granular matter or granulated matter is that which is formed by a set of solid macroscopic particles large enough so that the force of interaction between them is that of friction. Collectively, this type of matter presents properties that can resemble, depending on the type of forces to which it is subjected, those of the solid state, the liquid state or a gas.[1] An important characteristic is that granular matter tends to quickly dissipate the energy of its particles due to the force of friction. This gives rise to phenomena of great importance such as avalanches, blockages in silo discharges, among others. Examples of granular matter are grains and seeds, snow, sand, etc.[2].
Although granular matter has been known since ancient times, the appearance of phenomena that seem counterintuitive, such as the Brazil nut effect, has led to increased study by physicists in recent years. The study of this type of materials is of utmost importance because it is the type of matter most used by man only after water.
History
The study of granular matter began in ancient times, despite not having received the same attention as other areas of physics. The oldest record on this subject comes from the Roman poet Lucretius. Around the year 55 BC. C. wrote:[3].
In the Renaissance, Leonardo da Vinci carried out experiments that demonstrated the laws of dry friction"). Charles de Coulomb, in the 19th century, wrote the article "Essay on maxima and minima applied to balance problems related to architecture", where he exposes observations and experiments on the balance of embankments and structures composed of rocks.[4].
Later, Ernst Chladni used granular materials placed on metal plates to study the vibration modes of the latter. From his work, what is known today as Chladni figures was discovered.[5] A problem related to the previous one was studied by Michael Faraday, who investigated the formation of sand piles when subjected to vibrations.[6] William Rankine studied friction in granular materials and, based on Coulomb's work, established what in soil mechanics is known as Rankine active states").[7]
Later, various researchers studied the way in which the forces of the grains stored in a silo are distributed. I. Roberts") studied the dependence of the pressure of grains against the walls of the silo.[8] H. Janssen") described how the pressure forces changed direction towards the walls.[9] Lord Rayleigh established an analogy between this problem and the tensile strength of a rope wrapped around a pole.[10][11].
granular layers
Introduction
granular matter or granulated matter is that which is formed by a set of solid macroscopic particles large enough so that the force of interaction between them is that of friction. Collectively, this type of matter presents properties that can resemble, depending on the type of forces to which it is subjected, those of the solid state, the liquid state or a gas.[1] An important characteristic is that granular matter tends to quickly dissipate the energy of its particles due to the force of friction. This gives rise to phenomena of great importance such as avalanches, blockages in silo discharges, among others. Examples of granular matter are grains and seeds, snow, sand, etc.[2].
Although granular matter has been known since ancient times, the appearance of phenomena that seem counterintuitive, such as the Brazil nut effect, has led to increased study by physicists in recent years. The study of this type of materials is of utmost importance because it is the type of matter most used by man only after water.
History
The study of granular matter began in ancient times, despite not having received the same attention as other areas of physics. The oldest record on this subject comes from the Roman poet Lucretius. Around the year 55 BC. C. wrote:[3].
In the Renaissance, Leonardo da Vinci carried out experiments that demonstrated the laws of dry friction"). Charles de Coulomb, in the 19th century, wrote the article "Essay on maxima and minima applied to balance problems related to architecture", where he exposes observations and experiments on the balance of embankments and structures composed of rocks.[4].
Later, Ernst Chladni used granular materials placed on metal plates to study the vibration modes of the latter. From his work, what is known today as Chladni figures was discovered.[5] A problem related to the previous one was studied by Michael Faraday, who investigated the formation of sand piles when subjected to vibrations.[6] William Rankine studied friction in granular materials and, based on Coulomb's work, established what in soil mechanics is known as Rankine active states").[7]
Towards the end of the century Osborne Reynolds made important discoveries that contributed to the study of granular matter.[12] From that moment on, during the course of the century and to date, the number of scientists dedicated to the study of granular matter has been increasing. Among them, one of the most important has been Ralph Bagnold"), who between 1940 and 1970 made observations of desert sands.[13][14].
Properties
particle size
Granular materials are composed of a large number of solid particles, which are discernible with the naked eye. The size of the particles usually ranges from a few microns to the order of meters or larger. As examples of the above, there is the case of powders where their particles are so small that they can barely be distinguished with the naked eye. In the opposite case, you can have particles as large as rocks that can measure several meters, and even asteroids, with sizes of several hundred meters.
Forces, accelerations and energies
The main property of granular matter is that the only interaction force that exists between the particles that compose it is static friction. An exception to this occurs in the case of finer powders, in which electrostatic interactions may appear when their particles become electrically charged. The existence of static friction as the predominant force between the particles of these materials gives rise to a rapid dissipation of the kinetic energy of the particles, because it generates inelastic collisions between them. For this reason it is not possible to study granular matter with statistical mechanics models for systems where there is conservation of energy. As a consequence of this, it can be said that the effective temperature of a granular material is zero and the only energy of relevance in this type of systems is the potential energy, due to its position with respect to a gravitational field.[15].
Different external forces can act on granular media, which are capable of substantially modifying their global behavior. The main external force to which granular materials are usually subjected is the force of gravity. This force generates a distribution of stresses through the particles of the material. These tensions support the material and allow it to maintain a defined shape. On the other hand, when the material is allowed to slide or fall, gravity forces it to behave similarly to a fluid, as seen in hourglasses.
If granular matter is subjected to periodic shocks, various types of phenomena usually occur, such as convection, particle segregation, among others.[16] The strength of these shocks can be measured in terms of the acceleration created by them. In the case of a shock consisting of a periodic sinusoidal oscillation, the average acceleration, , in one cycle is:.
where T is the period of oscillation, A is the amplitude of oscillation, is the angular frequency of the oscillation and t is the time. The above can be expressed in terms of a Froude number, which gives an idea of the magnitude of the inertial forces with respect to the forces of gravity. In this case, said number is taken as a dimensionless acceleration denoted by:.
being g the value of the acceleration of gravity.
Temperature
As already mentioned, due to the almost instantaneous loss of kinetic energy of the particles in the granular matter, its effective temperature has a value of zero. However, if the granular material is continually subjected to oscillating forces, such as shaking, the particles acquire a velocity. From this velocity and obtaining its quadratic mean, a "granular temperature" can be calculated, just as it would be done with an ideal gas:[17].
Where is the root mean square of the velocity, is the Boltzmann constant, is the temperature and is the mass of the material.
However, it is important to note that, when the force that generates the movement of the particles ceases, the granular medium loses its kinetic energy almost immediately, which is why the temperature returns to its value of zero. It is for this reason that it is not possible to apply the principles of classical thermodynamics to granular matter. That is, classically (see Laws of Thermodynamics) one would expect that energy would be conserved, the entropy of the system would increase naturally, and zero temperature could not be reached. However, none of the above situations occur with granular matter.
To take into account the temperature of a granular medium, it is necessary to use thermodynamic models for non-equilibrium systems. Many researchers of granular matter have tended not to take granular temperature into account or to neglect it, eliminating it from the equations of motion. However, other authors have tried to show that said temperature is necessary to describe this type of materials.[18].
Polydispersity
In polymer science, when you have a set of polymer molecules such that certain molecules are larger than others, the set is said to be polydispersed. In the case of polymers, it is more convenient to pay attention to the difference between the masses of the molecules than to their size. The polydispersity index") or polydispersity (PDI, from English: Polydispersity index) gives us an idea of the diversity of molecules existing in a mixture. This index is calculated by dividing the average mass per molecular weight by the average mass per number of molecules. That is to say:
, the average mass times molecular weight, is calculated by adding the products of the mass of the total number of molecules of a given species and the mass of one molecule of said species until all types of molecules are taken into account, and dividing that sum by the mass of all the molecules. It is simply the sum of the masses of each molecule divided by the total number of molecules. When , all the molecules are of the same type and the mixture is then said to be monodisperse.[19].
By analogy, in granular matter a granular polydispersity is defined. However, in this case the difference in particle size is taken into account instead of the difference in mass. To calculate polydispersity in the study of granular matter, it is necessary to count the number of particles that have a certain diameter in order to obtain the diameter distribution. The polydispersity is then obtained by calculating the variance of said distribution:
with the diameter of a given particle, the average diameter of the particles, the total number of particles and the variance of the distribution.[20].
Polydispersion in granular mixtures is of utmost importance because, in polydisperse materials subject to vertical oscillatory movements, particle segregation phenomena usually appear in which they are separated by size.[21][22].
Compaction
The particles that make up a granular material can be distributed in different ways within it. When you have spherical particles, a percentage of the volume of the granular material corresponds to the spheres themselves, while another percentage of the volume corresponds to the voids that form between the particles. The ratio between the volume occupied by the particles and the total volume of the material—particles and voids—is known as volume fraction"), represented by .
