Some Remarks on the Equilibrium of Granular Materials Described by Constitutive Relations That Depend on the Gradients of the Density or Volume Fraction

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
S. P. Atul Narayan ◽  
K. R. Rajagopal
Keyword(s):  
Granular Materials ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Short Note ◽  
The Body ◽  
Principal Stresses ◽  
Static State ◽  
State Of Stress ◽  
Axial State ◽  
Reasonable Model

Abstract One of the ways of determining the properties of granular materials is to subject the material to a tri-axial state of stress with three distinct eigenvalues, in a cubical tri-axial testing equipment, the material tested being in a static state. A constitutive relation for granular materials that does not allow the body to be in tri-axial static equilibrium would not be a reasonable model as experiments clearly show that granular materials can exist in such tri-axial states. In this short note, we observe that a large class of models that have been developed and used to describe the response of granular materials will not allow them to be in a static tri-axial state of stress with three different principal stresses.

Numerical Study of a Non-Linear Model for the Heat Flux Vector for Granular Materials

2012 ◽  
Author(s):  
Hyunjin Yang ◽  
Nadine Aubry ◽  
Mehrdad Massoudi
Keyword(s):  
Heat Flux ◽  
Granular Materials ◽  
Volume Fraction ◽  
Numerical Study ◽  
Constitutive Relations ◽  
Flux Vector ◽  
Heat Flux Vector ◽  
Non Linear ◽  
Effects Of Radiation ◽  
The One

The two important constitutive relations needed for the study of flow and heat transfer in granular materials, where the effects of radiation are ignored, are the stress tensor and the heat flux vector. Massoudi [1, 2] derived a constitutive model that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient and the velocity gradient, in an appropriate frame-invariant formulation. In this paper we use a simplified version of this model and consider the one dimensional fully developed flow of granular materials down a heated inclined plane, subject to a constant temperature boundary condition. The equations are made dimensionless and a parametric study is performed in order to examine the effects of the additional parameters on the heat flux vector. The derived governing equations are coupled non-linear second order ordinary differential equations which are solved numerically and the results are shown for the temperature, volume fraction and velocity profiles.

scholarly journals Phase-field gradient theory

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Luis Espath ◽  
Victor Calo
Keyword(s):  
Free Energy ◽  
Field Equation ◽  
Field Gradient ◽  
Phase Field ◽  
Constitutive Relations ◽  
The Body ◽  
Second Grade ◽  
Gradient Theory ◽  
Boundary Edge ◽  
Field Equations

AbstractWe propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

scholarly journals Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows

2015 ◽  
Vol 766 ◽  
pp. 337-367 ◽  
Author(s):  
Bartosz Protas ◽  
Bernd R. Noack ◽  
Jan Östh
Keyword(s):  
Turbulent Flows ◽  
High Speed ◽  
Eddy Viscosity ◽  
Broad Class ◽  
Constitutive Relations ◽  
Body Height ◽  
Mixing Layer ◽  
The Body ◽  
Stream Velocity ◽  
Initial Vorticity

AbstractWe propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

Errata

1993 ◽  
Vol 443 (1919) ◽  
pp. 625-627
Keyword(s):  
Granular Materials ◽  
Constitutive Relations

Proc. R. Soc. Lond . A 441, 433-463 (1993) Incremental constitutive relations for granular materials based on micromechanics By M. M. Mehra Badi, B. Loret and S. Nemat - Nasser Figures 6 and 10 in this paper were originally printed with an incorrect layout. They are printed below, with their correct layout, complete with captions, which remain unchanged.

Modeling and Experimental Validations of a Piezoelectric Energy Harvester Under Combined Galloping and Base Excitations

2014 ◽  
Author(s):  
Amin Bibo ◽  
Abdessattar Abdelkefi ◽  
Mohammed F. Daqaq
Keyword(s):  
Bluff Body ◽  
Energy Harvester ◽  
Constitutive Relations ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Beam Theory ◽  
Excitation Amplitude ◽  
The Body ◽  
Base Excitation ◽  
Piezoelectric Energy

This paper develops an experimentally validated model of a piezoelectric energy harvester under combined aeroelastic-galloping and base excitations. To that end, an energy harvester consisting of a thin piezoelectric cantilever beam subjected to vibratory base excitation is considered. To permit galloping excitation, a bluff body is rigidly attached at the free end such that a net aerodynamic lift is generated as the incoming airflow separates on both sides of the body giving rise to limit cycle oscillations when the flow velocity exceeds a critical value. A nonlinear electromechanical distributed-parameter model of the harvester under the combined excitation is derived using the energy approach and by adopting the nonlinear Euler-Bernoulli beam theory, linear constitutive relations for the piezoelectric transduction, and the quasi-steady assumption for the aerodynamic loading. The partial differential equations of the system are discretized and a reduced-order-model is obtained. The mathematical model is validated by conducting a series of experiments with different loading conditions represented by wind speed, base excitation amplitude, and excitation frequency around the primary resonance.

