(22) 



88 BELL S YSTEAf TECH NIC A L JOURNA L 



T\ = CnSi + C12S2 -f- C13S3 -\- CuSi -\- Ci^Si -\- CioSe 



T2 = C21S1 + C22S2 + C23S3 + C24S4 -\- C2bSs + ^26^ 6 



7^3 = ^31'5*1 + CS2S2 + ^33^3 + €3484 + ^35^6 + ^36-^6 



Ti = C41S1 + €4282 + r43'5'3 + CiiSi -\- €4^3 f, -\- ^46^6 



Tt = Cr,iSi + f52^2 + ^53^3 + C^Si + Ci^S;, + ^56.5 6 



7^6 = CeiSl -\- f 62'?2 + f e3'S'3 + C64Si + f 65^5 + ^66^6 



where Cn for example is an elastic constant expressing the proportionality 

 between the Si strain and the Ti stress in the absence of any other strains. 

 It can be shown that the law of conservation of energy, it is a necessary 

 consequence that 



C12 = C21 and in general c,,- = Cji. (23) 



This reduces the number of independent elastic constants for the most 

 unsymmetrical medium to 21. As shown in a later section, any symmetry 

 existing in the crystal will reduce the possible number of elastic constants 

 and simplify the stress strain relationship of equation (22). 



Introducing the values of the stresses from (22) in the expression for the 

 potential energy (21), this can be written in the form 



2PE = cnSl + 2C12S1S2 + IcnSiSs + 2fi4^i54 + 2cuSiS\ + Ici&SiS^ 



+ ^22^2 + 2r23^2'S'3 + 2C24'S'25'4 + 2f25'S'2^5 + 2C2oS'26'6 



+ C33S3 -{- IcsiSsSi -\- IczffSzSi, -f- IcsgS^S^ 



+ f44'^4 + 2r45^4^'5 + ICi^'iSfi (24) 



The relations (22) thus can be obtained by differentiating the potential 

 energy according to the relation 



c)PF c)PF 



It is sometimes ad\antageous to exi)ress the strains in terms of the stresses. 

 This can be done by solving the equations (22) simultaneously for the 

 strains resulting in the equations 



