PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



83 



We call the components 



r 



T. 



T 



T21, 7^22, T,, 

 T31 , Tz2 , T33 



(7) 



the stress components exerted on the elementary cube which tend to deform 

 it. The rate of change of these stresses determines the resultant force on 

 the cube. The second form of (7) is commonly used when the stresses are 

 considered as a second rank tensor. 



Fiff. 2. — Shearing stresses exerted on a cube. 



It can be shown that there is a relation between 3 pairs of these compo- 

 nents, namely 



T = T 



1 TV 1 ■ 



T = T 



T = T 



(8) 



To show this consider P'ig. 2 which shows the stresses tending to rotate the 



elementary cube about the Zaxis 



the cube about the Z axis by producing the couple 



The stresses Ty^^'dnd Ty^^ tend to rotate 



Tyx dx dy dz 



(9) 



The stresses Tjy^ and T^y.^ produce a couple tending to cause a rotation in 

 the opposite direction so that 



^ {Tyj, — T:ry) dx dy dz = couple 



I (hi 



(10) 



is the total couj^ie ter.ding to produce a rotation around the Z axis. 

 But from dynamics, it is known that tliis cou])le is equal to the product of 

 the moment of inertia of the section times the angular acceleration. This 

 moment of inertia of the section is proportional to the fourth power of the 

 cube edge and the angular acceleration is fmite. Hence as the cube edge 



M 



