312
20 Problem Setting
From the comparison of sub-integral expressions in formulas (20.30)–(20.32), it
follows that 1
ε z =
1
4
π
0
r zz dβ 0 ,
γ xy =
1
2
π
0
(r x x sin 2β 0 + r x y cos 2ββ 0 )dβ 0 .
(20.33)
The following is proved.
Lemma If the directions ν and λ are located in the plane of the principal axes
(1, 2) of the strain tensor and the third principal axis coincides with the symmetry
axis of the half-sphere, principal strains are defined through the three (r x y , r x x ,
and r zz ) components of the tensor slip rate using formulas (20.29) and (20.33).
References
1. S. Batdorf, B. Budyanskii, Mekhanika: Sb. Perev., No. 5(33) (Mechanics: a collection
of translations, no. 5(33)) (Inostrannaya literatura Publ., Moscow, 1955). Zavisimost’
mezhdu napryazheniyami i deformatsiyami dlya uprochnyayushchegosya metalla pri slozhnom
napryazhennom sostoyanii (The relationship between stress and deformations for the hardening
of metals under complex stress state), Mekhanika: sb. perev., no. 5(33) (Mechanics: a collection
of translations), pp. 120–127
2. A. Kottrell, Teoriya Dislokatsii (Dislocation theory) (Mir Publ., Moscow, 1969)
3. 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)
4. M. Leonov, V. Molotnikov, B. Rychkov, XIII International Congr. According to the Theory.
and Appl. Mekh-ks (IUTAM) (IUTAM Publ., Moscow, 1972). The development of the concept
the slip theory of plasticity, pp. 31–32
5. A. Lyav, Matematicheskaya teoriya uprugosti(Mathematical theory of elasticity) (ONTI NKTP
SSSR Publ., Moscow, 1935)
1 In formulas (20.33), sub-integral functions can turn zero at some segments of the interval 0
β 0 π .
20 Problem Setting
From the comparison of sub-integral expressions in formulas (20.30)–(20.32), it
follows that 1
ε z =
1
4
π
0
r zz dβ 0 ,
γ xy =
1
2
π
0
(r x x sin 2β 0 + r x y cos 2ββ 0 )dβ 0 .
(20.33)
The following is proved.
Lemma If the directions ν and λ are located in the plane of the principal axes
(1, 2) of the strain tensor and the third principal axis coincides with the symmetry
axis of the half-sphere, principal strains are defined through the three (r x y , r x x ,
and r zz ) components of the tensor slip rate using formulas (20.29) and (20.33).
References
1. S. Batdorf, B. Budyanskii, Mekhanika: Sb. Perev., No. 5(33) (Mechanics: a collection
of translations, no. 5(33)) (Inostrannaya literatura Publ., Moscow, 1955). Zavisimost’
mezhdu napryazheniyami i deformatsiyami dlya uprochnyayushchegosya metalla pri slozhnom
napryazhennom sostoyanii (The relationship between stress and deformations for the hardening
of metals under complex stress state), Mekhanika: sb. perev., no. 5(33) (Mechanics: a collection
of translations), pp. 120–127
2. A. Kottrell, Teoriya Dislokatsii (Dislocation theory) (Mir Publ., Moscow, 1969)
3. 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)
4. M. Leonov, V. Molotnikov, B. Rychkov, XIII International Congr. According to the Theory.
and Appl. Mekh-ks (IUTAM) (IUTAM Publ., Moscow, 1972). The development of the concept
the slip theory of plasticity, pp. 31–32
5. A. Lyav, Matematicheskaya teoriya uprugosti(Mathematical theory of elasticity) (ONTI NKTP
SSSR Publ., Moscow, 1935)
1 In formulas (20.33), sub-integral functions can turn zero at some segments of the interval 0
β 0 π .