Singularities at the Tip of a Crack Terminating Normally at an Interface Between Two Orthotropic Media

1996 ◽  
Vol 63 (2) ◽  
pp. 264-270 ◽  
Author(s):  
J. C. Sung ◽  
J. Y. Liou
Keyword(s):  
Characteristic Equation ◽  
Complex Form ◽  
Stress Singularities ◽  
Real Form ◽  
Antisymmetric Mode ◽  
Real Roots ◽  
Complex Roots ◽  
Principal Axes ◽  
Orthotropic Media ◽  
Set Up

The order of stress singularities at the tip of a crack terminating normally at an interface between two orthotropic media is analyzed. Characteristic equation in complex form for the power of singularity s, where 0 < Re{s} < 1, is first set up for two general anisotropic materials. Attention is then focused on the problem that is composed by two orthotropic media where one of them (say, material #2 ) the material principal axes are aligned while the other one (say, material #1) the principal axes can have an angle γ relative to the interface. For such a problem, a real form of the characteristic equation is obtained. The roots are functions of γ in general. Two real roots exist for most values of γ; however, there are possible ranges of γ that the complex roots will occur. The roots s are found to be independent of γ when material #1 has the property that δ(1) = 1. When γ = 0, two roots are always real. Furthermore, each of these two roots is associated with symmetric or antisymmetric mode and they become equal when Δ = 1. Many other features of the effects of the material parameters on the behaviors of the roots s are further investigated in the present work, where the six generalized Dundurs’ constants, expressed in terms of Krenk’s parameters, play an important role in the analysis.

Stress Singularities at the Apex of a Dissimilar Anisotropic Wedge

1998 ◽  
Vol 65 (2) ◽  
pp. 454-463 ◽  
Author(s):  
Y. Y. Lin ◽  
J. C. Sung
Keyword(s):  
Characteristic Equation ◽  
Complex Form ◽  
Numerical Results ◽  
Orthotropic Materials ◽  
Stress Singularities ◽  
Material Parameters ◽  
Real Forms

The complex form of the characteristic equation for the stress singularities of the order rλ-1(0<Re[λ]<1) for the dissimilar anisotropic wedges is derived. Special attention is then focused on the problems that are composed by two orthotropic materials. For such problems the characteristic equation is expressed in real forms from which the dependence of the singularities on the material parameters and wedge angles is investigated. The case of a single free-fixed wedge problem is particularly studied in detail. Numerical results for several special wedge geometries are also presented.

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

scholarly journals An enhanced set of displacement parameter restraints in CRYSTALS

2018 ◽  
Vol 51 (4) ◽  
pp. 1059-1068 ◽  
Author(s):  
Pascal Parois ◽  
James Arnold ◽  
Richard Cooper
Keyword(s):  
Structural Model ◽  
Rigid Bodies ◽  
Model Parameters ◽  
Coordinate Systems ◽  
Principal Axes ◽  
Anisotropic Displacement Parameters ◽  
Set Up ◽  
Meaningful Relationships ◽  
Parameter Values ◽  
Total Model

Crystallographic restraints are widely used during refinement of small-molecule and macromolecular crystal structures. They can be especially useful for introducing additional observations and information into structure refinements against low-quality or low-resolution data (e.g. data obtained at high pressure) or to retain physically meaningful parameter values in disordered or unstable refinements. However, despite the fact that the anisotropic displacement parameters (ADPs) often constitute more than half of the total model parameters determined in a structure analysis, there are relatively few useful restraints for them, examples being Hirshfeld rigid-bond restraints, direct equivalence of parameters and SHELXL RIGU-type restraints. Conversely, geometric parameters can be subject to a multitude of restraints (e.g. absolute or relative distance, angle, planarity, chiral volume, and geometric similarity). This article presents a series of new ADP restraints implemented in CRYSTALS [Parois, Cooper & Thompson (2015), Chem. Cent. J. 9, 30] to give more control over ADPs by restraining, in a variety of ways, the directions and magnitudes of the principal axes of the ellipsoids in locally defined coordinate systems. The use of these new ADPs results in more realistic models, as well as a better user experience, through restraints that are more efficient and faster to set up. The use of these restraints is recommended to preserve physically meaningful relationships between displacement parameters in a structural model for rigid bodies, rotationally disordered groups and low-completeness data.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

