The Runge-Kutta Taylor-SPH model, a new improved model for soil dynamics problems
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(3) UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE CAMINOS, CANALES Y PUERTOS DEPARTAMENTO DE INGENIERÍA Y MORFOLOGÍA DEL TERRENO. THE RUNGE-KUTTA TAYLOR-SPH MODEL, A NEW IMPROVED MODEL FOR SOIL DYNAMICS PROBLEMS. TESIS DOCTORAL. THOMAS BLANC Ingeniero de Montes. DIRECTOR DE TESIS: MANUEL PASTOR PÉREZ Dr. Ingeniero de Caminos, Canales y Puertos. MADRID 2011.
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(5) Título de la tesis:. “THE RUNGE-KUTTA TAYLOR-SPH MODEL, A NEW IMPROVED MODEL FOR SOIL DYNAMICS PROBLEMS”. Autor:. D. Thomas Blanc. Director: D. Manuel Pastor Pérez. Tribunal nombrado por el Mgfco. y Excmo. Sr. Rector de la Universidad Politécnica de Madrid, el día …….. de ……………………2011. Presidente D. ..……………………………………………………………….. Vocal 1º D. ..…………………………………………………………………. Vocal 2º D. ..…………………………………………………………………. Vocal 3º D. ..…………………………………………………………………. Secretario D. ..………………………………………………………………... Realizado el acto de defensa y lectura de la tesis el día …… de ……………. de 2011 en……………, los miembros del tribunal acuerdan otorgar la calificación de:………………………………………………………………... EL PRESIDENTE. LOS VOCALES. EL SECRETARIO.
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(7) Acknowledgements First I would like to thank Manuel Pastor Pérez for his invaluable ideas brought to this thesis, his advice, his encouragements, his support and the trust that he has manifested during the realization of this work. I would like to tell him that beyond his explanations on scientific topics, I appreciate his availability and his good mood.. I would also like to thank Pablo Mira who has ever been available for answering to my questions and for giving me some advice on how improving this work.. I would like to thank the Spanish ministry, Ministerio de Educación, for granting me the scholarship Formación del Profesorado Universitario.. I am very grateful to have had the opportunity to work in the M2i group belonging to CEDEX (Centro de Estudios y Experimentación de Obras Públicas) and UPM (Universidad Politécnica de Madrid). The members of this group have accompanied me in the development of this work. They have always let me share in their friendly atmosphere. Thus I would particularly thank to Ana Sofía, Bouchra, Cristina, Diego, Federico, Honghen, Jose Antonio, Miguel, Mila, Paola, Silvia, and Valentina.. I would like to thank my family and in particular to my parents who have supported me and encouraged me in doing this work.. Finally I thank Cristina a lot to accompany me everyday life.. I.
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(9) Abstract The study developed in this thesis is focused on numerical modeling of dynamics behavior of dry soils in the framework of small and large deformation theory and of dynamics behavior of saturated soils in the framework of small deformation theory. A numerical code has been developed to analyze such behaviors and presents three versions: -. The Runge-Kutta Taylor-SPH for analysis of dry soils behavior in dynamic (small deformation theory). -. The new Runge-Kutta Taylor-SPH for analysis of dry soils behavior in dynamic (large deformation theory). -. The v pw Runge-Kutta Taylor-SPH for analysis of saturated soils behavior in dynamic (small deformation theory). Both of them are based on a mathematical model which describes the dynamic of viscoplastic solids. The governing equations of the mathematical model behind the Runge-Kutta Taylor-SPH and the new Runge-Kutta Taylor-SPH are: i) the balance of momentum equation, ii) the constitutive equation and iii) the kinetic relation. The mathematical model used to describe the dynamics behavior of saturated soil is based on the u pw model of Zienkiewicz and his team. This model represents the behavior of the mixture of solid particles and pore water. It is based on the following governing equations: i) the mass balance equations of the solid phase, of the pore water and of the mixture, ii) the momentum balance equations of the solid phase, of the pore water and of the mixture, iii) the constitutive equation and iv) the kinetic relation. In the thesis we used the Perzyna‟s model as the constitutive equation. It is a simple viscoplastic model. This constitutive equation is completed by the Von Mises yield criterion or the yield surface of the modified Cam-Clay model depending on the problem studied. The mathematical model is formulated in terms of velocity and stress for the dry soils behavior and in terms of velocity, effective stress and pore water pressure for the saturated soils behavior. This formulation avoids the shortcomings of the classical formulation of solid dynamics.. III.
(10) IV.
(11) The numerical tool to discretize the equations of the mathematical model is the Smoothed Particle Hydrodynamics method (SPH). This work presents the theory and the limitations of this method. The classical SPH method presents a tensile instability when it is applied to solids. In the different versions of the Runge-Kutta Taylor-SPH model, the equations of the mathematical model are first discretized in time using the Taylor-Galerkin method. In a second step the equations are discretized in space with the SPH method. The mixing between the SPH method and the Taylor-Galerkin method allows avoiding the development of the SPH tensile instability and in consequence, the new model can be used for solids modeling. The numerical code proposed in this work has been tested with 17 case studies. The results obtained with the numerical code have been compared to analytical solutions when it was possible. The results show that the proposed numerical models are accurate and useful to predict localized failures of dry and saturated soils.. V.
(12) VI.
(13) Resumen El estudio desarrollado en esta tesis está centrado en la modelización numérica del comportamiento dinámico de los suelos secos en el marco de las teorías de pequeñas y grandes deformaciones y del comportamiento dinámico de los suelos saturados en el marco de la teoría de pequeñas deformaciones. Un código numérico ha sido desarrollado y presenta las tres versiones siguientes: -. El modelo Runge-Kutta Taylor-SPH para el análisis del comportamiento de los suelos secos en dinámica (pequeñas deformaciones). -. El “nuevo” modelo Runge-Kutta Taylor-SPH para el análisis del comportamiento de los suelos secos en dinámica (grandes deformaciones). -. El modelo. v pw. Runge-Kutta Taylor-SPH para el análisis del. comportamiento de los suelos saturados en dinámica (pequeñas deformaciones) Las tres versiones se basan en un modelo matemático capaz de describir la dinámica de los sólidos viscoplásticos. Las ecuaciones del modelo matemático en el cual se basan el modelo Runge-Kutta Taylor-SPH y el “nuevo” modelo RungeKutta Taylor-SPH son: i) la ecuación de balance del momento lineal, ii) la ecuación constitutiva y iii) la relación cinética. El modelo matemático utilizado para describir el comportamiento dinámico de los suelos saturados viene del modelo u pw introducido por Zienkiewicz y su equipo. Este modelo representa el. comportamiento de la mezcla de partículas solidas con agua intersticial. Está basado en las ecuaciones siguientes: i) las ecuaciones de balance de masa de la fase solida, del agua intersticial y de la mezcla, ii) las ecuaciones de balance del momento lineal de la fase solida, del agua intersticial y de la mezcla, iii) la ecuación constitutiva y iv) la relación cinética. En este trabajo la ecuación constitutiva elegida es el modelo viscoplástico de Perzyna. La ecuación constitutiva está completada por el criterio de fluencia de Von Mises o por la superficie de fluencia del modelo Cam-Clay modificada según el problema estudiado. El modelo matemático está formulado en velocidades y tensiones para el comportamiento de los suelos secos y en velocidades, tensiones efectivas y presión intersticial para los suelos saturados.. VII.
(14) VIII.
