r - The Hong Kong Polytechnic University

As the main difference with the linear problem, the Eshelby tensor depends now on the tangent operator of the
clay matrix L m . Depending on the local constitutive model used for clay matrix, the tangent operator can
become anisotropic in nature as it is the case for non-associated plastic model. As a consequence, the Eshelby
(or Hill) tensor can not be evaluated in analytical way and a suitable numerical integration method should be
adopted. Further, as shown in some previous works (Doghri, and Ouaar, 2003; Chaboche et al. 2005;
Abou-Charka et al., 2008), the use of anisotropic local tangent operator in the determination of incremental
strain concentration tensors general leads to too stiff responses of composites. The physical reason of such a
result is mainly the fact that the heterogeneity of local fields is neglected in the proposed method. As a
pragmatic corrective method, it is proposed to decompose the local tangent operator into an isotropic part and
anisotropic one and then to use the isotropic part in the evaluation of the macroscopic tangent operator. As a
example, we propose here to use the isotropic part of local tangent operator of the clay matrix only for the
evaluation of Eshelby tensor. The isotropic part is determined by (Bornert et al. 2001):
δij
are components of the second order unit tensor,
iso
1
L m = ( J :: L m ) J + ( K:: L m ) K= = 3k
t J + 2μ
t K
5
(29)
⎧ 1
Iijkl = ( δikδ jl +δilδjk
)
⎪ 2
⎨
⎪ 1
Jijkl = δijδ kl ; Kijkl = Iijkl −Jijkl
⎪⎩ 3
(30)
k t and μ t denote respectively the tangent bulk and shear
modulus related to the isotropic part of the tangent operator of the clay matrix. Considering now spherical
mineral inclusions, the Eshelby tensor is given by:
0 iso 3kt 6 kt + 2μt
SI( Lm
) = J + K (31)
3k
+ 4μ 53k
+ 4μ
t t t t
And the Hill tensor for the inclusion phase becomes:
0 0 iso 1
P = S ( L ): L (32)
Example of application
I I m m −
For the argillite studied here, the clay matrix is described by an elastoplastic model coupled with damage due to
microcracks. The mineral grains (quartz and calcite) are described by the linear elastic model. An important
challenge in micromechanical modeling of heterogeneous materials is the determination of local behavior of
different mineral phases. Direct identification remains a serious technical challenge even if significant advances
have been made such as nano indentation tests. In the present studies, an indirect method is adopted. The
model’s parameters are determined for a given mineral composition by numerical fitting. And then the obtained
set of parameters is checked for other mineral compositions. Note that the main advantage of micromechanical
modeling is that the mineral composition becomes direct input data.
In Figure 9, we show the simulation of a triaxial compression test with 10MPa confining pressure. The
macroscopic responses of argillite are compared with experimental data and also with those of clay matrix alone.
We can see that there is a good agreement between numerical results and test data. The main features of argillite
mechanical behavior are well reproduced by the micromechanical model. Further, the effects of mineral
inclusions are clearly illustrated. In Figure 10, a series of simulations are presented. These triaxial compression
tests are performed under different confining pressures and on the argillites samples with different mineral
compositions. Again, the numerical results are in good agreement with experimental data. In most important
feature of the micromechanical model is that the influences of mineral composition are inherently taken into
account in the formulation of the model.
-194-