Nonhydrostatic Stress

1993 ◽  
Author(s):  
Brian Bayly
Keyword(s):  
Normal Stress ◽  
Chemical Potential ◽  
Normal Force ◽  
Tangential Force ◽  
Unit Vector ◽  
Small Areas ◽  
Left Handedness ◽  
Principal Directions ◽  
Unit Of Length ◽  
Set Up

As in the chapters on chemical potential, it will again be assumed that the reader has thought about the topic before, so that our task is to select rather than to build. The interior of a continuous sample contains many small volumes and small areas, on any of which attention can be focused. A small internal area has the property that, across it, the material on one side exerts a normal force and a tangential force on the material on the other side. Let the normal force be F and the area A; then the ratio F/A approaches a limit as the size of A approaches zero. Thus we define the magnitude of the normal stress at a point across an infinitesimal area of a particular orientation. If we set up Cartesian coordinates so that the orientation of the area can be specified by the direction of its normal then, at a point, for every direction vector there is a normal-stress magnitude. The stress may be compressive or tensile, and in this text we treat compressions as positive. It is possible to imagine a universe where space itself has an attribute of left-handedness or right-handedness, or where space does not but materials do. But if we set these possibilities aside and use ordinary ideas about symmetry, it follows that at any point where stresses exist inside a continuum, there are three orthogonal planes across which the tangential stress is zero; these planes suffer only normal stresses. The planes themselves are principal planes, their normals are the three principal directions at the point and the normal-stress magnitudes are the principal stress magnitudes. The largest, intermediate, and smallest normal compressions will be designated σ 1, σ 2 and σ 3, respectively; for most of what follows we shall designate the directions along which these compressions act as x1, x2, and x3 (so that the plane compressed by stress σ 1 has x1 for its normal), and we shall use x1, x2, and x3 as axes for a local Cartesian system with which other planes and directions at the point can be specified. In particular, for any direction through the point, a unit vector can be imagined (magnitude = 1 unit of length); its components along the three axes will be called n1, n2, and n3, combining to give the unit vector n.

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

scholarly journals Effect of Water-Based Nanolubricants in Ultrasonic Vibration Assisted Grinding

2018 ◽  
Vol 2 (4) ◽  
pp. 80 ◽  
Author(s):  
Mir Molaie ◽  
Ali Zahedi ◽  
Javad Akbari
Keyword(s):  
Surface Roughness ◽  
Specific Energy ◽  
Material Removal ◽  
Normal Force ◽  
Tangential Force ◽  
Process Efficiency ◽  
Multiwall Carbon Nanotube ◽  
Ultrasonic Vibrations ◽  
Mql Grinding ◽  
Water Based

Currently, because of stricter environmental standards and highly competitive markets, machining operations, as the main part of the manufacturing cycle, need to be rigorously optimized. In order to simultaneously maximize the production quality and minimize the environmental issues related to the grinding process, this research study evaluates the performance of minimum quantity lubrication (MQL) grinding using water-based nanofluids in the presence of horizontal ultrasonic vibrations (UV). In spite of the positive impacts of MQL using nanofluids and UV which are extensively reported in the literature, there is only a handful of studies on concurrent utilization of these two techniques. To this end, for this paper, five kinds of water-based nanofluids including multiwall carbon nanotube (MWCNT), graphite, Al2O3, graphene oxide (GO) nanoparticles, and hybrid Al2O3/graphite were employed as MQL coolants, and the workpiece was oscillated along the feed direction with 21.9 kHz frequency and 10 µm amplitude. Machining forces, specific energy, and surface quality were measured for determining the process efficiency. As specified by experimental results, the variation in the material removal nature made by ultrasonic vibrations resulted in a drastic reduction of the grinding normal force and surface roughness. In addition, the type of nanoparticles dispersed in water had a strong effect on the grinding tangential force. Hybrid Al2O3/graphite nanofluid through two different kinds of lubrication mechanisms—third body and slider layers—generated better lubrication than the other coolants, thereby having the lowest grinding forces and specific energy (40.13 J/mm3). It was also found that chemically exfoliating the graphene layers via oxidation and then purification prior to dispersion in water promoted their effectiveness. In conclusion, UV assisted MQL grinding increases operation efficiency by facilitating the material removal and reducing the use of coolants, frictional losses, and energy consumption in the grinding zone. Improvements up to 52%, 47%, and 61%, respectively, can be achieved in grinding normal force, specific energy, and surface roughness compared with conventional dry grinding.

