On the Anisotropic Elastic Inclusions in Plane Elastostatics

1993 ◽  
Vol 60 (3) ◽  
pp. 626-632 ◽  
Author(s):  
Chyanbin Hwu ◽  
Wen J. Yen
Keyword(s):  
Major Axis ◽  
Analytical Continuation ◽  
Complex Matrix ◽  
Arbitrary Loading ◽  
Special Cases ◽  
The Matrix ◽  
Point Forces ◽  
Elastic Inclusions ◽  
Anisotropic Elastic ◽  
Plane Elastostatics

By combining the method of Stroh’s formalism, the concept of perturbation, the technique of conformal mapping and the method of analytical continuation, a general analytical solution for the elliptical anisotropic elastic inclusions embedded in an infinite anisotropic matrix subjected to an arbitrary loading has been obtained in this paper. The inclusion as well as the matrix are of general anisotropic elastic materials which do not imply any material symmetry. The special cases when the inclusion is rigid or a hole are also studied. The arbitrary loadings include in-plane and antiplane loadings. The shapes of ellipses cover the lines or circles when the minor axis is taken to be zero or equal to the major axis. The solutions of the stresses and deformations in the entire domain are expressed in complex matrix notation. Simplified results are provided for the interfacial stresses along the inclusion boundary. Some interesting examples are solved explicitly, such as point forces or dislocations in the matrix and uniform loadings at infinity. Since the general solutions have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our results are exact and universal.

Thermal Stresses in a Generally Anisotropic Body With an Elliptic Inclusion Subject to Uniform Heat Flow

1998 ◽  
Vol 65 (1) ◽  
pp. 51-58 ◽  
Author(s):  
C. K. Chao ◽  
M. H. Shen
Keyword(s):  
Heat Flow ◽  
Thermal Stresses ◽  
Anisotropic Body ◽  
Analytical Continuation ◽  
Complex Matrix ◽  
Elliptic Inclusion ◽  
Uniform Heat ◽  
Special Cases ◽  
The Matrix ◽  
Stress Functions

A general analytical solution for the elliptical anisotropic inclusion embedded in an infinite anisotropic matrix subjected to uniform heat flow is provided in this paper. Based upon the method of Lekhnitskii formulation, the technique of conformal mapping, the method of analytical continuation, and the concept of superposition, both the solutions of the temperature and stress, functions either in the matrix or in the inclusion are expressed in complex matrix notation. Numerical results are carried out and provided in graphic form to elucidate the effect of material and geometric parameters on the interfacial stresses. Since the general solutions have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our solutions presented here are exact and general.

Interactions Between Dislocations and Anisotropic Elastic Elliptical Inclusions

1994 ◽  
Vol 61 (3) ◽  
pp. 548-554 ◽  
Author(s):  
Wen J. Yen ◽  
Chyanbin Hwu
Keyword(s):  
Anisotropic Media ◽  
Analytical Continuation ◽  
Two Dimensional ◽  
Closed Form Solutions ◽  
Special Cases ◽  
General Field ◽  
Isotropic Media ◽  
In Series ◽  
Elastic Inclusions ◽  
Anisotropic Elastic

A general field solution for the stresses and displacements of the interactions between dislocations and inclusions has been derived in this paper by applying the Stroh’s formalism and the Muskhelishvili’s method of analytical continuation. The solutions are valid for general elastic anisotropic media under two-dimensional deformation. The interaction energy between dislocations and elastic inclusions is obtained explicitly. The solutions in general are expressed in series form for elastic inclusions. However, for the special cases when the elastic inclusions are replaced by a hole or rigid inclusion, simple closed-form solutions are derived. The general solutions are verified by considering the isotropic media since it is the only solution available in the literature. For the general anisotropic media, a series of contour diagrams for the glide component of the force on a dislocation are provided in this paper to study the effects of inclusion hardness, shape, and matrix anisotropy.

Aspects of Heterogenization

1994 ◽  
Vol 116 (3) ◽  
pp. 298-304 ◽  
Author(s):  
E. Honein ◽  
T. Honein ◽  
G. Herrmann
Keyword(s):  
Group Structure ◽  
Infinite Matrix ◽  
Real Numbers ◽  
Arbitrary Loading ◽  
Infinite Extent ◽  
The Matrix ◽  
Recent Developments ◽  
Elastic Inclusions

In this paper, we briefly review some of the recent developments in the methodology of heterogenization. A connection between a group structure on the set (−1, 1) of real numbers t such that −1 < t < 1, and the elastostatics of a multilayered fiber perfectly bonded to an infinite matrix is pointed out. Also, universal formulae, pertaining to the solution of two circular elastic inclusions perfectly bonded to a matrix, of infinite extent, which is subjected to arbitrary loading, are discussed. As a novel illustration of the heterogenization procedure, we study here the case where the inclusions are elastically (i.e., “imperfectly”) embedded in the matrix. Several cases are presented and discussed.

