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

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Fig. 2.1. Yield curves with compaction and dilation regimes
The critical line represents the state of stress in the material that causes yielding
without changes in density, therefore corresponds to steady flow. The above model
comprises hardening, softening, or flow in the critical state of stresses. The hardening
or softening are determined by the sign of the partial derivative ∂F/∂p. If the
following relations occur in the process under study:
F(p,q,ρ) = 0,
dF(p,q,ρ) = 0,
(2.6)
then the yield condition is fulfilled. If we also have the inequality:
∂F
∂F
dp + dq > 0,
∂p
∂q
(2.7)
which means that the angle between the direction of stress increment (dp, dq) and
the direction of the normal to the yield curve is less than 90 o and
∂F
> 0 ,
∂p
(2.8)
then, on the grounds of the assumption of coaxiality of the principal stresses and
strain increments, the increase of volumetric deformation is positive (dε p > 0).
In such a case material hardening takes place. Density increases (dρ > 0), and the
yield curve expands. Such plastic strain is called stable strain. In this case the
flow rule presents a unique description of plastic strain of the material.