The volume fraction gives us an idea of how compact a granular material is. In the case of monodisperse materials, those that tend to have lower compaction have a volume fraction of around 0.56. By shaking the materials, greater compactions are usually achieved; the maximum of them reached by this way is 0.68 (in which case it is known as random close packing or RCP, from English: Random Close Packing). The maximum compaction possible in monodisperse materials is achieved by accommodating the hexagonal close packing (HCP) particles. When this is the case, the volume fraction reaches 0.74.[23].
Ratios between dissipative forces
When a granular material flows, different dissipative forces—friction between particles, air resistance, etc.—occur and alter its behavior. There are different ways to analyze these behaviors. One of them is separating the dissipative forces into four classes: collisional forces, friction forces, viscosity and pore pressure.[24] If ratios are made between these forces, the following dimensionless numbers are obtained:
• - Bagnold number: ratio between collisional forces and viscous forces.
• - Savage number"): quotient between collisional and friction forces.
• - Friction number"): ratio between friction forces and viscous forces.
• - Darcy number: ratio between friction and pore pressure forces.
Depending on the way in which each of these forces are calculated, different formulas are obtained for each of these numbers, although all of them, in one way or another, depend on the density of the solid particles.[25].
Particle shape
Although the simplest way to study granular matter is to assume that the particles that compose it are spherical, in many cases this is not the case. In a large number of situations particles can have shapes other than spheres. For example, lentil grains are shaped like oblate spheroids, rice grains are shaped like prolate spheroids, salt grains are cubic shaped, etc.
When studying granular media it is important to take into account the shape of its particles. It has been discovered that the shape of the grains can modify the stress distribution in granular materials at rest.[26] Grains with an elongated shape can modify friction and make the flow of the granular material more difficult because energy is lost when they rotate.[27] On the other hand, a material composed of oblate or prolate spheroids can reach a greater volume fraction than one composed of spheres.[28].
Behavior
Contenido
Como ya se ha explicado anteriormente, la materia granular exhibe diferentes comportamientos dependiendo del tipo de fuerzas externas a las que esté sujeta. Dichos comportamientos pueden semejar el de un sólido, el de un líquido o el de un gas. Cuando el material se encuentra en reposo, se comporta como un sólido. Si el material se encuentra bajo la acción de la gravedad, su comportamiento es similar al de un fluido viscoso. Bajo la acción de oscilaciones periódicas de baja aceleración, el material presenta comportamientos similares a los que presentan los fluidos en convección. En el caso en el que el material es sujeto a oscilaciones de alta aceleración, éste asemeja a un gas cuyas partículas sufren colisiones inelásticas.
No obstante, la descripción de los medios granulados no es simple; una gran cantidad de fenómenos que aparentan desafiar la intuición se presentan, debido a la naturaleza disipativa de las fuerzas existen en ellos.
Granular matter at rest
A granular material is at rest when the sum of forces acting on it and on each of the particles that compose it is equal to zero. When this occurs, the behavior of the granular medium resembles that of a solid. This similarity, however, is usually very easily lost by simply applying a small force to the material. A pile of grains, for example, can lose its solidity and begin to flow by simply tilting the material.[29] Depending on the circumstances in which a granular material is at rest—for example, the way it is stored—different behaviors are observed that have been studied to a greater or lesser extent by granular media physicists, engineers, geologists, among others.
A granular material is at rest only if the sum of forces on each of its particles is equal to zero (according to Newton's first law). For this to happen, the weight of a given particle must be balanced by the normal force and static friction due to neighboring particles. Put another way, a grain must be supported by the particles below and to the sides of it to avoid falling. In turn, the particles below it must be supported by others below, and so on until they reach the bottom or walls of the container. This succession of forces can be seen as a chain of efforts; Each part of the granular material is supported by forces transmitted from particle to particle until reaching the base of the container.[30] Likewise, if a force is applied to the surface of the medium, said force will be transmitted downwards and to the sides within the material, being distributed among all the grains. This explains why a person can remain standing on sand: although the force due to his weight is great, it is distributed among many grains.[31].
The transmission of forces from particle to particle can only occur through the point of contact between the grains. The number of contact points that particles have with each other depends largely on the volume fraction of the granular material. The further apart the grains are from each other—that is, if the volume fraction is smaller—there will be fewer contact points per particle and the transmission of stress will be less efficient. The way the stress chains are created then depends largely on the way the particles are arranged in the material. A slight change in the compaction of the medium will cause the chains to adopt another shape.[30].
A phenomenon associated with the formation of stress chains is the formation of arches. When sufficient pressure is exerted on a granular medium, the stress chains take the shape of an arc. Thanks to this, the material can have sufficient support. The reason why the arches are formed can be explained using variational calculus: it can be demonstrated mathematically that by placing a sequence of spheres supported by static friction, the most stable possible way to arrange them is that described by an inverted catenary.[31].
When a set of particles are stored, with no other structure to support them except the ground, the static friction forces between them force said set to form a conical structure "Cone (geometry)"). In mechanics, the static friction of a material can be calculated experimentally by placing two objects—for example, two blocks with flat surfaces—made of the same material on top of each other. If you begin to slowly tilt this system, there will come a time when the upper block will slide, overcoming that friction force. The theoretical angle of inclination, , at which this force is overcome is calculated as follows:[32].
The symbol represents the coefficient of static friction which depends mainly on the roughness of the material.
In granular materials this angle is known as the angle of repose. This angle defines the maximum slope "Slope (geography)") that a pile of particles can have without them precipitating in the form of an avalanche and is the angle formed between the ground and the surface of the mound. Because granular matter is not a continuous medium, but is made up of discrete particles and voids, the friction force is not constant over the entire surface of the material. The volume fraction of the material, the shape of the particles, among other factors, influence the way friction acts. For this reason, an angle of inclination equal to is not a guarantee of stability in the material. A small force on it can cause the grains to slide, similar to what is observed in snow avalanches. Despite the above, no pile of granular material can exist if the angle of inclination of its walls is greater than the angle of repose.
When other types of forces exist between the particles in the granular material—which can collectively be considered cohesive forces "Cohesion (force)")—such as electrical charges, the particles have greater difficulty sliding downward, so the stack of particles may have a greater angle of inclination and, therefore, the angle of repose increases. When this occurs, an angle of internal friction is defined as the angle that the mound would have if only static friction forces acted within it. In this case, this last angle is always less than the angle of repose and, only when the cohesive forces are zero, both angles coincide.[33].
When a fluid is placed in a cylindrical container, it is well known that the pressure at the bottom of said container increases as the height to which it is filled increases. Hydrostatic pressure can be calculated through Stevin's law as follows:
where is the hydrostatic pressure, is the density of the fluid, is the value of the acceleration of gravity and is the height of the fluid column.[34].
In the case of granular matter, it would be expected that when filling a silo—or any cylindrical container—with grains, the pressure at the bottom would increase in the same way that occurs for simple fluids. However, a granular material stops increasing pressure on the bottom of its container once a certain height is reached. H. A. Janssen discovered that the pressure on the walls of a container containing a granular material follows the following relationship:[9].
In this case, it is a parameter that depends on the static friction between the walls of the silo and the grains and its value is usually the order of magnitude of the radius of the container. This behavior is known as Janssen effect").[11].
The explanation for this effect lies in the way in which the stresses are transmitted between the grains: depending on the way in which the particles are distributed, the stress chains tend to direct the force due to the weight of the material towards the walls of the container. In simple fluids, the pressure at a certain point is directed in all directions (obeying Pascal's principle). On the other hand, in granular media, the pressure can follow different contact paths until reaching the walls. For this reason there is no equitable distribution horizontally and vertically; More pressure is directed towards the walls than towards the bottom.
The Janssen effect represents a problem for engineers, since, if they calculate the pressure of a granulated medium on the wall of a silo as if it were hydrostatic pressure, they can underestimate the resistance that said walls would need to have, even causing an explosion in the silo.[35].
Granular materials undergo a change in their volume fraction when they are subjected to pressure. The phenomenon was first described by Osborne Reynolds in 1885.[12] Reynolds verified this phenomenon by filling a rubber container with sand and water, adding a glass tube to the mouth of the container, so that the water reached a certain level inside the tube. By compressing the rubber container with his hands, the water level in the tube dropped, contrary to what would be expected. This phenomenon is known as *Reynolds dilation").[36].
The explanation of this phenomenon, given by Reynolds himself, consists of the change in the volume fraction of the granular material. When sand is compressed, its grains are rearranged in their positions, in such a way that the empty space between the particles increases. When this occurs, the water occupies these new spaces and its level drops. This phenomenon is also observed on beaches: when a person walks on wet sand, the footprints seem to dry out. The explanation is the same: the pressure due to the weight of the person walking on the beach generates a change in the volume fraction of the sand and the water within it drops in level, making the surface of the sand appear dry.[37].
