Evaluation of Plasticity Models Ability to Analyze Typical Earth Dams Soil Materials

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1 Geotech Geol Eng (2009) 27:71 80 DOI /s ORIGINAL PAPER Evaluation of Plasticity Models Ability to Analyze Typical Earth Dams Soil Materials C. J. Loupasakis Æ B. G. Christaras Æ G. Ch. Dimopoulos Æ T. N. Hatzigogos Received: 19 April 2006 / Accepted: 20 April 2008 / Published online: 6 May 2008 Ó Springer Science+Business Media B.V Abstract The selection of the constitutive laws reproducing the response of geomaterials usually becomes a very complicated procedure; especially during the analysis of geostructures containing coarse grained materials. The objective of this study, is the evaluation of models based on the perfect and hardening plasticity theory, regarding their ability to analyze the behaviour of soil materials, used for the construction of earth dams. The evaluated constitutive laws are the elastic perfect plastic model based on the Mohr Coulomb failure criterion and the isotropic hardening CAP model based on the same criterion. The data used for the evaluation of the models, originate from the body of the Sfikia earth dam. This dam is considered suitable, because of the specialized laboratory tests available data, regarding the materials of every zone. The availability of those data, as well as the variety of the construction geomaterials, allows the deduction of generalized conclusions, concerning the ability of the C. J. Loupasakis (&) Engineering Geology Department, Institute of Geology and Mineral Exploration of Greece, Messogion Avenue 70, Athens 11527, Greece cloupas@igme.gr B. G. Christaras G. Ch. Dimopoulos Department of Geology, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece T. N. Hatzigogos Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece specific models to simulate a variety of similar earth constructions. Keywords Clay soils Coarse-grained soils Elastoplasticity Constitutive models Numerical analysis Earth dams 1 Introduction The estimation of the mechanical parameters and the selection of the constitutive laws that reproduce the response of geostructures containing coarse grained materials (e.g. earth dams or embankments), usually becomes a very complicated procedure. The large diameter of the coarse grains and gravels makes the conduction of standard laboratory tests impossible. So, the only way to get data concerning the stress strain relationships of those materials is to conduct specialized laboratory tests on oversized samples or in situ tests on pilot embankments. Besides the difficulties on the conduction of in situ and laboratory tests, the selection of the constitutive laws that reproduce the behavior of the coarse grain materials in static and dynamic loading can also become a very difficult procedure, because of the numerous available models. Although the most frequently used models are those based on the perfect and hardening plasticity theory, the combination of the theory s principals with the variant failure criteria

2 72 Geotech Geol Eng (2009) 27:71 80 (Desai and Abel 1972; Chen 1982; Bazant et al. 1982; Desai and Siriwardane 1984; Chen and Baladi 1986; Roland and Bernard 1987; Comodromos 2001) has led to the expression of a sufficient number of constitutive laws. Those laws are distinguished in elastic perfect plastic and hardening elasto-plastic models. The elastic perfect plastic model based on the Mohr Coulomb failure criterion, the group of CAP hardening laws and the Cam Clay elasto-plastic model are some of the widely used models based on that particular theory. The main objective of the present study is the evaluation of specific constitutive laws for their ability to analyze the behaviour of typical soil materials used for the construction of earth dams. The evaluated constitutive laws are the elastic perfect plastic model based on the Mohr Coulomb failure criterion and the isotropic hardening CAP model, based on the same failure criterion. For the purposes of this study, the readily available data from specialized laboratory tests, of the materials composing the earth dam in the hydroelectric power plant of Sfikia, were chosen. The availability of the data, as well as the variety of the construction geomaterials, allows the expression of generalized conclusions concerning the ability of the specific models to simulate many other similar geostructures (e.g. embankments, landfills etc.). 