.^•v^W^ %n«sV(l m »^ [ 539 ] iM 



^ife b' . LXX. Proceedings of Learned Societies, ^q fan md 



.jjbinn ^ ROYAL SOCIETY. 



rcfOTi amu to » 



J«ii [Continued from p. 481.] 



'■•-' April 24, 1856.— The Lord Wrottesley, President, in the Chair. 

 nPHE following communications were read : — 

 -■- " Elements of a Mathematical Theory of Elasticity." By 

 Professor William Thomson, F.R.S. 



This paper consists of two parts : Part I. on Stresses and Strains ; 

 P^rt II. on the Mechanical Conditions of Relation between Stresses 

 and Strains experienced by an Elastic Solid. 



Part I. — The terms Stress and Strain are used in accordance with 

 the valuable definitions by which they were first distinctively intro- 

 duced into the Theory of Elasticity by Mr. Rankine*; with only 

 this deviation ; that instead of defining a stress as the reactive force 

 exerted by an elastic body when in a condition of strain, the author 

 of the present paper defines stress as " a definite external applicatioii 

 of force to a body." 



Various well-known theorems regarding the geometrical relations of 

 the displacements among the parts of a body in a state of strain, and 

 the geometrical representation of stresses and strains are enunciated, 

 and briefly demonstrated, for the sake of convenience. A mode of 

 expressing in absolute measure the magnitude of a stress or a strain, 

 which the author believes to be new, is laid down nearly in the follow- 

 ing terms. The amount of work done by a stress applied to a body 

 of unit volume^ while acquiring a strain of the same type as the stress, 

 is measured by the product of the magnitude of the stress into the 

 piiagnitude of the strain. 



J'' When a stress and a strain are of the same type, they are said to 

 lie concurrent ; or, if directly opposed, they are said to be negatively 

 concurrent. When a stress and a strain are of any different types, 

 the degree of their concurrence, or simply " their concurrence,*' is 

 measured by the work done by the stress applied to a body of unit 

 volume acquiring the strain, divided by the product of the magnitude 

 of the stress into the magnitude of the strain. The measure of per- 

 fect concurrence is therefore -J- 1, and that of perfect opposition —1. 

 When work is neither spent nor gained in the application of a cer- 

 tain stress to a body while acquiring a certain strain, that stress and 

 that strain, or any stresses or strains of the same types respectively, 

 are said to be orthogonal to one another. The measure of their con- 

 currence is zero. 



A system of stress or strain coordinates involving symmetrically 

 six independent variables, perfectly analogous to the system of triple 

 coordinates for specifying the position of a point in space, is laid 

 down. The concurrence of a stress or strain with six orthogonal 

 types of reference being denoted by /, m^n, A, fx, v, it is demonstrated 



*^»^ Z2-f-m2 + w2 + \2 ^2 + ^2^:1^ 



and it is proved that if cos d denote the mutual concurrence between 



two stress or strain types, whose concurrences with six orthogonal 



* ** On iUes Qf {llasticity and Crystalline Forms," Phil. Mag. vol. xi. p. 301, 



