modelling sand instability within the framework of critical state soil mechanics

MODELLING SAND INSTABILITY WITHIN THE FRAMEWORK OF CRITICAL STATE SOIL MECHANICS CATARINA PAISANA FERREIRA CORREIA RAMOS Dissertação submetida para satisfação parcial dos requisitos do grau de MESTRE EM ENGENHARIA CIVIL — ESPECIALIZAÇÃO EM GEOTECNIA Orientador: Professor Doutor António Joaquim Pereira Viana da Fonseca Coorientador: Professor Doutor Jean Vaunat JULHO DE 2013 MESTRADO INTEGRADO EM ENGENHARIA CIVIL 2012/2013 DEPARTAMENTO DE ENGENHARIA CIVIL Tel. +351-22-508 1901 Fax +351-22-508 1446  [email protected] Editado por FACULDADE DE ENGENHARIA DA UNIVERSIDADE DO PORTO Rua Dr. Roberto Frias 4200-465 PORTO Portugal Tel. +351-22-508 1400 Fax +351-22-508 1440  [email protected]  http://www.fe.up.pt Reproduções parciais deste documento serão autorizadas na condição que seja mencionado o Autor e feita referência a Mestrado Integrado em Engenharia Civil - 2012/2013 - Departamento de Engenharia Civil, Faculdade de Engenharia da Universidade do Porto, Porto, Portugal, 2013. As opiniões e informações incluídas neste documento representam unicamente o ponto de vista do respetivo Autor, não podendo o Editor aceitar qualquer responsabilidade legal ou outra em relação a erros ou omissões que possam existir. Este documento foi produzido a partir de versão eletrónica fornecida pelo respetivo Autor. Modelling Sand Instability within the Framework of Critical State Soil Mechanics To my Mother and Father Intelligence without ambition is a bird without wings Salvador Dalí Modelling Sand Instability within the Framework of Critical State Soil Mechanics i ACKNOWLEDGEMENTS During the process of doing and writing this Dissertation the author had the help and support of some people that deserve to be acknowledged:  To my Portuguese supervisor, Professor António Viana da Fonseca, for the proposition of such a challenging work, for his constant help and advice during this period, for always recommending new approaches for the work and for helping me with the bibliography and final revision of the work.  To my co-supervisor, Professor Jean Vaunat, for all the patience and help during these months, for the suggestion of new ideas and for the guidance throughout the elaboration of this work.  To my mother and my father for always believing in me, supporting me and loving me.  To my sister, family and friends for always being there for me and for all the good moments, the care and enjoyment they bring to my life.  To my man Luís, for all the conversations, the motivation and support but especially for the unconditional love. Modelling Sand Instability within the Framework of Critical State Soil Mechanics ii Modelling Sand Instability within the Framework of Critical State Soil Mechanics iii ABSTRACT The aim of the present work was the study of a sand from Les Dunes beach in Ain Beninan, Algeria, where the Boumerdès earthquake occurred in 2003. This earthquake caused a lot of structural damages and claimed the lives of many people. Many damages caused to infrastructures are related to the phenomenon of liquefaction. A brief introduction to the concepts of liquefaction, elasto-plasticity and critical state theory is done in the attempt to have a framework. These concepts are of main importance as they are the base of this study and essential for the interpretation of the results presented. The model used in this work was the unified critical state constitutive model for clays and sands (CASM), developed by Yu (1998), which is implemented in Code_Bright. The study was based on the results of two drained and six undrained triaxial tests, performed by Pinheiro (2009) and Rocha (2010) at LabGeo. All the tests performed followed compression stresspaths in monotonic conditions and the specimens were isotropically consolidated as the objective was to study the instability and liquefaction phenomenon due to static loading. The results were compared with simulations done in Code_Bright, a finite element program for numerical calculations. The parameters that characterize the sand were obtained from laboratory data or through calibration for a suitable curve fitting. CASM is implemented in Code_Bright and the comparison between the experimental results and the numerical solution allows for the verification of the applicability of the model. The instability is studied with the second-order work increment criterion, with which the instability line for this sand is determined. The dilatancy rate defined by Rowe (1962) is studied in the points where instability begins and it is shown once more that CASM can model liquefaction in materials where the instability line is straight in p’‒q space. It is shown that specimens that have been performed under drained conditions are always stable, even when their stress-paths are inside the region of potential instability. On the other hand, under undrained conditions, specimens may become unstable after crossing the instability line. Some tests were also simulated, which started with drained conditions but then change to undrained conditions at different time steps. Finally, the Normal Compression Line, the Critical State Line and the Instability Line were defined in the e-lnp’ space. KEYWORDS: constitutive model, critical state, elastoplasticity, flow liquefaction, instability of sands Modelling Sand Instability within the Framework of Critical State Soil Mechanics iv Modelling Sand Instability within the Framework of Critical State Soil Mechanics v RESUMO O objetivo deste trabalho foi o estudo de uma areia proveniente da praia Les Dunes, em Ain Beninan na Argélia, onde, em 2003, o terramoto de Boumerdès ocorreu. Este terramoto causou vários danos estruturais e tirou a vida a muitas pessoas. Muitos dos danos causados em infra-estruturas estão relacionados com o fenómeno da liquefação. Uma breve introdução e enquadramento sobre os conceitos de liquefação, elasto-plasticidade e teoria dos estados críticos são apresentados. Estes conceitos são muito importantes pois são a base deste estudo e essenciais para a interpretação dos resultados apresentados. O modelo utilizado neste trabalho é um modelo constitutivo do estado crítico unificado para argilas e areias (CASM), desenvolvido por Yu (1998) e está implementado no programa Code_Bright. O estudo foi baseado nos resultados de ensaios triaxiais, dois drenados e seis não drenados, realizados por Pinheiro (2009) e Rocha (2010) no LabGeo, da FEUP. Todos os ensaios foram realizados em carregamento de compressão e monotónicos e todas as amostras foram consolidadas isotropicamente, uma vez que o objetivo deste trabalho é o estudo da instabilidade e do fenómeno de liquefação devido a carregamentos estáticos. Os resultados foram comparados com simulações realizadas no Code_Bright, um programa de elementos finitos para cálculo numérico. Os parâmetros que caracterizam a areia foram obtidos através de ensaios de laboratório ou por calibração até as curvas se ajustarem adequadamente. O CASM está implementado no Code_Bright e a comparação entre os resultados experimentais e a solução numérica permite a verificação da aplicabilidade do modelo. A instabilidade foi estudada segundo o critério do incremento de trabalho de segunda ordem e a linha de instabilidade da areia é determinada. A taxa de dilatância definida por Rowe (1962) é estudada nos pontos de início de instabilidade e é demostrado mais uma vez que o modelo CASM permite a modelação da liquefação nos materiais em que a linha de instabilidade é reta no espaço p’‒q. É demonstrado que as amostras realizadas em condições drenadas são sempre estáveis, mesmo quando as tensões se encontram dentro da região de potencial instabilidade. Por outro lado, quando submetidas a condições não drenadas, as amostras podem tornar-se instáveis após passar a linha de instabilidade. Também foram simulados com sucesso ensaios que começam drenados mas passam a não drenados em diferentes intervalos de tempo. Finalmente foram definidas a linha normalmente consolidada, a linha dos estados críticos e a linha de instabilidade no espaço e-lnp’. PALAVRAS-CHAVE: elasto-plasticidade, estado crítico, instabilidade nas areias, liquefação por fluxo, modelo constitutivo Modelling Sand Instability within the Framework of Critical State Soil Mechanics xii Figure 2.26 – Normalized CSL and NCL (Atkinson, 1993) .................................................................... 24 Figure 2.27 – Yield curve for original Cam-clay .................................................................................... 25 Figure 2.28 – Definition of state parameter, critical state constants and reference state parameter (Yu, 1998) ...................................................................................................................................................... 26 Figure 3.1 – a) Total displacement of the first floor; b) Elementary school in Corso, insufficient lateral resisting system; c) Damage to Highway 5 Bridge; d) Several floors pancaked (EERI, 2003) ............ 29 Figure 3.2 – Distribution of liquefaction during the Bourmedès earthquake (Bouhadad et al., 2004)... 30 Figure 3.3 – Sand flow on Isser River, 22 May 2003 (Google) ............................................................. 30 Figure 3.4 – a) Cracking; b) Sand boils; c) and d) damage on roads due to liquefaction (EERI, 2003) 31 Figure 3.5 – Particles of Les Dunes sand.............................................................................................. 31 Figure 3.6 – Soil gradation of Les Dunes Sand ..................................................................................... 32 Figure 3.7 – Definition of the Kf line (Fonseca, 2009) ........................................................................... 33 Figure 3.8 – The triaxial apparatus (adapted from Head and Epps, 2011) ........................................... 34 Figure 3.9 – Stress-path for undrained tests with more than 400kPa of confining pressure ................ 36 Figure 3.10 – Stress-path for undrained tests with less than 200kPa of confining pressure ................ 37 Figure 3.11 – Stress-path for drained tests ........................................................................................... 38 Figure 3.12 – Deviatoric stress versus axial strain for undrained tests with more than 400kPa of confining pressure ................................................................................................................................. 38 Figure 3.13 – Deviatoric stress versus axial strain for undrained tests with less than 200kPa of confining pressure ................................................................................................................................. 39 Figure 3.14 – Deviatoric stress versus axial strain for drained tests ..................................................... 39 Figure 3.15 – Excess pore pressure versus axial strain for undrained tests with more than 400kPa of confining pressure ................................................................................................................................. 40 Figure 3.16 – Excess pore pressure versus axial strain for undrained tests with less than 200kPa of confining pressure ................................................................................................................................. 40 Figure 3.17 – Volumetric strain versus axial strain for drained tests ..................................................... 41 Figure 4.1– Main steps to solve a problem with GiD ............................................................................. 44 Figure 4.2 – Oedometer test results (50 mm diameter) for determination of κ ..................................... 46 Figure 4.3 – Critical State Line in υ-ln p' space (Viana da Fonseca and Soares, 2012) ....................... 47 Figure 4.4 – Oedometer test results for determination of λ; a) 50mm; b) 75mm .................................. 47 Figure 4.5 – Evolution of the shape parameter n .................................................................................. 48 Figure 4.6 – Evolution of the spacing ratio r .......................................................................................... 