Behavior at low accelerations
A granular medium that ceases to be at rest, either due to the action of gravity or periodic shaking, usually behaves, in most cases, in a manner very similar to that of a fluid. When a granular material moves thanks to the force of gravity through a hole (for example, when unloading a silo or in an hourglass) a flow of grains is generated, which depending on the size and shape of the particles can be continuous or interrupted by particle clogging. On the other hand, when a granular material is subjected to periodic shaking, a phenomenon similar to the convection that simple fluids present is usually present.[16] If the material is also polydispersed, a segregation of particles by size is observed, giving rise to phenomena such as the Brazil nut effect.[38].
The transition between a static granular medium and a flowing one usually occurs when an external oscillatory force is initiated, giving rise to fluidization.[39] This effect generates the loss of solidity of the material, causing an object that is on the surface of the medium to sink.[40] Fluidization has an extremely destructive effect in earthquakes, since when the apparently stable but water-saturated earth moves, liquefaction occurs "Liquefaction (instability)") and the constructions on its surface lose support and collapse when sinking into it.[41].
When a set of grains are poured into a container, the material usually has a low compaction, with a volume fraction of around 0.55. To reduce the volume occupied by the empty spaces, so that the material as a whole occupies less space, the medium must be subjected to horizontal vibrations. In this way, the volume fraction increases and can reach higher values. This makes it easier to store grains, since smaller containers are required.[42].
Usually, the volume fraction does not exceed a value of 0.64.[43] However, using different configurations, such as combined horizontal and vertical vibrations,[44] or systems with few particles,[45] vibration-induced compactions can be achieved with fractions greater than this value. The disadvantage of the above is that the shape of the container as well as the configuration of the system itself affect the behavior of the granular material, so they cannot be considered general cases.[42].
To achieve the maximum volume fraction, 0.74, which corresponds to a granular crystal arranged in a hexagonal shape by means of vibration, it has been necessary to resort to more sophisticated methods. A layer of granular material is placed in a vacuum on a perforated metal plate. In this way, the particles are forced to fall on the perforations and form an ordered arrangement. When the layer is complete, the next layer is placed. If any particle was disordered, that "imperfection" would be removed by hand.[46] The first perfect arrangement, without defects, achieved by purely mechanical means and without the need for manual intervention was reported by Nahmad-Molinari and Ruiz-Suárez in 2002, who used a type of epitaxial growth. In their method, they use a triangular prism-shaped container into which they throw steel spheres one by one. The container is shaken vertically with accelerations slightly greater than gravity. When a certain amount of particles are in the container, they agglomerate due to inelastic collisions between them, in such a way that a nucleus of spheres is generated in constant contact with each other. The new particles that are most recently ejected bind to this nucleus until the first layer is formed. In the next layer, the particles occupy the rest positions between the gaps in the spheres below, similar to the method before this one. Because the process is "one at a time" and the low acceleration to which the system is subjected, each sphere "searches" for its assumption and the second layer is completed. Finally the rest of the layers are formed until the container is filled.[23].
Granular gases
Granular matter subjected to high accelerations usually behaves in a similar way to a theoretical molecular gas. In the latter the molecules that form them undergo elastic collisions and a conservation of energy can be assumed as occurs in the theory of ideal gases. However, this is not the case for granular media. Each particle loses a part of its kinetic energy when colliding with another, converting said energy into heat, sound, vibration, rotation or another form of energy (that is, an inelastic collision occurs). The amount of energy lost in each collision depends on the coefficient of restitution of the material that makes up the grain.[70] When the granular system is considered as a whole, the energy loss depends on the number of collisions that occur in a given time, as an example, if a marble is dropped into a glass container, it will bounce a certain number of times until it finally stops. However, if a large number of these objects are dropped at the same time into the same container, the system as a whole will stop almost instantaneously. because the number of collisions is much greater.[71].
For a granulated medium to behave like a gas, it must be subjected to a sufficiently large constant force. If the force that maintains it in this state is suddenly stopped, the material will come to rest almost immediately. Furthermore, if the acceleration is not high enough, the material will have a certain time to relax, and its behavior will resemble that of a liquid (see the section "Granular convection").[71].
The dissipative nature of collisions makes granular gases systems out of thermodynamic equilibrium. This fact generates certain phenomena that at first glance would seem to violate the laws of thermodynamics, if the system is studied in a simple way without considering these energy losses. Among the phenomena that appear in these media we can mention granular agglomeration, the rupture of the equipotentialization of energy and inelastic collapse.[70].
Because in granular matter the collisions between particles are essentially inelastic, the amount of energy dissipated during a certain time depends on the number of collisions that occur in that period. The greater the number of collisions, the greater the energy that the system loses. In a granular gas, collisions occur randomly. Normally it would be expected that, on average, the number of collisions per unit of time in a given area of the gas is equal to the number of collisions in another area of the same size as the first in the same time. However, this number is subject to statistical fluctuations"), so there is a probability that in a certain area, for a moment a slightly greater number of collisions will occur than in another. This will produce a greater loss of energy in that region, resulting in a decrease in the speed of the particles. When this occurs, the pressure exerted by them will decrease, causing other particles that were not in that area to enter it, further increasing the number of collisions and the loss of energy. In the end, the gas granular will present a non-homogeneous appearance"), with some regions with very low densities with particles moving at high speed and others populated with a large number of grains agglomerated against each other.[72].
General
Literature
• - Duran, J., Reisinger A., Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials. November 1999, Springer-Verlag New York, Inc., New York, ISBN 0-387-98656-1.
• - Aste, Tomaso; Di Matteo, T.; Tordesillas, A. Granular and complex materials. 2007, World Scientific, ISBN 981-277-198-0.
References
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[2] ↑ Duran, p. 1-5.
[3] ↑ Duran, p. 16.
[4] ↑ Coulomb, Charles Augustin (1773). Academie Royal des Sciences Mem. Mat. et Phys. par Diver Savants, 7, 34.
[5] ↑ Rossing, Thomas D (1982). «Chladni's Law for Vibrating Plates» American Journal of Physics, 50, 3.
[6] ↑ a b Faraday, Michael (1831). «On a peculiar class of acoustic figures, and on certain forms assumed by groups of particles upon vibrating surfaces». Phylos. Trans. Roy. Soc., 52, 299. Londres.
[7] ↑ Rankine, William J. W. (1857). Phil. Trans. Roy. Soc., 147, 9.
[8] ↑ Roberts, I. (1857). Proccedings of the Royal Society, 147, 9.
[9] ↑ a b Janssen, H. A. (1895). Zeitschr. d. Vereines deutscher Ingenieure, 39, 1045.
[10] ↑ Lord Rayleigh (1906). «On an instrument for compounding vibrations, with application to the drawing of curves such as might represent white light» Philosophical Magazine Series 6, 11-61, 127.
[11] ↑ a b c d Duran, p. 17.
[12] ↑ a b Reynolds, Osborne (1885). «On the dilatancy of media composed of rigid particles in contact. With experimental illustrations». Philosophical Magazine Series 5, 20-127, 469. doi: 10.1080/14786448508627791.
[13] ↑ Bagnold, Ralph A. (1954). Procceedings of the Royal Society, London Series A, 225, 49.
[18] ↑ Serero, D.; Goldenberg, C.; Noskowicz, S. H.; Goldhirsch, I. (2007). «The classical granular temperature and slightly beyond». arXiv:cond-mat/0702545.: http://arxiv.org/abs/cond-mat/0702545v1
[26] ↑ Zuriguel, I.; Mullin, T. y Rotter, J. M. (2007). «Effect of particle shape on the stress dip under a sandpile», Physical Review Letters, 98, 0280001.
[27] ↑ Cleary, P. W. (2008). «The effect of particle shape on simple shear flows» Powder technology, World Conference of Particle Technology No5, 179, 3 p. 97. Ed. Elsevier, Suiza.
[30] ↑ a b Aste, T.; Di Matteo, T.; Galleani d'Agliano, E. (2001). «Stress transmission in granular matter». Journal of Physics: Condensed Matter, 14, 9, pp. 2391-2402.
[31] ↑ a b Duran, p. 11.
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[33] ↑ Anthony, S. p. 73.
[34] ↑ Resnick et al., p. 430.
[35] ↑ Knowlton, T. M.; Carson, J. W.; Klinzing, G. E.; y Yang, W.-C. (1994). «The importance of storage, transfer, and collection».
[42] ↑ a b Pouliquen, O.; Nicolas, M.; y Weidman, P. D. (1997), «Crystallization of non-Brownian Spheres under Horizontal Shaking», Physical Review Letters, 79, 19.
[43] ↑ Anónimo (1972), «What is Random Packing?», Nature, 239, 488.
[44] ↑ Owe Berg, T. G.; McDonald, R. L.; y Trainor Jr., R. J. (1969), «The packing of spheres» Powder Technology, 3, 183.
[45] ↑ Vanel,L.; Rosato, A. D.; y Dave, R. N. (1997), «Rise-Time Regimes of a Large Sphere in Vibrated Bulk Solids», Physical Review Letters, 78, 1255.