(15) Esta formulación elimina los defectos de la formulación clásica de la dinámica de sólidos. La herramienta numérica para discretizar las ecuaciones del modelo matemático es el método Smoothed Particle Hydrodynamics (SPH). Este trabajo presenta la teoría y las limitaciones de este método. El método clásico SPH presenta una inestabilidad de tensión cuando se aplica a sólidos. En las varias versionas del modelo Runge-Kutta Taylor-SPH, las ecuaciones del modelo matemático se discretizan en el tiempo basándose en el método Taylor-Galerkin. En una segunda etapa las ecuaciones se discretizan en el espacio con el método SPH. La mezcla entre el SPH clásico y el método Taylor-Galerkin permite evitar el desarrollo de la inestabilidad de tensión y en consecuencia el nuevo modelo se puede utilizar para modelizar el comportamiento dinamico de sólidos. El código numérico propuesto en esta tesis ha sido probado con 17 casos. Los resultados obtenidos han sido comparados con soluciones analíticas cuando era posible y han mostrado que el modelo Runge-Kutta Taylor-SPH es exacto y es útil para predecir roturas localizadas en suelos secos y saturados.. IX.
(16) X.
(17) Table of contents ACKNOWLEDGEMENTS .................................................................... I ABSTRACT .................................................................................... III RESUMEN ................................................................................... VII TABLE OF CONTENTS .................................................................... XI LIST OF FIGURES .......................................................................XVII LIST OF TABLES ...................................................................... XXIII NOTATIONS ............................................................................ XXIII CHAPTER 1 INTRODUCTION ............................................................ 1 1.1. Motivations ............................................................................................. 1 1.2. Objectives ................................................................................................ 2 1.3. Methodology and structure of the work .................................................. 3. CHAPTER 2 MATHEMATICAL MODEL .............................................. 9 2.1. Shortcomings of the classical approach .................................................. 9 2.2. Basics of continuum mechanics ............................................................ 10 2.2.1. Lagrangian description versus eulerian description .................. 10 2.2.2. Displacement and velocity vectors ............................................ 12 2.2.3. The stress tensor ........................................................................ 12 2.2.4. Principal stress........................................................................... 13 2.2.5. Decomposition of the stress tensor into a hydrostatic and a deviatoric component ................................................................ 14 2.2.6. Invariants of the stress tensor .................................................... 15 2.2.7. Infinitesimal strain tensor .......................................................... 18 2.3. Balance of momentum .......................................................................... 19 2.4. Constitutive equation ............................................................................ 19 2.4.1. Ill posing of the mathematical model for elastoplastic materials .................................................................................... 19 2.4.2. Time dependent behavior of soils under dynamics conditions .................................................................................. 20 2.4.3. The Perzyna model .................................................................... 21 2.4.4. Yield criterion of Von Mises ..................................................... 22 2.4.5. The modified Cam-Clay model ................................................. 25. XI.
(18) 2.4.6. Elasticity matrix in plane stress condition and in plane strain condition .................................................................................... 28 2.4.7. Summary of the Perzyna model ................................................ 30 2.5. Kinetic relation ...................................................................................... 31 2.6. System of equations of the mathematical model in two dimensions .... 31 2.6.1. System of equations in plane stress condition ........................... 31 2.6.2. System of equations in plane strain condition ........................... 33 2.6.3. Conservation form of the equations .......................................... 34 2.7. Conclusions ........................................................................................... 35. CHAPTER 3 PROPOSED MODEL: THE TWO-STEP RUNGE-KUTTA TAYLOR-SPH ALGORITHM ........................................................... 37 3.1.Introduction ............................................................................................ 37 3.2. The Smoothed Particle Hydrodynamics method: Backgrounds and Theory .......................................................................................................... 37 3.2.1. Background ............................................................................... 37 3.2.2. General aspects .......................................................................... 38 3.2.3. Integral approximation of a function ......................................... 39 3.2.4. Smoothing kernel function: properties and examples ............... 40 3.2.5. Integral approximation of the spatial derivative of a function .. 44 3.2.6. Particle approximation .............................................................. 45 3.3. SPH pitfalls ........................................................................................... 48 3.3.1. Boundary deficiency problem ................................................... 48 3.3.2. SPH Tensile instability and its solutions proposed in the literature .................................................................................... 53 3.4. Discretization in time by the classical two-step Taylor-Galerkin method .......................................................................................................... 55 3.4.1. Where does the two-step Taylor-Galerkin method come from? ......................................................................................... 55 3.4.2. The two-step Taylor Galerkin method ...................................... 57 3.4.3. Limitations of the two-step Taylor-Galerkin Method ............... 60 3.5. Discretization in space of the equation of the Runge-Kutta TaylorGalerkin model ............................................................................................. 64 3.5.1. SPH grid used by the Runge-Kutta Taylor-SPH model ............ 64 3.5.2. Time discretization of the governing equations ........................ 65 3.5.3. Spatial discretization by the corrective smoothed particle hydrodynamics method ............................................................. 66 3.5.4. Boundary conditions ................................................................. 68 3.5.5. Calculation steps of the algorithm ............................................. 68. XII.
(19) 3.6. Conclusions ........................................................................................... 70. CHAPTER 4 VALIDATION. OF THE RUNGE-KUTTA TAYLOR-SPH MODEL WITH SMALL DEFORMATION PROBLEMS ............................ 73. 4.1. Purpose .................................................................................................. 73 4.2. 1D Elastic bar: Test of stability............................................................. 73 4.3. Propagation of a shock wave in a 1D viscoplastic bar .......................... 78 4.4. Strain localization in a 2D soil sample .................................................. 85 4.5. Vertical slope under gravity .................................................................. 93 4.5.1. Elastic vertical slope.................................................................. 94 4.5.2. Viscoplastic vertical slope without softening.......................... 100 4.5.3. Vertical slope with softening................................................... 105 4.6. Conclusions ......................................................................................... 107. CHAPTER 5 ADAPTATION AND VALIDATION OF THE RUNGE-KUTTA TAYLOR-SPH FOR LARGE DEFORMATION PROBLEMS ................. 109 5.1. Introduction ......................................................................................... 109 5.2. Modifications of the algorithm of the Runge-Kutta Taylor-SPH ....... 110 5.2.1. Problem of the zero-energy mode: Hourglass deformation .... 110 5.2.2. From small deformation to large deformation theory ............. 119 5.2.3. Updating of the smoothing length ........................................... 120 5.2.4. Nearest neighboring particle searching ................................... 122 5.2.5. Free-surface detection and calculation of the normal to the free-surface .............................................................................. 124 5.2.6. Adapting the boundary conditions .......................................... 133 5.3. Validation of the Runge-Kutta Taylor-SPH with large deformation problems ..................................................................................................... 135 5.3.1. Purpose .................................................................................... 135 5.3.2. Propagation of a wave on an elastic one-dimensional bar ...... 136 5.3.3. Propagation of a wave on a viscoplastic bi-dimensional bar .. 140 5.3.4. Failure of an extremely low cohesive vertical slope under gravity...................................................................................... 144 5.3.5. Failure of a cohesive vertical slope under gravity................... 148 5.3.6. Failure of a vertical cut under constant loading ...................... 151 5.3.7. Failure of a shallow stratum under a wide strip footing: bearing capacity test ................................................................ 157 5.4. Conclusions ......................................................................................... 163. XIII.
