712 WALTER H. BUCHER 



The most irregular pattern of Liiders' lines results when the 

 lines of stress are not parallel to the axis of the test piece, but inter- 

 sect with it at varying angles. In that case, the angle formed 

 by planes of yielding may be cut by the surface in all possible 

 directions and the apparent angle of intersection of Liiders' lines 

 as seen on the surface varies from point to point and must not 

 be mistaken for the true constant angle bisected by the line of 

 maximum stress of which it is only the oblique outcrop. 



In 1900, O. Mohr published a mathematical study which led 

 him to views practically identical with those of Hartmann. They 

 may be summarized as follows:^ 



a) In all hard materials (except the most brittle ones), under 

 tensional as well as compressional stresses, deformation by shearing 

 takes place in two systems of intersecting planes of shearing. 



h) Adjoining planes of one system are parallel. 



c) The angle at which the two systems intersect is constant 

 for any given material, that is, it is independent of the nature or 

 intensity of the stresses involved. 



d) For the same kind of material, this angle differs the more 

 from 90° the harder and the more brittle the material is (e.g., 

 hard or soft steel) . 



e) If we consider tension as negative compression, the law 

 governing the arrangement of the yield planes with reference to 

 the principal axes of stress which will be referred to as Hartmann' s 

 Law, can be expressed as follows: In brittle materials, the acute 

 angle formed by the shearing planes is bisected by the axis of maximum 

 compression, and the obtuse angle by the axis of minimum compression 

 which is generally negative, representing tension. 



f) If the position of the principal axes changes from point to 

 point, the shearing surfaces are warped. The less this is the case, 

 i.e., the more nearly homogeneous a material is, the more regular 

 are the shearing planes. 



g) The shearing planes do not originate simultaneously, and are 

 not uniformly distributed. 



■ F. Rinne, "Vergleichende Untersuchungen uber die Methoden zur Bestimmimg 

 der Druckfestigkeit von Gesteinen," N. Jahrb.f. Min., etc., Vol. I (1907), p. 45. 



