86 



BELL SYSTEM TECIINICA L JOI 'UNA L 



which represents the most general type of disj^lacement that the Hne P Q 

 can undergo. 



As discussed in section 4 the definition of the shearing strains given by 

 equation (16) does not allow them to be represented as part of a tensor. 

 If however we defined the shearing strains as 



25,3 = S, = 



\dy dzj 



25|3 — Si, 



= i^ + ^i • 



dz dx ' 



25. = S. = p + 'J 

 dx ay 



(19) 



they can be expressed in the form of a symmetrical tensor 



S(, 65 



^11 



S\2 



Si 



(20) 



For an element suffering a shearing strain S^ — 2Si2 only, the displace- 

 ment along X is proportional to y, while the displacement along y is propor- 

 tional to the X dimiension. A cubic element of volum.e will be strained into 

 a rhombic form, as shown by Fig. 4, and the cosine of the resulting angle 6 



Fig. 4. — Distortion due to ;i shear! iig strain. 



measures the shearing deformation. For an element suffering a rotation 

 ccz only, the dis])lacement along x is proj;ortional to y and in the negative 

 y direction, while the dis])laccmcnt along y is in the ]>ositive .v direction. 

 Hence a rectangle has the displacement shown by lig. 5, which is a pure 

 rotation of the body without change of form, about the z axis. For any 



