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

11
Density ρ is strictly related to volumetric deformation and dependent on the major
principal stress ρ(σ 1 ). The most important contribution in the development of the
model of granular material with hardening and softening is that by Roscoe [143].
In the model, for the particular values of density ρ we obtain, in the plane (τ,σ),
yielding conditions separating the plastic states of the material from its elastic or
rigid states. As higher density is related to higher strength, the yield condition is
a monotonically increasing function of density. For a fixed density ρ the yield
condition represents in the stress space an enclosed surface that, in the case of
a cohesionless material, passes through the origin of the system of coordinates
whose axis of symmetry is the axis of isotropic stress. In axial-symmetric state of
stress the yielding condition can be written in the system of coordinates (p,q):
where:
(2.4)
In figure 2.1 the critical line separates the area of compaction where plastic
strain is accompanied by an increase in density ρ > ρ 1 and therefore expansion of
the yield curve from the area of dilation in which strain is accompanied by
volume increase of the material, decrease in density ρ < ρ 2 i.e. in effect shrinking
of the yield curve. The change in density is defined by the law of mass
conservation:
where:
1
p = (σ1
+ 2σ
3
q = σ − σ ,
σ
1
1
≠ σ
2
2
= σ
3
.
2
),
F( p,q, ρ)
= 0,
dρ = ρdε p
,
(2.5)
dV
dε
p
= = dε1
+ 2dε
2,
V
2
dε
q
= ( dε1
− dε
2).
3