2 Earth Dam of Hydroelectric Power Plant in Sfikia site Description & Construction Geo-materials The earth dam at the Hydroelectric Power Plant of Sfikia (PPC Power Plant Company of Greece) is constructed along Aliakmon River and has been operating since The plant is located within the limits of Imathea prefecture, 20 km south of the city of Veria in the northern Greece. The dam of Sfikia is a rockfill dam with an inclined, centred clay core. The maximum height of the construction is 82 m and the length of its crest is 220 m. (Liakouris 1995). The body of the dam is divided in six zones of materials with dissimilar grain distribution (Fig. 1). Zone 1 constitutes the impermeable clay core, zones 2 constitute the filters that enclose the core and the rest of the zones compose the rockfill shell of the dam. The impermeable clay core (zone 1) consists of sandy clays, with less than 15% gravels, while the filters and rockfill shell zones consist of sand with gravels. The percentage of sand decreases in proportion to the distance from the core. So, the filters (zones 2) consist of sand in a percentage up to 55%, the transition zones 3 & 3 a consist of 35 & 22% sand correspondingly and zone 5 contains sand in a percentage less than 27%. In addition to sand and gravels, the rockfill shell zones contain up to 10% clay. Figure 2 presents the grain distribution acceptance limits of the materials used for the construction of the dam. In order to control the quality of the under construction core, PPC conducted a full scale program of laboratory tests on undisturbed samples coming from different levels of the core. A number of 275 undisturbed samples were selected and each was subjected to tests concerning the determination of physical and mechanical parameters. Those tests provided vital and sufficient data for the evaluation of the plasticity models ability to simulate the core. The materials of the rockfill shell zones were subjected to two sets of laboratory tests. The first group of tests defined the mechanical properties of soil materials, coming from several locations surrounding the site, in order to determine the most suitable for construction. The tests took place in the University of Kaslsruhe s laboratories of soil and rock mechanics. The second group of tests took place during the construction of the dam, in order to reassure the quality standards of the selected construction materials. Of the above mentioned tests, the obtained data from the specialized consolidation and triaxial tests Fig. 1 Cross section of the Sfikia earth dam across the maximum height position (height 82 m)

3 Geotech Geol Eng (2009) 27: Fig. 2 The grain distribution acceptance limits of the materials used for the construction of the dam of the University of Kaslsruhe, were especially useful. The tests were conducted on a variety of samples, which resulted from primary materials by the gradual removal of selected grain diameters. Triaxial tests were carried out on saturated disturbed samples of 100, 20 and 10 cm diameters while the consolidation tests on disturbed samples of a 50 cm diameter. The compaction procedure of the disturbed samples was standardized. The mechanical properties and the experimental stress strain relationships for the materials of each zone, were accurately correlated by comparing the grain distribution of the materials, that were finally used for the construction of the rockfill shell zones, with the grain distribution of the samples tested at the University of Kaslsruhe. as a non-even hexagonal cone. Because of the cone s complicated form, the surface does not coincide with the plastic potential surface. This fact classifies the Mohr Coulomb elastoplastic model in the category of the non-associated constitutive laws. The constitutive equations of the perfect elastoplastic theory do not cover a multi surface yield contour as presented in the Mohr Coulomb model. For such a yield surface, the theory of plasticity has been extended by Koiter (1960) and others, in order to consider the flow vertices of two or more plastic potential functions. According to that theory, the full Mohr Coulomb yield condition can be defined by the three following yield functions (1), (2) and (3), formulated in terms of principal stresses (Smith and Griffith 1982; Brinkgreve et al. 1998): 3 Concise Presentation of the Evaluated Constitutive Laws f 1 ¼ 1 j 2 r 2 r 3 jþ 1 j 2 r 2 r 3 jsin / c cos / 0 ð1þ According to the elastic perfect plastic theory (Desai and Abel 1972; Oden 1972; Desai 1979, 1980; Hinton and Owen 1979; Owen and Hinton 1980; Smith 1982; Chen 1984; Zienkiewich and Morz 1984), when the yield limit of a soil material coincides with the failure limit, the hardening function does not occur. In that case the stress strain curves consist of two independent linear branches: the first branch referring to the elastic deformations, while the second to plastic. The yield surface of the Mohr Coulomb elastoplastic model is presented in the principal stress space f 2 ¼ 1 j 2 r 3 r 1 jþ 1 j 2 r 3 r 1 jsin / c cos / 0 f 3 ¼ 1 j 2 r 1 r 2 jþ 1 j 2 r 1 r 2 jsin / c cos / 0 ð2þ ð3þ where: r 1, r 2, r 3 = the principal stresses; / = the friction angle; c = the cohesion. The values of the friction angle, /, the cohesion, c, and the dilatancy angle, w, are required for the