(20) CHAPTER 6 A STABILIZED ALGORITHM FOR COUPLED ANALYSIS OF SATURATED SOIL PROBLEMS: THE V-PW RUNGE-KUTTA TAYLORSPH ALGORITHM ........................................................................ 165 6.1. Introduction ......................................................................................... 165 6.2. Mathematical model: the v-pw model .................................................. 166 6.2.1. Physical properties of geomaterials......................................... 166 6.2.2. Balance of mass equations for general coupled models .......... 171 6.2.3. Balance of linear momentum equations for general coupled models ..................................................................................... 172 6.2.4. The u-pw-w model: a model for saturated and non-saturated soils.......................................................................................... 175 6.2.5. Equations of the v-pw model .................................................... 176 6.3. Numerical model: the v-pw Runge-Kutta Taylor-SPH algorithm ....... 179 6.3.1. Equations of the fractional step algorithm .............................. 179 6.3.2. Calculation of the intermediate velocity v* ............................ 182 6.3.3. Calculation of the pore pressure pwn 1 ...................................... 183 6.3.4. Resolution of the system of equations Klaplacian pwn1 RHS by the preconditioned conjugate gradient method .................. 185 6.3.5. Steps of the algorithm of the v-pw Runge-Kutta Taylor-SPH model ....................................................................................... 188 6.4. Validation of the v-pw Runge-Kutta Taylor-SPH model for coupled problems ..................................................................................................... 190 6.4.1. Purpose .................................................................................... 190 6.4.2. Consolidation of a saturated soil column ................................ 190 6.4.3. Saturated soil column under harmonic loading ....................... 195 6.4.4. Strip foundation on elastic saturated soil stratum ................... 200 6.4.5. Strain localization in saturated soil ......................................... 203 6.4.6. Strip foundation on viscoplastic saturated soil stratum: vertical cut ............................................................................... 206 6.5. Conclusions ......................................................................................... 210. CHAPTER 7 CONCLUSIONS AND FUTURE INVESTIGATIONS .......... 213 7.1. Conclusions ......................................................................................... 213 7.1.1. Main contributions of this thesis ............................................. 213 7.1.2. Conclusions on the mathematical and numerical models ....... 216 7.1.3. Conclusion on the validation of the numerical models ........... 222 7.2. Future investigations ........................................................................... 224. XIV.
(21) REFERENCES............................................................................... 227 APPENDIX 1 SUBROUTINE OF THE RUNGE-KUTTA ...................... 237 APPENDIX 2 SUBROUTINE OF THE FREE-SURFACE DETECTION .... 239 APPENDIX 3 SUBROUTINE TO APPLY THE BOUNDARY CONDITIONS ON THE FREE-SURFACE ............................................................... 243 APPENDIX 4 ANALYTICAL SOLUTION FOR THE FOOTING ON THE VERTICAL CUT ............................................................................ 245 APPENDIX 5 ANALYTICAL SOLUTION FOR BEARING CAPACITY TEST ........................................................................................... 247 APPENDIX 6 BALANCE OF MASS EQUATIONS FOR SOILS COMPOSED OF SOLID PARTICLES AND SEVERAL FLUID PHASES ...................... 250 APPENDIX 7 SUBROUTINE TO BUILD THE MATRIX K Laplacian ........... 256 APPENDIX 8 ALGORITHM OF THE PRECONDITIONED CONJUGATE GRADIENT ................................................................................... 258. XV.
(22) XVI.
(23) List of figures Figure 1-1 Landslide of Shum Wan (Hong-Kong, 1995) which travelled across two lane-roads, three shipyards, a factory and causing two fatalities and five injuries (Knill 2006)............................................................................................................... 1 Figure 1-2 Sketch of the methodology used to develop the thesis ............................ 4 Figure 2-1 Reference and current configurations.................................................... 10 Figure 2-2 Stresses on an elementary parallelepiped .............................................. 13 Figure 2-3 The principal stresses ............................................................................ 14 Figure 2-4 Invariants in principal stress space ........................................................ 18 Figure 2-5 Variation of the strength of clays and sands (Perzyna 1966) ................ 20 Figure 2-6 Von Mises yield criterion – a) In the principal stress space ; b) Section by the - plane....................................................................................................... 23 Figure 2-7 Yield surface of the modified Cam Clay model .................................... 25 Figure 2-8 Hydrostatic compression test on a normally consolidated clay – a) Experimental results ; b) Idealized behavior (Zienkiewicz et al. 1999).................. 26 Figure 3-1 The Gaussian kernel and its first derivative .......................................... 41 Figure 3-2 The cubic spline kernel and its first derivative ...................................... 42 Figure 3-3 The quintic smoothing function and its first derivative ........................ 43 Figure 3-4 Nodes and numerical integration in a SPH mesh .................................. 46 Figure 3-5 The support domain of the smoothing function and problem domain .. 49 Figure 3-6 Rubber rings shortly after maximum compression showing fracture in a collision (Monaghan 2000) ..................................................................................... 53 Figure 3-7 Particle clustering during the fracture of cohesive soil (Bui et al. 2008) ................................................................................................................................ 53 Figure 3-8 SPH nodes arrangement for the 1D-SPH model based on stress point method (Dyka et al. 1997) ...................................................................................... 54 Figure 3-9 Bi-dimensional SPH nodes arrangement for stress point method (Randles and Libersky 2000) .................................................................................. 55. XVII.
(24) Figure 3-10 Comparison of the velocity obtained with the analytical solution and with the two-step Taylor-Galerkin scheme at time t 2.5 103 s - a) 1 ; b). 50 (Mabssout et al. 2006b) .............................................................................. 60 Figure 3-11 Comparison of the stress obtained with the analytical solution and with the two-step Taylor-Galerkin scheme at time t 2.5 103 s - a) 1 ; b) 50 (Mabssout et al. 2006b)........................................................................................... 60 Figure 3-12 SPH grid for the Taylor-SPH model ................................................... 65 Figure 4-1 1D elastic bar......................................................................................... 74 Figure 4-2 Discretization of the 1D elastic bar ....................................................... 74 Figure 4-3 Velocity history for the end left f the bar (Courant = 0.6) .................... 75 Figure 4-4 Extended stress history for SPH node 11 (Courant = 0.6) .................... 76 Figure 4-5 Early stress history at SPH node 11 (Courant = 0.6)............................. 76 Figure 4-6 Velocity history for the end left bar (Courant=1).................................. 77 Figure 4-7 Extended stress history at SPH node 11 (Courant = 1) ......................... 77 Figure 4-8 1D viscoplastic bar ................................................................................ 78 Figure 4-9 Imposed velocity on the right end of the bar ......................................... 79 Figure 4-10 Discretization of the 1D viscoplastic bar ............................................ 79 Figure 4-11 a) Evolution in time of the velocity along the bar ; b) Evolution in time of the stress along the bar........................................................................................ 81 Figure 4-12 Stress history at the left end of the bar ................................................ 82 Figure 4-13 Stress history and accumulated viscoplastic strain at SPH node 26 .... 83 Figure 4-14 Velocity history and accumulated viscoplastic strain at SPH node 26 83 Figure 4-15 Viscoplastic strain along the bar at different times ............................. 84 Figure 4-16 Soil sample: 3D representation and plane strain representation .......... 85 Figure 4-17 Imposed vertical velocity on the top of the soil sample ...................... 86 Figure 4-18 Boundary conditions for the soil sample compression test ................. 87 Figure 4-19 First apparition of the shear band at time t 0.023 sec ...................... 89 Figure 4-20 Strain localization in shear bands ........................................................ 90 Figure 4-21 Inclination of the shear bands.............................................................. 91 Figure 4-22 a) Displacement a time t 0.1sec among the soil sample ; b) Deformed SPH grid................................................................................................. 92 Figure 4-23 Vertical slope: 3D and plane strain representations ............................ 93 Figure 4-24 Time factor for application of the gravity forces................................. 93. XVIII.
