Gravity-Driven Granular Flows of Smooth, Inelastic Spheres Down Bumpy Inclines

1990 ◽  
Vol 57 (4) ◽  
pp. 1036-1043 ◽  
Author(s):  
M. W. Richman ◽  
R. P. Marciniec
Keyword(s):  
Solid Fraction ◽  
Constitutive Relations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Shear Stresses ◽  
Steady Flows ◽  
Mean Velocity ◽  
Flow Rates ◽  
Gravity Driven ◽  
Fast Flow

We employ a kinetic constitutive theory that includes the effects of particle transport and collisions to analyze steady, fully developed, gravity-driven flows of smooth, inelastic spheres down bumpy inclines. By replacing the solid fraction by its depth-averaged value wherever it occurs in the balance equations, we obtain a closed-form solution for the granular temperature profile, which when employed in the constitutive relations for the normal and shear stresses determines the solid fraction and mean velocity profiles. The solutions ensure that the stresses and energy flux vanish at the free surface and that the balance of momentum and energy are satisfied at the base. We also determine for fixed flow rates the ranges of inclinations within which steady flows are possible, and for fixed inclinations the depths of the resulting flows. Most striking are the inclinations and flow rates at which both a dilute, fast flow and a dense, slow flow are possible.

scholarly journals Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. The closed form solution is validated by both direct numerical simulation and experiments. The studies confirm that the van Driest mixing length theory is suitable for smooth walls, and the Prandtl mixing length theory is suitable for rough walls. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.

scholarly journals Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this letter, a century-old problem is studied; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. Considering the Prandtl mixing length model, a closed form solution of the mean velocity profile of plane turbulent flow is obtained, and approximate analytical solution of the van Driest mixing length theory is proposed. The profiles of several useful quantities are given, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. It is shown that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. The closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is given in implicit form.

Effect of temperature variation on the plate-end debonding of FRP-strengthened beams: A theoretical study

2021 ◽  
pp. 136943322110463
Author(s):  
Dong Guo ◽  
Wan-Yang Gao ◽  
Dilum Fernando ◽  
Jian-Guo Dai
Keyword(s):  
Temperature Variation ◽  
Thermal Loading ◽  
Closed Form Solution ◽  
Form Solution ◽  
Effect Of Temperature ◽  
Shear Stresses ◽  
Limited Information ◽  
Element Analysis ◽  
Bond Slip ◽  
Loading Effects

Steel/concrete structures strengthened with externally bonded FRP plates may be subjected to significant temperature variations during their service time. Such temperature variation (i.e., thermal loading) may significantly influence the debonding mechanism in FRP-strengthened structures due to the thermal incompatibility between the FRP plate and the substrate as well as the temperature-induced bond degradation at the FRP-to-steel/concrete interface. However, limited information is available on the effect of temperature variation on the debonding failure in FRP-strengthened beams. This paper presents a new and closed-form solution to investigate the plate-end debonding failure of the FRP-strengthened beam subjected to combined thermal and mechanical (i.e., flexural) loading. A bilinear bond-slip model is used to describe the bond behavior of the FRP-to-substrate interface. The analytical solution is validated through comparisons with finite element analysis results regarding the distributions of the interfacial shear stresses, the interfacial slips and the axial stresses of the FRP plate. Given that a constant bond-slip relationship is adopted, it is observed that an increase in service temperature will lead to an increased interfacial slip at the plate end and consequently a reduced plate-end debonding load, and vice versa. Further parametric studies have indicated that the thermal loading effects become more significant when shorter and stiffer FRP plates are applied for strengthening.

A Circular Inclusion With Imperfect Interface: Eshelby’s Tensor and Related Problems

1995 ◽  
Vol 62 (4) ◽  
pp. 860-866 ◽  
Author(s):  
Zhanjun Gao
Keyword(s):  
Closed Form Solution ◽  
Subsequent Development ◽  
Physical Nature ◽  
Circular Inclusion ◽  
Imperfect Interface ◽  
Form Solution ◽  
Normal Displacement ◽  
Tangential Displacement ◽  
Shear Stresses ◽  
Eshelby’S Tensor

Eshelby’s tensor for an ellipsoidal inclusion with perfect bonding at interface has proven to have a far-reaching influence on the subsequent development of micromechanics of solids. However, the condition of perfect interface is often inadequate in describing the physical nature of the interface for many materials in various loading situations. In this paper, Airy stress functions are used to derive Eshelby’s tensor for a circular inclusion with imperfect interface. The interface is modeled as a spring layer with vanishing thickness. The normal and tangential displacement discontinuities at the interface are proportional to the normal and shear stresses at the interface. Unlike the case of the perfectly bonded inclusion, the Eshelby’s tensor is, in general, not constant for an inclusion with the spring layer interface. The normal stresses are dependent on the shear eigenstrain. A closed-form solution for a circular inclusion with imperfect interface under general two-dimensional eigenstrain and uniform tension is obtained. The possible normal displacement overlapping at the interface is discussed. The conditions for nonoverlapping are established.

scholarly journals Analysis of the Fully Developed Chute Flow of Granular Materials

1992 ◽  
Vol 59 (1) ◽  
pp. 109-119 ◽  
Author(s):  
Hojin Ahn ◽  
Christopher E. Brennen ◽  
Rolf H. Sabersky
Keyword(s):  
Boundary Value Problem ◽  
Granular Materials ◽  
Solid Fraction ◽  
Shooting Method ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Granular Temperature ◽  
Mean Velocity ◽  
Chute Flow ◽  
Two Parameters

Existing constitutive relations and governing equations have been used to solve for fully developed chute flows of granular materials. In particular, the results of Lun et al. (1984) have been employed and the boundary value problem has been formulated with two parameters (the coefficient of restitution between particles, and the chute inclination), and three boundary values at the chute base wall, namely the values of solid fraction, granular temperature, and mean velocity at the wall. The boundary value problem has been numerically solved by the “shooting method.” The results show the significant role played by granular conduction in determining the profiles of granular temperature, solid fraction, and mean velocity in chute flows. These analytical results are also compared with experimental measurements of velocity fluctuation, solid fraction, and mean velocity made by Ahn et al. (1989), and with the computer simulations by Campbell and Brennen (1985b).

