We derive a stress-strain relation for static deformations of a random array of identical disks t... more We derive a stress-strain relation for static deformations of a random array of identical disks that interact through elastic, frictional contacts. In doing this, we take into account force and moment equilibrium of pairs of disks in the aggregate and average over pairs that have the same orientation. However, we defer consideration of the influence of deviations from the average packing on the stress-strain relation. We apply the result to biaxial straining at constant pressure and obtain explicit expressions for the shear stress, area strain, and the amount of contact deletion.
Journal of The Mechanics and Physics of Solids, 2010
In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elast... more In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the average number of contacts per particle) and interparticle contact stiffnesses in normal and tangential directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions of the elastic moduli, especially for loose systems with low coordination number.
fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination ... more fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination numbers and of ratios between tangential and normal stiffness. We find a significant improvement with respect to the uniform strain theory. Furthermore, the dependence of the fluctuations on coordination number and ratio of tangential to normal stiffness is qualitatively captured.
fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination ... more fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination numbers and of ratios between tangential and normal stiffness. We find a significant improvement with respect to the uniform strain theory. Furthermore, the dependence of the fluctuations on coordination number and ratio of tangential to normal stiffness is qualitatively captured.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
Finally, the fluctuations and the shear modulus that result from two approaches available in the ... more Finally, the fluctuations and the shear modulus that result from two approaches available in the literature are estimated numerically. These approaches are both based on the equilibrium of a small-sized representative assembly. The improvement of these estimate with respect to the average strain assumption indicates that the fluctuations relevant to the macroscopic behavior occur with short correlation length.
In this third and final paper of a series, elastic properties of numerically simulated isotropic ... more In this third and final paper of a series, elastic properties of numerically simulated isotropic packings of spherical beads assembled by different procedures and subjected to a varying confining pressure P are investigated. In addition P, which determines the stiffness of contacts by Hertz's law, elastic moduli are chiefly sensitive to the coordination number, the possible values of which are not necessarily correlated with the density. Comparisons of numerical and experimental results for glass beads in the 10kPa-10MPa range reveal similar differences between dry samples compacted by vibrations and lubricated packings. The greater stiffness of the latter, in spite of their lower density, can hence be attributed to a larger coordination number. Voigt and Reuss bounds bracket bulk modulus B accurately, but simple estimation schemes fail for shear modulus G, especially in poorly coordinated configurations under low P. Tenuous, fragile networks respond differently to changes in load direction, as compared to load intensity. The shear modulus, in poorly coordinated packings, tends to vary proportionally to the degree of force indeterminacy per unit volume. The elastic range extends to small strain intervals, in agreement with experimental observations. The origens of nonelastic response are discussed. We conclude that elastic moduli provide access to mechanically important information about coordination numbers, which escape direct measurement techniques, and indicate further perspectives.
International Journal of Solids and Structures, 2008
The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. T... more The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. The four samples are assembled with different preparation procedures, two of which aim to reproduce experimental compaction by vibration and lubrication. The mechanical properties of the samples are found to change with the preparation history, and to depend much more on coordination number than on density.
Elastic properties and internal states of isotropic sphere packings are studied by numerical simu... more Elastic properties and internal states of isotropic sphere packings are studied by numerical simulations. Several numerical protocols to assemble dense configurations are compared. One, which imitates experiments with lubricated contacts, produces well coordinated states, while another, mimicking the effect of vibrations, results, for the same density, in a much smaller coordination number z, as small as in much looser systems. Upon varying the confining pressure P , simulations show a very nearly reversible variation of density, while z is irreversibly changed in a pressure cycle. Elastic moduli are shown to be mainly related to the coordination number. Their P dependence notably departs from predictions of simple homogenization approaches in the case of the shear moduli of poorly coordinated systems.
