1.1 Stresses

In order to understand well the problem we have to show several concepts which are helping. We choose an octhaedrical plane and its normal (see Fig. 1.1.1). For a point of stress ( ? ₁ , ? ₂ , ? ₃ ) we have the stress vector t _n which must be projected on the hydrostatic line and on a normal to the hydrostatic line. Since

its components are

so that the projection of this vector on the hydrostatic line is

where ? is the mean stress. The projection on the hydrostatic line is now

with absolute value

Fig. 1.1.1: Octahedral plane and all other associated concepts.

The projection normal to the octhaedrical line is

with the absolute value

We got thus the interpretation of Ro and Eichinger [1926]: the absolute value of the vector tensor normal to the linear hydrostatic axes is equal to the square root of the second invariant of the deviatoric stress tensor. For notation we give

and = ? _ij ??? _ij is the stress deviator.

Since the vector t _? in entirely in the octahedral plane, one can try to project it on various directions in this plane. Thus projecting on the ? plane is giving (see Fig. 1.1.2):

Fig. 1.1.2: Octahedral plane with the assumption .

Now projecting on the direction we have,

Thus we can write:

Taking into account that

we arrive at the relation

? [0, ?/3] defines the orientation in...