Chemical Potential Under Nonhydrostatic Stress

1993 ◽  
Author(s):  
Brian Bayly
Keyword(s):  
Equilibrium State ◽  
Normal Stress ◽  
Compressive Stress ◽  
Chemical Potential ◽  
Classical Mechanics ◽  
Compressive Stresses ◽  
Second Derivatives ◽  
Derivatives Of

The purpose of this chapter is to continue the unification that was begun in Chapter 8. There, first and second derivatives of normal stress with respect to orientation were used; we now examine the idea that the chemical potential of a component at a point can be a multivalued direction-dependent scalar like the normal-stress magnitude, and that it too can have a gradient with respect to orientation. The essence of a nonhydrostatic stress is that different planes through a point are subject to different normal compressive stresses: σn varies with the orientation of the plane considered. Let us focus on a plane i across which the normal compressive stress is σi: then we put forward the assertion that an equilibrium state that can be associated with plane i is a hydrostatic state whose pressure has the same magnitude as σi. For illustration, see Figure 9.1. (For the present, we take a cautious stance: each hydrostatic state in the figure is certainly an equilibrium state, and each is certainly associated with a plane, but is it the associated equilibrium state that properly belongs with that plane according to the precepts of, for example, de Groot (1951, p. 11)? For now, we make no attempt to prove that it is so: we simply use the assertion and explore its consequences. Fortunately its consequences include large amounts of classical mechanics so that it counts as a "successful assertion" on those grounds, but at least for now it lacks any underpinnings.) An immediate consequence of the assertion illustrated in Figure 9.1 is the relation in Figure 9.2.

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

"New Donors" in Czochralski Grown Silicon Annealed at T≥ 600°C under Compressive Stress

2005 ◽  
Vol 108-109 ◽  
pp. 181-186 ◽  
Author(s):  
Valentin V. Emtsev ◽  
Boris A. Andreev ◽  
Gagik A. Oganesyan ◽  
D.I. Kryzhkov ◽  
Andrzej Misiuk ◽  
...  
Keyword(s):  
Mechanical Stress ◽  
Compressive Stress ◽  
Reference Temperature ◽  
Enhancement Effect ◽  
Shallow Acceptor ◽  
Thermal Donor ◽  
Compressive Stresses ◽  
Heat Treated ◽  
Donor Formation ◽  
New Donors

Effects of compressive stress on oxygen agglomeration processes in Czochralski grown silicon heat treated at T= 450OC, used as a reference temperature, and T= 600OC to 800OC are investigated in some detail. Compressive stresses of about P= 1 GPa lead to enhanced formation of Thermal Double Donors in materials annealed over a temperature range of T= 450OC – 600OC. It has been shown that the formation of thermal donors at T= 450OC under normal conditions and compressive stress is accompanied with loss of substitutional boron. In contrast, the concentration of the shallow acceptor states of substitutional boron in silicon annealed under stress at T≥ 600OC remains constant. An enhancement effect of thermal donor formation is gradually weakened at T≥ 700OC. The oxygen diffusivity sensitive to mechanical stress is believed to be responsible for the observed effects in heat-treated silicon.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Investigation of a Technique for the Differentiation of Empirical Data

1983 ◽  
Vol 105 (3) ◽  
pp. 200-202 ◽  
Author(s):  
D. M. Trujillo ◽  
H. R. Busby
Keyword(s):  
Dynamic Programming ◽  
Differentiable Function ◽  
Empirical Data ◽  
Numerical Experiments ◽  
Random Noise ◽  
Second Derivatives ◽  
Derivatives Of

A dynamic programming filter is derived to estimate the first and second derivatives of empirical data. A series of numerical experiments are conducted using a known differentiable function with various amounts of added random noise.

Automatic 3-D interpretation of potential field data using analytic signal derivatives

Geophysics ◽  
1997 ◽  
Vol 62 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Nicole Debeglia ◽  
Jacques Corpel
Keyword(s):  
Signal Amplitude ◽  
Vertical Gradient ◽  
Analytic Signal ◽  
Magnetic Data ◽  
Practical Implementation ◽  
Potential Field Data ◽  
Gravity And Magnetic ◽  
Second Derivatives ◽  
Gravity And Magnetic Data ◽  
Derivatives Of

A new method has been developed for the automatic and general interpretation of gravity and magnetic data. This technique, based on the analysis of 3-D analytic signal derivatives, involves as few assumptions as possible on the magnetization or density properties and on the geometry of the structures. It is therefore particularly well suited to preliminary interpretation and model initialization. Processing the derivatives of the analytic signal amplitude, instead of the original analytic signal amplitude, gives a more efficient separation of anomalies caused by close structures. Moreover, gravity and magnetic data can be taken into account by the same procedure merely through using the gravity vertical gradient. The main advantage of derivatives, however, is that any source geometry can be considered as the sum of only two types of model: contact and thin‐dike models. In a first step, depths are estimated using a double interpretation of the analytic signal amplitude function for these two basic models. Second, the most suitable solution is defined at each estimation location through analysis of the vertical and horizontal gradients. Practical implementation of the method involves accurate frequency‐domain algorithms for computing derivatives with an automatic control of noise effects by appropriate filtering and upward continuation operations. Tests on theoretical magnetic fields give good depth evaluations for derivative orders ranging from 0 to 3. For actual magnetic data with borehole controls, the first and second derivatives seem to provide the most satisfactory depth estimations.

Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

2015 ◽  
Vol 113 (13-14) ◽  
pp. 1961-1977 ◽  
Author(s):  
Dmytro Bykov ◽  
Taras Petrenko ◽  
Róbert Izsák ◽  
Simone Kossmann ◽  
Ute Becker ◽  
...  
Keyword(s):  
Detailed Analysis ◽  
Efficient Implementation ◽  
Hartree Fock ◽  
Second Derivatives ◽  
Derivatives Of
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