4 74 Geotech Geol Eng (2009) 27:71 80 determination of the plastic behaviour of the stress strain curve. Respectively, the values of the stiffness modulus, E, and the Poisson s ratio, v are necessary for the determination of the elastic branch. The hardening elastoplastic constitutive laws enable the simulation of the hardening performance. This performance is observed, when the yield limit of a soil material differs from the failure limit. The isotropic hardening CAP model, based on the Mohr Coulomb failure criterion, is an advanced model for simulating the behaviour of different types of soil, both soft and stiff soils (Schanz 1998). This model derives from the development of the hyperbolic model (Kondner 1963; Duncan and Chang 1970). The basic idea for the formulation of the hardening model, is the hyperbolic relationship between the vertical strain, e 1, and the deviatoric stress, q = (r 1 - r 3 ) in a primary triaxial loading. According to that model, the standard drained triaxial tests tend to yield curves that can be described by the following Eq. 4: e 1 ¼ 1 q ð4þ 2E 50 1 q=q a where: q a = the asymptotic value of the shear strength; q f = the value of the shear strength and E 50 = the confining stress dependent stiffness modulus. The values of the modulus are given by the Eq. 5: E 50 ¼ E ref c cot / r 0 m 3 50 c cot / p ref ð5þ where: P ref = the reference pressure; E ref 50 = the stiffness modulus defined for a reference minor principal stress of r 0 3 = pref ;m= the parameter that defines depends between P ref and minor principal stress, r 0 3 : Janbu (1963) reports values of m around 0.5 for Norwegian sands and silts, while Von Soos (1990) reports various different values in the range 0.5 \ m \ 1.0. The asymptotic value of the shear strength, q a,is defined as (6): q a ¼ q f Rf ð6þ where: R f = the failure ratio. The oedometer stiffness modulus E oed is used as a stress-dependent stiffness modulus (7) for one-dimensional compression stress paths. E oed ¼ E ref oed r0 p ref m ð7þ where: E ref oed : the oedometer stiffness modulus defined for a reference vertical stress of r 0 = p ref. Both E ref 50 and Eref oed are necessary for the formulation of the hardening model. The triaxial modulus controls the shear yield surface and the oedometer modulus controls the cap yield surface. In fact, E ref 50 controls the magnitude of the plastic strains associated with the shear yield surface. Similarly, E ref oed is used to control the magnitude of plastic strains originating from the yield cap. 4 Evaluation Procedure of the Constitutive Laws 4.1 Evaluation of the Constitutive Laws, Regarding their Ability to Simulate the Behaviour of the Core s Compacted Sandy Clays In order to determine which of the models simulates the elastoplastic behaviour of the materials composing the core, a selection of experimental stress strain curves from the triaxial tests, were compared with theoretical curves produced by the constitutive laws. The experimental curves came from CD triaxial tests, conducted on undisturbed samples of the core. Correspondingly, the theoretical stress strain curves, were yielded from the constitutive equations using characteristic values of the mechanical parameters. Specifically, for each experimental curve, two theoretical curves were constructed based on the elastic perfect plastic model and twelve based on the hardening CAP model. The main difference between the two elastic perfect plastic model theoretical curves is the inclination of their elastic branch. The first curve is designed by using the stiffness modulus value E 0, while the second by the stiffness modulus value E 50. The theoretical stress strain curves, resulting for each experimental curve of the hardening model, are practically innumerable. This fact can be easily understood by taking into consideration the numerous combinations of the parameters composing the constitutive equations of the model (4, 5, 6 and 7). The main parameters effecting the form of the stress strain curves are: (a) the failure ratio, R f,

