324 REPORTS ON THE STATE OF SCIENCE, ETC. 
(4) ‘The Phenomena of Rupture and Flow in Solids.’ A. A. Griffith, Phil. 
Trans. A, vol. 221, pp. 163-198, Oct. 21, 1920. 
(5) ‘ The Hard and Soft States in Metals.’ Sir G. Beilby, May Lecture, Inst. of 
Metals, 1911. 
(6) ‘ Fracture of Metals under Alternating Stress.’ Sir J. A. Ewing and 
J. C. W. Humftrey, PAil. Trans. A., 200, 1902. 
We 
The Strain Energy-Function and the Elastic Limit. 
By Professor B. P. Hatcu, D.Sc., M.B.L., Royal Naval College, Greenwich. 
In a contribution to the report of this Committee, for 1919, the author 
analysed published data, with the object of comparing, for ductile metals, 
the different elastic limits under simple and complex stresses. Experimental 
values of the ratios between the elastic limits for different kinds of stress 
were compared with the corresponding values predicted by applying three 
alternative hypotheses, due to Lamé and Rankine, to Darwin, Tresca and 
Guest, and to Saint Venant; and with values deduced from a noval hypothesis, 
viz. that the relation between the elastic limits is governed by the strain energy 
per unit volume which, at the elastic limit, reaches a definite limiting value 
independent of the simple or complex nature of the applied stress. 
Diagrams were plotted in which the marked points represented experimental 
determinations of the ratios between the elastic limits. On comparing the loci 
of these experimental points with the graphs representing the alternative 
hypotheses, it was observed that the strain-energy graph was in fair agreement 
with experiment throughout the field of investigation, and that the graphs ~ 
representing other hypotheses were in agreement in narrower fields, e.g. in the 
case of Saint Venant’s hypothesis, when the ratio between the principal stresses 
was less than + 0.30. 
The diagram shown in fig. 13 illustrates the method of comparison adopted — 
for two-dimensional stresses; the axes OX and OY representing the principal — 
stresses on the material and the lengths OA and OB the elastic limits in simple 
tension. The co-ordinates of points in the quadrants represent the principal 
stresses for complex combinations. The strain-energy hypothesis is represented 
by an ellipse whose eccentricity varies slightly for different values of Poisson’s 
Ratio o. Other hypotheses are represented by figures composed of intersecting — 
straight lines. The diagram includes also a fifth hypothesis, published in 1916 — 
by Dr. Albert Becker,! viz. that the elastic limit is determined by a dual — 
condition—limiting maximum shear stress and limiting maximum strain. This 
is represented by a ten-sided figure which, for combinations approximating to 
shear stress, nearly coincides with the ellipse for the strain-energy hypothesis. — 
Where two nearly equal like stresses are combined, as in turbine discs, Dr. 
Becker’s hypothesis appears to overestimate the elastic limit, although not so — 
greatly as does Saint Venant’s. 
In the earlier report it was explained that the strain-energy hypothesis 
represents not merely an arbitrary assumption, roughly endorsed by published — 
data, but an attempt to apply established thermodynamic principles to the 
now generally accepted theory that the production of non-elastic strain is 
associated with a change in the state of the metal, from the crystalline to the 
‘vitreous’ or ‘amorphous’ phase. The earlier report, however, was chiefly 
an analysis of published data, without any attempt at explanation. In what 
follows, the object is to trace the theoretical bearing of thermodynamic prin-— 
ciples on the relation between the elastic limits under different kinds of stress. 
Physical Theory of Non-elastic Strain. 
The current theory for permanent strain rests on two main experimental — 
observations : (1) that permanent strain is the cumulative result of numerous 
gliding movements in individual grains of metal, the reality of these movements 
being demonstrated by the slip-bands observed when polished faces of a block 
strained beyond the elastic limit are examined under the microscope; and 
1A. J. Becker, Bulletin 85, University of Illinois. 
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