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.

Further Aspects of the Elastic Field for Two Circular Inclusions in Antiplane Elastostatics

1992 ◽  
Vol 59 (4) ◽  
pp. 774-779 ◽  
Author(s):  
E. Honein ◽  
T. Honein ◽  
G. Herrmann
Keyword(s):  
Stress Field ◽  
Elastic Field ◽  
Shear Moduli ◽  
Two Circular Holes ◽  
Universal Formula ◽  
Arbitrary Loading ◽  
Infinite Extent ◽  
Circular Holes ◽  
Elastic Inclusions ◽  
The Difference

The heterogenization technique, recently developed by the authors, is applied to the problem, in antiplane elastostatics, of two circular inclusions of arbitrary radii and of different shear moduli, and perfectly bonded to a matrix, of infinite extent, subjected to arbitrary loading. The solution is formulated in a manner which leads to some exact results. Universal formulae are derived for the stress field at the point of contact between two elastic inclusions. It is also discovered that the difference in the displacement field, at the limit points of the Apollonius family of circles to which the boundaries of the inclusions belong, is the same for the heterogeneous problem as for the corresponding homogeneous one. This discovery leads to a universal formula for the average stress between two circular holes or rigid inclusions. Moreover, the asymptotic behavior of the stress field at the closest points of two circular holes or rigid inclusions approaching each other is also studied and given by universal formulae, i.e., formulae which are independent of the loading being considered.

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.

Cosserat Modeling of Size Effects in Crystalline Solids

2000 ◽  
Vol 653 ◽  
Author(s):  
Samuel Forest
Keyword(s):  
Stress State ◽  
Size Effects ◽  
Elastic Inclusion ◽  
Characteristic Size ◽  
Infinite Matrix ◽  
Crystalline Solids ◽  
Generalized Continua ◽  
Absolute Size ◽  
Kink Bands ◽  
The Matrix

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

A UNIFIED FORMALISM OF THERMAL QUANTUM FIELD THEORY

1994 ◽  
Vol 09 (14) ◽  
pp. 2363-2409 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vacuum Polarization ◽  
Time Dependent ◽  
Quantum Field ◽  
Bogoliubov Transformation ◽  
Transformation Matrices ◽  
Thermo Field Dynamics ◽  
The Matrix ◽  
Recent Developments

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

A Spherical Inclusion with Inhomogeneous Interface in Conduction

2003 ◽  
Vol 19 (1) ◽  
pp. 1-8
Author(s):  
T. Chen ◽  
C. H. Hsieh ◽  
P. C. Chuang
Keyword(s):  
Infinite Number ◽  
Effective Conductivity ◽  
Spherical Inclusion ◽  
Cone Angle ◽  
Imperfect Interface ◽  
Infinite Matrix ◽  
Algebraic Equations ◽  
Legendre Series ◽  
Linear Set ◽  
The Matrix

ABSTRACTA series solution is presented for a spherical inclusion embedded in an infinite matrix under a remotely applied uniform intensity. Particularly, the interface between the inclusion and the matrix is considered to be inhomegeneously bonded. We examine the axisymmetric case in which the interface parameter varies with the cone angle θ. Two kinds of imperfect interfaces are considered: an imperfect interface which models a thin interphase of low conductivity and an imperfect interface which models a thin interphase of high conductivity. We show that, by expanding the solutions of terms of Legendre polynomials, the field solution is governed by a linear set of algebraic equations with an infinite number of unknowns. The key step of the formulation relies on algebraic identities between coefficients of products of Legendre series. Some numerical illustrations are presented to show the correctness of the presented procedures. Further, solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of spherical inclusions with equal size. The effective conductivity solely depends on one particular constant among an infinite number of unknowns.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

scholarly journals Development of a Hydrogeological Model of the Borrowdale Volcanics at Sellafield

1997 ◽  
Vol 1 (1) ◽  
pp. 35-46 ◽  
Author(s):  
R. J. Lunn ◽  
A. D. Lunn ◽  
R. Mackay
Keyword(s):  
Radioactive Waste ◽  
Transport Model ◽  
Research Programme ◽  
Maximum Principal Stress ◽  
Hydrogeological Model ◽  
Borehole Data ◽  
The Matrix ◽  
Recent Developments ◽  
Conceptual Hydrogeological Model ◽  
Current Maximum