Transport properties of concrete-like granular materials interacted by their microstructures and particle components

2018 ◽  
Vol 32 (18) ◽  
pp. 1840011 ◽  
Author(s):  
Wenxiang Xu ◽  
Hongguang Sun ◽  
Wen Chen ◽  
Huisu Chen
Keyword(s):  
Transport Properties ◽  
Granular Materials ◽  
Soft Matter ◽  
Volume Fraction ◽  
Structural Characteristics ◽  
Effective Diffusion ◽  
Nonspherical Particles ◽  
Structure Property ◽  
Engineering Structures ◽  
Component Structure

Granular materials as typical soft matter, their transport properties play significant roles in durability and service life in relevant practical engineering structures. Physico-mechanical properties of materials are generally dependent of their microstructures including interfacial and porous characteristics. The formation of such microstructures is directly related to particle components in granular materials. Understanding the interactive mechanism of particle components, microstructures, and transport properties is a problem of great interest in materials research community. The resulting rigorous component-structure-property relations are also valuable for material design and microstructure optimization. This review article describes state-of-the-art progresses on modeling particle components, interfacial and porous configurations and incorporating these internal structural characteristics into modeling transport properties of granular materials. We mainly focus on three issues involving the simulation for geometrical components, the quantitative characterization for interfacial and porous microstructures, and the modeling strategies for diffusive behaviors of granular materials. In the first aspect, in-depth reviews are presented to realize complex morphologies of geometrical particles, to detect the overlap between adjacent nonspherical particles, and to simulate the random packings of nonspherical particles. In the second aspect, we emphasize the development progresses on the interfacial thickness and porosity distribution, the interfacial volume fraction, and the continuum percolation of soft particles representing compliant interfaces and discrete pores. In the final aspect, a literature review is also provided on modeling of transport properties on the forefront of the effective diffusion and anomalous diffusion in multiphase granular materials. Finally, some conclusions and perspectives for future studies are provided.

Review of an experiment to study the propane discharge in a closed production area

2021 ◽  
pp. 25-30
Author(s):  
Евгений Евгеньевич Простов
Keyword(s):  
Volume Fraction ◽  
The Body ◽  
Concentration Limit ◽  
Internal Diameter ◽  
Plastic Film ◽  
Confined Space ◽  
Gas Analyzer ◽  
Ansys Fluent ◽  
Gas Source ◽  
Gas Supply

В статье представлены результаты экспериментальных исследований истечения пропана в различных направлениях в закрытом помещении. Рассматривался случай, когда источник истечения находился в багажнике автомобиля - имитация нахождения автомобиля с газомоторным топливом на станции технического обслуживания. Целью эксперимента являлось изучение механизма пространственного распространения газа в закрытом помещении для валидации математических моделей, используемых в программном комплексе ANSYS Fluent при моделировании поступления пропана в закрытое помещение. This scientific work describes a test conducted in a multidisciplinary test box on the testing training ground of the Orenburg branch of the All-Russian Research Institute for Fire Protection of EMERCOM of Russia. For the experiment there was built a room to simulate a service station (or parking box) for two cars. The frame was made of wooden bars and a plastic film was used to isolate the internal volume. The experimental installation consisted of a gas source with an internal diameter of 5 mm, located in the center of the room, and a system for gas supply and registration of experimental data from six gas analyzers SGOES-2 with a measurement range of pre-explosive concentrations from 0 to 100% of the lower concentration limit of flame propagation (NKPR) or a volume fraction from 0 to 1.7% with absolute ± 5% NKPR (in the range from 0 to 50% NKPR) and relative ± 10% NKPR (in the range from 50 to 100% NKPR) errors. In the center of the experimental room there was placed a car with the gas source in the trunk. All openings to the interior were insulated with plastic film and mounting foam. Natural cracks were left between the trunk lid and the body. The gas source is located in the trunk of the car and is directed towards the wide part of the trunk at an angle of 30 degrees relative to the floor (simulating the location of the gas cylinder used in cars). The gas analyzers were located along the wall, where the outflow is directed along the perimeter of the trunk, and one gas analyzer was located directly in the trunk behind the gas analyzer to control gas contamination. Propane has been released into the trunk with a constant flow rate of 2.8 m/h for 5 minutes. There were 8 test starts of the gas supply system (the flow vertically down), and then there were carried out 3 experiments per 3 series of tests in each. The purpose of the test was to study the mechanism of spatial gas propagation in an confined space for validation of mathematical models used in the ANSYS Fluent software package when modeling the propane intake into the confined space

