504 



Mr. C. A. Carus- Wilson on the Distribution 



after a certain amount of flow has taken place, and lines 

 drawn on the sides respectively at right angles to and parallel 

 to o x will remain so, since the displacement parallel to o x will 

 be proportional to the distance from the plane y oz. 



The question I propose to discuss is how to determine the 

 distribution of flow when the section at right angles to o x 

 varies in any way as we pass along o x. 



I will for simplicity consider a plate in which the thickness 

 parallel to oy is considerable compared with the thickness 

 parallel to o z, and I will consider the displacements in the 

 plane xoy. 



We may take a plate in which there is a gradual reduction 

 of section, in which there is what is commonly called a 

 " shoulder," as in fig. 2. 



It is required to determine the displacements in the plane 

 of the paper — the plane xoy: in other words, supposing the 

 plate covered with a series of lines parallel to ox and to oy 

 respectively, it is required to determine the curvature of these 

 lines due to flow. 



The distortion of the elements is caused by the shearing- 

 strain induced in the bar when subjected to longitudinal stress. 

 Since this shearing-strain is a maximum at 45° to the axis o x 

 of the bar, it will be convenient to consider the bar divided 



Fig. 1. 



into a number of small elements with sides making angles of 

 45° to the axis o x. Every such element as a b c d (fig. 1) will 



