A Microstructural Model for Soils and Granular Materials - Multiscale Geomechanics: From Soil to Engineering Projects - page 186

α α p α

= k

[7.11]

i j

Detailed expression of the elasto-plastic stiffness tensor is given in [CHA 05].

7.2.2. Stress − strain relationship

7.2.2.1. Macro − micro relationship

The stress−strain relationship for an assembly can be determined from
integrating the behavior of interparticle contacts in all orientations. During the
integration process, a relationship is required to link the macro and micro variables.
The stress increment ij

σ  can be related to the contact forces and branch vectors for
contacts in all orientations [CHR 81, ROT 81]:

1 N

=  



αα

σ f l

[7.12]

ij j i

V

α = 1

Using the principle of energy balance, the mean force on the contact plane of
each orientation can be written

α

= σ AlV

− 1 α

[7.13]

i j i k k

The fabric tensor in equation [7.13] is defined by:

N

= 

A l l

αα

[7.14]

ik i k

α= 1

Using the static hypotheses proposed by Liao et al. [LIA 97], we obtain the
relation between the macro strain and interparticle displacement (here, we do not
consider the finite strain condition):

N

 

uA l  =

− 1

α δ

αα

[7.15]

ji , ik j k

= 1

δ  is the relative displacement between two contact particles and the branch
vector k l is the vector joining the centers of two contact particles. It is noted that
contact particles include both direct and indirect contact of neighboring particles

where j