Implementation of an Isotropic Elastic-Viscoplastic Model for Soft Soils using COMSOL Multiphysics

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1 Imlementation of an Isotroic Elastic-Viscolastic Model for Soft Soils using COMSOL Multihysics M. Olsson 1,, T. Wood 1,, C. Alén 1 1 Division of GeoEngineering, Chalmers University of Technology, Gothenburg, Sweden NCC Construction Sverige AB, Gothenburg, Sweden Corresonding author: Sven Hultins gata 8, SE-41 96, Gothenburg, Sweden, mats.olsson@chalmers.se Abstract: In this aer a elastic-viscolastic (cree) model is imlemented in COMSOL v.4.a and v.4.3 and behmarked against another commercial finite element software ackage with a very similar material model. It is also validated against commonly erformed laboratory tests such as Constant Rate of Strain oedometer tests (CRS) and K 0 -Consolidated Undrained triaxial tests (K 0 CU). The imlementation in COMSOL is conducted by using a fully couled analysis between the Solid Mechanics node, Darcy s Law node and a distributed ODE node. The ODE node is used to simulate the cree contribution. The model imlemented rovides a material model that works as anticiated and can cature many imortant features of soft soil behaviour. Keywords: Soft soil, cree, elastic-viscolastic model, imlementation, behmark. 1. Introduction Time-deendent behaviour in clay constitutes an engineering challenge in road design and construction in areas with dee deosits of soft clay. Soil imrovement and construction of building foundations or embankments can be quite comlicated and exensive in such areas. Construction costs need to be balaed against high maintenae costs. In order to do this otimally, there is a need to redict long-term settlement with a higher degree of accuracy. To be able to redict long-term settlements of building foundations or embankments constructed on soft soils it is necessary to ilude the effect of cree. In this aer a elastic-viscolastic (EVP) model is imlemented in COMSOL. The imlemented material model EVP is behmarked against another commercial finite element software ackage, Plaxis BV, with a very similar material model called the Soft Soil Cree model (SSC). It is also validated against commonly erformed laboratory tests such as Constant Rate of Strain oedometer tests (CRS) and K 0 -Consolidated Undrained triaxial tests (K 0 CU).. The isotroic elastic-viscolastic model The imlemented material model in COMSOL is based on a material model resented by [1]. Some basic characteristics of the EVP model are: Stress-deendent stiffness (logarithmic comression behaviour) Distition between rimary loading and unloading-reloading Cree behaviour Memory of reconsolidation ressure Failure behaviour according to Matsuoka- Nakai (EMN) criterion The EVP model is, like the SSC model, based on the modified cam-clay tye ellises. The wellknown stress invariants for the mean effective stress, ', and deviatoric stress, q, are adoted, []. These stress invariants are used to define the size of the ellise and stress-strain relation, see. (1)-(4) below. The EVP model is an elastic-viscolastic model, formulated as a relationshi between stress rates and total strain rates. Total strain rates are decomosed into elastic strain rates and cree strain rates as formulated in. (1). r s e c D (1) r s In. (1) 1 D is Hooke s law of isotroic elasticity with a linear stress-deendent stiffness where the Youngs modulus for unloading/reloading, E ur, is defined as: E ur 31 ur () Excert from the Proceedings of the 01 COMSOL Conferee in Milan

2 and: (3) Figure 1 suggests that tensile stresses are ossible but this could be revented by using a tension cut-off otion. The critical state line M cs is calculated according to. (5), see []. This gives the shae of the ca as shown in Figure 1. q M cs is the uivalent isotroic effective stress based on the Modified Cam-Clay yield fution. The material model comrises of the following arameters: Modified swelling index Modified comression index rs cree number ur Poission s ratio for unloading/reloading = Referee time (normally set to 1 day) = Isotroic reconsolidation stress (4) M 3 1 K 0 1 K 0 1 K 1 ur 0 1K0 1ur 1 K0 1ur 1 (5) In Figure 1 the soil arameter M CS and M MC are shown and reresent the so-called critical state line and the Mohr-Coulomb failure line. In this imlementation the Mohr-Coulomb failure criteria has been substituted with the extended Matsuoka-Nakai (EMN) failure criterion, see [8]. This has been done to get a smooth failure surface and also that the EMN also uses the intermediate stress unlike the Mohr-Coulomb failure criteria. The imlemeted failure criteria is Lode angle deendent, see [3]. The arameters needed to describe the failure criteria is the friction angle (φ), dilatay angle (ψ) and effective cohesion (c ). The failure surface and referee surface (ca) are visualized in riial stress sace in Figure. Figure. The failure surface (green) and referee surface (red) visualized in riial stress sace. The evolution of the reconsolidation stress,, is controlled by the volumetric cree strains: c v 0 ex (6) Figure 1. Diagram of ellise in a -q lane, [4]. Where r s c 1 v (7) r s Excert from the Proceedings of the 01 COMSOL Conferee in Milan

