The Coupled Theory of Mixtures in Geomechanics with Applications

The Coupled Theory of Mixtures in Geomechanics with Applications George Z. Voyiadjis and Chung R. Song The Coupled Th...

Author: George Z. Voyiadjis | C.R. Song

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The Coupled Theory of Mixtures in Geomechanics with Applications

George Z. Voyiadjis and Chung R. Song

The Coupled Theory of Mixtures in Geomechanics with Applications

ABC

Dr. George Z. Voyiadjis

Dr. Chung R. Song

Department of Civil and Environmental Engineering Louisiana State University CEBA Building, Room 3508-B Baton Rouge, LA 70803-6405 U.S.A. E-mail: [email protected]

218 Carrier Department of Civil Engineering University of Mississippi University, MS 38677 U.S.A. E-mail: [email protected]

Library of Congress Control Number: 2005936354 ISBN-10 3-540-25130-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25130-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11347415

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543210

The authors dedicate this book to the victims and those displaced due to the Hurricane Katrina in the states of Louisiana and Mississippi.

Preface

Geomaterials consist of a mixture of solid particles and void space that may be filled with fluid and gas. The solid particles may be different in sizes, shapes, and behavior; and the pore liquid may have various physical and chemical properties. Hence, physical, chemical or electrical interaction between the solid particles and pore fluid or gas may take place. Therefore, the geomaterials in general must be considered a mixture or a multiphase material whose state is described by physical quantities in each phase. The stresses carried by the solid skeleton are typically termed “effective stress” while the stresses carried by the pore liquid are termed “pore pressure.” The summation of the effective stress and pore pressure is termed “total stress” (Terzaghi, 1943). For a free drainage condition or completely undrained condition, the pore pressure change is zero or depends only on the initial stress condition; it does not depend on the skeleton response to external forces. Therefore, a single phase description of soil behavior is adequate. For an intermediate condition, however, some flow (pore pressure leak) may take place while the force is applied and the skeleton is under deformation. Due to the leak of pore pressure, the pore pressure changes with time, and the effective stress changes and the skeleton deforms with time accordingly. The solution of this intermediate condition, therefore, requires a multi-phase continuum formulations that may address the interaction of solid skeleton and pore liquid interaction. The pore pressure leak(flow) is related to the hydraulic conductivity, it implies that the hydraulic conductivity affects the behavior of geomaterials. Conversely, when the behavior of a soil is known, the hydraulic characteristics of the soil can be known from the hydro-mechanical analysis – the so-called “coupled theory of mixtures”. Biot (1955, 1978) was one of the first scientists to develop a coupled theory for an elastic porous medium. However, experiments have shown that the stress-strain-strength behavior of the soil skeleton is strongly non-linear, anisotropic, and elasto-plastic. An extension of Biot’s theory into the nonlinear, anisotropic range is therefore necessary in order to analyze the transient response of soil deposits. This extension has acquired considerable importance in recent years because of the increased concern with the dynamic behavior of saturated soil deposits and associated liquefaction of saturated

VIII

Preface

sand deposits under seismic loading conditions. Such an extension of Biot’s formulation was proposed by Prevost (1980). Prevost (1980)’s theory of mixture was further extended for the updated Lagrangian reference frame by Kiousis and Voyiadjis (1985), and Voyiadjis and Abu-Farsakh (1996), thus the application to the large strain behavior becomes possible. This book addresses the coupled theory of mixtures for geo-materials. It also presents the formulation and the numerical procedures using the coupled theory of mixtures for geo-materials for the solution of a variety of problems in geo-mechanics, including the applicability and use of the cone penetrometer (for evaluation of soil properties), soft soil tunneling, Implementation of Nano-mechanics to geo-materials, Application of Geo-acoustics, and so forth.

Baton Rouge Oxford December 29, 2005

George Z. Voyiadjis Chung R. Song

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Historical Review of the Theory of Mixtures for Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Early Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Classical Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The Modern Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 7

2

Flow in Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nature of Geo-materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Flow of Water in Geo-Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Transient or Non-steady Flow . . . . . . . . . . . . . . . . . . . . . .

11 11 13 13 13

3

Coupled Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Traditional Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Terzaghi’s Consolidation Theory . . . . . . . . . . . . . . . . . . . 3.1.2 Biot’s Consolidation Theory . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modern Theory of Mixtures for Finite Strain . . . . . . . . . . . . . . . 3.2.1 Balance of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Coupling of Mass Balance Equation and Darcy’s Law .

15 15 15 16 17 18 20

4

Coupling Yield Criteria and Micro-mechanics with the Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Soil Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Anisotropic Soil Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Anisotropic Modified Cam Clay Model . . . . . . . . . . . . . . 4.3 Elasto-Plastic Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . 4.4 Micro-Mechanical Considerations/Bridging Different Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 RVE Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Characteristic Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Bridging Different Length Scales . . . . . . . . . . . . . . . . . . . 4.5 Micro-mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Back Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Rotation of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

23 23 30 33 36 38 39 40 45 50 50 55

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Contents

4.5.3 Grain Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Rate Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Damage of Solid Grains . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 Summary of Micro-mechanisms . . . . . . . . . . . . . . . . . . . . 4.6 Equation of Equilibrium of the External and Internal Forces in an Updated Lagrangian Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 68 73 74

5

Finite Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Updated Lagrangian Reference Sheme . . . . . . . . . . . . . . . . . . . . . 5.2 Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Remeshing and Return to Yield Surface . . . . . . . . . . . . . . . . . . .

