450 



IX. DYNAMICS OF DEFORMABLE BODIES. 



These are the exact analogues of equations 17). In other words, the 

 stress -vector is a self - conjugate linear vector -function of the normal 



to the stress -plane. The stress -vector F n occupies the place of * 



in 17). Accordingly the whole geometry of the linear vector func- 

 tion may be applied to the consideration of stress as follows. 



171. Geometrical Representation of Stress. If we construct 

 the quadric 



96) cp EEE P x x 2 + P y y* + P z s* + 2T x yz + 2T y zx + 2T z xy = E 2 



any stress -vector F n is perpendicular to the tangent plane drawn at 



the point where the normal to the stress- 

 plane cuts the quadric cp (Fig. 148). This 

 is known as Cauchy's stress -quadric. Let 

 its equation, referred to its principal axes, 

 which are known as the axes of the stress, be 



97) cp = P x 2 + P 2 y 2 + P 3 * 2 = E 2 . 



P 1; P 2 , P 3 are called the principal tractions, 

 being the normal stresses on the planes 

 perpendicular to the axes, these planes 

 being subject to no tangential stresses. 

 Thus, as for any strain we may find three 

 planes for which the slides vanish, so for 

 stress we may find three planes for which 

 the shearing stress vanishes. 

 In the reciprocal quadric, 



.2 ,2 *2 



QQ\ ' _|_ " _|_ i 7)2 



the stress -vector is conjugate to its stress -plane, for the normal to 

 the stress -plane is parallel to the normal to cp' where it is cut by 

 the stress -vector. The quadric cp' is known as Lame's stress -director 



quadric. In equations 17) and 14) putting F n for -- we obtain 



Fig. 148. 



99) 



p == _}_ ^L __ _j i 



pr r 2 cos(?*r') 



or 



100) 



So that the traction or component of the stress -vector normal to its 

 stress -plane is inversely proportional to the square of the radius- 

 vector of the quadric (p in the direction of that normal, or is directly 

 proportional to the square of the perpendicular upon the tangent 

 plane to the quadric cp r parallel to the stress -plane. 