48 Figure 4.7 – Yield surfaces for undrained triaxial tests: a) LD12; b) LD42 (to be continued) ............... 49 Figure 4.8 – Specimen geometry .......................................................................................................... 51 Figure 4.10 – Representation of the boundary conditions imposed ...................................................... 52 Figure 4.11 – Representation of the mesh ............................................................................................ 54 Figure 4.12 – Stress-strain-volumetric curves for drained triaxial test LD63 ......................................... 55 Figure 4.13 – Stress-strain-volumetric curves for drained triaxial test LD65 ......................................... 55 Figure 4.14 – LD12: Stress-strain curve and Stress-path ..................................................................... 56 Modelling Sand Instability within the Framework of Critical State Soil Mechanics xiii Figure 4.15 – LD42: Stress-strain curve and Stress-path ..................................................................... 56 Figure 4.16 – LD44: Stress-strain curve and Stress-path ..................................................................... 56 Figure 4.17 – LD45: Stress-strain curve and Stress-path ..................................................................... 57 Figure 4.18 – LD46: Stress-strain curve and Stress-path ..................................................................... 57 Figure 4.19 – LD48: Stress-strain curve and Stress-path ..................................................................... 57 Figure 4.20 – LD12: Stress-strain curve and Stress-path ..................................................................... 58 Figure 4.21 – LD42: Stress-strain curve and Stress-path ..................................................................... 58 Figure 4.22 – LD44: Stress-strain curve and Stress-path ..................................................................... 59 Figure 4.23 – LD45: Stress-strain curve and Stress-path ..................................................................... 59 Figure 4.24 – LD46: Stress-strain curve and Stress-path ..................................................................... 60 Figure 4.25 – LD48: Stress-strain curve and Stress-path ..................................................................... 60 Figure 4.26 – Pore pressure versus axial strain: a) LD12; b) LD42; c) LD44; d) LD45; e) LD46; f) LD48 ............................................................................................................................................................... 61 Figure 4.27 – Deviatoric stress versus axial strain for undrained tests................................................. 62 Figure 4.28 – Pore pressure versus axial strain for undrained tests ..................................................... 63 Figure 4.29 – Stress-path for undrained tests ....................................................................................... 64 Figure 5.1 – Drucker’s stability postulate for solid metals (Lade, 1994) ............................................... 66 Figure 5.2 – Definition of the Instability Line (Lade, 1992) .................................................................... 68 Figure 5.3 – Stress-path and stress-strain relations for loose sand in undrained conditions (Lade, 1994) ..................................................................................................................................................... 69 Figure 5.4 – Definition of the Instability Line ......................................................................................... 70 Figure 5.5 – Stress-paths of Drained Tests and Instability Line ........................................................... 70 Figure 5.6 – Stress-paths of Drained-Undrained simulations ............................................................... 72 Figure 5.7 – Final points of the stress-paths and Critical State Line ..................................................... 73 Figure 5.8 – LD64: Stress-strain curve and Stress-path ....................................................................... 74 Figure 5.9 – LD64, CSL and instability line ........................................................................................... 75 Figure 5.10 – LD64 and yield surfaces ................................................................................................. 76 Figure 5.11 – LD63 stress-path and instability line ............................................................................... 77 Figure 5.12 – LD65 stress-path and instability line ............................................................................... 78 Figure 5.13 – LD64 stress-path and instability line ............................................................................... 79 Figure 5.14 – Second-order work increment versus axial strain for drained tests ................................ 80 Figure 5.15 – Second-order work increment versus axial strain and time for LD64 ............................. 80 Figure 5.16 – Second-order work increment and deviatoric stress versus axial strain for LD12 .......... 81 Figure 5.17 – Second-order work increment and deviatoric stress versus axial strain for LD42 .......... 81 Figure 5.18 – Second-order work increment and deviatoric stress versus axial strain for LD44 .......... 82 Figure 5.19 – Second-order work increment and deviatoric stress versus axial strain for LD45 .......... 82 Figure 5.20 – Second-order work increment and deviatoric stress versus axial strain for LD46 .......... 82 Figure 5.21 – Second-order work increment and deviatoric stress versus axial strain for LD48 .......... 83 Figure 5.22 – Second-order work increment and deviatoric stress versus time for 30 h ...................... 83 Modelling Sand Instability within the Framework of Critical State Soil Mechanics xiv Figure 5.23 – Second-order work increment and deviatoric stress versus time for 15 h ...................... 84 Figure 5.24 – Second-order work increment and deviatoric stress versus time for 10 h ...................... 84 Figure 5.25 – Second-order work increment and deviatoric stress versus time for 5 h ........................ 84 Figure 5.26 – Second-order work increment and deviatoric stress versus time for 2 h ........................ 85 Figure 5.27 – Second-order work increment and deviatoric stress versus time for 1 h ........................ 85 Figure 5.28 – Second-order work increment and deviatoric stress versus time for 30 min .................. 85 Figure 5.29 – Instability points for each test .......................................................................................... 86 Figure 5.30 – Relation between p’ in CSL, NCL and instability line ...................................................... 87 Figure 5.31 – CSL, NCL and Instability line in elnp’ space ................................................................. 89 Modelling Sand Instability within the Framework of Critical State Soil Mechanics xv LIST OF TABLES Table 3.1 – Densities and void ratios .................................................................................................... 33 Table 3.2 – Summary of triaxial tests performed .................................................................................. 36 Table 4.1 – Description of constitutive model parameters .................................................................... 45 Table 4.2 – Description of initial state values and history variables ...................................................... 45 Table 4.3 - Values of p’0 for undrained triaxial tests .............................................................................. 49 Table 4.4 – Values of n and initial stress for each triaxial test .............................................................. 52 Table 4.5 – Mechanical Data (parameters for CASM general) ............................................................. 53 Table 4.6 – Hydraulic Data (parameters for intrinsic permeability) ....................................................... 53 Table 4.7 – Value of e0 and p’0 for drained triaxial tests ....................................................................... 54 Table 5.1 – Experimental Observations of Conditions for Stability and Instability inside Failure Surface (Lade, 1992) .......................................................................................................................................... 67 Table 5.2 – Definition of the Instability Line .......................................................................................... 69 Table 5.3 – Values of e0, n, p’0 and initial stress for the test ................................................................. 71 Table 5.4 – Final values of q and p’ for each test ................................................................................. 73 Table 5.5 – Values of e0, n, p’0 and initial stresses for LD64 ................................................................ 74 Table 5.6 – Value of p’0 ......................................................................................................................... 75 Table 5.7 – Dilatancy rate for undrained tests at the instability point.................................................... 76 Table 5.8 – Dilatancy rate for the intersection point of LD63 with the instability line ............................ 77 Table 5.9 – Dilatancy rate for the intersection point of LD65 with the instability line ............................ 78 Table 5.10 – Dilatancy rate for the intersection point of LD64 with the instability line .......................... 79 Table 5.11 – Time step and q and p’ values for the instability points ................................................... 86 Modelling Sand Instability within the Framework of Critical State Soil Mechanics xvi Modelling Sand Instability within the Framework of Critical State Soil Mechanics xvii NOMENCLATURE Latin characters B – Skempton parameter CU – coefficient of uniformity CC – coefficient of curvature D – dry Dr – relative density D10 – diameter correspondent to 10% passing D30 – diameter correspondent to 30% passing D60 – diameter correspondent to 60% passing D100 – diameter correspondent to 100% passing d – dilatancy rate dq – increment of deviatoric stress dp’ – increment of effective mean stress d2W – second order work increment E – Young’s modulus e – void ratio e0 – initial void ratio ef – void ratio at ultimate state emax – maximum void ratio emin – minimum void ratio f – yield function G – shear modulus g – plastic potential function K – bulk modulus K0 – coefficient of earth pressure at rest (k11)0 – intrinsic permeability, 1st principal direction M – slope of the Critical State line (in q-p' space) N – specific volume of the NCL when p’ = 1 kPa n – shape parameter n – porosity Modelling Sand Instability within the Framework of Critical State Soil Mechanics xviii p’ – effective mean stress p’0 – preconsolidation pressure p’e – equivalent pressure on the NCL p’c – critical pressure q – deviatoric stress r – spacing ratio S – degree of saturation u – pore pressure VP – P wave velocity VS – S wave velocity W – wet w – moisture content Greek characters β – size parameter Γ – specific volume of the critical state line at p’=1kPa γ – total unit weight γsat – saturated unit weight γd – density δ – small increment of εa – axial strain εp – volumetric strain εq – shear strain εr – radial strain εs – shear strain εv – volumetric strain η – stress ratio κ – slope of the isotropic swelling line (in υ-lnp' space) λ – slope of the CSL (in υ-lnp' space) ν – Poisson’s ratio ξ – state parameter ξR – reference state parameter Modelling Sand Instability within the Framework of Critical State Soil Mechanics xix σ – total stress σa – total axial stress σr – total radial stress υ – specific volume Φ’ – friction angle Φc – critical friction angle Abbreviations and Acronyms BP – Back Pressure CAD – Computer Aided Design CASM – Clay And Sand Model CID – Consolidated Isotropically Drained CIMNE – International Center for Numerical Methods in Engineering CIU – Consolidated Isotropically Undrained CP – Cell Pressure CSL – Critical State Line FEUP – Faculdade de Engenharia da Universidade do Porto LD – Les Dunes NCL – Normal Compression Line NRC – National Research Council OCR – Overconsolidation ratio UPC – Universitat Politècnica de Catalunya Modelling Sand Instability within the Framework of Critical State Soil Mechanics xx Modelling Sand Instability within the Framework of Critical State Soil Mechanics 1 1 INTRODUCTION 1.1. PROLOGUE Every day, geotechnical engineers have to deal with problems related to failure of granular soils. One of these problems is the liquefaction of loose sands under undrained conditions, which can cause devastating damages. Constitutive models can reproduce the response of soils to applied loads. One of the widely used frameworks to formulate constitutive models is the elasto-plastic theory. Elasto-plastic models based on the critical state concept have been used to simulate real soil behaviour. Some examples are the Cam-clay model (Roscoe et al., 1958), the modified Cam-clay model (Roscoe and Burland, 1968) and the Clay and Sand model (Yu, 1998). But the constitutive models formulations alone are not sufficient to solving the practical engineering problems. It is necessary to implement the models in numerical tools, such as finite element codes and checking and validate those implementations. For that, the comparison of numerical solutions with results of tests performed in real soils, that show the real soil behaviour, is essential. The choice of the model that is going to be used to simulate the material behaviour is very important. The quality of the results obtained depends on the selected model and on its implementation. So it is important to choose a model based on the kind of material to be modelled, the number of information available and the type of results being pursued. The applicability of the model for a specific material is of extreme importance, inview of using the numerical results with sufficient reliability. The study of instability is a huge concern when dealing with the safety of dams, structures, excavations or slopes because instability eventually leads to failure. In this work, the instability that leads to flow liquefaction under undrained conditions is studied according to the second-order work increment criterion and according to Lade (1994), that explains that, for materials isotropically consolidated under undrained conditions, the top of the yield surface is the onset of flow liquefaction. 1.2. SCOPE AND OBJECTIVES This work is part of a study that the LabGeo from FEUP is developing, under the coordination of Professor Viana da Fonseca. It is scope of this study of soils susceptible to liquefaction, to evaluate in situ and in the laboratory the risks and their mitigation. This work complements the studies that have been done in a particular sand, from Les Dunes beach in Ain Benian in Algeria, where in 2003 the Boumerdès earthquake occurred. This earthquake caused many liquefaction related occurrences so the study of the behaviour of this sand is very important for Modelling Sand Instability within the Framework of Critical State Soil Mechanics 8 2.1.4. SOIL SUSCEPTIBILITY FOR LIQUEFACTION One approach for evaluating if a soil is susceptible to suffer liquefaction is based on its composition, i. e., the grain size of the particles. According to Terzaghi et al. (1996), a well graded soil is less susceptible to liquefaction because the smaller particles fill in the voids. This results in lower volumetric changes in drained conditions and in lower values of pore pressure generation in undrained conditions. It is safe to say that the grain size of the particles that constitute the solid part of the soil can be determinant for the analyses of soil liquefaction. As it is shown in Figure 2.7, where the boundaries in the gradation curves for liquefiable soil and potentially liquefiable soil proposed by Tsuchida (1970) are represented, saturated granular soils are the most susceptible to suffer liquefaction. On the other hand, soils with great content of fine particles such as clays or fine silts, or coarse materials such as gravel or coarse sands are less susceptible to the liquefaction phenomenon occurrence. In the first type of soils, the plasticity of clays and fines prevent the rearrangement of particles. In the second type of soils, the space between the particles allows the dissipation of the pore water. It is also notable that a soil is more susceptible to liquefy if it is poorly graded, i. e., if the particles have approximately the same size and the soil is uniformly graded. Figure 2.7 – Boundaries in the gradation curves for soils susceptible to liquefaction (adapted from Tsuchida, 1970) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 9 2.2. ELASTO-PLASTICITY Elasto-plastic theory provides one of the best frameworks to formulate constitutive models that can describe some of the most important features of soil behaviour. Elasto-plastic models based on the critical state concept have been used to simulate real soil behaviour. Drucker et al. (1957) proposed the existence of a cap yield surface controlled by volume change and Roscoe et al. (1958) suggested a behavioural framework based on the concepts of critical state and the existence of a state boundary surface. These works on soil hardening and soil yielding established the basis for critical state theory. The elastic constitutive models are unable to simulate some of the most important characteristics of real soil behaviour (Potts and Zdravković, 1999). By using the theory of plasticity, those models can be improved in order to describe those features. To explain and introduce the ideas of plastic yield, hardening and softening, it is considered the uniaxial behaviour of a linear elasto-plastic material. 2.2.1. UNIAXIAL BEHAVIOUR OF A LINEAR ELASTIC PERFECTLY PLASTIC MATERIAL Considering a bar of an ideal linear elasto-plastic material loaded by the application of a compressive axial strain, ε, the stress-strain curve represented in Figure 2.8 illustrates this behaviour. Figure 2.8 – Uniaxial loading for linear elastic perfectly plastic material (Potts and Zdravković, 1999) As the strain is applied, the bar behaves elastically and the stress-strain curve follows the path AB. The slope of the line AB is the Young’s modulus, E. As the material is linear elastic, if the bar is unloaded or unstrained before reaching point B, the stress-strain response moves down on the line AB and there is no permanent deformations. If the strain applied passes , the material becomes plastic because it reached the yield stress, σy. The stress remains constant, with the value of σy, and if the bar is unloaded it becomes elastic and the stress-strain curve travels along CD, a line parallel to AB. When the bar is unloaded until point D, where the stress is zero, there is a permanent strain (shortening) in the bar, equal to . On reloading the bar, the stress-strain curve follows the path DC until point C, in which the axial stress is the same as the yield stress and the bar is plastic. After reaching point C, the curve moves along CF. The behaviour on the paths AB and CD is reversible whereas on the path BCF is irreversible. If the bar is loaded by an increasing stress, it is not possible to apply a stress greater than the yield stress otherwise it would result in infinite strains. This type of behaviour is known as linear elastic perfectly plastic. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 10 2.2.2. UNIAXIAL BEHAVIOUR OF A LINEAR ELASTIC STRAIN HARDENING PLASTIC MATERIAL Considering a bar made of a linear elastic strain hardening plastic material, the stress-strain curve is represented in Figure 2.9. Figure 2.9 – Uniaxial loading for linear elastic strain hardening plastic material (Potts and Zdravković, 1999) While the bar is loaded or strained until point B it behaves elastically (all along path AB). When the path reaches the point B, the stress is equal to the yield stress, σyB and if the strain goes beyond point B to point C, the initial yield stress is exceeded, without remaining constant. Instead, the stress increases to σyC. At this point, if the bar is unloaded, it becomes elastic and follows the path CD, a line parallel to BA. When the bar is unloaded to zero stress (point D) there is a permanent strain, . If the bar is reloaded, it follows the path DC, behaving elastically until the curve reaches point C where the bar is again plastic. The yield stress σyC is greater than σyB and this increase is due to the plastic straining from B to C. If the bar continues to be strained, eventually at some point (for example point F), the stress-strain curve becomes horizontal and the stress becomes constant. 2.2.3. UNIAXIAL BEHAVIOUR OF A LINEAR ELASTIC STRAIN SOFTENING PLASTIC MATERIAL If the bar is now made of a linear elastic strain softening plastic material, the stress-strain curve is represented in Figure 2.10. Figure 2.10 – Uniaxial loading for linear elastic strain softening plastic material (Potts and Zdravković, 1999) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 11 When straining to point B, the bar has the same behaviour as the two presented before, both following a linear initial behaviour. However, when the stress-strain curve passes point B, instead of increasing with plastic straining, it decreases. The behaviour during unloading and reloading is the same as in the examples presented before. The fact that the yield stress decreases beyond point B is a concern because the material resistance to load diminishes. 2.2.4. BASIC CONCEPTS OF THE ELASTO-PLASTIC THEORY As the model and results used in this work are obtained from triaxial tests, the model will be described in terms of triaxial stress variables p’ and q and strain variables εp and εq. There are four basic components to formulate an elasto-plastic model:  Elastic properties The more usual elastic parameters are Young’s modulus, E’, and Poisson’s ratio, ν’. The bulk and shear moduli, for isotropic materials, can be written as a function of these two parameters as shown in equations (2.1) and (2.2) respectively. ( ) (2.1) ( ) (2.2) It is preferable to use bulk and shear moduli instead of Young’s modulus and Poisson’s ratio because in this way, the deviatoric or shear stress-strain relation and the volumetric strain variation due to isotropic stress change are considered separately. Assuming that the soil behaves isotropically and elastically within the yield surface, the elastic response of the soil can be written as in (2.3). The linear elastic material behaviour is represented in Figure 2.11. In Figure 2.11 a) shearing mode is represented while in Figure 2.11 b) volumetric compression and expansion are drawn. [ ] [ ][ ] (2.3) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 12 Figure 2.11 – Behaviour of linear elastic material (adapted from Atkinson, 1993)  Yield function In the multi-axial case, the stress has many components. So in order to define the onset of plastic straining, it is best to define a yield function, f, which depends on the stress and the state parameter. ( ) (2.4) The value of this function identifies the type of material behaviour. F<0 represents a material with purely elastic behaviour (recoverable deformations) and F=0 represents elasto-plastic behaviour. It is impossible to have F>0 as above this boundary there is overall failure, therefore, no stress-strain equilibrium. The surface is a function of the stress state while the size is controlled by the state parameter. In case of perfect plasticity, the state parameter is constant. If the state parameter varies with plastic straining there is hardening or softening plasticity. Figure 2.12 shows the yield function plotted in σ1, σ2 and σ3. In a), σ2 is equal to zero so the function is a curve while in b) σ2 is allowed to vary so the yield function is plotted as a surface. The elastic domain is the space enclosed by this surface. Figure 2.12 – Yield function (Potts and Zdravković, 1999) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 13  Plastic potential function According to Muir Wood (1990), the plastic deformations depend on the stress state at which the yielding occurs. The yielding of a soil is associated with some plastic volumetric strain increment, and some plastic shear strain increment, . If these components are plotted at a stress state where yielding occurs (e. g. point Y), with axes parallel to p’ and q, a plastic strain increment vector (YS) is formed, as shown in Figure 2.13. Figure 2.13 – Plastic strain increment vectors normal to a family of plastic potential curves (Muir Wood, 1990) A line orthogonal to the plastic strain increment vector can be drawn (AB). Each combination of stress that causes yielding have a plastic strain vector associated and an orthogonal line can be drawn at each of those points. These lines can be joined to form a family of curves, known as plastic potentials. The relative magnitudes of the components of plastic deformation are defined by the direction of the outward normal to the plastic potential surface. The plastic potential is defined by equation (2.5), where the parameter β controls the size of the plastic potential. ( ) (2.5) In the multi-axial situation, it is necessary to specify the direction of plastic straining at every stress state which is done by the flow rule. If the yield function and the plastic potential are the same, the flow rule is associated, otherwise it is non-associated. In the case of associated flow rule, the vector that defines the increment of plastic strain is normal to the yield surface. In constitutive modeling, flow rules govern, or in other view, are conditioned by dilatancy effects, which influence volume changes and strength of the soil (Potts and Zdravković, 1999).  