83 83 86 89

6

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Piezocone Penetration Test (PCPT) . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Current Practice of Determining Hydraulic Properties from the Piezocone Penetrometer . . . . . . . . . . . . . . . . . . . 6.1.3 New Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 (Shield) Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Finite Element Numerical Simulation . . . . . . . . . . . . . . . 6.2.4 Remeshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Mapping Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Modeling of Interface Friction . . . . . . . . . . . . . . . . . . . . . . 6.2.7 Case Study of N-2 Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Estimation of Hydraulic Conductivity using Acoustic Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Basics of Wave Propagation in Saturated Media . . . . . . 6.3.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 91

7

Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Nano-mechanics for Geotechnical Engineering . . . . . . . . . . . . . . 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Brief History of Nano-mechanics . . . . . . . . . . . . . . . . . . . 7.1.3 Nano-mechanics as a General Platform for Studying Detailed Behavior of Geo-materials . . . . . . 7.1.4 Nano-mechanics as a Tool to Study Macro-level Material Properties Through Continuumization . . . . . . 7.1.5 Space Science Application . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Coupled Behavior of Micro-Mechanisms . . . . . . . . . . . . . . . . . . .

76

93 95 115 116 116 119 122 126 128 134 138 150 150 151 155 157 161 161 161 162 163 167 171 172

Contents

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Simplification of Equations Incorporating the Physical Behavior of Soils . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Rate Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 (Strain) Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.7 Anisotropy and Plastic Spin . . . . . . . . . . . . . . . . . . . . . . . 7.2.8 Experimental Verification of Coupled Pore Pressure Around a Cone Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.9 Back Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.10 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

172 173 175 179 180 182 184 188 201 204

Appendix: Fortran Codes of CS-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

1 Introduction

Geomaterials consist of assemblages of particles with different sizes and shapes that form a skeleton (porous matrix) whose voids are filled with water or other liquids and air or gas. The word ‘soil’ therefore implies a mixture of assorted mineral grains with various fluids and must for that reason be considered a multi phase material. The study of this type of material is of great importance in applied civil engineering, particularly in geotechnical engineering. The analysis of the response of multi phase materials (i.e. soils or porous media) has a multitude of important applications: analyzing the settlement of underlying soil deposits, determining the dissipation of excess pore water pressures resulting from foundation loading, designing foundations for vibrating machines, analyzing the vulnerability of offshore structures under wave loading, studying the propagation of earthquake impulses in geologic materials, and measuring the catastrophic loss of strength of a soil associated with the increase of pore water pressure (the phenomenon known as “liquefaction”). The importance of understanding these phenomena has been recognized since the early 1940’s, when the colossal devastation caused by major earthquakes heightened the need to better understand the behavior of soils in such events in order to mitigate higher losses that could be avoided by adequate engineering design. During the last three decades, significant progress has been made in understanding the behavior of a porous matrix interacting with one or more fluids. In the area of geotechnical engineering, three approaches may be identified as a result of this process: (1) field observations prior, during, and after earthquakes; (2) laboratory experimentation; and (3) theoretical/numerical studies. While many researchers have focused on what could be observed from laboratory experiments and field evidence, others have emphasized their work in the development of appropriate theoretical procedures to describe these observations. Theoretical studies have yielded three different philosophies: decoupled methods (or the total stress approach), indirectly coupled methods (the quasi-effective stress approach), and fully coupled methods (or the effective stress approach). This book focuses on the fully coupled methods of analysis, which are based on the theory of mixtures. The premise of these theories is that a mixture may be viewed as a superposition of “n” individual continua, each

2

1 Introduction

following its own motion. It is also assumed that at any time t, the entire space in the mixture is occupied simultaneously by several different particles, one from each constituent. In addition, each component of a mixture is regarded as an open system for which local balance relations are postulated. In this condition, their production densities differ from zero because of the fact that the interaction between the individual constituents must be incorporated. Furthermore, because it is natural to expect that the mixture as a whole will act as one-component single body, restrictive conditions must be postulated in terms of conservation principles. In the light of these theories, the behavior of porous media cannot be described directly because in general it is not possible to know a priori which spatial position will be occupied by which particular constituent. For this reason, substitute continuum models must be used to replace the particulate structure, where a single material component is assumed to occupy the total region. The use of volume fraction theories constitutes the necessary connection between global bulk average quantities and the local effective quantities. Within this framework, the problem generally leads to geometrical and physical nonlinear relations. In order to obtain a simple and practical theory, further simplifications must be introduced. For this reason, a mixture composed of only two constituents, i.e. the porous matrix and a saturating fluid, frequently is considered. The different components are assumed to share a common temperature with a vanishing temperature gradient. In addition, the material under consideration frequently assumed to be isotropic, and it is assumed that the compressibility of the solid particles is much smaller than the compressibility of the body as a whole. The deviator stress in the media fluid is assumed to be negligible in comparison to that in the solids skeleton. Finally, the development of the constitutive equations is restricted to the case of small strains initially: it is extended to the case of large strains later. From this analysis, the differential equations that govern the motion of a soil mixture become coupled with the global mass balance equation, an operation that leads to a set of coupled differential equations. To obtain a solution, the set of differential equations may be approximately be a weighted residual method such as Galerkin’s method, and the approximated integral may be discretized using finite element techniques and robust temporal integration methods. This approach results in a full coupling of the pore pressure generation/dissipation with the deformation of the soil skeleton. In this sense, the ability that the selected constitutive model may have to predict permanent volume changes during loading becomes a major factor in the global performance of the algorithm. The historical review in Sect. 1.1 is obtained from the reference by Arduino (1996) with the author’s permission. This section is mostly verbatim from this reference with minor updates on some additional research works.