scholarly journals The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 746
Author(s):  
Ilija Tanackov ◽  
Ivan Pavkov ◽  
Željko Stević
Keyword(s):  
Impossibility Theorem ◽  
Second Phase ◽  
Polynomial Roots ◽  
Real Roots ◽  
Quintic Equation ◽  
Complex Roots ◽  
Abel Polynomial ◽  
Quintic Polynomial ◽  
Univariate Polynomial ◽  
Newton’S Identities

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

Three-Dimensional Elasticity Solution for Sandwich Plates With Orthotropic Phases: The Positive Discriminant Case

2008 ◽  
Vol 76 (1) ◽  
Author(s):  
George A. Kardomateas
Keyword(s):  
Characteristic Equation ◽  
Orthotropic Material ◽  
Three Dimensional ◽  
Sandwich Plates ◽  
Elasticity Solution ◽  
Material Constants ◽  
Real Roots ◽  
Three Dimensional Elasticity ◽  
Rectangular Sandwich Plates ◽  
Positive Discriminant

A three-dimensional elasticity solution for rectangular sandwich plates exists only under restrictive assumptions on the orthotropic material constants of the constitutive phases (i.e., face sheets and core). In particular, only for negative or zero discriminant of the cubic characteristic equation, which is formed from these constants (case of three real roots). The purpose of the present paper is to present the corresponding solution for the more challenging case of positive discriminant, in which two of the roots are complex conjugates.

scholarly journals Computation of the Negative Damping Associated to the Hunting Motion of the Railway Wheelset, without Using Geometrical or Tribological Restrictions into the Model

2020 ◽  
Vol 5 (2) ◽  
pp. 210-217
Author(s):  
Barenten Suciu
Keyword(s):  
Characteristic Equation ◽  
Opposite Sign ◽  
Damping Ratio ◽  
Complex Conjugate ◽  
Railway Vehicle ◽  
Real Roots ◽  
Vibration Model ◽  
Hunting Motion ◽  
Negative Damping ◽  
Analytical Expressions

Recently, analytical expressions for the damped natural frequency and damping ratio were proposed for the so-called dynamical hunting, either by assuming that the wheel conicity can be neglected, or by imposing restrictions on the ratio between the lateral and longitudinal creep coefficients, and also, on the ratio of the track span to the yawing diameter. However, instead of a pair of complex conjugate roots, and two real roots, of opposite sign, two pairs of complex conjugate roots were obtained for the characteristic equation. Purpose of this work is to achieve accurate expressions for the damping associated to the hunting motion, without imposing geometrical or tribological limitations into the vibration model, and to evaluate the error on the damping ratio, introduced by the simplified models. Also, nature of the roots of the characteristic equation is discussed, relative to the critical speed of the railway vehicle.

scholarly journals Increasing the interval of convergence for a generalized Newton's method of solving nonlinear equations

2016 ◽  
pp. 7-12
Author(s):  
А.Н. Громов
Keyword(s):  
Continuous Function ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
The Real ◽  
Real Roots ◽  
Complex Roots ◽  
Transcendental Equations ◽  
Generalized Newton’S Method

Рассмотрен подход к построению расширения промежутка сходимости ранее предложенного обобщения метода Ньютона для решения нелинейных уравнений одного переменного. Подход основан на использовании свойства ограниченности непрерывной функции, определенной на отрезке. Доказано, что для поиска действительных корней вещественнозначного многочлена с комплексными корнями предложенный подход дает итерации с нелокальной сходимостью. Результат обобщен на случай трансцендентных уравнений. An approach to the construction of an extended interval of convergence for a previously proposed generalization of Newton's method to solve nonlinear equations of one variable. This approach is based on the boundedness of a continuous function defined on a segment. It is proved that, for the search for the real roots of a real-valued polynomial with complex roots, the proposed approach provides iterations with nonlocal convergence. This result is generalized to the case transcendental equations.

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