scholarly journals Flavour effects in gravitational leptogenesis

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Rome Samanta ◽  
Satyabrata Datta
Keyword(s):  
Mass Spectrum ◽  
Chemical Potential ◽  
Mass Scale ◽  
Lepton Number ◽  
Type I ◽  
Lepton Asymmetry ◽  
Seesaw Mechanism ◽  
Set Up ◽  
The Right ◽  
Lepton Number Violating

Abstract Within the Type-I seesaw mechanism, quantum effects of the right-handed (RH) neutrinos in the gravitational background lead to an asymmetric propagation of lepton and anti-leptons which induces a Ricci scalar and neutrino Dirac-Yukawa coupling dependent chemical potential and therefore a lepton asymmetry in equilibrium. At high temperature, lepton number violating scattering processes try to maintain a dynamically generated lepton asymmetry in equilibrium. However, when the temperature drops down, the interactions become weaker, and the asymmetry freezes out. The frozen out asymmetry can act as a pre-existing asymmetry prior to the standard Fukugita-Yanagida leptogenesis phase (Ti ∼ Mi, where Mi is the mass of ith RH neutrino). It is then natural to consider the viability of gravitational leptogenesis for a given RH mass spectrum which is not consistent with successful leptogenesis from decays. Primary threat to this gravity-induced lepton asymmetry to be able to successfully reproduce the observed baryon-to-photon ratio is the lepton number violating washout processes at Ti ∼ Mi. In a minimal seesaw set up with two RH neutrinos, these washout processes are strong enough to erase a pre-existing asymmetry of significant magnitude. We show that when effects of flavour on the washout processes are taken into account, the mechanism opens up the possibility of successful leptogenesis (gravitational) for a mass spectrum M2 » 109GeV » M1 with M1 ≳ 6.3 × 106 GeV. We then briefly discuss how, in general, the mechanism leaves its imprints on the low energy CP phases and absolute light neutrino mass scale.

Quantifying the influence of non-hydrostatic stress on polymorph equilibria

2021 ◽  
Author(s):  
Benjamin Hess ◽  
Jay Ague
Keyword(s):  
High Pressure ◽  
Normal Stress ◽  
Chemical Potential ◽  
Hydrostatic Stress ◽  
Low Pressure ◽  
Surprising Result ◽  
Metamorphic Petrology ◽  
Equilibrium Thermodynamics ◽  
Stress Variation ◽  
Mineral Assemblages

