General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan's one-dimensional(1D) consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations(PDEs), which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domain. The Crump's method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.     

Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with semi-permeable drainage boundary

The semi-analytical solutions to Fredlund and Hasan's one-dimensional(1 D)consolidation for unsaturated soils with a semi-permeable drainage boundary are presented. Two variables are introduced to transform the two coupled governing equations of pore-water and pore-air pressures into an equivalent set of partial differential equations(PDFs), which are easily solved by the Laplace transform method. Then, the pore-water pressure, pore-air pressure, and soil settlement are obtained in the Laplace domain. The Crump method is adopted to perform the inverse Laplace transform in order to obtain the semi-analytical solutions in the time domain. It is shown that the proposed solutions are more applicable to various types of boundary conditions and agree well with the existing solutions from the literature. Several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with single, double, mixed, and semi-permeable drainage boundaries. The changes in the pore-air and pore-water pressures and the soil settlement with the time factor at different values of the semi-permeable drainage boundary parameters are illustrated. In addition, parametric studies are conducted on the pore-air and pore-water pressures at different ratios(the air permeability coefficient to the water permeability coefficient) and depths.     

Rheological behavior of concentrated suspensions as affected by the dynamics of the mixing process

The rheological characterizations of concentrated suspensions are generally carried out assuming “well-mixed” suspensions. However, the variation of the concentration distributions of the ingredients of the formulation, i.e., the “goodness of mixing”, the size and shape distributions of the particle clusters and the rheological behavior of the suspension all depend on the thermo-mechanical history that the suspension is exposed to during the mixing process. Here, various experimental tools are used for the characterization of the degree of mixedness (concentration distributions) of various ingredients along with the characterization of rheological material functions, wall slip behavior and the maximum packing fraction of a graphite/elastomer suspension. The degree of mixedness values of the ingredients of the suspensions processed using batch and continuous processes and under differing operating conditions were characterized quantitatively using wide-angle X-ray diffraction and thermo gravimetric analysis and were elucidated under the light of the electrical properties of the suspension as affected by the mixing process. Upon achieving better homogeneity of the graphite particles and the binder and decreases in the size and breadth of the size distributions of particle clusters (as inferred from electrical measurements and maximum packing fraction values), the elasticity (storage modulus) and the shear viscosity (magnitude of the complex viscosity from small-amplitude oscillatory shear and shear viscosity from steady torsional and capillary rheometry) of the suspension decreased significantly and the wall slip velocity values increased. These findings demonstrate the intimate relationships that exist between the rheological behavior of concentrated suspensions and the thermo-mechanical history that they are exposed to during the processing stage and suggest that the preparation conditions for suspensions should be carefully selected and well documented to achieve reproducible characterization of rheological material functions.     

A vanishing diffusion process in unsaturated soils

A model for simulating water flow and air flow in unsaturated soils is presented herein. Drainage from a one-dimensional soil column is specifically investigated. A mathematical analysis of the equations reveals that, due to a vanishing diffusion process, the gradient of the water degree of saturation is infinite at the water table. Moreover, this discontinuity propagates at a velocity that satisfies a local non-linear equation involving only the properties of the material. A numerical simulation of this problem serves to confirm the results obtained.     

Rheological modeling of the diffusion process and the interphase of symmetrical bilayers based on PVDF and PMMA with varying molecular weights