[46] ↑ Blair, D. L.; Mueggenburg, N.W.; Marshall, A. H.; Jaeger, H.M.
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[52] ↑ Duran, J.; Mazozi, T.; Clément, E. y Rajchenbach, J. (1994), «Decompaction modes of a two-dimensional “sandpile” under vibration: Model and experiments», Physical Review E, 50, 3092.
[53] ↑ Ehrichs, E. et al. (1995), «Granular convection observed by magnetic resonance imaging», Science, 267, 1632.
[54] ↑ Duran, p. 96.
[55] ↑ Knight, J. B.; Jaeger, H. M. y Nagel, S. R. (1993), «Vibration-induced size separation in granular media: The convection connection», Physical Review Letters, 70, 3728.
[56] ↑ Knight, J. B. (1997), «External boundaries and internal shear bands in granular convection», Physical Review E, 55, 6016.
[58] ↑ Rosato, A., Strandburg, K. J., Prinz, F. y Swendsen, R. H. (1987), «Why the Brazil Nuts are On Top: Size Segregation of Particulate Matter by Shaking», Physical Review Letters, 58, 1038.
[59] ↑ Knight, J. B., Jaeger, H. M. y Nagel, S. (1993), «Vibration-Induced Size Separation in Granular Media: The Convection Connection», Physical Review Letters, 70, 3728.
[60] ↑ Möbius, M. E.; Lauderdale B. E.; Nagel, S. R. y Jaeger, H. R. (2001), «Size Separation of Granular Particles», Nature, 414, 270.
[61] ↑ Nahmad-Molinari, Y.; Canul-Chay, G.; Ruiz-Suárez, J. C. (2003), «Inertia in the Brazil nut problem», Physical Review E, 68, 041301.
[62] ↑ a b Möbius, M. E.; Cheng, X.; Karczmar, G. S.; Nagel, S. R.; Jaeger, H. M. (2004), «Intruders in the Dust: Air-Driven Granular Size Separation», Physical Review Letters, 93, 198001.
[63] ↑ a b Möbius, M. E. et al. (2005), «Effect of air on granular size separation in a vibrated granular bed», Physical Review E, 72, 011304.
[64] ↑ Hong, D. C.; Quinn, P. V.; Luding, S. (2001), «Reverse Brazil Nut Problem: Competition between Percolation and Condensation», Physical Review Letters, 86, 3423.
[65] ↑ Canul-Chay, G. A.; Belmont, P. A.; Nahmad-Molinari,
[66] ↑ Shinbrot, T. (2004), «The brazil nut effect - in reverse», Nature, 429, 352.
[67] ↑ Resnick et al., p. 459.
[68] ↑ Zuriguel, I.; Pugnaloni, L. A.; Garcimartín, A.; Maza, D. (2003), «Jamming during the discharge of grains from a silo described as a percolating transition», Physical Review E, 68, 030301.
[69] ↑ Zuriguel, I.; Garcimartín, A.; Maza, D.; Pugnaloni, L. A.; Pastor, J. M. (2005), «Jamming during the discharge of granular matter from a silo», Physical Review E, 71, 051303.
[70] ↑ a b Barrat, A.; Trizac, E. y Ernst, M. H. (2005), «Granular gases: dynamics and collective effects», Journal of Physics: Condensed Matter, 24, S2429.
[71] ↑ a b c Jaeger, H. M.; Nagel, S. y Behringer, R. P (1996), «Granular solids, liquids, and gases», Reviews of Modern Physics, 8, 1259.
[72] ↑ a b Goldhirsch, I. y Zanetti, G. (1993), «Clustering instability in dissipative gases», Physical Review Letters, 70, 1619.
[73] ↑ Hopkins, M. A. y Louge, M. Y. (1991), «Inelastic microstructure in rapid granular flows of smooth disks», Physics of Fluids A: Fluid Dynamics, 3, 47.
[74] ↑ a b McNamara, S y Young W. R. (1992), «Inelastic collapse and clumping in a one-dimensional granular medium», Physics of Fluids A, 4, 496.
[75] ↑ Trizac, E. y Barrat, A. (2000), «Free cooling and inelastic collapse of granular gases in high dimensions», The European Physical Journal E, 3, 291.
Later, various researchers studied the way in which the forces of the grains stored in a silo are distributed. I. Roberts") studied the dependence of the pressure of grains against the walls of the silo.[8] H. Janssen") described how the pressure forces changed direction towards the walls.[9] Lord Rayleigh established an analogy between this problem and the tensile strength of a rope wrapped around a pole.[10][11].
Towards the end of the century Osborne Reynolds made important discoveries that contributed to the study of granular matter.[12] From that moment on, during the course of the century and to date, the number of scientists dedicated to the study of granular matter has been increasing. Among them, one of the most important has been Ralph Bagnold"), who between 1940 and 1970 made observations of desert sands.[13][14].
Properties
particle size
Granular materials are composed of a large number of solid particles, which are discernible with the naked eye. The size of the particles usually ranges from a few microns to the order of meters or larger. As examples of the above, there is the case of powders where their particles are so small that they can barely be distinguished with the naked eye. In the opposite case, you can have particles as large as rocks that can measure several meters, and even asteroids, with sizes of several hundred meters.
Forces, accelerations and energies
The main property of granular matter is that the only interaction force that exists between the particles that compose it is static friction. An exception to this occurs in the case of finer powders, in which electrostatic interactions may appear when their particles become electrically charged. The existence of static friction as the predominant force between the particles of these materials gives rise to a rapid dissipation of the kinetic energy of the particles, because it generates inelastic collisions between them. For this reason it is not possible to study granular matter with statistical mechanics models for systems where there is conservation of energy. As a consequence of this, it can be said that the effective temperature of a granular material is zero and the only energy of relevance in this type of systems is the potential energy, due to its position with respect to a gravitational field.[15].
Different external forces can act on granular media, which are capable of substantially modifying their global behavior. The main external force to which granular materials are usually subjected is the force of gravity. This force generates a distribution of stresses through the particles of the material. These tensions support the material and allow it to maintain a defined shape. On the other hand, when the material is allowed to slide or fall, gravity forces it to behave similarly to a fluid, as seen in hourglasses.
If granular matter is subjected to periodic shocks, various types of phenomena usually occur, such as convection, particle segregation, among others.[16] The strength of these shocks can be measured in terms of the acceleration created by them. In the case of a shock consisting of a periodic sinusoidal oscillation, the average acceleration, , in one cycle is:.
where T is the period of oscillation, A is the amplitude of oscillation, is the angular frequency of the oscillation and t is the time. The above can be expressed in terms of a Froude number, which gives an idea of the magnitude of the inertial forces with respect to the forces of gravity. In this case, said number is taken as a dimensionless acceleration denoted by:.
being g the value of the acceleration of gravity.
Temperature
As already mentioned, due to the almost instantaneous loss of kinetic energy of the particles in the granular matter, its effective temperature has a value of zero. However, if the granular material is continually subjected to oscillating forces, such as shaking, the particles acquire a velocity. From this velocity and obtaining its quadratic mean, a "granular temperature" can be calculated, just as it would be done with an ideal gas:[17].
Where is the root mean square of the velocity, is the Boltzmann constant, is the temperature and is the mass of the material.
However, it is important to note that, when the force that generates the movement of the particles ceases, the granular medium loses its kinetic energy almost immediately, which is why the temperature returns to its value of zero. It is for this reason that it is not possible to apply the principles of classical thermodynamics to granular matter. That is, classically (see Laws of Thermodynamics) one would expect that energy would be conserved, the entropy of the system would increase naturally, and zero temperature could not be reached. However, none of the above situations occur with granular matter.
To take into account the temperature of a granular medium, it is necessary to use thermodynamic models for non-equilibrium systems. Many researchers of granular matter have tended not to take granular temperature into account or to neglect it, eliminating it from the equations of motion. However, other authors have tried to show that said temperature is necessary to describe this type of materials.[18].
Polydispersity
In polymer science, when you have a set of polymer molecules such that certain molecules are larger than others, the set is said to be polydispersed. In the case of polymers, it is more convenient to pay attention to the difference between the masses of the molecules than to their size. The polydispersity index") or polydispersity (PDI, from English: Polydispersity index) gives us an idea of the diversity of molecules existing in a mixture. This index is calculated by dividing the average mass per molecular weight by the average mass per number of molecules. That is to say:
, the average mass times molecular weight, is calculated by adding the products of the mass of the total number of molecules of a given species and the mass of one molecule of said species until all types of molecules are taken into account, and dividing that sum by the mass of all the molecules. It is simply the sum of the masses of each molecule divided by the total number of molecules. When , all the molecules are of the same type and the mixture is then said to be monodisperse.[19].