(25) Figure 4-25 Boundary condition for the vertical slope under gravity test .............. 94 Figure 4-26 History of the vertical stress (a) and vertical velocity (b) (without damping of the gravity forces) ................................................................................ 95 Figure 4-27 History of the vertical stress (a) and vertical velocity (b) (without damping: 20 ) ................................................................................................... 96 Figure 4-28 History of the vertical stress (a) and vertical velocity (b) (without damping: 50 ) ................................................................................................... 97 Figure 4-29 Vertical stress (a), horizontal stress (b), displacement (c) among the vertical slope at time t 2 sec ................................................................................ 98 Figure 4-30 Vertical stress among the vertical slope: contour lines representation 99 Figure 4-31 Vertical stress history at SPH node 16 and 102 ................................ 100 Figure 4-32 Beginning of the shear band formation ............................................. 102 Figure 4-33 Factor of stability determined with the Taylor-SPH model .............. 103 Figure 4-34 Taylor‟s abacus: factor of stability for vertical slope (Taylor 1937). 104 Figure 4-35 Shear band development at time: a) t 1.17 sec ; b) t 3.1sec ; c) t 3.41sec and d) t 3.67 sec ............................................................................ 106. Figure 4-36 Displacement of the SPH nodes and deformed mesh at time t 5 sec .............................................................................................................................. 106 Figure 5-1 A highly oscillating 2D pressure distribution indicated by the numbers. The pressure distribution leads to a zero pressure gradients and hence leads to a spurious velocity contribution (Liu and Liu 2003) ............................................... 110 Figure 5-2 A 2D velocity field as a consequence of the zero-energy mode ......... 111 Figure 5-3 Hourglass deformation ........................................................................ 111 Figure 5-5 Regular distribution of the SPH nodes ................................................ 112 Figure 5-6 Sketch of the hanged vertical beam and boundary conditions ............ 115 Figure 5-7 Time factor for application of the gravity forces ................................ 116 Figure 5-8 Vertical velocity field and deformed mesh according to the velocity variable (The factor of deformation of the mesh is 200) ...................................... 116 Figure 5-9 2D elastic bar....................................................................................... 117 Figure 5-10 Velocity shock wave imposed on the left extremity of the beam...... 117 Figure 5-11 Velocity evolution in the middle of the bi-dimensional beam .......... 118 Figure 5-12 Update of the spatial configuration of the SPH particles .................. 119 Figure 5-13 Updating of the smoothing length ..................................................... 121. XIX.
(26) Figure 5-14 Temporary mesh employed in the linked-list NNPS algorithm ........ 123 Figure 5-15 Sketch of the scan region .................................................................. 125 Figure 5-16 Method to calculate the normal to the free-surface ........................... 127 Figure 5-17 Orientation of the vector f IJ ............................................................. 128 Figure 5-18 Geometry of the validation test for free-surface detection ................ 129 Figure 5-19 Approximation of the normal with the method presented by Marrone (Marrone et al. 2010) ............................................................................................ 129 Figure 5-20 Number of particles J which are situated in the scan region of each particle I .............................................................................................................. 130 Figure 5-21 Classification of all the SPH nodes into two groups (Subset 1 and 2) .............................................................................................................................. 131 Figure 5-22 Function f x, y .............................................................................. 131 Figure 5-23 Gradient g f of the function f x, y ................................................ 132 Figure 5-24 Unit vector n I normal to the free-surface ......................................... 132 Figure 5-25 Boundary conditions on the free-surface........................................... 133 Figure 5-26 Sketch of the elastic one-dimensional bar ......................................... 136 Figure 5-27 Imposed velocity on the right extremity of the bar ........................... 136 Figure 5-28 History of the velocity at SPH node 11 ............................................. 137 Figure 5-29 Horizontal stress history at SPH node 11 .......................................... 137 Figure 5-30 Horizontal displacement history at SPH node 11 .............................. 138 Figure 5-31 Deformed bar (Factor x20) ................................................................ 139 Figure 5-32 Sketch of the bi-dimensional bar ....................................................... 140 Figure 5-33 Boundary conditions and discretized bar .......................................... 140 Figure 5-34 Horizontal velocity along the bi-dimensional bar at time t 0.0016 m/s ....................................................................................................... 141. Figure 5-35 Horizontal stress along the bi-dimensional bar at time t 0.0016 m/s .............................................................................................................................. 141 Figure 5-36 Deviatoric viscoplastic strain at time t 0.0032 sec ........................ 142 Figure 5-37 Evolution of the deviatoric viscoplastic deformations over time ...... 142 Figure 5-38 Displacement field plotted on the deformed mesh (Factor of deformation x30) ................................................................................................... 143 Figure 5-39 Sketch of the vertical slope and the applied boundary conditions .... 144. XX.
(27) Figure 5-40 Deviatoric viscoplastic strain at the end of the calculation ............... 145 Figure 5-41 Displacement of the vertical slope plotted on the deformed mesh (Factor of deformation x1) .................................................................................... 147 Figure 5-42 Sketch of the vertical slope and of the applied boundary conditions 148 Figure 5-43 Viscoplastic strains evolution over time ........................................... 149 Figure 5-44 Displacements plotted on the deformed mesh (factor of deformation x5) ......................................................................................................................... 150 Figure 5-45 Sketch of the vertical cut and its discretization ................................. 151 Figure 5-46 Boundary conditions of the vertical cut ............................................ 152 Figure 5-47 Velocity Vt in function of time ......................................................... 152 Figure 5-48 Load-displacement curves: Analytical and computational solutions 154 Figure 5-49 Evolution of the viscoplastic deformation in the vertical cut: a) perfect viscoplastic case ; b) case with softening.............................................................. 155 Figure 5-50 Displacement contours on the deformed mesh (Factor of deformation x2). : a) perfect viscoplastic case ; b) case with softening ................................ 156. Figure 5-51 Sketch of the shallow stratum ........................................................... 157 Figure 5-52 Boundary condition on the shallow stratum: bearing capacity test ... 158 Figure 5-53 Imposed velocity on the area situated under the footing ................... 158 Figure 5-54 Load-displacement curve: Analytical and computational solutions .. 159 Figure 5-55 Evolution of the viscoplastic deformation in the shallow stratum .... 161 Figure 5-56 Displacement contours on the deformed mesh (Factor of deformation x2) ......................................................................................................................... 162 Figure 6-1 Sketch of tri-phases soil ...................................................................... 166 Figure 6-2 Sketch of different saturation states: a) Saturated soil ; b) Dry soil ; c) Unsaturated soil .................................................................................................... 168 Figure 6-3 Imposed vertical stress on the top of the soil column ......................... 191 Figure 6-4 Boundary conditions on the soil column and discretized mesh .......... 192 Figure 6-5 Evolution of the pore pressure and of the vertical effective and total stresses at the SPH node with coordinates x 1 m and y 23 m ......................... 193 Figure 6-6 Evolution of the pore pressure and of the vertical effective and total stresses in the soil column ..................................................................................... 194 Figure 6-7 Influence of the permeability on the pore pressure: numerical versus analytical solution ................................................................................................. 199. XXI.
