der magnetostriktiven Spannungen werden explizit angegeben.Es ist seit langem bekannt, daß die nichtmetrischen Spannungen im Rahmen einer linearen Theorie mit Hilfe des Begriffs Quasiversetzungsdichte in die Kontinuumstheorie der Versetzungen einbezogen werden können. Durch eine Verallgemeinerung dieser Gedankengänge wird gezeigt, daß in der nichtlinearen Kontinuumstheorie außer der Quasiversetzungsdichte auch eine Quasidisklinationsdichte eingeführt werden muß. Mit Hilfe dieser beiden Begriffe erreicht man erne vollkommene Analogie der nichtmetrischen Spannungen zu den von Kristallversetzungen und Kristalldisklinationen verursachten Spannungen. Es wird eine einheitliche, nichtlineare Kontinuumstheorie der quasiplastischen Deformationen, der Kristallversetzungen und der Kristalldisklinationen angegeben. Man kann zeigen, daß nichtmetrische Spannungen i. allg. durch Versetzungsbewegungen nicht vollständig abgebaut werden können. Vorgelegt von J. Meixner Herrn Prof. Dr. A. Seeger danke ich für die verständnisvolle Förderung dieser Arbeit. Herrn Dr. C. Teodosiu möchte ich für zahlreiche nützliche Diskussionen danken.

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On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Historical deformation analysis of elastoplastic structures as a parametric linear complementarity problem

Summary Discrete models of elastoplastic beams and frames are considered. The plastic deformability laws (bending moments versus plastic rotations) are piecewise linearized. Plastic deformations are sought as they develop along a proportional loading process. The historical analysis in this sense, is shown to be ameneable to the solution of a parametric linear complementarity problem. Recent mathematical results on this problem, due to R. W. Cottle, are used to obtain numerical solutions. Extensions are pointed out to stepwise proportional loading paths, allowing for the irreversible nature of plastic deformations.

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Sommario Si studiano modelli discreti di telai elastoplastici. I legami tra momento flettente e rotazione plastica sono lineari a tratti. Si determina l'intera evoluzione delle deformazioni plastiche al crescere proporzionale e monotono dei carichi, formulando e risolvendo un problema parametrico lineare di complementarietà. La tecnica numerica alquanto efficiente adottata, è fornita da recenti risultati di R. W. Cottle. Il metodo è esteso a processi di caricamento proporzionali a stadi, tenendo conto, tra stadio e stadio, della irreversibilità della deformazione plastica.

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All other symbols are defined when used for the first time.     

Description of a Flow of a Gas‐Condensate Mixture in an Axisymmetric Capillary Tube by the Density‐Functional Method

An isothermal flow of a twophase multicomponent mixture through a smalldiameter capillary tube is examined by the densityfunctional method. For low ratios of the characteristic radius of the capillary to its length, a general form of the dominating term in the asymptotic solution is found. An improved version of the law of mixture transfer is obtained. The form of possible corrections to the Darcy law for the filtration rates of the phases is discussed.     

Comparison of a new internal viscosity model with other constrained-connector theories of dilute polymer solution rheology

Complex viscosity ^* = -i predictions of the Dasbach-Manke-Williams (DMW) internal viscosity (IV) model for dilute polymer solutions, which employs a mathematically rigorous formulation of the IV forces, are examined in the limit of infinite IV over the full range of frequency number of submolecules N, and hydrodynamic interaction h ^*. Although the DMW model employs linear entropic spring forces, infinite IV makes the submolecules rigid by suppressing spring deformations, thereby emulating the dynamics of a freely jointed chain of rigid links. The DMW () and () predictions are in close agreement with results for true freely jointed chain models obtained by Hassager (1974) and Fixman and Kovac (1974 a, b) with far more complicated formalisms. The infinite-frequency dynamic viscosity predicted by the DMW infinite-IV model is also found to be in remarkable agreement with the calculations of Doi et al. (1975). In contrast to the other freely jointed chain models cited above, however, the DMW model yields a simple closed-form solution for complex viscosity expressed in terms of Rouse-Zimm relaxation times.