1.1 Historical Review of the Theory of Mixtures for Geo-materials

3

1.1 Historical Review of the Theory of Mixtures for Geo-materials Three major periods may be identified in the historical development of the porous media theory (De Boer, 1996): The early era (18th and 19th centuries), the classical era (1910–1960), and the modern era (1960-present). In the early era, the concept of volume fractions was stated, some fundamental laws were discovered, and the mixture theory was founded. In the classical era, scientists first attempted to clarify the mechanical interaction of liquids, gases, and rigid porous solids to explore the behavior of deformable saturated porous solids. In the modern era, researchers developed and continue to study theories of immiscible mixtures. 1.1.1 The Early Era By 1794, R. Woltman had made the first contributions to the theory of porous media. Motivated by a competition at the Imperial Academy of Sciences in St. Petersburg, Woltman developed a sophisticated earth pressure theory, introducing a concept for the calculation of failure conditions in soil mechanics – the angle of internal friction. Woltman had already separated soils into four types: sand, lime, clay, and compost-earth. At that time he pointed out that friction is a common property in all kinds of earth and that this friction effect differentiates soils from fluids. He stated that only by total saturation with water is the friction effect lost. Woltman called this state of the mixture “quicksand” or “mud”. In connection with his discussion of the mechanical behavior of “mud”, Woltman spoke of a mixture and (surprisingly) introduced the concept of volume fractions. He was probably the first scientist to formulate this concept. After the development of the concept of volume fractions by Woltman, no substantial contributions to the theory of heterogeneously composed bodies were published for a long time. Around the mid-nineteenth century, important findings in the theory of mixtures and in the theory of porous bodies were made principally by Delesse, Fick, and Darcy (De Boer, 1996). Delessian Law With the concept of volume fractions in hand, scientists were able to treat heterogeneously composed bodies with existing continuum mechanical models. However, it was also necessary to develop a corresponding concept for the surface element on a saturated porous body. In 1848, Delesse made a decisive contribution. As a mining engineer, he was interested in determining the ratios of single minerals in valuable ores without destroying samples, and after some theoretical considerations, he concluded that the surface rations are equal to the volume fractions. Although this concept was, and must be,

4

1 Introduction

considered as a statistical necessity, it was extensively used in the following developments of the porous media theory. Fick’s Law The first attempts to develop a phenomenological theory of mixtures were made by Fick (1855), who studied the problem of diffusion. Fick, who was a physician, was inspired by hydro-diffusion through membranes. He remarked that this problem is not only important for organic life, but also for other important physical processes. Following the development of the Fourier equation of heat propagation, Fick arrived at the differential equation of the diffusion stream which in the case of a constant cross-section takes the form; ∂2y ∂y = −k 2 ∂t ∂x

(1.1)

where y is concentration, t is time, k is a constant which depends on the nature of the constituents, and x is a measure of height. Today, this relation is known as Fick’s second law of diffusion. Darcy’s Law Darcy (1856) observed in tests with natural sand both the proportionality of the total volume of running water through the sand and the loss of pressure. Although his investigations were purely of experimental nature, his results are essential for a continuum mechanical treatment of the motion of a liquid in a porous solid. In Darcy’s work, the interaction of different constituents in a multi-phase continuum was studied for the first time. Today, Darcy’s law is theoretically well founded with thermodynamic principles. However, Darcy’s law becomes invalid for liquids at high velocities, and for gases at very low and at very high velocities. Although Delesse, Fick and Darcy discovered their fundamental laws more or less heuristically rather than by development from the fundamental relations of mechanics and thermodynamics, they established fundamental statements that have been extensively applied in the theories that have followed, such as the mixture and the porous media theories. At the end of the nineteenth century, the development of a new branch of mechanics began that is of essential importance for the creation of a consistent porous media theory: the theory of mixtures. As previously stated, it was Fick who discovered the differential equation for the diffusion problem of liquids with different concentrations. However, Maxwell was the first scientist who, starting from the basic principles of mechanics, developed the hydrodynamic equations for gas mixtures (Stefan, 1871). His investigations finally led to the creation of the kinetic gas theory. A decisive step to the continuum mechanical theory of mixtures was accomplished by Stefan (1871):


5

If the true processes in a mixture should be calculated, it is not sufficient anymore, to consider the mixture as a uniform body as common mechanics does; equations must be set up which should contain the condition of equilibrium and the laws of motion for every individual constituent in the mixture. He then introduced the main assumption concerning the interaction forces between constituents: “In a mixture, each particle of a gas, if it is in motion, suffers from each of the other gases a resistance which is proportional to the density of this gas and the relative velocity of both.” Stefan subsequently formulated the equations of equilibrium for a mixture of two gases and the balance of mass excluding any mass exchange. Following the derivation of the differential equations, Stefan discussed solutions for special initial conditions cases. In the eighth section of his valuable paper, he treated the problem of the diffusion of a gas through a porous diaphragm. This study can be considered as the first time that the mixture theory restricted by the volume fractions concept was applied to a binary model within the framework of continuum mechanics. 1.1.2 The Classical Era From the second to the fourth decades of the current century, decisive progress was made towards creating a consistent porous media theory. There were two notable steps in this progression. First, scientists discovered important mechanical effects in a liquid saturated rigid porous solid. Second, in the 1920’s and 1930’s, the scientists first attempted to investigate saturated deformable porous solids. The discovery of fundamental mechanical effects in saturated porous solids and the formulation of the first porous media theories are mainly due to two distinguished professors at the Technische Hocheschule of Vienna: Paul Fillunger and Karl von Terzaghi. From 1913 to 1934, these two professors described for the first time important physical phenomena in rigid liquid-saturated porous media, specifically with reference to the effects of uplift, friction, capillarity, and effective stress. It was Fillunger (De Boer, 1996) who pioneered the porous media theory of liquid-saturated porous solids, investigating the problem of uplift and friction forces acting on heavy weight masonry dams. At the same time, von Terzaghi addressed the theoretical problem of capillarity and formally established the concept of effective stresses. The effective stress concept had already been presented at the beginning of the century, but definitive evidence was provided by von Terzaghi and later by Fillunger, (Bjerrum et al., 1960; De Boer and Ehlers, 1990a,b; De Boer, 1996). Although much success had been gained with the model of a saturated rigid porous solid, the theory of porous media remained incomplete because the description of the deformation and the determination of the stress state in saturated deformable porous bodies were excluded from Terzaghi’s and

