Statistical Mechanics of Stress Transmission in Static Arrays of Rigid Grains

2000 ◽  
Vol 627 ◽  
Author(s):  
Dmitri V. Grinev ◽  
Sam F. Edwards
Keyword(s):  
Stress Tensor ◽  
Coordination Number ◽  
Complete System ◽  
External Loading ◽  
Contact Network ◽  
Cauchy Equation ◽  
Stress Equilibrium ◽  
Stress Transmission ◽  
Statistical Mechanical ◽  
Fundamental Equations

ABSTRACTWe develop the statistical-mechanical theory that delivers the fundamental equations of stress equilibrium for static arrays of rigid grains. The random geometry of static granular packing composed of rigid cohesionless particles can be visualised as a network of intergranular contacts. The contact network and external loading determine the network of intergranular forces. In general, the contact network can have an arbitrary coordination number varying within the system. It follows then that the network of intergranular forces is indeterminate i.e. the number of unknown forces is larger than the number of Newton's equations of mechanical equilibrium. Thus, in order for the network of intergranular forces to be determined, the number of equations must equal the number of unknowns. We argue that this determines the contact network with a certain fixed coordination number. The complete system of equations for the stress tensor is derived from the equations of intergranular force and torque balance, given the geometric specification of the packing. The granular material fabric gives rise to corrections to the Euler-Cauchy equation that become significant at mesoscopic lengthscales. The stress-geometry equation establishes the relation between various components of the stress tensor, and depends on the topology of the granular array.

Towards a Unified Solution Method for Fluid-Structure Interaction Problems: Progress and Challenges

2006 ◽  
Author(s):  
G. Papadakis ◽  
C. G. Giannopapa
Keyword(s):  
Stress Tensor ◽  
Solution Method ◽  
Constitutive Relations ◽  
Fluid Structure Interaction ◽  
Strain Tensor ◽  
Internal Stresses ◽  
Fluid Structure ◽  
Structure Interaction ◽  
The Difference ◽  
Fundamental Equations

The paper presents the progress in the development of a novel unified method for solving coupled fluid-structure interaction problems as well as the associated major challenges. The new approach is based on the fact that there are four fundamental equations in continuum mechanics: the continuity equation and the three momentum equations that describe Newton’s second law in three directions. These equations are valid for fluids and solids, the difference being in the constitutive relations that provide the internal stresses in the momentum equations: in solids the stress tensor is a function of the strain tensor while in fluids the viscous stress tensor depends on the rate of strain tensor. The equations are written in such a way that both media have the same unknown variables, namely the three velocity components and pressure. The same discretisation technique (finite volume) and solution method (segregated approach) are used irrespective of the medium. Also the same methodology to handle the pressure-velocity coupling is employed. A common set of variables as well as a unified discretisation and solution method leads to a strong coupling between the two media and is very beneficial for the robustness of the algorithm. Significant challenges include the derivation of consistent boundary conditions for the pressure equation in boundaries with prescribed traction as well as the handling of discontinuity of pressure at the fluid-structure interface.

scholarly journals Geometrical network of granular materials under isochoric cyclic shearing

2021 ◽  
Vol 249 ◽  
pp. 11004
Author(s):  
Ming Yang ◽  
Mahdi Taiebat ◽  
Patrick Mutabaruka ◽  
Farhang Radjai
Keyword(s):  
Coordination Number ◽  
Three Dimensional ◽  
Particle Dynamics ◽  
Network Evolution ◽  
Contact Network ◽  
Fabric Anisotropy ◽  
Sudden Drop ◽  
Physical Experiments ◽  
State Evolution ◽  
Dynamics Simulations

We use three-dimensional particle dynamics simulations to investigate the microstructure evolution of granular material subjected to isochoric cyclic shearing, driving the system to a liquefaction state. The cyclically sheared assembly presents a realistic macroscopic response as observed in physical experiments. By analyzing the contact network evolution in the post-liquefaction period, we show that the onset of liquefaction state is characterized by a sudden drop of coordination number and a fragile particle connectivity network. The simulation suggests a critical coordination number for exiting the liquefaction state. Evolution of fabric anisotropy combined with coordination number implies the isotropic and anisotropic gain or loss of contacts at certain durations of a post-liquefaction loading cycle.

scholarly journals Influence of stress anisotropy on stress distributions in gap-graded soils

2019 ◽  
Vol 92 ◽  
pp. 14007
Author(s):  
Marion Artigaut ◽  
Adnan Sufian ◽  
Xiaoxiao Ding ◽  
Tom Shire ◽  
Catherine O'Sullivan
Keyword(s):  
Applied Stress ◽  
Significant Proportion ◽  
Internal Erosion ◽  
Contact Network ◽  
Stress Distributions ◽  
Strong Force ◽  
Internal Instability ◽  
Sample Density ◽  
Stress Transmission ◽  
Stress Heterogeneity

The behaviour of gap graded soils comprising non-plastic fines (sand or silt) mixed with a coarser sand or gravel fraction has received attention from researchers interested in internal instability under seepage loading (a form of internal erosion) as well as researchers interested in load:deformation responses. Skempton and Brogan [1] postulated that resistance to seepage induced instability depends upon the proportion of the overall applied stress that is transmitted by the finer fraction. Shire et al. [2] explored Skempton and Brogan’s hypothesis using DEM simulations to look at the proportion of the applied stress transmitted by the finer fractions (α) in ideal isotropic samples. They showed that at low fines contents (FC< FC*) the average stress transmitted by the finer grains is less than the applied stress (α<1), while for FC>FC+ the fines play a key role in stress transmission (α>1); for FC*<FC< FC+, α depends on the sample density. The current contribution describes a series of constant p’ DEM triaxial test simulations carried out to assess the evolution of stress heterogeneity with shearing. The simulation data generated indicate that a sample can transition from being fines dominated (with the fines transmitting a significant proportion of the applied stress and α ≥1) to coarse or sand- dominated (with α <1) as the material dilates during shear deformation. While α reduces as the samples dilate, the relationship between the α and the sample void ratio is non-trivial. The anisotropy of the coarse-coarse contact network exceeds the overall contact force anisotropy; this indicates that the deviator stress is transmitted through a strong force network passing through the coarse-coarse contacts supported by the fine-coarse contacts.

