PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



87 



body in equilibrium or in nonrotational vibration, the co's can be set equal 



to zero. 



The total potential energy stored in a general distortion can be calculated 



as the sum of the energies due to the distortion of the various modes. For 



fih 

 example in expanding the cube in the x direction by an amount — dx = 



ox 



Si dx, the work done is the force times the displacement. The force wil 



Fig. 5. — A rotation of a solid body. 



be the force Ti and will be Ti dy dz. Hence the potential energy stored in 

 this distortion is 



T\ dSi dx dy dz 

 For a shearing stress T^ of the type shown by Fig. 4 the displacement 



dS(,dx 



7r» T 



times the force T^ dy dz and the displacement — ^-^ times the force T(, dx dz 



equals the stored energy or 



AP^e = \ (dS^Te + dSeT^) dx dy dz = dS^T^ dx dy dz. 



Hence for all modes of motion the stored potential ener gy is equal to 



APE = [Ti dSi +■ Ti dS2 + Ti dSi + Ti dSi + T^, dSs 



(21) 

 + Tt dSe] dx dy dz. 



1 .3 Generalized Hookers Law 



Having specified stresses and strains, we next consider the relationship 

 ; between them. For small displacements, it is a consequence of Hooke's 

 I Law that the stresses are proportional to the strains. For the most un- 

 I symmetrical medium, this proportionality can be written in the form 



