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.

Disequilibrium 3: Internal Variables

1993 ◽  
Author(s):  
Brian Bayly
Keyword(s):  
Vinyl Chloride ◽  
Chemical Potential ◽  
Potential Gradient ◽  
Material Behavior ◽  
Internal Variables ◽  
Starting State ◽  
Separate Point ◽  
Point To Point ◽  
New Location ◽  
Rate Of Response

In Chapters 2, 3, and 4, the usefulness of the concept chemical potential has been explored for describing and predicting movement of material from point to point in space—from a location where a component's potential is high to a location where its potential is lower. But chemical potential influences another type of material behavior as well, as in the example at the end of Chapter 2, the polymerization of vinyl chloride. The polymerization is a process that runs at a certain rate, like diffusion of salt, and the rate depends on the potential difference between the starting state and the end state; but unlike diffusion of salt, there is no overall movement from one location to a new location—the vinyl chloride simply polymerizes where it is. There are movements, of course, on the scale of the interatomic distances, but nothing corresponding to the 4 m of travel that appears in the discussion of the dike. If no travel is involved, it is not so easy to calculate a potential gradient along the travel path and go on to predict a rate of response. Yet there definitely is a rate of response, even with PVC polymerizing. The purpose of this chapter is to consider this matter; we shall then be equipped to begin considering nonhydrostatic conditions. The essential idea is to represent all possible degrees of polymerization along an axis, as in Figure 5.1. The figure is drawn to represent a condition where the chemical potential per kilogram is greater in the monomer form than in the dimer form, i.e., a condition where the material polymerizes spontaneously. Suppose we know the chemical potential per kilogram for all degrees of polymerization and also, at some temperature, the rates at which 2 forms from 1, 3 forms from 2, etc. (per kg of the starting form in a pure state). Then we arbitrarily pick a distance on the horizontal axis to separate point 1 from point 2.

scholarly journals Principal directions in a gravitational field

1939 ◽  
Vol 6 (1) ◽  
pp. 12-16 ◽  
Author(s):  
A. G. D. Watson ◽  
H. W. Turnbull
Keyword(s):  
Gravitational Field ◽  
Metric Space ◽  
Four Dimensions ◽  
Principal Axes ◽  
Principal Directions

In a general metric space of four dimensions, with an interval given by , where – ds2 = gμνdxμdxν, where g = ║gμν║<0, we can choose locally galilean coordinates at any point. The initial directions of the axes can be fixed in an absolute fashion as the directions of the principal axes of the quadric Gμνdxμdxν = const., is the contracted Riemann-Christoffel tensor.

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.

Property determination and wave propagation in a block of Barre granite

1977 ◽  
Vol 67 (1) ◽  
pp. 87-102 ◽  
Author(s):  
Werner Goldsmith ◽  
J. L. Sackman ◽  
R. L. Taylor
Keyword(s):  
Wave Propagation ◽  
Finite Element Program ◽  
In Situ Calibration ◽  
Propagation Analysis ◽  
Principal Axes ◽  
Element Program ◽  
Principal Directions ◽  
Calibration Techniques ◽  
Barre Granite ◽  
The Impact

abstract The principal axes of a 666.8 by 609.6 by 489.0 mm (2614 in by 24 in by 1914 in) block of Barre granite, treated as an orthotropic elastic material were determined from measured pulse velocities along directions connecting 160 pairs of surface points, encompassing the entire spectrum of possible orientations. The elastic moduli of the rock were ascertained by Hopkinson bar tests involving rods cored from other samples along their principal directions; this was required for the execution of a wave-propagation analysis in the block treated as a half-space. Construction and insertion techniques were developed for transducers to be embedded in the rock at 14 locations. External and internal calibration procedures were devised to permit interpretation of the data transmitted from the interior of the sample. Transients in the block were generated by the impact of 6.35-mm (14 in) diameter steel spheres on loading bars sandwiching a thin quartz disk, serving as an input transducer, against the specimen. The wave patterns sensed by the transducers were displayed and photographed on oscillographic screens. A finite element program capable of handling arbitrary anisotropy was developed and employed for comparing the experimental results with analytical predictions based on the measured input as the boundary condition. For those stations where computations were performed, the correlation ranged from good to qualitative. It is concluded that better transducer embedment and in situ calibration techniques are required for internal transducers used in hard rocks of this type.