By analogy, in granular matter a granular polydispersity is defined. However, in this case the difference in particle size is taken into account instead of the difference in mass. To calculate polydispersity in the study of granular matter, it is necessary to count the number of particles that have a certain diameter in order to obtain the diameter distribution. The polydispersity is then obtained by calculating the variance of said distribution:
with the diameter of a given particle, the average diameter of the particles, the total number of particles and the variance of the distribution.[20].
Polydispersion in granular mixtures is of utmost importance because, in polydisperse materials subject to vertical oscillatory movements, particle segregation phenomena usually appear in which they are separated by size.[21][22].
Compaction
The particles that make up a granular material can be distributed in different ways within it. When you have spherical particles, a percentage of the volume of the granular material corresponds to the spheres themselves, while another percentage of the volume corresponds to the voids that form between the particles. The ratio between the volume occupied by the particles and the total volume of the material—particles and voids—is known as volume fraction"), represented by .
The volume fraction gives us an idea of how compact a granular material is. In the case of monodisperse materials, those that tend to have lower compaction have a volume fraction of around 0.56. By shaking the materials, greater compactions are usually achieved; the maximum of them reached by this way is 0.68 (in which case it is known as random close packing or RCP, from English: Random Close Packing). The maximum compaction possible in monodisperse materials is achieved by accommodating the hexagonal close packing (HCP) particles. When this is the case, the volume fraction reaches 0.74.[23].
Ratios between dissipative forces
When a granular material flows, different dissipative forces—friction between particles, air resistance, etc.—occur and alter its behavior. There are different ways to analyze these behaviors. One of them is separating the dissipative forces into four classes: collisional forces, friction forces, viscosity and pore pressure.[24] If ratios are made between these forces, the following dimensionless numbers are obtained:
• - Bagnold number: ratio between collisional forces and viscous forces.
• - Savage number"): quotient between collisional and friction forces.
• - Friction number"): ratio between friction forces and viscous forces.
• - Darcy number: ratio between friction and pore pressure forces.
Depending on the way in which each of these forces are calculated, different formulas are obtained for each of these numbers, although all of them, in one way or another, depend on the density of the solid particles.[25].
Particle shape
Although the simplest way to study granular matter is to assume that the particles that compose it are spherical, in many cases this is not the case. In a large number of situations particles can have shapes other than spheres. For example, lentil grains are shaped like oblate spheroids, rice grains are shaped like prolate spheroids, salt grains are cubic shaped, etc.
When studying granular media it is important to take into account the shape of its particles. It has been discovered that the shape of the grains can modify the stress distribution in granular materials at rest.[26] Grains with an elongated shape can modify friction and make the flow of the granular material more difficult because energy is lost when they rotate.[27] On the other hand, a material composed of oblate or prolate spheroids can reach a greater volume fraction than one composed of spheres.[28].
Behavior
Contenido
Como ya se ha explicado anteriormente, la materia granular exhibe diferentes comportamientos dependiendo del tipo de fuerzas externas a las que esté sujeta. Dichos comportamientos pueden semejar el de un sólido, el de un líquido o el de un gas. Cuando el material se encuentra en reposo, se comporta como un sólido. Si el material se encuentra bajo la acción de la gravedad, su comportamiento es similar al de un fluido viscoso. Bajo la acción de oscilaciones periódicas de baja aceleración, el material presenta comportamientos similares a los que presentan los fluidos en convección. En el caso en el que el material es sujeto a oscilaciones de alta aceleración, éste asemeja a un gas cuyas partículas sufren colisiones inelásticas.
No obstante, la descripción de los medios granulados no es simple; una gran cantidad de fenómenos que aparentan desafiar la intuición se presentan, debido a la naturaleza disipativa de las fuerzas existen en ellos.
Granular matter at rest
A granular material is at rest when the sum of forces acting on it and on each of the particles that compose it is equal to zero. When this occurs, the behavior of the granular medium resembles that of a solid. This similarity, however, is usually very easily lost by simply applying a small force to the material. A pile of grains, for example, can lose its solidity and begin to flow by simply tilting the material.[29] Depending on the circumstances in which a granular material is at rest—for example, the way it is stored—different behaviors are observed that have been studied to a greater or lesser extent by granular media physicists, engineers, geologists, among others.
A granular material is at rest only if the sum of forces on each of its particles is equal to zero (according to Newton's first law). For this to happen, the weight of a given particle must be balanced by the normal force and static friction due to neighboring particles. Put another way, a grain must be supported by the particles below and to the sides of it to avoid falling. In turn, the particles below it must be supported by others below, and so on until they reach the bottom or walls of the container. This succession of forces can be seen as a chain of efforts; Each part of the granular material is supported by forces transmitted from particle to particle until reaching the base of the container.[30] Likewise, if a force is applied to the surface of the medium, said force will be transmitted downwards and to the sides within the material, being distributed among all the grains. This explains why a person can remain standing on sand: although the force due to his weight is great, it is distributed among many grains.[31].
The transmission of forces from particle to particle can only occur through the point of contact between the grains. The number of contact points that particles have with each other depends largely on the volume fraction of the granular material. The further apart the grains are from each other—that is, if the volume fraction is smaller—there will be fewer contact points per particle and the transmission of stress will be less efficient. The way the stress chains are created then depends largely on the way the particles are arranged in the material. A slight change in the compaction of the medium will cause the chains to adopt another shape.[30].
A phenomenon associated with the formation of stress chains is the formation of arches. When sufficient pressure is exerted on a granular medium, the stress chains take the shape of an arc. Thanks to this, the material can have sufficient support. The reason why the arches are formed can be explained using variational calculus: it can be demonstrated mathematically that by placing a sequence of spheres supported by static friction, the most stable possible way to arrange them is that described by an inverted catenary.[31].
When a set of particles are stored, with no other structure to support them except the ground, the static friction forces between them force said set to form a conical structure "Cone (geometry)"). In mechanics, the static friction of a material can be calculated experimentally by placing two objects—for example, two blocks with flat surfaces—made of the same material on top of each other. If you begin to slowly tilt this system, there will come a time when the upper block will slide, overcoming that friction force. The theoretical angle of inclination, , at which this force is overcome is calculated as follows:[32].
The symbol represents the coefficient of static friction which depends mainly on the roughness of the material.
In granular materials this angle is known as the angle of repose. This angle defines the maximum slope "Slope (geography)") that a pile of particles can have without them precipitating in the form of an avalanche and is the angle formed between the ground and the surface of the mound. Because granular matter is not a continuous medium, but is made up of discrete particles and voids, the friction force is not constant over the entire surface of the material. The volume fraction of the material, the shape of the particles, among other factors, influence the way friction acts. For this reason, an angle of inclination equal to is not a guarantee of stability in the material. A small force on it can cause the grains to slide, similar to what is observed in snow avalanches. Despite the above, no pile of granular material can exist if the angle of inclination of its walls is greater than the angle of repose.
When other types of forces exist between the particles in the granular material—which can collectively be considered cohesive forces "Cohesion (force)")—such as electrical charges, the particles have greater difficulty sliding downward, so the stack of particles may have a greater angle of inclination and, therefore, the angle of repose increases. When this occurs, an angle of internal friction is defined as the angle that the mound would have if only static friction forces acted within it. In this case, this last angle is always less than the angle of repose and, only when the cohesive forces are zero, both angles coincide.[33].
When a fluid is placed in a cylindrical container, it is well known that the pressure at the bottom of said container increases as the height to which it is filled increases. Hydrostatic pressure can be calculated through Stevin's law as follows:
where is the hydrostatic pressure, is the density of the fluid, is the value of the acceleration of gravity and is the height of the fluid column.[34].
In the case of granular matter, it would be expected that when filling a silo—or any cylindrical container—with grains, the pressure at the bottom would increase in the same way that occurs for simple fluids. However, a granular material stops increasing pressure on the bottom of its container once a certain height is reached. H. A. Janssen discovered that the pressure on the walls of a container containing a granular material follows the following relationship:[9].
In this case, it is a parameter that depends on the static friction between the walls of the silo and the grains and its value is usually the order of magnitude of the radius of the container. This behavior is known as Janssen effect").[11].
The explanation for this effect lies in the way in which the stresses are transmitted between the grains: depending on the way in which the particles are distributed, the stress chains tend to direct the force due to the weight of the material towards the walls of the container. In simple fluids, the pressure at a certain point is directed in all directions (obeying Pascal's principle). On the other hand, in granular media, the pressure can follow different contact paths until reaching the walls. For this reason there is no equitable distribution horizontally and vertically; More pressure is directed towards the walls than towards the bottom.
The Janssen effect represents a problem for engineers, since, if they calculate the pressure of a granulated medium on the wall of a silo as if it were hydrostatic pressure, they can underestimate the resistance that said walls would need to have, even causing an explosion in the silo.[35].