6

1 Introduction

Fillunger’s research. The first author to deal with the important problems of fluid-filled deformable porous solids was von Terzaghi. He recognized that a water-filled and deformable soil body, despite being of great relevance for the foundation of buildings, had still not been scientifically treated. In a famous paper presented to the Academy of Sciences in Vienna in June 1923, von Terzaghi showed the derivation of his consolidation theory. This theory, which brought him immediate fame, was later published in an internationally acclaimed expanded study that is considered the first substantial book in soil mechanics (Terzaghi, 1943). Subsequently, Fillunger also studied in detail the consolidation problem and published his results in a pamphlet called “Erdbaumechanik?” where Karl Terzaghi’s work was strongly criticized (De Boer, 1996). He proceeded from a two-phase system which he described with sound mechanical axioms and principles. Certain statements of inexorable criticism and personal defamatory attacks against von Terzaghi led to a personal controversy between von Terzaghi and Fillunger known as the “Viennese Affair” which ended with the tragic suicide of Fillunger and his wife. (For more details see Bjerrum et al., 1960; De Boer, 1996). Despite this tragic episode, the discovery of fundamental mechanical effects in liquid-filled porous solids by these two Viennese professors in the first half of this century represents a brilliant achievement in engineering. At that time, thermodynamics, in the modern sense, was not yet developed and the constitutive theory with the procedure to gain restrictions from the entropy inequality was completely unknown. Therefore it is not surprising that in the sequel of the discoveries, many errors and incorrect proofs appeared. After the fundamental contributions of von Terzaghi and Fillunger, the theory of porous media was further developed, in particular, by Maurice Biot who followed, at the beginning of his career, von Terzaghi’s direction. In the early 40’s Biot (1941) generalized von Terzaghi’s theory of consolidation by extending it to the three dimensional case and by establishing equations valid for any arbitrary load varied with time. In the following years, Biot generalized his theory to include properties of anisotropy, variable permeability, linear visco-elasticity, and the propagation of elastic waves in a fluid saturated porous solid (Biot, 1941, 1955, 1956b). The three-dimensional theory of wave propagation developed by Biot is rather intuitive (but consistent) and shows the existence of two dilational and one rotational body waves. A chronological review of Biot’s work is discussed in his publications on the mechanics of deformation and acoustic propagation in porous media (Biot, 1956a,c,d). The main disadvantage of Biot’s model, however, is that the corresponding theory is not developed from the fundamental axioms and principles of mechanics and thermodynamics. Thus, some derivations are complicated and obscure. Finally, Biot (1965) developed, within the framework of quasi-static and isothermal deformations, a theory of finite deformations of porous media.


7

1.1.3 The Modern Era Since the beginning of the 1960’s the study of porous media has advanced in several directions: A. In the geotechnical field, the necessity of obtaining practical solutions led to the development of simplified theories and empirical relations. Scientists developed experimental and sometimes incomplete formulations to acquire a better understanding of the soil-fluid interaction problem. This work is important, for it sheds light on the physical behavior of saturated soils. With von Terzaghi’s consolidation theory in hand, the problem of the static (or quasi-static) analysis of saturated porous media found a practical solution that was applied to many “real life” problems. However, the colossal devastation caused by earthquakes required the study of the dynamic problem as well. Prior to 1975, dynamic analyses in geotechnical engineering were based on total stresses because of the deficiency in practical models that could predict pore water pressures. A widely used method was the Equivalent Linear Method (ELM), which provided an approximate solution based on elastic soil stiffness and damping that are compatible with induced strains in the soil. In this method, results from laboratory tests relate the damping ratio and the shear modulus to cyclic shear strain levels. A linear solution is based on initial values of the shear modulus and the damping ratio. From the time variation of the shear strain, an equivalent strain magnitude is estimated and used to obtain new values of the shear modulus and damping ratio. A new solution is calculated, and the procedure is repeated until convergence is achieved. The method became popular after Seed and co-workers and was implemented in computer applications such as SHAKE (Schanabel et al. 1972), QUAD (Idriss et al., 1983), and FLUSH (Lysmer et al., 1975). However, none of these models could predict either the increase of pore water pressure or its effect in the effective stresses. In 1975, (Martin et al., 1975) presented a model that clarified the densification mechanism that occurs during liquefaction. They stated that for a saturated sand, if drainage cannot occur during the time span of the loading sequence, the tendency for volume reduction during each cycle of loading results in a corresponding progressive increase in pore-water pressure. In this model, it was assumed that the plastic volumetric strain that occurs during one cycle of uniform shear strain in an undrained simple shear test is equal to the total volumetric strain in a drained simple shear test. The plastic volumetric strains during undrained cyclic loading are absorbed by the elastic rebound in the soil skeleton due to the decrease in effective stress and a constant volume is retained, which is a basic assumption in critical state theory as applied to undrained loading. Because of its ability to estimate pore pressures, the model was implemented in non-linear dynamic effective stress analyses which originated