A New Method for Analyzing Thick Walled Shape Memory Alloy Cylinders Subjected to Internal Pressure

2009 ◽  
Author(s):  
Reza Mirzaeifar
Keyword(s):  
Shape Memory Alloy ◽  
Internal Pressure ◽  
Shape Memory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
New Method ◽  
Plane Stress Condition ◽  
Three Dimensional Finite Element

In this paper, a new method is proposed for analyzing thick walled shape memory alloy cylinders subjected to internal pressure. The plane stress condition is assumed for the cylinder and three-dimensional phenomenological macroscopic SMA constitutive model presented by Boyd and Lagoudas is simplified to obtain the required two-dimensional constitutive relations. The cylinder is divided to some narrow annular regions and appropriate assumptions are made in order to find a closed-form solution for the equilibrium equation in terms of radial displacements in each region. Numerical examples are presented for demonstrating the performance of the proposed method and the results are compared with three-dimensional finite element method.

Transient Response of an Infinite Elastic Medium Containing a Spherical Cavity Subjected to Torsion

1999 ◽  
Vol 67 (2) ◽  
pp. 282-287 ◽  
Author(s):  
U. Zakout ◽  
Z. Akkas ◽  
G. E. Tupholme
Keyword(s):  
Elastic Medium ◽  
Transient Response ◽  
Spherical Cavity ◽  
Laplace Transformation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Shear Stresses ◽  
Surface Loading ◽  
Transformation Techniques ◽  
Isotropic Elastic Medium

An exact closed-form solution is obtained for the transient response of an infinite isotropic elastic medium containing a spherical cavity subjected to torsional surface loading using the residual variable method. The main advantage of the present approach is that it eliminates the computational problems arising in the existing methods which are primarily based on Fourier or Laplace transformation techniques. Extensive numerical results for the circumferential displacements and shear stresses at various locations are presented graphically for Heaviside loadings. [S0021-8936(00)01102-8]

scholarly journals Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface

2022 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Energy Dissipation ◽  
Turbulent Flow ◽  
Closed Form ◽  
Scaling Law ◽  
Closed Form Solution ◽  
Form Solution ◽  
Analytic Description ◽  
Mixing Length ◽  
Mean Velocity ◽  
Mixing Length Theory

In this paper, a century-old problem is solved; namely, to find a unified analytic description of the non-uniform distribution of mean velocity across the entire domain of turbulent flow for all Reynolds numbers within the framework of the Prandtl mixing length theory. This study obtains a closed form solution of the mean velocity profile of plane turbulent flow for the Prandtl theory, and as well an approximate analytical solution for the van Driest mixing length theory. The profiles of several useful quantities are given based the closed form solution, such as turbulent viscosity, Reynolds turbulent stress, Kolmogorov's scaling law, and energy dissipation density. The investigation shows that the energy dissipation density at the surface is finite, whereas Landau's energy dissipation density is infinite. Strictly speaking, the closed form solution reveals that the universality of the turbulent velocity logarithmic profile no longer holds, but the von K\'arm\'an constant is still universal. Furthermore, a new formulation of the resistance coefficient of turbulent flow in pipes is formulated in implicit form.

Micropolar Constitutive Relations for Cellular Solids

2016 ◽  
Vol 83 (4) ◽  
Author(s):  
Armanj D. Hasanyan ◽  
Anthony M. Waas
Keyword(s):  
Closed Form ◽  
Material Properties ◽  
Constitutive Relations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Constitutive Relationship ◽  
Cellular Solids ◽  
Multiplicative Constant ◽  
Cellular Solid ◽  
Numerical Predictions

With the recent development of micromechanics in micropolar solids, it is now possible to characterize the macroscopic mechanical behavior of cellular solids as a micropolar continuum. The aim of the present article is to apply these methods to determine the micropolar constitutive relation of various cellular solids. The main focus will be on the hexagonal packed circular honeycomb to demonstrate how its constitutive relationship is obtained. In addition, the same method will be applied to determine the material properties of a grid structure and a regular hexagon honeycomb. Since we model the cellular solid as an assembly of Euler–Bernoulli beams, we know that the macroscopic material properties will depend on the cell wall thickness, length, and Young's modulus. From this, and in conjunction with nondimensional analysis, we can provide a closed form solution, up to a multiplicative constant, without resorting to analyzing the governing equations. The multiplicative constant is evaluated through a single numerical simulation. The predicted values are then compared against assemblies with different local properties, using the numerical result as a benchmark since it takes into account higher order thickness effects. It is concluded that our closed form expressions vary from the numerical predictions only when the thickness of the beams increase, as expected since shear effects must be taken into account. However, for most engineering applications, these expressions are practical since our closed form solution with the Euler–Bernoulli assumption only produces about 10% error for most extreme cases. Our results are also verified by comparing them against those reported in the literature.

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