Chapter 27
Non-elastic Uniaxial
Elongation–Compression
27.1 Calculating Slip Intensity
Assume that only the expression σ z (t) differs from zero from all stress tensor
components. Taking into account that the tangential stress component τ νλ is defined
using formula (20.8), and the shear component γ νλ is calculated [6] using the
formula:
γ νλ = 2(ν x λ x ε x + ν y λ y ε y + ν z λ z ε z ) + (λ x ν y + λ y ν x )γ xy +
+ (λ y ν z + λ z ν y )γ yz + (λ z ν x + x ν z )γ xz ,
(27.1)
and for uniaxial elongation–compression, we have
τ m =
σ z
2
, τ νλ =
σ z
2
sin 2α cos ω, γ νλ =
3
2
ε z sin 2α cos ω, τ i =
√
2
3
σ z .
According to the definition, plastic strain resistance in the slip area equals the
corresponding component of tangential stress: S νλ ϕ nl = τ νλ . By substituting the
previous formulas into this equation, as well as the representations (25.2) and (25.3),
we obtain 1
0.5σ z (t) sin 2α cos ω = ψ(t) + (t)
1 + ε
∂
∂t
L νλ ϕ nl
−
− AA sin 2α cos ω − BB,
(27.2)
(ψ(t) = ψ[τ i (t), τ m (t)], ,(t) = i (t), L NL ϕ nl (t)]) ,
1 Due to the symmetry of stress and strain states in the considered case of uniaxial elongation, the
components τ νλ and γ νλ , as well as the function ϕ, do not depend on the variable β.
© 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_27
351