3 This exression defines the time-deendent cree behaviour and imlies that the isotroic overconsolidation ratio (OCR = ) has a considerable influee on the cree rate. If the stress remains uhanged the cree rocess continues and the reconsolidation stress kees on ireasing, but at a decresing rate. For more details on the SSC model see i.e. [1,, 5, 7] Table. Inut for the failure criterion. φ (deg) ψ (deg) c (kpa) Imlementation in COMSOL Sie COMSOL has the linear elastic model built in, with the otion of isotroic, orthotroic and full anisotroy, the only uation that needs to be imlemented in COMSOL is the second art of. (1) i.e. the cree art. This has been imlemented in Comsol with the use of a distributed ODE node. The uations,.(3)-(6), in addition to derivitives are defined under local varibles. A Darcy law node is added to coule the ore ressure resonse that is derived from the volumtric strains or used to set boundary conditions. The inut arameters are given as constants or as user defined futions that could deend on stress or strain for examle. This otion gives the user total freedom, e.g. over how the reconsolidation stress changes with deth. Figure 3. -q stress lot comarison between EVPmodel in COMSOL and SSC model in Plaxis. 4. Behmark To validate the imlementation of the EVP model a behmark with the SSC model has been done. The behmark consists of two K 0 CU 1 comression tests, ositive in deviatoric stress, and two K 0 CU extension tests, negative in deviatoric stress. Inut arameters that have been used are resented in Table 1 and Table. Table 1. Inut arameters for behmark simulations. κ λ r s υ ur OCR v K 0 K K 0 CU = K 0 -Consolidated triaxial test with secified vertical and horisontal stress and then sheared undrained. Figure 4. Stress-strain lot comarison between EVPmodel in COMSOL and SSC model in Plaxis. Excert from the Proceedings of the 01 COMSOL Conferee in Milan

4 4.1 Discussion As shown in Figure 3 and Figure 4 the imlemented material model shows very similar results to the SSC model in Plaxis, as exected. The small differees that could be seen in Figure 4 could be one of many reasons e.g. the imlementation is not identical, the solver is different etc. 5. Validation of model In this chater a comarison with some common laboratory tests will be conducted and some other imortant features will be shown. The laboratory tests are conducted on soft clay from a site located just north of Gothenburg, Sweden. The comarison comrises of four K 0 CU triaxial tests, one comression and one extension test for clay from 1m and 18m deth resectively, and a constant rate of strain test (CRS) from 4 m deth. The simulations are conducted by using axi-symmetry which reduces the number of nodes of the model tremendously. The soil samles are modeled with a height of 100 mm for the triaxial test and 0 mm for the CRS test. The diameter is the same for both tests and is 50 mm. Figure 5. -q stress lot comarison between EVPmodel in COMSOL and triaxial laboratory tests. 5.1 Comarision with laboratory tests The evaluated arameters used as inut in the EVP model for the triaxial test simulations are summarized in Table 3 and Table 4. The vertical stress was set to 7 kpa and 108 kpa for the 1m and 18 m soil samle resectively. Table 3. Inut arameters for triaxial laboratory simulations. κ λ r s υ ur OCR v K 0 K Table 4. Inut for the failure criterion. φ (deg) ψ (deg) c (kpa) The results from the EVP model comared with the triaxaial tests are resented in Figure 5 and Figure 6. Figure 6. Stress-strain lot comarison between EVPmodel in COMSOL and triaxial laboratory tests. Excert from the Proceedings of the 01 COMSOL Conferee in Milan