Abstract. This work has arisen out of recent developments within the radioactive waste research programme managed by Her Majesty's Inspectorate of Pollution, UK (HMIP)*, to develop an integrated flow and transport model for the potential deep radioactive waste repository at Sellafield. One of the largest sources of uncertainty in model predictions, is the characterisation of the hydrogeological properties of the underlying strata, in particular, of the Borrowdale Volcanic Group (BVG) within which the repository is to be located. Analysis of the available borehole data (that released by the proponent company, Nirex, by December 1995) for the BVG formation has indicated a dual regime consisting of flow within faults and flow within the matrix (or an equivalent porous medium containing micro-fractures). Significant relationships between permeability, depth and the presence and orientation of faults have been identified; they account for a variation of up to 6 orders of magnitude in mean permeability measurements. This can be explained in part by the effect of the orientation of the current maximum principal stress directions within the BVG: however, it is likely that permeability is also dependent on the existence of fracture families, which cannot be effectively identified from the data currently available. These analyses have enabled considerable insight to be gained into the dominant features of flow within the BVG. The conceptual hydrogeological model derived here will have a significant effect on the outcome and reliability of future radionuclide transport predictions in the Sellafield area.

On Multiple Inhomogeneities in Plane Elasticity

2019 ◽  
Author(s):  
Elie Honein ◽  
Tony Honein ◽  
Michel Najjar ◽  
Habib Rai
Keyword(s):  
Homogeneous Problem ◽  
Analytical Techniques ◽  
Plane Elasticity ◽  
Plane Case ◽  
Schwarz Alternating Method ◽  
Innovative Methods ◽  
Arbitrary Loading ◽  
The Matrix ◽  
Homogeneous Matrix ◽  
Made In

Abstract In this paper we present some new analytical techniques which have been recently developed to solve for problems of circular elastic inhomogeneities in anti-plane and plane elasticity. The inhomogeneities may be composed of different materials and have different radii. The matrix may be subjected to arbitrary loadings or singularities. The solution to this heterogeneous problem is sought as a transformation performed on the solution of the corresponding homogeneous problem, i.e., the problem when all the inhomogeneities are removed and the homogeneous matrix is subjected to the same loading/singularities, a procedure which has been dubbed ‘heterogenization’. In previous works, a single inhomogeneity or hole has been considered and the transformation has been shown to be purely algebraic in the antiplane case and involves differentiation of the Kolosov-Mushkelishvili complex potentials in the plane case. Universal formulas, i.e., formulas which are independent of the loading/singularities, that express the stresses at the inter-face of the inhomogeneity in terms of the stresses that would have existed at the same interface had the inhomogeneity been absent, have been be derived. The solution for a single inhomogeneity bonded to a matrix which is subjected to arbitrary loading/singularities can then in principle be used systematically in a Schwarz alternating method to obtain the solution for multiple inhomogeneities to any degree of accuracy. However alternative and innovative methods have been sought which lead to a much faster convergence and in some cases to exact expressions in terms of infinite series. The aim of this paper is to present some of the progress that has been made in this direction.

Weighted Averaging Techniques in Oscillation Theory for Second Order Difference Equations

1992 ◽  
Vol 35 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Lynn H. Erbe ◽  
Pengxiang Yan
Keyword(s):  
Oscillation Theory ◽  
Second Order ◽  
Weighted Averaging ◽  
Real Numbers ◽  
Order Difference ◽  
The Matrix ◽  
System 2 ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation ◽  
Averaging Techniques

AbstractWe consider the self-adjoint second-order scalar difference equation (1) Δ(rnΔxn) +pnXn+1 = 0 and the matrix system (2) Δ(RnΔXn) + PnXn+1 = 0, where are seQuences of real numbers (d x d Hermitian matrices) with rn > 0(Rn > 0). The oscillation and nonoscillation criteria for solutions of (1) and (2), obtained in [3, 4, 10], are extended to a much wider class of equations by Riccati and averaging techniques.

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