260
17 Additions and Generalizations to the Strain Theory of Plasticity
we consider a change of elastic constants of the material in the strain process, the
gradientality principle has no sense. Ilyushin believes that the issue of convexity of
the loading surface is also speculative.
17.9 On the Applicability Limits of the Strain Theory of
Plasticity
The applicability of the ratios of the strain theory of plasticity in the case of loading
different from proportional ones is studied in detail by Budiansky [1]. According
to Budiansky, we assume consistency with the Drucker postulate as a criterion of
validity of using these ratios.
If we consider the material to be non-compressible for simplicity, let us write the
principal equations of the Hencky–Nadai–Ilyushin theory
ε
ij =
1
2G s
σ
ij ,
(17.22)
setting links between deviators of stresses (σ
ij ) and full (elastic–plastic) strains
(ε
ij ).
If the hardening curve is known
τ o = G s γ o ,
τ o =
1
3
σ
ij σ
ij
1/2
(17.23)
(Fig. 17.10), the secant (plastic) modulus G s is known in each point K. The elastic
components of strain are defined similarly to formula (17.22) where the plastic
modulus must be replaced by the shear modulus G:
Fig. 17.10 Hardening
diagram and its parameters
17 Additions and Generalizations to the Strain Theory of Plasticity
we consider a change of elastic constants of the material in the strain process, the
gradientality principle has no sense. Ilyushin believes that the issue of convexity of
the loading surface is also speculative.
17.9 On the Applicability Limits of the Strain Theory of
Plasticity
The applicability of the ratios of the strain theory of plasticity in the case of loading
different from proportional ones is studied in detail by Budiansky [1]. According
to Budiansky, we assume consistency with the Drucker postulate as a criterion of
validity of using these ratios.
If we consider the material to be non-compressible for simplicity, let us write the
principal equations of the Hencky–Nadai–Ilyushin theory
ε
ij =
1
2G s
σ
ij ,
(17.22)
setting links between deviators of stresses (σ
ij ) and full (elastic–plastic) strains
(ε
ij ).
If the hardening curve is known
τ o = G s γ o ,
τ o =
1
3
σ
ij σ
ij
1/2
(17.23)
(Fig. 17.10), the secant (plastic) modulus G s is known in each point K. The elastic
components of strain are defined similarly to formula (17.22) where the plastic
modulus must be replaced by the shear modulus G:
Fig. 17.10 Hardening
diagram and its parameters