A thermo-hydro-mechanically coupled analysis of clay using a thermo-elasto-viscoplastic model

JHUWS05 A thermo-hydro-mechanically coupled analysis of clay using a thermo-elasto-viscoplastic model by F. Oka, S. Kimoto, Y.-S. Kim, N. Takada Department of Civil & Earth Resources Engineering, Kyoto University and Y. Higo Geo-Research Institute 1

Outline Background, Objective & Scope Thermo-Elasto-Viscoplastic Model for Clay Thermo-Hydro-Mechanically Coupled FEM Numerical Analysis of Thermal Consolidation Instability analysis Conclusion 2

Background THM coupled analysis Environmental, GeoHazard and Energy resources problems Bentonite is used for buffer and backfill material of nuclear waste material (Loret, Heuckel and Gajo, 2004) Hydraulic properties and swelling characteristics of compacted bentonite is temperature dependent. Diffusion of contamination in the ground is dependent on temperature Thermal softening plays a key role in the Dynamic thermo-poro-mechanical analysis of landslides (Vardoulakis, 2002) Thermal consolidation is caused by the pore water pressure development due to volume expansion of soil particles and pore fluid, and thermal viscoplastic softening due to temperature change. (Mitchell et al., 1968, Adachi et al., 1998, Delage, Sultan and Cui, 2000) Methane hydrate has been attracted as a new energy resources Deformation characteristics of geomaterials containing methane hydrate are significantly affected by temperature variation (Kimoto and Oka, 2005). 3

Objective Prediction method of THM coupled behavior of clay considering Objective By Thermo-Hydro-Mechanically coupled FEM using Thermo-Elasto-Viscoplastic model Method Temperature dependency and stain softening as well as strain rate sensitivity of clay Volume expansion of soil and water due to an increase of temperature Temperature changes due to plastic stress power Temperature dependent viscosity of pore water (i.e., temperature dependent permeability) 4

JHUWS05 Background, Objectives & Scope Thermo-Elasto-Viscoplastic Model for Clay Thermo-Hydro-Mechanically Coupled FEM Instability analysis Example: Numerical Analysis of Thermal Consolidation Conclusions 5

Elasto-viscoplastic model considering microstructure changes Overstress type of viscoplasticity (Perzyna 1963; Adachi & Oka 1982,Kimoto 2003) Strain rate sensitivity, creep, stress relaxation, etc. : Viscoplastic stretching tensor : Terzhagi s effective stress : Static yield function : Plastic potential function Overconsolidation boundary surface (Oka 1982) Viscoplastic strain inside overconsolidated region N.C. region : Stress ratio at maximum compression : OC boundary surface :Second invariant of relative stress ratio tensor : Hardening-softening parameter O.C. region Overconsolidation boundary surface 6

Elasto-viscoplastic model considering microstructure changes Cam-clay type of static yield function σ σ ' σ ' 1+ e exp( ε λ κ ( s) ma ( s) 0 vp ' my = σ ' myi kk mai ) :dilatancy coefficient : Hardening-softening parameter Viscoplastic flow rule σ ' mb = σ :Mean effective stress ' ma 1+ e0 exp( ε λ κ vp kk ) :viscoplastic parameter :viscoplastic parameter :dilatancy coefficient :Hardening-softening parameter dependent on microstructure changes 7

Elasto-viscoplastic model considering microstructure changes Material Instability = Degradation of Microstructure Shrinkage of Static yield function & OC boundary surface Hardening-Softening rule (Kimoto, Oka, & Higo 2004) ( s) σ ' ma ( s) 1+ e0 vp ' ' 1 + e0 vp σ ' ' exp( ) σ exp( ) my = σ myi ε kk mb = σ ma ε kk σ ' mai λ κ λ κ Softening rule Softening parameter of yield function & OC boundary surface, :Structural parameters Accumulated viscoplastic strain (Volumetric strain component+ Deviatoric strain component) :rate of viscoplastic strain 8

Consolidation yield stress with higher temperatures and low strain rates is smaller Boudali, Leroueil and Murthy (1994) 9

Temperature-dependent viscoplastic parameter Temperature Dependency of Viscoplastic Parameter (Yashima, Leroueil, Oka and Guntro, 1998) based on the results of CRS (constant rate of strain tests) (Boudali et al. 1994) Viscoplastic parameters of the model : is not dependent on temperature :Strain rate :Consolidation yield stress (Strain rate sensitivity by m is not affected by temperature changes) Viscoplastic parameter may depend on Temperature Relation between logarithm of strain rate & consolidation yield stress (Yashima et al.(1998)) 10

Temperature-dependent viscoplastic parameter ' σ p Relation between Consolidation Yield stress & Temperature based on the results of CRS (Boudali et al. 1994) is equivalent to '( s) σ myi (Yashima, Leroueil, Oka and Guntro, 1998) :Consolidation yield stress :Consolidation yield stress at the referential temperature :Temperature :Referential temperature : Slope of Relation between logarithm of temperature & consolidation yield stress (Yashima, Leroueil, Oka and Guntro, 1998) 11