scholarly journals Pairs of matrices with a non-zero commutator

1955 ◽  
Vol 51 (4) ◽  
pp. 551-553 ◽  
Author(s):  
L. S. Goddard ◽  
H. Schneider
Keyword(s):  
Characteristic Root ◽  
Column Vector ◽  
Closed Field ◽  
Complex Matrix ◽  
Algebraically Closed Field ◽  
Characteristic Roots ◽  
Linearly Independent ◽  
Special Cases ◽  
The Matrix ◽  
Square Complex

1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

The Influence of the Shape and Rigidity of an Elastic Inclusion on the Transverse Flexure of Thin Plates

1943 ◽  
Vol 10 (2) ◽  
pp. A69-A75
Author(s):  
Martin Goland
Keyword(s):  
Stress Distribution ◽  
Future Study ◽  
Rigid Inclusion ◽  
Elastic Inclusion ◽  
Thin Plates ◽  
Elastic Plates ◽  
Perforated Plates ◽  
Circular Rigid Inclusion ◽  
Special Cases ◽  
Elastic Inclusions

Abstract The purpose of this paper is to investigate the influence of several types of inclusions on the stress distribution in elastic plates under transverse flexure. An “inclusion” is defined as a close-fitting plate of some second material cemented into a hole cut in the interior of the elastic plate. Depending upon the properties of the material of which it is composed, the inclusion is described as rigid or elastic. In particular, the solutions presented will deal with the effects of circular inclusions of differing degrees of elasticity and rigid inclusions of varying elliptical form. Since the rigid inclusion and the hole are limiting types of elastic inclusions, and the circular shape is a special form of the ellipse, plates with either a circular hole or a circular rigid inclusion are important special cases of this discussion. It is hoped that the present analysis of several types of inclusions will aid in a future study of perforated plates stiffened by means of reinforcing rings fitted into the holes.

A Rapid Approach for Calculating the Damped Eigenvalues of a Gas Turbine on a Minicomputer: Theory

1984 ◽  
Vol 106 (2) ◽  
pp. 239-249 ◽  
Author(s):  
E. J. Gunter ◽  
R. R. Humphris ◽  
H. Springer
Keyword(s):  
Gas Turbine ◽  
Characteristic Polynomial ◽  
Normal Modes ◽  
Complex Matrix ◽  
Large Computer ◽  
Mainframe Computer ◽  
The Matrix ◽  
Rigid Body Modes ◽  
Transfer Method ◽  
Matrix Transfer Method

The calculation of the damped eigenvalues of a large multistation gas turbine by the complex matrix transfer procedure may encounter numerical difficulties, even on a large computer due to numerical round-off errors. In this paper, a procedure is presented in which the damped eigenvalues may be rapidly and accurately calculated on a minicomputer with accuracy which rivals that of a mainframe computer using the matrix transfer method. The method presented in this paper is based upon the use of constrained normal modes plus the rigid body modes in order to generate the characteristic polynomial of the system. The constrained undamped modes, using the matrix transfer process with scaling, may be very accurately calculated for a multistation turbine on a minicomputer. In this paper, a five station rotor is evaluated to demonstrate the procedure. A method is presented in which the characteristic polynomial may be automatically generated by Leverrier’s algorithm. The characteristic polynomial may be directly solved or the coefficients of the polynomial may be examined by the Routh criteria to determine stability. The method is accurate and easy to implement on a 16 bit minicomputer.

scholarly journals Volumetric Constraint Models for Anisotropic Elastic Solids

2004 ◽  
Vol 71 (5) ◽  
pp. 731-734 ◽  
Author(s):  
Carlos A. Felippa ◽  
Eugenio On˜ate
Keyword(s):  
Hydrostatic Stress ◽  
Stress Pattern ◽  
Stress System ◽  
Anisotropic Behavior ◽  
Elastic Solids ◽  
Physical Behavior ◽  
Special Cases ◽  
Anisotropic Elastic ◽  
Arbitrary Anisotropy ◽  
Physical Materials

We study three “incompressibility flavors” of linearly elastic anisotropic solids that exhibit volumetric constraints: isochoric, hydroisochoric, and rigidtropic. An isochoric material deforms without volume change under any stress system. An hydroisochoric material does so under hydrostatic stress. A rigidtropic material undergoes zero deformations under a certain stress pattern. Whereas the three models coalesce for isotropic materials, important differences appear for anisotropic behavior. We find that isochoric and hydroisochoric models under certain conditions may be hampered by unstable physical behavior. Rigidtropic models can represent semistable physical materials of arbitrary anisotropy while including isochoric and hydroisochoric behavior as special cases.

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