scholarly journals Development of the Rolling Texture in Titanium

1988 ◽  
Vol 7 (4) ◽  
pp. 317-337 ◽  
Author(s):  
H. P. Lee ◽  
C. Esling ◽  
H. J. Bunge
Keyword(s):  
Series Expansion ◽  
Characteristic Zero ◽  
Minor Component ◽  
Rolling Texture ◽  
Expansion Method ◽  
Zero Range ◽  
Different Types ◽  
Cold Rolled ◽  
A Minor ◽  
Series Expansion Method

The complete ODF of titanium, cold rolled up to 80% deformation, was calculated using the series expansion method, including the zero range method. The rolling texture obtained after 80% deformation is mainly characterized by the well-known orientation {0001}〈101¯0〉 ± 40°TD but with distinct spread ranges about it. At about 40% deformation several other texture components are found of which the component {0001}〈112¯0〉 must be mentioned. Further features of the obtained textures are a minor component as well as characteristic zero ranges. Texture development as function of the rolling degree can be divided into three ranges judged by increase or decrease of various texture components. In the early stages twinning in two different types of twinning systems is assumed whereas at higher deformation degrees the formation of the rolling texture is ascribed to glide deformation only.

scholarly journals The Use of a Quadratic Form for the Determination of Non-negative Texture Functions

1983 ◽  
Vol 6 (1) ◽  
pp. 1-19 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Series Expansion ◽  
Quadratic Forms ◽  
Expansion Method ◽  
Experimental Methods ◽  
Classical Analysis ◽  
Pole Figures ◽  
Zero Range ◽  
Euler Space ◽  
Series Expansion Method

The classical analysis of measured pole figures of textured polycrystals by the series expansion method does not necessarily produce a non-negative texture function. The main reason for this is, that the method is unable to find the terms of odd rank l of the series expansion.A new method is proposed, which introduces the non-negativity condition into the series expansion method by the use of quadratic forms. The method is found to be successful when treating sharp textures, which have a considerable zero range in Euler space. The preliminary determination of this zero range by experimental methods is however not necessary.

scholarly journals Approximation of the Orientation Distribution of Grains in Polycrystalline Samples by Means of Gaussians

1992 ◽  
Vol 19 (1-2) ◽  
pp. 9-27 ◽  
Author(s):  
D. I. Nikolayev ◽  
T. I. Savyolova ◽  
K. Feldmann
Keyword(s):  
Rolling Texture ◽  
Expansion Method ◽  
Operating Mode ◽  
Orientation Distribution ◽  
New Method ◽  
Gaussian Distributions ◽  
Positivity Condition ◽  
The Central Limit Theorem ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) obtained by classical spherical harmonics analysis may be falsified by ghost influences as well as series truncation effects. The ghosts are a consequence of the inversion symmetry of experimental pole figures which leads to the loss of information on the “odd” part of ODF.In the present paper a new method for ODF reproduction is proposed. It is based on the superposition of Gaussian distributions satisfying the central limit theorem in the SO(3)-space as well as the ODF positivity condition. The kind of ODF determination offered here is restricted to the fit of Gaussian parameters and weights with respect to the experimental pole figures. The operating mode of the new method is demonstrated for a rolling texture of copper. The results are compared with the corresponding ones obtained by the series expansion method.

scholarly journals Introduction of the Phone-Concept Into Pole Figure Inversion Using the Iterative Series Expansion Method

1992 ◽  
Vol 19 (3) ◽  
pp. 169-174 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Texture Analysis ◽  
Pole Figure ◽  
Series Expansion ◽  
Expansion Method ◽  
Probabilistic Methods ◽  
Series Expansion Method

The phone-concept as it is used in the various kinds of probabilistic methods can easily be applied to the iterative series expansion method for quantitative texture analysis. Only slight modifications of the existing routines are necessary. The advantages of this concept are demonstrated by a mathematical and an experimental example.

scholarly journals COMPARISON OF A GENERAL SERIES EXPANSION METHOD AND THE HOMOTOPY ANALYSIS METHOD

2010 ◽  
Vol 24 (15) ◽  
pp. 1699-1706 ◽  
Author(s):  
CHENG-SHI LIU ◽  
YANG LIU
Keyword(s):  
Series Expansion ◽  
Homotopy Analysis Method ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Basis Functions ◽  
Analysis Method ◽  
Convergence Region ◽  
General Series ◽  
Homotopy Analysis ◽  
Series Expansion Method

A simple analytic tool, namely the general series expansion method, is proposed to find the solutions for nonlinear differential equations. A set of suitable basis functions [Formula: see text] is chosen such that the solution to the equation can be expressed by [Formula: see text]. In general, t0 can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is simpler and clearer. As a result, we show that the secret parameter h in the homotopy analysis methods can be explained by using our parameter t0. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.

Series Expansion Method

2015 ◽  
pp. 63-83
Author(s):  
Matthew N. O. Sadiku ◽  
Sudarshan R. Nelatury
Keyword(s):  
Series Expansion ◽  
Expansion Method ◽  
Series Expansion Method

scholarly journals Laplace transform series expansion method for solving the local fractional heat-transfer equation defined on Cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 777-780
Author(s):  
Huan Sun ◽  
Xing-Hua Liu
Keyword(s):  
Heat Transfer ◽  
Laplace Transform ◽  
Series Expansion ◽  
Transfer Equation ◽  
Expansion Method ◽  
Cantor Sets ◽  
Heat Transfer Equation ◽  
Local Fractional Calculus ◽  
The Laplace Transform ◽  
Series Expansion Method

In this paper, we use the Laplace transform series expansion method to find the analytical solution for the local fractional heat-transfer equation defined on Cantor sets via local fractional calculus.

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