An analytical theory has been developed for properties of a steady, axisymmetric liquid–gas capillary bridge that is present between two identical, perfectly wettable, rigid spheres. In this theory ...the meridional profile of the capillary bridge surface is represented by a part of an ellipse. Parameters in this geometrical description are determined from the boundary conditions at the three-phase contact circle at the sphere and at the neck (i.e. in the middle between the two spheres) and by the condition that the mean curvature be equal at the three-phase contact circle and at the neck. Thus, the current theory takes into account properties of the governing Young–Laplace equation, contrary to the often-used toroidal approximation. Expressions have been developed analytically that give the geometrical parameters of the elliptical meridional profile as a function of the capillary bridge volume and the separation between the spheres. A rupture criterion has been obtained analytically that provides the maximum separation between the spheres as a function of the capillary bridge volume. This rupture criterion agrees well with a rupture criterion from the literature that is based on many numerical solutions of the Young–Laplace equation. An expression has been formulated analytically for the capillary force as a function of the capillary bridge volume and the separation between the spheres. The theoretical predictions for the capillary force agree well with the capillary forces obtained from the numerical solutions of the Young–Laplace equation and with those according to a comprehensive fit from the literature (that is based on many numerical solutions of the Young–Laplace equation), especially for smaller capillary bridge volumes.
The behaviour of nearly neutrally-buoyant suspensions has been studied experimentally, using a concentric-cylinder rheometer. The effect on the suspension viscosity of: (i) solid fraction, (ii) ...diameter of the solid, spherical particles, (iii) viscosity of the suspending liquid, and (iv) shear rate has been determined experimentally, where all suspensions have been created using the same particulate and liquid materials. The suspension behaviour can well be described by the Ostwald-deWaele model, i.e. a power-law relation between shear stress and shear rate. For low solid fractions, the behaviour is nearly Newtonian, while for larger solid fractions shear thinning behaviour is observed. The relative viscosity, i.e. the ratio between suspension and liquid viscosities, is a rapidly increasing function of solid fraction. For suspensions based on a liquid with high viscosity, the effect of the particle diameter is small, while for those with a low liquid viscosity, the suspension viscosity decreases weakly with increasing particle diameter. The relative viscosity is higher for systems based on a liquid with low viscosity, in comparison with systems with a high liquid viscosity. The current investigation shows that commonly used relations for the relative viscosity of suspensions, that only include solid fraction and a maximum solid fraction, are not generally valid.
In an influential study on the micromechanical origin of stress transmission in granular materials, Radjaï et al. (1998) have proposed a division of the network of interparticle contacts into ‘weak’ ...and ‘strong’ contacts. This division is based on a comparison of the force at contacts with the average (over all contacts) force. They observed, from the results of a two-dimensional computer simulation of the behaviour of a system of particles, that the shear stress of the granular material is mainly carried by the strong contacts and that the anisotropy in the orientational distribution of the weak contacts is in the direction perpendicular to that of the full contact network.
These findings are analytically predicted here in a qualitative sense, within a statistical framework that is based on a simple, self-similar expression for the conditional probability function for the normal force at contacts with given contact orientation.
An alternative definition of weak and strong contacts is proposed here, in which the division of the contacts is based on a comparison of the force at contacts with the average force corresponding to the contact orientation. Contrary to the finding based on the definition of weak and strong contacts by Radjaï et al. (1998), with this alternative definition the pressure and the shear stress are (almost) equally carried by the weak contact network.
In micromechanics of granular materials, relationships are investigated between micro-scale characteristics of particles and contacts and macro-scale continuum characteristics. For three-dimensional ...isotropic assemblies the macro-scale elastic characteristics are described by the bulk and the shear modulus, which depend on the micro-scale characteristics of the coordination number (i.e. the average number of contacts per particle) and the interparticle contact stiffnesses in directions normal and tangential to the contact.
It is well-known that the uniform-strain theory (or mean-field theory) overpredicts the elastic moduli. To find improved predictions, approximations of the particle displacement and rotations fields are obtained here by solving the equilibrium equations for small subassemblies that are centred around particles. At the boundary of these subassemblies, the particle displacements and rotations are prescribed such that they conform to the mean field.
Employing this approach, improved predictions of bulk and shear moduli are obtained, in comparison with those according to the uniform-strain assumption, especially when the size is increased of the subassemblies for which equilibrium equations are solved.
