198
15 Plasticity Theory of Henky–Nadai–Ilyushin
Fig. 15.3 Change of the
lateral strain coefficient
(Poisson ratio) of steel
σ i = σ 1 , ε i =
2
3
(1 + ν p )ε 1 .
For the hardening function, we obtain as follows:
σ i = 2
3
2
ε i
(1 + ν p )
≡ i ).
For the developed plastic strains, we can assume ν p =
1
2
, (see Fig. 15.3). For this
condition
σ i = 2 (ε 1 ).
Let us note that the dependency σ i ∼ ε i can be built without measuring
lateral strains and we can use the dependency σ 1 ∼ ε 1 involving Hooke’s law for
volumetric strain (see [4, p. 171]).
15.3 Some Properties of the Hardening Function
In the elastic stage of the material work, Hooke’s law is correct (14.32)
σ i = 3Gε i .
This means that the tangent of the angle M OM (Fig. 15.4) equals the triple shear
modulus, e.g. tg M OM = 3G.
Let us take an arbitrary point M (Fig. 15.4) on the non-elastic section of the
hardening diagram. Let us draw a beam OM from the reference point O to this
point, and draw a tangent line to the curve σ i ∼ ε i in the point M. The incline angle
tangent of the beam OM to the coordinate axis is called a secant modulus E s
E s = tg MOM
=
σ i
ε i
,