<p>Thermodynamic modeling in active tectonic settings typically makes the assumption that stress is equal in all directions. This allows for the application of classical equilibrium thermodynamics. In contrast, geodynamic modeling indicates that differential, or non-hydrostatic, stresses are widespread. Non-hydrostatic equilibrium thermodynamics have been developed by past workers [1], but their application to geological systems has generated controversy in recent years [2-5]. Therefore, we seek to clarify how stress influences the chemical potential of non-hydrostatically stressed elastic solids. To quantify this, we consider the effects of stress variation on the equilibrium between the single-component polymorph pairs of kyanite/sillimanite, quartz/coesite, calcite/aragonite, and diamond/graphite.</p><p>The stress on each interface of a solid can be decomposed into components normal to the interface and parallel to the interface. In our work, we determine the shift in the temperature of equilibrium on fixed interfaces between polymorph pairs as a function of both interface-normal and interface-parallel stress variation. We find that the influence of normal stress variation on the equilibrium temperature of polymorphs is approximately two orders of magnitude greater than interface-parallel stress variation. Thus, at a fixed temperature, normal stress determines the chemical potential of a given interface to first order. Consequently, high-pressure polymorphs will preferentially form normal to the maximum stress, while low-pressure polymorphs, normal to the minimum stress.</p><p>Nonetheless, interface-parallel stress variations can meaningfully affect the stability of phases that are at or near equilibrium. We demonstrate the surprising result that for a given polymorph pair, a decrease in interface-parallel stresses can make a high-pressure polymorph more stable relative to a low-pressure polymorph on the given interface.</p><p>The effects of non-hydrostatic stress on mineral assemblages are most likely to be seen in dry systems. Many reactions in metamorphic systems are fluid-mediated, and fluids cannot sustain non-hydrostatic stress. Consequently, in systems with interconnected, fluid-filled porosity, mineral assemblages will tend to form at a pressure approximately equal to the fluid pressure. In contrast, in dry systems all reactions occur directly between solids which can sustain non-hydrostatic stress. This facilitates the application of non-hydrostatic thermodynamics. Consequently, dry rocks containing polymorphs such as such as quartzites, marbles, and peridotites represent ideal lithologies for the testing and application of these concepts. By influencing the chemical potential of solid interfaces, non-hydrostatic stress alters the thermodynamic driving force and subsequent kinetics of polymorphic reactions. This likely results in preferential orientations of polymorphs which could influence seismic anisotropy and potentially generate seismicity.</p><p>[1] Larché, F., & Cahn, J. W. (1985). Acta Metallurgica, 33(3), 331-357. https://doi.org/10.1016/0001-6160(85)90077-X</p><p>[2] Hobbs, B. E., & Ord, A. (2016). Earth-Science Reviews, 163, 190-233. https://doi.org/10.1016/j.earscirev.2016.08.013</p><p>[3] Powell, R., Evans, K. A., Green, E. C. R., & White, R. W. (2018). Journal of Metamorphic Petrology, 36(4), 419-438. https://doi.org/10.1111/jmg.12298</p><p>[4] Tajčmanová, L., Podladchikov, Y., Powell, R., Moulas, E., Vrijmoed, J. C., & Connolly, J. A. D. (2014). Journal of Metamorphic Petrology, 32(2), 195-207. https://doi.org/10.1111/jmg.12066</p><p>[5] Wheeler, J. (2018). Journal of Metamorphic Petrology, 36(4), 439-461. https://doi.org/10.1111/jmg.12299</p>

scholarly journals Modeling chemical reactions in porous media: a review

2021 ◽  
Vol 33 (6) ◽  
pp. 2279-2300
Author(s):  
Bettina Detmann
Keyword(s):  
Porous Media ◽  
Chemical Reactions ◽  
Chemical Potential ◽  
Mixture Theory ◽  
Reaction Diffusion ◽  
Second Law Of Thermodynamics ◽  
Reaction Diffusion Equations ◽  
Set Up ◽  
The 1960S ◽  
In The Beginning

AbstractFirst, different porous media theories are presented. Some approaches are based on the classical mixture theory for fluids introduced in the 1960s by Truesdell and Coworkers. One of the first researchers who extended the theory to porous media (thus mixtures containing at least one solid constituent) and also accounting for chemical reactions was Bowen. Another important branch of porous media theory goes back to Biot. In the beginning, he dealt with classical geotechnical problems and set up his model empirically. Mathematicians often use reaction–diffusion equations which are limited in comparison with continuum models by several restrictive assumptions and very often only applicable to special problems. In this paper, the focus lies on approaches based on the mixture theory which incorporate chemical reactions. Different strategies to describe the chemical potential for mixtures are presented, and different opinions about the exploitation of the second law of thermodynamics for mixtures are put forward. Finally, several works of different types including chemical reactions in porous media are summarized.

