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Numerical Study of Strain Rate Effects on Stress Strain

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Numerical Study of Strain Rate Effects on Stress Strain NUMERICAL STUDY OF STRAIN RATE EFFECTS ON STRESS STRAIN RESPONSE OF SOILS. A thesis presented for the degree of Doctor of Philosophy in the Faculty of Engineering of the University of London by JOSE CARLOS ZIOLKOWSKI B.Sc. Eng, M. Sc.. Department of Civil Engineering Imperial College of Science and Technology London, October - 1964 To Riva and our parents. i ACKNOWLEDGEMENTS The work described in this thesis has been carried out in the Soil Mechanics Section of the Civil Engineering Department, Imperial College, London. This thesis would not have been possible without the kind help of several people. The author wishes to thank Dr. P.R.Vaughan, interim head of the Soil Mechanics Section, When this research began in 1979, and later head of the section Prof. J. B. Burland, for taking interest in my work and for allowing it to be carried out in the academic enviroment of the Imperial College of Science and Technology. The author wishes to thank the Brazilian Government and the Conselho Nacional de Pesquisa (CNPq) for the financial support given. The author would very much like to express his sincere gratitude to the extraordinary person, Dr. A. Skinner, appointed supervisor to this research, for his immense kindness, patience and understanding, particularly for supporting the author in every circumstance during the long period of contact. A great deal of help, encouragement, equilibrium and confidence was received from him during the course of this research to which the author is greatly in debt. Much useful help and constructive suggestions were given by Prof. J. B. Burland, David Potts, David Hight, Richard Jardine, and Dr. S. Cavoudinis. Special thanks to Eileen Gibbs, Lou Spall, Fred Evans, Steven Ackerley and all the staff of the Soil Laboratory for the most friendly reception during tea time for during those years. David Toll, a person whom the author greatly admires for his ii experience, for his nature, devoted friendship, who made life in England more enjoyable and most of all easier, for his prompt and unescrupulous help in all matters and for the devotion of a great deal of his precious time painstakingly going through the whole thesis, suggesting, contributing and correcting the English use of language. There are no words to express the author's gratitude for what he has dene in contributing to the carpletion of this thesis. The author very much appreciated the valuable discussion and suggestion frem his many colleagues in the Soil Mechanics Section, T. de Campos, M. Chandler, A. Fourie, L. Lemos, M. Martins, J. Maswoswe, S. Shibuya, M. Tfckahashi, D. Rinaldis, L. Oosta-Filho, F. Lupini, P. Martins, and M. Symes, J. Hellings, E. Ovando-Shelley, M. Maccarini. Malccm Clark, who made himself always available whenever help was required, specially When it concerned equations and computing, which the author greatly appreciated. The author also wishes to thank his oollegues, U. Trueb, D. Bates, N. Agelidis, fran the Structure section of Imperial College, for their help, discussion, and advice in caiputing method and finite elements affairs. Many thanks are due to Mrs Majorie Carter, Mrs. Kay Crooks and Ms. Stringer for their assistance in the library. Thanks are also due to Mr. H. Catanhede, for his friendship, willingness to help and for undertaking the task of draughting the figures for this thesis, work of a craftsmn, and Whose help Vvas indispensable. The author also wishes to extend his thanks to Prof. E. Osvaldo Cruz, representative of the Brazilian Institution CNPq here in London for his tireless support during the carpletion of this thesis. I l l Last, but not least, the author would like to thank Riva for her continuous encouragement throughout the research programme, for her uninterrupted and tireless assistance in typing the draft manuscript, and in making several of the drawings included in the report. Her support is most appreciated and valued by the author. iv ABSTRACT A discussion of consolidation of saturated clay is presented. The governing differential equations are reached using the concepts of continuum mechanics of a mixture, where one phase represents the deformable clay skeleton, and the other represents the pore fluid Which fills the pores of the skeleton. A geometrically non-linear- system and a elasto-visccplastic-plastic material are accounted for. For the gecmetrically non-linearity the up-dated Lagrange method is applied. Two independent yield surfaces have been used to describe the viscoplastic- plastic constitutive relationship, and an associative and/or a non-associative flew rule have been assumed. Darcy's law for a deformable skeleton and a permeability matrix dependent on the void ratio have been