A Thermo-Hydro-Mechanical Damage Model for Unsaturated Geomaterials

Transcription

1 A Thermo-Hydro-Mechanical Damage Model for Unsaturated Geomaterials Chloé Arson ALERT PhD Prize PhD Advisor : Behrouz Gatmiri Paris-Est University, U.R. Navier, geotechnical group (CERMES)

2 This research was funded by TIMODAZ project (Thermal Impact on the Damaged Zone Around a Radioactive Waste Disposal in Clay Host Rocks) launched by EURATOM (European Office of Atomic Energy)

3 Motivations and Objectives Nuclear Waste Storage

4 Motivations and Objectives Nuclear Waste Storage

5 Motivations and Objectives Excavation Damaged Zone, EDZ

6 Motivations and Objectives Unsaturated Porous Media

7 1 Assumptions 2 Model Outline 3 Numerical Validation (Θ-Stock) 4 Parametric Studies (Θ-Stock)

8 Damage Variable: Crack Density Tensor [Kachanov 1992] Ω ij = 1 V VER NX K =1 Assumption 1: cracks do not interact l 3 K n K i n K j Assumption 2 (result of assumption 1): the loss of deformation energy due to shear cracking is negligible, i.e. cracks open due to tensile stress (opening orthogonal to the crack plane) Assumption 3: at the scale of the REV, damage may be represented by three equivalent cracks (homogenization)

9 Damage Variable: Crack Density Tensor [Kachanov 1992] Ω ij = 1 V VER NX K =1 Assumption 1: cracks not interact " 8(1 ν 2 W e = 0) 3(1 ν 0 /2)E 0 l 3 K n K i n K j (σ ij σ jl ) Ω li ν 0 2 σ ji 1 X V VER K # l 3 K n K in K jn K kn K l σ lk Assumption 2 (result of assumption 1): the loss of deformation energy due to shear cracking is negligible, i.e. cracks open due to tensile stress (opening orthogonal to the crack plane) Assumption 3: at the scale of the REV, damage may be represented by three equivalent cracks (homogenization)

10 Damage Variable: Crack Density Tensor [Kachanov 1992] Ω ij = 1 V VER NX K =1 Assumption 1: cracks do not interact l 3 K n K i n K j Assumption 2 (result of assumption 1): the loss of deformation energy due to shear cracking is negligible, i.e. cracks open due to tensile stress (opening orthogonal to the crack plane) W e = 8(1 ν 2 0) 3(1 ν 0 /2)E 0 (σ ij σ jl ) Ω li Assumption 3: at the scale of the REV, damage may be represented by three equivalent cracks (homogenization)

11 Damage Variable: Crack Density Tensor [Kachanov 1992] Ω ij = 1 NX l 3 K n K i n K j V VER K =1 Assumption 1: cracks do not interact Assumption 2 (result of assumption 1): the loss of deformation energy due to shear cracking is negligible, i.e. cracks open due to tensile stress (opening orthogonal to the crack plane) Assumption 3: at the scale of the REV, damage may be represented by three equivalent cracks (homogenization) 3X Ω ij = d k n k i n k j k=1

12 Independent State Variables 1 Assumption: incompressible solid phase. Clausius-Duhem Inequality: (σ ij p aδ ij ) ɛ ji +(p a p w) ( ns w) η T Ψ s (ɛ ij, ns w, T, Ω ij ) independent strain variables: mechanical strain ɛ Mij, capillary strain ɛ Sv and thermal strain ɛ Tv......conjugate to 3 independent stress variables: net stress σ ij = σ ij p aδ ij, suction s = p w p a, and thermal stress p T : 8 < : σ ij ɛ M ij s ɛ Sv p T ɛ Tv 3 Thermodynamic decomposition of the total strain tensor: dɛ ij = dɛ e M ij + dɛ d M ij δ ij dɛ e Sv + dɛ d Sv δ ij dɛ e Tv + dɛ d Tv e: elastic, d: non-elastic 4 Clausius-Duhem Inequality written in terms of stress/strain products: σ ij ɛ Mij +s ɛ Sv +p T ɛ Tv Ψ s `ɛmij, ɛ Sv, ɛ Tv, Ω ij 0