(28) Figure 6-8 Sketch of the soil stratum .................................................................... 200 Figure 6-9 Discretization of the soil stratum ........................................................ 201 Figure 6-10 Oscillation of the pore pressure field got with a finite element code with standard formulation of 4 nodes in displacement and 4 nodes in pore pressure (Mira 2001) ........................................................................................................... 202 Figure 6-11 Pore pressure developed in the soils stratum..................................... 202 Figure 6-12 Sketch of the soil sample ................................................................... 203 Figure 6-13 Discretization of the soil sample ....................................................... 205 Figure 6-14 Strain localization on the soil sample: a)contour fills of strain, b) velocity vectors and c) deformed mesh (factor 1) ................................................. 205 Figure 6-15 Sketch of the vertical cut and of the discretization of the vertical cut .............................................................................................................................. 206 Figure 6-16 Velocity Vt in function of time ......................................................... 207 Figure 6-17 Localization on the vertical cut: a) velocity vectors and b) deformed mesh (factor x4) .................................................................................................... 208 Figure 6-18 Development of a clearly defined shear band ................................... 209 Figure 7-1 Sketch of the mathematical and numerical models presented in the thesis ..................................................................................................................... 217 Figure 7-2 Accelerogram of the earthquake and plastic deformation produced by the seismic acceleration (Fernandez Merodo 2001).............................................. 226. XXII.
(29) List of tables Table 1 Steps of the Runge-Kutta Taylor-SPH algorithm ...................................... 69 Table 2 Steps of the new algorithm ...................................................................... 114 Table 3 Steps for calculating the intermediate velocity ........................................ 182 Table 4 Steps of the algorithm of the v pw Runge-Kutta Taylor-SPH model ... 189 Table 5 Parameters chosen for the calculations .................................................... 197 Table 6 Summary of all the case studies presented in the thesis .......................... 222. XXIII.
(30) XXIV.
(31) Notations In this work the following notations have been used: -. The four-order tensors are written with capital letters of latin alphabet in boldface.. -. The second-order tensor are written with lower case letters of the latin or greek alphabet in boldface. -. The vectors are written with italics lower and upper case letters of the latin or greek alphabet in boldface. -. The matrix are written with italics upper case letters of the latin alphabet.. -. The scalar are written with italics lower and upper case letters of the latin or greek alphabet.. In the next table the rules of notations are summarized: Upper Lower Latin Greek case case Examples alphabet Alphabet letters letters. Boldface. Italics. Tensor of 4th order. Yes. No. Yes. No. Tensor of 2nd order. Yes. No. No. Yes. Vector. Yes. Yes. Matrix. No. Yes. Scalar. No. Yes. Yes. a ; A. Both No. Both. XXV. Yes. λ ; Γ. No Both. A. a ;δ. Both. Both Yes. No. A. a ; A. .
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(33) CHAPTER 1 INTRODUCTION. Chapter 1 Introduction. 1.1. Motivations Slope stability is an important topic of research in the field of geotechnical engineering. Slope instability causes lot of damages and remains a problem which has not been entirely solved yet. For instance if embankment fails, infrastructures, such as railways and highways, can be damaged or destroyed. More disastrous if a landslide initiates upstream of a densely populated area, human people are in danger and human lives are threatened.. Figure 1-1 Landslide of Shum Wan (Hong-Kong, 1995) which travelled across two laneroads, three shipyards, a factory and causing two fatalities and five injuries (Knill 2006). Therefore engineers try to reduce the risk of such disasters by implementing mitigation measures in endangered areas to decrease the probability of occurrence. 1.
(34) CHAPTER 1 INTRODUCTION. or by analyzing the potential hazard. Powerful and successful tools to analyze landslides hazard are numerical models. However numerical modeling of slope failure and of landslide propagation is complex and it is still possible to develop models and to improve their utilization. Landslide processes can be divided in three phases: -. Initiation phase. -. Propagation phase. -. Deposition phase.. The initiation of landslides is of a great importance because the magnitude of events will depend on the volume mobilized by the landslide. The initiation of landslides can occur after an increase of pore water pressure due to an important rainfall, after earthquakes, or after perturbations by human activity for instance mining activities. The propagation and the deposition of landslides must be studied to know where infrastructures and settlements can be built. Nowadays numerical tools allow predicting the failure of slopes and the propagation of landslides. Thus they are used for risk management and principally for risk hazard mapping. However, generally, the numerical methods used to forecast the initiation of landslides, framework of small deformation theory, are not the same as those used to predict the propagation, framework of large deformation theory. Developing new numerical methods able to model geomaterials behavior in small and large deformation theory is a great challenge.. 1.2. Objectives The main objective of this PhD thesis consists on the development of a new numerical model able to reproduce dry and saturated soil dynamic behaviors in small and large theory framework. From this main objective, the following sub-objectives are derived: -. Choose a mathematical model for dry soil dynamics problems which assures good shockwave propagation properties and a good failure prediction of geomaterials.. -. Develop a numerical model to discretize in time and in space the equations of the mathematical model. The numerical model should be adapted to. 2.
(35) CHAPTER 1 INTRODUCTION. small and large deformation theory and should present improvements of the classical SPH method in order to overcome the problems related to this method. -. Implement the numerical model in a code written in FORTRAN 90.. -. Carry out a sufficient number of numerical simulations to validate the numerical code developed in FORTRAN 90. Different kinds of numerical simulations should be done: i) Case studies with an analytical solution, ii) Case studies of localized failure in geomaterials, ii) Case studies of complicated mechanism of failure of geomaterials. -. Choose of a mathematical model able to represent the coupled behavior of saturated soils.. -. Discretize the equations of the coupled mathematical model by the proposed numerical model.. -. Validate the new numerical code for coupled behavior of saturated soil with a sufficient number of numerical case studies.. 1.3. Methodology and structure of the work The thesis is divided in three main parts: -. The development and validation of a numerical model for dry soil dynamic behavior in the framework of small deformation theory. -. The improvement of the first model to develop a new version of the model adapted to large deformation theory.. -. The development and validation of a numerical model for saturated soil dynamic behavior in the framework of small deformation theory.. For each part of the thesis the methodology described in the Figure 1-2 has been employed. Based on the state of the art, mathematical and numerical models have been developed in order to model the behavior of geomaterials. Then the equations of the proposed model have been implemented in a numerical code. Once the model was programmed, it entered in validation phases: -. Phase 1: One-dimensional case-studies. -. Phase 2 : Bi-dimensional case studies with elastic material. -. Phase 3 : Bi-dimensional case studies with viscoplastic material. 3.
(36) CHAPTER 1 INTRODUCTION. When the results obtained at the end of each validation phase were not correct, we went back to the previous step in order to find where the problems were. This methodology allows us reaching our objectives.. Figure 1-2 Sketch of the methodology used to develop the thesis. 4.