A Continuum Theory That Couples Creep and Self-Diffusion

2004 ◽  
Vol 71 (5) ◽  
pp. 646-651 ◽  
Author(s):  
Z. Suo
Keyword(s):  
Film Thickness ◽  
Chemical Potential ◽  
Relative Rate ◽  
Diffusion Flux ◽  
Stress Gradient ◽  
Continuum Theory ◽  
Potential Gradient ◽  
Self Diffusion ◽  
Component Material ◽  
Coupled Theory

In a single-component material, a chemical potential gradient or a wind force drives self-diffusion. If the self-diffusion flux has a divergence, the material deforms. We formulate a continuum theory to be consistent with this kinematic constraint. When the diffusion flux is divergence-free, the theory decouples into Stokes’s theory for creep and Herring’s theory for self-diffusion. A length emerges from the coupled theory to characterize the relative rate of self-diffusion and creep. For a flow in a film driven by a stress gradient, creep dominates in thick films, and self-diffusion dominates in thin films. Depending on the film thickness, either stress-driven creep or stress-driven diffusion prevails to counterbalance electromigration. The transition occurs when the film thickness is comparable to the characteristic length of the material.

scholarly journals Modeling of Catalytic Centers Formation Processes during Annealing of Multilayer Nanosized Metal Films for Carbon Nanotubes Growth

Nanomaterials ◽  
2020 ◽  
Vol 10 (3) ◽  
pp. 554
Author(s):  
Oleg Il’in ◽  
Nikolay Rudyk ◽  
Alexandr Fedotov ◽  
Marina Il’ina ◽  
Dmitriy Cherednichenko ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Mass Transfer ◽  
Chemical Potential ◽  
Stress Gradient ◽  
Metal Films ◽  
Film Structure ◽  
Formation Processes ◽  
Thermal Expansion Coefficients ◽  
Mass Transfer Processes ◽  
The Difference

The paper presents a theoretical model of the catalytic centers formation processes during annealing of multilayer nanosized metal films for carbon nanotubes growth. The approach to the description of the model is based on the mass transfer processes under the influence of mechanical thermoelastic stresses, which arise due to the difference in the thermal expansion coefficients of the substrate materials and nanosized metal layers. The thermal stress gradient resulting from annealing creates a drop in the chemical potential over the thickness of the film structure. This leads to the initiation of diffusion mass transfer between the inner and outer surfaces of the films. As a result, the outer surface begins to corrugate and fragment, creating separate islands, which serve as the basis for the catalytic centers formation. Experimental research on the formation of catalytic centers in the structure of Ni/Cr/Si was carried out. It is demonstrated that the proposed model allows to predict the geometric dimensions of the catalytic centers before growing carbon nanotubes. The results can be used to create micro- and nanoelectronics devices based on carbon nanotube arrays.

Coarse-grained potential for interaction with a spherical colloidal particle and planar wall

2010 ◽  
Vol 75 (5) ◽  
pp. 527-545 ◽  
Author(s):  
Milan Předota ◽  
Ivo Nezbeda ◽  
Stanislav Pařez
Keyword(s):  
Colloidal Particle ◽  
Point Interaction ◽  
Point Particle ◽  
Coarse Grained ◽  
Large Sphere ◽  
Interaction Sites ◽  
Inverse Power Law ◽  
Graphite Sphere ◽  
Point To Point ◽  
Spherical Colloidal Particle

An effective coarse-grained interaction potential between a point particle and a spherical colloidal particle with continuously distributed inverse power-law interaction sites is derived. The potential covers all ranges of spherical particle size, from a point particle up to an infinitely large particle forming a planar surface. In the small size limit, the point-to-point interaction is recovered, while in the limit of an infinitely large sphere the potential comes over to the known particle–wall potentials as, e.g., the 9–3 potential in the case of the Lennard–Jones interaction. Correctness and usefulness of the derived potential is exemplified by its application to SPC/E water at a graphite sphere and wall.