5 Geotech Geol Eng (2009) 27: (0.9 \ R f \ 0.95), (b) the exponent m (0.5 \ m \ 1.0) and (c) the reference pressure, P ref, that defines the stiffness modulus E ref 50 and Eref oed : The characteristic values of the above mentioned main parameters, used for the construction of the twelve theoretical curves, based on the hardening model, are: (a) the borderline values 0.9 and 0.95 for the failure ratio, R f, (b) the borderline values 0.5 and 1.0 for the exponent m and (c) the values 75,300 and 450 kn/m 2 for the reference pressure, P ref. These reference pressure values correspond to the minor principal active stresses, r 0 3 ; of the conducted CD triaxial tests. The selection of the values, enable the accurate calculation of the stiffness module values, E ref 50 ; by using the actual laboratory results. Moreover, these values correspond to the vertical stresses, r 0,of the conducted one-dimensional compression tests. So, their selection also enables the direct calculation of oedometer stiffness module values, E ref oed : As an example of the applied procedure, Figs. 3 6 present the theoretical stress strain curves plotted in order to be compared with an experimental curve. The selected experimental curve was produced by a CD triaxial test, conducted with a minor principal Fig. 4 The experimental curve resulting from the CD triaxial test (r 0 3 = 150 KN/m2 ), compared to the hardening model s theoretical curves, using a P ref value of 300 KN/m 2 Fig. 3 The experimental curve resulting from the CD triaxial test (r 0 3 = 150 KN/m2 ), compared to the hardening model s theoretical curves, using a P ref value of 75 KN/m 2 Fig. 5 The experimental curve resulting from the CD triaxial test (r 0 3 = 150 KN/m2 ), compared to the hardening model s theoretical curves, using a P ref value of 450 KN/m 2

6 76 Geotech Geol Eng (2009) 27:71 80 Fig. 6 The experimental curve resulting from the CD triaxial test (r 0 3 = 150 KN/m2 ), compared to the theoretical curves, based on the M-C model & best fitting curves based on the hardening model active stress, r 0 3 = 150 KN/m2, on an undisturbed sample coming from the level of 75 m. So, Figs. 3 5 present the comparison between the experimental curve and the theoretical curves resulting from the reference pressure values P ref = 75,300 and 450 KN/ m 2, correspondingly using all the selected combinations of the borderline values of R f and m. By comparing the above mentioned curves, the theoretical curves that fit the best with the experimental, are those resulting from the reference pressures values P ref = 300 KN/m 2 and P ref = 450 KN/m 2 and from the borderline value m = 0.5 (Fig. 6). Further more, as presented in Fig. 6, the two curves resulting from value R f = 0.9, coincide along the entire length of the experimental curve. On the contrary, the two curves resulting from value R f = 0.95, match only along the part expanding from 0% to 7.5% of the strain and then diverge along the failure limit. In this figure the curves resulting from the elastoplastic model are also presented. These curves coincide along the elastic part of the experimental curve and diverge completely along the hardening part of the curve. In conclusion, the theoretical curves that are the best match for the experimental, are those resulting from the hardening CAP model (especially from the value sets P ref = 300 KN/m 2, R f = 0.9, m = 0.5 and P ref = 450 KN/m 2, R f = 0.9, m = 0.5). This procedure was conducted for the majority of the available CD triaxial test data. The final conclusion arising from the evaluation of those results, is that the theoretical curves simulating the stress strain relationship of the core s sandy clay are those resulting from the isotropic hardening CAP model, based on the M C failure criterion (especially those resulting from the values P ref = 300 KN/m 2, R f = 0.9, m = 0.5). It is notable that, even though the necessary correlation was succeeded in some cases (including the example), by more than one combinations of the parameters P ref - m - R f, the values P ref = 300 KN/m 2, R f = 0.9, m = 0.5 eventually provided the highest matching percentages. The above mentioned parameters were also evaluated for their ability to simulate the experimental data of the one-dimensional compression tests. In order to conduct this evaluation, the theoretical oedometer stiffness module values, E ref oed ; were calculated