Dynamic bulk and shear moduli due to grain-scale local fluid flow in fluid-saturated cracked poroelastic rocks: Theoretical model

Grain-scale local fluid flow is an important loss mechanism for attenuating waves in cracked fluid-saturated poroelastic rocks. In this study, a dynamic elastic modulus model is developed to quantify local flow effect on wave attenuation and velocity dispersion in porous isotropic rocks. The Eshelby transform technique, inclusion-based effective medium model (the Mori–Tanaka scheme), fluid dynamics and mass conservation principle are combined to analyze pore-fluid pressure relaxation and its influences on overall elastic properties. The derivation gives fully analytic, frequency-dependent effective bulk and shear moduli of a fluid-saturated porous rock. It is shown that the derived bulk and shear moduli rigorously satisfy the Biot-Gassmann relationship of poroelasticity in the low-frequency limit, while they are consistent with isolated-pore effective medium theory in the high-frequency limit. In particular, a simplified model is proposed to quantify the squirt-flow dispersion for frequencies lower than stiff-pore relaxation frequency. The main advantage of the proposed model over previous models is its ability to predict the dispersion due to squirt flow between pores and cracks with distributed aspect ratio instead of flow in a simply conceptual double-porosity structure. Independent input parameters include pore aspect ratio distribution, fluid bulk modulus and viscosity, and bulk and shear moduli of the solid grain. Physical assumptions made in this model include (1) pores are inter-connected and (2) crack thickness is smaller than the viscous skin depth. This study is restricted to linear elastic, well-consolidated granular rocks.     

Rheology of oil-in-water emulsions

The effect of interfacial tension on the steady-flow and dynamic viscoelastic behavior of emulsions are studied experimentally. At very low inter-facial tensions and low volume fractions, the viscosity decreases with increasing shear rate and becomes constant at high shear rates. The high-shear-rate Newtonian viscosity is not affected by interfacial tension, but the transition from pseudoplastic to Newtonian flow shifts to lower shear rates as the interfacial tension decreases. At an interfacial tension of 5 × 10^–3 Nm^–1, the viscosity decreases, passes through a minimum, and then increases as the shear rate is increased. The dilatant behavior may be attributed to elastic responses of interfaces during collision of drops. At high volume fractions, the emulsions show remarkable elasticity resulting from the interfacial energy associated with deformation of liquid films. The modulus and viscosity are proportional to interfacial tension and inversely proportional to drop size.     

Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading

The linear viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading is investigated in the glass transition range. Using the time-temperature superposition technique, the master curves of the shear dynamic relaxation moduli are obtained at a reference temperature of 566°C. A method to determine the viscoelastic constants from dynamic relaxation moduli is proposed. However, some viscoelastic constants cannot be directly measured from the experimental curves and others cannot be precisely obtained due to non-linearity effects at very low frequencies. The generalized Maxwell model is investigated from the experimental dynamic moduli without fixing the viscoelastic constants. A set of parameters is shown to be in good agreement with the experimental dynamic relaxation moduli, but does not give the correct values of the viscoelastic constants of the investigated glass. The soda-lime-silica glass exhibits a non-linear viscoelastic behavior at very low stress level which is usually observed for organic glasses. This non-linear behavior is questioned.     

Free vibration analysis of the axially moving Levy-type viscoelastic plate

In this paper, the viscoelastic theory is applied to the axially moving Levy-type plate with two simply supported and two free edges. On the basis of the elastic – viscoelastic equivalence, a linear mathematical model in the form of the equilibrium state equation of the moving plate is derived in the complex frequency domain. Numerical calculations of dynamic stability were conducted for a steel plate. The effects of transport speed and relaxation times modeled with two-parameter Kelvin–Voigt and three-parameter Zener rheological models on the dynamic behavior of the axially moving viscoelastic plate are analyzed.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers II: oblique anchoring angles