8

1 Introduction

the programs DESRA (Lee and Finn, 1978) and TARA (Finn et al. 1986). In 1976 Bazant and Krizek developed an endochronic constitutive law for the study of the liquefaction of sand. The development of the endochronic theory is attributed to Valanis (1971) and is based on an intrinsic time parameter which is an independent scalar variable that depends on both time and deformation increments. Bazant and Krizek used endochronic variables in order to express the densification of loose sand that develops during cyclic shearing. Lee and Finn (1978) investigated the possibility of expressing the pore water pressures and the volumetric strains in terms of endochronic variables. They presented an efficient response function to represent the pore pressure data by simplifying the family of curves used in previous methods to a single curve. B. The work of Biot also received great attention and was extensively used. The literature is replete with publications pertaining to the analytical solution of the general governing equations of motion for two-phase media based on the work of Biot. Deresiewicz (1960, 1962) solved Biot’s governing equations of motion for an elastic half-space under harmonic time variations using displacement potentials. Derski (1978) used velocity terms to express the relative motion of different phases. Burridge and Vargas (1979) obtained the time domain fundamental solution (Green’s function) for an infinite space; they also studied the disturbance in a poro-elastic infinite space due to application of an instantaneous point body force. Simon et al. (1984) presented an analytical one-dimensional solution for the transient response of an infinite domain by using Laplace transformations. Gazetas and Petrakis (1981) evaluated the compliance of a poroelastic half-space for swaying and rocking motions of an infinitely long, rigid and pervious strip that permitted complete drainage at the contact surface. Finally, Halpern and Christiano (1986) evaluated the compliances of three-dimensional square footings considering pervious as well as impervious cases. C. At the same time, the development of new continuum theories of mixtures opened once again the dilemma about the validity of many of the approaches that had been used in the past. In 1960, Truesdell and Toupin presented a treatise on the classical field theories in which they developed in detail the properties of motion and the fundamental physical principles of balance. They also listed mathematical principles that help to formulate definite constitutive models. The treatise contains a wide reference to previous work, and constitutes a rational work based on methods of modern continuum mechanics. In 1965, Green and Naghdi (1965b, 1967, 1968) developed a dynamic theory for the relative flow of two continua based on an energy equation and an entropy production inequality for the entire continuum. This approach takes into account the use of invariance conditions under superposed rigid body motions, theoretical approach that is more in line with the work of Green and Rivlin (1964), which examines a


9

new type of continuum theory. The new theory, known as multi-polar continuum mechanics, is based on some concepts developed by Truesdell and Toupin, who introduced generalized forces, body and surface forces, and generalized stresses. Green and Rivlin also discussed Truesdell’s theory and related it to their new approach. The theories of mixtures were actually studied by many researchers, who developed their variations to gain further understanding of the behavior of mixtures. The work of Bowen (1976), which contributed to the formulation of rational mixture theories and the understanding of wave propagation through porous media, was studied, by Garg et al. (1974) and Derski (1978), among others. Aifantis (1980) and co-workers also contributed useful works, including studies of porous media. Atkin and Craine (1976) outlined the historical development of mixture theories and demonstrated the applicability of mixture theory to the case of chemically inert mixtures of ideal gases. It seems that Morland (1972) was the first scientist to use the volume fraction concept in connection with modern mixture theories to describe the behavior of porous media. Drumheller (1978) presented a theoretical treatment of porous solid using a mixture theory in which the volume fraction concept was introduced. The key point in his derivation was in the formulation of pore collapse relations to express the rate of change of the volume fractions. Bowen (1982) summarized all findings of the mixture theory and introduced the volume fractions concept for the saturated condition. Bowen substituted the free Helmholtz energy function per unit mass in the Clausius-Duhem inequality by a free energy review concerning these and other thoeories for structured mixtures (e.g. porous media). Katsube and Carroll (1987) modified the mixture theory of Green and Naghdi by introducing a micro-mechanical constitutive theory. Ehlers (1989) presented a macroscopic description of fluid-saturated porous media via mixture theories extended by the volume fractions concept where elasto-plasticity was taken into account by means of a multiplicative decomposition of the deformation gradient. Following this line of research, Ehlers (1993) was able to describe the behavior of compressible, incompressible, and hybrid porous materials. D. With the advances in modern computational science and the development of rigorous numerical techniques such as the finite element method and numerical implementations of the consolidation theory, Biot’s equations and mixture theories found wide applications. A variational formulation of the dynamics of fluid-saturated porous solids was the basis of a numerical method that Ghaboussi and Dikmen (1978) developed for the purpose of discretizing a partial media into finite elements. Shandu and Wilson (1969) first applied the finite element method to study fluid flow in saturated porous media. With the introduction of the FEM as a sound numerical technique, it became possible to extend the mixture theory to encompass elasto-plastic non-linear constitutive models and obtain reliable

10

1 Introduction

solutions of the field displacements and pressures. Prevost (1980) presented a general analytical procedure that accounts for non-linear effects. In his work, Prevost focused on the integration of the discretized field equations based on the mixture theories of Green and Naghdi (1965b). Later, he worked on several numerical applications to study the consolidation of inelastic porous media (Prevost, 1981) and on the non-linear transient phenomena and wave propagation effects in saturated porous media (Prevost, 1982, 1985; Lacy and Prevost, 1987). Because of the increasing necessity of non-linear applications, Zienkiewicz, and other researchers published a series of papers that elucidated various numerical solutions for pore–fluid interaction analysis. Zienkiewicz and Shiomi (1984) classified different methods of analysis in a comprehensive paper on numerical solutions of the Biot formulation. These numerical solutions were further studied and used in several numerical applications related to the undrained, consolidating, and dynamic behavior of saturated soils (Simon et al., 1986; Zienkiewicz et al., 1990; Simon et al., 1986). A continuum theory for saturated porous media that is applicable for soils exhibiting large strains was formulated later by Kiousis and Voyiadjis (1985) using a Lagrangian reference frame. E. In their later references, the porous solid was considered as being completely saturated with the fluid. In reality, problems of soil mechanics often involve partially saturated soils, where the pores are filled with the fluid and air. The mechanics of a solid-fluid-gas medium was considered by Raats and Klute (1968) and De Boer and Ehlers (1986) for establishing field equations based on mixture theories. Schrefler et al. (1989) followed Biot’s equations to establish equivalent relations for partially saturated soils (Zienkiewicz et al., 1990). Nevertheless, no complete and totally accepted theory for partially saturated media is yet available. The modern version of the porous media theory is based on the mixture theory developed in the 1960’s. The research in porous media theories is now mainly focused on three directions: first, the implementation and validation of the developed porous media models into numerical algorithms; second, the incorporation of different material behavior into the developed mathematical models; and third, the investigation of special phenomena which appear in saturated and empty porous solids based on available theoretical solutions and experimental results. This book presents the fundamental pore water flow in geo-materials in Chap. 2, the coupled theory of mixtures in Chap. 3, the yield criteria in Chap. 4, the finite element formulations in Chap. 5, applications in Chap. 6, advanced topics in Chap. 7, and Fortran Code in Appendix.