Granular materials undergo a change in their volume fraction when they are subjected to pressure. The phenomenon was first described by Osborne Reynolds in 1885.[12] Reynolds verified this phenomenon by filling a rubber container with sand and water, adding a glass tube to the mouth of the container, so that the water reached a certain level inside the tube. By compressing the rubber container with his hands, the water level in the tube dropped, contrary to what would be expected. This phenomenon is known as *Reynolds dilation").[36].
The explanation of this phenomenon, given by Reynolds himself, consists of the change in the volume fraction of the granular material. When sand is compressed, its grains are rearranged in their positions, in such a way that the empty space between the particles increases. When this occurs, the water occupies these new spaces and its level drops. This phenomenon is also observed on beaches: when a person walks on wet sand, the footprints seem to dry out. The explanation is the same: the pressure due to the weight of the person walking on the beach generates a change in the volume fraction of the sand and the water within it drops in level, making the surface of the sand appear dry.[37].
Behavior at low accelerations
A granular medium that ceases to be at rest, either due to the action of gravity or periodic shaking, usually behaves, in most cases, in a manner very similar to that of a fluid. When a granular material moves thanks to the force of gravity through a hole (for example, when unloading a silo or in an hourglass) a flow of grains is generated, which depending on the size and shape of the particles can be continuous or interrupted by particle clogging. On the other hand, when a granular material is subjected to periodic shaking, a phenomenon similar to the convection that simple fluids present is usually present.[16] If the material is also polydispersed, a segregation of particles by size is observed, giving rise to phenomena such as the Brazil nut effect.[38].
The transition between a static granular medium and a flowing one usually occurs when an external oscillatory force is initiated, giving rise to fluidization.[39] This effect generates the loss of solidity of the material, causing an object that is on the surface of the medium to sink.[40] Fluidization has an extremely destructive effect in earthquakes, since when the apparently stable but water-saturated earth moves, liquefaction occurs "Liquefaction (instability)") and the constructions on its surface lose support and collapse when sinking into it.[41].
When a set of grains are poured into a container, the material usually has a low compaction, with a volume fraction of around 0.55. To reduce the volume occupied by the empty spaces, so that the material as a whole occupies less space, the medium must be subjected to horizontal vibrations. In this way, the volume fraction increases and can reach higher values. This makes it easier to store grains, since smaller containers are required.[42].
Usually, the volume fraction does not exceed a value of 0.64.[43] However, using different configurations, such as combined horizontal and vertical vibrations,[44] or systems with few particles,[45] vibration-induced compactions can be achieved with fractions greater than this value. The disadvantage of the above is that the shape of the container as well as the configuration of the system itself affect the behavior of the granular material, so they cannot be considered general cases.[42].
To achieve the maximum volume fraction, 0.74, which corresponds to a granular crystal arranged in a hexagonal shape by means of vibration, it has been necessary to resort to more sophisticated methods. A layer of granular material is placed in a vacuum on a perforated metal plate. In this way, the particles are forced to fall on the perforations and form an ordered arrangement. When the layer is complete, the next layer is placed. If any particle was disordered, that "imperfection" would be removed by hand.[46] The first perfect arrangement, without defects, achieved by purely mechanical means and without the need for manual intervention was reported by Nahmad-Molinari and Ruiz-Suárez in 2002, who used a type of epitaxial growth. In their method, they use a triangular prism-shaped container into which they throw steel spheres one by one. The container is shaken vertically with accelerations slightly greater than gravity. When a certain amount of particles are in the container, they agglomerate due to inelastic collisions between them, in such a way that a nucleus of spheres is generated in constant contact with each other. The new particles that are most recently ejected bind to this nucleus until the first layer is formed. In the next layer, the particles occupy the rest positions between the gaps in the spheres below, similar to the method before this one. Because the process is "one at a time" and the low acceleration to which the system is subjected, each sphere "searches" for its assumption and the second layer is completed. Finally the rest of the layers are formed until the container is filled.[23].
Granular gases
Granular matter subjected to high accelerations usually behaves in a similar way to a theoretical molecular gas. In the latter the molecules that form them undergo elastic collisions and a conservation of energy can be assumed as occurs in the theory of ideal gases. However, this is not the case for granular media. Each particle loses a part of its kinetic energy when colliding with another, converting said energy into heat, sound, vibration, rotation or another form of energy (that is, an inelastic collision occurs). The amount of energy lost in each collision depends on the coefficient of restitution of the material that makes up the grain.[70] When the granular system is considered as a whole, the energy loss depends on the number of collisions that occur in a given time, as an example, if a marble is dropped into a glass container, it will bounce a certain number of times until it finally stops. However, if a large number of these objects are dropped at the same time into the same container, the system as a whole will stop almost instantaneously. because the number of collisions is much greater.[71].
For a granulated medium to behave like a gas, it must be subjected to a sufficiently large constant force. If the force that maintains it in this state is suddenly stopped, the material will come to rest almost immediately. Furthermore, if the acceleration is not high enough, the material will have a certain time to relax, and its behavior will resemble that of a liquid (see the section "Granular convection").[71].
The dissipative nature of collisions makes granular gases systems out of thermodynamic equilibrium. This fact generates certain phenomena that at first glance would seem to violate the laws of thermodynamics, if the system is studied in a simple way without considering these energy losses. Among the phenomena that appear in these media we can mention granular agglomeration, the rupture of the equipotentialization of energy and inelastic collapse.[70].
Because in granular matter the collisions between particles are essentially inelastic, the amount of energy dissipated during a certain time depends on the number of collisions that occur in that period. The greater the number of collisions, the greater the energy that the system loses. In a granular gas, collisions occur randomly. Normally it would be expected that, on average, the number of collisions per unit of time in a given area of the gas is equal to the number of collisions in another area of the same size as the first in the same time. However, this number is subject to statistical fluctuations"), so there is a probability that in a certain area, for a moment a slightly greater number of collisions will occur than in another. This will produce a greater loss of energy in that region, resulting in a decrease in the speed of the particles. When this occurs, the pressure exerted by them will decrease, causing other particles that were not in that area to enter it, further increasing the number of collisions and the loss of energy. In the end, the gas granular will present a non-homogeneous appearance"), with some regions with very low densities with particles moving at high speed and others populated with a large number of grains agglomerated against each other.[72].
General
Literature
• - Duran, J., Reisinger A., Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials. November 1999, Springer-Verlag New York, Inc., New York, ISBN 0-387-98656-1.
• - Aste, Tomaso; Di Matteo, T.; Tordesillas, A. Granular and complex materials. 2007, World Scientific, ISBN 981-277-198-0.
References
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[2] ↑ Duran, p. 1-5.
[3] ↑ Duran, p. 16.
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[5] ↑ Rossing, Thomas D (1982). «Chladni's Law for Vibrating Plates» American Journal of Physics, 50, 3.
[6] ↑ a b Faraday, Michael (1831). «On a peculiar class of acoustic figures, and on certain forms assumed by groups of particles upon vibrating surfaces». Phylos. Trans. Roy. Soc., 52, 299. Londres.
[7] ↑ Rankine, William J. W. (1857). Phil. Trans. Roy. Soc., 147, 9.
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[10] ↑ Lord Rayleigh (1906). «On an instrument for compounding vibrations, with application to the drawing of curves such as might represent white light» Philosophical Magazine Series 6, 11-61, 127.
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[12] ↑ a b Reynolds, Osborne (1885). «On the dilatancy of media composed of rigid particles in contact. With experimental illustrations». Philosophical Magazine Series 5, 20-127, 469. doi: 10.1080/14786448508627791.
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[18] ↑ Serero, D.; Goldenberg, C.; Noskowicz, S. H.; Goldhirsch, I. (2007). «The classical granular temperature and slightly beyond». arXiv:cond-mat/0702545.: http://arxiv.org/abs/cond-mat/0702545v1
[26] ↑ Zuriguel, I.; Mullin, T. y Rotter, J. M. (2007). «Effect of particle shape on the stress dip under a sandpile», Physical Review Letters, 98, 0280001.
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[33] ↑ Anthony, S. p. 73.
[34] ↑ Resnick et al., p. 430.
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[43] ↑ Anónimo (1972), «What is Random Packing?», Nature, 239, 488.
[44] ↑ Owe Berg, T. G.; McDonald, R. L.; y Trainor Jr., R. J. (1969), «The packing of spheres» Powder Technology, 3, 183.
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[46] ↑ Blair, D. L.; Mueggenburg, N.W.; Marshall, A. H.; Jaeger, H.M.
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[52] ↑ Duran, J.; Mazozi, T.; Clément, E. y Rajchenbach, J. (1994), «Decompaction modes of a two-dimensional “sandpile” under vibration: Model and experiments», Physical Review E, 50, 3092.
[53] ↑ Ehrichs, E. et al. (1995), «Granular convection observed by magnetic resonance imaging», Science, 267, 1632.
