STRAINED ELASTIC SOLIDS 429 



clearly the three principal planes ol the quadric surface whose 

 equation has been written down above. Were we to choose as 

 axes of reference the three principal axes of this quadric, we 

 know that the equation would only involve terms in ^^, 7?^, f ^, 

 but not in Tjf , f^, ^r). In short, with such a choice of axes of 

 reference only three of the stress-components would have a 

 finite value, viz., those corresponding to Xx, Yy, Zz. The 

 remaining six (actually only three) would be zero, and as Gibbs 

 states in equation [392] the stress action across any plane 

 (a, /?, 7) would have as its components aXx, ^Yy, yZz. These 

 three special axes are called the principal axes of stress, and their 

 existence is a point of considerable importance in the discussion 

 in Gibbs, I, 195 et seq. 



Special cases arise if the quadric surface at a point referred to 

 above is one of revolution, i.e., if the section by one of the 

 principal planes is a circle. In this event, assuming that it is 

 the plane perpendicular to that one of the principal axes of 

 stress designated as OX, it is clear that Yy = Zz, and the stress 

 action across any plane containing the local axis of x at P is 

 normal to this plane. Or it may happen that the "stress- 

 quadric" is actually a sphere, so that Xx = Yy = Zz. Any 

 triad of perpendicular lines will serve as principal axes of 

 stress if this be so, and the stress-components which do not 

 vanish have one numerical value, the stress across any plane 

 being normal to it and having a value independent of direction. 

 This is in fact the general state of affairs for a fluid at rest and 

 Xx = Yy = Zy = —p where p is the fluid pressure. It is clear 

 that the equations of equilibrium (29) then degenerate to those 

 for a fluid quoted on page 422. 



5. Stress-Strain Relations and Strain-Energy. We have now 

 considered at some length the mathematical methods by which 

 the strains and stresses in a body are analyzed into their most 

 convenient constituents, and it is clear that the differences of 

 behavior observed in various elastic media when subject to 

 given external forces arise from the different "constitutive" re- 

 lations which exist between the constituents of stress and the co- 

 efficients of strain in these different media. We know for instance 

 that the same pull will elongate a wire of brass of given section 



