Report. Unified enhanced soft clay creep model user manual and documentation PIAG_GA_ R4. For. by: Gustav Grimstad. (sign.

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1 PIAG_GA_ R4 Date: Rev. Date: Rev. 2 Date: Report Unified enhanced soft clay creep model user manual and documentation For by: Gustav Grimstad (sign.) Project manager: Gustav Grimstad (sign.)

2 GG JR Page: of 22 Summary This report gives a short description of the unified enhanced model for creep in soft clays developed under the framework of the CREEP project. The report contains examples of use and a user manual for the model. The model is a synthesis of the different models developed at the various partners of the project in the past. Much of this development has basis in the S CLAYS line of models.

3 GG JR Page: 2 of 22 Content Summary... Background Definitions Engineering parameters and current practice Time resistance concept for volumetric creep in oedometer condition Extending the D creep model to general stress/strain space Anisotropy and structure Potential surface and rotational hardening Elastic properties Implementation User manual Parameters Examples Benchmarking Bibliography... 2

4 GG JR Page: 3 of 22 Background Under the EU CREEP project Creep of geomaterials an enhanced unified soil model for creep in soft clay was formulated and implemented as a user defined material model, to be used in FEA. Creep in clay is a well described phenomenon. This is especially the case for volumetric creep. Šuklje (957) described volumetric creep in clay using the isotache framework. This form the basis for the creep description in many recently developed advanced constitutive models for clay. From the different partners of the CREEP project the CREEP SCLAY model see e.g. Sivasithamparam et al. (205), the EVP SCLAY S model described in e.g. Yin and Karstunen (20) and the n SAC model (Grimstad and Degago 200) have been used as a basis for the development of a unified creep model for soft clays. The differences in the above mentioned models come from the underlying elastoplastic model, as all essentially have a similar extension for including creep. The underlying model of CREEP SCLAY model is a model with a rotated and distorted ellipsoid as yield surface (i.e. the ACCM surface after Dafalias (986)). The hardening rules are adopted from Wheeler et al. (2003). The EVP SCLAYS model has on extra state parameter describing structure with a destructuration rule from Gens and Nova (993). The n SAC model is different from the extended SCLAY models in the flow rule. The n SAC model uses a non associated flow rule in a similar manner as the SANICLAY model from (Dafalias et al. 2006). The destructuration rule and kinematic hardening rule (rotational hardening rule) in the n SAC model have the same form as the SCLAYS models. In the EU CREEP project a unified model for creep in clay was developed. The model is a synthesis of these models into a single model.

5 GG JR Page: 4 of 22 2 Definitions Stress vector: σ () Mean effective stress: p Deviatoric stress vector: σ d p 22 p 33 p (2) (3) Potential rotational vector: α d (4) Reference surface rotational vector: β d (5)

6 GG JR Page: 5 of 22 3 Engineering parameters and current practice Figure gives a graphical description of the commonly used clay parameters for compressibility and creep. The interrelations and descriptions are given in Table. The main differences comes from whether the parameters are determined based on void ratio or strain and if mean stress or vertical stress are used as stress variable. In normally consolidated range, the interrelationship between the parameters is exact when the compressibility parameters are assumed to be constants, i.e. independent of stress/deformation level. Normally the change in void ration, e, is small such that it can be replaced by the initial value. For large volumetric deformation the assumption of small strain is not valid and the concept should be modified to ensure positive void ratio for compression. However, for practical engineering cases in soft clay, the volumetric strains are small enough to not consider limiting the reduction in void ratio. For some soft organic materials like peat, this effect might be more important. Figure Graphical view of conventional parameter determination following international practice Table Soil parameters for creep and compressibility commonly used in engineering practice 3. Time resistance concept for volumetric creep in oedometer condition Generally, resistance is defined as: Resistance = Cause/Effect (Janbu 969). Hence, time resistance is defined as in Equation (6). This description is adopted from Grimstad et al. (200).