Hardening/softening laws As it was discussed before, when the soil reaches the yield stress and is strained further that point, it suffers irrecoverable plastic strains. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 14 The hardening/softening rules define the variations on the state parameters with plastic straining. As it was explained before, if the material is perfectly plastic there is no need for hardening or softening rules because when the stress reaches yield it strains indefinitely. However, if the material suffers from hardening or softening, some rules have to be defined to evaluate the changes in the yield surface. Observing Figure 2.9, it is notable that the yield stress increases with plastic strain (BCF). The definition of the increase of the yield stress with the plastic strain is known as hardening. On the other hand, in Figure 2.10 the yield stress decreases with plastic strain (softening). The relation that defines the decrease of the yield stress with plastic strained is called softening rule. In Figure 2.14 two examples of these relationships are represented. To sum up, the hardening/softening law is the relationship between the changing in the yield locus and the plastic deformation and it describes the expansion/contraction of the yield surface. A model that simulates the behaviour of a real soil usually involves strain hardening and strain softening. Figure 2.14 – Examples of hardening and softening rules (Potts and Zdravković, 1999) 2.2.5. TWO DIMENSIONAL BEHAVIOUR To explain better these concepts, in this section it is considered a two dimensional situation. It is assumed that the yield and plastic potential functions are the same, in order to simplify the explanation. If the material behaviour is linear elastic perfectly plastic, the yield surface position does not change when the material is loaded. Such behaviour is illustrated in Figure 2.15. If the stress state is below the yield surface the behaviour is elastic and if it reaches the yield surface plastic straining takes place. The stress state cannot go further the yield surface so when the stress reaches it (point b), it stays constant and plastic straining occurs. The ratio between and is fixed by the gradient of the yield surface (the same as the plastic potential because of the associated conditions considered) at point b so the element of soil failed (Potts and Zdravković, 1999). Modelling Sand Instability within the Framework of Critical State Soil Mechanics 15 Figure 2.15 – Two dimensional behaviour of a linear elastic perfectly plastic material (Potts and Zdravković, 1999) In the case of a linear elastic hardening plastic material, the yield surface changes size with plastic strain. In isotropic hardening, it stays centred in the same point but if it is kinematic hardening, the center point changes position but the yield surface does not change size, as it is exemplified in Figure 2.16. Hardening can include both types at the same time. Figure 2.16 – Hardening types (Potts and Zdravković, 1999) As it is shown in Figure 2.17, until the stress state reaches the yield surface, the material has an elastic behaviour. If the stress increases beyond point b, plastic deformation takes place, the yield surface changes size and expands (isotropic hardening). The soil behaviour is elasto-plastic so there is the development of both elastic and plastic strains. The ratio between and can change with the continued loading. Failure happens when the yield surface stops hardening. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 16 Figure 2.17 – Two dimensional behaviour of a linear elastic hardening plastic material (Potts and Zdravković, 1999) In the case of a linear elastic softening plastic material, the yield surface reduces size with increasing of plastic strain (Figure 2.18). Once the point b is reached and plastic deformations occur, isotropic softening happens and the yield surface decreases its size. In that way, the stress σY has to reduce. The strain softening stress-strain curve can be followed if the action is made in strain (εY) control, which will be mostly impossible if the action is made under stress (σY) control (Potts and Zdravković, 1999). Failure will also be defined when yield surface stops to shrink, which can be the extreme position in zero effective stress (i. e., liquefaction). Figure 2.18 – Two dimensional behaviour of a linear elastic softening plastic material (Potts and Zdravković, 1999) 2.3. CRITICAL STATE THEORY 2.3.1. ISOTROPIC COMPRESSION According to Atkinson (1993), the rearrangement of the grains is what causes soil compression or dilation. Initial stiffness can decrease when state is loos, while it will increase in dense states. During the loading, the stress-strain line is not linear therefore the mechanisms of volume change due to Modelling Sand Instability within the Framework of Critical State Soil Mechanics 17 rearrangement of the grains accounts for non-linear bulk stiffness behaviour. On the unloadingreloading, the stiffness is higher than for the first loading as the grains cannot invert the rearrangement they suffered. Figure 2.19 shows the curve of isotropic compression and swelling on both linear and logarithmic horizontal scale and it represents the specific volume, υ, plotted against the mean effective stress, p’. Figure 2.19 – Isotropic compression and swelling (adapted from Atkinson, 1993) The line OACD is the normal compression line (NCL) and it represents the first loading. Its equation is given by expression (2.6). (2.6) The λ is the gradient and N is the value of υ when p’=1.0 kPa. The swelling line (also called unloading line) is ABC and it is given by equation (2.7). (2.7) Where κ is the gradient and υκ is the value of υ when p’=1.0 kPa. The parameters λ, κ and N are constant for a certain soil, except if there is an evolution of particle size, due for instance to particle breakage. There are many swelling lines as the soil can be unloaded from any point of the NCL. 2.3.2. WET AND DRY SIDE OF CRITICAL STATE Figure 2.20 represent the types of behaviour a soil has depending on whether it is on the wet or dry side of the critical state before shearing. A combination of pressure and specific volume define the state of the soil. The state of a conventional soil (that is, without interparticle cementation (bounding)) can never be above and to the right of the NCL (Roscoe et al., 1958). Heavily overconsolidated clays Modelling Sand Instability within the Framework of Critical State Soil Mechanics 24 The normalization consists in dividing the q and p’ by the normalizing parameters. In this way the lines such as NCL and CSL or another parallel to these will be represented by a point in the new normalized plot, shown in Figure 2.26. Figure 2.26 – Normalized CSL and NCL (Atkinson, 1993) 2.4. CAM-CLAY MODELS Roscoe et al. (1958) developed the original Cam-clay model, named by the authors in reference to the river flowing in the vicinity of Geotech Lab in Cambridge University. Later, the modified Cam-clay model was proposed by Roscoe and Burland (1968). These were the first critical state models, developed at the University of Cambridge. Afterwards, a large number of new models have been proposed based on the Cam-clay models with the purpose of achieving a better adjustment between predicted and observed behaviour in different soils, such as sands, residual and cemented soils, etc. A unified model for clays and sand was proposed by Yu (1998), with the name CASM (Clay and Sand Model). Its main feature is that a single yield function and plastic potential are used for both clay and sand under both drained and undrained conditions. 2.4.1. ORIGINAL CAM-CLAY The Original Cam-clay model was described by Schofield and Wroth (1968). This model combines the Critical State Soil Mechanics and the idea of a state boundary surface essential to the theory of plasticity. The equation that defines the yield curve is shown in (2.26). (2.26) Figure 2.27 represents the yield curve for the original Cam-clay model. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 25 Figure 2.27 – Yield curve for original Cam-clay This model is associated, so the yield function is identical to the plastic potential. The elastic law is defined by equations (2.27) and (2.28). The flow rule is expressed in (2.29) and the hardening law in (2.30). ( ) (2.27) (2.28) (2.29) ( ) (2.30) 2.4.2. CASM – CLAY AND SAND MODEL The Clay and Sand model (CASM), developed by Yu (1998), is a unified critical state constitutive model and it is based on the state parameter. The state parameter is defined as the difference between specific volume (or void ratio) and the specific volume (or void ratio) at the critical state at the same mean effective stress (Been and Jefferies, 1985). This parameter is very important when modelling sand behaviour. It depends on the specific volume ( , being the void ratio), two critical state constants (λ and Γ) and the mean effective stress (p’). Its expression is given by (2.31). Figure 2.28 shows the definition of some of these parameters. q p' CSL p' 0 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 26 (2.31) Figure 2.28 – Definition of state parameter, critical state constants and reference state parameter (Yu, 1998)  Elastic component CASM assumes the same elastic behaviour as the standard Cam-clay models, being the bulk and shear moduli defined by expressions (2.32) and (2.33). A constant value of Poisson’s ratio (ν) is specified, so that the shear modulus varies with the bulk modulus and therefore with the stress level. (2.32) ( ) ( ) (2.33)  Yield function The yield function of CASM is expressed in equation (2.34), where p’ is the mean effective stress, q is the deviatoric stress and p’0 is the preconsolidation pressure. The constant n is the stress-state coefficient and specifies the shape of the yield surface and r is the spacing ratio. ( ) ( ) (2.34) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 27 Knowing that parameter ξR is the reference state parameter obtained by equation (2.35) and the spacing ratio is defined by equation (2.36), the geometry shown in Figure (2.28) allows to write the equation (2.37) (Yu, 1998). ( ) (2.35) (2.36) ( ) ( ⁄ ) ( ) ( ⁄ ) (2.37) Replacing equation (2.37) into the equation of the yield function (2.34), the yield function can be defined in terms of state parameter, as follows: ( ) (2.38) If the parameters n and r are replaced by 1 and 2.718, respectively, the yield function reduces to the original Cam-clay yield surface.  