The elastic moduli are evaluated from the particle displacement and rotations fields by two methods. In the first, stress-based method the micromechanical expression for the average stress tensor, in terms of the forces at contacts and the branch vectors that connect particles in contacts, is employed. In the second, energy-based method the minimum potential-energy principle is used to obtain rigorous upper bounds to the moduli. It is generally observed that the moduli obtained from the stress-based method give closer agreement with the results from Discrete Element Method simulations than those from the energy-based method.
These improvements in the predictions of the elastic moduli are observed over the range of coordination numbers and interparticle stiffnesses considered here.
In this micromechanical study of the behaviour of granular materials, relationships are investigated between deformation at the continuum macro-scale and at the micro-scale of interparticle contacts. ...Special attention is paid to the role of the microstructure, or fabric, as it is well known to have a strong influence on the behaviour of granular materials. Two-dimensional Discrete Element Method simulations of isobaric tests have been used to formulate truncated Fourier series representations for suitably-averaged relative displacement increment vectors at interparticle contacts and of parameters that describe the fabric. Based on a micromechanical expression for the average strain tensor that is valid in the two-dimensional case considered here and on these Fourier series representations, a Strain–Displacement–Fabric relationship has been derived that links the macro-scale dilatancy rate to the micro-scale relative displacements and fabric. Results of the Discrete Element Method simulations, using samples with different densities, have been employed to verify the accuracy of the proposed relationship for the dilatancy rate.
•Wide range of measurements of the forced-convective rate of heat transfer from spheres at high Reynolds numbers.•Sudden increase in the Nusselt number at the critical Reynolds number, analogous to ...the drag crisis for the drag force.•New correlation for the Nusselt number as a function of the Reynolds number, valid for large subcritical Reynolds numbers.
Forced convective heat transfer from smooth, solid and isothermal spheres of various diameters has been studied experimentally in air flows with various free-stream velocities. The average heat transfer coefficient has been determined from the steady measured power input to a heating element inside the spheres and the steady measured temperatures of the flowing air and of the surface of the spheres, employing corrections to account for heat transfer due to thermal radiation and due to natural convection. The current data for the average heat transfer coefficient, expressed as a relationship between the Nusselt number and the Reynolds number, complement data in literature with respect to the range of large Reynolds numbers that have been considered: here the Reynolds numbers were between 7.8×103 and 3.3×105. The experimental results show a sudden increase in the Nusselt number above a critical Reynolds number of approximately 2.9×105, analogous to the “drag crisis” for the drag force on the sphere. A correlation for the Nusselt number as a function of the Reynolds number has been formulated for air flows that describes these data well for Reynolds numbers below the critical Reynolds number.
A theory of critical state and of the evolution of coordination number during quasi-static deformations of granular materials is developed, based on the interpretation of several discrete element ...simulations of plane granular assemblies with a range of interparticle friction from nearly frictionless to infinitely rough. The theory is based on the concept that shear deformations tend to destroy interparticle contacts and create locally unstable configurations that regain stability by forming new or restoring old interparticle contacts. The process is operating in such a way that in a dense state the rate of contact disintegration exceeds the rate of contact creation, while in the critical state both rates are equalized. In a loose state more contacts are created than disintegrated until rates are equalized. Interparticle friction is viewed as an essential element that affects stability of local configurations. This explains the pronounced dependence of critical coordination number on interparticle friction as observed in two-dimensional discrete element simulations. The derived differential equation for the evolution of coordination number in biaxial tests is shown to describe the results of the discrete element simulations remarkably well. The paper also presents analyses of simulation data to investigate a relationship between packing fraction of granular assemblies and coordination number. The data suggest that the packing fraction is affected by the anisotropy of contact orientations as well as by the coordination number. The latter is the primary variable. Limited data also suggest that critical state is characterized by both critical coordination number and by critical anisotropy induced by shear deformations.
In micromechanics of granular materials, relationships are investigated between micro-scale characteristics of particles and contacts and macro-scale, continuum characteristics. Dilatancy is an ...important property of granular materials, defined as volume changes (dilative or compressive) induced by shear deformation. To obtain detailed information at the micro-scale, two-dimensional Discrete Element Method simulations of isobaric tests with disk-shaped particles have been performed. The required information includes the fabric tensor which characterizes statistical properties of the contact network.