We extend previous work on the linear viscoelastic moduli of heterogeneous nematic polymers in a small-amplitude oscillatory shear flow, focusing on the role of the orientational anchoring conditions at the plates. When tangential or normal anchoring conditions are applied, the Doi–Marrucci–Greco orientation tensor-flow model effectively reduces to the Leslie–Ericksen director-flow model, predicting that director distortions control the dynamic moduli with negligible contributions from tensor-order parameters. In this paper, we examine oblique anchoring angles. We use a combination of analysis and numerical simulation on the generalized tensor-flow system for arbitrary anchoring conditions to show that any oblique anchoring condition induces a nontrivial order parameter contribution to the dynamic moduli, which vanishes only in the limit of tangential or normal anchoring. Our approach reveals that the storage and loss moduli admit an approximate decomposition in terms of two reduced problems that are exactly solvable: the heterogeneous director–flow response plus the monodomain tensor response to an imposed shear. The importance of this result is that we gain scaling properties of the moduli with respect to material parameters and experimental conditions without having to compute and assimilate across the full parameter space. These results provide insight into the relative importance of the distortional vs bulk nematic elastic stress in determining the viscoelastic moduli, predicting that anchoring conditions tune the relative contributions.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

The total creep of viscoelastic composites under hydrostatic or antiplane loading

The problem of bounding the total creep (or total stress relaxation) of a composite made of two linear viscoelastic materials and subjected to a constant hydrostatic or antiplane loading is considered. It is done by coupling the immediate and the relaxed responses of the composite, which are pure elastic. The coupled bounds provide the possible range of the total deformation at infinite time as a function of the initial deformation of the composite. For antiplane shear existing bounds for coupled two-dimensional conductivity yield the required coupled bounds, and these are attained by doubly coated cylinder assemblages. The translation method is used to couple the effective bulk moduli of a viscoelastic composite at zero and infinite time. A number of microgeometries are found to attain the bulk modulus bounds. It is shown that the Hashin's composite sphere assemblage does not necessarily correspond to the maximum or minimum overall creep, although it necessarily attains the bounds for effective bulk moduli. For instance, there are cases when the doubly coated sphere microstructure or some special polycrystal arrangements attain the bounds on the total creep.     

Rheology of polymer blends: linear model for viscoelastic emulsions

 Kerner's model for flow of composite elastic media is extended to an emulsion of viscoelastic phases with interfacial tension undergoing deformations of small amplitude. A privileged internal structure inside the suspended drops is discussed in terms of fluid circulation across the interface. It is shown that for usual drop radius and interfacial tension values of emulsions, the rheological behavior predicted by the model, with very simple expression for the complex shear modulus, is quantitatively similar to that predicted by Palierne's model. Predictions of the model are compared with experimental data obtained on a polystyrene/polyethylene blend sheared in a small-amplitude oscillatory mode. Received: 10 August 1998 Accepted: 18 December 1998     

Modified Zener Theory to Accurately Predict Impact Force Histories for Soft Impactors Employing Spiral Sensing

While predicting the impact force histories from the Zener impact model with different material properties of impactors, several discrepancies were observed and reported in this article. To overcome these discrepancies, a modified Zener model is proposed to accurately calculate impact force histories. In the original Zener theory, nonlinear Hertzian contact law was used, and it was assumed that impact forces are transmitted through natural intensity factors depending on coupled physical properties of the plate and the impactor. However, when the force histories were predicted, a diverging trend appeared for softer materials with elastic moduli below 20 GPa. It is hypothesized that the primary reasons for this divergence are due to the contact time delay and the viscoelastic dissipation of energy, which are not considered in current Zener models. Several modifications of the model have been proposed since its inception, but it has been found that they are not independently sufficient to accurately predict impact force histories. In this article, a modified Zener theory is proposed introducing two new parameters in the governing differential equation derived from the sensor phase lag index and the dominant frequency band through a set of experiments employing a spiral sensing mechanism followed by an optimization process. The spiral lag index shows an unexpected peculiar trend with soft impactors (< 20 GPa), which are distinctly different from hard impactors and are judicially incorporated in the model. Furthermore, the force histories are accurately reconstructed with the proposed modifications.     

