13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- ( ) 0 d m T h σ σ + − ≤ . (1) The parameters T and h respectively characterize the friction coefficient and the tensile strength of the solid phase of the clay matrix. The result of the first homogenization step is the derivation of the strength properties of the porous clay matrix at the mesoscopic scale where it is described as a homogeneous material. These properties were estimated successfully in [6] by means of the modified secant method. Now 1 d mσ σ =σ− denotes the deviatoric part of the stress tensor σ at the mesoscopic scale and we introduce : : dσ = σ σ K and ( : : ) /3 1 mσ = J σ . In the situation of associated plasticity, the domain of admissible stress states is an ellipse in the (σm, σd)-plane: 2 2 2 2 2 2 1 2 /3 3 ( , ) ( 1) 2(1 ) (1 ) 0 2 meso d m m f f F f T f h f h T T σ σ σ + = + − + − − − ≤ σ, . (2) Note that 0 3 2 T f < ≤ / (see [6]), f is the porosity. At the mesoscopic scale, the clay matrix is described by the elliptic criterion (2). In the framework of limit analysis theory (see e.g. [10]), a dual characterization of the strength criterion ( ) 0 F σ ≤ is the support function ( ) sup( ( ) 0} d d F F π σ σ = : , ≤ of the convex set of admissible stress states. The support function ( )dπ associated with (2) of the porous matrix finally takes the form 0 2 ( ) with 3 d d d F EQ EQ v d d d H π σ λ = − = : : . (3) 2 2 0 2 2 3 3 2 (1 ) (1 ) 2 3 2 1 2 3 3 2 f T T f h f h f T f f T σ λ = − ; = − − + / − . (4) where 1 α = + H J K is a fourth order tensor, with 2 3 2 3 2 f T f α / − = + . (5) 3. Overall dissipation at the mesoscopic scale We now focus on the transition from the mesoscopic scale to the macroscopic scale which constitutes the second homogenization step and is the main subject of the present paper. We seek the macroscopic criterion by means of a Gurson-type approach. As already stated, the microstructure at the mesoscopic scale is described by a composite sphere Ωwith a rigid core surrounded by the homogenized clay resulting from the micro-to-meso transition. The external (resp. internal) radius is denoted by er (resp. ir ). The shell mΩ ( i e r r r ≤ ≤ ) around the core represents the clay. The volume fraction 3 ( ) i e r r ρ= / of the core in the composite sphere is equal to the volume fraction of the rigid inclusions in a representative volume element of argillite. 3.1 Velocity field at the mesoscopic scale For geomaterials, we define here a family of cinematically admissible (k.a.) velocity fields withD, depending on one compressible parameter A: 3 2 ( ) D e i m d A r r r r Ax D A x v e r ≥ : = + − + ⋅ . (6) The strain rate in the clay ( i e r r r ≤ ≤ ) can be determined from (6):
RkJQdWJsaXNoZXIy MjM0NDE=