for all P ref and m borderline parameter combinations (Eq. 7) and then got compared with the experimental data. As an example of the applied procedure, Fig. 7 presents the curves resulting from the correlation of the theoretical & experimental oedometer stiffness module values, to the reference pressures values. These curves refer to the previously mentioned undisturbed sample from the level of 75 m. The comparison of the curves revealed that the theoretical curves, that best fit the experimental, are those resulting from the reference pressures P ref = 300 & 450 KN/m 2 and from the borderline value m = 0.5. From the application of the previous procedure to the sum of the selected samples, the theoretical oedometer stiffness module values that simulate the response of the core materials, are those resulting from the reference pressure value P ref = 300 KN/m 2 and from the borderline value m = 0.5. Concluding, the procedure described in this paragraph proves that the constitutive law which simulates the behaviour of the dam core s compacted sandy clays, is the isotropic hardening CAP model based on the Mohr Coulomb failure criterion. This constitution law can be safely used for the numerical analysis of compacted sandy clays with similar grain distribution (Fig. 2).

7 Geotech Geol Eng (2009) 27: Fig. 7 The curves resulting by the correlation of the theoretical and the experimental oedometer stiffness module values, with the reference pressures values, P ref, for the selected undisturbed sample Also, the estimation of the parameters that provide the highest coincidence percentages between the theoretical and the experimental curves, led to a very interesting conclusion. According to the principles of the hardening model (Schanz et al. 1999), the reference stress P ref is defined as the minor principal stress, r 0 3 ; used as a confining pressure to a set of triaxial tests (usually 100 KN/m 2 ). In this study, the minor principal stresses, r 0 3, used as confining pressures in the CD triaxial tests are 75 or 150 KN/m 2, depending on the sample. Unlike the hardening model s principles, the theoretical curves that fit best with the experimental, are those resulting from the reference stress value P ref = 300 KN/m 2. Note that, from the evaluation of the one-dimensional compression tests data, the resulting mean pre-consolidation stress of the core materials is 254 KN/m 2. The main result arising from this observation is that, for pre-consolidated sandy clays, the reference stress P ref must be defined as the minor principal stress, r 0 3 ; exceeding the pre-consolidation stress and not as the minor principal stress resulting from experimental data in general. 4.2 Evaluation of the Constitutive Laws, Regarding their Ability to Simulate the Behaviour of the Rockfill Shell s Coarse Grained Materials The experimental data from the specialized tests conducted at the University of Kaslsruhe, were evaluated in order to determine which of the models simulates the behaviour of the rockfill shell materials. Due to the increased sample sizes, the tests provided reliable experimental stress strain curves even for the coarse materials of the zones 4 and 5. In order to determine which of the models simulates the behaviour of the rockfill shell materials, the experimental stress strain curves are compared with those produced by the constitutive laws. This procedure is almost the same as the one previously described, applied on the core s materials. In this case, two theoretical curves based on the elastic perfect plastic model and four based on the hardening CAP model were plotted for each experimental curve. The main difference between the two elastic perfect plastic model s theoretical curves are the stiffness modulus values (E 0 or E 50 ) affecting the inclination of the elastic part of the curve. For the construction of the four theoretical curves based on the hardening model, all possible combinations of the reference pressure value P ref = 200 kn/m 2 with the borderline values R f (0.9 and 0.95) and m (0.5 and 1.0), were used. The reference pressure value P ref = 200 kn/m 2 corresponds to the minor principal active stress, r 0 3 ; of the conducted CD triaxial tests. As an example of the applied procedure, in Figs. 8, 9, 10 and 11 the theoretical stress strain curves are plotted in order to be compared to the experimental curves produced by the triaxial tests, conducted on a sample of zone 2. The triaxial tests were carried out using minor principal active stresses, r 0 3 = 200, 400, 600, and 800 KN/m2.

