13.5 On the Criterion of Similarity of Stress and Strain Deviators
175
Let us designate the area of the triangle 123 using ω and areas of triangles lying
in the coordinate planes with normal lines 1, 2, 3 using ω 1 , ω 2 , ω 3 . We have
ω 1 = ω 2 = ω 3 = ω ·
1
√
3
.
(13.35)
Assume that #» n is a normal line to the octahedral plane. Its guiding cosines are
defined by formula (13.25). As above (p. 173),
#»
β in Fig. 13.2 designates a unit
vector of octahedral tangential stress. Let us write equilibrium equations of the
considered triangle pyramid in projections on the principal axes of the stress tensor
as follows:
σ
i =
√
3τ 0 β i , (i ∼ 1, 2, 3),
(13.36)
where σ
i = σ i − σ 0 , and σ 0 is the mean (hydrostatic) stress. Let us connect the point
C, Fig. 13.2 with a point of crossing between the axis 1 and octahedral plane 123
using a straight line and designate the unit vector of that line as
#»
k . We have
k 1 =
2
√
6
, k 2 = k 3 =
1
√
6
,
#»
k
#»
β = cos ,
(13.37)
where is the angle between the vectors
#»
k and
#»
β .
Let us express the guiding cosines of the unit vector
#»
β through the angle . We
can write
β 2
1 + β 2
2 + β 2
3 = 1,
β 1 + β 2 + β 3 = 0,
2β 1 − β 2 − β 3 =
√
6 cos .
(13.38)
The second equation of the system (13.38) is a consequence of orthogonality of the
vectors #» n and
#»
β , and the third one represents a result of the scalar product of the
unit vectors
#»
k and
#»
β .
The system (13.38) is solved by the formulas
β 1 =
2
3
cos ,
β 2 =
2
3
cos
−
2π
3
,
β 3 =
2
3
cos
−
4π
3
.
(13.39)
By substituting Eqs. (13.39) into equilibrium equations (13.36), we obtain