[54] ↑ Duran, p. 96.
[55] ↑ Knight, J. B.; Jaeger, H. M. y Nagel, S. R. (1993), «Vibration-induced size separation in granular media: The convection connection», Physical Review Letters, 70, 3728.
[56] ↑ Knight, J. B. (1997), «External boundaries and internal shear bands in granular convection», Physical Review E, 55, 6016.
[58] ↑ Rosato, A., Strandburg, K. J., Prinz, F. y Swendsen, R. H. (1987), «Why the Brazil Nuts are On Top: Size Segregation of Particulate Matter by Shaking», Physical Review Letters, 58, 1038.
[59] ↑ Knight, J. B., Jaeger, H. M. y Nagel, S. (1993), «Vibration-Induced Size Separation in Granular Media: The Convection Connection», Physical Review Letters, 70, 3728.
[60] ↑ Möbius, M. E.; Lauderdale B. E.; Nagel, S. R. y Jaeger, H. R. (2001), «Size Separation of Granular Particles», Nature, 414, 270.
[61] ↑ Nahmad-Molinari, Y.; Canul-Chay, G.; Ruiz-Suárez, J. C. (2003), «Inertia in the Brazil nut problem», Physical Review E, 68, 041301.
[62] ↑ a b Möbius, M. E.; Cheng, X.; Karczmar, G. S.; Nagel, S. R.; Jaeger, H. M. (2004), «Intruders in the Dust: Air-Driven Granular Size Separation», Physical Review Letters, 93, 198001.
[63] ↑ a b Möbius, M. E. et al. (2005), «Effect of air on granular size separation in a vibrated granular bed», Physical Review E, 72, 011304.
[64] ↑ Hong, D. C.; Quinn, P. V.; Luding, S. (2001), «Reverse Brazil Nut Problem: Competition between Percolation and Condensation», Physical Review Letters, 86, 3423.
[65] ↑ Canul-Chay, G. A.; Belmont, P. A.; Nahmad-Molinari,
[66] ↑ Shinbrot, T. (2004), «The brazil nut effect - in reverse», Nature, 429, 352.
[67] ↑ Resnick et al., p. 459.
[68] ↑ Zuriguel, I.; Pugnaloni, L. A.; Garcimartín, A.; Maza, D. (2003), «Jamming during the discharge of grains from a silo described as a percolating transition», Physical Review E, 68, 030301.
[69] ↑ Zuriguel, I.; Garcimartín, A.; Maza, D.; Pugnaloni, L. A.; Pastor, J. M. (2005), «Jamming during the discharge of granular matter from a silo», Physical Review E, 71, 051303.
[70] ↑ a b Barrat, A.; Trizac, E. y Ernst, M. H. (2005), «Granular gases: dynamics and collective effects», Journal of Physics: Condensed Matter, 24, S2429.
[71] ↑ a b c Jaeger, H. M.; Nagel, S. y Behringer, R. P (1996), «Granular solids, liquids, and gases», Reviews of Modern Physics, 8, 1259.
[72] ↑ a b Goldhirsch, I. y Zanetti, G. (1993), «Clustering instability in dissipative gases», Physical Review Letters, 70, 1619.
[73] ↑ Hopkins, M. A. y Louge, M. Y. (1991), «Inelastic microstructure in rapid granular flows of smooth disks», Physics of Fluids A: Fluid Dynamics, 3, 47.
[74] ↑ a b McNamara, S y Young W. R. (1992), «Inelastic collapse and clumping in a one-dimensional granular medium», Physics of Fluids A, 4, 496.
[75] ↑ Trizac, E. y Barrat, A. (2000), «Free cooling and inelastic collapse of granular gases in high dimensions», The European Physical Journal E, 3, 291.
In simple fluids, convection occurs when the lower part of the medium is heated, generating a decrease in the density of the fluid in this region. This generates an instability, where the gravity acting on the medium comes into competition with the thrust force due to this change in density. The result is the creation of a cyclic movement where the hot fluid rises and the cold fluid falls.[47].
Convection in granular matter "appears when it is subjected to vertical vibrations. This phenomenon has an appearance very similar to that observed in simple fluids: a part of the material rises, while another descends, establishing a continuous circulation. However, the mechanism that gives rise to granular convection is a little different from the convective instability in simple fluids. A large number of authors have dedicated time to studying this phenomenon, proposing various mechanisms that generate this phenomenon. convection.[48].
Michael Faraday was the first person to report this phenomenon, studying the formation of mounds in granular materials under vibration. Since then, a large number of studies have been done on this behavior.[6] It has been demonstrated through experimental work that the limiting acceleration at which the collective movement of the granular medium begins is slightly higher than the acceleration of gravity.[49] It was originally proposed that the origin of granular convection was due to the circulation of air between the grains, causing the latter to ascend through the center of the system and descend along the walls of the container.[50] Later, it was found that the walls of the container can generate enough force to give rise to convection.[51][52] The above was finally confirmed by the team of Edward Ehrichs") and collaborators from the University of Chicago, who experimented with a granular material subjected to vertical agitations and observed the collective movement of the grains through nuclear magnetic resonance images.[53] The explanation of the phenomenon is based on the fact that, as the grains are thrown upwards thanks to the vertical vibration, those that are close to the walls suffer a greater force of convection. friction due to them, which prevents them from rising higher than the particles in the center of the container. As the central particles have risen higher, a hole is generated in the bottom of the container that is occupied by the outer grains. In this way, in each cycle the effective movement is a circulation where the grains ascend through the center and descend along the walls.[54].
The geometry of the container also influences the way granular convection occurs. While in a cylindrical container the process occurs in the way explained above, in a container whose walls have a certain inclination, convection reverses its direction of movement. It has been observed that if a set of grains is placed in a cone-shaped container "Inverted Cone (geometry)") and subjected to vertical vibrations, the particles descend through the center and ascend along the edges.[55] This inversion occurs because the inclination of the walls reduces the contact and, therefore, the friction between them and the grains. When thrown upward, the grains in the outer areas separate from each other, returning to the wall at a higher point than where they started. In this way, the outer material is pushed further outward and upward than the central material, generating this "reverse convection".[56].
An extremely important effect on granular matter is granular segregation. When a polydisperse mixture of grains is shaken vertically, the particles are separated by size, with the largest particles remaining at the top and the smallest particles remaining at the bottom. This occurs even if the larger ones have a greater mass than the smaller ones. The above seems to defy physical principles; Particles with greater mass would be expected to fall, while those with lower mass would remain at the top, thus reducing the potential energy. This phenomenon was called the Brazil nut effect because in a nut mixture, the Brazil nuts tend to be the ones with the greatest mass and, therefore, always appear on the surface of the mixture after it has been shaken.[57].
Several authors have dedicated themselves to the explanation of this phenomenon. The first explanation, given by Anthony Rosato"), maintained that the rise of large particles was due to the infiltration of small particles below it. At the moment in which the system moves upward, the large particle generates a gap below it that is occupied by the small ones. When the movement of the system changes direction, the grains that recently occupied this space prevent the larger particle from descending. This generates a net upward movement.[58] Other authors proposed that the granular segregation was due to the convection, which drags the large particle upward. Due to its size, the latter cannot descend as the small ones would since the downward flow is carried out only through a very thin space near the walls.[59].
Mathias Möbius") and collaborators from the University of Chicago demonstrated that the rise time depends on the density of the larger particles. When the latter have a density similar to that of the smaller grains, the time it takes them to reach the surface is longer. This time is reduced if their density increases or decreases with respect to the density of the small grains.[60] This fact gave a complete turn to the problem: convection and infiltration were insufficient to explain the granular segregation. From this it was established proposed models based on the inertia of the particles: Those grains with greater mass would have greater kinetic energy and, as a consequence, could do more work against the friction of the granules, penetrating a greater length. For those particles with greater densities, the phenomenon could be explained as simply due to a buoyant force.[61].
A new complication arose when it was discovered that, if a granular mixture was placed in a vacuum, the rise time of the largest particles became the same for all of them. As the air pressure in the granulate is reduced, the difference between the rise times is reduced, until they become equal in a vacuum. From this it was suggested that the pressure gradient within the granular medium played an important role in the phenomenon of segregation.[62] In order to correctly describe the latter, it is necessary to take into account all the variables described by the different authors.[63].
In certain circumstances you can have a reverse Brazil nut effect. In this case the larger particles precipitate to the bottom of the container. This effect was first predicted through computer simulations.[64] However, some authors questioned its existence due to the lack of experimental evidence,[65] until it could finally be confirmed definitively.[66] The reverse Brazil nut effect usually occurs when a particle of larger size, but lower density, is introduced at a depth very close to the bottom of the container. This phenomenon has been explained, like the conventional effect, with a pressure gradient.[62][63].