The diffusion process in the molten state at a polymer/polymer interface of symmetrical and model bilayers has been investigated using a small-amplitude oscillatory shear measurement. The polymers employed in this study were poly (vinylidene fluoride) (PVDF) and poly (methyl methacrylate) (PMMA) of varying molecular weights and polydispersities. The measurements were conducted in the linear viscoelastic regime (small deformations) so as to decouple the effect of flow from the diffusion. The focus of this paper has been to investigate the effects of healing time, angular frequency (ω), temperature, and molecular weight on the inter-diffusion and the triggered interphase between the neighboring layers. The kinetics of diffusion, based on the evolution of the apparent diffusion coefficient (D _a) versus the healing time, was experimentally obtained. The transition from the non-Fickian to the normal Fickian region for the inter-diffusion at the interface was clearly observed, qualitatively consistent with the reptation model, but it occurred at a critical time greater than the reptation time (τ _rep). In non-Fickian region, effects of frequency and temperature were studied with regard to the ratio of the apparent diffusion coefficient to the self-diffusion coefficient (D _a/D _s). The D _s determined in the Fickian region was found to be consistent with Graessley’s model as well as with the literatures. And the dependence of the D_s on the frequency agreed well with the Doi–Edwards theory, in particular, scaling as $D_{\rm s} \sim \omega^{1/2}$ at ω?>?1/τ _e and $D_{\rm s} \sim \omega^{0}$ at ω?<?1/τ _rep. Our experimental results also confirmed that the dependence of the D _s on the temperature for PMMA and PVDF can be well described by the Arrhenius law. Moreover, blends of PMMAs have been proposed in order to be able to change the $\overline M_\emph{w} $ . The rheological investigations of these corresponding bilayers rendered it possible to monitor the effect of $\overline M_\emph{w} $ on the diffusion process. The obtained results gave $D_{\rm s} \sim \overline M_\emph{w}^{-1}$ , thus corroborating some earlier studies and some experimental results recently reported by Time-Resolved Neutron Reflectivity Measurements. Lastly, the thickness of the interphase and its corresponding viscoelastic properties could be theoretically determined as a function of the healing?time.     

Elasto-plastic consolidation of soil

A theory of the behaviour of an ideal mass of two-phase soil is developed. This theory unites the usually separately considered aspects of analytical soil mechanics of settlement and deformation; time dependent consolidation; and yielding leading to collapse of the mass. The theory assumes the soil to be an elasto-plastic permeable material and a finite element approximation of the resulting equations is derived through the principle of virtual work. Two example problems are treated numerically.     

Rheological behaviour of the fiber-forming mixture viscose-hydroxyethylcellulose

This paper presents the results of a rheological study of the fiber-forming mixture viscose-hydroxyethylcellulose. For describing the behaviour over the large range of shear stresses a four-parameter Meter model was used. The dependence of the parameters of the model are described for technologically important ranges of temperature and concentration of the mixture. The correlation of the parameters is described in the form of the regression equations resulting from the statistical evaluation of planed experiments.     

Rheological behaviour of asphalts

Summary Stiffness moduli of asphalts of different origins and penetration grades were measured at various temperatures by the sliding plate rheometer. The creep curves obtained were best explained on the basis of Burgers mechanical model. For small loading times and at low temperatures, the stiffness modulus curves converged with the horizontal elastic asymptote and showed maximum values of 5 10⁸ pascals. For longer loading times and at high temperatures the stiffness modulus converged with the viscous asymptote. The stiffness moduli calculated from the combined equations ofSchweyer et al. and ofWilliams-Landell-Ferry for temperatures ranging from –10 to 20°C and those obtained experimentally agreed well only at 20°C. For short loading times (< 50 sec) the experimental stiffiness modulus values agree well with those read from theVan Der Poel nomograph.

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Zusammenfassung Mittels eines Gleit-Platten-Rheometers wurden an einer Reihe von Asphaltproben verschiedener Herkunft und mit verschiedenen Penetrationsgraden bei unterschiedlichen Temperaturen Steifheitsmoduln bestimmt. Die erhaltenen Kriechkurven ließen sich am besten mit Hilfe eines rheologischen Modells nachBurgers darstellen. Bei kurzen Belastungszeiten und niedrigen Temperaturen strebten die Steifheitsmodul-Kurven gegen die das elastische Verhalten beschreibende horizontale Asymptote mit Maximalwerten von 5 10⁸ Pa. Bei längeren Belastungszeiten und höheren Temperaturen strebten diese Kurven jedoch gegen die das viskose Verhalten kennzeichnende Asymptote. Die mittels der kombinierten Gleichungen nachSchweyer et al. und nachWilliams-Landel-Ferry für Temperaturen zwischen –10 und 20°C berechneten Steifheitsmoduln stimmten nur für 20°C gut mit den experimentell ermittelten Werten überein. Für kurze Belastungszeiten (<50 sec.) decken sich die experimentellen Steifheitsmoduln recht genau mit den aus dem Nomogramm nachVan Der Poel abgelesenen Werten.

