346
26 Full Bauschinger Effect
26.2 Proportional Primary Loading
Let us consider the proportional loading of the body element beyond the yield stress.
Assume that N and L designate a normal line to the maximum tangential stress
plane and the direction of its action at some point in time t 0 until which proportional
loading took place. Starting with the moment t 0 characterized by the stress τ 0
i , we
will do proportional unloading of the element in the direction (−L) opposite to L,
(dτ i /dt < 0). The previously described (see p. 333) behavior of steel specimens
in elongation shows that the “freezing” of slips does not occur immediately in
unloading in the specimen, since the unloading process is at first accompanied by a
plastic strain increment of the same sign. Consequently, unloading is accompanied
by a change in the slip area and rate.
Finally, at some moment t ∗ at the stress τ ∗
i , the slip intensity rate (∂ϕ nl /∂t) will
turn zero. The latter means that at the moment t ∗ , the “freezing” of slips will occur,
and in the case of further unloading, the material will follow Hooke’s law.
Based on the definition (20.6), the shear resistance equals the corresponding
component of tangential stress: S νλ ϕ nl = τ νλ . By substituting into this equation the
representation (25.2), in the direction of the maximum tangential stress τ ∗
m (τ ∗
m =
τ ∗
NL ), we will find as follows at the moment t ∗1 :
τ
∗
NL = ψ
τ
∗
i , τ
∗
m
+
τ
∗
i , . . .
L NL ϕ
∗
nl − AA NL + BB.
(26.1)
Hereinafter all values and functions marked with an asterisk are defined at the time
moment t ∗ .
26.3 Proportional Loading of an Opposite Sign
Due to the antisymmetry (20.4) of slip intensity upon the argument l (ϕ nl ≡
−ϕ n(−l) ), the integral operator L νλ has the property of
L νλ ϕ nl = −L ν(−λ) ϕ nl .
(26.2)
For the same reason, a similar property is possessed by the aging tensor component:
νλ = − ν(−λ) .
(26.3)
Using the condition of the equation of shear resistance to the tangential stress and
properties (26.2)–(26.3) in the direction (−L) for the material with full Bauschinger
effect, we can write as follows:
1 It is taken into account that at the moment t = t ∗ , ∂L νλ ϕ nl /∂t = 0.
26 Full Bauschinger Effect
26.2 Proportional Primary Loading
Let us consider the proportional loading of the body element beyond the yield stress.
Assume that N and L designate a normal line to the maximum tangential stress
plane and the direction of its action at some point in time t 0 until which proportional
loading took place. Starting with the moment t 0 characterized by the stress τ 0
i , we
will do proportional unloading of the element in the direction (−L) opposite to L,
(dτ i /dt < 0). The previously described (see p. 333) behavior of steel specimens
in elongation shows that the “freezing” of slips does not occur immediately in
unloading in the specimen, since the unloading process is at first accompanied by a
plastic strain increment of the same sign. Consequently, unloading is accompanied
by a change in the slip area and rate.
Finally, at some moment t ∗ at the stress τ ∗
i , the slip intensity rate (∂ϕ nl /∂t) will
turn zero. The latter means that at the moment t ∗ , the “freezing” of slips will occur,
and in the case of further unloading, the material will follow Hooke’s law.
Based on the definition (20.6), the shear resistance equals the corresponding
component of tangential stress: S νλ ϕ nl = τ νλ . By substituting into this equation the
representation (25.2), in the direction of the maximum tangential stress τ ∗
m (τ ∗
m =
τ ∗
NL ), we will find as follows at the moment t ∗1 :
τ
∗
NL = ψ
τ
∗
i , τ
∗
m
+
τ
∗
i , . . .
L NL ϕ
∗
nl − AA NL + BB.
(26.1)
Hereinafter all values and functions marked with an asterisk are defined at the time
moment t ∗ .
26.3 Proportional Loading of an Opposite Sign
Due to the antisymmetry (20.4) of slip intensity upon the argument l (ϕ nl ≡
−ϕ n(−l) ), the integral operator L νλ has the property of
L νλ ϕ nl = −L ν(−λ) ϕ nl .
(26.2)
For the same reason, a similar property is possessed by the aging tensor component:
νλ = − ν(−λ) .
(26.3)
Using the condition of the equation of shear resistance to the tangential stress and
properties (26.2)–(26.3) in the direction (−L) for the material with full Bauschinger
effect, we can write as follows:
1 It is taken into account that at the moment t = t ∗ , ∂L νλ ϕ nl /∂t = 0.