270
18 Theories of Plastic Yield
λ
=
Γ
2τ s
,
where Γ is the intensity of shear strain rates of a non-compressible medium defined
by the formula
Γ =
2ε ij ε ij
1/2 .
Thus, Eqs. (18.10) can also be written as
˙
ε ij
Γ
=
σ
ij
2τ s
.
(18.11)
These ratios show that strains cannot be unambiguously defined when setting
strain rate stresses. The uncertainty of strain rate components related to an uncertainty of the multiplier λ is necessary for satisfying the conditions of joint strain. On
the opposite, for the defined rates of strain ˙
ε ij , the components of the stress deviator
σ
ij are defined unambiguously. In this case, the deviator components defined using
formulas (18.11) identically satisfy the Mises plasticity condition.
Equations (18.10) for the case of plane strain for the Tresca plasticity conditions
were given by Saint-Venant [9]. In a general case, they are defined by Levy and
Mises [9].
18.4 Plastic Yield in Isotropic Hardening
Assume that the loading surface undergoes even expansion during the plastic
strain of a material. This hardening is called isotropic. Assume that in this case the
equation has the following form:
f [J 2 (D σ ), J
3 (D σ )] = F (q),
(18.12)
where F is the ascending function of some parameter q characterizing the extent of
material hardening. In a simpler case, we can assume that the function f depends
only on the square invariant of the stress deviator, and we can write the dependency
(18.12) as
τ 0 = f (q).
(18.13)
The work of plastic strain can be taken as a measure of material hardening
A ? =
A ij dε
p
ij .
18 Theories of Plastic Yield
λ
=
Γ
2τ s
,
where Γ is the intensity of shear strain rates of a non-compressible medium defined
by the formula
Γ =
2ε ij ε ij
1/2 .
Thus, Eqs. (18.10) can also be written as
˙
ε ij
Γ
=
σ
ij
2τ s
.
(18.11)
These ratios show that strains cannot be unambiguously defined when setting
strain rate stresses. The uncertainty of strain rate components related to an uncertainty of the multiplier λ is necessary for satisfying the conditions of joint strain. On
the opposite, for the defined rates of strain ˙
ε ij , the components of the stress deviator
σ
ij are defined unambiguously. In this case, the deviator components defined using
formulas (18.11) identically satisfy the Mises plasticity condition.
Equations (18.10) for the case of plane strain for the Tresca plasticity conditions
were given by Saint-Venant [9]. In a general case, they are defined by Levy and
Mises [9].
18.4 Plastic Yield in Isotropic Hardening
Assume that the loading surface undergoes even expansion during the plastic
strain of a material. This hardening is called isotropic. Assume that in this case the
equation has the following form:
f [J 2 (D σ ), J
3 (D σ )] = F (q),
(18.12)
where F is the ascending function of some parameter q characterizing the extent of
material hardening. In a simpler case, we can assume that the function f depends
only on the square invariant of the stress deviator, and we can write the dependency
(18.12) as
τ 0 = f (q).
(18.13)
The work of plastic strain can be taken as a measure of material hardening
A ? =
A ij dε
p
ij .