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With 6 figures and 11 tables     

Rheological properties of sputum

Summary Sputum is a mixture of saliva with bronchial secretion which, in its turn, is a mixture of tissue fluid transudate through surface epithelium as well as acid glycoproteins from the specialised secretory cells, the goblet cells of surface epithelium, the serous and mucous cells of the submucosal glands. It has recently been shown (Lamb andReid, 1969) that the acid glycoproteins include several types of sialic acid and sulphate. Since sputum has a low dry weight yield its viscosity probably arises from the nature and interaction of its chemical constituents.Sputum is a non-Newtonian fluid. We have recently studied it over a wide shear rate range using aWeissenberg rheogoniometer with oscillation for lower rates and aFerranti-Shirley viscometer for higher levels. This combination has enabled the pattern of sputum viscosity and elasticity to be established. At very low shear rate a feature has been described for the first time for any biological fluid: it has been found in all sputum samples studied and at the same shear rate, regardless of the disease or of the absolute level of viscosity. It would seem, therefore, to arise from some essential feature of the physico-chemical arrangement of sputum. Elasticity over the same shear rates has also been calculated.The following additional features will be discussed: repeatability of results from different aliquots of the same specimen, time-dependent increase in viscosity, variation between certain diseases and the effect of dehydration.

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Zusammenfassung Sputum (Auswurf) ist eine Mischung von Speichel und Bronchialsekret, das seinerseits eine von dem Oberflächenepithel ausgeschiedene Mischung von Gewebeflüssigkeit ist, die auch saure Glykoproteine enthält, die von spezialisierten Sekretionszellen, den Gobletzellen des Oberflächenepithels, den Serösen- und Mukösenzellen der Subkösendrüsen ausgeschieden werden. Kürzlich wurde gezeigt (Lamb undReid, 1969), daß die sauren Glykoproteine verschiedene Arten von Sialinsäure und Sialinsulfat einschließen. Da Sputum ein geringes Trockengewicht hat, beruht seine Viskosität wahrscheinlich auf der Art und der Wechselwirkung seiner chemischen Bestandteile. Sputum ist eine Nicht-Newtonsche Flüssigkeit. Wir haben neulich über einen weiten Bereich von Schergeschwindigkeiten Untersuchungen durchgeführt; für geringe Geschwindigkeiten und oszillierende Bedingungen wurde einWeissenberg-Rheogoniometer verwendet und für höhere Geschwindigkeiten einFerranti-Shirley-Viskosimeter. Diese Kombination ergab ein Bild der Viskosität und Elastizität von Sputum. Für eine biologische Flüssigkeit wurde zum ersten Mal bei geringer Schubgeschwindigkeit ein Charakteristikum beschrieben: dieses wurde in allen untersuchten Sputum-Proben bei der gleichen Schubgeschwindigkeit gefunden und war unabhängig von Krankheiten oder den absoluten Viskositätswerten. Daraus könnte man schließen, daß dieses wesentliche Charakteristikum durch die physikalisch-chemische Anordnung des Sputum bedingt sei. Ebenfalls wurde die Elastizität über gleiche Schergeschwindigkeiten berechnet.Folgende zusätzliche Gesichtspunkte werden diskutiert: Wiederholbarkeit der Ergebnisse mit verschiedenen Teilen der gleichen Probe, Zeitabhängigkeit der Viskosität, Variation aufgrund verschiedener Krankheiten und Einfluß von Wasserentzug.

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Paper presented to the British Society of Rheology Conference on Rheology in Medicine and Pharmacy, London, April 14–15, 1970.     

A theory of anisotropic consolidation

Summary Within the tensor functions representations constitutive relations are studied for solids with a directional permeability. The overall anisotropy of an originally isotropie skeleton is due to the arrangement of microchannels and it varies with the deformation induced.The laws concerning deformation, fluid flow, couplings between fluid and solid motions as well as those regarding evolution of anisotropy are established.An application of the polynomial representations of tensor functions of scalar, vector and tensor independent variables allows to devise a consistent nonlinear theory of anisotropic consolidation and thus results in the general form of stress-strain-permeability relations.Anisotropic filtration and equations governing the mechanical response of linearly elastic, anisotropic, porous solids are recovered as special cases.