(37) CHAPTER 1 INTRODUCTION. The thesis contains 7 chapters. Chapter 1 is the present introduction. Chapters 2 and 3 consist on the methodology and the theory behind the RungeKutta Taylor-SPH model. The first part of chapter 2 presents the shortcomings of the classical approach of numerical analysis and some basics of continuum mechanics. Further this chapter details the equations of the mathematical model which has been chosen to solve dry soil dynamics problems in the framework of small deformation theory. In the beginning of chapter 3, the theory of the Smoothed Particle Hydrodynamics method (SPH) is introduced and the pitfalls of SPH are pointed out. Then the history of the two-step Taylor-Galerkin scheme is presented and the limitations of this scheme are given. Finally chapter 3 ends with description in depth of the new proposed model which is called the Runge-Kutta Taylor-SPH model. In chapter 4, applications in the framework of small deformation theory have been done with the Runge-Kutta Taylor-SPH model. First one-dimensional applications of the propagation of velocity shockwaves in a bar are presented. Secondly bidimensional cases are treated. In order to evaluate our model the numerical results are compared to some analytical results. Chapter 5 presents the modification done of the algorithm of the Runge-Kutta Taylor-SPH in order to adapt it to the large deformation theory. First the mathematical model has to be modified. Then the equations of the new mathematical model are discretized with the numerical model. The last part of this chapter consists on the validation of the new algorithm following our methodology. The new algorithm is called the new Runge-Kutta Taylor-SPH model. In chapter 6 a new mathematical model is introduced. This mathematical model will replace the mathematical model described in Chapter 2 in order to analyze the coupled behavior of saturated soil. Then a numerical model based on Chapter 3 and 5 is introduced to discretize the equations of the coupled mathematical model. Finally the new model is validated for small deformation theory applying it to a series of case studies. We called the new model the v pw Runge-Kutta- TaylorSPH model.. 5.
(38) CHAPTER 1 INTRODUCTION. The thesis ends with the conclusions of this work and recommendations for future investigations (Chapter 7).. 6.
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(41) CHAPTER 2 MATHEMATICAL MODEL. Chapter 2 Mathematical model. 2.1. Shortcomings of the classical approach The classical approach of numerical analysis is based on displacement formulations where the main variables are the stress and the displacement. Nevertheless the classical approach presents two weak points when the finite elements method is used to solve solid dynamics problems. The first shortcoming of the classical approach is related to the propagation of short wavelengths. Errors in the velocity propagation of waves together with numerical damping make unsuitable the displacement-based finite elements method for shocks propagation problems. The second drawback of the classical approach is that the low-order elements like triangles and tetrahedral elements cannot be used as they cause volumetric locking and they are not accurate in bending dominated situations. It has been shown that, in some cases, the low-order elements overestimate limit loads or predict wrong failure mechanisms (Pastor and Quecedo 1995; Quecedo et al. 2000b; Zienkiewicz et al. 1995). Moreover the results depend on the mesh alignment. A possible solution is to formulate the problem in terms of velocities and pressures (Pastor et al. 2000; Quecedo et al. 2000a; Zienkiewicz et al. 1998). Mabssout et Al. presented an alternative formulation for solid dynamics problems in terms of velocities and stresses which mitigates the locking and the mesh alignment problems (Mabssout and Pastor 2003a; Mabssout and Pastor 2003b; Mabssout et al. 2006b). Our mathematical model will be based on the model of these authors.. 9.
(42) CHAPTER 2 MATHEMATICAL MODEL. Before starting the description of the mathematical model, some basics of continuum mechanics will be presented.. 2.2. Basics of continuum mechanics 2.2.1. Lagrangian description versus eulerian description In continuum mechanics, the matter is supposed to be formed by an infinite of particles which occupy different positions in the space during its motion along the time. Suppose that a body B is formed by material which is in the region 0 at the time t 0 . Let X be the position vector, relative to O , of a particular point P0 of B. The configuration 0 at the time t 0 is called the reference configuration. (Figure 2-1). Figure 2-1 Reference and current configurations. Suppose now that the material which occupied 0 at t 0 moves so that at a subsequent time t it occupies a new continuous region of the space, . The configuration at the time t is called the current configuration. We can identify a point P of B with the position vector x , which is occupied at t by the particle which was in the point P0 at t 0 . X i are called material coordinates and xi are called spatial coordinates.. 10.
(43) CHAPTER 2 MATHEMATICAL MODEL. The motion can be described in two ways. Firstly it is possible to describe the motion by the evolution of the spatial coordinates along the time. x x X ,t . (2.1). Equation (2.1) gives the position xi at the time t of the particle which occupies the position X i in the reference configuration. Secondly the motion can also be described by the inverse of equation (2.1).. X X x,t . (2.2). Equation (2.2) gives the reference coordinates X i of the particle which occupies the position xi in the current configuration. In continuum mechanics, the description of a problem with spatial coordinates xi is an eulerian description and the description of a problem with material coordinates X i is a lagrangian description. In the lagrangian description, a certain property (for instance the density ) is described using a function ,t whose argument in ,t is the material coordinates X i .. X , t X1 , X 2 , X 3 , t . (2.3). If the material coordinates X i are fixed, a determined particle is followed. It is why the lagrangian description is also called material description. In the eulerian description, the attention is fixed on a point of the space. A certain property (for instance the density ) is described using a function ,t whose argument in ,t is the spatial coordinates xi .. x, t x1 , x2 , x3 , t . (2.4). If the material coordinates xi are fixed, the evolution of the density is got for a particular point of the space. If the time t is fixed, a distribution of the property is got for the entire space. It is why the eulerian description is also called spatial description.. 11.
(44) CHAPTER 2 MATHEMATICAL MODEL. 2.2.2. Displacement and velocity vectors The displacement vector u of a particular particle, from its position X in the reference configuration to its position x in the current configuration, is u xX. (2.5). The lagrangian description of the displacement is. u X ,t x X ,t X. (2.6). And the eulerian description of the displacement is. u x, t x X x, t . (2.7). The velocity vector v of a particle is the rate of change of its displacement. Since X i are constant in time, it is better to use the lagrangian description to express the. velocity v X ,t . u X , t t. . x X , t t. (2.8). 2.2.3. The stress tensor It is common to represent graphically the stress tensor on an elementary parallelepiped around the considered particle. Each face of the parallelepiped is a plane normal to x1 , x2 and x3 axis respectively. On each face, the traction vector is decomposed in one normal and two tangential components. The components of stress tensor can be written in the basis x1 , x2 , x3 as (Figure 2-2) 11 12 13 ζ 21 22 23 31 32 33 x1 , x2 , x3 . 12. (2.9).
(45) CHAPTER 2 MATHEMATICAL MODEL. Or. ζ ij. (2.10). Where the subscript i indicates that the stress actuates on the plane orthogonal to the xi axis and the subscript j shows that the direction of the stress is the direction of the x j axis.. Figure 2-2 Stresses on an elementary parallelepiped. One property of the stress tensor is to be symmetric, in consequence, the components of the stress tensor are 11 12 13 ζ 12 22 23 13 23 33 x1 , x2 , x3 . (2.11). 2.2.4. Principal stress As the stress tensor is a symmetric second order tensor, it has a diagonal form in an orthonormal basis x1, x2 , x3 (Figure 2-3): 1 0 0 ζ 0 2 0 0 0 3 x1 , x2 , x3 . 13. (2.12).
(46) CHAPTER 2 MATHEMATICAL MODEL. Figure 2-3 The principal stresses. The principal stresses are defined as the eigenvalues of the stress tensor ( 1 , 2 , 3 ). Each principal stress i is associated to a principal plane, which is orthogonal to the axis xi . On these planes, the shear stress is zero. The principal stresses are ordered as. 1 2 3. (2.13). With: -. 1 the major stress. -. 2 the intermediate stress. -. 3 the minor stress.. 2.2.5. Decomposition of the stress tensor into a hydrostatic and a deviatoric component The mean stress is defined as the average of the principal stresses. m . 1 1 2 3 3. 14. (2.14).
