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principal. Then the full stress (it is principal)  directed along the
normal , and its components p  l  , p  m  , p  n  .
x  y  z 
Substituting them in (3.5), we obtain

     l  yx m  zx n  0,
x


  xy l      m  zy n  0, (3.8)
y

   xz l  yz m      n  0.
z
The system(3.8) will have nontrivial solutions if the
determinant composed of its coefficients is equal to zero
     yx  zx
x
       0 . (3.9)
xy y zy
     
xz yz z
Having found the determinant, we obtain the equation:
3
2
  I   I   I  0, (3.10)
1 2 3

where
I       ;
1 x y z
 x  yx    y  zy
I   x zx  ;
2
     
xy y xz z yz z
  
x yx xz
I     .
3 xy y zy
  
xz yz z
The coefficients I , I , I are real numbers and are called first,
1 2 3
second and third invariants of the stress tensor (they do not depend
on the choice of coordinates). It is known that a cubic equation
with real coefficients has at least one real root. This root

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