MR. i;. V. sorTinvKU, ON THK (JKNERAL THKORY OF ELASTIC STABILITY. 193 



after distortion, and that if tli.-se stresses be expressed in terms of the extensions of 

 the sides we have complete relations between stress and strain. 



We shall therefore assume that these principal stresses and principal strains, 

 irtottever their magnitude, arc <-n,,n. </,-,/ I,,/ //,, ,,//,,, i-rfimf,',,,,.^ ../' I[O<>KK'S L\i-; 

 that is to say, if the extensions in the principal directions are e lt e e st and the 

 corresponding stresses are R,, H 3 . Tt :i , then 



where E is YOUNG'S Modulus, and is POISSON'S ratio for the material under 



m 



consideration. 



These relations may be written in the form 



20 



....... (2) 



m-2 L 

 where C is the Modulus of Rigidity. 



In these relations the measure of extension is assumed to be 



Increase in length of linear element 

 Length of the element tafore strain ' 

 and of stress* 



Total action over an element, of surface 

 Area of the element before strain 



We have then the usual expression! for the energy of strain, per unit volume of the 

 unstrained material, in terms of the principal extensions, viz.: 



i 



(3) 



The above assumptions yield sufficient data for the calculation of the stress system 

 in any configuration of equilibrium, even when the strains are not small. Assuming 

 that the calculation has been effected, we have to show how conditions for the 

 stability of the system may be obtained. 



We must distinguish three configurations : the unstrained configuration, in which 

 the co-ordinates of any point are given by x, y, z ; the configuration of equilibrium 

 under the stress-system, the stability of which we are investigating ; and a configu- 



' This assumption is open to the objection that it would render possible the compression of a material 

 to zero volume by means of a finite stress. It will not, however, introduce any serious error, and has the 

 advantage, which more probable assumptions do not possess, of leading to a definite energy -function. 

 Tin- definitions of stress and strain given alwve are generally employed in the construction of " straw- 

 ~tr:iin diagrams" from a tension test, the extensions of the specimen being taken as abscissa;, and the 

 total loads as ordinates of the plotted curve, 

 t LOVE, np. ci/., 68. 



VOL. CCXIII. A. 2 C 