A New Model of Local Elastic Deflections in Grinding

1984 ◽  
Vol 106 (1) ◽  
pp. 154-163 ◽  
Author(s):  
D. P. Saini
Keyword(s):  
Plastic Deformation ◽  
Mild Steel ◽  
Mathematical Models ◽  
Normal Force ◽  
Tangential Force ◽  
Force Component ◽  
Workpiece Material ◽  
Direct Function ◽  
New Model ◽  
Single Grain

Mathematical models describing the deflection behavior of the wheel-work contact presented so far are based on the assumption that contact deflections are a direct function of the normal force on the wheel or the grains during grinding. This paper presents experimental results showing the evidence of a new mechanism of contact deflections due to the rotation of grain as a result of the tangential force component. In this perspective, a new model which considers the deflections due to both the normal and the tangential force is proposed and developed with the assumption of elasto-plastic deformation of the workpiece material around the grain during cutting. The model is shown to be consistent with experimental deflections obtained from single grain cutting on mild steel and EN25 steel specimens.

Experimental Study of the Plastic Interaction of Model Surface Asperities during Sliding

1968 ◽  
Vol 10 (2) ◽  
pp. 121-132 ◽  
Author(s):  
C. M. Edwards ◽  
J. Halling
Keyword(s):  
Experimental Study ◽  
Friction And Wear ◽  
Normal Force ◽  
Tangential Force ◽  
Experimental Results ◽  
Interfacial Conditions ◽  
High Adhesion ◽  
Theoretical Predictions ◽  
Very High ◽  
Low Shear

The paper describes an experimental study of the plastic interaction of triangular shaped lead model asperities deformed under conditions of plane strain. The investigation yields values of the normal and tangential force variations throughout the junction interaction and details of the plastic deformation particularly in relation to junction growth. A number of asperity interfacial conditions are considered ranging from complete adhesion to very low shear strengths achieved using p.t.f.e. strip. The experimental results are compared with an earlier theoretical solution to this problem and show reasonable agreement with the theoretical predictions. In particular it is shown that the normal force, which is usually compressive, may become tensile for conditions of high adhesion between the asperities. This leads to very high values of the macroscopic friction coefficient such as occur in hard vacuum situations. The experimental results for the various surface conditions show sufficient agreement with theoretical predictions to justify the use of this type of theoretical approach for the wider study of the friction and wear of mating surfaces.

Investigation of Grinding Force and Surface Integrity of IC10 in Creep Feed Grinding

2020 ◽  
Author(s):  
Jun-chen Li ◽  
Wen-hu Wang ◽  
Rui-song Jiang ◽  
Xiao-fen Liu ◽  
Huang Bo ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Tool Wear ◽  
Material Removal ◽  
Normal Force ◽  
Removal Rate ◽  
Tangential Force ◽  
Grinding Force ◽  
Wheel Speed ◽  
Creep Feed Grinding ◽  
Creep Feed

Abstract The IC10 superalloy material is one of the most important materials for aero-engine turbine blade due to its excellent performances. However, it is difficult to be machined because of its special properties such as terrible tool wear and low machined efficiency. The creep feed grinding is widely used in machining IC10 superalloy due to the advance in reducing tool wear, improving material removal rate and surface quality. The creep feed grinding is a promising machining process with the advantages of high material removal rate due to large cutting depth, long cutting arc and very slow workpiece, and its predominant features might have significant influence on the grinding force and surface quality for the workpiece. Hence, it is of great importance to study the grinding force and surface integrity in creep feed grinding IC10 superalloy. In this paper, a series of orthogonal experiments have been carried out and the effects of grinding parameters on the grinding force and the surface roughness are analyzed. The topographies and defects of the machined surface were observed and analyzed using SEM. The results of the experiments show that the tangential force is decreased with the workpiece speed increasing. However, there is no significant change in tangential force with the increasing of grinding depth and wheel speed. The normal force is decreased with the workpiece speed increasing when the workpiece speed is less than 150 mm/min, but when the workpiece speed is more than 150 mm/min the normal force is increased tardily. Moreover, the normal force is increased sharply with the increase of grinding depth and is increased slowly with the increase of wheel speed. In general, the surface roughness is increased with workpiece speed and grinding depth increasing, while the trend of increase corresponding that of workpiece speed is more evident. The value of the surface roughness is decreased with wheel speed increasing. And it is found out that the main defect is burning of the IC10 superalloy material in creep feed grinding by energy spectrum analysis of some typical topography in this study.

