A Microstructural Model for Soils and Granular Materials - Multiscale Geomechanics: From Soil to Engineering Projects - page 182

7.2. The micro-structural model

In this model, we conceive a granular material as a collection of particles. The
deformation of a representative volume of the material is generated by mobilizing
contact particles in all orientations. Thus, the stress−strain relationship can be
derived as an average of the mobilization behavior of local contact planes in all
orientations. For a contact plane in the α

f

α th

orientation, the local forces i and the

local movements i α

δ can be denoted:

{ } { }

α

= f f f

ααα α

=

ααα

and

n n

where the subscripts n , s, and t represent the components in the three directions of
the local coordinate system. The direction normal to the plane is denoted n ; the other
two orthogonal directions, s and t , are tangential to the plane (see Figure 7.2).

Figure 7.2. Local coordinates

The forces and movements at the contact planes of all orientations are suitably
superimposed to obtain the macroscopic stress−strain tensors. The macroscopic
stiffness tensor is obtained on the condition that the rate of energy dissipation
expressed in terms of the macro stress and strain must be equivalent to that
expressed in terms of micro forces and movements. Under such a formulation, it has
usually been assumed that the microstructure is statically constrained, which means
that the forces on each contact plane are assumed to be equal to the resolve
components of the macroscopic stress tensor.