If a container of granular matter is punctured at the bottom, the grains inside it will flow out. Many factors intervene in the form of this flow, which can be constant or suddenly interrupted. As examples of the above, in an hourglass the flow is practically constant, while in a salt shaker it is necessary to shake it to extract the grains.[11].
For liquids escaping through an orifice, the flow rate depends primarily on the height to which the liquid reaches inside the container. The phenomenon is explained through Torricelli's theorem and is due to the increase in hydrostatic pressure at the bottom of the container as the height of the fluid increases.[67] In granular media, however, the pressure stops increasing when the material reaches a height of approximately twice the diameter of the container. This causes that, during most of the grain discharge, the flow comes out with the same speed, only reducing when the container is almost empty.[11].
In some cases the granular flow is interrupted by the clogging of particles in the exit orifice. When said opening has a very small diameter, although larger than that of the grains, the material gets stuck in it, interrupting the flow. The reason for these jams is that when several particles try to come out at the same time, an arc is formed in the hole. Since the arches have great stability, the grains are unable to move, thanks to static friction, obstructing the exit. The only way to restore the flow is to remove one of the grains in the arc, either manually or by applying a force to the system.[68] It has been demonstrated by experimental means that interruptions in the discharges of spherical grains occur when the exit orifice has a diameter less than approximately 4.5 times the diameter of the particles. With openings greater than this value the flow becomes constant. If the grains are not spherical, the diameter of the hole at which jams do not occur may be different.[69].
Inelastic collapse was first observed in computer simulations[73] in one dimension[74] and later in more dimensions.[75] The effect occurs in simulations by increasing the number of collisions per unit time between two particles; When these lose energy and are pushed against each other by the rest of the particles, they begin to bounce faster and faster until the collisions between them become infinite, causing a "collapse" in the simulation.[74] This phenomenon causes the formation of structures in the form of chains and filaments,[72] which have a great similarity to the structure of the universe on a large scale.[71].
• - Micropolar elasticity.
In simple fluids, convection occurs when the lower part of the medium is heated, generating a decrease in the density of the fluid in this region. This generates an instability, where the gravity acting on the medium comes into competition with the thrust force due to this change in density. The result is the creation of a cyclic movement where the hot fluid rises and the cold fluid falls.[47].
Convection in granular matter "appears when it is subjected to vertical vibrations. This phenomenon has an appearance very similar to that observed in simple fluids: a part of the material rises, while another descends, establishing a continuous circulation. However, the mechanism that gives rise to granular convection is a little different from the convective instability in simple fluids. A large number of authors have dedicated time to studying this phenomenon, proposing various mechanisms that generate this phenomenon. convection.[48].
Michael Faraday was the first person to report this phenomenon, studying the formation of mounds in granular materials under vibration. Since then, a large number of studies have been done on this behavior.[6] It has been demonstrated through experimental work that the limiting acceleration at which the collective movement of the granular medium begins is slightly higher than the acceleration of gravity.[49] It was originally proposed that the origin of granular convection was due to the circulation of air between the grains, causing the latter to ascend through the center of the system and descend along the walls of the container.[50] Later, it was found that the walls of the container can generate enough force to give rise to convection.[51][52] The above was finally confirmed by the team of Edward Ehrichs") and collaborators from the University of Chicago, who experimented with a granular material subjected to vertical agitations and observed the collective movement of the grains through nuclear magnetic resonance images.[53] The explanation of the phenomenon is based on the fact that, as the grains are thrown upwards thanks to the vertical vibration, those that are close to the walls suffer a greater force of convection. friction due to them, which prevents them from rising higher than the particles in the center of the container. As the central particles have risen higher, a hole is generated in the bottom of the container that is occupied by the outer grains. In this way, in each cycle the effective movement is a circulation where the grains ascend through the center and descend along the walls.[54].
The geometry of the container also influences the way granular convection occurs. While in a cylindrical container the process occurs in the way explained above, in a container whose walls have a certain inclination, convection reverses its direction of movement. It has been observed that if a set of grains is placed in a cone-shaped container "Inverted Cone (geometry)") and subjected to vertical vibrations, the particles descend through the center and ascend along the edges.[55] This inversion occurs because the inclination of the walls reduces the contact and, therefore, the friction between them and the grains. When thrown upward, the grains in the outer areas separate from each other, returning to the wall at a higher point than where they started. In this way, the outer material is pushed further outward and upward than the central material, generating this "reverse convection".[56].
An extremely important effect on granular matter is granular segregation. When a polydisperse mixture of grains is shaken vertically, the particles are separated by size, with the largest particles remaining at the top and the smallest particles remaining at the bottom. This occurs even if the larger ones have a greater mass than the smaller ones. The above seems to defy physical principles; Particles with greater mass would be expected to fall, while those with lower mass would remain at the top, thus reducing the potential energy. This phenomenon was called the Brazil nut effect because in a nut mixture, the Brazil nuts tend to be the ones with the greatest mass and, therefore, always appear on the surface of the mixture after it has been shaken.[57].
Several authors have dedicated themselves to the explanation of this phenomenon. The first explanation, given by Anthony Rosato"), maintained that the rise of large particles was due to the infiltration of small particles below it. At the moment in which the system moves upward, the large particle generates a gap below it that is occupied by the small ones. When the movement of the system changes direction, the grains that recently occupied this space prevent the larger particle from descending. This generates a net upward movement.[58] Other authors proposed that the granular segregation was due to the convection, which drags the large particle upward. Due to its size, the latter cannot descend as the small ones would since the downward flow is carried out only through a very thin space near the walls.[59].
Mathias Möbius") and collaborators from the University of Chicago demonstrated that the rise time depends on the density of the larger particles. When the latter have a density similar to that of the smaller grains, the time it takes them to reach the surface is longer. This time is reduced if their density increases or decreases with respect to the density of the small grains.[60] This fact gave a complete turn to the problem: convection and infiltration were insufficient to explain the granular segregation. From this it was established proposed models based on the inertia of the particles: Those grains with greater mass would have greater kinetic energy and, as a consequence, could do more work against the friction of the granules, penetrating a greater length. For those particles with greater densities, the phenomenon could be explained as simply due to a buoyant force.[61].
A new complication arose when it was discovered that, if a granular mixture was placed in a vacuum, the rise time of the largest particles became the same for all of them. As the air pressure in the granulate is reduced, the difference between the rise times is reduced, until they become equal in a vacuum. From this it was suggested that the pressure gradient within the granular medium played an important role in the phenomenon of segregation.[62] In order to correctly describe the latter, it is necessary to take into account all the variables described by the different authors.[63].
In certain circumstances you can have a reverse Brazil nut effect. In this case the larger particles precipitate to the bottom of the container. This effect was first predicted through computer simulations.[64] However, some authors questioned its existence due to the lack of experimental evidence,[65] until it could finally be confirmed definitively.[66] The reverse Brazil nut effect usually occurs when a particle of larger size, but lower density, is introduced at a depth very close to the bottom of the container. This phenomenon has been explained, like the conventional effect, with a pressure gradient.[62][63].
If a container of granular matter is punctured at the bottom, the grains inside it will flow out. Many factors intervene in the form of this flow, which can be constant or suddenly interrupted. As examples of the above, in an hourglass the flow is practically constant, while in a salt shaker it is necessary to shake it to extract the grains.[11].
For liquids escaping through an orifice, the flow rate depends primarily on the height to which the liquid reaches inside the container. The phenomenon is explained through Torricelli's theorem and is due to the increase in hydrostatic pressure at the bottom of the container as the height of the fluid increases.[67] In granular media, however, the pressure stops increasing when the material reaches a height of approximately twice the diameter of the container. This causes that, during most of the grain discharge, the flow comes out with the same speed, only reducing when the container is almost empty.[11].
In some cases the granular flow is interrupted by the clogging of particles in the exit orifice. When said opening has a very small diameter, although larger than that of the grains, the material gets stuck in it, interrupting the flow. The reason for these jams is that when several particles try to come out at the same time, an arc is formed in the hole. Since the arches have great stability, the grains are unable to move, thanks to static friction, obstructing the exit. The only way to restore the flow is to remove one of the grains in the arc, either manually or by applying a force to the system.[68] It has been demonstrated by experimental means that interruptions in the discharges of spherical grains occur when the exit orifice has a diameter less than approximately 4.5 times the diameter of the particles. With openings greater than this value the flow becomes constant. If the grains are not spherical, the diameter of the hole at which jams do not occur may be different.[69].
Inelastic collapse was first observed in computer simulations[73] in one dimension[74] and later in more dimensions.[75] The effect occurs in simulations by increasing the number of collisions per unit time between two particles; When these lose energy and are pushed against each other by the rest of the particles, they begin to bounce faster and faster until the collisions between them become infinite, causing a "collapse" in the simulation.[74] This phenomenon causes the formation of structures in the form of chains and filaments,[72] which have a great similarity to the structure of the universe on a large scale.[71].