452 IX. DYNAMICS OF DEFOBMABLE BODIES. 



2. Simple traction , 



P X =P, P y = P. = 0, 



105) T T T 



J-x J-y= -Lz M- 



The stress quadric is 



106) y = Px 2 = 1, 



a pair of planes perpendicular to the X-axis at a distance from 



the origin. The stress on any plane is parallel to the X-axis. The 

 stress -director quadric and the shear -cone reduce to the axis of X, 

 all planes tangent to which experience only shear. 

 Cauchy's ellipsoid, 



107) P V + f + - = 1, 



with axes, p oo, oo, is a pair of planes perpendicular to the X-axis, 



and Lame's ellipsoid with axes, P, 0, 0, becomes simply that part of 

 the axis of X from x = P to x = P. From the property of this 

 ellipsoid the stress -vector is proportional to the perpendicular on the 

 tangent plane parallel to the stress plane. Since the tangent plane 

 here always passes through one of the extremities we have 



108) , F n = Pcos(nx) 

 as is indeed evident from equations 95). 



3. Simple shearing stress. 



p _ p p o 

 , y * ' 



T z = T, T x = T y = 0. 



Equations 95) become 



110) 



The stress quadric is 

 111) 



which represents a pair of rectangular hyperbolic cylinders with the 

 semi -axes = The stress -director quadric is 



yr 



112) 



The shear cone xy = represents the coordinate planes of X. 

 and YZ. 



