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

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the tensor of plastic strain increments or the plastic strain rate and the tensor of
stress during yielding:
∂G(
σ
ij)
p
& ε
ij
= λ ,
(2.1)
∂σ
where:
p
ε&
ij
– tensor of plastic strain rate,
λ – non-negative coefficient.
The above relation means that coaxiality of the stress and strain rate tensors has
been assumed, which is an expression of isotropy of the material during yielding.
The plastic flow rule has the form of a potential rule. This means that the tensor of
plastic strain rate is normal to the surface representing the potential G. The plastic
potential G is frequently taken to be identical with the yield condition F which is
the limiting states of stress that must be reached for plastic strain to occur, F≡G.
In such a case we speak about so-called associated flow rule:
& ε
(2.2)
The plastic potential for an ideally plastic material can be chosen in various
manners, and associated or non-associated flow rule can be constructed. Such
relations, however, are never completely in agreement with the results of
experimental studies and usually cover only a certain aspect of yielding (e.g.
dilatation or steady flow without volume change). In reality, the principal directions
of the tensors of stress and of strain rate are not coaxial, and the dilatation of the
material as predicted by the models is much greater from that observed
experimentally. The process of plastic strain of granular materials is more realistically
approximated by models including material hardening and softening [118].
2.2. Plastic model with hardening and softening
p
ij
Models of plastic flow with material hardening and softening attempt to
predict overall change of the material state from any initial state to any other final
state or to critical state when material yield without volume change. Special
attention is payed in the models to important role of density ρ, which is treated as
hardening parameter [45]. It is assumed that the material has no single yield
condition but a whole family of such conditions:
F(σ ,ρ) = 0.
ji
ij
∂F(
σ
ij)
= λ .
∂σ
ij
(2.3)