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

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The micropolar elasto-plastic constitutive model of a granular material with
isotropic hardening and softening differs from the classical theory of plasticity by
the presence of rotations, couple stresses, and a characteristic length corresponding to
the mean grain diameter. Due to the introduction of rotations into the kinematics, each
material point in the 3D case has three translational and three rotational degrees of
freedom, while in 2D and in axis-symmetrical cases two translational and one
rotational degree of freedom. The gradient components of the rotation cause
curvatures that are associated with the couple stresses. This makes the stress and
strain tensors non-symmetric, and the constitutive equation contains the characteristic
length. The micropolar elasto-plastic model in Mühlhaus’s approach [119] was
formed by the extension of the non-associated elasto-plastic flow rule of Drucker-
Prager with isotropic hardening and softening by the Cosserats’ rotations, curvatures,
couple stresses, and mean grain diameter. As a result, the micropolar model includes
the characteristic length and at the same time retains the essence of the continuous
medium. The constitutive model of granular materials formulated by Mühlhaus
contains a number of constants and of material functions that have to be determined
experimentally. These include the modulus of elasticity, Poisson constant, cohesion,
dependence of internal friction angle on plastic strain, dependence of dilatation angle
on plastic strain, mean grain diameter, and micropolar constants.
a) b)
c) d)
Fig. 3.15. Comparison of the particle displacement (a, b) and rotation (c, d) fields obtained from the
discrete method (a, c) and the continuum method (b, d) for the case of loading by (a, b) normal and
shear stress (c, d) couple stresses [32]