ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 



487 



constants of the material. Under a single principal stress, X, the strain 

 energy depends only on Young's modulus E. Thus : 



Strain-energy=W,= Px '^=XV2E 



Similarly, under simple shear-stress Q with modulus of rigidity C ; or 

 under bulk-stress P with compressibility modulus K — the values are 



W, = Q2/2 C or Wp = P2/2 K 

 Under two-dimensional stress, represented by two finite principal stresses 

 X and Y, the elongation in the direction of either principal stress is the 

 algebraic sum of two components produced by the two stresses, thus 



e, = (X -crY)/E 

 where <r is Poisson's ratio. Integrating the sum of the products X.fZe^. 

 and Y.dey, to find the mean strain-energy W,^, we have 



2E.W,„ = (X2 4- Y2 - 20-.X.Y) 

 Taking 2E.W,,, as constant, this quadratic equation expresses the relation 

 between two principal stresses varied in such a manner that the strain- 

 energy W,^ is constant. This equation is plotted in fig. 16 for difierent 



FiQ. 16. 



N N 2 



