Strona 2_redak - Instytut Agrofizyki im. Bohdana Dobrzańskiego ...

13
In a case when the partial derivative equals zero:
∂F
∂p
= 0
(2.9)
the strain increment dε q tends to infinity, and increase of volumetric strain dε p is
indeterminate. This is the case of critical yielding. The material is in the state of
steady flow at constant material density. Therefore, neither hardening nor softening of
the material take place.
In the case of the inequality of:
∂F
∂p
< 0
(2.10)
it follows from the flow rule that the increase in the volumetric strain is negative
(dε p < 0), and therefore density decreases (dρ < 0) and material softening takes
place. The material yields, and the yield curve shrinks due to the decreasing
density ρ:
∂F
dρ > 0.
∂ ρ
(2.11)
As the total differential of the yield condition F(p,q,ρ) equals zero:
∂F
∂F
∂F
dp + dq + dρ = 0,
∂p
∂q
∂ ρ
(2.12)
therefore, taking into account relation (11), vector (dp, dq) must be pointed into
the interior of the initial yield curve:
∂F
∂F
dp + dq < 0.
∂p
∂q
(2.13)
This is a case of experimentally observable unstable yielding with softening.