7 GG JR Page: 6 of 22 R increment in time dt increment in strain d (6) Figure 2 shows determination of the time resistance, R, and time resistance number, r s, for an idealized incremental oedometer test, as illustrated by Janbu (969). The time resistance number could be found by numerical differentiation, following the procedure of Equation (7). t t d trref a dr a r s dt dt t (7) where ε a is the axial strain in the oedometer test and R ref is the time resistance at a certain reference time, τ. In similar manner (Janbu 963) defined the stiffness number, m, as shown in Equation (8). a d a deoed m d d a a (8) where σ a is the effective axial stress and E oed is the oedometer stiffness. In some cases the secant stiffness number has greater importance, since it can be linked to the compressibility parameter λ*. The secant stiffness number is defined as: m sec a ref a Eoed p * a ref (9) where E oed ref is the oedometer stiffness at the chosen reference stress, p ref (= 00 kpa) and *, is the compressibility in vertical strain ln(p ) space. Rearranging Equation (7) into the differential Equation (0) (valid for constant effective stress) and solving the differential equation from t = τ to t gives us Equation (). dt trst Rref rst d vp dv vp t v ln dt r t r s s (0) () The change in the equivalent one dimensional pre consolidation pressure, p c, due to change in volume strain can be found from integration of the hardening rule. Integration from the constant equivalent effective stress p eq, for which under the creep is acting at t = τ, to the stress (p c,ref which corresponds to the same increase in strain due to time, t, is given in equation (2).

8 GG JR Page: 7 of 22 dp, c p p c vp cref ln vp v eq dv p (2) Where: ζ is a parameter related to irrecoverable compressibility (ζ = λ* κ*) p c,ref is the effective D reference pre consolidation stress for corresponding R ref κ* is the modified swelling index Combining equations (0), () and (2) gives us Equation (3), where we have eliminated time from the equations. This D expression, so far valid for oedometer condition, gives us the volume/axial strain rate as a function of stress and reference pre consolidation pressure. The formulation used in the Soft Soil Creep (SSC) model, found in the finite element program PLAXIS (Stolle et. al. 999b) is identical to the time resistance concept, by assuming constant r s an ζ with stress level and in time. vp v * * r eq s eq * eq p * p vp p v, ref ref c, ref c, ref c, ref R p p p (3) β is here defined as the creep ratio. Equation (3) rewrites to Equation (4) to express apparent preconsolidation pressure, p c, as a function of strain rate. vp v pc p c, ref vp vref, (4) t R R = t/ ε Pure creep r s Figure 2 Graphical determination of time resistance number for oedometer tests t

9 GG JR Page: 8 of Extending the D creep model to general stress/strain space Researchers like Yin and Graham (999), Leoni et al. (2008) and Stolle et al. (999) assumed that eq. (3) was valid for any stress state, meaning that they assumed positive volumetric creep regardless of state. As a consequence, the following expression for general creep strains, ε ij vp, is found (eq. (5)): vp ij vp Q v Q ij p (5) Where Q is the plastic potential surface. Since, according to eq. (3), ε v vp always is positive such a formulation will be limited to contractive behavior and when approaching critical state, mean stress will go towards zero (instability). Yin et al. (2002) modified eq. (5) to eq. (6). vp ij vp Q v Q ij p (6) This improved the model by allowing for simulation of the dry side. However, the formulation would still cause instability when approaching critical state. Grimstad et al. (2008) suggested using the oedometer loading condition (loading with K 0 NC ) as the reference state (eq. (7)). vp ij vp Q v Q oed ij p NC K0 (7) Which using the AMCCM as potential surface rewrites: vp ij vp Q Mf v oed M 2 2 K0NC 2 2 ij f K 0NC (8) Where η K0NC is the q/p ratio in oedometer loading and α K0NC is the steady state rotation of the potential surface under oedometer loading. M f defines the critical state line in p q space for compression loading (TXC state). Figure 3 shows the consequence on the viscoplastic multiplier for different choices of formulations. As seen in the figure eq. (5) leads to a situation with no dry side and no critical state. Eq. (6) leads to a situation with two solutions (one going below and one going above the critical state line). It is clear that both these options lead to numerical instabilities close to the critical state line. On the other hand, eq. (8) is a consistent formulation without any instability and has possibility for reaching critical state.

