17.3 Vector Representation of Tensors
249
Let us match the stress deviator (σ
ij ) with the stress vector S with components
S i , (i = 1, 2, . . . , 9) and match the strain deviator (ε
ij ) with the strain vector
Э( i ), (i = 1, 2, . . . , 9) under the following law:
S 1 = σ
11 , S 2 = σ
22 , S 3 = σ
33 , S 4 = σ
13 , . . . , S 9 = σ
32 ;
˜ 1 = ε
11 , . . . , ˜ 9 = ε
32 , (1, 2, 3 ∼ x, y, z).
Let us note that since stress and strain deviators in the most general case have
only five independent components, then the vectors S and Э are always in the fivedimensional sub-space of the nine-dimensional space. Such five-dimensional spaces
were introduced by A. A. Ilyushin.
In the five-dimensional orthogonal space of Ilyushin (S 5 ), (˜ 5 ) the link between
the components of the deviators D σ , D ε and the components of the vectors S and
Э is defined by mutually unambiguous linear ratios so that the second invariants of
deviators are equal to the squares of the vector moduli |S| 2 and |Э| 2 , respectively:
J
2 = σ
ij σ
ij =
5
i=1
S 2
i = |S| 2 =
2
3
σ
2
i ;
I
2 = ε
ij ε
ij =
5
i=1
˜
2
i = |Э| 2 =
3
2
ε
2
i .
An example of such ratios can be setting the components of the vectors S and Э in
the form of
S 1 =
√
2
2
(σ x − σ y ), S 2 =
3
2
(σ z − σ 0 ),
S 3 =
√
2τ xy , S 4 =
√
2τ yz , S 5 =
√
2τ zx ;
˜ 1 =
√
2
2
(ε x − ε y ), ˜ 2 =
3
2
(ε z − ε 0 ), ˜ 3 =
√
2ε xy =
√
2
2
γ xy ,
˜ 4 =
√
2ε yz =
√
2
2
γ yz , ˜ 5 =
√
2ε zx =
√
2
2
γ zx .
(17.8)
While loading, the end of the vector Э circumscribes an arch in the space
˜ 5 (Fig. 17.2) that is called a strain trajectory. Proportional loading in the fiveFig. 17.2 Strain trajectory in
the space ˜ 5