252
17 Additions and Generalizations to the Strain Theory of Plasticity
These two parameters fully define the strain trajectory with an accuracy up to the
transformation of rotation of reflection.
The internal geometry is fully defined by a natural orthogonal frame [12] built
in any point of the trajectory. The block axes p i , (i = 1, 2, . . . , 5) satisfy the
conditions
p i p j = δ ij
and are calculated using the Frenet–Serret formulas.
17.5 Ilyushin’s Isotropy Postulate
If we switch from tensor values to vector ones in the law (17.1) of Hencky–Nadai–
Ilyushin, we will obtain
S = 2G s (¸)Э.
(17.11)
We have already said that the law (17.1) is rather precise in the case of simple
loading. The representation of the law as (17.11) indicates its dependency on the
vector direction Э. The latter means that the five-dimensional space of Ilyushin
is isotropic relative to the rotation of the vector Э. The generalization of this
provision onto complicated loadings was called Ilyushin’s isotropy postulate. For
this generalization, the law invariance relative to rotation is supplemented by the
invariance of the link (17.11) relative to reflection in all possible planes and
directions. As a result, the formulation of the isotropy postulate is given in the
following wording:
A link between S and Э is invariant relative to transformations of rotation and
reflection.
The postulate also permits a different formulation: the image of the strain process
is fully defined by the internal geometry of the strain trajectory, or else: the space
˜ 5 is isotropic relative to the process image.
Let us indicate some consequences resulting from the postulate.
Consequence 1 The link
S = L(Э)
must be vector-linear
Indeed, when transforming the rotation or reflection of the vector Э into Э
, the
vector S is transformed into S under the same law. Therefore, the angle α will not
look as a transformed dependency
S
= L(Э
)
17 Additions and Generalizations to the Strain Theory of Plasticity
These two parameters fully define the strain trajectory with an accuracy up to the
transformation of rotation of reflection.
The internal geometry is fully defined by a natural orthogonal frame [12] built
in any point of the trajectory. The block axes p i , (i = 1, 2, . . . , 5) satisfy the
conditions
p i p j = δ ij
and are calculated using the Frenet–Serret formulas.
17.5 Ilyushin’s Isotropy Postulate
If we switch from tensor values to vector ones in the law (17.1) of Hencky–Nadai–
Ilyushin, we will obtain
S = 2G s (¸)Э.
(17.11)
We have already said that the law (17.1) is rather precise in the case of simple
loading. The representation of the law as (17.11) indicates its dependency on the
vector direction Э. The latter means that the five-dimensional space of Ilyushin
is isotropic relative to the rotation of the vector Э. The generalization of this
provision onto complicated loadings was called Ilyushin’s isotropy postulate. For
this generalization, the law invariance relative to rotation is supplemented by the
invariance of the link (17.11) relative to reflection in all possible planes and
directions. As a result, the formulation of the isotropy postulate is given in the
following wording:
A link between S and Э is invariant relative to transformations of rotation and
reflection.
The postulate also permits a different formulation: the image of the strain process
is fully defined by the internal geometry of the strain trajectory, or else: the space
˜ 5 is isotropic relative to the process image.
Let us indicate some consequences resulting from the postulate.
Consequence 1 The link
S = L(Э)
must be vector-linear
Indeed, when transforming the rotation or reflection of the vector Э into Э
, the
vector S is transformed into S under the same law. Therefore, the angle α will not
look as a transformed dependency
S
= L(Э
)