Change of Shape and Change of Volume

1993 ◽  
Author(s):  
Brian Bayly
Keyword(s):  
Isotropic Material ◽  
Chemical Potential ◽  
Stress Gradient ◽  
Small Length ◽  
Large Sphere ◽  
Point Of Interest ◽  
Principal Axes ◽  
Principal Directions ◽  
Point To Point ◽  
Spatial Stress

In earlier chapters we first defined a material's chemical potential, and then went on to enquire how the material responds. And similarly with a state of nonhydrostatic stress: having reviewed what it is, we consider how a material might respond. For the sake of simplicity, we imagine an extensive sample, such as a cubic meter, and suppose that the stress state is the same in every cubic centimeter; that is to say, there are no gradients in stress from point to point. Thus we do not enquire yet how a material responds to a spatial stress gradient; that comes later. We first enquire how it responds to a homogeneous but nonhydrostatic stress. Inside the material, close to the point of interest, we define a small length l by means of the material particles at its two ends. If, at a later moment, we find the distance between the particles to be l — δl, then we envisage the limit of the ratio δl/l as l goes to zero, give the limit the symbol ε, and name it the linear strain at the point of interest in the direction of l, positive when δl is positive, i.e., for a shortening and negative for an elongation. Another mental operation that can be performed in the neighborhood of the point of interest is to define a small sphere by means of the material particles that form its surface. At a later moment the particles will form the surface of an ellipsoid. (For a large sphere and an inhomogeneous situation, the new shape can be something more complicated; but as the imagined original sphere approaches zero diameter, the shape of its deformed counter-part can only approach an ellipsoid). The axes of the ellipsoid are principal directions of strain, and the magnitudes of the strains along them are named ε1, ε2, and ε3, with ε1 the largest. In an isotropic material, the principal axes of stress and strain coincide, with ε1 lying along the direction of σ1 and correspondingly; see Figure 7.la. As with stresses, the three values of ε themselves define an ellipsoid if they are all positive—see Figure 7.1b.

The Return of the Exiled Elements

2016 ◽  
Author(s):  
Ben McFarland
Keyword(s):  
Periodic Table ◽  
Chemical Potential ◽  
Comic Book ◽  
Cambrian Explosion ◽  
Signaling Proteins ◽  
Dry Land ◽  
Rocky Slope ◽  
Late Cambrian ◽  
Fast Chemistry ◽  
Set Up

The last page in a comic book is often a cliffhanger, so you’ll be more inclined to buy the next issue. It happens so regularly that as I read through the comic (yes, I still read a comic or two), I find myself trying to anticipate what kind of twist will be on the last page. The best twists are the ones you could have seen coming, but didn’t. The story in this book also has a chemical twist here, near the end. This twist is innovative, expensive, and predictable from chemistry. For this twist, the periodic table plays spoiler. Before the Cambrian explosion, hidden in the nets of signaling proteins within cells and signaling molecules outside cells, the cells held a secret chemical potential that could send a much faster signal, built from four elements involved in two of the balances set up in Chapter 5. This form of signaling would be incredibly expensive, but also incredibly fast. It would be electric in its nature and in its effects, the basis of both muscles and brains. Like water flowing randomly down a rocky slope, this fast signaling built from fast chemistry spread out in many different ways in life. At certain points, evolution came together and converged, repeatedly finding that a particular shape or signal was the best solution to a particular problem. Because the liquid flow of life was increased, it could diverge and converge more quickly, while predictably fitting into the shape of its landscape and efficiently moving downhill. The fast chemistry that forms the basis of fast muscles and faster neurons developed with the Cambrian explosion, along with oxygen and calcium use. The explosion of life provided predators that ate and prey that was eaten. Oxygen’s energy (resulting from its place on the periodic table) allowed more complex food chains, with more predators and more prey. For example, some calculate that more oxygen in the late Cambrian made more predators evolve. In response to this oxygen, certain species moved onto dry land, where they had more contact with that element.

Sign in / Sign up

Export Citation Format

Share Document