The Effect of Two Rigid Spherical Inclusions on the Stresses in an Infinite Elastic Solid

1966 ◽  
Vol 33 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Joseph F. Shelley ◽  
Yi-Yuan Yu
Keyword(s):  
Stress Field ◽  
Uniaxial Tension ◽  
Elastic Solid ◽  
Displacement Function ◽  
Hydrostatic Tension ◽  
Series Form ◽  
Spherical Inclusions ◽  
Spherical Cavities ◽  
In Series ◽  
The Common

Presented in this paper is a solution in series form for the stresses in an infinite elastic solid which contains two rigid spherical inclusions of the same size. The stress field at infinity is assumed to be either hydrostatic tension or uniaxial tension in the direction of the common axis of the inclusions. The solution is based upon the Papkovich-Boussinesq displacement-function approach and makes use of the spherical dipolar harmonics developed by Sternberg and Sadowsky. The problem is closely related to, but turns out to be much more involved than, the corresponding problem of two spherical cavities solved by these authors.

On the Axisymmetric Problem of the Theory of Elasticity for an Infinite Region Containing Two Spherical Cavities

1952 ◽  
Vol 19 (1) ◽  
pp. 19-27
Author(s):  
E. Sternberg ◽  
M. A. Sadowsky
Keyword(s):  
Axisymmetric Problem ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Uniform Field ◽  
Dimensional Problem ◽  
Spherical Cavities ◽  
Infinite Region ◽  
In Series ◽  
The Common ◽  
Axis Of Symmetry

Abstract This paper contains a solution in series form for the stress distribution in an infinite elastic medium which possesses two spherical cavities of the same size. The loading consists of tractions applied to the cavities, as well as of a uniform field of tractions at infinity, and both are assumed to be symmetric with respect to the common axis of symmetry of the cavities and with respect to the plane of geometric symmetry perpendicular to this axis. The loading is otherwise unrestricted. The solution is based upon the Boussinesq stress-function approach and apparently constitutes the first application of spherical dipolar co-ordinates in the theory of elasticity. Numerical evaluations are given for the case in which the surfaces of the cavities are free from tractions and the stress field at infinity is hydrostatic. The results illustrate the interference of two sources of stress concentration in a three-dimensional problem. The approach used here may be extended to cope with the general equilibrium problem for a region bounded by two nonconcentric spheres.

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

scholarly journals Extension of the Sophomore’s Dream

2021 ◽  
Vol 29 (1) ◽  
pp. 211-218
Author(s):  
Gábor Román
Keyword(s):  
Convergence Properties ◽  
Series Form ◽  
In Series

Abstract In this article, we are going to look at the convergence properties of the integral ∫ 0 1 ( a x + b ) c x + d d x \int_0^1 {{{\left( {ax + b} \right)}^{cx + d}}dx} , and express it in series form, where a, b, c and d are real parameters.

Conventional Water Treatments

2020 ◽  
pp. 138-165
Keyword(s):  
Water Quality ◽  
Water Treatment ◽  
Rural Regions ◽  
Sludge Management ◽  
Treatment Processes ◽  
Water Treatment Plants ◽  
In Series ◽  
Treatment Facilities ◽  
The Common ◽  
Water Treatments

Conventional water treatments have several successive processes in series to produce potable water. This chapter talks about the conventional water treatment processes which are mainly used to treat water originated from freshwater sources. Besides, the discussion covers some typical water quality, both raw and treated, as well as the standards of water quality. One of the highlighted topics in this chapter is the common issues that are frequently happening in the conventional water treatment facilities around the rural regions experiencing tropical climate, which is centred on the issues affecting the raw water quality and treatment processes. The major issue during post-treatment which is on sludge management is also discussed by underlining some alternative to the traditional way of using sludge lagoons. Topics in this chapter provide a better perspective to the water treatment operators and students who are interested in this topic of major processes used in conventional water treatment plants as well as the common issues encountered.

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