Art. 18. 



SHEAR MODULUS. 



15 



in pairs at right angles to each other, if there is equilibrium. It 

 would not be possible, in Fig. 13, to have the tangential forces 

 acting on planes E H and F G without those on E F and HG, 

 for the former form a couple which would turn the cube in a 

 clockwise direction. It follows that, wherever, in an elastic solid, 

 a transverse shearing stress occurs, there is also a longitudinal 

 shearing stress of equal intensity; the resultants of these, upon 

 planes at angles of 45 degrees with the shearing stresses, are equal 

 tensile and compressive stresses. It may easily be shown (Fig. 12) 

 that the intensities of these three kinds of stress are the same (76) . 

 If the cubes of Figs. 12 and 13 be taken indefinitely small, 

 the stresses may, in any case, be considered as 

 uniformly distributed over the faces of the cube, 

 and the above principle holds even if there are 

 also tangential forces (Fig. 12) or normal forces 

 (Fig. 13) acting on the faces, or both kinds act- 

 ing on the front and rear faces. 

 In the beam of Fig. 9 we found a vertical shear, which 

 would evidently produce a deformation of the part between any 

 planes as pq and rs, as shown in Fig, 14. But according to the 

 above principle, there must also be horizontal shearing stresses; 

 that the deformation will be as shown in Fig. 14 is evident from 

 Fig. 13, which presents the condition of any small cube in the 

 beam, and shows that the diagonal EG will be lengthened and 

 the diagonal FH shortened. There are also bendiDg stresses in a 

 beam, as shown in Chapter VI, where it is again proven that 

 there are horizontal shearing stresses in beams. 



18. Shear Modulus. Since a shearing stress moves the 

 molecules in one section parallel to those in an adjacent section, 

 the resulting deformation, between any two parallel sections, will 

 be as shown in Fig. 14. The angle 8 S is the measure of the 

 deformation which is r'r in the distance pr' ; unit deformation is, 

 therefore, 



Fig .14. 



rr 



pr> 



tan 83 = S s (in circular measure), since the deformations, 



within the elastic limit, are very small. 



The relation between unit deformation and unit stress is 

 constant, within the elastic limit, and is determined by experi- 

 ment as in tension. 