10 GG JR Page: 9 of 22 Leoni et. al. (2008) Yin et al. (2002) Grimstad et al. (200) q/p 0 0 q/p 0 0 q/p p/p p/p p/p 0 Figure 3 Curves in normalized p q space of constant dλ/dt for eq. (5), (6) and (8) using AMCCM as reference surface 3.3 Anisotropy and structure Anisotropy (rotation of reference surface) is introduced, as mentioned above, from the formulation in AMCCM (Dafalias 986). Evolving anisotropy (stain/stress induced anisotropy) is introduced by including a kinematic rotational rule for the reference surface. This is done in addition to the isotropic hardening rule which is previously defined by eq. (2). The reference surface is given by an equivalent stress measure, calculated by equation (9). T eq 3 σd σd p p βd β dg (9) 2 p p where: p is the mean stress, σ d is the deviatoric stress vector, β d is the deviatoric rotational vector and: g 2 3 T M βd β 2 d (20) Where M defines peak in p q space which again is dependent on the modified Lode angle, θ β. Figure 4 shows how the surface looks like in principal stress space. The model used a modified form of the Lade Duncan dependency (Lade and Duncan 975) the exact equation is not presented here.

11 GG JR Page: 0 of 22 Figure 4 Reference surface in principal stress space showing an undrained triaxial compression stress path. The effect of structure is introduced to the model by a state variable, χ. Equation (3) can now be modified to an expression that gives the plastic multiplier, dλ/dt: rsi i eq 2 2 p M f K0NC rsi i 2 2 si mi OCRmax f K 0NC d dt r p M (2) where the index i is introduced as reference is made to an intrinsic value, meaning that one should use the remolded parameters. For χ = 0, p c,ref is equal to p mi consistent to previous notation of S CLAYS. Eq. (22) gives the isotropic hardening rule for the intrinsic reference stress. dpmi Q pmi d p i (22) The destructuration rule is given by eq. (23). The destructuration rule involves both the viscoplastic volumetric and shear strain components. d Q 2 2 Q Q av d p 3σd σd 2 T (23) where a v is a destructuration parameter and ω gives the relative portion of destructuration coming from shearing.

12 GG JR Page: of Potential surface and rotational hardening The potential surface has a similar formulation as eq. (9): T 3 σ d σd eq Qp αd α dgf p (24) 2 p p Where α d is the deviatoric rotational vector and g f (θ α ) is a load angle dependent function that defines critical state criterion in stress space. A non associated flow rule is activated when the state parameters for rotation α d is not equal to β d and g f (θ α ) is different to g(θ β ). The rotational hardening functions are as follows: For reference surface: dβd p d p β β (25) mi d,t d 2 η 3 K0NC MTXC β d,t M tanh atanh (26) 3 2 M TXC K 0NC M Where η and are defined as: σd η p 3 T ηη 2 For reference potential surface: dαd p d p α α (27) mi d,t d 2 η 3 M K0NC ftxc, α d,t M tanh atanh (28) 3 2M ftxc, K0NC Mf Where μ is a parameter that determines how fast the surfaces will rotate towards their steady values, α K0NC and β K0NC are the steady values obtained in a K 0NC condition.

13 GG JR Page: 2 of Elastic properties The isotropic elastic stiffness matrix is made stress dependent trough the following relation: D D2 D D2 D D D 2 D2 D e dσp dε (29) D D D3 Where: 4 2 D, D * * 2, D * * 3 * 3g 3g g (30) The parameters κ * and g * are related to the volumetric and deviatoric stiffness respectively and are non dimensional.

