Granular Materials in Civil Engineering: Recent Advances in the Physics of Their Mechanical Behavior and Applications to Engineering Works - Multiscale Geomechanics: From Soil to Engineering Projects - page 41

3.1.2.3. Equivalent continuum

Experimental results have shown that the macroscopic behavior of granular
media verifies an energy-dissipation relationship, provided that boundary conditions
are sufficiently regular, such as the ones applied during triaxial or plane strain tests,
for example [FRO 79, FRO 83, FRO 86]. This experimental relation links the
eigenvalues of the Eulerian stress and strain rate tensors, σ and 

ε − usually
assumed to be coaxial − together with a material constant, interpreted as an apparent
friction, ψ* :

 

sin * .

[3.10]

σσε = ψ σε

ii ii

It has been shown that equation [3.10] can be written with the sole eigenvalues
of a tensor, π , representing the “internal actions for the equivalent continuum”. This
tensor is the contracted product of the internal forces and the internal movements; it
is a second-order symmetrical tensor and its trace corresponds to the work rate of the
internal forces in the continuum (see Figure 3.2c):

{ }

π





=⊗+⊗
=

1
2

σσεεσ contracted

[3.11]

{}



π



then Tr σε

ij ij

ij

With this definition, the experimental dissipation relation (equation [3.10]) can
be written as follows:

{ } { }

π π π π

Tr = sin *. ψ N

[3.12]

Let us note that there is a formal identity between equations [3.12] and [3.9] that
corresponds to the expression of the energy dissipation within the discontinuous
granular mass.

3.1.2.4. Correspondence between equivalent continuum and discontinuous granular
mass: the equivalence of internal actions
Comparing discontinuous granular mass A with the equivalent continuum V(A) , it
appears that the work rate produced by the internal forces within the continuum
must be equal to the work rate developed within the discontinuous medium:

{ }

( { }



Tr π = Tr P [3.13]

dv

A

V