Particular Cases of States of Strain and Stress
The formulations of the fundamental problems by means of the potential functions is one of the most used methods in the theory of elasticity. In the case of the first fundamental problem (conditions in displacements on the boundary) one introduces displac
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Particular Cases of States of Strain and Stress
The formulations of the fundamental problems by means of the potential functions is one of the most used methods in the theory of elasticity. In the case of the first fundamental problem (conditions in displacements on the boundary) one introduces displacement functions, while in the case of the second problem (conditions in stresses on the boundary) one introduces stress functions. The case of the mixed problem may be studied by anyone of the two representations. Obviously, in the case of a dynamic problem, intervenes the temporal variable too, so that initial conditions must also be put. Hereafter, we shall consider homogeneous, isotropic and linearly elastic bodies in the case of infinitesimal deformations and in the absence of volume forces; the potential functions depend, in general, on three variables (xi ; i ¼ 1; 2; 3; in case of orthogonal Cartesian coordinates). The computation difficulties led to the study of more simple problems, in two variables only. Thus, one considered plane problems (plane state of stress or plane state of strain), as well as antiplane ones. In both mentioned problems, one assumes, from the very beginning, that some of the components of the stress or strain tensors vanish. Thus, in what follows, we shall deal with problems in which one or two of the occurring components (e.g. normal or tangential stress or linear or angular strain) are considered to be equal to zero. The plane and the antiplane problems will be taken into consideration too. A problem where the third variable (let be the variable x3 ) appears only by its powers and where the potential functions depend only on two variables will be called a two-dimensional problem. Such results have been given by J. H. Michell [6], by A.-E.-H. Love [2] for a finite cylinder and by E. Almansi [4] for a thick plate. Many results in this direction as well as various generalizations have been given by G. Supino [8–13]. Similar problems have been considered by Gr. C. Moisil [7], A. Davidescu-Moisil [5] and A. Clebsch [1]. We also dealt with these problems [3, 14–26], which we will use in what follows.
P. P. Teodorescu, Treatise on Classical Elasticity, Mathematical and Analytical Techniques with Applications to Engineering, DOI: 10.1007/978-94-007-2616-1_13, Ó Springer Science+Business Media Dordrecht 2013
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Particular Cases of States of Strain and Stress
13.1 Conditions for Stresses Hereafter we shall put conditions for normal or for tangential stresses, obtaining general stress functions for the states of stress thus put in evidence.
13.1.1 Case of a Zero Normal Stress First of all, we deal with the general problem of elastostatics which results by putting the condition that one normal stress vanishes, namely r33 ¼ 0:
ð13:1Þ
13.1.1.1 General Case Taking into account the condition (13.1), in the absence of volume forces, the third equation of equilibrium (3.62) leads to r31 ¼ K2 ; r23 ¼ K;1 ;
ð13:2Þ
the function K ¼ Kðx1 ; x2 ; x3 Þ being arbitrary. This function must
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