Plastic potential The plastic flow rule used in CASM is non-associated, so the yield function and the plastic potencial are different. According to Yu (1998), the stress-dilatancy relation used in the model is due to Rowe (1962), as it succeeded in describing the deformation of sands and other granular materials. The dilatancy rate is defined in equation (2.39), where η=q/p’. ( ) (2.39) The plastic potential is obtained by the integration of Rowe’s stress dilatancy relation and it is expressed on equation (2.40), where β is the size parameter and can be determined by solving that equation for any given stress state (p’, q). ( ) ( ) ( ) ( ) ( ) (2.40) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 28  Hardening rule CASM postulates that the hardening law is of the isotropic volumetric plastic strain hardening type (Yu, 1998). The change in size of the yield surface is related to the incremental plastic volumetric strain, as shown in equation (2.41). ( ) (2.41) In appendix A the equations for the implementation of CASM are exposed. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 29 3 CASE HISTORY: THE BOUMERDÈS EARTHQUAKE, ALGERIA (2003) 3.1. DESCRIPTION OF THE CASE On May 21, 2003, an earthquake occurred in Boumerdès, a province located in the north of Algeria, close to the capital, Algiers. It was one of the most destructive earthquakes seen in the past years with a magnitude of 6.8, Richter scale. Since January 1716, when a severe earthquake destroyed Algiers, this was the worst seismic event in the region and caused a lot of structural damages and claimed the lives of 2271 people, injured more than 10000 and left about 160000 homeless (Bouhadad et al., 2004). The shock was so violent that it was felt on the Spanish and French coasts and on Italy (Genoa). To evaluate the severity of the damages, and based on the observed effects and press reports, an intensity of IX (Destructive) on the MSK scale was attributed to the epicentral area. Foundations of buildings lying on sand where liquefaction occurred, suffer a sudden loss of support, giving rise to differential settlements and consequently inducing structural damages. In Figure 3.1 some damages in buildings and transportation systems are illustrated. Figure 3.1 – a) Total displacement of the first floor; b) Elementary school in Corso, insufficient lateral resisting system; c) Damage to Highway 5 Bridge; d) Several floors pancaked (EERI, 2003) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 30 Many liquefaction occurrences were documented especially on the coastline and areas near the river. The liquefaction features vary in size and morphology. Figure 3.2 shows some regions affected by this phenomenon. Figure 3.2 – Distribution of liquefaction during the Bourmedès earthquake (Bouhadad et al., 2004) The earthquake triggered the sand liquefaction and excess water came to the ground surface (sand boils) causing landslides and cracking near the Isser River. The cracking appeared on both banks of the river causing the scattering of soil into the river. As the Isser River has a large solid flow a lot of sand was transported downstream to the Mediterranean Sea, as it is observed in Figure 3.3, an image from Google-Earth®. Figure 3.3 – Sand flow on Isser River, 22 May 2003 (Google) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 31 Examples of these features of liquefaction are shown on Figure 3.4. The sand boils (Figure 3.4 b)) caused damage to roads and the collapse of houses. Big craters were also observed together with boiled sand and ground fissures due to differential settlements and soil collapse (Figure 3.4 a) and c)). Figure 3.4 d) shows the damage caused by loss of ground support when the soil liquefaction pushed down part of a pavement. Figure 3.4 – a) Cracking; b) Sand boils; c) and d) damage on roads due to liquefaction (EERI, 2003) 3.2. MATERIAL DESCRIPTION The material that was used in the laboratory studies is sand from Les Dunes beach, located in Ain Beninan in Algiers, where liquefaction was observed. The original soil is a siliceous medium sand with uniform particle size with grains fairly regular and slightly angled (Figure 3.5). Figure 3.5 – Particles of Les Dunes sand Modelling Sand Instability within the Framework of Critical State Soil Mechanics 32 Figure 3.6 shows two soil gradation curves for this sand. The light blue curve was developed by Ghili Tahar at Laboratoire GIENA, in Faculty of Civil Engineering of Argel, and the dark blue was obtained at LabGEO in Faculty of Engineering of the University of Porto (FEUP), being this confirmed several times during the experimental program developed herein. Figure 3.6 – Soil gradation of Les Dunes Sand According to the curve determined by Ghili Tahar, the effective diameter, D10, is 0.204 mm and represents the grain diameter of the sieve through which 10% of the weight of particles passes. The values of D30, D60 and D100 are 0.331 mm, 0.439 mm and 0.800 mm, respectively, and have the equivalent meaning as D10. The coefficient of uniformity, CU, gives an idea of the diversity of the particles sizes and it is given by the equation (3.1). A high value of CU corresponds to a well graded soil and a low value of CU corresponds to a poor graded soil. According to the Unified Soil Classification System, a well graded soil has CU values higher than 4 (for gravels) or 6 (for sands). When CU is close to the unit, the soil is said to be uniform. (3.1) The coefficient of curvature, CC, can be correlated to the particles’ shape and is given by the equation (3.2). A soil is considered well graded when the value of CC is between 1 and 3. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 33 ( ) (3.2) According to the light blue curve in Figure 3.6, the coefficient of uniformity is CU=2.15 and the coefficient of curvature is CC= 1.22 revealing a poorly graded soil. The values of these coefficients determined in LabGEO (dark blue curve) are CU=1.55 and CC=1.01. Comparing the sand gradation exposed in Figure 3.6 with the boundaries in the boundary curves for liquefiable soil and potentially liquefiable soil (Figure 2.7), it is possible to note that the gradation curves for Les Dunes sand are located between the boundaries for the most liquefiable soil. As the sand is poorly graded, one can expect that it is susceptible to liquefy under loading. The minimum and maximum density and void ratio are represented on Table 3.1. They were determined by Ghili (2003) according to the recommended procedure of the Japanese Society of Soil Mechanics for fine sands (D100 < 2 mm) with less than 5 % of fines (≈ #200, ASTM). Following studies performed at FEUP showed that it is possible to obtain a higher value of emáx, but that is not relevant for this particular study. Table 3.1 – Densities and void ratios γd,min (kN/m3) 13.96 emáx 0.890 γd,máx (kN/m3) 17.24 emin 0.531 Fonseca (2009) and Pinheiro (2009) determined the value of the friction angle at critical state with data from the drained and undrained tests that did not liquefied until an axial strain of 20%. In Figure 3.7 the results of those tests are plotted, in t-s’ space. By defining a linear regression with those points, the slope of that line is equal to the tangent of the line angle (α). The friction angle is obtained with the expression of equation (3.3). Figure 3.7 – Definition of the Kf line (Fonseca, 2009) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 40 Figure 3.15 – Excess pore pressure versus axial strain for undrained tests with more than 400kPa of confining pressure Figure 3.16 – Excess pore pressure versus axial strain for undrained tests with less than 200kPa of confining pressure 0 200 400 600 800 1000 1200 010 20 30 Δu (kPa) εa (%) LD46 LD48 2ºtest_2nd_stage 4ºtest_2nd_stage 0 50 100 150 200 250 0123456 Δu (kPa) εa (%) LD12 LD42 LD44 LD45 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 41 Figure 3.17 – Volumetric strain versus axial strain for drained tests For lower values of confining pressure, in the undrained tests (Figure 3.13) it is evident a reduction of the deviatoric stress until zero while the excess pore pressure increases (Figure 3.16) until reaching the value of the confining pressure, lowering the shear strength of the soil and causing the occurrence of liquefaction. Observing the undrained test results for higher values of confining pressure, it is noticeable a fast increase of the excess pore pressure but then it starts decreasing. As the pore pressure diminishes the effective mean stress is never going to decrease until zero so liquefaction will not occur. In the drained tests, the deviatoric stress quickly increases with the axial strain at first but then it becomes relatively constant. Figure 3.17 shows that the specimens contract. At the end of the test LD63, the specimen dilates a little but this is probably due to the fact that in this part of the laboratory test the variable control was not so accurate. In Figure 3.14, LD64 starts increasing the deviatoric stress with axial strain but when the test changes from drained to undrained it suffers softening and the deviatoric stress decreases. The 2º_2nd stage and the 3º_2nd stage tests were not used in the work developed in this thesis because the first suffered a lot of hardening and did not reach the Critical State Line and the second suffered grain crushing. 0 0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 20 25 εv (%) εa (%) LD63 LD65 3ºtest_2nd_stage Modelling Sand Instability within the Framework of Critical State Soil Mechanics 42 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 43 4 REPRODUCTION OF THE TRIAXIAL TESTS WITH THE MODEL 4.1. INTRODUCTION In this chapter, the results of both drained and undrained triaxial tests performed in the laboratory were compared with the numerical solution. The numerical solution is obtained using a finite element program, Code_Bright which uses GiD system for pre-processing and post-processing. The parameters that define the sand are obtained from laboratory data or through calibration. The input of the tests in Code_Bright is also briefly explained. CASM is implemented in the program so the comparison between the experimental results and the numerical solution allows the verification and validation of the model. 4.1.1. CODE_BRIGHT Code_Bright is the code used to obtain the numerical solutions of the tests in this thesis. It was developed by the Department of Geotechnical Engineering and Geosciences of the Universitat Politècnica de Catalunya and it consists on a finite element program that analyses thermo-hydromechanical problems. It is written in FORTRAN and it does not use external libraries therefore the user has to introduce most of the inputs. The version used in this thesis was version 4, which is available for free at http://www.etcg.upc.edu/recerca/webs/code_bright/downloads. The problem is solved considering as variables (unknowns): solid displacements (u, on three special directions), liquid pressure (Pl), gas pressure (Pg) and temperature (T). Stresses and strains are related by the mechanical constitutive model together with the equation of stress equilibrium. The strains are described in terms of displacements. The deformations are controlled by small strains and small strain rates. There are four main groups of equations that govern this problem: balance equations, constitutive equations, equilibrium relationships and constraints definition. The constitutive equations relate the independent variables (or unknowns) and the dependent variables. The governing equations are written in terms of the unknowns when the constitutive equations are replaced in the balance equations (Code_Bright User’s Guide, 2012). Modelling Sand Instability within the Framework of Critical State Soil Mechanics 44 4.1.2. GID GiD is a program developed by the International Center for Numerical Methods in Engineering (CIMNE) at the Universitat Politècnica de Catalunya. It is a universal pre and postprocessor for numerical simulations in engineering which is developed to work with different codes, such as RamSeries, Tdyn, Vulcan or Code_Bright. The author of this thesis obtained a license to work with the program, in the version 11.0.4. For the definition of the geometry, the program works in a similar way to CAD (Computer Aided Design) system. The main difference is that the geometry is created in a hierarchical mode, which means that the higher level entities are constructed over entities of lower level. In Figure 4.1 the principal steps necessary to solve a problem with GiD are exposed. Figure 4.1– Main steps to solve a problem with GiD The graphic visualization is flexible so that the user can analyse and interpret the results without difficulty. There are many options to display the results such as contour maps at each step of the calculation process, time evolution of variables graphs or graphs with the evolution of one variable as a function of other and vector distribution. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 45 4.2. MODEL PARAMETERS CASM requires the definition of six parameters (material constants) in order to correctly reproduce the results for a specific material. These parameters describe the elastic behaviour, the yield surface shape and the critical state. Table 4.1 summarizes the constitutive model parameters required and Table 4.2 lists the initial state values and history variables of the soil. Table 4.1 – Description of constitutive model parameters Group Symbol Description Elastic components ν Poisson's ratio κ Slope of the isotropic swelling line (in υ-lnp' space) Yield surface Shape r Spacing ratio n Shape parameter Critical State Constants M Slope of the Critical State line (in q-p' space) λ Slope of Critical State Line (in υ-lnp' space) Table 4.2 – Description of initial state values and history variables Group Symbol Description Initial state values e0 Initial void ratio OCR Overconsolidation ratio K0 Coefficient of earth pressure at rest History variables p'0 Preconsolidation pressure F Current value of the yield function The results of the laboratory tests performed allow the definition of most CASM parameters (ν, κ, M, λ, e0, OCR, K0 and p’0). The experimental data is simulated using mathematical models. Other parameters are obtained through calibration (such as n, r and F) of the tests and the model. In this section, these parameters are defined. 4.2.1. ELASTIC PARAMETERS  Poisson’s ratio, ν According to Yu (1998), the Poisson’s ratio is typically in the range of 0.15-0.35 for clays and sand. For this particular sand, the Poisson’s ratio was determined by Fonseca (2009) using the expression in equation (4.1) where VP and VS are the P and S wave velocities respectively. Its value is 0.362. ( ) ( ) (4.1) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 46  Slope of the isotropic swelling line (in υ-lnp’ space), κ The value of κ is highly variable and depends on the effect of loading and unloading in soil grains (Atkinson, 1993). The swelling line is defined by the path of the unloading-reloading during isotropic compression and swelling. This parameter was obtained from an oedometer test performed in Les Dunes sand in a sample with 50 mm of diameter As shown in Figure 4.2, the slope of the isotropic swelling line plotted in υ-lnp’ space, κ is 0.005. According to Yu (1998) this is a typical value for sands. Figure 4.2 – Oedometer test results (50 mm diameter) for determination of κ 4.2.2. CRITICAL STATE CONSTANTS  Slope of the Critical State line in q-p' space, M The slope of the critical state line, also known as critical stress ratio, was calculated using equation (4.2) and the friction angle at the critical state obtained in previous studies (Section 3.2), the parameter M is determined to be 1.3047. Typical values for sands lie between 1.1 and 1.4 (Yu, 1998). (4.2)  Slope of the Critical State Line (in υ-lnp' space), λ The critical state line for sands is difficult to determine clearly. However, using the results of some drained and undrained triaxial tests performed in Les Dunes sand, by Fonseca (2009), Rocha (2010) and along the experimental program of Soares (reported in Viana da Fonseca and Soares, 2012) it was possible to draw it. The parameter λ is the slope of the critical state line in υ-lnp' space and, as it is seen in Figure 4.3, has the value of 0.021. y = -0.005ln(x) + 1.756 R² = 0.997 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 110 100 1000 10000 υ =1+e ln p' (kPa) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 47 Figure 4.3 – Critical State Line in υ-ln p' space (Viana da Fonseca and Soares, 2012) In order to validate this value, the results of two oedometer tests were used. One test was performed in a sample with 50 mm of diameter and other in a sample with a diameter of 75 mm. The slopes of the Normal Compression Lines were determined to be 0.022 and 0.023, respectively, as shown in Figure 4.4. The line was identified before the grain crushing. The zone after the NCL corresponds to the evolutive behaviour after particle breakage. Knowing that the Critical State Line is admitted to be parallel to the Normal Compression Line, it can be said that the slope of the CSL was well determined and the value of λ=0.021 is acceptable. Figure 4.4 – Oedometer test results for determination of λ; a) 50mm; b) 75mm y = -0.021ln(x) + 1.9659 R² = 0.9642 1.75 1.8 1.85 1.9 1.95 110 100 1000 υ =1+e ln p' (kPa) y = -0.023ln(x) + 1.9671 R² = 0.9956 1.75 1.8 1.85 1.9 1.95 2 1 100 10000 1+e ln p' y = -0.022ln(x) + 1.9898 R² = 0.9951 1.7 1.75 1.8 1.85 1.9 1.95 2 1 100 10000 1+e ln p' b) a) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 48 4.2.3. YIELD SURFACE SHAPE PARAMETERS The r and n parameters are specific for the model. Therefore, they cannot be determined directly from laboratory data and have to be calibrated using test results. In Figure 4.5, the effects of changing the value of n are shown. When n=1.5, the yield surface is symmetrical, when its value increases the maximum deviatoric stress tends to move right and when its value decreases that peak moves left. Figure 4.6 illustrates how the yield function changes according to the value of r. As the value of r increases, the peak of the yield surface decreases. Figure 4.5 – Evolution of the shape parameter n Figure 4.6 – Evolution of the spacing ratio r 0 20 40 60 80 100 120 140 160 050 100 150 200 250 q (kPa) p' (kPa) n=1.5 ; r=2 n=1 ; r=2 n=3 ; r=2 0 20 40 60 80 100 120 140 050 100 150 200 250 q (kPa) p' (kPa) r=2 ; n=1.5 r=3 ; n=1.5 r=4 ; n=1.5 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 49 The calibration is based on the plot of the yield surface and the results of undrained triaxial tests. For each test, the values of e0 and p’0 are different but the yield surface parameters should be the same. The values of p’0 for each undrained triaxial test are listed in Table 4.3 and they are obtained from the experimental results. The equation (4.3) was used to define the yield surface of the CASM model and it was obtained by equating the yield function to zero. ⇒ ( ( ) ) (4.3) Table 4.3 - Values of p’0 for undrained triaxial tests Designation Triaxial Test p'0 (kPa) LD12 200 LD42 23 LD44 30 LD45 100 LD46 400 LD48 1000 4º_2nd stage 1500 After trying several possibilities, the best values of n and r found were 3 and 15 respectively. The results are shown in Figure 4.7, where it is seen the similarity between the yield surfaces computed using the yield function from CASM (red) and the laboratory test results (grey). The 4º_2nd stage test was not used on the following analysis because as it can be seen in Error! Reference source not ound. g), its stress-path was peculiar and the values of p’ increase at the beginning of the test. Since this test was performed with a high confining pressure (1500kPa), the material is already behaving with a higher compressibility (the CSL steepens its slope) due to the increase in fines resulting from grain crushing. Therefore, as its behaviour is not the same as the other tests, it was decided not to use it in the present work. a) b) Figure 4.7 – Yield surfaces for undrained triaxial tests: a) LD12; b) LD42 (to be continued) 0 20 40 60 80 100 -10 40 90 140 190 240 q (kPa) p' (kPa) LD12 Yield function 0 2 4 6 8 10 12 0 5 10 15 20 25 q (kPa) p' (kPa) LD42 Yield function Modelling Sand Instability within the Framework of Critical State Soil Mechanics 56 Figure 4.13 – LD12: Stress-strain curve and Stress-path Figure 4.14 – LD42: Stress-strain curve and Stress-path Figure 4.15 – LD44: Stress-strain curve and Stress-path 0 20 40 60 80 100 120 -0.5 1.5 3.5 5.5 q (kPa) εa (%) LD12 Code 0 20 40 60 80 100 120 0 100 200 q (kPa) p' (kPa) LD12 Code 0 2 4 6 8 10 12 0 0.5 1 1.5 q (kPa) εa (%) LD42 Code 0 2 4 6 8 10 12 0 5 10 15 20 25 q (kPa) p' (kPa) LD42 Code 0 4 8 12 16 -0.5 0.5 1.5 2.5 q (kPa) εa (%) LD44 Code 0 4 8 12 16 010 20 30 q (kPa) p' (kPa) LD44 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 57 Figure 4.16 – LD45: Stress-strain curve and Stress-path Figure 4.17 – LD46: Stress-strain curve and Stress-path Figure 4.18 – LD48: Stress-strain curve and Stress-path 0 10 20 30 40 50 60 0 1 2 3 q (kPa) εa (%) LD45 Code 0 10 20 30 40 50 60 050 100 q (kPa) p' (kPa) LD45 Code 0 50 100 150 200 250 300 0246 q (kPa) εa (%) LD46 Code 0 50 100 150 200 250 300 0 200 400 q (kPa) p' (kPa) LD48 Code 0 100 200 300 400 500 600 0 2 4 6 q (kPa) εa (%) LD48 Code 0 100 200 300 400 500 600 0 500 1000 q (kPa) p' (kPa) LD48 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 58 Analysing the stress-paths and the stress-strain curves presented in the Figures above, it is clearly expressed that the results obtained with the model adjust rather well the laboratory test results so it can be said that the CASM model is adequate to simulate the behaviour of Les Dunes sand until the instability point. With the first initial time step imposed on the Interval Data menu (1e-5), Code_Bright was not able to reproduce results after the instability point, i.e., the peak of the deviatoric stress, q. This is shown in the Figures above, where it is notable that the model follows the stress path of the laboratory tests but, when the peak is reached, the path follows erratic directions. This is due to numerical instability during the reproduction of the physical instability. Since this does not really happen, some regularization technics need to be implemented in Code_Bright. However, when the initial time step was increased to 0.1, Code_Bright was able to compute results after passing the instability point, as shown in Figure 4.19 to Figure 4.24. This is justified by the fact that with a higher initial time step, the number of points calculated is smaller and probably the stresspath avoids matching the instability point (this point may be between two time steps) and so it continues to give results after that point. Figure 4.19 – LD12: Stress-strain curve and Stress-path Figure 4.20 – LD42: Stress-strain curve and Stress-path 0 20 40 60 80 100 120 0 2 4 q (kPa) εa (%) LD12 Code 0 20 40 60 80 100 120 0 100 200 q (kPa) p' (kPa) LD12 Code 0 4 8 12 0 0.5 1 1.5 q (kPa) εa (%) LD42 Code 0 4 8 12 0 5 10 15 20 25 q (kPa) p' (kPa) LD42 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 59 Figure 4.21 – LD44: Stress-strain curve and Stress-path Figure 4.22 – LD45: Stress-strain curve and Stress-path Figure 4.23 and Figure 4.24 present two undrained tests where the results from the Code show that the stress-path decreases after the peak of deviatoric stress, but the samples experiment dilatancy afterwards, that is, an increase in deviatoric stress. This experimental change in behaviour is called phase transformation and occurs when the stress point is above a given threshold in the p’-q plane. This important issue has not been tackled in the present work and would require a modification in the flow rule of CASM model. 0 4 8 12 16 0 1 2 3 q (kPa) εa (%) LD44 Code 0 4 8 12 16 010 20 30 q (kPa) p' (kPa) LD44 Code 0 10 20 30 40 50 60 0 1 2 3 q (kPa) εa (%) LD45 Code 0 10 20 30 40 50 60 050 100 q (kPa) p' (kPa) LD45 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 60 Figure 4.23 – LD46: Stress-strain curve and Stress-path Figure 4.24 – LD48: Stress-strain curve and Stress-path Observing the tests that suffered liquefaction (LD12, LD42, LD44 and LD45), it should be noted that the peak of the deviatoric stress is a little higher in the results obtained from Code_Bright than in the experimental results. Therefore, the instability line concept that will be studied in a different approach in the next Chapter, although it does not preclude the use of the results given by Code_Bright. Figure 4.25 shows the pore pressure versus the axial strain curves for all the six previous tests. It is noticeable that the pore pressure increases fast in the beginning but then it stabilizes. The Code results are very similar to the laboratory results, evidencing once more of how well the CASM implemented in Code_Bright can simulate the behaviour of Les Dunes sand until the instability point. 