On Bonded Inclusions With Circular or Straight Boundaries in Plane Elastostatics

1990 ◽  
Vol 57 (4) ◽  
pp. 850-856 ◽  
Author(s):  
T. Honein ◽  
G. Herrmann
Keyword(s):  
Homogeneous Problem ◽  
Dissimilar Materials ◽  
Circular Inclusion ◽  
Algebraic Expression ◽  
Infinite Domain ◽  
Limiting Case ◽  
Plane Elastostatics

It is shown that the solution, in plane elastostatics, for an infinite domain with a bonded circular inclusion (heterogeneous problem), may be obtained from the solution of the corresponding homogeneous problem merely by substitution into a simple algebraic expression (heterogenization). This relation is universal in the sense of being independent of the loading considered. The case of two half-planes occupied by two dissimilar materials and bonded along a straight boundary is obtained as a limiting case.

Explicit Solutions of Plane Elastostatics Problems in Heterogeneous Solids

1998 ◽  
Vol 14 (2) ◽  
pp. 67-73 ◽  
Author(s):  
C. K. Chao ◽  
B. Gao
Keyword(s):  
Elastic Moduli ◽  
Homogeneous Problem ◽  
Circular Inclusion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Infinite Matrix ◽  
Heterogeneous Solids ◽  
Arbitrary Loading ◽  
Accuracy Result ◽  
Plane Elastostatics

AbstractThe problem of two circular inclusions of arbitrary radii and of different elastic moduli, which are perfectly bonded to an infinite matrix subjected to arbitrary loading, is solved by the heterogenization technique. This implies that the solution of the heterogeneous problem can be readily obtained from that of the corresponding homogeneous problem by a simple algebraic substitution. Based on the method of successive approximations and the technique of analytical continuation, the solution is formulated in a manner which leads to an approximate, but arbitrary accuracy, result. The present derived solution can be also applied to the problem with straight boundaries. Both the problem of two circular inclusions embedded in an infinite matrix and the problem of a circular inclusion embedded in a half-plane matrix are considered as our examples to demonstrate the use of the present approach.

Pressurized Cylindrical Shell With a Rigid Circular Inclusion

1978 ◽  
Vol 100 (2) ◽  
pp. 158-163 ◽  
Author(s):  
D. H. Bonde ◽  
K. P. Rao
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Shallow Shell ◽  
Circular Inclusion ◽  
Hankel Functions ◽  
Curvature Parameter ◽  
Shell Equations ◽  
Limiting Case ◽  
Rigid Circular Inclusion ◽  
Good Agreement

The effect of a rigid circular inclusion on stresses in a cylindrical shell subjected to internal pressure has been studied. The two linear shallow shell equations governing the behavior of a cylindrical shell are converted into a single differential equation involving a curvature parameter and a potential function in nondimensionalized form. The solution in terms of Hankel functions is used to find membrane and bending stressses. Boundary conditions at the inclusion shell junction are expressed in a simple form involving the in-plane strains and change of curvature. Good agreement has been obtained for the limiting case of a flat plate. The shell results are plotted in nondimensional form for ready use.

A Circular Inclusion With Circumferentially Inhomogeneous Sliding Interface in Plane Elastostatics

1998 ◽  
Vol 65 (1) ◽  
pp. 30-38 ◽  
Author(s):  
C. Q. Ru
Keyword(s):  
Circular Inclusion ◽  
Form Solution ◽  
Infinite Matrix ◽  
Rigorous Solution ◽  
First Order ◽  
Sliding Interface ◽  
The Mean ◽  
Plane Elastostatics ◽  
Basic Boundary ◽  
General Method

A general method is presented to obtain the rigorous solution for a circular inclusion embedded within an infinite matrix with a circumferentially inhomogeneous sliding interface in plane elastostatics. By virtue of analytic continuation, the basic boundary value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function inside the circular inclusion. The finite form solution is obtained that includes a finite number of unknown constants determined by the analyticity of the solution and certain other auxiliary conditions. With this method, the exact values of the average stresses within the circular inclusion can be calculated without solving the full problem. Several specific examples are used to illustrate the method. The effects of the circumferential variation of the interface parameter on the mean stress at the interface and the average stresses within the inclusion are discussed.

