. A treatise on the mathematical theory of elasticity . axis of x is given by theequations u=ex, v=0, w=0,where e is the amount of the extension. If e is negative there is contraction. The displacement in a simple shear of amount «(=2tano), by which lines parallel tothe axis of x slide along themselves, and particles in any plane parallel to the plane of(X, y) remain in that plane, is given by the equations u=sy, ?»=0, w=0. * The greater part of the theory is due to Cauohy (see Introduction). Somg^mprovementswere made by Clebsch in his treatise of 1862, and others were made by Kelvin and Tait,
. A treatise on the mathematical theory of elasticity . axis of x is given by theequations u=ex, v=0, w=0,where e is the amount of the extension. If e is negative there is contraction. The displacement in a simple shear of amount «(=2tano), by which lines parallel tothe axis of x slide along themselves, and particles in any plane parallel to the plane of(X, y) remain in that plane, is given by the equations u=sy, ?»=0, w=0. * The greater part of the theory is due to Cauohy (see Introduction). Somg^mprovementswere made by Clebsch in his treatise of 1862, and others were made by Kelvin and Tait, Nat. I. 3—2 36 SPECIFICATION OF STRAIN [CH. In Fig. 2, ^S is a segment of a line parallel to the axis of x, which subtends an angle2a at 0 and is bisected by Oy. By the simple shear particles lying on the line OA aredisplaced so as to he on OB. The particle at any point P on ABM displaced to § pn ABso that Pq=AB, and the particles on OP are displaced to points on 0§. A parallelogramsuch as OPNM becomes a parallelogram such as Fig. 2. If the angle xOP=6 we may prove that „_- 2tanatan2 5 ia,nPOQ= — iaxixOQ= tan 6 1 + 2 tan a tan 6 sec^ fl + 2 tan a tan 6 In particular, if 6 = ^17, cot xOQ=s, so that, if s is small, it is the complement of theangle in the strained state between two lines of particles which, in the unstrained state,were at right angles to each other. 6. Homogeneous strain. In the cases of simple extension and simple shear, the component dis-placements are expressed as linear functions of the coordinates. In general,if a body is strained so that the component displacement can be expressed inthis way, the strain is said to be homogeneous. Let the displacement corresponding with a homogeneous strain be givenby the equations u = a^x + ai22/ + (ht^^K ^ = ^^^ + ^? + w = C^ + o-^y + a^ x, y, z are changed into x+u,y + v, z + w, that is, are transformed bya linear substitution, any plane is transformed into a plane, and any ellipsoidi
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