First- and second- order analysis of axially loaded crystals in n-fold symmetry

1984 ◽  
Vol 311 (1518) ◽  
pp. 469-493 ◽  
Keyword(s):  
Second Order ◽  
Rate Of Change ◽  
Order Analysis ◽  
Hardening Rule ◽  
Small Load ◽  
Second Derivatives ◽  
Latent Hardening ◽  
Hardening Parameter ◽  
Derivatives Of ◽  
Minimum Work

A comprehensive theoretical investigation of multiple slip in axially tensile-loaded f.c.c. crystals in n -fold symmetry positions ( n = 4, 6, 8) is presented. The analysis is complete to second order in terms of series expansions of all variables in the prescribed small load increment. In the first part of the paper, general kinematic relations and slip-system inequalities are given, and several new results discovered that apply independently of hardening rule and degree of symmetry. Subsequent sections contain extensive firsthand second-order analyses corresponding to four specific hardening theories, including Taylor's classical isotropic rule and the ‘ simple theory’ of anisotropic latent hardening. For minimum work, unifying relations are found connecting a generic hardening parameter, its rate of change with load, and the first and second derivatives of axial stretch that hold for all four theories.

XV.—A Chapter in the Calculus of Variations: Maxima and Minima, for Weak Variations, of Integrals involving Ordinary Derivatives of the Second Order

1927 ◽  
Vol 46 ◽  
pp. 149-193
Author(s):  
A. R. Forsyth
Keyword(s):  
Calculus Of Variations ◽  
Second Order ◽  
Second Derivatives ◽  
Independent Variable ◽  
Derivatives Of

The present memoir is devoted to the determination of criteria for the possession of maxima and minima by an integral which involves one independent variable, one dependent variable, and the first and second derivatives of the latter. All the variations, leading to tests for the existence of maxima and minima, are of the class called “weak”; not only are the variations themselves small, but their derivatives of all orders are small also.

scholarly journals Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

2021 ◽  
Author(s):  
Benjamin B. McMillan ◽  
Keyword(s):  
Differential Equation ◽  
Parabolic Equation ◽  
Conservation Laws ◽  
Parabolic Equations ◽  
General Class ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Derivatives ◽  
Monge Ampere ◽  
Derivatives Of

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.

Local Error Analysis and Comparison of the Swept- and Intersection-Based Remapping Methods

2017 ◽  
Vol 21 (2) ◽  
pp. 526-558 ◽  
Author(s):  
Matej Klima ◽  
Milan Kucharik ◽  
Mikhail Shashkov
Keyword(s):  
Second Order ◽  
Order Estimate ◽  
Local Error ◽  
General Estimate ◽  
Practical Applications ◽  
Numerical Tests ◽  
Computational Mesh ◽  
Local Accuracy ◽  
Second Derivatives ◽  
Derivatives Of

AbstractIn this paper, the numerical error of two widely used methods for remapping of discrete quantities from one computational mesh to another is investigated. We compare the intuitive, but resource intensive method utilizing intersections of computational cells with the faster and simpler swept-region-based method. Both algorithms are formally second order accurate, however, they are known to produce slightly different quantity profiles in practical applications. The second-order estimate of the error formula is constructed algebraically for both algorithms so that their local accuracy can be evaluated. This general estimate is then used to assess the dependence of the performance of both methods on parameters such as the second derivatives of the remapped distribution, mesh geometry or mesh movement. Due to the complexity of such analysis, it is performed on a set of simplified elementary mesh patterns such as cell corner expansion, rotation or shear. On selected numerical tests it is demonstrated that the swept-based method can distort a symmetric quantity distribution more substantially than the intersection-based approach when the computational mesh moves in an unsuitable direction.

Discrete optimization of semi-rigid steel frames in second-order analysis

2019 ◽  
Author(s):  
Marcos Henrique Bossardi Borges ◽  
Adelano Esposito ◽  
Herbert Gomes
Keyword(s):  
Discrete Optimization ◽  
Steel Frames ◽  
Second Order ◽  
Order Analysis

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.

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