4
I. A. Volkov et al.
• components of strain tensors e i j and of strain rates ˙
e i j include elastic e
e
i j , ˙
e
e
i j and
plastic strains e
p
i j , ˙
e
p
i j , i.e., reversible and irreversible components;
• the initial yield surface for different temperatures is described by a Mises-type
surface. The evolution of the yield surface is described by the change of its radius
C p and by the displacement of its center ρ i j ;
• the body volume changes elastically;
• initially isotropic media are considered;
• only anisotropy due to the processes of plasticity is accounted for (anisotropy due
to the processes of damaged material is not accounted for);
• processes characterized by small deformations are considered.
In the elastic region, the relation between spherical and deviatoric components of
stress and strain tensors and their rates is described by Hooke’s law:
σ = 3K [e − α(T − T 0 )], σ i j = 2Ge
e
i j ,
˙
σ = 3K ( ˙
e − ˙
αT − α ˙
T ) +
˙
K
K
σ, ˙
σ
i j = 2G ˙
e
e
i j +
˙
G
G
σ
i j ,
(1.1)
where T is temperature, T 0 is initial temperature, K (T ) is volumetric compression
modulus, G(T ) is shear modulus, and α(T ) is linear thermal expansion coefficient
of the material.
To describe the effects of monotone and cyclic deformation, a yield surface is
introduced:
F s = S i j S i j − C
2
p = 0, S i j = σ
i j − ρ i j .
(1.2)
To describe complex cyclic deformation regimes, a cyclic ‘memory’ surface is
introduced in the stress space. The equation of the ‘memory’ surface is as follows:
F ρ = ρ i j ρ i j − ρ
2
max = 0,
(1.3)
where ρ max is maximal modulus ρ i j during the entire loading history.
In the range of temperatures T, at which annealing effects can be neglected, it is
assumed that isotropic hardening (evolution of C p ) may be of three types: monotone,
cyclic and that connected with the change in temperature T. The concretization of the
evolutionary equation for the yield surface radius has the form (Volkov and Korotkikh
2008; Mitenkov et al. 2015):
˙
C p = [q χ H (F ρ ) + a(Q s − C p ))(F ρ )] ˙
χ + q 3 ˙
T ,
(1.4)
I. A. Volkov et al.
• components of strain tensors e i j and of strain rates ˙
e i j include elastic e
e
i j , ˙
e
e
i j and
plastic strains e
p
i j , ˙
e
p
i j , i.e., reversible and irreversible components;
• the initial yield surface for different temperatures is described by a Mises-type
surface. The evolution of the yield surface is described by the change of its radius
C p and by the displacement of its center ρ i j ;
• the body volume changes elastically;
• initially isotropic media are considered;
• only anisotropy due to the processes of plasticity is accounted for (anisotropy due
to the processes of damaged material is not accounted for);
• processes characterized by small deformations are considered.
In the elastic region, the relation between spherical and deviatoric components of
stress and strain tensors and their rates is described by Hooke’s law:
σ = 3K [e − α(T − T 0 )], σ i j = 2Ge
e
i j ,
˙
σ = 3K ( ˙
e − ˙
αT − α ˙
T ) +
˙
K
K
σ, ˙
σ
i j = 2G ˙
e
e
i j +
˙
G
G
σ
i j ,
(1.1)
where T is temperature, T 0 is initial temperature, K (T ) is volumetric compression
modulus, G(T ) is shear modulus, and α(T ) is linear thermal expansion coefficient
of the material.
To describe the effects of monotone and cyclic deformation, a yield surface is
introduced:
F s = S i j S i j − C
2
p = 0, S i j = σ
i j − ρ i j .
(1.2)
To describe complex cyclic deformation regimes, a cyclic ‘memory’ surface is
introduced in the stress space. The equation of the ‘memory’ surface is as follows:
F ρ = ρ i j ρ i j − ρ
2
max = 0,
(1.3)
where ρ max is maximal modulus ρ i j during the entire loading history.
In the range of temperatures T, at which annealing effects can be neglected, it is
assumed that isotropic hardening (evolution of C p ) may be of three types: monotone,
cyclic and that connected with the change in temperature T. The concretization of the
evolutionary equation for the yield surface radius has the form (Volkov and Korotkikh
2008; Mitenkov et al. 2015):
˙
C p = [q χ H (F ρ ) + a(Q s − C p ))(F ρ )] ˙
χ + q 3 ˙
T ,
(1.4)