272 Cilley — Fundamental Propositions in the 



These, the actual or total stresses in all bodies at rest, whether 

 solid or liquid, whether elastic or inelastic, whether homo- 

 geneous or heterogeneous, whether seolotropic or isotropic, 

 must satisfy. But these equations have precisely the same 

 form as those for the changes in stress due to given applied 

 forces.* This has erroneously led to the conclusion that the 

 changes in stress are the total stresses, whereas actually the 

 total stresses consist of two parts, one the changes in stress due 

 to applied forces, the other the (often preexisting) self-balanc- 

 ing or primary stresses. The latter are ordinarily and incor- 

 rectly neglected. 



The primary stresses evidently must satisfy the condition 

 that the resultant stresses on any complete section shall be zero. 

 And in particular they must make the resultant stresses on any 

 elementary portion of the body zero. This leads to three dif- 

 ferential (interior) and three ordinary (surface) equations of 

 equilibrium, which primary stresses must satisfy.f These are 

 nothing (in form) but the ordinary general interior and surface 

 equations of equilibrium with all bodily forces and surface 

 tractions zero. It has incorrectly been assumed that these 

 equations had no solutions other than zero, but since primary 

 stresses are real quantities which must satisfy these equations 

 it follows that other solutions than zero for these equations 

 exist and have a real significance. Actually, as will later be 

 seen, the solutions are innumerable. 



Changes in stress due to applied forces — u the " stresses of 

 the ordinary theory of elasticity, — we know, theoretically, are 

 definitely determinable. This is rendered possible by these 

 changes in stress being definitely related to changes in strain 

 through stress-strain relations holding for the given body and 

 material, and by the changes in strain, if elastic, being related to 

 the displacements by certain geometrically necessary relations. 

 Wherever the stress-strain relations are definitely known, this 



* Equations limiting changes in stress. 

 For interior points. For surface points. 



6P dU dT 

 te~ + ~d^ + ~te = ~ pX > eta (1) lP+mU+nT= I\ etc. (2') 



where PQR are the intensities of the normal components of the changes in 

 stress or stresses due to the bodily forces XYZ and surface tractions FGH while 

 STUave the intensities of the tangential components of these changes. 



f Equations limiting primary stresses. 

 For interior points. For surface points. 



d(P) d(U) d(T) 



"^ + V + ~^~ °' etC ' ( } m + m{U) + n(T)=0, etc. (2") 



where (P)(Q)(R) are the intensities of the normal components of the primary 

 stresses aud (S)(T)(U) are the intensities of the tangential components of the 

 primary stresses. 



