TWO SCALE MODEL FOR HYDRO-MECHANICAL DAMAGE MODEL

1. TWO SCALE MODEL FOR HYDRO-MECHANICAL DAMAGE MODEL Ferdinando Marinelli, René Chambon et Yannick Sieffert Université de Grenoble, Laboratory 3S-R

2. INTRODUCTION Geomaterials and different observation scales AT WHICH SCALES ‘INTERESTING PHYSICS TAKING PLACE’ ? The answer depends on the phenomena we wish to model. The mechanical behavior of clay rocks is inherently multi-scale. This is true for localized damage, localized strain and fractures phenomena. Thesis of E. Ando (2011), Université de Grenoble Thesis of J.C. Robinet (2008),Université de Poitiers

3. CONSTITUTIVE LAW: homogenization method HOMOGENIZATION METHOD ALLOW TO BUILD CONSTITUTIVE EQUATIONS DIRECTLY FROM MICROSTRUCTURE FEATURES OF THE MATERIAL Analitycal method (asymptotic method) Numerical method (DEM-FEM) For each gauss point another numerical computation is performed to obtain an homogenized value of the stress field. Micro-structure solved with FEM Micro-structure solved with DEM In both case the follwing assumption are considered: • Finely periodic media • Periodic boundary conditions

4. FINITE SQUARE ELEMENT METHOD, FE² Periodic Boundary Conditions u3 - u1 = Δu for all x Є 1,3 u2 – u4 = Δu for all x Є 2,4 [ εmacro ]xd = ud [ εmacro ]xc = uc [ εmacro ]xb = ub p3 - p1 = Δp for all x Є 1,3 p2 – p4 = Δp for all x Є 2,4 FE² FE² method (Kouznetsova et al. 2002, Eindhoven University of technology)

5. FE² METHOD: microstructure GRAINS INTERFACES General isotropic hyperelastic material Linear damage constitutive law Energy Function Cauchy Stress A simple microstructure geometry is considered in order to obtain preliminar results and validate the implementation INTERFACES GRAINS

6. FE² METHOD: solution of hydraulic system Fluid system: network of channel Conservation of fluid mass ω in fluid network channel Constitutive laws for a fluid system

7. FE² METHOD: hydraulic characterization

8. FE² METHOD: hydraulic characterization • THE CONNECTIVITY OF FLUID NETWORK HAS ALWAYS TO BE HOLD • TWO SYSTEMS OF INTERFACES OPENING ARE CONSIDERED. Even if Δun tends to zero there is always a minimal opening of the interface for hydraulic system, in this way the fluid can always flows. The minimal opening of the interface Δmin represents a constitutive feature of the hydraulic network Mechanical system Δun= Δn Δmin := Δfluid - Δn Hydraulic system Δufluid = Δn+ Δmin

9. FE² METHOD: coupling behaviour of microstructure Mechanical part Hydraulic part 1. Opening interface: Δun, Δut 2. Coupling term: k(s) = Δufluid^3/12η 3. Conservation mass equations ω 4.Computation of pressure fields Pw distribution 5. Fluid forces: Fp, t Cohesive force: Tn, Tn

10. SECOND GRADIENT CONTINUUM MEDIA Continuum with micro-structure ( Germain, 1974 ) Second gradient continuum in coupled problems Objectivity of solution Collin, Chambon, Charlier 2006

11. PERUTURBATION METHOD ...... • If εis too small consistent stiffness matrix is not well computed because of some numerical noise issues DRAWBACK OF THE METHOD: Choice of perturbation parameter ε • if εis too big, the components of the tensor are far from their theoretical value

12. PERTURBATION PARAMETER Grains Mechanic Parameters λ = 1450 mPa μ = 950 mPa u = 0,001 mm 1 mm Interface Mechanic Parameters Tmax = 5 mPa Δelast = 0,015 mm Δrupt= 0,33 mm

13. PERTURBATION PARAMETER

14. PERTURBATION PARAMETER Grains Mechanic Parameters λ = 1442 mPa μ = 961 mPa u = -0,001 mm 1 mm Interface Mechanic Parameters Tmax = 5 mPa Δelast = 0,015 mm Δrupt= 0,33 mm

15. PERTURBATION PARAMETER

16. RESULTS HOMOGEINITY OF STRESS FIELDS all components of the stress are equals for all Gauss points ISOTROPIC TRACTION CASE σxx = 0,76961 mPa σxx = 0,76862 mPa ERRx = 0,0013 σyy = 0,46560 mPa σyy = 0,46493 mPa ERRy = 0,0014 σxy = 0,3e-9 mPa σxy = -1,4e-9 mPa

17. COMPUTATION TIME Number of step at macro level: 10 Number of step at micro level: 10 CPU Intel Core 2 Duo E8400 3.00 GHz RAM 3.21 Gb

18. HYDRO-MECHANICAL COMPUTATIONS: results DRAINED MONODIMENSIONAL TEST • Geometry of the problem: H=4mm, L=0,5mm • Imposed displacement of traction: u=0,001 mm • Boundary drainage systems are opened at the top and the bottom: pw = 0 • Vertical sides are impermeable: mx=0 WATER PRESSURE Δmin = 0.1 mm WATER PRESSURE Δmin = 0.05 mm H [mm] H [mm]

19. HYDRO-MECHANICAL COMPUTATIONS: results DRAINED MONODIMENSIONAL TEST • Geometry of the problem: H=4mm, L=0,5mm • Imposed displacement of traction: u=0,001 mm • Boundary drainage systems are opened at the top and the bottom: pw = 0 • Vertical sides are impermeable: mx=0 HORIZONTAL STRESS Δmin = 0.1 mm HORIZONTAL STRESS Δmin = 0.05 mm Horizzontal stress [mPa] Horizzontal stress [mPa]

20. HYDRO-MECHANICAL COMPUTATIONS: results DRAINED MONODIMENSIONAL TEST • Geometry of the problem: H=4mm, L=0,5mm • Imposed displacement of traction: u=0,001 mm • Boundary drainage systems are opened at the top and the bottom: pw = 0 • Vertical sides are impermeable: mx=0 VERTICAL STRESS Δmin = 0.05 mm VERTICAL STRESS Δmin = 0.1 mm Vertical stress [mPa] Vertical stress [mPa]

21. HYDRO-MECHANICAL COMPUTATIONS: results DRAINED MONODIMENSIONAL TEST • Geometry of the problem: H=4mm, L=0,5mm • Imposed displacement of traction: u=0,001 mm • Boundary drainage systems are opened at the top and the bottom: pw = 0 • Vertical sides are impermeable: mx=0 FLUX MASS Δmin = 0.1 mm FLUX MASS Δmin = 0.05 mm Floux mass [g/s] Floux mass [g/s]

22. CONCLUSIONS • Some computations with a finite element square method are presented in this study in order to model an hydro-mechanic behaviour of soft rocks • The mean feature of FE² is that a constitutive law can be built take into account the micro-structure of the material • Different and more complex constitutive law can be added to the model to describe the behaviour of the interfaces and the grains. • Computational effort is the mean drawback of the method so parallelization of the finite element code is needed to perform calculations with more and more elements and to study the behaviour of the material in the softening field