18.5 Handelman–Lin–Prager Plasticity Theory
271
The hardening condition (18.13) then looks as follows:
τ 0 = f (A p ).
(18.14)
If the universal hardening function τ 0 = ˜
(γ 0 ) is found from experiments, we
can write γ 0 = 0 ) as a result of its inversion and represent as follows:
dA p =
dd(τ 0 )
dτ 0
dτ 0 =
(τ 0 )dτ 0 .
By introducing this representation in formula (18.5) and designating
(τ 0 )
2τ 2
0
= (τ 0 ),
we obtain
dλ = (τ 0 )dτ 0 .
(18.15)
Substituting formulas (18.15) into Eqs. (18.6) allows calculating the gains of the
full strain components
dε ij = dε
y
ij + σ
ij 0 )dτ 0 .
(18.16)
The ratios (18.16) are true for active strain dτ 0 0. If dτ 0 = 0, there is neutral
loading, and in the case of dτ 0 < 0, unloading occurs under the elastic law. When
switching from active loading to neutral one and to unloading, the strain component
gains are continuously changed. Therefore, the ratios of yield theory are free from
the above distortions of continuity suffered by strain theory.
We shall note that these ratios set an unambiguous dependency of strain component gains on stresses and their gains if the hardening is present. The hardening state
has no condition binding the stress components (as in the case of ideal plasticity),
and the multiplier λ is well defined by formula (18.15).
18.5 Handelman–Lin–Prager Plasticity Theory
This variant of plasticity theory is sometimes called the simplest yield theory. The
following pre-requisites are taken as fundamental.
1. The gain of the strain deviator is well defined by the stress deviator and its gain.
2. The link of the strain deviator gain is linear relative to the stress deviator and its
gain.
3. The continuity condition is true.
271
The hardening condition (18.13) then looks as follows:
τ 0 = f (A p ).
(18.14)
If the universal hardening function τ 0 = ˜
(γ 0 ) is found from experiments, we
can write γ 0 = 0 ) as a result of its inversion and represent as follows:
dA p =
dd(τ 0 )
dτ 0
dτ 0 =
(τ 0 )dτ 0 .
By introducing this representation in formula (18.5) and designating
(τ 0 )
2τ 2
0
= (τ 0 ),
we obtain
dλ = (τ 0 )dτ 0 .
(18.15)
Substituting formulas (18.15) into Eqs. (18.6) allows calculating the gains of the
full strain components
dε ij = dε
y
ij + σ
ij 0 )dτ 0 .
(18.16)
The ratios (18.16) are true for active strain dτ 0 0. If dτ 0 = 0, there is neutral
loading, and in the case of dτ 0 < 0, unloading occurs under the elastic law. When
switching from active loading to neutral one and to unloading, the strain component
gains are continuously changed. Therefore, the ratios of yield theory are free from
the above distortions of continuity suffered by strain theory.
We shall note that these ratios set an unambiguous dependency of strain component gains on stresses and their gains if the hardening is present. The hardening state
has no condition binding the stress components (as in the case of ideal plasticity),
and the multiplier λ is well defined by formula (18.15).
18.5 Handelman–Lin–Prager Plasticity Theory
This variant of plasticity theory is sometimes called the simplest yield theory. The
following pre-requisites are taken as fundamental.
1. The gain of the strain deviator is well defined by the stress deviator and its gain.
2. The link of the strain deviator gain is linear relative to the stress deviator and its
gain.
3. The continuity condition is true.