2 Flow in Geo-materials

2.1 Nature of Geo-materials Soils are Mother Nature’s products and their composition is quite complex. Typically soils are composed of three phases: solid, water, and air. When soils are under the water table, they are inherently saturated. Under the water table, air voids in the soils are completely filled with water, and the soils are essentially two phase materials. Above the water table, there are mixtures of air and water in the voids and the soils are three phase materials. This multi-phase nature of soils develops the coupled behavior of their constituents. A simple example of coupled behavior is the stress distribution of a composite beam such as a reinforced concrete beam. The reinforcing steel carries greater stress than the concrete due to its higher modulus. In soils, the phases as well as moduli of their various constituents are different. Therefore, additional coupled behavior from different phases is expected in addition to that from the different moduli. The coupled behavior from the different phases involves the flow characteristics of pore liquids (e.g. air and water). Traditional coupled theories of mixtures (Biot 1956b, Prevost 1980) address the effects of pore liquid flow; however, more recent coupled theories of mixtures address the additional coupling with rate dependency and micro-mechanical mechanisms such as grain rotations, grain interactions, and damages Voyiadjis and Song (2005a). This chapter, however, focuses primarily on the flow of water in soils and fundamental coupling concepts. When the soils are not saturated (above the water table), the air reserves some volume in the pore spaces and the soils are three phase materials as shown in Fig. 2.1. For three phase conditions (unsaturated soils), coupled behavior operates in a tri-valent relationship involving solids, water and air. When soils are saturated (under the water table), the volume of air is zero and the soils are essentially two phase materials (e.g. solid and water) as shown in Fig. 2.1. For two phase conditions (saturated soils), coupled behavior exists only between the two constituents. The classical effective stress equation for solids and water (two phase materials) is expressed by Terzaghi (1943) as follows: (2.1) σ = σ − u

12

2 Flow in Geo-materials

Fig. 2.1. (a) Soil element In natural state; (b) three phases of soil element (Das, 2005)

where, σ is the effective stress, σ is the total stress and u is the pore (water) pressure. An expanded version of (2.1), that includes the air pressure, is derived by Bishop et al. (1960) as shown in (2.2): σ = σ − ua + χ(ua − uw )

(2.2)

where, ua is the pore air pressure, χ is an experimental parameter and uw is the pore water pressure. Equations (2.1) and (2.2) deal with the effective stress, pore water pressure, and pore air pressure as three separate quantities. Therefore, these two equations represent weakly-coupled relations among those three quantities. The concept of weakly or uncoupled behavior is generally correct when there is no flow in the soils (static condition). When there is some kind of flow, it affects the pore water pressure and ultimately the effective stress is affected. Flow of water in soils is controlled by many factors, but the most important factor governing soil flow property is the hydraulic conductivity (permeability) of soils. Ultimately, hydraulic conductivity comes into play in (2.1) and (2.2) for dynamic (or pseudo static) condition. Incorporating the hydraulic conductivity into the effective stress equations is the key concept of the coupled theory of mixtures.

2.2 Flow of Water in Geo-Materials

13

2.2 Flow of Water in Geo-Materials Flow of water in the geo-materials is divided into steady flow and transient (or non-steady) flow. Steady flow is the condition in which flow rate is constant with time. Therefore one can apply (2.1) and (2.2) without sacrificing theoretical integrity. Transient or non-steady flow is the condition in which flow rate is not constant with time. In this condition, continuous changes in pore pressure are present, and using (2.1) and (2.2) may result in a severe theoretical compromise. (Both for steady flow and transient flow, Darcy’s law is assumed to be valid. For some cases, fluid flow may be caused by chemical potential difference or electrical potential difference; nevertheless, only the Darcy’s law is considered in this book.) 2.2.1 Steady Flow Steady flow in the soil cause the seepage pressure in the soils and affects the pore water pressure. Steady flow can increase or decrease the measured pore water pressure depending on the direction of the seepage as shown in (2.3). σ = σ ± us = σ ± izγw

(2.3)

where, σ is the effective stress considering seepage, us is the pore water pressure caused by seepage, i is the hydraulic gradient, z is the depth and γw is the unit weight of water. The ± sign is positive for downward flow and negative for upward flow. The seepage pressure us affects the pore water pressure; however, it does not change with time in the steady flow while it changes with time in the transient flow. 2.2.2 Transient or Non-steady Flow Transient flow occurs when the flow condition is forced to change by external energy. Common examples of transient flow are the vibration-induced pore water pressure and its associated flow, stress (deformation)-induced flow, consolidation- induced flow, and many others. During transient flow, the flow rate will depend on the hydraulic conductivity and time, as described by Darcy’s law. Therefore one may expect the incorporation of the hydraulic conductivity and time in (2.3). One can anticipate that the effective stress also will be affected by the pore water pressure as expressed in (2.4). ∆u = B[∆s1 + A(∆s1 − ∆σ 3 )]

(2.4)

where, ∆u is the generated (excess) pore water pressure, A and B are pore pressure parameters, ∆σ1 is the change in the deviator stress, and ∆σ3 is the change in the confining stress. From (2.3) and (2.4) we can see that the pore water pressure and the effective stresses are inter-related. Consequently, we may say that the effective stress, the pore water pressure, and the hydraulic conductivity are all coupled in the transient flow; thus these relations show the fundamental idea of the coupled theory of mixtures.

