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

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Two phases of the unloading can be observed (see figure 6.4). The first phase
is characterized by a purely elastic deformation and was used in the determination
of elastic constants, the modulus of elasticity E and Poisson’s ratio ν. The second
stage of unloading is characterized by both elastic and plastic deformations. It was
assumed that the material reversible response is governed by Hooke’s law:
e 1
ε x = [( 1 −ν
) σ x −νσ
z0]
E
(6.8)
e 1
ε z = [ σ z0 − 2νσ
x]
E
(6.9)
During the first phase of unloading (path BC) granular material exhibits
a linear relationship which is characteristic for elastic deformation. Assuming that
ε e x = 0 from (Eq. 6.8) σ x /σ z0 = (ν/1−ν) is obtained and applying the assumption
that ε z = ε e z to (Eq. 6.9) ε z may be expressed as below:
ε
z
σ
=
z0
E
⎛
2
⎞
⎜
2ν
1−
⎟
⎝ 1−ν
⎠
(6.10)
Fig. 6.4. Uniaxial compression tester and determination of modulus of elasticity [154]
Elastic constants were determined using experimental results measured during
the linear phase of unloading. The ratio of horizontal stress σ x to vertical stress σ z0
was assumed constant (elastic state of stress) and the slope of the straight portion