STRAINED ELASTIC SOLIDS 431 



Zi for Xx, /i for en - 1, 



X2 for Yy, J2 for 622 — 1, 



X3 for Zz, fz for 633 - 1, 



Xi for Fz or Zy, /4 for 633 + 632, 



Xi for Zx or Xz, ft, for 631 + e^, 



Xe for Xy or Yx, /e for 612 + 621- 



(fh h} h are the fractions of elongation along the axes and 

 fi, fh, /e are the shears or changes in the angles between the 

 axes.) A complete experimental knowledge of the elastic 

 properties of any material would therefore be embodied in the 

 ascertained values of the 36 elastic constants Crs in six consti- 

 tutive ''stress-strain" equations such as 



Xi = Cu/i + C12/2 + C13/3 + C14/4 + C15/5 + Cifi/e,! 



j (31) 



X2 = C21/1 + C22/2 + C23/3 + C24/4 + C25/5 + C26/6,J 



and four similar equations. These equations are the expression 

 of a general Hooke's law, a natural extension of the famous 

 law concerning extension of strings and wires due to that 

 English natural philosopher. 



This apparently presents an appallingly complex problem for 

 the experimental physicist; however, there are important 

 simplifications in practice. To begin with, it will appear from 

 energy considerations to be discussed presently, that even in 

 the most general case the 36 constants must only involve 21 

 different numerical values at most, and actually for a great 

 variety of materials still further reductions are involved. 

 Indeed, for isotropic bodies all the elastic constants of such a 

 material are calculable from the numerical values of two 

 "elastic moduh," the well-known "bulk modulus" (or "elasticity 

 of volume") and the "modulus of rigidity." For various crys- 

 talline bodies conditions of symmetry also involve a material 

 reduction of the number of independent constants below the 

 number 21. 



The two moduli for isotropic bodies are referred to by 



