4.86 REPORTS ON THE STATE OF SCIENCE. — 1919. 



The Strain-energy Function and the Elastic-limit. 

 By B. P. Haigh, M.B.E., D.Sc. 



Introduction. 



The elastic-limit of a ductile metal is usually defined as that limiting 

 intensity of stress beyond which permanent strain is first produced ; or 

 alternatively, beyond which stress and strain are no longer proportional. 

 The same definition may be adopted for complex stresses, the intensities 

 of all three principal stresses being then required to specify the elastic- 

 limit. 



Several hypotheses, based on direct experiment and empirical in nature, 

 have been advanced with the object of establishing a direct relation 

 between the elastic-limits for simple and complex stresses ; and certain of 

 these are now in general use, yielding reliable results in particular circum- 

 stances. In what follows, reference will be made to three hypotheses — 

 due to Lame and Rankine, to de Saint Venant and to Tresca, Darwin, 

 and Guest. According to different hypotheses, the maximum principal 

 stress or strain, or the maximum tangential stress, or the latter together 

 with internal friction, may be regarded as the criterion of elastic failure. 



In studying these alternative hypotheses, it occurred to the author 

 that a more general relation between the elastic-limits imder simple and 

 complex stresses might be found by directing attention to the energy 

 absorbed by the material in virtue of its elastic strain. This view was 

 based upon thermo-dynamic considerations ; but, in this paper, only the 

 experimental aspect of the question will be discussed. On another occasion 

 the author hopes to deal with the theoretical results obtained by applying 

 the principles of thermodynamics to the current theory of ductile strain ; 

 the process of strain being regarded as one in which crystalline metal is 

 converted to the harder vitreous state in a manner that is thermodyna- 

 mically reversible although associated (like every thermal process) with 

 actions that are irreversible. 



The mechanical energy absorbed by a body stressed within the elastic- 

 range is variously termed the 'strain-energy' or 'resilience.' The term 

 ' limiting strain-energy ' will now be used to signify the quantities of energy 

 that can be absorbed per unit volume of material uniformly strained to 

 its elastic limit by the application of specified combinations of principal 

 stresses. The immediate object of the investigation is to find how nearly 

 constant is this quantity, for any one material, independently of the 

 nature of the simple or complex stress applied. No new experiments will 

 be quoted because the data available in published records afEord ample 

 scope for a preliminary investigation ; but it is hoped that the results 

 may encourage other investigators to study the strain-energy function 

 when analysing further experiments. 



Strain-energy Functions. 



The mean strain-energy of a test-piece, initially free from internal 

 stress and then uniformly strained under complex stress, is readily ex- 

 pressed in terms of the three principal stresses, X, Y, and Z, and the elastic 



