Some anomalies of the curved bar problem in classical elasticity

The classical formulation of the homogeneous problem of a curved bar loaded only by and end force involves the assumption of an appropriate stress function with four arbitrary constants and the determination of these constants from the boundary conditions. Since there are five boundary conditions, four on the curved edge and one at the end, the solution is only possible because the coefficient matrix of the resulting algebraic equations is singular. This in turn means that certain inhomogeneous problems in which the curved edges are loaded by sinusoidally varying tractions cannot be solved using apparently appropriate stress functions.Additional stress functions which resolve this difficulty are introduced and an example problem is solved, which exhibits qualitatively different behavior from that in more general cases of loading. The problem is then reconsidered as a limiting case of sinusoidal loading of arbitrary wavelength. It is shown that the solution of the latter problem appears to become unbounded as the special case is approached, but that when the end conditions have been correctly satisfied by superposing an appropriate multiple of the end-loaded solution, the limit can be approached regularly and the correct special solution is recovered. The limiting process reveals a general procedure for determining the additional stress functions required for the special case.Similar relationships between homogeneous and inhomogeneous solutions for other common geometries are discussed.     

A unilateral contact problem in linear elasticity

In this paper we study the equilibrium of an elastic body which is simply supported, without friction, by a soft elastic plane. We prove that, if the exterior loads satisfy certain conditions of compatibility, the solution exists and is unique. Moreover, we find which regularity properties of the solution hold, provided the data are sufficiently smooth.Istituto di Scienza delle Costruzioni, Università di Pisa     

On steady plane flows in classical elasticity

A new class of boundary-value problems in mathematical elasticity is proposed, wherein the medium flows steadily relative to a non-embedded surface over which tractions or velocities are prescribed. Such flows are seen in metal forming operations where purely elastic streams enter and leave the working zone. The deformations are assumed here to be plane and isochoric. A general solution is formulated in terms of two complex potentials. Residual stress is accounted for in detail and a uniqueness theorem is proved. Some simple flows are examined, but it remains to develop a systematic procedure for matching the general solution to arbitrary boundary data.     

On the first boundary value problem for the classical theory of elasticity in a three-dimensional domain with a singular boundary

An existence and uniqueness theorem is proved for the first boundary value problem of classical elasticity, relating to a broad class of three-dimensional domains whose boundaries may have edges and vertexes.The qualitative properties, up to the boundary, of the solution are investigated.

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Résumé On démontre un théorème d'existence et unicité pour le premier problème de valeurs au contour de l'élasticité classique, concernant une large classe de domaines de l'éspace à trois dimensions, avec des frontières présentant des arètes et des sommets.On étudie les propriétés qualitatives, jusque'au contour, de la solution.

     

On uniqueness for the traction problem in finite elasticity

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered, and within the context of the fraction problem in finite elasticity, a number of uniqueness results are established. This work extends results obtained for the mixed problem by Gurtin and Spector.

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Resumé Dans plusieurs proble\`mes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment géerale, est considerée et un nombre des résultats d'unicité est établit dans le cadre du probleme de traction de l'elasticité non linéaire. Cet ouure prolonge des résultats pour le proble\`me mixte obtenus par Gurtin et Spector.

     

Nonstationary plane contact problem in theory of elasticity for conformal cylindrical surfaces

A numerical–analytical approach is described to investigate the process of impact interaction between a long smooth rigid body and the surface of a circular cylindrical cavity in elastic space. A non-stationary mixed initial boundary value problem is formulated with a priori unknown boundaries moving with variable velocity. The problem is solved using the methods of the theory of integral transforms, expansion of desired variables into a Fourier series, and the quadrature method to reduce the problem to solving a system of linear algebraic equations at each time step. Some concrete numerical computations are presented.The cylindrical body mass and radius impact on the proile of the transient process of contact interaction has been analysed.     

Critical velocity in a contact elasticity problem for the case of transonic punch velocity

The paper deals with a dynamic contact problem in the presence of friction forces in the transonic range of punch velocities, where the punch velocity exceeds the transverse wave velocity but is still less than the longitudinal wave velocity. It is shown that there exists a critical velocity at which the solution structure and the character of its behavior on the boundary of the contact region change. This velocity is $\sqrt 2 $ times the transverse wave velocity. The existence of this velocity is possibly related to the surface wave velocity under restricted deformation conditions.     

On uniqueness for the displacement problem in the dynamical theory of elasticity

By means of a weighted form of Garding's inequality, we prove a uniqueness theorem for the initial boundary value problem of finite elastodynamics in unbounded domains.

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Sommario Mediante un'opportuna forma pesata della disuguaglianza di Garding, che ricaviamo preliminarmente, dimostriamo un teorema di unicità per il problema degli spostamenti in elastodinamica non lineare relativo a corpi occupanti regioni non limitate dello spazio.