00024___62fc0285f4cb12f54789ab9f2b2ee99b

# 00024___62fc0285f4cb12f54789ab9f2b2ee99b - denoted by PI Pa...

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Sec. 1.1 HOOKE’SLAW 3 Under this hypothesis the atomistic structure of the body is ignored and the body is idealized into a geometrical copy in Euclidean space whose Fig. 1.1:l. Static equilibrium of a body under external forces. points are identified with the material particles of the body. Continuity is de- fined in mathematical sense as an iso- morphism of the real number system. Neighboring points remain as neigh- bors under any loading condition. No cracks or holes may open up in the in- terior of the body under the action of external load. A material satisfying this hypoth- esis is said to be a continuum. The study of the deformation or motion of a continuum under the action of forces is called the continuum mechanics. To introduce the second hypothesis, let us consider the action of a set of forces on the body. Let every force be fixed in direction and in point of application, and let the magnitude of all the forces be increased or decreased together: always bearing the same ratio to each other. Let the forces be
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Unformatted text preview: denoted by PI, Pa,. . . , P n and their magnitude by PI, P2, . . . , Pn. Then the ratios PI : P2 : . .. : P,, remain fixed. When such a set of forces is applied on the body, the body deforms. Let the displacement at an arbitrary point in an arbitrary direction be measured with respect to a rectangular Cartesian frame of reference fixed with the supports. Let this displacement be denoted by u. Then our second hypothesis is (H2) Hooke’s law; where al, a2,. . . ,a, are constants independent of the magnitude of PI, Pz,. . . , P,. The constants all a2,. . . ,a, depend, of course, on the loca- tion of the point at which the displacement component is measured and on the directions and points of application of the individual forces of the loading. Hooke’s law in the form (H2) is one that can be subjected readily to direct experimental examination. To complete the formulation of the theory of elasticity, we need a third hypothesis:...
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