Chapter 29
Plane-Plastic Strain
29.1 Theorem of Strain in Pure Shear
Assume that all stress tensor components except for τ xy equal zero. Based on the
previously proved lemma (p. 312), the plastic strain component (γ xy ) is expressed
via the components (r x x , r x y ) of slip tensor intensity using formula (20.33):
γ xy =
1
2
π
0
(r x x sin 2β 0 + r x y cos 2β 0 )dβ 0 .
By using this formula, we will prove the following conclusion.
Theorem If the opening of the slip plane fan in the angle α 0 is small, the following
formula takes place for the plastic strain component (γ xy ) in pure shift:
γ xy ≈
1
2
π
0
r x y cos 2β 0 dβ 0 .
(29.1)
Proving From the definitions (20.16)–(20.17) and formulas (20.31) in the considered case, it follows:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_29
377
Plane-Plastic Strain
29.1 Theorem of Strain in Pure Shear
Assume that all stress tensor components except for τ xy equal zero. Based on the
previously proved lemma (p. 312), the plastic strain component (γ xy ) is expressed
via the components (r x x , r x y ) of slip tensor intensity using formula (20.33):
γ xy =
1
2
π
0
(r x x sin 2β 0 + r x y cos 2β 0 )dβ 0 .
By using this formula, we will prove the following conclusion.
Theorem If the opening of the slip plane fan in the angle α 0 is small, the following
formula takes place for the plastic strain component (γ xy ) in pure shift:
γ xy ≈
1
2
π
0
r x y cos 2β 0 dβ 0 .
(29.1)
Proving From the definitions (20.16)–(20.17) and formulas (20.31) in the considered case, it follows:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_29
377