Finding the Right Problems: Some HREM Applications to Interfaces

1984 ◽  
Vol 42 ◽  
pp. 376-379
Author(s):  
D. Cherns
Keyword(s):  
Grain Boundaries ◽  
High Resolution Electron Microscopy ◽  
Boundary Structure ◽  
Atomic Structures ◽  
Grain Boundary Structure ◽  
Resolution Electron ◽  
Point To Point ◽  
The Right ◽  
New Generation ◽  
Better Than

The use of high resolution electron microscopy (HREM) to determine the atomic structure of grain boundaries and interfaces is a topic of great current interest. Grain boundary structure has been considered for many years as central to an understanding of the mechanical and transport properties of materials. Some more recent attention has focussed on the atomic structures of metalsemiconductor interfaces which are believed to control electrical properties of contacts. The atomic structures of interfaces in semiconductor or metal multilayers is an area of growing interest for understanding the unusual electrical or mechanical properties which these new materials possess. However, although the point-to-point resolutions of currently available HREMs, ∼2-3Å, appear sufficient to solve many of these problems, few atomic models of grain boundaries and interfaces have been derived. Moreover, with a new generation of 300-400kV instruments promising resolutions in the 1.6-2.0 Å range, and resolutions better than 1.5Å expected from specialist instruments, it is an appropriate time to consider the usefulness of HREM for interface studies.

An x-ray microprobe based upon a close-coupled focal-spot / glass capillary arrangement

1992 ◽  
Vol 50 (2) ◽  
pp. 1758-1759
Author(s):  
D. A. Carpenter ◽  
M. A. Taylor
Keyword(s):  
Imaging Techniques ◽  
Fluorescence Analysis ◽  
High Sensitivity ◽  
Specimen Preparation ◽  
X Rays ◽  
Glass Capillary ◽  
Close Coupling ◽  
X Ray ◽  
Point To Point ◽  
Beam Damage

The development of intense sources of x rays has led to renewed interest in the use of microbeams of x rays in x-ray fluorescence analysis. Sparks pointed out that the use of x rays as a probe offered the advantages of high sensitivity, low detection limits, low beam damage, and large penetration depths with minimal specimen preparation or perturbation. In addition, the option of air operation provided special advantages for examination of hydrated systems or for nondestructive microanalysis of large specimens.The disadvantages of synchrotron sources prompted the development of laboratory-based instrumentation with various schemes to maximize the beam flux while maintaining small point-to-point resolution. Nichols and Ryon developed a microprobe using a rotating anode source and a modified microdiffractometer. Cross and Wherry showed that by close-coupling the x-ray source, specimen, and detector, good intensities could be obtained for beam sizes between 30 and 100μm. More importantly, both groups combined specimen scanning with modern imaging techniques for rapid element mapping.

Atomic structure imaging of the Si/SiO2 interface with high-resolution electron microscopy

1986 ◽  
Vol 44 ◽  
pp. 390-391
Author(s):  
J.L. Batstone ◽  
J.M. Gibson ◽  
Alice.E. White ◽  
K.T. Short
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Atomic Structure ◽  
High Resolution Electron Microscopy ◽  
Medium Voltage ◽  
Voltage Electron ◽  
Resolution Electron ◽  
Structural Image ◽  
Electron Microscopes ◽  
Point To Point

High resolution electron microscopy (HREM) is a powerful tool for the determination of interface atomic structure. With the previous generation of HREM's of point-to-point resolution (rpp) >2.5Å, imaging of semiconductors in only <110> directions was possible. Useful imaging of other important zone axes became available with the advent of high voltage, high resolution microscopes with rpp <1.8Å, leading to a study of the NiSi2 interface. More recently, it was shown that images in <100>, <111> and <112> directions are easily obtainable from Si in the new medium voltage electron microscopes. We report here the examination of the important Si/Si02 interface with the use of a JEOL 4000EX HREM with rpp <1.8Å, in a <100> orientation. This represents a true structural image of this interface.

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