3 Coupled Theory of Mixtures

3.1 Traditional Theory of Mixtures The consolidation is a well known time dependent volume contraction behavior of saturated clayey soils. The consolidation is triggered by external loading, void spaces are compressed, pore water pressure is increased, and the pore water pressure starts to dissipate (flow out). In turn, the pore water pressure is decreased and the effective stress is increased. This consolidation is the classical example of the coupled behavior of soils though it was not recognized as one of them in the past. This chapter addresses Terzaghi’s and Biot’s consolidation theories in terms of the coupled theory of mixtures for two phase (saturated) materials. 3.1.1 Terzaghi’s Consolidation Theory Terzaghi’s one-dimensional consolidation theory is a classical consolidation theory as expressed in the following equation; ∂2u ∂u = cv 2 ∂t ∂t

(3.1)

where, u is the excess pore water pressure, t is the time, and cv is the consolidation coefficient. Terzaghi (1943) obtained (3.1) intuitively from the analogy of the consolidation and heat diffusion phenomena. Equation (3.1) is readily found in thermodynamics or partial differential equation text books. A comparison of the consolidation and heat diffusion equation is expressed as follows by Terzaghi (1943). From (3.1) and Table 3.1 we can see that (3.1) is exactly the same as the heat diffusion equation; therefore, the assumptions in the heat diffusion equation may be transferred into (3.1). One of assumptions is that uncoupled behavior exists between the effective stress and excess pore water pressure. This assumption is implicitly shown in (3.1) through the constant cv . cv is a constant in the heat diffusion equation, and therefore, as we have seen, it is also a constant in the consolidation equation. One of the solutions of (3.1) may be expressed (Craig, 1979) as follows:

16

3 Coupled Theory of Mixtures Table 3.1. Analogy of consolidation and heat diffusion Conclusion

Symbol

Excess pore water pressure Time Coefficient of consolidation

u=

m=∞ m=0

2ui M

u t cv

Mz sin d

Heat Diffusion Temperature Time Diffusivity

exp −M 2 Tv

(3.2)

where, u is the excess pore pressure at certain time t, ui is the initial excess pressure, M = (π/2)(2m + 1), z is the depth, d is the length of longest drainage path, and Tv is the time factor that is expressed in Tv = (cv t/d2 ). In reality, because of the coupling of the effective stress and the excess pore water pressure, cv is not a constant. Terzaghi (1943) assumed it to be a constant for simplicity. This simplification made the calculation easier, but it also lost some details as a trade-off. One example of the consequences of such a significant loss of detail is the fact that Terzaghi’s consolidation is not able to predict the ‘Mandel-Cryer’ effect, which can be predicted only by coupled multi-dimensional consolidation theory. Biot’s more recent consolidation theory has the advantage of considering the coupled behavior of the effective stress and the excess pore pressure, but Biot’s equations are not convenient to use. 3.1.2 Biot’s Consolidation Theory Biot (1955) assumed that the soil is a linearly elastic isotropic solid experiencing small strain (Miga et al., 1998). The equations incorporating twodimensional plane strain mechanical equilibrium in Cartesian coordinates for a homogeneous linear elastic media is described by the following equations; G∇2 u +

∂p G ∂ε −α =0 1 − 2ν ∂x ∂x

(3.3)

G∇2 v +

∂p G ∂ε −α =0 1 − 2ν ∂y ∂y

(3.4)

where, G is shear modulus, ν is Poisson ratio, α is the ratio of water volume extracted to the volume change of the soil, u, v are the x, y displacements in the Cartesian plane, p is the pore water pressure, and ε is the volumetric strain. Biot (1955) added a constitutive relationship relating volumetric strain and fluid pressure in order to complete the continuum model. The constitutive equation describing this relationship is shown as follows: ∇ · k∇p − α

1 ∂p ∂ε − =0 ∂t S ∂t

(3.5)

3.2 Modern Theory of Mixtures for Finite Strain

17

Fig. 3.1. Comparison of Terzaghi’s one dimensional consolidation and Biot’s two dimensional consolidation

where, k is the coefficient of hydraulic conductivity and 1/S is the amount of water that can be forced into the soil under pressure while the volume of the soil is kept constant. The first two terms in this equation provide the coupling relationships among volumetric strain, pore fluid pressure, and hydraulic conductivity. Generalization of (3.3) through (3.5) using a displacement tensor u and pressure p will produce the following relations; G∇ · ∇u +

G ∇(∇ · u) − α∇p = 0 1 − 2ν

(3.6)

∂ 1 ∂p (∇ · u) + − ∇ · k∇p = 0 (3.7) ∂t S ∂t Equations (3.6) and (3.7) are Biot’s coupled consolidation equations. Biot’s consolidation equations, of course, are more difficult to use than Terzaghi’s consolidation equations. Biot’s consolidation equations, however, are more rational and are able to predict so called ‘Mandel-Cryer’ effect (Lambe and Whitman, 1979), the fact that the excess pore water pressure can be bigger than the applied stress at the initial stage of the consolidation. The MandelCryer effect was observed in the laboratory and in the field, and it could be predicted by Biot’s consolidation equations. Figure 3.1 shows the comparison of Terzaghi’s one-dimensional consolidation and Biot’s two-dimensional coupled consolidation to exemplify the Mandel Cryer effects predicted by Abu-Farsakh (1997). α

3.2 Modern Theory of Mixtures for Finite Strain Biot (1955, 1978) first developed the foundation of modern coupled theory of mixtures for an elastic porous medium. However, experiments have

18


shown that the stress-strain-strength behavior of the soil skeleton may be strongly non-linear, anisotropic, and elastoplatic characteristics that Biot’s theory does not adequately account for. An extension of Biot’s theory into the non-linear, anisotropic range is, therefore, necessary in order to analyze the transient response of soil deposits. This extension has acquired considerable importance in recent years due to the increased concern with the dynamic behavior of saturated soil deposits and the associated liquefaction of saturated sand deposits under seismic loading conditions. Such an extension of Biot’s formulation was proposed by Prevost (1980). Prevost (1980)’s theory of mixture was subsequently coupled with Terzaghi (1943)’s effective stress theory for finite strain by Kiousis and Voyiadjis (1985) and Voyiadjis and Abu-Farsakh (1997), and therefore a coupled theory of mixtures for finite strain condition is obtained. This book further extends the work of Voyiadjis and Abu-Farsakh (1997) for the anisotropic stress condition with the micro-mechanical consideration such as grain rotations. The main goal of this book is to illuminate the coupled behavior of soils. This method uses the hydro-mechanical analysis by the coupled theory of mixtures at finite strains. To perform this analysis, the formulation of the coupled field equations for soils using the theory of mixtures in an updated Lagrangian frame based on the principle of virtual work and implemented in a finite element program is used. When it is saturated, soil consists of two deformable media, the solid grains and water as in shown in Fig. 3.1. Based on this figure, the following quantities are defined; γw = ρw = nw = ρw = ρs =