14 GG JR Page: 3 of 22 4 Implementation The model is implemented into PLAXIS using an implicit implementation scheme following the principles given in (Grimstad and Degago 200). The user defined subroutine takes in strain and time increments and gives out the stresses. The state of the soil is defined in part by the stress state, but also by a set of state parameters. In total there are 2 state variables, and 2 equations providing the constitutive behavior. These variables and equations are contained in two vectors called v and r, respectively: σ pmi α d v, βd Q σ Dε σ dpmi pmi d dαd αd r d dβd β d d d t dt d d (3) The infinitesimal operator d has been replaced by Δ to indicate that the first derivatives have been replaced by finite difference approximations. This system of equations r is solved by the Newton Raphson method (figure 5) until a sufficiently small tolerance is obtained. In boundary value problems, convergence issues are marked in two ways. If a stress point does not converge to a desired residual of e 2 within 5 iterations, then it is marked with white color. If a more serious error should occur, such as the inability to provide a new stress state regardless of accuracy, the stress point is marked with red color. Convergence is typically obtained within one to two iterations. Figure 5 Illustration of the Newton Raphson method. The derivative of the functions are used to predict better solutions in an iteratively manner.

15 GG JR Page: 4 of 22 5 User manual 5. Installation For recent versions of PLAXIS the two DLL files, usdm_creep.dll and udsm_creep64.dll, can be copied to the udsm folder within the main PLAXIS folder. The model can then be accessed by selecting userdefined when a new material is created. 5.2 Parameters In total there are 5 input parameters which must be provided. Some of them may be set to zero to simplify the constitutive behavior. If the viscous component of strain is to be disregarded (i.e. only elastic plastic behavior) a high value of r si can be given. Table 2 Input parameters Symbol Units Description φ cs The internal critical state friction angle in triaxial compression NC K 0 The asymptotic value of horizontal stress over vertical stress in an oedometer condition λ * i Intrinsic volumetric compressibility parameter r si The intrinsic creep number Χ 0 Initial value of structure φ p Mobilized internal frictional angle at peak deviatoric stress κ * Elastic volumetric compressibility parameter g * Elastic deviatoric compressibility parameter a v Rate of destructuration ω Relative contribution to destructuration from plastic shear strains OCR Initial over consolidation ratio μ The rate of rotation of reference and potential surfaces Β K0NC Initial rotation of reference surface τ day(s) Reference time OCR max Limit for creep induced OCR 5.3 Initialization The initialization of the model is done in a separate phase by PLAXIS at every stress point, where the initial K 0 stress state together with the some of the input parameters defines the initial state.

16 GG JR Page: 5 of 22 6 Examples Benchmarking A test case is performed to verify the model formulation, figure 6. The crust and fill are modeled using the built in Mohr Coulomb model in PLAXIS with the following input parameters: Table 3 Material properties of the fill and dry crust Parameter Fill material Dry crust (upper layer) Type of behavior Drained Drained Soil weight 2 kn/m 3 8 kn/m 3 Permeability m/day 0.0 m/day Young s modulus, E kpa 5000 kpa Poisson s ratio, v Cohesion, c 0 kpa kpa Friction angle Dilatancy angle 0 0 Coeff. of earth pressure Default Default Figure 6 Calculation example.

17 GG JR Page: 6 of 22 For the soft clay the following input parameters are used (Table 4): Table 4 Input parameters used for the example test case Symbol Units Value φ cs 32 NC K λ * i 0.06 r si 500 Χ 0 9 φ p 25 κ * 0.0 g * a v 20 ω 0.3 OCR.6 μ 40 Β K0NC 0.25 τ day(s) OCR max.7 Figure 7 Undrained triaxial compression test (q Ɛ ) Permeability: 0 4 m/day Soil weight: 20 kn/m 3 Initial K 0 : 0.55 Figure 7 and Figure 8 show results of a simulated undrained trixial compression test starting from the K 0 condition. Figure 8 Undrained triaxial compression test (q p).