13 Equivalent Mechanical State [Swoboda et Yang 1999] micro-stresses (opening the cracks in tension): 3X τ ij = τ k n k i n k j k=1 in the equivalent mechanical state: homogenization: eτ ij = g Ω ij

14 Equivalent Mechanical State [Swoboda et Yang 1999] equivalent stress: eσ ij = σ ij + eτ ij = σ ij + g Ω ij, eσ ij = Ψe(ɛpq, Ωpq) ɛ ij Helmholtz free energy for an isothermal solid: σ ij = Ψs(ɛpq, Ωpq, ) ɛ ij = Ψe(ɛpq, Ωpq) ɛ ij gω ij Ψ s(ɛ pq, Ω pq) = Ψ e(ɛ pq, Ω pq) gω ij ɛ ji = 1 2 ɛ ji D eijkl (Ω pq) ɛ lk gω ij ɛ ji damaged elastic energy + energy required to maintain cracks closed

15 Equivalent Mechanical State [Swoboda et Yang 1999] equivalent stress: eσ ij = σ ij + eτ ij = σ ij + g Ω ij, eσ ij = Ψe(ɛpq, Ωpq) ɛ ij Helmholtz free energy for an isothermal solid: σ ij = Ψs(ɛpq, Ωpq, ) ɛ ij = Ψe(ɛpq, Ωpq) ɛ ij gω ij

16 Stress/Strain Relationships 3 components of Helmholtz free energy: Ψ s(ɛ Mkl, ɛ Sv, ɛ Tv, Ω kl ) = 1 2 ɛ M ji D eijkl (Ω pq) ɛ Mlk ɛ Svβ s (Ω pq) ɛ Sv ɛ Tvβ T (Ω pq) ɛ Tv g M Ω ij ɛ Mji g S 3 δ ij Ω ji ɛ Sv g T 3 δ ij Ω ji ɛ Tv Conjugation Relationships: σ ij = Ψs(ɛ M kl,ɛ Sv,ɛ Tv,Ω kl ) ɛ Mij, s = Ψs(ɛ M kl,ɛ Sv,ɛ Tv,Ω kl ) ɛ Sv p T = Ψs(ɛ M kl,ɛ Sv,ɛ Tv,Ω kl ) ɛ Tv, Y d ij = Ψs(ɛ M kl,ɛ Sv,ɛ Tv,Ω kl ) Ω ij Damaged Rigidities Computed by the Principle of Equivalent Elastic Energy [Codebois et Sidoroff 1982]

17 Damage Evolution Law influence of tensile mechanical stress, thermal expansion and capillary pore shrinkage : Y + d1ij = g M ɛ + M ij + g S 3 ɛ Sv δ ij + g T 3 ɛ+ Tvδ ij a unique damage criterion [Dragon et Halm 1996] : r 1 f d `Yd pq, Ω pq = 2 Tr Y + d 1 Y + ij d 1 C 0 C 1 δ ij Ω ji ji after applying the consistency rules: dω ij = dλ d f d `Yd pq, Ω pq Y + d 1 ji

18 Transfer Equations Liquid Water Flow: V wi = Ψ R (θ w) σ(t ref ) dσ(t ) dt Kw ij (T ) j + 1 γ w σ(t ) σ(t ref ) Kw ij (s) j Influence of Damage on the intrinsic permeability : K wij = k T (T ) k r (S w) K intij (n, Ω pq) K wij = k T (T ) k r (S w) h K intact ij (n rev ) + K dg `nfrac ij, Ω pq i K wij (z) j Assumption: laminar flow in the 3 equivalent cracks of the REV [Shao et al. 2005] : K dg ij n frac, Ω rs = π 2/3 γ w 12 µ w (T ref ) χ4/3 b 2 b : internal length parameter 3X k=1 d k 5/3 δ ij ni k nj k

19 Transfer Equations Liquid Water Flow: K dg ij n frac, Ω rs = π 2/3 γ w 12 µ w (T ref ) χ4/3 b 2 b : internal length parameter 3X k=1 d k 5/3 δ ij ni k nj k Vapor Flow: damaged intrinsic diffusivity computed from the mean of K dg ij Introduction of a second internal length parameter b* Air and Heat Flows: same transfer equations as for the undamaged material damage accounted for in porosity and in the degree of saturation