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Zur anisotropen Verdichtung von Werkstoffen

Übersicht Es werden Stoffgleichungen für feste Körper mit richtungsabhängiger Permeabilität untersucht, wobei sich die Anisotropie mit der Verformung ändert. Dabei werden u. a. Gesetze aufgestellt, die die Bewegung von Flüssigkeiten und die Kopplung zwischen der Bewegung flüssiger und fester Körper beschreiben.Anwendung der Polynomdarstellung von Tensorfunktionen ermöglicht es, eine konsistente nichtlineare Theorie anisotroper Verfestigung aufzustellen. Auf diese Weise erhält man allgemeine Beziehungen zwischen Spannungen, Verzerrungen und Permeabilität.

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This paper was presented at the Second Symposium on Inelastic Solids and Structures, Bad Honnef, September 1981

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on leave from the Institute of Fundamental Technological Research IPPT-PAN, Warsaw, Poland     

Rheological aspects of the rejuvenation of aged bitumen

Rheological techniques are used to investigate the rejuvenation of aged bitumen. The thermal transition associated with the collapse of the compact structure constituted by asphaltene is determined by Dynamic Mechanical Thermal Analysis. For aged bitumen, this transition shifts to a higher temperature but when rejuvenating agents are added, the transition returns to its original value. The “rutting factor,” G*/sin δ allows to define the maximum temperature the binder can reach without permanent deformation. The employed rejuvenating agents are suitable because permanent deformation is postponed. Viscosity results reveal that aged bitumen needs a high mixing temperature (>200°C) to behave like a fluid material able to wet, adhere, and envelop aggregates. The addition of rejuvenating agents considerably reduces mixing and compaction temperatures. The mixture of 80% aged bitumen—20% recycled motor oil, obtained exclusively from waste materials is an apt binder that can compete satisfactorily with new 60/70 bitumen.Paper presented at the AERC2005. This paper will be part of a special issue of AERC2005.     

Rheological model of concrete in the dynamic field

Summary The definition of the rheological model of concrete, or uncracked reinforced concrete, in the field of dynamic stresses sufficiently far from breaking, is here attempted; an aim towards which we had already recently directed a series of investigations to clarify the different aspects of experimental behaviour.Even with gaps in detail, the model found seems to correctely interpret the dissipativemechanism of our material, and it is — contrary to current viscous schematizations — of a prevalently plasto-elastic characteristic, with dampers of asolid-friction type.This work concludes dealing with the experimental determination of the parameters of the model and with particular approximate solutions of motion.

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Sommario Viene qui tentata la definizione del modello reologico del calcestruzzo, o cemento armato non parzializzato, in campo di sforzi dinamici sufficientemente lontani dalla rottura; fine al quale avevamo di recente già indirizzato una serie di indagini, volte a chiarire i diversi aspetti del comportamento sperimentale.Pur con lacune di dettaglio, il modello trovato sembra correttamente interpretare ilmeccanismo dissipativo del nostro materiale, ed è — contrariamente a correnti schematizzazioni viscose — a caratteristica prevalentemente elastoplastica, con smorzatori di tiposolid-friction.Il lavoro si conclude trattando della determinazione sperimentale dei parametri del modello e di particolari soluzioni approssimate del moto.

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The first results of this work were presented at 2° Convegno Nazionale Gruppo di Studio C.N.R. sulle Costruzioni Antisismiche — Camogli (Genova), Nov. 1966 [1].     