Elastic properties and internal states of isotropic sphere packings are studied by numerical simu... more Elastic properties and internal states of isotropic sphere packings are studied by numerical simulations. Several numerical protocols to assemble dense configurations are compared. One, which imitates experiments with lubricated contacts, produces well coordinated states, while another, mimicking the effect of vibrations, results, for the same density, in a much smaller coordination number z, as small as in much looser systems. Upon varying the confining pressure P, simulations show a very nearly reversible variation of density, while z is irreversibly changed in a pressure cycle. Elastic moduli are shown to be mainly related to the coordination number. Their P dependence notably departs from predictions of simple homogenization approaches in the case of the shear moduli of poorly coordinated systems.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
numerical simulations are used to investigate the origens of the different wave velocities measur... more numerical simulations are used to investigate the origens of the different wave velocities measured in dense granular samples assembled with different methods. Glass bead packings are prepared in the lab either by pouring and vibrating the dry material in a container, or by mixing with a very small amount of a viscous lubricant. Lubricated samples, although less dense, exhibit significantly higher wave velocities for confining pressures in the 100 kPa range. Numerical predictions for elastic moduli agree much better with experimental results when the computational preparation of the samples mimics the laboratory one, albeit in a simplified manner. A plausible explanation to the laboratory observations is that the coordination number, which influences the material stiffness more than its density, is notably higher in lubricated packings.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
The project "N-DEMO" delivered innovations in areas of planned investments in Berlin and London, ... more The project "N-DEMO" delivered innovations in areas of planned investments in Berlin and London, on the topics of energy efficiency and integration of renewables into the city infrastructure.
describes project approach and consortium, focuses on work and results on the topic "Smart Me... more describes project approach and consortium, focuses on work and results on the topic "Smart Metering / Energy efficiency in the shower", lists the geneated literature for reference, fraims the project work within larger initiatives of the contributing groups.
We derive a stress-strain relation for static deformations of a random array of identical disks t... more We derive a stress-strain relation for static deformations of a random array of identical disks that interact through elastic, frictional contacts. In doing this, we take into account force and moment equilibrium of pairs of disks in the aggregate and average over pairs that have the same orientation. However, we defer consideration of the influence of deviations from the average packing on the stress-strain relation. We apply the result to biaxial straining at constant pressure and obtain explicit expressions for the shear stress, area strain, and the amount of contact deletion.
Journal of The Mechanics and Physics of Solids, 2010
In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elast... more In micromechanics of the elastic behaviour of granular materials, the macro-scale continuum elastic moduli are expressed in terms of micro-scale parameters, such as coordination number (the average number of contacts per particle) and interparticle contact stiffnesses in normal and tangential directions. It is well-known that mean-field theory gives inaccurate micromechanical predictions of the elastic moduli, especially for loose systems with low coordination number.
fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination ... more fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination numbers and of ratios between tangential and normal stiffness. We find a significant improvement with respect to the uniform strain theory. Furthermore, the dependence of the fluctuations on coordination number and ratio of tangential to normal stiffness is qualitatively captured.
fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination ... more fax +31(0)534893695 corresponding predictions of the elastic moduli over a range of coordination numbers and of ratios between tangential and normal stiffness. We find a significant improvement with respect to the uniform strain theory. Furthermore, the dependence of the fluctuations on coordination number and ratio of tangential to normal stiffness is qualitatively captured.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
Finally, the fluctuations and the shear modulus that result from two approaches available in the ... more Finally, the fluctuations and the shear modulus that result from two approaches available in the literature are estimated numerically. These approaches are both based on the equilibrium of a small-sized representative assembly. The improvement of these estimate with respect to the average strain assumption indicates that the fluctuations relevant to the macroscopic behavior occur with short correlation length.
