36 



G. F. BECKER FINITE STRAIN IN ROCKS. 



The relations between finite stress and displacement lack satisfactory 

 experimental basis and cannot therefore be fully developed, but it is 

 desirable to show just where knowledge cuds and ignorance begins. 



Stresses in n Shear. — From the discussion of the properties of shear, it 

 follows that the imdistorted planes are necessarily subjected to purely 

 tangential stresses ; for they are neither elongated nor drawn apart during 

 strain, while normal forces acting upon them would produce such effects. 



The stress phenomena in a shear can be examined as a case of equi- 

 librium, and such an examination reveals the somewhat important fact 

 that the planes of maximum tangential stress do not coincide with the 

 planes of maximum tangential strain.* It also teaches how the two 

 component forces involved in a finite shear are related, and thus, in spite 

 of ignorance of the direct relations between stress and strain, the inquiry 

 is by no means fruitless. 



Figure 3.— Stresses infinite Shear. 



Let the rectangle o b represent one-quarter of a strained cube and let 

 — ■ Q and Pbe the stresses (or forces per unit area) holding it in this state 

 of strain. Then it is easy to find the stress on any plane cutting the x y 

 plane at right angles along the line a e. Let the normal to the plane 

 make an angle # with o x. Then — 



ab = ac sin >'J ; be = ac cos >>. 



If i^and G are the component stresses on ac parallel to ox and o y, 

 these components must hold the stresses on a h and a c in equilibrium. 

 Now, the total force on a c in the direction of o x is — Fa c and the whole 

 force on b c is Pb c. Q and G are similarly related, so that — 



— Fac = Pbc = Pac cos <v. 



Gac = — Qab = — Qac sin ! >, ■ 



or — 



— F=Pcos <'/; G = — Qsin &. 



*In at least some treatises on elasticity and geological mechanics it seems to have been assumed 

 that these planes do coincide. 