18 GG JR Page: 7 of 22 The analysis considers large displacement and updated water pressures (update mesh and water pressures). Table 5 gives the phases and durations. Table 5 The phases which have been simulated Phase Duration (days) Initialization First part of fill 0 Consolidation phase 00 Second part of fill 0 Consolidation phase The following results are obtained (see figure text for explanations to each plot): Figure 9 Deformation contours after first fill

19 GG JR Page: 8 of 22 Figure 0 Excess pore pressure after first fill Figure Deformation contours after second fill

20 GG JR Page: 9 of Chart 6 N68( A) u y [m] Time [day] Figure 2 Time deformation curve for point C -0 Chart -20 S5595(K) -30 s yy [kn/m²] Time [day] Figure 3 Vertical effective stress versus time for point C

21 GG JR Page: 20 of 22 Figure 4 Deformation contours after 0 20 days

22 GG JR Page: 2 of 22 7 Bibliography Y. F. Dafalias, An anisotropic critical state soil plasticity model. Mechanics Research Communications 986; 3: DOI: 643(86) Y. F. Dafalias, M. T. Manzari, A. G. Papadimitriou, SANICLAY: simple anisotropic clay plasticity model. International Journal for Numerical and Analytical Methods in Geomechanics 2006; 30: DOI: 0.002/nag.524 A. Gens, R. Nova, Conceptual bases for a constitutive model for bonded soils and weak rocks. Geotech Eng of hard soils soft rocks., 993. G. Grimstad, S. A. Degago, A non associated creep model for structured anisotropic clay (n SAC) Numerical Methods in Geotechnical Engineering, 200. G. Grimstad, S. A. Degago, S. Nordal, M. Karstunen, in Nordisk Geoteknikermøte nr.5, Proceedings, (Norsk Geoteknisk Forening og Tekna, 2008), pp G. Grimstad, S. A. Degago, S. Nordal, M. Karstunen, Modeling creep and rate effects in structured anisotropic soft clays. Acta Geotech. 200; 5: DOI: DOI 0.007/s y N. Janbu, Soil compressibility as determined by oedometer and triaxial tests. Proc. ECSMFE Wiesbaden 963; : N. Janbu, The resistance concept applied to deformations of soils 7th International Conference Soil Mechanics Foundation Engineering, Mexico city, 969. P. V. Lade, J. M. Duncan, Elastoplastic stress strain theory for cohesionless soil. ASCE Journal of the Geotechnical Engineering Division 975; 0: M. Leoni, M. Karstunen, P. A. Vermeer, Anisotropic creep model for soft soils. Géotechnique 2008; 58: N. Sivasithamparam, M. Karstunen, P. Bonnier, Modelling creep behaviour of anisotropic soft soils. Computers and Geotechnics 205; 69: DOI: D. F. E. Stolle, P. A. Vermeer, P. G. Bonnier, A consolidation model for a creeping clay. Canadian Geotechnical Journal 999; 36: DOI: 0.39/t L. Šuklje, The analysis of the consolidation process by the Isotaches method 4th Int. Conf. Soil Mech. Found. Engng, London, 957. S. J. Wheeler, A. Näätänen, M. Karstunen, M. Lojander, An anisotropic elastoplastic model for soft clays. Canadian Geotechnical Journal 2003; 40: DOI: 0.39/t02 9 J. H. Yin, J. Graham, Elastic viscoplastic modelling of the time dependent stress strain behaviour of soils. Canadian Geotechnical Journal 999; 36: DOI: 0.39/t J. H. Yin, J. G. Zhu, J. Graham, A new elastic viscoplastic model for time dependent behaviour of normally and overconsolidated clays: theory and verification. Canadian Geotechnical Journal 2002; 39: DOI: 0.39/t0 074 Z. Y. Yin, M. Karstunen, Modelling strain rate dependency of natural soft clays combined with anisotropy and destructuration. Acta Mechanica Solida Sinica 20; 24: DOI: 966()

23 page: 22 av 22 Document information Document title Enhanced clay model user manual Document type X Report Technical note Distribution X Public Limited Document no: PIAG_GA_ R4 Date: Rev.no. None Client EU CREEP project Keywords creep, clay Place: Country, province Norway Municipality Trondheim Location NTNU Map sheet UTM coordinates Document control Quality control after own QC system Rev. Rev. on basis of Self check: Internal control Independent control: 0 Original document GG JR Updated description and references GG JR 2 Updated results of analysis due to bugfix GG JR Document approved for publishing GG Date Sign. PM. GG