169, 170] 



STRESS. 



447 



exert certain forces on those on the other side, and the resultant of 

 the forces which pass through an element dS of the plane will be 

 a single force , proportional to the area dS, which we will write 

 F n dS. The stress at a point P is completely determined when we 

 know the direction and magnitude of the force F n for every possible 

 direction of the normal to the element dS constructed at P. The 

 stress -vector F nt which is in general not normal to dS, may be 



resolved into its components. X n , Y n , Z n , so that its direction -cosines 



X n Y n Z n 

 are -=r> -^=r> -=- The normal component 



F n F n F n 



83) F nn = I n cos (F n ri) = X n cos(nx) -\- Y n cos (ny) -\- Z n cos (ne). 



If we draw the normal in either direction from the element dS, and 

 if we understand by F n the force exerted through dS by the portion 

 of the body lying on the side toward which n is drawn on the 

 portion lying on the other side, then if the normal component 

 F nn = F n cos (F n ri) is positive it is called a traction, if negative, a 

 pressure. In other words it is a traction if its effect is to cause the 

 portions of the body to approach each other, a pressure if it is to 

 make them recede. 



The force upon any element dS can be expressed in terms of 

 the forces upon three mutually perpendicular plane elements at the 

 same point. Construct, enclosing the 

 point P, an infinitesimal tetrahedron 

 bounded by the element dS and three 

 planes parallel to the coordinate planes 

 (Fig. 146). Let the areas of the four 

 triangular faces be dS, dS x , dS y , dS~, the 

 suffix in each case denoting the direc- 

 tion of the normal to the face. Further 

 denote the stress -vector for any face by 

 a suffix giving the normal to that face, 

 and let the stress -vectors be those for 

 the portion of the body within the 

 tetrahedron. Suppose that forces are 

 applied to every portion of matter in 

 proportion to its mass, such, for instance, 



as gravity, the components being X, Y, Z per unit mass. If d-c 

 denote the volume of the tetrahedron the X- component of these 

 external forces is accordingly Xgdt. 



Let us now form the equations for equilibrium of the matter 

 contained in the tetrahedron under the influence of the external 

 forces and the stresses developed. The first of these is 



84) Xgdr + X n dS - X x dS x - X y dS y - X 3 dS z = 0. 



Fig. 146. 



