Page 53 - 4749

Having written by a similar way the dependence for p and
 y
p we will obtain the system of equilibrium equations of an
 z
elementary tetrahedron
 p   x l  yx m  zx ; n
x


p   xy l  y m  zy ; n (3.5)


y

 p   xz l  yz m  z . n
z

Using the equations (3.5) for any area which is defined by the
direction cosines l , m , n , the components of full tension can be
defined.
The components p , p , p , form with the normal the angles
x  y  z 
which cosines are respectively equal to l , m , n . Therefore,
normal to the inclined edge stress is easily determined by the

formula   p l  p m  p n, or
 x  y  z 
2
2
2
   l   m   n  2 lm  2 ln  2 mn. (3.6)
 x y z xy xz yz
Tangential stresses are on the inclined edge
2
  p 2   , (3.7)
  
where
2
2
2
2
p  p  p  p .
 x  y  z 

3.3 Main areas and Main stresses

Tensor as mathematical concept is a generalization of the
concept of vector. Therefore, there are certain analogies in the
properties of vector and tensor. Only for vector these properties are
obvious and for tensor they should be proved.

Consider the vector M  a , a y  (fig. 3.4) on the plane. By
x
turning the coordinate axes the components of the vector M are
53