18.7 Yield for Some Loading Surfaces
275
The fact that the second invariant of the stress deviator J
2 differs from the square
of stress intensity σ 2
? and octahedral tangential stress τ 2
0 by constant multipliers only
results in the proportionality of values
∂J
2
∂t
∼ σ ? ˙
σ i ∼ τ 0 ˙
τ 0 .
By substituting the last results to the law (18.28), we obtain
˙
e ij = H 1 (σ i ) ˙
σ i σ
ij ,
(18.29)
or
˙
e ij = H 2 (τ 0 ) ˙
τ 0 σ
ij ,
(18.30)
where
H 1 (σ i ) = H (σ i )σ i , H 2 (τ 0 ) = H (τ 0 )τ 0 .
By comparing the expressions (18.29) and (18.30) with formula (18.24), we
conclude that in the considered case, the law (18.28) results in ratios of the Laning
law. Let us remind that in this case for simple loading, the law (18.29) (or (18.30))
can be integrated and is converted into the Hencky–Nadai–Ilyushin law.
18.7 Yield for Some Loading Surfaces
Let us consider a material that finds the ideal Baushinger effect. The diagram of
the alternating-sign uniaxial stressed state beyond the yield stress of this material
is shown [5] in Fig. 18.1. 1 is the yield start in the case of primary loading of the
initially isotropic material; 2 is partial unloading with further loading of the same
sign; 3 is loading start; 4 is full unloading; 5 is the start of yield in the case of loading
of opposite sign. The reduction of the yield stress in point 5 equals hardening in the
direction of initial unloading that numerically equals the difference of ordinates of
points 3 and 1.
In a general case of loading, when hardening this material in some direction,
an equal softening occurs in the opposite direction. The loading surface in each
successive moment in time t 1 , t 2 , t 3 progressively moves as a rigid whole following
the loading point as shown in Fig. 18.2. This hardening is called translational.
If the initial loading surface equation is represented by formula (18.25)
f (σ ij ) = k
2 ,
275
The fact that the second invariant of the stress deviator J
2 differs from the square
of stress intensity σ 2
? and octahedral tangential stress τ 2
0 by constant multipliers only
results in the proportionality of values
∂J
2
∂t
∼ σ ? ˙
σ i ∼ τ 0 ˙
τ 0 .
By substituting the last results to the law (18.28), we obtain
˙
e ij = H 1 (σ i ) ˙
σ i σ
ij ,
(18.29)
or
˙
e ij = H 2 (τ 0 ) ˙
τ 0 σ
ij ,
(18.30)
where
H 1 (σ i ) = H (σ i )σ i , H 2 (τ 0 ) = H (τ 0 )τ 0 .
By comparing the expressions (18.29) and (18.30) with formula (18.24), we
conclude that in the considered case, the law (18.28) results in ratios of the Laning
law. Let us remind that in this case for simple loading, the law (18.29) (or (18.30))
can be integrated and is converted into the Hencky–Nadai–Ilyushin law.
18.7 Yield for Some Loading Surfaces
Let us consider a material that finds the ideal Baushinger effect. The diagram of
the alternating-sign uniaxial stressed state beyond the yield stress of this material
is shown [5] in Fig. 18.1. 1 is the yield start in the case of primary loading of the
initially isotropic material; 2 is partial unloading with further loading of the same
sign; 3 is loading start; 4 is full unloading; 5 is the start of yield in the case of loading
of opposite sign. The reduction of the yield stress in point 5 equals hardening in the
direction of initial unloading that numerically equals the difference of ordinates of
points 3 and 1.
In a general case of loading, when hardening this material in some direction,
an equal softening occurs in the opposite direction. The loading surface in each
successive moment in time t 1 , t 2 , t 3 progressively moves as a rigid whole following
the loading point as shown in Fig. 18.2. This hardening is called translational.
If the initial loading surface equation is represented by formula (18.25)
f (σ ij ) = k
2 ,