0 50 100 150 200 250 300 0 5 10 q (kPa) εa (%) LD46 Code 0 50 100 150 200 250 300 0 200 400 q (kPa) p' (kPa) LD46 Code 0 100 200 300 400 500 600 0 5 10 q (kPa) εa (%) LD48 Code 0 100 200 300 400 500 600 0 500 1000 q (kPa) p' (kPa) LD48 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 61 a) b) c) d) e) f) Figure 4.25 – Pore pressure versus axial strain: a) LD12; b) LD42; c) LD44; d) LD45; e) LD46; f) LD48 0 50 100 150 200 250 0 2 4 u (kPa) εa (%) LD12 Code 0 5 10 15 20 25 30 0 0.5 1 1.5 u (kPa) εa (%) LD42 Code 0 5 10 15 20 25 30 35 0123 u (kPa) εa (%) LD44 Code 0 20 40 60 80 100 120 0 1 2 3 u (kPa) εa (%) LD45 Code 0 100 200 300 400 02468 u (kPa) εa (%) LD46 Code 0 200 400 600 800 1000 0 1 2 3 4 5 u (kPa) εa (%) LD48 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 62 Figures 4.26, 4.27 and 4.28 represent,respectively, the deviatoric stress versus the axial straincurves, the pore pressure versus the axial strain curves and the stress-paths for all the undrained tests performed in the laboratory and their simulation with Code_Bright. Figure 4.26 – Deviatoric stress versus axial strain for undrained tests 0 100 200 300 400 500 0123456 q (kPa) εa (%) Test Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 63 Figure 4.27 – Pore pressure versus axial strain for undrained tests 0 200 400 600 800 1000 012345678 u (kPa) εa (%) Test Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 64 Figure 4.28 – Stress-path for undrained tests 0 100 200 300 400 500 0 100 200 300 400 500 600 700 800 900 1000 q (kPa) p' (kPa) CSL Test Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 65 5 STUDY OF INSTABILITY IN LES DUNES SAND 5.1. BACKGROUND ON INSTABILITY OF SANDS 5.1.1. PLASTIC WORK There are two types of work, based on the stresses and the plastic strain increment. The plastic work (first order work increment) is defined by equation (5.1). It defines the condition of irreversibility that requires that this plastic work is positive whenever a change in plastic strain occurs (Kim and Lade, 1988). (5.1) The second-order work increment is the one used to study the instability in the present work and will be explained further. 5.1.2. STABILITY POSTULATES Drucker (1959) and Hill (1958) proposed some stability postulates that provide conditions that are sufficient to ensure stability and guarantee uniqueness in both dynamic and static problems. The stability postulate proposed by Drucker for solid metals demands an associated plastic flow, where the plastic potential surface coincides with the yield surface. This postulate requires that the second increment of plastic work is greater than or equal to zero, which is represented in equation (5.2), where is the increment of stress and is the resulting increment in plastic strain. (5.2) For metals, when the second increment of plastic work is positive, the stress-strain relation is ascending which is associated with stability. However, when it is negative, it means that the stressstrain curve is descending, it has passed the peak (where the second increment of plastic work is zero) and it is unstable (Figure 5.1). Modelling Sand Instability within the Framework of Critical State Soil Mechanics 72 Figure 5.6 – Stress-paths of Drained-Undrained simulations The stress-path for 250 hours represents the fully drained test. As it is shown in Figure 5.6, all tests performed follow the drained path until the moment they start to be undrained. For tests that start to be undrained after their stress-path passes the instability line, the moment they are under undrained conditions they are already instable. If a test starts to be undrained before reaching the instability line, it follows a typical undrained stress-path, with increase of the deviatoric stress while the effective mean stress decreases. The moment it reaches that instability line, the deviatoric stress begins to reduce. The peak of the deviatoric stress is really close to that point where the stress-path crosses the instability line, so it can be concluded that if a test changes from drained to undrained conditions before its path reaches the instability line, it stays stable. From the time when the two curves intersect the specimen is unstable and the deviatoric stress starts decreasing. Observing Figure 5.6, it is also notable that the calculation process stops when the stress-paths reach the critical state line, except for the one for 30 hours in Flux B. C. condition. Table 5.4 presents the values of the deviatoric and effective mean stresses of the final point of each stress-path. 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 q (kPa) p' (kPa) CSL Instability line 250 hours 30 hours 15 hours 10 hours 5 hours 2 hours 1 hour 30 min Modelling Sand Instability within the Framework of Critical State Soil Mechanics 73 Table 5.4 – Final values of q and p’ for each test Time (hours) q (kPa) p' (kPa) 30 539.75 452.11 15 228.59 186.52 10 164.20 125.85 5 119.34 91.47 2 94.55 72.47 1 87.21 66.85 0.5 83.62 64.09 In Figure 5.7, the final point of each stress-path is represented along the Critical State Line. As it is shown, the stress-paths do not finish all in the same point. As the drained to undrained process occurs at different steps, the p’0 value for each simulation is different and so the final point of each stress-path is different. Nevertheless, all the stress-paths finish at or near the Critical State Line, i. e., the failure line. Figure 5.7 – Final points of the stress-paths and Critical State Line 5.2.4. SIMULATION OF LD64 TEST In Subsection 3.3.2, some tests performed in the laboratory were described. As referred, one of the LD64 started to be performed under drained conditions but at a certain point, the BP valves had to be closed and the test became undrained. In this section, this test will be reproduced with Code_Bright to study the response of the code to this type of test and validate the simulations aforementioned. The parameters of CASM, the geometry and the conditions are the same as the ones mentioned above. The specific parameters for the test are presented in Table 5.5. 0 50 100 150 200 250 300 050 100 150 200 250 q (kPa) p' (kPa) CSL Modelling Sand Instability within the Framework of Critical State Soil Mechanics 74 Table 5.5 – Values of e0, n, p’0 and initial stresses for LD64 e0 n X stress (kPa) Y stress (kPa) Z stress (kPa) p'0 (kPa) 0.859 0.462 1000 1000 1000 1000 In order to find the time when the conditions change from drained to undrained, a complete drained test was simulated. The peak in the deviatoric stress in the test performed in the laboratory was at 1293.80 kPa. The results of the deviatoric stress versus time were exported from Code_Bright. After exporting the results of the drained test, it was found that the peak of the deviatoric stress occurred after 17.75 hours, so the test was turned to undrained at that time. Therefore, the computation was developed on two intervals. The first until 17.75 hours where the specimen was exposed to drained conditions and the second from 17.75 hours to 250 hours where the specimen was submitted to undrained conditions. Figure 5.8 shows the results from Code_Bright compared with the experimental data. Figure 5.8 – LD64: Stress-strain curve and Stress-path As it is observed in Figure 5.8, both the stress-strain curves and the stress-paths are similar. This means that Code_Bright is able to reproduce well this type of test control with CASM. When the test changes from drained to undrained conditions, the deviatoric stress decreases but the plastic strain does not stop. Softening is observed and the soil deforms plastically with decreasing levels. In Figure 5.9, the stress-path is represented as well as the critical state line and the instability line defined for this sand. It is observed that when the test becomes undrained, the stress-path is already inside the region of potential instability. After that point, both the deviatoric and effective mean stresses decrease and the stress-path stops at the Critical State Line. As the Code cannot compute instability, the stress-path just follows the yield surface, as it is shown in Figure 5.10, where both the initial and final yield surfaces are drawn. As the test is, at first, drained, the yield surface expands. The yield surfaces were defined by equation (5.6). The value of p’0 for the first yield surface is 1000 kPa and the value for the second is 3570.28 kPa. The last value of p’0 was obtained by replacing the values of q and p’ at the end of the drained part of the test in equation (5.7) (Table 5.6). 0 200 400 600 800 1000 1200 1400 0 5 10 15 q (kPa) εa (%) LD64 Code 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 q (kPa) p' (kPa) LD64 Code Modelling Sand Instability within the Framework of Critical State Soil Mechanics 75 ⇒ ( ( ) ) (5.6) ( ) (5.7) Table 5.6 – Value of p’0 q (kPa) p' (kPa) p'0 (kPa) 1301.50 1433.56 3570.28 Figure 5.9 – LD64, CSL and instability line 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 q (kPa) p' (kPa) LD64 Instability Line CSL Modelling Sand Instability within the Framework of Critical State Soil Mechanics 76 Figure 5.10 – LD64 and yield surfaces 5.3. DILATANCY RATE According to Rowe (1962), the dilatancy and strength of a group of particles in contact when subjected to a deviatoric stress system depend on three main parameters: the friction angle between the surface of the particles, the geometrical angle of packing and the degree of energy loss during remolding. The stress-dilatancy relation used in this work is due to Rowe (1962). As it was explained in Chapter 2, it defines the relation between stress ratio (η) and dilatancy rate (d) which is expressed on equation (5.8). ( ) (5.8) For each undrained test, equation (5.8) was applied, using the deviatoric and effective mean stress values corresponding to the instability point. Table 5.7 – Dilatancy rate for undrained tests at the instability point Designation Triaxial Test q (kPa) p' (kPa) η = q/p' d LD12 101.59 152.43 0.666 0.514 LD42 11.55 17.86 0.647 0.528 LD44 15.54 23.42 0.664 0.516 LD45 51.18 74.03 0.691 0.497 LD46 204.18 297.95 0.685 0.501 LD48 512.12 731.08 0.701 0.490 0 200 400 600 800 1000 1200 1400 1600 1800 0 1000 2000 3000 4000 q (kPa) p' (kPa) YS1 YS2 LD64 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 77 Analysing the values of the dilatancy rate obtained on each undrained test (Table 5.7), it can be concluded that the value of dilatancy for Les Dunes sand at the instability point is about 0.5. That means that when the dilatancy rate reaches the value of 0.5, the soil may experience instability. Drained tests remain stable after their stress-path crosses the instability line. The dilatancy rate in the intersection point of the instability line and the drained test stress-path was studied. In Figure 5.11 the two lines for test LD63 are represented. Figure 5.11 – LD63 stress-path and instability line Equation (5.9) defines the stress-path while equation (5.10) defines the instability line. (5.9) (5.10) Solving the equation system, the values of the deviatoric and effective mean stress for the intersection point were found. Table 5.8 shows the results along with the value of the dilatancy rate calculated. Table 5.8 – Dilatancy rate for the intersection point of LD63 with the instability line q (kPa) p' (kPa) η d 454.22 651.58 0.697 0.493 The same process was applied on the results of LD65 test. Figure 5.12 shows the intersection between the stress-path and the instability line, equation (5.11) defines the stress-path. In Table 5.9 the values 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 q (kPa) p' (kPa) LD63 Instability Line (651.58 ; 454.22) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 78 of the deviatoric and effective mean stress at the intersection point and the dilatancy rate are represented. The equation of the instability line is the same presented on equation (5.10). Figure 5.12 – LD65 stress-path and instability line (5.11) Table 5.9 – Dilatancy rate for the intersection point of LD65 with the instability line q (kPa) p' (kPa) η d 13.63 19.55 0.697 0.493 As for the test LD64, the change from drained to undrained conditions happened after the stress-path crossed the instability line. That way, the dilatancy rate at the point where the drained path intersects the instability line was studied. The process aforementioned was repeated for LD64 results. In Figure 5.13 the intersection between the stress-path and the instability line is shown, equation (5.12) defines the stress-path and in Table 5.10 the values of the deviatoric and effective mean stress at the intersection point and the dilatancy rate are represented. The equation of the instability line is the same presented on equation (5.10). 