Rhéologie des matériaux en cours de prise : Approche théorique

The rheological behavior of setting heterogeneous materials is studied from a theoretical approach by means of the homogenization technique of periodic medium. These materials considered as suspensions of gas bubbles at finite concentration in a viscoelastic matrix with low compressibility, present the macroscopic behavior of a compressible viscoelastic medium. The shear and volume macroscopic moduli are of the same order of magnitude and directly proportional to that of the fluid. The effective compressibility of the gas (out of thermal equilibrium) is added to these fluid contributions.     

Bending of a fiber-reinforced viscoelastic composite plate resting on elastic foundations

Composite structures on an elastic foundation are being widely used in engineering applications. Bending response of inhomogeneous viscoelastic plate as a composite structure on a two-parameter (Pasternak’s type) elastic foundation is investigated. The formulations are based on sinusoidal shear deformation plate theory. Trigonometric terms are used in the present theory for the displacements in addition to the initial terms of a power series through the thickness. The transverse shear correction factors are not needed because a correct representation of the transverse shear strain is given. The interaction between the plate and the foundation is included in the formulation with a two-parameter Pasternak’s model. The effective moduli and Illyushin’s approximation methods are used to derive the viscoelastic solution. The effects played by foundation stiffness, plate aspect ratio, and other parameters are presented.     

Spectral element technique for efficient parameter identification of layered media. Part III: viscoelastic aspects

This article addresses the issues of wave propagation in elastic–viscoelastic layered systems and viscous parameter identification from non-destructive dynamic tests. A methodology that combines the spectral element technique, for the simulation of wave propagation, with the differential operator technique, for stress–strain relationship in viscoelastic materials, is adopted. The compatibility between the two techniques stems from the fact that both can be treated in the frequency domain, which enables naturally the adoption of Fourier superposition. The mathematical formulation of spectral elements for Burger's viscoelastic material model is highlighted. Also, an inverse procedure for the identification of the material's Young's moduli and complex moduli for layer systems is described. It is shown that the proposed methodology enables the substitution of an expensive laboratory testing procedure for the determination of material complex moduli with non-destructive dynamic testing.     

Linear rheology of viscoelastic emulsions with interfacial tension

Emulsions of incompressible viscoelastic materials are considered, in which the addition of an interfacial agent causes the interfacial tension to depend on shear deformation and variation of area. The average complex shear modulus of the medium accounts for the mechanical interactions between inclusions by a self consistent treatment similar to the Lorentz sphere method in electricity. The resulting expression of the average modulus includes as special cases the Kerner formula for incompressible elastic materials and the Oldroyd expression of the complex viscosity of emulsions of Newtonian liquids in time-dependent flow.     

The elastic–viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties

This paper revisits the elastic–viscoelastic correspondence principle for non-homogeneous materials. Several recent publications discussed this principle for functionally graded materials (FGMs) with time translation invariant viscoelastic properties. It was demonstrated that the correspondence principle is valid only for the FGMs with separable relaxation moduli (moduli in separable form in space and time). This paper reconsiders this issue. It shows that the correspondence principle is valid even for non-homogeneous materials with separable relaxation moduli even if the time-dependences of the relaxation moduli in shear and dilatation are not necessarily time translation invariant. The property of similarity of Volterra operators is used to obtain the corresponding elastic solution. The correspondence is established between the elastic solution and the operator-transformed viscoelastic solution. The transformation operators are combinations of the Laplace transform operator and additional integral operators.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Dynamic shear and compressional behavior of polydimethylsiloxanes: Ultrasonic and low frequency characterization