8 78 Geotech Geol Eng (2009) 27:71 80 Fig. 8 The triaxial test s experimental curve (r 0 3 = 200 KN/m2 ) and the theoretical curves based on the applied models Fig. 10 The triaxial test s experimental curve (r 0 3 = 600 KN/m2 ) and the theoretical curves based on the applied models Fig. 9 The triaxial test s experimental curve (r 0 3 = 400 KN/m2 ) and the theoretical curves based on the applied models A comparison of these stress strain curves, resulted in the conclusion that the theoretical curves best matching to the experimental, are those yielding Fig. 11 The triaxial test s experimental curve (r 0 3 = 800 KN/m2 ) and the theoretical curves based on the applied models from the elastic perfect plastic model based on the Mohr Coulomb failure criterion. Specifically, the curves resulting from the hardening model, are in

9 Geotech Geol Eng (2009) 27: complete divergence to the entire length of the experimental curve. On the contrary, the curves resulting from the elastoplastic model, especially those resulting from the stiffness modulus values E 50, coincide with the elastic, the initial plastic part and the failure limit of the curves. In fact, the resulting curves diverge only at the part referring to the hardening behaviour. The procedure mentioned above was applied to all rockfill shell zones materials (Loupasakis 2002), resulting to the same conclusions. 5 Conclusions From the evaluation of the constitutive laws, regarding their ability to simulate the behaviour of the coarse and fine grained materials, used for the construction of the rockfill shell dam of Sfikia, it can be deduced that: (1) The constitutive law, simulating the behaviour of the core s compacted sandy clay, is the isotropic hardening CAP model based on the Mohr Coulomb failure criterion. This model can be safely used for the numerical analysis of compacted sandy clays with similar grain distribution (Fig. 2), used in any earth construction. (2) Despite the principles of the hardening model, for pre-consolidated sandy clays, the applied reference stiffness modulus values, E ref 50 ; must be defined as the values estimated for a reference stress, P ref, exceeding the pre-consolidation stress and not for a lower minor principal stress, r 0 3 : (3) The constitutive law reproducing the stress strain relations of the material used for the construction of the rockfill shell zones, is the elastic perfect plastic model based on the Mohr Coulomb failure criterion. Based on the variety of the grain distribution in the rockfill shell materials, this model can be safely suggested for the numerical analysis of similar coarse grain materials used in any earth construction. (4) Regarding the elastic perfect plastic model s stiffness modulus values, the theoretical curves that best match the experimental, are those resulting from the stiffness modulus values E 50. It is important to use the experimental data referring to the minor principal stresses, r 0 3 ; approaching the real values developed in the stress field of the analyzed construction. The conclusions arising from this study were used for the numerical analysis of deformations at the Sfikia earth dam. Numerous back analyses were conducted to evaluate the constitutive laws, using PLAXIS finite element code (Brinkgreve et al. 1998) and the available in situ instruments recordings. These analyses proved that by using the evaluated constitutive laws and the properly estimated set of mechanical parameters, the response of the construction can be simulated accurately. Overall, the above mentioned conclusions can be safely used for the simulation of deformations in similar earth constructions. Acknowledgments The first writer