What is the modulus of elasticity

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Shear elasticity

[818]Shear elasticity, the elasticity (s.d.) of the body against stresses that try to move two adjacent or very close surfaces along each other (see shear, pressure, stresses, shear stresses, shear strength). Designated f the shift in question for the initially small sizel distant flat surfacesF. by a resting force evenly distributed over themP = F τ in the direction of f and it creates a change d P of P. a change d f of f, so is called in [818]

d P = G F df / l, d f = l / GF d P = l / G d τ

1.


G the Modulus of elasticity, sometimes short shear modulus, for the direction of f (cf. modulus of elasticity, vol. 3, p. 393).

Hereafter the same means the relation of Voltage changed τ to Change in slipdf / l, under Sliding the longitudinal movement per unit distance of the displaced surfaces, f / l = daily γ (see the figure) or with the ordinary little ones f the angleγ connected as an arc of radius 1. Is G constant or becomes instead of variable G introduced a constant mean, that of P. = 0 to P = P the same after all f conditional, then follow from 1:

P = G F f / l, f = Pl / GF = l / G τ,

2.


so in this caseG the imaginary tensionτ for the slide f / l = 1 and vice versa 1 /G the slide f / l for the tensionτ = 1 represents. Given the difficulty G to determine from shear tests, its value is mostly obtained by torsion tests or from the relationship between isotropic bodies G and the tensile modulus of elasticity E. (cf. elasticity quotient, vol. 3, pp. 393, 395). However, especially with fibrous materials, a variability of G with the direction of stress is to be expected. On the influence of shear elasticity in bending, see d. (Vol. 1, p. 797) and Bending work, bending elasticity, beams (simple, vol. 1, p. 520), Elastic line (Vol. 3, p. 375), modulus of elasticity (Vol. 3, p. 394) etc .; General information about their influence s. Elasticity theory, general (Vol. 3, p. 389). It should be noted that in 1., 2. f only from the evenly distributed shear forces mentioned, and not from bending stresses to any part of the equation. For the case indicated in the figure above, one would have according to the bending theory with regard to the influence of the shear forces (vol. 1, p. 521; vol. 2, p. 3):



which only agrees with 2. when the second term disappears in the brackets, as with a sufficiently small one l, simultaneously k = 1 is, as with evenly on the cross sectionF. distributed P, while the shear stresses resulting from the bending are not evenly distributed (see Shear stresses and Vol. 1, p. 792). With regard to exceeding the elastic limit (sd) during thrust, there are similar conditions as in the corresponding case with tension (vol. 3, p. 386) and pressure (vol. 3, p. 116), in which considerable permanent deformations and, depending on the circumstances, cuts, Cracks etc and finally the break occurs (cf. yield point, shear strength).


Literature: [1] Navier, Résumé des leçons etc. sur l'application de la mécanique, avec des notes et des appendices de Saint-Venant, Paris 1864, pp. CXXV, 36, 185, 329 etc. - [2] Winkler, The theory of elasticity and skill, Prague 1867, pp. 20, 45 etc. - [3] Grashof, Theory of Elasticity and Strength, Berlin 1878, pp. 123, 199, 203, 283. - [4] Weyrauch, Theory of elastic bodies, Leipzig 1884, pp. 6, 32, 43, 123, 124, etc. - [5] Foeppl, lectures on technical mechanics, III, strength theory, Leipzig 1900, pp. 59, 74, 128. - [6] v. Tetmajer, The applied elasticity and strength theory, Leipzig and Vienna 1904, p. 294. - [7] Bach, elasticity and skill, Leipzig 1905, p. 288, 357, 395, 424. - [8] Keck-Hotop, lectures on elasticity theory, I, Hannover 1905, pp. 69, 89, 225.

Weyrauch.