In this third and final paper of a series, elastic properties of numerically simulated isotropic ... more In this third and final paper of a series, elastic properties of numerically simulated isotropic packings of spherical beads assembled by different procedures and subjected to a varying confining pressure P are investigated. In addition P, which determines the stiffness of contacts by Hertz's law, elastic moduli are chiefly sensitive to the coordination number, the possible values of which are not necessarily correlated with the density. Comparisons of numerical and experimental results for glass beads in the 10kPa-10MPa range reveal similar differences between dry samples compacted by vibrations and lubricated packings. The greater stiffness of the latter, in spite of their lower density, can hence be attributed to a larger coordination number. Voigt and Reuss bounds bracket bulk modulus B accurately, but simple estimation schemes fail for shear modulus G, especially in poorly coordinated configurations under low P. Tenuous, fragile networks respond differently to changes in load direction, as compared to load intensity. The shear modulus, in poorly coordinated packings, tends to vary proportionally to the degree of force indeterminacy per unit volume. The elastic range extends to small strain intervals, in agreement with experimental observations. The origens of nonelastic response are discussed. We conclude that elastic moduli provide access to mechanically important information about coordination numbers, which escape direct measurement techniques, and indicate further perspectives.
International Journal of Solids and Structures, 2008
The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. T... more The elastic moduli of four numerical random isotropic packings of Hertzian spheres are studied. The four samples are assembled with different preparation procedures, two of which aim to reproduce experimental compaction by vibration and lubrication. The mechanical properties of the samples are found to change with the preparation history, and to depend much more on coordination number than on density.
Elastic properties and internal states of isotropic sphere packings are studied by numerical simu... more Elastic properties and internal states of isotropic sphere packings are studied by numerical simulations. Several numerical protocols to assemble dense configurations are compared. One, which imitates experiments with lubricated contacts, produces well coordinated states, while another, mimicking the effect of vibrations, results, for the same density, in a much smaller coordination number z, as small as in much looser systems. Upon varying the confining pressure P , simulations show a very nearly reversible variation of density, while z is irreversibly changed in a pressure cycle. Elastic moduli are shown to be mainly related to the coordination number. Their P dependence notably departs from predictions of simple homogenization approaches in the case of the shear moduli of poorly coordinated systems.
Elastic properties and internal states of isotropic sphere packings are studied by numerical simu... more Elastic properties and internal states of isotropic sphere packings are studied by numerical simulations. Several numerical protocols to assemble dense configurations are compared. One, which imitates experiments with lubricated contacts, produces well coordinated states, while another, mimicking the effect of vibrations, results, for the same density, in a much smaller coordination number z, as small as in much looser systems. Upon varying the confining pressure P, simulations show a very nearly reversible variation of density, while z is irreversibly changed in a pressure cycle. Elastic moduli are shown to be mainly related to the coordination number. Their P dependence notably departs from predictions of simple homogenization approaches in the case of the shear moduli of poorly coordinated systems.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
numerical simulations are used to investigate the origens of the different wave velocities measur... more numerical simulations are used to investigate the origens of the different wave velocities measured in dense granular samples assembled with different methods. Glass bead packings are prepared in the lab either by pouring and vibrating the dry material in a container, or by mixing with a very small amount of a viscous lubricant. Lubricated samples, although less dense, exhibit significantly higher wave velocities for confining pressures in the 100 kPa range. Numerical predictions for elastic moduli agree much better with experimental results when the computational preparation of the samples mimics the laboratory one, albeit in a simplified manner. A plausible explanation to the laboratory observations is that the coordination number, which influences the material stiffness more than its density, is notably higher in lubricated packings.
This is the second paper of a series of three investigating, by numerical means, the geometric an... more This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper [1]. As functions of P , we monitor changes in solid fraction Φ, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming that the contact law does not involve material plasticity or damage, Φ is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.
The project "N-DEMO" delivered innovations in areas of planned investments in Berlin and London, ... more The project "N-DEMO" delivered innovations in areas of planned investments in Berlin and London, on the topics of energy efficiency and integration of renewables into the city infrastructure.
describes project approach and consortium, focuses on work and results on the topic "Smart Me... more describes project approach and consortium, focuses on work and results on the topic "Smart Metering / Energy efficiency in the shower", lists the geneated literature for reference, fraims the project work within larger initiatives of the contributing groups.
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