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Μ(I) Rheology
In granular mechanics, the ''μ(I)'' rheology is one model of the rheology of a granular flow. Details The inertial number of a granular flow is a dimensionless quantity defined as I = \frac, where \dot\gamma is the shear rate tensor, , , \dot\gamma, , is its magnitude, ''d'' is the average particle diameter, ''P'' is the isotropic pressure and ''ρ'' is the density. It is a local quantity and may take different values at different locations in the flow. The ''μ''(''I'') rheology asserts a constitutive relationship between the stress tensor of the flow and the rate of strain tensor: \sigma_ = -P\delta_ + \mu(I)P \frac where the eponymous ''μ''(''I'') is a dimensionless function of ''I''. As with Newtonian fluids, the first term -''Pδ''''ij'' represents the effect of pressure. The second term represents a shear stress: it acts in the direction of the shear, and its magnitude is equal to the pressure multiplied by a coefficient of friction ''μ''(''I''). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Granular Mechanics
A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compose granular material are large enough such that they are not subject to thermal motion fluctuations. Thus, the lower size limit for grains in granular material is about 1 μm. On the upper size limit, the physics of granular materials may be applied to ice floes where the individual grains are icebergs and to asteroid belts of the Solar System with individual grains being asteroids. Some examples of granular materials are snow, nuts, coal, sand, rice, coffee, corn flakes, fertilizer, and bearing balls. Research into granular materials is thus directly applicable and goes back at least to Charles-Augustin de Coulomb, whose law of friction was originally stated for granular materials. Granular materials are commercially important in applic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid ( liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force. Rheology is a branch of physics, and it is the science that deals with the deformation and flow of materials, both solids and liquids.W. R. Schowalter (1978) Mechanics of Non-Newtonian Fluids Pergamon The term ''rheology'' was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920, from a suggestion by a colleague, Markus Reiner.The Deborah Number The term was inspired by the of |
Granular Flow
A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains collide). The constituents that compose granular material are large enough such that they are not subject to thermal motion fluctuations. Thus, the lower size limit for grains in granular material is about 1 μm. On the upper size limit, the physics of granular materials may be applied to ice floes where the individual grains are icebergs and to asteroid belts of the Solar System with individual grains being asteroids. Some examples of granular materials are snow, nuts, coal, sand, rice, coffee, corn flakes, fertilizer, and bearing balls. Research into granular materials is thus directly applicable and goes back at least to Charles-Augustin de Coulomb, whose law of friction was originally stated for granular materials. Granular materials are commercially important in applic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inertial Number
The Inertial number I is a dimensionless quantity which quantifies the significance of dynamic effects on the flow of a granular material. It measures the ratio of inertial forces of grains to imposed forces: a small value corresponds to the quasi-static state, while a high value corresponds to the inertial state or even the "dynamic" state. It is given by: I = \frac, where \dot\gamma is the shear rate, d the average particle diameter, P is the pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ... and \rho is the density. Generally three regimes are distinguished: * I<10^: quasi static flow * |
Dimensionless Quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1), ISBN 978-92-822-2272-0. which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds). Dimensionless units are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians. History Quantities having dimension one, ''dimensionless quantities'', regularly occur in sciences, and are formally treated within the field of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constitutive Relationship
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations. Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Stress Tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: :\mathbf^ = \mathbf e \cdot\boldsymbol\quad \text \quad T_^= \sigma_e_i, or, :\leftright\leftrightcdot \leftright The SI units of both stress tensor and traction vector are N/m2, corresponding to the stress scalar. The unit vector is dimensionless. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress. The Cauchy stress tensor is used for stress analysis of materi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newtonian Fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of change of the fluid's velocity vector. A fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic (mechanical properties are the same along any direction), the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion, respectively. Newtonian fluids are the simplest mathematical models of fluids that account for viscosity. While no real fluid fits the definition perfectly, many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. General shear stress The formula to calculate average shear stress is force per unit area.: : \tau = , where: : = the shear stress; : = the force applied; : = the cross-sectional area of material with area parallel to the applied force vector. Other forms Wall shear stress Wall shear stress expresses the retarding force (per unit area) from a wall in the layers of a fluid flowing next to the wall. It is defined as: \tau_w:=\mu\left(\frac\right)_ Where \mu is the dynamic viscosity, u the flow velocity and y the distance from the wall. It is used, for example, in the description of arterial blood flow in which case which ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Dry friction is subdivided into ''static friction'' ("stiction") between non-moving surfaces, and ''kinetic friction'' between moving surfaces. With the exception of atomic or molecular friction, dry friction generally arises from the interaction of surface features, known as asperities (see Figure 1). *Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. *Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces. *Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. *Internal friction is the force resisting motion between the elements making up a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hysteresis
Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction; and dissipation is a common side effect. Hysteresis can be found in physics, chemistry, engineering, biology, and economics. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Element Method
A discrete element method (DEM), also called a distinct element method, is any of a family of numerical methods for computing the motion and effect of a large number of small particles. Though DEM is very closely related to molecular dynamics, the method is generally distinguished by its inclusion of rotational degrees-of-freedom as well as stateful contact and often complicated geometries (including polyhedra). With advances in computing power and numerical algorithms for nearest neighbor sorting, it has become possible to numerically simulate millions of particles on a single processor. Today DEM is becoming widely accepted as an effective method of addressing engineering problems in granular and discontinuous materials, especially in granular flows, powder mechanics, and rock mechanics. DEM has been extended into the Extended Discrete Element Method taking heat transfer, chemical reaction and coupling to CFD and FEM into account. Discrete element methods are relatively com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
