References
389
If principal stresses rotate clockwise, we will find as follows from the condition (29.42) and formula (29.40):
1
2λ
2
0
λ
2
0 − λ 2
4 cos 2 2
λ 0 ( ˙
λ 0 cos 2 + 2λ 0 ˙
sin 2) (λ 4 −
− 2
λ
2
0 − λ 2
4 cos 2 2
+ ( ˙
λ 4 λ
2
0 + λ
2
4
˙
λ 4 cos
2 2) cos 2
+
+
˙
λ 0
λ 0
cos 2 ˙
χ(t)
1
2λ
2
0
λ
2
0 − λ 2
4 cos 2 2
×
×
λ 0 ( ˙
λ 0 cos 2 + 2λ 0 ˙
sin 2)(λ 4 + 2
λ
2
0 − λ 2
4 cos 2 2) +
+( ˙
λ 4 λ
2
0 + λ
2
4
˙
λ 4 cos
2 2) cos 2
−
˙
λ 0
λ
cos 2.
(29.44)
Thus the conditions (29.43)–(29.44) limit the rotation velocity of principal
stresses ( ˙
χ(t)) when monotonous plastic strain takes place.
Let us note that if in the shear resistance operator (29.7) we bring the constant
value g to zero, we will obtain based on formulas (29.32)–(29.33) and (29.29)
that ζ ≡ ±, = 0, e.g. slips will take place at any moment in time in a single
direction only. In this case, formulas (29.39) show that ˙
χ (t) = 0. This means that
for g = 0 monotonous strain occurs only in the case of proportional loading. Using
the operator (29.7) for g = 0 is rather efficient to solve problems at sufficiently
non-monotonous strain due to the extreme simplicity of such operator.
References
1. M. Leonov, V. Molotnikov, K teorii deformatsii metallov s yarko vyrazhennym predelom
tekuchesti (On the theory of deformations of metals with bright expressed yield strength) Izv.
AN Kirg. SSR [Izv. Academy of Sciences of Kyrghyz. SSR] 6, 3–10 (1974)
2. M. Leonov, N. Shvaiko, Slozhnaya ploskaya deformatsiya (Complex plane deformation), dan
sssr (reports of the USSR Academy of Sciences) DAN SSSR [Reports of the USSR Academy
of Sciences] (5) 159, 1007–1010 (1964)
3. V. Molotnikov, A. Molotnikova, Plosko-plasticheskaya deformatsiya (flat-plastic deformation),
in Sostoyanie i perspektivy razvitiya sel’skokhozyaistvennogo mashinostroeniya: materialy
mezhdunar. nauchn.-pr. konf. “Interagromash” (State and prospects of agricultural engineering
development: materials of the international conference. nauchn. - Ave. Conf. “Interagromash”)
(2009), pp. 186–189
389
If principal stresses rotate clockwise, we will find as follows from the condition (29.42) and formula (29.40):
1
2λ
2
0
λ
2
0 − λ 2
4 cos 2 2
λ 0 ( ˙
λ 0 cos 2 + 2λ 0 ˙
sin 2) (λ 4 −
− 2
λ
2
0 − λ 2
4 cos 2 2
+ ( ˙
λ 4 λ
2
0 + λ
2
4
˙
λ 4 cos
2 2) cos 2
+
+
˙
λ 0
λ 0
cos 2 ˙
χ(t)
1
2λ
2
0
λ
2
0 − λ 2
4 cos 2 2
×
×
λ 0 ( ˙
λ 0 cos 2 + 2λ 0 ˙
sin 2)(λ 4 + 2
λ
2
0 − λ 2
4 cos 2 2) +
+( ˙
λ 4 λ
2
0 + λ
2
4
˙
λ 4 cos
2 2) cos 2
−
˙
λ 0
λ
cos 2.
(29.44)
Thus the conditions (29.43)–(29.44) limit the rotation velocity of principal
stresses ( ˙
χ(t)) when monotonous plastic strain takes place.
Let us note that if in the shear resistance operator (29.7) we bring the constant
value g to zero, we will obtain based on formulas (29.32)–(29.33) and (29.29)
that ζ ≡ ±, = 0, e.g. slips will take place at any moment in time in a single
direction only. In this case, formulas (29.39) show that ˙
χ (t) = 0. This means that
for g = 0 monotonous strain occurs only in the case of proportional loading. Using
the operator (29.7) for g = 0 is rather efficient to solve problems at sufficiently
non-monotonous strain due to the extreme simplicity of such operator.
References
1. M. Leonov, V. Molotnikov, K teorii deformatsii metallov s yarko vyrazhennym predelom
tekuchesti (On the theory of deformations of metals with bright expressed yield strength) Izv.
AN Kirg. SSR [Izv. Academy of Sciences of Kyrghyz. SSR] 6, 3–10 (1974)
2. M. Leonov, N. Shvaiko, Slozhnaya ploskaya deformatsiya (Complex plane deformation), dan
sssr (reports of the USSR Academy of Sciences) DAN SSSR [Reports of the USSR Academy
of Sciences] (5) 159, 1007–1010 (1964)
3. V. Molotnikov, A. Molotnikova, Plosko-plasticheskaya deformatsiya (flat-plastic deformation),
in Sostoyanie i perspektivy razvitiya sel’skokhozyaistvennogo mashinostroeniya: materialy
mezhdunar. nauchn.-pr. konf. “Interagromash” (State and prospects of agricultural engineering
development: materials of the international conference. nauchn. - Ave. Conf. “Interagromash”)
(2009), pp. 186–189