0 10 20 30 40 010 20 30 40 50 q (kPa) p' (kPa) LD65 Instability Line (19.55 ; 13.63) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 79 Figure 5.13 – LD64 stress-path and instability line (5.12) Table 5.10 – Dilatancy rate for the intersection point of LD64 with the instability line q (kPa) p' (kPa) η d 907.95 1302.46 0.697 0.493 The instability line appears to be both related to a constant value of dilatancy (0.493) or a constant value of stress ratio (0.697). According to instability concept, it is the direction of plastic strain that initiate the instability and thus it is more rational to link the instability line to the value of dilatancy. However, as in CASM model the dilatancy depends only on the stress ratio (η), both concepts are equivalent. A conclusion of that is that the shape of the instability line in p’‒ q diagram (straight line) validates the assumption made for the dilatancy rule in CASM model. The model appears then to allow for modeling liquefaction in materials where instability line is a straight line in the p’‒ q space. 5.4. SECOND-ORDER WORK In order to validate the described trends and to continue the study of instability in Les Dunes sand, the second-order work criterion was applied to the tests. Using the expression of equation (5.4) the second-order work increment was calculated for each step of the simulation. The values of the variables used in the expression for the definition of the second-order work increment were exported from Code_Bright. 5.4.1. DRAINED AND LD64 TESTS The second-order work increment values were compared with the evolution of the axial strain for the drained tests. Figure 5.14 shows these relations for LD63 and LD65. 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 q (kPa) p' (kPa) LD64 Instability Line (1302.46 ; 907.95) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 80 Figure 5.14 – Second-order work increment versus axial strain for drained tests These results confirm what was defended above, for drained tests the second-order work increment is always positive so they are stable. As LD64 stress-path crosses the instability line while under drained conditions, its analysis was considered in this subchapter. Figure 5.15 represent the evolution of the second-order work increment with axial strain and with time. In this case, it is important to reproduce the second-order work with time because, as it was explained before, this test starts to be drained and at 17.75 hours became undrained. As the stress-path has passed the instability line when the test starts to be undrained, it is inside the region of potential instability. As it is shown in Figure 5.15, when the time is 17.75 hours the second-order work is zero and starts to be negative, showing the instability of the specimen when it evolves in undrained conditions. Figure 5.15 – Second-order work increment versus axial strain and time for LD64 5.4.2. UNDRAINED TESTS The same procedure was applied to the undrained test results. In this case, the deviatoric stress was also represented, to show that when the second-order work increment is zero (starting to be negative) the deviatoric stress reaches a peak. Figures 5.16 to 5.21 present the results of the second-order work 0 1 2 3 4 5 6 010 20 30 40 d2W (kPa) εa (%) LD63 0 0.01 0.02 0.03 0.04 0.05 0.06 010 20 30 d2W (kPa) εa (%) LD65 -0.15 0.35 0.85 1.35 1.85 024 d2W (kPa) εa (%) LD64 -0.15 0.35 0.85 1.35 1.85 0 8.875 17.75 d2W (kPa) time (h) LD64 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 81 increment (orange curve) and the deviatoric stress (red curve) versus the axial strain. It can be concluded that when the second-order work increment is zero and begins to be negative, the deviatoric stress is maximum (and starts decreasing). This confirms the instability criterion explained above. The evolution and path of the second-order work increment was not an object of this study. However it would be interesting to study it in the future. Figure 5.16 – Second-order work increment and deviatoric stress versus axial strain for LD12 Figure 5.17 – Second-order work increment and deviatoric stress versus axial strain for LD42 -450 -375 -300 -225 -150 -75 0 75 150 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0 2 4 6 8 10 d2W (kPa) q (kPa) εa (%) -22.5 -15 -7.5 0 7.5 15 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0 0.5 1 d2W (kPa) q (kPa) εa (%) Modelling Sand Instability within the Framework of Critical State Soil Mechanics 88 To determine this value, the expression of the yield surface, defined in equation (2.32) of Subchapter 2.4.2. was equated to the expression of the CSL in the Critical State Theory (equation (2.14) in subchapter 2.3.4.), as it is shown in the expression (5.16). ( ( ) ) (5.17) The result is an expression that relates p’ with p’0 (equation (5.18)). (5.18) Replacing the value of p’ in equation (5.17), the NCL is defined in e-lnp' space as being (5.19). ( ) (5.19) To define the instability line, its equation is equated by the yield function (equation (5.20)). ( ( ) ) (5.20) The result is an expression that relates ln p’ with a function containing p’0 (equation (5.21)). ( ) (5.21) Replacing the value of ln p’ in equation (5.17), the instability line is defined in e-lnp' space as being (5.22). ( ) (5.22) In Figure 5.31 the three lines are represented. The parameters of the sand were replaced in the expressions above. It can be concluded that the instability line is also parallel to the CSL and NCL and it is situated between them. The instability line is closer to the CSL than to the NCL. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 89 Figure 5.31 – CSL, NCL and Instability line in elnp’ space 0.8 0.85 0.9 0.95 1 1.05 110 100 1000 e ln p' (kPa) NCL Instability Line CSL Modelling Sand Instability within the Framework of Critical State Soil Mechanics 90 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 91 6 CONCLUSIONS 6.1. CONCLUSIONS The aim of this work was to continue the study of Les Dunes sand, from a beach in Ain Benian in Algeria, where liquefaction occurred during the 2003 Boumerdès earthquake. This study was done based on a critical state model for clays and sand (CASM), developed by Yu (1998). This was chosen because it is a simple and unified formulation that has been proven suitable for modelling sand behaviour. First, a comparison between experimental data and the CASM model was done. CASM is implemented in Code_Bright, so in order to compare the results of the model with the laboratory data, some tests with the same characteristics were simulated. The results were very satisfactory, therefore, it can be concluded that CASM is a valid and good model to compute and simulate the real Les Dunes sand behaviour observed under both drained and undrained conditions. The n and r parameters are essential to define the yield surface shape. They are specific for the model and depend on the type of soil that is being study. Comparing the yield surface from CASM with the stress path obtained from the laboratory tests, the value of these parameters were found to be n = 3 and r = 15. When dealing with Code_Bright, some numerical instability was observed during the reproduction of the physical instability. To avoid this problem, the initial time step was increased. The numerical results are once again very similar to the experimental ones. For the study of instability, the second order work increment criterion was applied. The instability line was proven to be a straight line, passing in the peak of the deviatoric stress in q ‒p’ space, for the undrained tests. According to this criterion, a specimen under drained conditions is never unstable. When under undrained conditions, the specimen may become unstable after its stress-path crosses the instability line and enters the region of potential instability. This was also confirmed with the simulation of drained-undrained tests. These tests start to be drained and at different time the conditions were changed to undrained. It was concluded that if the undrained conditions start when the stress-path has already cross the instability line, the specimen may already be unstable (second order work increment is zero in that point). If a test starts to be undrained before reaching the instability line, it stays stable. It is only when the path crosses the instability line that the second order work increment is zero and instability may occur. As for the dilatancy rate, proposed by Rowe (1962), it was proven to be around 0.49 in the point where the stress-paths intersect the instability line. In CASM the dilatancy depends on the stress ratio, which controls the dilatancy rate, so in this case, the value of the stress ratio is the slope of the instability Modelling Sand Instability within the Framework of Critical State Soil Mechanics 92 line. The shape of the instability line in p’‒ q diagram (straight line) validates the assumption made for the dilatancy rule in CASM model. In conclusion, the results found can lead to a better understanding and modelling of instabilities in saturated granular soils and can help the continuation of the study of Les Dunes sand and the soil susceptibility for liquefaction. 6.2. FUTURE DEVELOPMENTS As in any study of model validation, the quantity and quality of the experimental results are very important. To assure that a model is valid and can be used to simulate a specific soil behaviour, the results of the simulations have to be compared with further experimental data on other soils. In this work, there were used two drained tests and six undrained test. It would be important to perform more undrained and drained triaxial tests, to better define the critical state line and the instability line as well as the sand parameters. When dealing with this type of study, the more extensive experimental data the more reliable will be the reproduction of real soil behaviour. Other future development is to compute the instability in Code_Bright. A lot of numerical and programing work has to be done in order to improve the abilities of this Code. It would be a significant improvement if Code_Bright could simulate the results until the annulment of the mean effective stress (liquefaction). Further developments would be the reproduction of the point of dilation (phase transformation point) with the Code. This would be very important to correctly simulate liquefaction and distinguish the perfect liquefaction (with annulment of the stresses) from the temporary instability. Modelling Sand Instability within the Framework of Critical State Soil Mechanics 93 REFERENCES Alonso, E. E., Gens, A., Josa, A. (1990). A constitutive model for partially saturated soils. Géotechnique 40(3): 405-430. Andrade, J. E. (2009). A predictive framework for liquefaction instability. Géotechnique 59 (8), 673682. Andrade, J. E., Ramos, A. M. and Lizcano, A. (2013). Criterion for flow liquefaction instability. Acta Geotechnica Atkinson, J. (1993). Introduction to the Mechanics of Soils and Foundations. 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Modelling Sand Instability within the Framework of Critical State Soil Mechanics 96 Modelling Sand Instability within the Framework of Critical State Soil Mechanics 97 APPENDIX A IMPLEMENTATION OF CASM ELASTO-PLASTIC MODEL FOR TRIAXIAL PATH A.1. ELASTIC LAW [ ] ( )( ) [ ] [ ] (A.1) [ ] [ ( ) ( ) ( ) ] [ ] (A.2) With: ( ) (A.3) And ( ) (A.4)