The Involution Correspondence in Plane Elastostatics for Regions Bounded by a Circle

1988 ◽  
Vol 55 (3) ◽  
pp. 566-573 ◽  
Author(s):  
T. Honein ◽  
G. Herrmann
Keyword(s):  
Outer Region ◽  
Elastic Field ◽  
Circular Disk ◽  
Infinite Plane ◽  
Infinite Domain ◽  
Circular Boundary ◽  
Inner Region ◽  
Plane Elastostatics

It is shown that the elastic field induced by prescribed displacements or surface tractions acting on a circular disk (inner region) can be expressed in terms of the elastic field induced by the same quantities acting on the circular boundary (hole) of an infinite plane (outer region), and vice versa. It is shown further that this correspondence is an involution. This novel representation permits one to express the elastic field in a disk with either vanishing displacements or tractions along the boundary in terms of the elastic field of an infinite domain, provided all singularities are in the inner region. Similarly, the elastic field in the outer region can be expressed in terms of the elastic field of the infinite domain, provided all singularities reside in the outer region. The expressions so-derived possess simple algebraic character and are universal in the sense of being independent of the applied loads (singularities) in the two problems.

scholarly journals On electrostatics in a gravitational field

1928 ◽  
Vol 118 (779) ◽  
pp. 184-194 ◽  
Keyword(s):  
Differential Equation ◽  
Harmonic Functions ◽  
Gravitational Force ◽  
Algebraic Expression ◽  
Uniform Field ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Gravitational Fields ◽  
Limiting Case ◽  
General Method

In a recent paper, Prof. Whittaker has discussed the effect, according to the general theory of relativity, of gravitation on electromagnetic phenomena. In particular, he has discussed electrostatics in gravitational fields of two kinds, namely (i) the field due to a single gravitating mass, in which case space-time has the metric discovered by Schwarzschild, and (ii) a limiting case of this, called a quasi-uniform field, in which the gravitational force is, in the neighbourhood of the origin, uniform. Whittaker’s general method, so far as electrostatical problems were concerned, was to solve the partial differential equation satisfied by the electrostatic potential in terms of generalised harmonic functions, and then, from these, to build up other solutions. In this way, he succeeded in finding an algebraic expression which represents the potential of a single electron in the quasi-uniform field; he did not, however, obtain a corresponding algebraic expression for the potential of an electron in the Schwarzschild field.

Multiple Circular Inclusion Problems in Plane Elastostatics

1974 ◽  
Vol 41 (1) ◽  
pp. 215-221 ◽  
Author(s):  
I.-W. Yu ◽  
G. P. Sendeckyj
Keyword(s):  
Uniaxial Tension ◽  
Elastic Moduli ◽  
Circular Inclusion ◽  
Elastic Matrix ◽  
Schwarz Alternating Method ◽  
Inclusion Problems ◽  
Alternating Method ◽  
Elastic Inclusions ◽  
Plane Bending ◽  
Plane Elastostatics

The problem of an unbounded elastic matrix containing any number of elastic inclusions is considered. The inclusions can have any radii and elastic moduli. Furthermore, the spacing of the inclusions can be arbitrary. The solution for the cases of uniaxial tension and in-plane bending is found by the Schwarz alternating method. Graphical results are presented for a number of examples.

scholarly journals The Inversion and Kelvin's Transformation in Plane Thermoelasticity with Circular or Straight Boundaries

2017 ◽  
Vol 34 (5) ◽  
pp. 617-627 ◽  
Author(s):  
C. K. Chao ◽  
C. H. Wu ◽  
K. Ting
Keyword(s):  
Thermal Loading ◽  
Homogeneous Problem ◽  
Elastic Inclusion ◽  
Infinite Extent ◽  
Stress Functions ◽  
Circular Elastic Inclusion ◽  
Limiting Case

AbstractThe problem of a circular elastic inclusion perfectly bonded to a matrix of infinite extent and subjected to arbitrarily thermal loading has been solved explicitly in terms of the corresponding homogeneous problem based on the inversion and Kelvin's transformation. It is to be noted that the relations established in this paper between the stress functions are algebraic and do not involve integration or solution of some other equations. Furthermore, the transformation leading from the solution for the homogeneous problem to that for the heterogeneous one is very simple, algebraic and universal in the sense of being independent of loading considered. The case of two bonded half-planes is obtained as a limiting case.

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