246
17 Additions and Generalizations to the Strain Theory of Plasticity
experimental data is found when the main axes of the stress tensor are turned during
loading.
As formulas (17.1) show, the Hencky–Nadai law is based on the proportionality
of stress and strain deviators. This provision can be replaced by a generalized one:
the strain deviator is the function (F ) of the stress deviator.
ε
ij = F (σ
ij ).
(17.2)
The dependency (17.2) must be invariant relative to coordinates since the material
is isotropic in the initial condition. Taking into account the considerations of tensor
dimensionality, I. I. Goldenblatt [4] and V. Prager later [13] suggested that in a
general case (17.2) might look as follows:
ε
ij = F (J
2 , J
2
3 )
P (J
2 , J
2
3 )σ
ij + Q(J
2 , J
2
3 )t ij
,
(17.3)
where
J
2 =
1
2
σ
ij · σ
ij , J
3 =
1
3
σ
ij σ
jk σ
kl ;
t ij =
∂J
3
∂σ
ij
= σ
ik σ
kj −
2
3
J
2 σ
ij ,
and the symbols P and Q designate some functions of these arguments.
If we assume Q = 0, P = 1, and F = F (J
2 ) in formula (17.3), the Hencky–
Nadai law (17.1) is obtained from the ratios (17.3) as a partial case. We shall also
note that the Prager law (17.3) includes not only the second invariant of the stress
deviator into the ratios of the link between stresses and strains but also the third one,
which is rather significant for some materials.
Both the Hencky–Nadai law and the Prager law are distinctive by the fact that
stresses and strains in them are related by finite ratios. Plasticity theories with such
links are usually called theories of strain type.
17.2 Tensor–Linear Ratios in Plasticity Theories
Stresses and strains can be linked by some integration–differentiation operations.
The simplest class of such relations are tensor–linear relations where tensor matrices
are related by some linear operators.
If we remain within the tensor–linear ratios, any plasticity law, as noted by
Prager, can be represented as follows:
L
σ
ij
= L
ε
ij
,
(17.4)
17 Additions and Generalizations to the Strain Theory of Plasticity
experimental data is found when the main axes of the stress tensor are turned during
loading.
As formulas (17.1) show, the Hencky–Nadai law is based on the proportionality
of stress and strain deviators. This provision can be replaced by a generalized one:
the strain deviator is the function (F ) of the stress deviator.
ε
ij = F (σ
ij ).
(17.2)
The dependency (17.2) must be invariant relative to coordinates since the material
is isotropic in the initial condition. Taking into account the considerations of tensor
dimensionality, I. I. Goldenblatt [4] and V. Prager later [13] suggested that in a
general case (17.2) might look as follows:
ε
ij = F (J
2 , J
2
3 )
P (J
2 , J
2
3 )σ
ij + Q(J
2 , J
2
3 )t ij
,
(17.3)
where
J
2 =
1
2
σ
ij · σ
ij , J
3 =
1
3
σ
ij σ
jk σ
kl ;
t ij =
∂J
3
∂σ
ij
= σ
ik σ
kj −
2
3
J
2 σ
ij ,
and the symbols P and Q designate some functions of these arguments.
If we assume Q = 0, P = 1, and F = F (J
2 ) in formula (17.3), the Hencky–
Nadai law (17.1) is obtained from the ratios (17.3) as a partial case. We shall also
note that the Prager law (17.3) includes not only the second invariant of the stress
deviator into the ratios of the link between stresses and strains but also the third one,
which is rather significant for some materials.
Both the Hencky–Nadai law and the Prager law are distinctive by the fact that
stresses and strains in them are related by finite ratios. Plasticity theories with such
links are usually called theories of strain type.
17.2 Tensor–Linear Ratios in Plasticity Theories
Stresses and strains can be linked by some integration–differentiation operations.
The simplest class of such relations are tensor–linear relations where tensor matrices
are related by some linear operators.
If we remain within the tensor–linear ratios, any plasticity law, as noted by
Prager, can be represented as follows:
L
σ
ij
= L
ε
ij
,
(17.4)