Temperature-dependent viscoplastic parameter In the one-dimensional consolidation, stress ratio can be assumed to be constant Consolidation yield stress = the initial value of hardening parameter σ σ '( s) ' σ myi p = ' ' mai σ0 :Initial stress :Dilatancy coefficient :Viscoplastic parameter 12

Temperature-dependent elastic stretching Additive decomposition of stretching tensor e vp 1 D ij = D ij + D ij + β θ θδ & : Viscoplastic stretching tensor ij 3 : Temperature volume expansion of soil particle due to temperature change : thermal coefficient of expansion for soil particles Elastic stretching due to stress change : Elastic shear modulus : Deviator stress tensor : Mean effective stress : Void ratio : Swelling index 13

Numerical simulation of CRS tests under different temperatures (Yashima,Leroueil, Oka, Guntoro, 1998, S&F) 14

Numerical simulation of CRS tests under different temperatures (Yashima,Leroueil, Oka, Guntoro, 1998, S&F) 15

Governing equation ~ Equilibrium equations ~ Biot s type of two-phase mixture theory Updated Lagrangian formulation with Jaumann rate of effective Cauchy stress W : spin tensor T : Cauchy stress Equilibrium equations (rate type) :Nominal stress rate (e.g.yatomi et al. 1989) 17

Governing equation ~ Continuity equation ~ Continuity equation : volumetric strain rate for solid sleleton : pore water pressure : unit weight of water : permeability coefficient : porosity : fluid density Compressiveness of pore fluid u w = f f K Dii : bulk modulus for pore fluid : volumetric strain rate for pore fluid Volume expansion of pore fluid due to temperature change Assumption: time derivative of fluid density depends only on temperature : volume of fluid : thermal coefficient of fluid 18

Governing equation ~ Continuity equation ~ Compressiveness of fluid u w = f f K Dii Volume expansion of pore fluid due to temperature change V& f / V f = D f ii Volume change Pore water pressure Temperature Continuity equation considering temperature change 19

Temperature dependency of permeability Temperature dependency of viscosity of water changes coefficient of permeability : Intrinsic permeability : Viscosity of water Experimental formula (Delage et al.1988) µ ( θ ) = 0.00046575ln( θ ) + 0.00239138 : Unit weight of water Assumption: constant (temperature dependency is very small) 20

Governing equation ~ Energy balance equation ~ Energy balance equation : Effective stress tensor : Stretching tensor : Density : Heat flux vector Constitutive equation for ρe& = ρθ c & + T ( D + βθδ) 3 ' e 1 θ & ij ij ij : Rate of internal energy :Temperature : specific heat capacity : elastic stretching tensor Additive decomposition of stretching tensor : Viscoplastic stretching tensor e vp 1 θ Dij = Dij + Dij + β θδ & : thermal coefficient of expansion ij 3 for soil particles 21

Governing equation ~ Energy balance equation ~ Energy balance equation e : Internal energy density : Effective stress tensor : Stretching tensor : Density : Heat flux vector Constitutive equation for ρe& = ρθ c & + T ( D + βθδ) 3 ' e 1 θ & ij ij ij : Rate of internal energy :Temperature : specific heat capacity : elastic stretching tensor Additive decomposition of stretching tensor : Viscoplastic stretching tensor e vp 1 θ Dij = Dij + Dij + β θδ & : thermal coefficient of expansion ij 3 for soil particles 22

Thermo-Elasto-Viscoplastic Coupling Thermo-Elastic Coupling Continuity equation considering temperature change: Volume expansion of pore fluid Temperature dependent elastic stretching: Volume expansion of soil skeleton Energy balance equation: Thermo-Viscoplastic Coupling Plastic stress power causes temperature change Temperature dependent viscoplastic parameter Temperature changes affect mechanical characteristics of clay 23

JHU-WS05 Background, Objectives & Scope Thermo-Elasto-Viscoplastic Model for Clay Thermo-Hydro-Mechanically Coupled FEM Instability analysis Example: Numerical Analysis of Thermal Consolidation Conclusions 24

Thermal consolidation Dissipation of pore water pressure induced by both Heating and Viscoplastic thermal softening Consolidation Temperature increase Temperature decrease Consolidation Swelling Relation between height of specimen & temperature (Campanella & Mitchell 1968) 25

Material parameters NC clay, (Rikanenpyo(J. Chronological table) 2003) 26

Boundary conditions for thermal consolidation Permeable boundary Isothermal boundary (atmospheric temperature: 20 Temperature at heat source Isothermal boundary 60 Pore water & heat are allowed to flow 10 10 = 100elms Impermeable & adiabatic boundaries 2m : Fixed 2m 60 20 10 increase per 1 hour 0 4 hours 148 hours Time : Horizontally fixed ~ Boundary conditions for pore water pressure & temperature ~ ~ increase in temperature at the heat source ~ 27