Physics of Granular Systems

2015 ◽  
Author(s):  
Raphael Blumenfeld ◽  
Sam F. Edwards ◽  
Stephen M. Walley
Keyword(s):  
Degrees Of Freedom ◽  
Granular Matter ◽  
Granular Systems ◽  
Mechanical Theory ◽  
Micro Structure ◽  
Fundamental Physics ◽  
Granular Assemblies ◽  
Stress Transmission ◽  
Statistical Mechanical ◽  
The Relationship

This article discusses the fundamental physics of granular systems. It begins with an overview of the science of granular matter, followed by a description of the ‘micro’-structure on the granular level. It then considers stress transmission in mechanically equilibrated granular assemblies, focusing on conditions for marginal rigidity, isostaticity theory, and limitations of linear stress theories. It also examines the use of statistical mechanics to analyse and classify granular materials, taking into account the micro-canonical volume ensemble, structural degrees of freedom, the canonical volume ensemble and the quasi-particles of the volume ensemble, the stress ensemble, and the relationship between the volume and stress ensembles. The article concludes with an assessment of recent advances in the ongoing attempt to construct a statistical mechanical theory of granular systems.

Statistical mechanics of inhomogeneous fluids

1982 ◽  
Vol 379 (1776) ◽  
pp. 231-246 ◽  
Keyword(s):  
Surface Tension ◽  
Free Energy ◽  
Angular Momentum ◽  
Statistical Mechanics ◽  
Stress Tensor ◽  
Pressure Difference ◽  
Strain Tensor ◽  
Pressure Tensor ◽  
Conservation Of Momentum ◽  
Statistical Mechanical

The nature of the microscopic stress tensor in an inhomogeneous fluid is discussed, with emphasis on the statistical mechanics of drops. Changes in free energy for isothermal deformations of a fluid are expressible as volume integrals of the stress tensor ‘times’ a strain tensor. A particular radial distortion of a drop leads to statistical mechanical expressions for the pressure difference across the surface of the drop. We find that the stress tensor is not uniquely defined by the microscopic laws embodying the conservation of momentum and angular momentum and that the am­biguity remains in the ensemble average, or pressure tensor, in regions of inhomogeneity. This leads to difficulties in defining statistical mechanical expressions for the surface tension of a drop.

scholarly journals Thermodynamic efficiency of contagions: a statistical mechanical analysis of the SIS epidemic model

2018 ◽  
Vol 8 (6) ◽  
pp. 20180036 ◽  
Author(s):  
Nathan Harding ◽  
Ramil Nigmatullin ◽  
Mikhail Prokopenko
Keyword(s):  
Mechanical Analysis ◽  
Contact Network ◽  
Thermodynamic Efficiency ◽  
Thermodynamic Quantities ◽  
Sis Model ◽  
Closed Form Solutions ◽  
Sis Epidemic Model ◽  
Novel Approach ◽  
Statistical Mechanical ◽  
Computational Processes

We present a novel approach to the study of epidemics on networks as thermodynamic phenomena, quantifying the thermodynamic efficiency of contagions, considered as distributed computational processes. Modelling SIS dynamics on a contact network statistical-mechanically, we follow the maximum entropy (MaxEnt) principle to obtain steady-state distributions and derive, under certain assumptions, relevant thermodynamic quantities both analytically and numerically. In particular, we obtain closed-form solutions for some cases, while interpreting key epidemic variables, such as the reproductive ratio of a SIS model, in a statistical mechanical setting. On the other hand, we consider configuration and free entropy, as well as the Fisher information, in the epidemiological context. This allowed us to identify criticality and distinct phases of epidemic processes. For each of the considered thermodynamic quantities, we compare the analytical solutions informed by the MaxEnt principle with the numerical estimates for SIS epidemics simulated on Watts–Strogatz random graphs.

scholarly journals Anisotropic nature of the capillary stress tensor

2021 ◽  
Vol 249 ◽  
pp. 11010
Author(s):  
Mojtaba Farahnak ◽  
Richard Wan ◽  
Mehdi Pouragha
Keyword(s):  
Stress Tensor ◽  
Relative Magnitude ◽  
Triaxial Tests ◽  
Mean Stress ◽  
Discrete Element Modeling ◽  
Capillary Stress ◽  
Micromechanical Approach ◽  
Characteristic Particle ◽  
Stress Transmission ◽  
Change In Mean

The paper describes a micromechanical approach that explores the anisotropic nature of the capillary stress tensor and its evolution in pendular granular materials via Discrete Element Modeling (DEM) simulations. Dimensionless parameters are used to address the conditions under which the contribution of capillarity (or cohesive interparticle forces) to the stress transmission within a Representative Elementary Volume (REV) is expected to be considerable. From a series of suction-controlled conventional triaxial tests, numerical results show that the significance of the capillary stress and the relative magnitude of its mean to deviatoric components is directly connected to the characteristic particle size and applied stress. In addition, it is shown that the anisotropic character of the capillary stress tensor intensifies with increasing suction. Furthermore, a simple shear test is conducted at constant mean stress to reveal the development of deviatoric capillary stresses in the absence of any change in mean stress, which cannot be captured by the commonly used Bishop’s stress expression.

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