7IO WALTER H. BUCHER 



The lines in this case are depressions only on one side, with the 

 corresponding lines on the reverse side forming delicate ridges. 

 Final rupturing, under compression, always follows these lines. 



The great importance of these lines of Liiders for our purposes 

 lies in the fact that they represent the outcrops of internal planes 

 of yielding, differing largely in scale and degree of deformation, 

 not in origin, from the planes of shearing observed on a large scale 

 in nature. 



In fact, in the small test piece as on a gigantic scale in nature, 

 we see that the stress acts not uniformly on every unit of the 

 mass undergoing deformation, but that it reaches a maximum 

 along these geometrically distributed surfaces, while maintaining 

 lower values in the volume between. We seem to be dealing here 

 with a sharply defined application of the principle of least work.^ 

 At every point along every imaginary line of stress within a body 

 undergoing elastic deformation there exists the tendency to shear 

 in any one of an infinite number of directions all inclined to the 

 direction of stress at the same angle, the sum of which forms 

 two infinitely small cones joined by their apices.^ Out of the 

 infinite number of surfaces which may be obtained by connect- 

 ing any two of such adjoining possible directions of shearing, those 

 only will form which involve the expenditure of a minimum of 

 energy. 



When the lines of stress are not parallel, owing to the unequal 

 distribution of stresses, the resulting surfaces of yielding may be 

 very complex and the pattern of lines formed by their traces on 

 the surface may be far from regular (Fig. 2A-O). 



In such cases Liiders' lines can be used to reconstruct the lines 

 of maximum stress on any given test piece, by drawing the lines 

 bisecting the angle of shear at every point of intersection of the 

 shearing planes. Figure 2C represents the lines of stress derived 

 in that way from Luders' lines as obtained in the experiment 

 illustrated in Figure 2A and B. 



1 H. von Helmholtz, "tJber die physikalische Bedeutung des Princips der klein- 

 sten Wirkung," Wissenschqftliche Abhandlungen, Vol. Ill, pp. 209-10. 



2 L. Hartmann, op. cit., pp. 18-19. 

 3Hartman, loc. cit., Figs. 48-50. 