(47) CHAPTER 2 MATHEMATICAL MODEL. The hydrostatic pressure is defined as the additive inverse of the mean stress p m . 1 1 2 3 3. (2.15). The stress tensor can be decomposed into a spherical component and a deviatoric component. The spherical component of the stress tensor is p p pδ 0 0 . 0 p 0. 0 0 p . (2.16). Where δ is the unit tensor of second order. The spherical component can be expressed by the mean pressure 0 m 0 p 0 m 0 0 0 m . (2.17). The deviatoric component of the stress tensor is by deduction. s ζ p ζ pδ. 12 13 11 m s 12 22 m 23 23 33 m 13 . (2.18). The spherical component p of the stress tensor is an isotropic tensor and defines a hydrostatical stress state. Therefore a change of orthogonal base does not modify the hydrostatic tensor. The deviatoric component indicates how far the stress state is from the hydrostatical stress state. s is called the deviator of ζ .. 2.2.6. Invariants of the stress tensor The invariants of a tensor are values which are independent of the choice of the coordinate system. The eigenvalues of a second order tensor are invariants of the tensor. In case of a symmetric tensor, its eigenvalues or principal values are basic invariants in the sense that any invariant of this tensor can be expressed by them.. 15.
(48) CHAPTER 2 MATHEMATICAL MODEL. In many applications it is more convenient to choose, as fundamental invariants of the stress tensor, symmetric functions of the principal values rather than its principal values themselves. Generally in continuum mechanics, we define ( I1 , I 2 , I 3 ) respectively the first, second and third invariant of the stress tensor ζ as: I1 tr ζ 1 2 3 ii 2 1 1 (2.19) ζ : ζ tr ζ 2 3 3 1 1 2 ij ij ii jj 2 2 I 3 det ζ 1 2 3. I2 . We can remark that the first invariant of the stress tensor and the hydrostatic pressure are related by p. I1 3. (2.20). Generally it is useful to consider the following set of stress invariants: I1 tr ζ 2 1 s : s tr s.s 2 1 J 3 tr s.s.s 3. J2 . (2.21). Where: -. I1 is the first invariant of the stress tensor. -. J 2 is the second invariant of the deviatoric stress tensor. -. J 3 is the third invariant of the deviatoric stress tensor.. Taking into account that tr s.s 0. and. sζ. I1 δ 3. I I 1 J2 ζ 1 δ : ζ 1 δ 2 3 3 . 16. (2.22). (2.23).
(49) CHAPTER 2 MATHEMATICAL MODEL. We can express J 2 as. I2 1 2 J 2 ζ : ζ I1δ : ζ 1 δ : δ 2 3 9 . (2.24). I2 1 J2 ζ : ζ 1 2 3 1 2 2 2 J 2 1 3 1 2 2 3 6. (2.25). And. Here we can add the definitions of: -. The hydrostatic axis which is the axis in the space of principal stresses where the condition 1 2 3 is verified. The points of this axis represent different hydrostatic stress states.. -. The deviatoric planes (or planes) which are all the planes orthogonal to the hydrostatic axis. The equation of the deviatoric planes is. 1 2 3 constante . -. The deviatoric stress. q 3J 2 -. 3 1 2 2 12 2 sij s ji 1 3 1 2 2 3 (2.26) 2 2. The Lode‟s angle 1. 3 3 J . 3 sin 1 32 3 2 J 2 . (2.27). A stress state, represented by a point P, in the principal stress space is directly characterized by the set of the three invariants I1 , J 2 , (Figure 2-4) -. The first invariant I1 of the stress tensor represents the distance between the origin of the principal stresses space and the plane where the point P locates. This distance is. -. I1 . 3. The second invariant J 2 of the deviatoric stress tensor characterizes the distance d from the point P to the hydrostatic axis: d 2 J 2. 17.
(50) CHAPTER 2 MATHEMATICAL MODEL. -. defines the orientation of the stress state in the plane.. Figure 2-4 Invariants in principal stress space. The description of our mathematical model will use the set of the stress invariants. I1 , J 2 , . 2.2.7. Infinitesimal strain tensor The infinitesimal strain tensor, ε , is defined as u1 x1 u 1 u ε 2 1 2 x1 x2 1 u3 u1 2 x1 x3 . 1 u1 u2 2 x2 x1 u2 x2 1 u3 u2 2 x2 x3 . 1 u1 u3 2 x3 x1 1 u2 u3 2 x3 x2 u3 x2 . (2.28). Or using the tensor component notation. 1 ui u j 2 x j xi . ij . Where ui are the components of the displacement vector u. 18. (2.29).
(51) CHAPTER 2 MATHEMATICAL MODEL. 2.3. Balance of momentum The first governing equation in soil dynamics problems is the balance of momentum. div ζ b . v t. (2.30). vi t. (2.31). Or using the tensor component notation ij x j. bi . Where ζ is the stress tensor, b is the body forces, the density and v the velocity vector.. 2.4. Constitutive equation The equation of balance of momentum has to be completed by a constitutive equation which gives the relation between the stress tensor and the strain tensor. The material behavior can be described in several alternative ways depending both on the type of material and on the velocity at which load is applied.. 2.4.1. Ill posing of the mathematical model for elastoplastic materials Solid dynamics problems are problems of wave propagation. In the case of elastoplastic materials, the equation of wave propagation in one dimension can be written as 0 1 t v . ET v 0 0 x 0 . (2.32). Equation (2.32) is a system of first order hyperbolic equations where v is the velocity, the density and ET the tangent strain modulus which corresponds to the slope of the tangent of the stress – strain curve of the material.. 19.
(52) CHAPTER 2 MATHEMATICAL MODEL. The propagation velocities of waves are the eigenvalues of the matrix of the system (2.32): c ET. . These velocities correspond, whenever ET is positive,. to two waves propagating in opposite directions. In case that the material behavior is described by an elastoplastic law with softening, the tangent strain modulus can get negative when the material starts to localize and in consequence the velocities of propagation, c , become imaginary. This problem with imaginary velocity of propagation corresponds to an elliptic problem which is in contradiction with the imposed boundary conditions of the solid dynamics problems (Pastor 1994; Sluys 1992). If the material considered has an elastoplastic behavior, the ill posing of the mathematical model will rise due to imaginary velocities of propagation. Therefore it is better to choose another kind of material behavior.. 2.4.2. Time dependent behavior of soils under dynamics conditions P. Chadwick, A.D. Cox and H.G. Hopkins (Chadwick et al. 1964) discussed data from measurements of plastic constants of soils under dynamics conditions. These data provided by A.W. Skempton and A.W. Bishop (Skempton and Bishop 1954) are presented in the Figure 2-5. Figure 2-5 Variation of the strength of clays and sands (Perzyna 1966). 20.
(53) CHAPTER 2 MATHEMATICAL MODEL. As the rate of strain increases from zero to about 1.5 sec-1, the strength of clays and sands increase by factors of about 2 and 1.2, respectively. This study shows that the behavior of soils, especially in their dynamic response, is sensible to the change of strain rate and therefore is time-dependent to the deformation process. Viscoplastic constitutive laws are the most adapted to represent this property.. 2.4.3. The Perzyna model The Perzyna model is a simple viscoplastic model. In this model, the relation between the stress tensor and the strain tensor is given by. ε ε vp ζ De : t t t. (2.33). Where De is the elastic constitutive tensor and ε the strain tensor. The superscript. vp indicates the viscoplastic component of the strain tensor which is given by (Perzyna 1966) ε vp m F t. In equation (2.34): -. if 0 the sign ... represents the Macaulay brackets: 0 otherwise. -. is the fluidity parameter. -. m characterizes the direction of the plastic flow. -. F is an arbitrary function. 21. (2.34).