The dependence of the dilatancy rate on the shear strength and the fabric tensor has been investigated, based on the results of the simulations employing a dense and a loose initial system. The dilatancy rate depends in a complex, non-unique way on the shear strength, while the dependence on the fabric tensor is more amenable to analytical description. Two micromechanical mechanisms of dilatancy have been identified: (i) dilatancy due to deformation of loops that are determined by the interparticle contact network and (ii) dilatancy due to topological changes in the interparticle contact network that correspond to the creation or disruption of contacts. For the first mechanism the anisotropy in the contact network is the primary parameter, while for the second mechanism the average number of contacts per particle is the primary parameter.
A fabric-based micromechanical relation for the dilatancy rate has been formulated that describes these identified mechanisms. Parameters present in this relation are determined by fitting this relation to the results of the Discrete Element Method simulations, using combined data for the dense and the loose initial system. Employing these fitted coefficients, good agreement is obtained between the results of the simulations and the predictions of the micromechanical dilatancy relation.
In micromechanics of quasi-static deformation of granular materials, relationships are investigated between the macro-scale, continuum-mechanical characteristics, and the micro-scale characteristics ...at the particle and interparticle contact level. An important micromechanical quantity is the fabric tensor that reflects the distribution of contact orientations. It also contains information on the coordination number, i.e. the average number of contacts per particle. Here, the focus is on characteristics of the critical state in the two-dimensional case. Critical state soil mechanics is reviewed from the micromechanical viewpoint. Two-dimensional discrete element method (DEM) simulations have been performed with discs from a fairly narrow particle-size distribution. Various values for the interparticle friction coefficient and for the confining pressure have been considered to investigate the effect of these quantities on critical state characteristics (shear strength, packing fraction, coordination number and fabric anisotropy). Results from these DEM simulations show that a limiting fabric state exists at the critical state, which is geometrical in origin. The contact network tessellates the assembly into loops that are formed by contacts. For each loop, a symmetrical loop tensor is defined, based on its contact normals. This loop tensor reflects the shape of the loop. An orientation is associated with each loop, based on its loop tensor. At the critical state, the frequencies with which loops with different number of sides occur depend on the coordination number. At the critical state, these loops have, on average, the following universal characteristics, i.e. independent of the coordination number: (1) loops with the same number of sides and orientation have identical anisotropy of the loop tensor, (2) the anisotropy of the loop tensor depends linearly on the number of sides of the loop, (3) the distribution of loop orientations is identical, (4) Lewis’s law for the loop areas, which is a linear relation between the number of sides of loops and their area, is satisfied (not exclusively at the critical state) and (5) the areas of the loops do not depend on their orientation.
In micromechanical studies of granular materials, relations are investigated between macro-level, continuum characteristics and micro-level, (inter) particle characteristics. For quasi-static ...deformation of granular materials, the fabric tensor is an important micromechanical characteristic that describes the orientational distribution of contacts between particles.
The evolution of the fabric tensor during quasi-static deformation has been investigated, using two-dimensional Discrete Element Method simulations of constant-pressure tests of a dense and a loose isotropic initial assembly. Invariants of the fabric tensor are the coordination number, i.e. the average number of contacts per particle, and the fabric anisotropy, i.e. the difference between the principal values of the fabric tensor. At large strains, both coordination number and fabric anisotropy become constant. This constant value is the same for the dense and the loose initial assembly, i.e. a critical fabric state is attained. The evolution towards the critical state value for coordination number is much faster than for fabric anisotropy.
The three micromechanical mechanisms through which the fabric tensor can change are contact creation, contact disruption and contact reorientation. Contact disruption is the dominant mechanism (except at the steady state) that is strongly anisotropic. The rate of contact disruption decreases rapidly with increasing imposed strain. Contact creation is approximately an isotropic mechanism. The rate of contact creation is fairly constant over the range of imposed strains. Fabric change due to contact reorientation is small, except at the steady state.
An evolution relation is proposed for the changes in the fabric tensor as a function of the fabric tensor and the strain increment tensor. With its coefficients obtained by fitting with the data for the fabric tensor from the Discrete Element Method simulations, good agreement is observed between results from these simulations and from the proposed fabric-evolution relation.
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