PIEZOELECTRIC CR VST A LS IN" TENSOR FORM 93 



to piezoelectric cn^stals, the energy stored in any unit volume of the crystal is 



dU = Ti dSi + T2 dS2 + T3 dS^ + Ti dS, + Ts dS, + Te dSe 



, J, dD, , ^ dD, , _, dDi ,^, (42) 

 ■iir 47r ■iir 



where Ei , E2 and £3 are the components of the field existing in the crystal 

 and Di , A and D3 the components of the electric displacement. In order 

 to avoid using the factor l/4ir we make the substitution 



The normal component of 5 at any bounding surface is fo the surface charge. 

 On the other hand if we employ the MKS systems of units the energy of 

 any component is given by Zn^/^^n directly and in the following formulation 5 

 can be replaced by D. 



There are two logical methods of writing the elastic, piezoelectric, pyro- 

 electric and dielectric relations. One considers the independent variables 

 as the stresses, fields, and temperature, and the dependent variables as the 

 strains, displacements and entropy. The other system considers the strains, 

 displacements and entropy as the fundamental independent variables and 

 the stresses, fields, and temperature as the independent variables. The 

 first system appears to be more fundamental for ferroelectric types of 

 crystals. 



If we develop the stresses, fields, and temperature in terms of their partial 

 derivatives, we can write 





i/d.<t 0^2/ D.a OCis/D.tr OOi/ D,a 



\/ s,a da /a 



Obz/S.a Off Js.D 



T, = ^-^\ dS,^^-^") dS2-\-^-^^ ^-^3 + ^^^^ dS, 



(44 A) 



O'Jl/D.a 002/ D.a OOs/D.a 004/0,0 



a^)5/D,o O0(,/D,a O0\ / s.a 002/ S,a 



003 /s.a dcr /S,D 



