18.9 Singular Loading Surfaces
279
σ 1 − σ 3 = ± 2k,
(18.34)
and the constant value k is selected such that the plasticity surface goes through the
loading point.
In a general case, the ratio between principal stresses during loading can change.
In the space of principal stresses, we cannot associate the names of axes with
the values of principal stresses without violating the inequation σ 1 σ 2 σ 3 .
Therefore, let us assume the designations σ ξ , σ η , σ ζ for principal stresses. In
various points of the space of principal stresses, the ratio between them is different,
and, in this connection, we assign the values of 1, 2, or 3 to the indexes ξ, η, ζ . For
this reason, we have six similar conditions (18.34) instead of two conditions:
σ ξ − σ η = ± 2k,
σ η − σ ζ = ± 2k,
σ ζ − σ ξ = ± 2k.
(18.35)
As we already know (p. 185), the conditions (18.35) are equations of six planes
that form the loading surface as a hexagonal prism. Let us assume that the loading
point belongs to the prism face f = σ ξ − σ η corresponding to the first of equations
(18.35). Derivatives from the function f will be
∂f
∂σ ξ
= 1,
∂f
∂σ η
= −1,
∂f
∂σ ζ
= 0.
Formula (18.28) gives
˙
e ξ = H ( ˙
σ ξ − ˙
σ η ),
˙
e η = −H ( ˙
σ ξ − ˙
σ η ),
˙
e ζ = 0.
(18.36)
If we assume that the function H depends on the value of the highest tangential
stress only, e.g. H = H (σ ξ − σ η ), the ratios (18.36) can be integrated; we have
e ξ = h(σ ξ − σ η ),
e η = −h(σ ξ − σ η ),
e ζ = 0,
(18.37)
where
h(σ ξ − σ η ) =
H (σ ξ − σ η )d(σ ξ − σ η ).
Though the ratios of the yield theory were initial, the adopted assumption of the
function form H leads to the final dependencies between plastic strains and stresses
as in the strain theory of plasticity. By comparing the dependencies (18.37) with