NUMERICAL EXAMPLE. 35 



being l/« 2 = 1.0908 ; and lastly, a scission for which s l = 0.0708. Since 

 << x and bj have the same sign, the plane undilational strain might have 

 been regarded as due to the combination of two shears without any scis- 

 sion, but these shears would not be at 45° to one another. 



The value of en is given by tan en = Ija — 0.7545, so that en = 37° 2'. 

 Had only a x , b v c y and/\ been given, en could have been obtained from 

 (9), which, of course, gives the same angle. 



Idie first fiber to occupy the position of major axis at the inception of 

 strain made an angle with o x, which was ^ = (y -f />-)/2 = 19° 20', and at 

 this same time the positions of the lines of maximum strain were at 

 v ± 45° ; i. e., at 04° 20' or — 25° 40'. The original position of the fiber 

 winch eventually constitutes the final major axis was at an angle <>. or 

 20° 20]' to ox. The original position of the fibers which at the end of 

 the strain undergo maximum strain was at /.« ± (90° — 03) ; i. c, 75° 18 V 

 and — 30° 37 ¥. The angles in the unstrained mass bounding the fibers 

 which subsequently undergo maximum strain on the side from wdiich 

 rotation takes place are thus, <>■ 4- 90° — w and v + 45°, and these differ 

 by 10° ^Y. On the other side the limiting angles are ^ — 45° and 

 /i — (90° — m), which differ by only 4° 57*'. Thus the fibers on the 

 positive side of the major axis pass through the condition of maximum 

 strain more than twice as rapidly as do those on the negative side of the 

 major axis. If the resistance wdiich the mass offers to deformation varies 

 with the rapidity of deformation (as is the case with real substances), this 

 difference w r ill somewhat affect the results. Had a and b different signs, 

 this difference would be far greater. 



The angle m for this example is by formula (10) 33° 25', so that the ,3 

 shear changes the direction of the lines of maximum strain by some o\ 

 degrees, though without tending to produce any further relative motion 

 upon them. 



Figure 2 is drawn for the displacements a u b x , <\ and / n and illustrates 

 the range of planes of maximum strain for this example. 



Finite Stress. 

 relations of steess and strain. 



In the foregoing discussion the geometrical properties of homogeneous 

 strain due to given displacements as exhibited on any principal plane of 

 a strain ellipsoid have been developed, and I am aware of no important 

 property of such strain which has been omitted. If the relations of dis- 

 placement to stress (or force per unit area) could be as fully developed, 

 we should have a substantial basis for a theory of finite distortion, since 

 however heterogeneous a strain may be, any infinitesimal portion of the 

 mass is homogeneously strained. 