(54) CHAPTER 2 MATHEMATICAL MODEL. We will choose for F as. F F0 F F0 . N. (2.35). Where N is a model parameter and F a function describing a convex surface in the stress space. The value F0 characterizes the stress below which no viscoplastic flow occurs. To complete the Perzyna model, the function F has to be defined. In this work, we have chosen two functions F : -. The surface determined by the Von Mises yield criterion. -. The yield surface of the modified Cam-Clay model. 2.4.4. Yield criterion of Von Mises The Von Mises yield criterion assumes that plastic strains appear whenever the deviatoric stress q reaches a critical value Y f q Y 0. (2.36). Where Y is generally the tensile strength. The function f can be expressed with the set of invariants I1 , J 2 , f 3J 2 Y 0. (2.37). An alternative expression in term of principal stresses is: f . 1 2 2 12 2 1 2 2 3 1 3 Y 0 2. (2.38). Taking into consideration that the second invariant of the deviatoric stress tensor is constant corresponds to stress states such that the distance to the hydrostatic axis ( 1 2 3 ) is constant, the yield surface corresponding to Von Mises criterion can be graphically represented in the principal stress space as a cylinder whose axis is the hydrostatic axis and whose radius is. 22. 2J 2 (Figure 2-6).
(55) CHAPTER 2 MATHEMATICAL MODEL. Figure 2-6 Von Mises yield criterion – a) In the principal stress space ; b) Section by the - plane. . Doing the connection between the equation (2.35) and (2.37), we will choose. F 3J 2 . 3 sij s ji q 2. (2.39). And the initial size of the yield surface F0 Y0. (2.40). In the Von Mises criterion the initial size of the yield surface corresponds to the cohesion of the material. The size of the yield surface will vary according to a suitable hardening/softening law. Here we will assume that the change of size of the yield surface will be proportional to the increase of the equivalent deviatoric plastic strain, vp : Y0 vp H t t. (2.41). Where H is the hardening modulus. We will assume an associated flow rule, which means that the plastic potential surface g ζ 0 coincides with the yield surface F ζ,internal variables 0 defined in equation (2.37) (Zienkiewicz et al. 1999). Therefore the plastic potential surface is. g I1 , J 2 , F I1 , J 2 , . (2.42). g I1 , J 2 , 3J 2. (2.43). 23.
(56) CHAPTER 2 MATHEMATICAL MODEL. By definition, the direction of the plastic flow passing through a stress point is: m. g ζ. (2.44). As the plastic potential surface is a function of the invariants I1 , J 2 , , we have. m. g I1 g J 2 g . . . I1 ζ J 2 ζ ζ. (2.45). Equation (2.45) can be written in a more compact manner as m C1m1 C2m2 C3m3. (2.46). Where. C1 . g I1. g tan 3 g C2 J J 2 2 C3 . (2.47). 3 1 g 2cos3 J 23 2 . And m1 m2 . I1 ζ J2. ζ J m3 3 ζ. (2.48). It can be observed that the constants C1 , C2 , C3 depend only on the yield criterion and that the tensors m i are independent of the yield criterion. Using equation (2.43), the constants C1 , C2 , C3 can be determined: C1 0 C2 3 C3 0. 24. (2.49).
(57) CHAPTER 2 MATHEMATICAL MODEL. 2.4.5. The modified Cam-Clay model The yield surface of the modified Cam-Clay model has the form of an ellipsoid in the p - q plane (Figure 2-7) and it is defined as (Burland 1965):. q 2 M 2 p p pc 0. (2.50). Where -. p is the hydrostatic pressure defined in equation (2.15),. -. q is the deviatoric stress defined in equation (2.26),. -. M is the slope of the failure line in the p - q plane (Figure 2-7) and is related to the angle friction and the Lode‟s angle by M. -. 6sin 3 sin sin 3. (2.51). and pc is a hardening parameter characterizing the size of the ellipsoid.. Figure 2-7 Yield surface of the modified Cam Clay model. Doing the connection between equation (2.35) and equation (2.50), we will choose. F. . 1 2 q M 2 p p pc pc. . (2.52). And F0 pc. 25. (2.53).
(58) CHAPTER 2 MATHEMATICAL MODEL. The hardening/softening rule is given by the relation between the size of the yield surface pc and the plastic volumetric strain, vp :. dpc 1 e pc d vp . (2.54). Where -. e is the void ratio of the material. -. is the slope of the „Normal Consolidation line‟, which is the line. observed in the ln p e plot during a hydrostatic compression test (Figure 2-8) -. and is the slope of the line observed in the ln p e plot at the beginning of the unloading process (Figure 2-8).. The parameter depends on the type of soil, and it can be related to the Plasticity index PI by the empirical relation (Atkinson and Bransby 1978):. . PI 171. (2.55). The parameter is a constant characterizing the elastic volumetric response and can be related to the bulk modulus, K v , by Kv . 1 e. . p. (2.56). Figure 2-8 Hydrostatic compression test on a normally consolidated clay – a) Experimental results ; b) Idealized behavior (Zienkiewicz et al. 1999). 26.
(59) CHAPTER 2 MATHEMATICAL MODEL. As for the Von Mises yield criterion, we will assume an associated flow rule. Therefore the plastic potential surface is. g p, q, F p, q, g p, q, . . 1 2 q M 2 . p. p pc pc. (2.57). . (2.58). As in the section 2.4.4. , the plastic flow passing through a stress point is m C1m1 C2m2 C3m3. (2.59). Where. C1 . g I1. g tan 3 g C2 J J 2 2 C3 . (2.60). 3 1 g 2cos3 J 23 2 . And m1 m2 . I1 ζ J2. ζ J m3 3 ζ. (2.61). In equation (2.58), the plastic potential is expressed in function of the invariants. p, q, M . therefore in order to easy the calculation, it is possible to use the. chain rule of derivative. g g p . I1 p I1 g J2. . g q . q J 2. g g M . M . 27. (2.62).
(60) CHAPTER 2 MATHEMATICAL MODEL. And to introduce these products of derivatives in the expression (2.60) as. C1 . g p . p I1. g q tan 3 g M C2 . . q J J 2 M 2 C3 . (2.63). 3 1 g M . 2cos3 J 23 2 M . Using equations (2.20), (2.26), (2.51) and (2.58), the following derivatives can be calculated:. g M 2 2 p pc p pc. p 1 I1 3. ;. g 2q q pc. q. ;. g 2Mp p pc M pc. J2 ;. 3. (2.64). M 18sin 2 cos3 3 sin sin 3 2. Introducing these derivatives in the expressions (2.63), we got C1 C2 . 1 M2 2 p pc 3 pc. 6 J2. C3 . pc. . tan 3 2 Mp p pc 18sin 2 cos3 . . 2 pc J2 3 sin sin 3 . (2.65). 3 1 2Mp p pc 18sin 2 cos3 . . 2 2cos3 J 23 2 pc 3 sin sin 3 . 2.4.6. Elasticity matrix in plane stress condition and in plane strain condition The last term of the Perzyna‟s viscoplastic model which has not been defined yet is the elastic constitutive tensor De which appears in equation (2.33). The component of this tensor depends on the dimensions and on the choice between plane stress and plane strain condition. In two dimensional problems, the velocity field is uniquely given by the velocity along the x1 and x2 axis. The velocity along the x3 axis is not considered.. 28.
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