On the Concept of Concentrated Loads and an Extension of Uniqueness Theorem in the Linear Theory of Elasticity

Sternberg, Eli; Eubanks, R. A.

doi:10.1512/iumj.1955.4.54003

Cited by 27 publications

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“…A comprehensive study aiming at a clarification and resolution of various fundamental questions connected with concentrated loads in elastostatics, was published by Sternberg and Eubanks [2] (1955).…”

Section: Introductionmentioning

confidence: 99%

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On concentrated loads and green's functions in elastostatics

Turteltaub

Sternberg

1968

Arch. Rational Mech. Anal.

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This investigation is concerned with the notion of concentrated loads in classical elastostati cs and related issues. Following a li1nit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and str e ss singularities, as well as the stress resultants of th e latter , are dete rmined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theor em. In addition, the pr esent work supplies a r ecipr ocal theorem and an integral representation-theorem applicable to singular problems of the type unde r consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended .

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Section: Introductionmentioning

confidence: 99%

“…[2] may briefly be outlined as follows . To begin with, the solution to a problem involving concentrated internal or surface loads is d efined as the limit of a sequence of regular solutions, corresponding to distributed body-forces or surface tractions.…”

Section: Introductionmentioning

confidence: 99%

On concentrated loads and green's functions in elastostatics

Turteltaub

Sternberg

1968

Arch. Rational Mech. Anal.

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“…13The difference between two solutions possessing these four properties evidently meets conditions (A), (B), (C), (D) of [9], p. 163, and hence must vanish identically. Requirement (c) precludes the presence of a concentrated load applied at the origin.…”

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“…"See [9] for the definition of a "regular" region of space employed here. Note that the region need not be bounded or simply connected.…”

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On the steady-state thermoelastic problem for the half-space

Sternberg¹,

McDowell²

1957

Quart. Appl. Math.

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Summary.This paper deals with the determination of the steady-state thermal stresses and displacements in a semi-infinite elastic medium which is bounded by a plane. The problem is treated within the classical theory of elasticity and is approached by the method of Green. It is shown that the stress field induced by an arbitrary distribution of surface temperatures is plane and parallel to the boundary. If the surface temperature is prescribed arbitrarily over a bounded "region of exposure" and is otherwise constant, the problem reduces to the determination of Boussinesq's three-dimensional logarithmic potential for a disk in the shape of the region of exposure, whose mass density is equal to the given temperature. Moreover, it is found that there exists a useful connection between the solutions to Boussinesq's and to the present problem for the half-space. An exact closed solution, in terms of complete and incomplete elliptic integrals of the first and second kind, is given for a circular region of exposure at uniform temperature. Exact solutions in terms of elementary functions are presented for a hemispherical distribution of temperature over a circular region, as well as for a rectangle at constant temperature.Introduction. Recent years have seen a revival of interest in the thermoelastic problem which has received repeated previous attention in the theory of elasticity.1 Nevertheless, the existing literature on this subject is largely confined to two-dimensional problems and to problems characterized by polar or rotational symmetry.A significant advance in connection with the general three-dimensional thermoelastic problem was made by Goodier [4], who considered a medium occupying the entire space; he reduced the computation of the thermal stresses due to an arbitrarily prescribed temperature distribution to the determination of the Newtonian potential for a mass distribution whose density coincides with the given temperature field. For domains other than the entire space, Goodier's approach merely supplies a particular solution of the thermoelastic equations and still necessitates the solution of an ordinary boundary-value problem in the theory of elasticity. A particular integral of the same form was employed earlier by Borchardt [5] in dealing with the special problem of the sphere.Goodier's method was extended by Mindlin and Cheng [6] to the problem of the half-space with a traction-free boundary, subjected to an arbitrary temperature distribution in its interior. As an application of the extended scheme, the problem of the half-space with a uniformly heated (or cooled) spherical core is solved in [6].In the present paper we return to the problem of the half-space but limit our attention

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