488 KEPORTS ON THE STATE OP SCIENCE. — 1919. 



values of Poisson's ratio from 0.50" to 0.25 (m ^ l/o- = 2, 2A, 3^, and 4). 

 The ellipses have been plotted to pass through the points (X = +1, 

 Y = and X = 0, Y = ± 1) representing the elastic-limits under sirople 

 longitudinal pull and pushes in different directions. Ihe major and 

 minor axes of the ellipses, at 45° to the axes of X and Y, are as given 

 below : — 



Value of m 2-0 25 3-0 3^ 3-5 4-0 



Semi-major axis 1-414 1-291 1-225 1195 1-183 1-153 

 Semi-minor axis 0-817 0-845 0-867 0-879 0-882 0-895 



On the hypothesis that the limiting strain-energy is constant and inde- 

 pendent of the nature of the applied stress, these ellipses indicate the varia- 

 tions of the limiting principal stresses for the elastic-limit imder two-dimen- 

 sional stress. For example, the elastic-limit under simple shear-stress 

 (X = — Y) should be somewhat more than half the tensile elastic-limit. 

 Also, imder combiaations of similar principal stresses, e.g., two pulls in 

 ]jerpendicular directions, the elastic-limit should be increased by the in- 

 fluence of the second stress when the latter is of moderate intensity ; and 

 reduced when the two stresses approach equality. The square A, B, C, D, 

 in the same diagram, represents Eankine's hypothesis that the elastic- 

 limit depends solely on the principal stress — the limits under pull and 

 push being here assumed to be equal. The six-sided figure A, H, G, C, 

 F, E, A likewise represents Guest's law, that the elastic-limit is determined 

 by the maximum tangential stress. Parallelograms, e.g., T, W, V, U, 

 represent de Saint Venant's hypothesis that elastic failure depends on 

 the maximum principal strain. These latter parallelograms circumscribe 

 the ellipses representing constant strain-energy, being tangent to the ellipses 

 at the points E, F, G, H. It will be observed that this diagram affords a 

 convenient means of comparing the results of experiment with the indica- 

 tions of the several hypotheses : sections of the diagram will be used for 

 this purpose in subsequent figures. 



For combinations of three finite principal stresses, the mean strain- 

 energy may likewise be expressed, 



2E.W^, = (X2 M- Y2 + Z2) — 2cr.(Y.Z + Z.X + X.Y) 



Other forms of the energy function, based on the three jJrincipal stresses 

 and two independent variables expressing the elasticity of isotropic mate- 

 rials, have been given by Lame (1852) ; but the above is well known and, 

 probably, the most convenient and fundamental. 



For the more general case of non-isotropic crystals, Green (1837) estab- 

 lished an expression in which 21 independent variables appear. It is 

 important to observe that the simpler expressions are rigorously applicable 

 only for isotropic materials ; and that, being applied to ordinary crystalline 

 metal composed of large numbers of non-isotropic grains with axes oriented 

 in different directions, they give only the mean strain-energy for the gross 

 mass under uniformly applied stress. The quantities of energy absorbed 

 by individual crystalline grains may vary above or below this mean value ; 

 and may vary for different parts of indi-vidual grains according to the con- 

 figuration of the boundaries. Evidence of this non-uniformity may be 

 observed by studying the disposition of slip-bands within particular grains. 



