ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 489 



Instead of expressing the strain-energy in terms of the three mutually 

 perpendicular principal stresses, we may adopt the method introduced by 

 Kelvin, and express the state of stress in terms of a single bulk stress P 

 and two mutually perpendicular shear stresses, Q^ and Q.^. The bulk 

 stress is the algebraic mean of the three principal stresses, 



P = 3 (X + Y + Z) 



while the two shear-stresses are equal to the two smaller of the three 

 residuals obtained by subtracting the bulk stress from the principals. For 

 example, if the residual (Y — P) is numerically greater than (X — P) 

 and (Z — P) — being equal and opposite to their sum (X + Z — 2P) — the 

 two shear-stresses are 



Q., = (X - P) = (2X/3 - Y/3 - Z/3) 

 and Q,, = (Z — P) = (2Z/3 — X/3 — Y/3). 



In terms of these fundamental stresses, the mean strain-energy is 

 simply the sum of the quantities due to the three independent stresses, 

 viz. : — 



W..,. = (PV2K + Q\,I2C + Q%/2C) 



On substituting K = E/3 (1 — 2cr) and C = E/2 (1 -|- o-), and the values 

 of the principal stresses, this expression naturally reduces to the form 

 already quoted, giving the strain-energy in terms of X, Y, Z, E, and a. 



Figure 17 shows a model constructed to visualise the quadratic equation 

 in the three stresses X, Y, and Z, every poiiit on the surface having the same 

 strain-energy. The symmetrical ellipsoid passes through the points 

 X = ±l, Y = Z = 0, etc., representing the elastic-limits under simple 

 pull and push in different directions and is oriented so that its major 

 axis coincides with the line X = Y = Z. The form of the model varies 

 for different values of Poisson's ratio, the ratios between the semi-major 

 axis and the elastic-limit for simple pull or push being as given in the 

 following table : — 



Value of ratio semi- 2-0 2-5 30 3^ 3-5 40 



major axis to 

 elastic-limit in 

 tension Infinity 2-235 1-732 1-582 1-528 1-414 



The equal minor axes are identical with those of the ellipses shown in 

 fig. 16; these being the traces of the model with the plane of projection, 

 Z = 0. In the case of an incompressible substance (K = infinity, m = 2"0) 

 the model becomes a cylinder. 



The lines drawn on the (glass) planes of projection represent the co- 

 ordinates of the elastic-limits under simple pidl and push. The markings 

 on the surface of the ellipsoid represent particular combinations of stress, 

 e.g., the three-dimensional stresses in thick and thin tubes subjected to 

 internal fluid -pressure. The circle drawn round the minor axis is the 

 locus of points representing combinations of shear-stresses without bulk 

 stress. Bulk stress alone is represented by the major axis, X = Y = Z. 



