Simplifications (Contd...)

Consider the equilibrium of a tetrahedral portion of the body which has three edges along the three axes of the co-ordinate and the vertex at which they meet at the origin O. Say is the direction normal to the plane ABC drawn away from the interior of the plane. Then the direction cosines of this plane are and . Let's say that the plane is at a distance from origin O with . If is the area of this plane then the areas of the remaining faces are and respectively.

When the tetrahedron is small, the traction across the face be and those on the remaining faces to be and .

(2.9)

by this equation we express traction across any plan through O in terms of the traction across planes parallel to the coordinate planes.

Now, consider a very small cube with edges parallel to the coordinate axes. The value for the cube can be taken as , where is the length of any edge. Hence from the results of the last paragraph, we have,

and . Thus, the number of components that must be specified in order that the stress at a point may be specified is six, three normal components and three tangential tractions.