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

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In turn, the model of Lade, also applied by Zhang et al. [175] for the description
of wheat grain during loading, assumes that the plastic strain increment dε p ij is the
sum of two independent components: the plastic strain increment related with the
compaction of the material dε c ij and the plastic strain increment related with the
dilation of the material dε d ij:
dε = dε + dε
p
ij
c
ij
(2.19)
The division of the plastic strain increment into two independent components entails
the necessity of adopting also two independent yield functions and two flow rules.
For the description of the behaviour of wheat Zhang et al. [175] adopted the
following yield functions F c and F d , and plastic potentials G c and G d :
d
ij
.
F
c
= I
2
1
+ 2⋅
I
2
− P ⋅(
W
a
c
/ C ⋅ P )
a
1
q
,
(2.20)
F
d
= ( I
3
1
/ I
3
− 27) ⋅(
I / P )
1
a
m
− ae
d
−bW
⋅(
W
d
/ P )
a
1
q
,
(2.21)
G c
2
= I 1
+ 2 ⋅ I 2
,
(2.22)
3
m
G
(2.23)
d
= I1 − ( 27 + η ⋅(
Pa
/ I1)
) ⋅ I3,
where:
a, b, c, m, q, η – material constants,
I 1 , I 2 , I 3 – first, second and third invariant of stress tensor,
P a – atmospheric pressure,
W c , W d – collapse and expansive plastic work.
The yield functions F c , responsible for irreversible compaction of material,
represents in the space of principal stresses a concave surface with axis of
symmetry lined with the axis of isotropic stresses. This condition confirms the
known rule that a granular material compacts the easiest under the isotropic stress.
In the plastic potential G d , responsible for material expansion, the material
constant η represents the slope of the plastic potential surface, and the exponent m
represents the curvature of the meridian of the surface.
Figure 2.3 presents examples of the application of the models of Ghaboussi
and Momen and of Lade for the approximation of the experimental stress-strain
relations obtained during tests of monotonic loading of wheat grain samples in triaxial
compression apparatus. The presented comparison shows that the model of Lade more
accurately approximates the course of the stress-strain relation during monotonic