lO BROOMALL : 



For steel its value is about 1/3, so that a bar of steel 12" long 

 and 2" square in section, if stretched i/ioo", producing a 

 unit elongation of 1/12 . i/ioo" ^ 1/1200", would suffer a unit 

 lateral contraction in all directions of 1/3 . 1/1200 = 1/3600", 

 making a total lateral contraction of 2 X 1/3600 = 1/1800". 

 The longitudinal stress brought into play by a given length- 

 ening or shortening of the bar is known from experimental 

 values of Young's Modulus. But it is evident that this mod- 

 ulus is not a modulus of pure stress and strain, but must be 

 a function of the cubical elasticity of the material and its 

 rigidity. In other words, the actual state of deformation 

 which results may be produced by the combination of cubical 

 expansion and shears. By a simple geometrical demonstra- 

 tion it can be shown that the value of Young's Modulus in 

 terms of the coefficient of bulk elasticity, k, and the coeffici- 

 ent of rigidity, n, is given by the expression 



gnk 



3k + n 

 It may also be shown the Poisson's Ratio has the value 



3k — 2n 



R 



2 (3k + n) 



Upon substitution of the coefficients of pure strain in these 

 expressions, results surprisingly close to the directly deter- 

 mined experimental value of Young's Modulus and Poisson's 

 Ratio are obtained. 



The existence of lateral contraction shows that the mate- 

 rial is subjected to an internal lateral unit stress in all direc- 

 tions at right angles to the axis, the value of which stress 

 equals the coefficient of elasticity multiplied by the unit late- 

 ral strain. That this lateral strain brings into existence a 

 very real compressive stress is evident from the fact that a 