20 Triaxial Compression Tests [Halm and Dragon 2002] Mechanical Tests - dry granite 16 x p c =0MPa p c =5MPa p c =20MPa σ zz σ rr (Pa) ε rr ε zz Ω rr ε x 10 3 p c = 0 MPa σ σ (Pa) x 10 8 zz rr Ω rr = Ω θθ, Ω zz = 0 Granite Main Material Parameters E 0 (Pa) ν 0 β S 0 (Pa) β0 T (Pa) C 0 (Pa) C 1 (Pa) g M (Pa) g S (Pa) g T (Pa)

21 Triaxial Compression Tests [Halm and Dragon 2002] Mechanical Tests - dry granite 2.5 x p c =0MPa p c =5MPa p c =20MPa σ zz σ rr (Pa) Ω rr ε rr ε zz ε x 10 3 p c = 5 MPa σ σ (Pa) x 10 8 zz rr Ω rr = Ω θθ, Ω zz = 0 Granite Main Material Parameters E 0 (Pa) ν 0 β S 0 (Pa) β0 T (Pa) C 0 (Pa) C 1 (Pa) g M (Pa) g S (Pa) g T (Pa)

22 Triaxial Compression Tests [Halm and Dragon 2002] Mechanical Tests - dry granite x p c =0MPa p c =5MPa p c =20MPa σ zz σ rr (Pa) Ω rr ε rr ε zz ε x 10 3 p c = 20 MPa σ σ (Pa) x 10 8 zz rr Ω rr = Ω θθ, Ω zz = 0 Granite Main Material Parameters E 0 (Pa) ν 0 β S 0 (Pa) β0 T (Pa) C 0 (Pa) C 1 (Pa) g M (Pa) g S (Pa) g T (Pa)

23 Elastic THM Couplings Kamaishi Experimental Site [Rutqvist et al. 2001] depth : 250m T 0 = 12.3 o C granite: S w0 = 1 bentonite: S w0 = heating source at 100 o C during 8.5 months (259 days) 2 relaxation period of 6 months (183 days)

24 Elastic THM Couplings Kamaishi Experimental Site [Rutqvist et al. 2001]

25 Elastic THM Couplings Kamaishi Experimental Site [Rutqvist et al. 2001]

26 Influence of the Initial Damage Field [Rutqvist et al. 2001]

27 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 10 days of heating INITIALLY UNDAMAGED INITIALLY CRACKED

28 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 20 days of heating INITIALLY UNDAMAGED INITIALLY CRACKED

29 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 30 days of heating INITIALLY UNDAMAGED INITIALLY CRACKED

30 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 50 days of heating INITIALLY UNDAMAGED INITIALLY CRACKED

31 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 259 days of heating INITIALLY UNDAMAGED INITIALLY CRACKED

32 Influence of the Initial Damage Field [Rutqvist et al. 2001] After 259 days of heating and 183 days of relaxation INITIALLY UNDAMAGED INITIALLY CRACKED

33 Influence of the Damage Parameters [Pollock 1986] S w0 = 0.15

34 Influence of the Internal Length Parameter [Pollock 1986] K dg 2 ij n frac, Ω rs = π 2/3 γw 12 µw (T ref ) χ4/3 b 2 P 3 k=1 d k 5/3 δ ij n k i n k j K dg ij = K max w, dg δ ij for Ω ij = 0.95δ ij computation of b

35 Summary a tensorial damage variable independent strain state variables (conjugate to net stress, suction and thermal stress) a phenomenological model, based on Continuum Damage Mechanics introduction of internal length parameters in water transfer equations 8 new damage parameters: damage evolution law: C 0, C 1 residual strains: g M, g S, g T internal length parameters: b, b* dilatancy parameter: χ

36 Summary model programmed in Θ-Stock FEM code [Gatmiri and Arson 2008] numerical validation of the mechanical aspects of the model parametric studies justifying the model for THM problems

37 Research Prospects chemical couplings in cracked porous media (corrosion, precipitation, radionuclide transport...) modeling of healing investigation of the potential potential rotation of the principal directions of the damage tensor regularization techniques to study damage localization: micro-structure-enriched model?

38 To my parents, friends and colleagues...