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17 Additions and Generalizations to the Strain Theory of Plasticity
dimensional space corresponds to a straight-line trajectory coming from the reference point.
Since it is assumed in plasticity theory that the link between stresses and strains
does not substantially depend on time, we can take the length of the strain trajectory
arch as a parameter characterizing the loading sequence:
d¸ =
d˜
2
1 + d˜
2
2 + . . . + d˜
2
5 ;
¸ =
t
t 0
d˜
2
1 + d˜
2
2 + . . . + d˜
2
5 ,
(17.9)
where ¸ is the length of the strain trajectory arch from the moment t 0 to the moment
t and d¸ is its differential.
An image of the strain process is an aggregate of the strain trajectories Э = Э
(t) and the law of conformity between S and Э that permits building a stress vector
S in each point of the strain trajectory.
Similarly to formulas (17.9) for the space S 5 , we can write as follows:
ds =
dS 2
1 + . . . + dS 2
5 ,
s =
t
t 0
dS 2
1 + . . . + dS 2
5 ,
(17.10)
where s is the length of the arch and ds is an element on the vector trajectory S in
the space S 5 .
In the space S 5 , we will call the process image an aggregate of the trajectories
S = S(t) and the law of conformity ∼ S.
The primary problem of plasticity consists in finding the law that permits building
a vector S in each point of the trajectory Э and vice versa.
17.4 Transformations of Rotation and Reflection
Let us at first consider transformation of the strain trajectory in a two-dimensional
case. In particular, in the case of axial elongation and torsion of a thin-wall tube as
per formulas (17.8), we have
S 1 =
3
2
(σ x − σ 0 ) =
2
3
σ x ; S 2 = 0; S 3 =
√
2 τ xy ;