A Method for Tracking a Solid Body in a Fluid Field in Immersed Boundary Methods

2018 ◽  
Author(s):  
Guangfa Yao
Keyword(s):  
Level Set ◽  
Solid Body ◽  
Volume Fraction ◽  
Transformation Matrix ◽  
The Body ◽  
New Method ◽  
Immersed Boundary ◽  
Body Interaction ◽  
Level Set Function ◽  
Set Function

Immersed boundary method has got increasing attention in modeling fluid-solid body interaction using computational fluid dynamics due to its robustness and simplicity. It usually simulates fluid-solid body interaction by adding a body force in the momentum equation. This eliminates the body conforming mesh generation that frequently requires a very labor-intensive and challenging task. But accurately tracking an arbitrary solid body is required to simulate most real world problems. In this paper, a few methods that are used to track a rigid solid body in a fluid domain are briefly reviewed. A new method is presented to track an arbitrary rigid solid body by solving a transformation matrix and identifying it using a level set function. Knowing level set function, the solid volume fraction can be derived if needed. A three-dimensional example is used to study a few methods used to represent and solve the transformation matrix, and demonstrate the presented new method.

Fully Integrating a Hydraulic Fracturing, Reservoir, and Wellbore Simulator into a Practical Engineering Tool

2021 ◽  
Author(s):  
Mark McClure
Keyword(s):  
User Interface ◽  
Hydraulic Fracturing ◽  
Volume Fraction ◽  
3D Visualization ◽  
Constitutive Relations ◽  
Coupled Processes ◽  
Tight Coupling ◽  
Member States ◽  
Proppant Transport ◽  
Fully Coupled

<p>In this talk, I give an overview of our software ResFrac, which fully integrates a ‘true’ hydraulic fracturing simulator and a multiphase reservoir simulator (McClure et al., 2020a). Conventionally, these processes have been described with separate codes, using separate meshes, and with different physics. Integrating these two categories of software is advantageous because it enables seamless description of the entire lifecycle of a well. It is possible to seamlessly integrate wells with complex histories such as frac hits from offset wells, refracs, and huff and puff EOR injection.</p><p>ResFrac has been applied on 25+ studies for operators optimizing development of oil and gas resources in shale and has been commercially licensed by 15+ companies (https://www.resfrac.com/case-studies; https://www.resfrac.com/publications; https://www.resfrac.com/about-us/our-team). The simulator has a modern user-interface with embedded help-documentation, wizards to help set up simulations, automated validators to identify issues with the setup prior to submitting, and plotting capabilities to preview 3D and tabular inputs. Simulations are run on the cloud and results are continuously downloaded to the user’s computer. This allows a user to easily run a large number of simultaneous simulations from their personal computer. The user-interface includes a custom-built and fully-featured visualization tool for 3D visualization and 2D plotting.</p><p>Hydraulic fracturing simulators must handle a diverse set of coupled physics: mechanics of crack propagation and stress shadowing, fluid flow in the fractures, leakoff, transport of fluid additives that impart non-Newtonian flow characteristics, and proppant transport. Proppant transport is particularly complex because proppant settles out into an immobile bed and may screen out at the tip. Many fracturing simulators approximate wellbore flow effects. However, because these effects are closely coupled to fracturing processes (especially in horizontal wells that have multiple simultaneously propagating fractures), we include a fully meshed, detailed wellbore model in the code, along with treatment of perforation pressure drop and near-wellbore tortuosity.</p><p>In the literature, separate constitutive relations are available to describe transport in open cracks, closed unpropped cracks, and closed propped cracks. However, there were not relations in the literature designed to describe transport under conditions transitional between these end-member states. A general numerical simulator must be able to describe all conditions (and avoid discontinuous changes between equations). To address this limitation, we developed a new set of constitutive equations that can smoothly transition between these end-member states – smoothly handling any general combination of aperture, effective normal stress, saturation, proppant volume fraction, and non-Newtonian fluid rheology (McClure et al., 2020).</p><p>The code solves all equations in a fully coupled way, using an adaptive implicit method. The fully coupled approach is chosen because of the tight coupling between many of the key physical processes. Iterative coupling converges very slowly and/or forces excessively small timesteps when tightly coupled processes are handled with iterative or explicit coupling.</p><p>McClure, Kang, Hewson, and Medam. 2020. ResFrac Technical Writeup (v5). arXiv.</p>

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