The dynamic rheological properties of a series of entangled polydimethylsiloxanes are investigated. Rheometrical shear measurements are first reported for three different molecular weight samples, together with ultrasonic tests, thus allowing to obtain dynamic shear master curves over ten decades. Winter‘s model for slightly polydisperse polymers is used and works well in this case. This provides an example of the applicability of this model. The propagation of longitudinal waves in the MHz range for the same PDMSs is studied next. The results are analyzed and combined with static measurements of the compressibility. They indicate that different relaxation mechanisms have to be considered in shear and compression, and that the ratio between volume and shear viscosity is frequency dependent, with a preponderance for shear effects at higher frequencies. Shear and compressional relaxational mechanisms are also well separated. Moreover, it is verified that the shear and compressional moduli are independent of the molecular weight in the transition region, above a certain frequency. This frequency corresponds to a wavelength comparable to the distance between entanglements, in the case of shear waves. Received: 9 December 1997 Accepted: 30 March 1998     

Spring-network model of red blood cell: From membrane mechanics to validation

Red blood cell membrane is highly elastic and proper modeling of this elasticity is essential for biomedical applications that involve computational experiments with blood flow. Inseparable and often some of the most difficult parts of modeling process are verification and validation. In this work, we present a revised model, which uses a spring network to represent the cell membrane immersed in a fluid and has been successfully used in blood flow simulations. We demonstrate the validation steps by first deriving the theoretical relations between the bulk properties of elastic membranes—shear modulus and area compressibility modulus—and parameters of the model that enter the nonlinear stretching and local area conservation computational moduli. We verify the theoretically derived relations using computer simulations of deformable triangular mesh. We calibrate the model by performing a computational version of the optical tweezers experiment. And finally, we validate the modeled cell behavior by investigating the cell rotation frequency when it is subjected to shear flow and cell deformation in narrow channels. The supplementary material contains an extensive dataset that can be used for setting different elastic properties for each cell in simulations of dense suspensions, while still conforming to the biological data. This work contains a complete model development process: From modelling of basic mechanical concepts (the spring network) and advanced biomechanical concepts (such as elasticity of the membrane), through calibration process towards the final stage of model validation.     

Role of droplet bridging on the stability of particle-containing immiscible polymer blends

The effect of micron-sized hydrophobic calcium carbonate particles on the stabilization of polydimethylsiloxane (PDMS)/polyisobutylene (PIB) immiscible model blends is investigated in this study. The analytical splitting of bulk and liquid–liquid interface contributions from the droplet bridging one is successfully performed due to the negligible contribution of hydrophobic microparticles to the bulk rheology of phases. The presence of particles at the fluid–fluid interface is supported by wetting parameter calculation and verified by optical microscopy observations. Moreover, direct visualizations shows that particles are able to form clusters of droplets by simultaneously adsorbing at two fluid–fluid interfaces and glue-dispersed droplets together, probably due to the patchy interactions induced by heterogeneous distribution of particles along the interface. Rheological studies show that the flow-induced coalescence is slowed down upon addition of particles and almost suppressed with the addition of 4 wt% particles. The linear viscoelastic response is modeled to estimate interfacial tension by considering the contribution of particle-induced droplet aggregation in addition to bulk and droplet deformation ones. From linear and nonlinear viscoelastic responses, the improved stability of filled polymer blends is attributed to the interfacial rheology and/or the bridged structure of droplets, even though the interfacial area is not fully covered by particles. Furthermore, Doi–Ohta scaling relations are investigated by employing stress growth response upon step-up of shear flow.     

Thin interphase/imperfect interface in elasticity with application to coated fiber composites

The imperfect interface conditions which are equivalent to the effect of a thin elastic interphase are derived by a Taylor expansion method in terms of interface displacement and traction jumps. Plane and cylindrical interfaces are analyzed as special cases. The effective elastic moduli of a unidirectional coated fiber composite are obtained on the basis of the derived imperfect interface conditions. High accuracy of the method is demonstrated by comparison of solutions of several problems in terms of the imperfect interface conditions or explicit presence of interphase as a third phase. The problems considered are transverse shear of a coated infinite fiber in infinite matrix and effective transverse bulk and shear moduli and effective axial shear modulus of a coated fiber composite. Unlike previous elastic imperfect interface conditions in the literature, the present ones are valid for the entire range of interphase stiffness, from very small to very large.