5 The evaluated arameters used as inut in the EVP model for the CRS test simulations are summarized in Table 5 and the comarision between the EVP model and the CRS laboratory test is resented in Figure 7. Table 5. Inut arameters for CRS laboratory simulations. κ λ r s υ ur OCR v K 0 K Figure 8. Effect of different strain rates lotted in a -q stress lot. Figure 7. Stress-strain lot comarision between EVPmodel in COMSOL and CRS laboratory test from 4 m deth. 5. Effect of strain rate To demonstrate some of the caabilites of the model a K 0 CU comression test is conducted with three different strain rates, namely 0.1, 0.01 and mm/hr. The inut arameters are the same as for the behmark above with the only differee that the isotroic reconsolidation stress is set to 90 kpa. The result are resented in Figure 8 and Figure 9. Figure 9. Effect of different strain rates lotted in stress-strain lot for samle of h=100 mm and d=50 mm. 5.3 Discussion The validation of the model shows good agreement for the triaxial comression tests but a very oor agreement for the triaxial extension tests. The latter is as exected as the model has an isotroic formulation of the referee surface (ca). The same alys to SCC model used in the behmarking. For the simulation of the CRS test there is a good agreement to about 14% strain where the simulation starts to diverge from the laboratory Excert from the Proceedings of the 01 COMSOL Conferee in Milan

6 test curve. This is also as exected due to the formulation that originates from the Modified Cam-Clay, see e.g. [6]. 6. Colusions This aer describes the imlementation of an isotroic Elastic Visco-Plastic (EVP) material model for soft soils in COMSOL. The behmark against a very similar material model (SCC) showed good agreement. Even so, one could argue that the EVP model needs to be behmarked against real tests and not just other material models. The imlemtation in COMSOL seems however to be accetable. The validation of the model by comarison with laboratory tests showed somewhat varying results. There was a very good agreement for the develoment of stresses in the comression triaxial test, even if the decrease of the deviatoric stress beyond the eak stress was somewhat underestimated. Also for the CRS test the agreement was very good. The overestimation of strains >14% would be of less imortae in real design as such large strains robably would not be accetable. Even so imortant features of soft soil behaviour such as strain rates effects can be catured. However, for the extension triaxial tests the agreement was far from satisfactory. The eak strength was highly overestimated. Nor does the model cature the viscous behavior in the extension tests. But, this is as anticiated. The alied model has, as said reviously, an isotroic formulation of the referee surface, and it is a well-known fact that such models have roblems of caturing the behavior of extension tests. Thus, the oor agreement is a result of the draw-backs of the EVP model and not the imlementation in COMSOL. As a consuee, future develoment an anisotroic formulation of the referee ca has to be iororated in the EVP-model. Using COMSOL for such a develoment would be a great advantage as it offers a user friendly interface for adding comlexity to a model in a ste-by-ste manner without the need of any in deth skills of rogramming language. 7. Referees 1. P. A. Vermeer and H. Neher, Beyond 000 in Comutational Geotechnics - A soft soil model that accounts for cree, A.A. Balkema, Rotterdam (1999). R. B. J. Brinkgreve, W. M. Swolfs and E. Engin, PLAXIS Manual D 011, Netherlands (011) 3. W. F. Chen and E. Mizuno, Nonlinear analysis in soil mechanics, Elsevier Sciee, Amsterdam (1990) 4. S. Satibi, Numerical Analysis and Design Criteria of Embankments on Floating Piles, Phd- Thesis, University of Stuttgart (009) 5. P. A. Vermeer, D. F. E. Stolle and P. G. Bonnier, From the classical theory of secondary comression to modern cree analysis, Proc. Comuter Methods and advaes in Geomechanics, (1998) 6. D. M. Wood, Soil behaviour and critical state soil mechanics, Cambridge University Press, Cambridge (1990) 7. D. F. E. Stolle, P. A. Vermeer and P. G. Bonnier, Time integration of a constitutive law for soft clays, Communications in Numerical Methods in Engineering, 15, (1999) 8. D. V. Griffiths and J. Huang, Observations on the extended Matsuoka Nakai failure criterion, International Journal for Numerical and Analytical Methods in Geomechanics, 33, (009) Excert from the Proceedings of the 01 COMSOL Conferee in Milan