Settlement time relation at node A 4 hour Temperature 60 0.0005 Displacement (m) 0.0000-0.0005-0.0010-0.0015 1 hour 40 hour 150 hour 20 148 hours 0 4 hours Time Node A 2m -0.0020 48.90 0.0 1.0 2.0 3.0 4.0 Time (s) 5.0 6.0 ( 10 5 ) 2m 37.60 26.30 15.00 1 hour 4 hour 40 hour Distribution of temperature ( ) 150hour 28

48.87 37.58 0 hr 1 hr 4 hr. 10 hr. 40 hr. 80 hr. 150 hr 26.29 15.00 27.28 19.10 0 hr (a) Distribution of temperture ( ) 1 hr 4 hr 10 hr 40 hr 80 hr 150 hr 8.19 0.00 (b) Distribution of excess pore water pressure (kpa) 201.51 198.11 0 hr 1 hr 4 hr 10 hr 40 hr 80 hr 150 hr 193.58 190.19 0.0053 0.0037 0 hr (c) Distribution of mean effective stress (kpa) 1 hr 4 hr 10 hr 40 hr 80 hr 150 hr 0.0016 0.0000 Fig.7 (d) Distribution of viscoplastic volumetric strain Distribution of Temp., PW, Mean Effective Stress, viscoplastic volumetric strain 29

Conclusions We have derived governing equations of thermo-elasto-viscoplastic water-saturated clay by considering thermal viscoplastic softening, volume expansion of soil skeleton and pore water and the temperature dependence of permeability We have proposed a thermo-hydro-mechanically coupled FEM with a thermo-elasto-viscoplastic constitutive model using the finite deformation theory The proposed method is applicable for simulating the thermal consolidation phenomena of clay From the linear instability analysis, strain softening and temperature softening are main causes of the instability 30

JHU-WS05 Preliminary stability analysis 31

One-dimensional governing equations x ' σ u + w = 0 x σ = Hε + µε& ' vp vp & µ = αθθ ( )& & kθ θ = θ, xx + σε& ρc ρc 1 ' vp & k u u & ε vp w = wxx, α w θ α k w γ w & H: strain hardening softening parameter α : temperature dependent viscous softening parameter 32

Thermal softening of viscosity & µ = αθ & 1 µ = µ = aθ b C( θ ) % µ = abθ ( b+ 1) θ% = αθ% α : temperature dependent viscous softening parameter 33

Linear instability analysis Perturbation of the strain, the temperature and the pore water pressure in periodic form as: vp ε = ε vp* exp( iqx + ωt) θ = θ * exp( iqx + ωt) u = u * exp( iqx+ωt) w w ω : growth rate of fluctuation q : wave number Fluctuation of viscosity % µ = αθ % 34

Governing equation in perturbed state vp *, H, 1 θ 0 αε & µω + kθ 2 1 q ' ω+, σω, 0 vp* = ρc ρc ε 0 kw 2 αω w, ω, q αω k * γ w u w 0 [ A]{ X } = {0} 35

Characteristics equation vp αε &, µω + H, 1 k 1 = ω+ σω ρc ρc k,, q θ 2 ' [ A] q,, 0 w 2 αwω ω αkω γ w * θ { X} = vp* ε * uw [ A]{ X } = {0} ω is obtained by the characteristic equation as: det [ A ] = 0 36

Stability Characteristics equation k k ασ σ µα ω + + µα + α αε + α ω + ' ' 3 w 2 θ 2 vp k 2 k { q k q kh & 1 w } γw ρc ρc ρc k k k k k σ k k { µ α α αε } ω ρ γ γ ρ ρ ρ γ ρ ' θ 2 w 2 w 2 θ 2 θ 2 vp w 2 θ 2 q q + H q + kh q + kh q & q + q c w w c c c w c k k H q q ρc γ θ 2 w 2 + = w 0 3 + a 2 + a + a = 0 1 2 3 ω ω ω 37

Instability 3 + a 2 + a + a = 0 1 2 3 ω ω ω ' ' 1 k { w 2 kθ 2 vp αkσ σ µαk αk αε& 1 α w } k w c c c a1 = q + q + H + µα γ ρ ρ ρ ' 1 k { θ 2 kw 2 kw 2 kθ 2 kθ 2 vp σ kw 2 kθ 2 µ αk αk αε& } k c w w c c c w c a2 = q q + H q + H q + H q q + q µα ρ γ γ ρ ρ ρ γ ρ ω kθ k a3 = H q q ρc γ 2 w 2 w From Routh-Hurwiz s theorem the negative real parts exist if and only if a > 0, a > 0, a > 0 1 2 3 and a a a > 0 1 2 3 38

Instability ω > 0 On the contrary, from Routh-Hurwitz s criteria the positive real parts may exist if a or 1 < 0 a 2 < 0 or a 3 < 0 or aa a < 1 2 3 0 These conditions are possible if H is negative, namely strain softening, and/or the thermal softening parameter is large enough σ = Hε + µε& ' vp vp µ θ & α b = a µ αθ = & 39