192 MB. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 



Any of these methods is valid for the investigation of elastic stability, and all have 

 in fact been employed, the displacement considered being that of the central-line or 

 mi. Idle-surface of the rod or shell, and the resultant actions over cross-sections being 

 derived in terms of this displacement, by the approximate theory first suggested by 

 KIROHHOFF. The third method is generally found to be preferable, and is the basis 

 of the investigation to be described below, but the actual procedure will be found to 

 possess one or two novel features. 



In the first place, an endeavour will be made to dispense with the assumption that 

 elastic break-down occurs at very small values of the strains ; instead, we shall deal 

 with an ideal material possessing perfect elasticity combined with unlimited strength. 

 Such a material could not fail, unless by instability, and our problems will no longer 

 be confined to thin rods, plates, or shells. It follows that we can only obtain sufficient 

 accuracy in our conditions for neutral stability by deriving them with reference to a 

 volume-element of the material. 



Further, since instability will in some cases not occur until the strains in the 

 material have reached finite values, we shall have to introduce an unusual precision 

 into our ideas of stress and strain. The discussion of finite strain is merely a problem 

 in kinematics, and has been worked out with some completeness* ; but the corre- 

 sponding stress-strain relations in our ideal material are necessarily less certain, since 

 they must be based upon experiments in which only small strains are permissible. 



For example, if we assume that HOOKE'S Law is satisfied at all stresses, we must 

 decide whether our definition of stress is to be 



T . FTotal action over an element of surface"! 



or 



Original area of that surface J 



T . [" Total action over the surface ~l 

 ' LArea of that surface after distortionj ' 



For the ordinary purposes of elastic theory the two definitions may be regarded as 

 equivalent, and the distinction is too fine to be settled experimentally. In the 

 absence of any generally-accepted molecular theory which might indicate the correct 

 result, it seems legitimate to make the simplest possible assumptions which do not 

 involve self-contradictions, and which yield the usual results when the strains are 

 very small. 



It may be shown t that in a distortion of any magnitude three orthogonal linear 

 elements issue from any point after distortion, which were also orthogonal in the 

 unstrained configuration, and that these linear elements undergo stationary (maximum 

 or minimum or minimax) extension. Hence an elementary parallelepiped constructed 

 at the point, with sides parallel to these linear elements, undergoes no change of 

 angle in the distortion. It is clear that only normal stresses will act upon its faces 



For a discussion of the theory, with references, see LOVE, op. at., Appendix to Chapter I. 

 t LOVE, op. cii., $ 26, 27. 