taken into account. An algorithm based on finite element discretization and numerical integration in time is adopted for the numerical treatment of the transient process. The finite element type chosen is a variable eight noded isoparametric one where the same number of nodes for displacement and pore water pressure have been adopted. A semi- implicit type method for time integration is used. For each time and/or load step a tolerable equilibrium condition is achieved iterativelly to take into account the material and geometric non- linearities . A sample of numerical examples have been calculated to show the V general abilities of the computer program. Key words - Consolidation, finite elements, plasticity, creep, large deformation. vi CONTENTS CHAPTER I - INTRODUCTION 1.1 - Need for More Realistic Analysis.............................. 1 1.2 - Purpose of this Research....................................... 2 1.3 - Original Characteristics........................................ 3 1.4 - Summary of the Contents and Scope of this Research............4 CHAPTER II - PHENOMENOLOGICAL CONCEPTS 11.1 - Introduction.................................................. 7 11.2 - Consolidation Phenomena........................................ 8 11.3 - Initial Settlement............................................. 9 11.4 - Primary Consolidation........................................ 10 11.5 - Secondary Consolidation...................................... 11 11.6 - Terminology................................................... 12 11.7 - Basic Principles: Discussion................................. 13 CHAPTER III - CONSOLIDATION THEORY : BRIEF SURVEY 111.1 - Introduction................................................. 16 111.2 - Basic Characteristics and Applicability.................... 16 111.3 - Non-Oonsistent Theory: Brief Discussion.................... 18 vii 111.4 - Self Consistent Theory: Biot's Theory ...................... 20 111.5 - Gontinumn Theory Approach...................................21 1 1 1 . 6 - Solution Methods ............................................ 23 CHAPTER IV - CONTINUOUS MECHANICS OF MIXTURE REVIEW IV. 1 - Introduction................................................ 24 IV.2 - Body Motion .................................................. 25 IV.3 - Independent Variables ......................................... 27 IV.4 - Strain Definitions ............................................ 30 TV.5 - Strain Invariants ............................................ 35 TV .6 - Ccnpatibility Conditions ...................................... 40 IV.7 - Force Distribution, Mass Density, Internal Energy Density... 42 IV .8 - Global Balance Law ............................................ 43 IV.9 - Local Balance Laws ............................................ 46 IV. 10 - Definition of Stress........................................ 49 1. Stress Vector .............................................. 49 2. Stress Tensor ............................................. 51 CHAPTER V - VARIATIONAL METHOD V. l - Introduction ................................................ 60 V.2 - A Brief Survey of Different Approaches ....................... 61 viii V. 3 - Principle of Virtual Work ................................... 65 CHAPTER VI - FIELD EQUATION IN INCREMENTAL FORM VI. 1 - Introduction...................................................69 VI.2 - Velocity Increment ..... ..................................... 73 VI.3 - Strain Increment ............................................. 74 VI.4 - Stress Increment ............................................. 76 VI.5 - Stress-Strain Increment Relationship for the Solid Skeleton. 78 VI . 6 - Stress-Strain Increment Relationship for the Fluid Phase and Darcy Law ................................................. 81 VI.7 - Bernoulli's Theorem - Darcy's Law ........................... 83 VI . 8 - Total Lagrange Formulation...................................85 VI.9 - Updated Lagrange .............................................. 88 VI. 10 - Linearization of Equilibrium Equations ..................... 90 CHAPTER VII - FINITE ELEMENT SOLUTION VII. 1 - Introduction............................................... 93 VII.2 - Finite Element Solution.....................................93 VII.3 - Finite Element Matrices ...................................... 95 VI1.4 - Numerical Integration ........................................ 99 VII.5 - Equilibrium Iteration 101 ix CHAPTER VIII - CONSTITUTIVE LOCAL STRESS-STRAIN RELATIONSHIPS VIII. 1 - Preliminaries............................................107 VIII.2 - Brief Description of Soil Properties ..................... 108 VIII.2.1 - Soil Properties in the Quasi-Static Region.............Ill 1. Triaxial Test Conditions .............................. Ill 2. Third Stress Invariant .................................. 125 VIII.2.2
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