would like to express his appreciation to the Greek State Scholarship s Foundation for granting financial support in the form of post graduation scholarship for his studies. References Bazant FQ, Chang T-P, Chen WF (1982) Constitutive relations and failure theories. In Finite Element Analysis of Reinforced Concrete, State of Art, ASCE Brinkgreve RBJ, Vermeer PA, Bakker KJ, Bonnier PG, Brand PJW, Burd HJ, Termaat RJ (1998) Plaxis, fine element code for soil and rock analyses, Version 7, manual. A.A. Balkema, Rotterdam Brookfield Chen WF (1982) Plasticity in reinforced concrete. McGraw- Hill Book Co., New York Chen WF (1984) Constitutive modeling in soil mechanics. In: Desai CS, Gallagher RH (eds) Mechanics of engineering materials. John Wiley, London, pp Chen WF, Baladi GY (1986) Soil plasticity Theory and implementation. Elsevier, New York Comodromos E (2001) Numerical geotechnical engineering, Linear No linear analysis. Zitis Press, Thessaloniki Desai CS (1979) Some aspects of constitutive laws of geologic media. In: Wittke W (ed) Proceedings of the 3rd international conference on numerical methods Geomech, 1979, vol 1, Balkema Press, Roterdam, 299 pp Desai CS (1980) A general basis for yield, failure and potential functions in plasticity. Int J Numer Anal Methods Geomech, pp Desai CS, Abel FJ (1972) Introduction to the finite element method. A numerical method for engineering analysis. Van Nostrand Reinhold Company, New York Desai CS, Siriwardane HJ (1984) Constitutive laws for engineering materials with emphasis on geological materials. Prentice Hall, Englewood Cliffs, NJ

10 80 Geotech Geol Eng (2009) 27:71 80 Duncan JM, Chang CY (1970) Nonlinear analysis of stress and strain in soil. ASCE J Soil Mech Found, pp Hinton E, Owen DRJ (1979) An introduction to the finite element computations. Pineridge Press Ltd, Swansea, UK Janbu J (1963) Soil compressibility as determined by oedometer and triaxial tests. In: Abstracts of the ECSMFE, vol 1, Wiesbaden, pp Koiter WT (1960) General theorems for elastic-plastic solids. In: Sneddon IN, Hill R (eds) Progress in solid mechanics, vol 1. North-Holland, Amsterdam, pp Kondner RL (1963) A hyperbolic stress strain formulation for sands. In: Abstracts of the 2nd Pan Am ICOSFE vol 1. Brazil, pp Liakouris D (1995) Geology and the earth dams of PPC. PPC - Public Power Corporation of Greece, Department of education, Athens Loupasakis C (2002) Study of the behaviour of earth dams bodies by the use of numerical analysis methods. Dissertation, Aristotle University of Thessaloniki Oden JT (1972) Finite elements of continua. McGraw-Hill Co., New York Owen DRJ, Hinton E (1980) Finite elements in plasticity, theory and practice. Pineridge Press Ltd, Swansea, UK Roland WL, Bernard AS (1987) The finite element method in the deformation and consolidation of porous media. John Wiley & Sons Ltd, Manchester Schanz T (1998) Zur Modellierung des Mechanischen Verhaltens von Reibungsmaterialen. Habilitation, Stuttgart Universität Schanz T, Vermeer PA, Bonnier PG (1999) Formulation and verification of the hardening Soil model. In: Brinkgreve RBJ (ed) Beyond 2000 in computational geotechnics. Balkema, Rotterdam, pp 281 Smith IL (1982) Programming the finite element method with applications to geomechanics. John Wiley & Sons Ltd, Manchester Smith IM, Griffith DV (1982) Programming the finite element method. 2nd ed. John Wiley & Sons, Chisester, UK Von Soos P (1990) Properties of soil and rock. In: Grundbautaschenbuch, Ernst & Sohn, Berlin Zienkiewich OC, Morz Z (1984) Generalized plasticity formulation and applications to geomechanics. In: Desai CS, Gallagher RH (eds) Mechanics of engineering materials. John Wiley, London, pp