38 G. F. BECKER FINITE STRAIN IN ROOKS. 



to a thickness l/« 2 any of the sides of the mass having a length a. The 

 upper surface has an area « 2 and the side an area, 1/a. 



The tensile stress on sides of the mass is /' in each direction, so that 

 the two tensile forces are each Pja. When only one shear acted on the 

 mass the contractile stress was Q, but the second shear increased each' 

 unit area to «, so that the contractile stress of the first shear was thereby 

 reduced to Q J a. The stress due to the second shear is of precisely the 

 same amount, so that the total contractile stress becomes 2 Q/aonan 

 area « 2 . Thus the total force acting on this surface is 2 Qa } which, as 

 has heen shown, is equal to 2 Pja in absolute value. 



Let the mass thus strained he subjected to an hydrostatic pressure equal 

 to PI a. Then the tensile forces would be balanced and the pressure on 

 the upper, surface would become 3 Q a. 



Thus, two equal shears combined with an hydrostatic pressure equal 

 to either component of either shear, applied to the unit cube, reduce to 

 a simple pressure acting on one surface of the cube. Had the shears 

 been so combined that their tensile axes coincided, a dilational stress 

 equal to either component of either shear would have been needful to 

 reduce the system to a simple traction. 



Conversely, it is evident that a finite traction or pressure is resoluble 

 into a dilational stress (positive or negative) and two shearing stresses, 

 just one-third of the force being employed in each of the three component 

 stresses. It is well known that precisely this resolution takes place for 

 infinitesimal tractions, but the analysis of such tractions is usually stated 

 as if the conclusions were true only for the limiting case of infinitesimal 

 forces. 



These results seem to exhaust what can be known of the relations of 

 finite stress and strain without a further knowledge of the actual value 

 of a in terms of Q. No two different pressures or different shears or dila- 

 tions can be compared without a law relating to stress and strain. 



Meaning of Hooke's Law. — It was to fill this gap that the famous law of 

 Hooke was proposed. This is Ut tensio sic vis, which is now translated, 

 Strain is proportional to stress. The brevity of Hooke's law has often 

 been admired. The fact is that it is too brief folly to express the mean- 

 ing really attached to it. It does not appear in this form of the law 

 whether the stress (or pressure per unit area) is to be reckoned for the 

 solid in an unstrained state or after the mass bus reached a condition of 

 equilibrium under the action of the external forces tending to deform it. 

 But since the purpose of the mathematical theory of elasticity is to find 

 equations expressing equilibrium of elastic masses, it is clear that this 

 equilibrium must be supposed established before one can reason on the 

 system of stresses which will maintain it, As a matter of fact, the lunda- 