Rheological behaviour of vitreous humour

The vitreous humour (VH) is a complex biofluid that occupies a large portion of the eyeball between the lens and the retina, and exhibits non-Newtonian rheological properties that are key for its function in the eye. It is often possible to distinguish two different phases in VH, known as liquid and gel phases (Sebag J Eye 1: 254–262, 1987). In this work, we present a detailed rheological characterisation of the two phases of the VH under shear and extensional flow conditions. Healthy New Zealand rabbit eyes were used to measure the surface tension and the shear and extensional rheological properties of VH in different phase conformations and at different times after dissection. The results show that VH liquid phase exhibits a surface tension of 47.8 mN/m, a shear thinning behaviour reaching a viscosity plateau around 10^?3 Pa s for shear rates above ~1000 s^?1, and an average relaxation time of 9.7 ms in extensional flow. Interestingly, both VH phases present higher storage modulus than loss modulus, and the measurements performed with VH gel phase 4?±?1 h after dissection exhibit the highest moduli values. The compliance measurements for the gel phase show a viscoelastic gel behaviour and that compliance values decrease substantially with time after dissection. Our results show that the two VH phases exhibit viscoelastic behaviour, but with distinct rheological characteristics, consistent with a gel phase mostly composed of collagen entangled by hyaluronan and a second phase mainly composed of hyaluronan in aqueous solution.     

Rheological properties of peanut butter

The rheological properties of two types of commercial peanut butter have been studied. Both products are concentrated suspensions, and differ by the presence of additives. The first type, referred to as “100% peanuts,” is an unstabilized suspension consisting of solid peanut particles in peanut oil which is a Newtonian fluid. The second type, referred to as “smooth,” consists of the same suspension stabilized with a vegetable oil and contains other ingredients such as salt and sugar in very small quantities. A mean volume particle diameter of 6.6 μm has been determined, the particle diameter distribution was found to be narrow, and the solids volume fraction was estimated to be 0.6. Slip encountered in rheometry was greatly reduced by gluing sandpaper to the parallel plates of the rheometer. Both samples behaved like plastic materials and apparent yield stresses of 24 Pa and 370 Pa have been determined for the unstabilized and the stabilized suspensions, respectively. No linear domain was found for both suspensions and the non-linearity was confirmed by deformed Lissajous curves and higher odd harmonics in the output signal of small amplitude oscillatory shear experiments. The stabilized suspension behaved more like a solid, the elastic modulus being larger than the loss modulus and almost independent of the frequency. This solid-like behavior is supposedly caused by strong repulsive (steric) forces induced by the stabilizing agent. Received: 29 September 1999 Accepted: 9 August 2000     

Rheological properties of polystyrene melts

Summary Measurements were taken of both viscous and elastic properties of polystyrene melts, using a capillary rheometer recently constructed byHan. The viscous property is determined from the slope of the axial pressure distribution, and the elastic property is determined from the exit pressure, which is obtained at the tube exit when the straight line portion of the axial pressure profile is extrapolated to the exit of the tube. Previous studies ofHan demonstrated that the exit pressure thus obtained can be correlated with die swell ratios, and that the exit pressure is equal to the normal stress difference for polymer melts in fully developed viscometric flow. The present study of polystyrene melts corroborates all previous correlations obtained with high density polyethylene and polypropylene melts as regard the exit pressure measurements. A salient feature of the method employed here is that it requires only a single capillary, long enough to assure that flow is fully developed.

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Zusammenfassung Mit einem Kapillar-Viskosimeter, das kürzlich vonHan entworfen worden ist, sind Messungen sowohl der viskosen als auch der elastischen Eigenschaften von Polystyrol-Schmelzen durchgeführt worden. Die viskosen Eigenschaften werden aus dem Verlauf der axialen Druekverteilung ermittelt, während die elastischen Eigenschaften aus dem Austrittsdruck bestimmt werden, der sich am Bohrende ergibt, wenn der geradlinige Teil des Axialdruckprofils bis zum Rohraustritt extrapoliert wird. Frühere Untersuchungen vonHan zeigten, daß der so ermittelte Austrittsdruck mit Strangaufweitungsverhältnissen korreliert werden kann und daß der Austrittsdruck bei Polymer-Schmelzen bei voll ausgebildetem viskosimetrischen Fließen gleich der Normalspannungsdifferenz ist. Die vorliegende Untersuchung von Polystyrol-Schmelzen bekräftigt alle früheren Beziehungen, die durch Messungen des Austrittsdrucks an HD-Polyäthylen und Polypropylen-Schmelzen erhalten worden waren. Der Vorteil der hier entwickelten Methode liegt darin, daß nur eine einzige Kapillare benötigt wird, die so lang gewählt sein muß, daß die Strömung voll ausgebildet ist.