Ww Ws ; γs = Vw Vs γw γs ; ρs = g g Vw VT Ww = nw ρw gVT ws = ns ρs = (1 − nw )ρs gVT

(3.8) (3.9) (3.10) (3.11) (3.12)

where, Ww and Ws are the weight of the water and solid phase respectively; Vw and Vs are the volume of the water and solids respectively; γw and γs are the unit weights of the water and the solid phase respectively; g is the acceleration due to gravity; ρw and ρs are the intrinsic mass densities of the water and solid phase respectively; nw is the porosity of the soil; ρw and ρs are the apparent mass densities of the water and the solid phase respectively. 3.2.1 Balance of Mass The amount of mass (mα ) of the constituent (α) in the continuum occupying the spatial volume V at time t is given by;


19

mα =

ρα dV

(α = s or w)

(3.13)

v α

where, ρ is the macroscopic mass density of the α-constituent. If the soil mixture is assumed to be chemically inert, then the law of conservation of mass requires that the rate of change of the mass of each constituent in the continuum to be equal to zero and hence the material derivate of mα be zero. Therefore the balance of mass for each constituent α implies that α α ∂ρ D D α α α ∂ρ + vi m dV = ρ dV = dV = 0 (3.14) Dt Dt ∂t ∂zi v

v

v

α

where v is the velocity of the α constituent. Applying the divergence theorem to (3.14), it becomes ∂ ∂ ∂ α α α α ρ dV + ρ v · ndA = ρ dV + div(ρα v α )dV = 0 (3.15) ∂t ∂t ∂t v

A

v

A

Making use of the following relation, Div(ρα v α ) = ρα div(v α ) + ∇ρα · v α Equation (3.14) then becomes α ∂ρ + ∇ρα · v α + ρα div(v α ) dV = 0 ∂t

(3.16)

(3.17)

v

Because (3.17) is valid for any arbitrary volume V , the balance of mass for each constituent (α) can be written as follows: D α ρ + ρα div(v α ) = 0 Dt

(3.18)

ρ˙ α + ρα div(v α ) = 0

(3.19)

or Similarly, the balance of mass for the mixture leads to ρ˙ + ρ div(v) = 0

(3.20)

In the Lagrangian configuration, the balance of mass becomes ρα J α = ρao ; ρJ = ρo

(3.21)

where ρao is the apparent mass density of the α-constituent in its reference configuration and ρ is the mass density of the mixture expressed as follows: α (3.22) ρ= αρ

20


Using the relation between the apparent mass densities and the intrinsic mass densities of the water and solid grains, the material derivatives then becomes (3.23) ρ˙ α = nα ρα + nα ρα Equation (3.19) may then be reduced to n˙ α + nα div(v α ) = −nα

ρα ρα

(3.24)

If one assumes the water phase is incompressible (ρ˙ w ), one obtains n˙ w + nw div(v w ) = 0

(3.25)

If the solid grains are taken incompressible (ρ˙ s ) then n˙ s + ns div(v s ) = 0

(3.26)

nw + ns = 1

(3.27)

∇ns = −∇nw

(3.28)

but Therefore, one obtains w

s

∂n ∂n + =0 ∂t ∂t n˙ s − v s ∇ns + n˙ w − v w ∇nw = 0

(3.29) (3.30)

Substituting n˙ s and ns in (3.26) and subtracting (3.25) from the resulting expression, the balance of mass can be written in terms of the soil porosity nw , solid velocity v s , and water velocity v w as follows; div(v s ) − div(v w ) =

1 [div(v s ) + (v w − v s )grad(nw )] nw

(3.31)

Equation (3.31) is the mass balance equation for saturated porous media derived by Prevost (1980). 3.2.2 Coupling of Mass Balance Equation and Darcy’s Law The general form of Darcy’s law for the flow of water through porous medium is given by 1 (3.32) (v w − v s ) = − K ws (grad(Pw ) − ρw b) γw where b is the body force vector, γw is the unit weight of water, and K ws is the permeability tensor in (m/sec). Taking the divergence of both sides of (3.32) and substitute the result into (3.31) one obtains 1 ws s w K (grad(Pw ) − ρw b) +(v w −v s )grad(nw ) = 0 (3.33) div(v )−n div γw


Substitute (3.32) into (3.33) one obtains w n K ws (grad(Pw ) − ρw b = 0 div(v s ) − div γw

21

(3.34)

Making use of the following relation, s s div(v s ) = dskk = Xk,a XK,B εAB

(3.35)

(3.34) can be rewritten as follows: dskk −

∂ [K ws (grad(ρw ) − ρw bb )] = 0 ∂Za ab

(3.36)

Making use of the following relations ws s s WS Kab = Xa,A Xb,B KAB

(3.37)

s Xb,B BB

(3.38)

bb = ∂Pw s ∂Pw = XC,b ∂Zb ∂Xc ∂ ∂ s = XD,a ∂Za ∂Xd

(3.39) (3.40)

one obtains the coupled equation in an updated Lagrangian formulation as follows; w n ∂Pw ∂ s s−1 s s−1 s−1 s WS s Cij εij − J Cij Cij XD,a K X − ρw BB =0 ∂Xd γ w AB a,A ∂XB (3.41) s s s = Xk,i Xk,j . where, Cij If one assumes incompressibility of the solid grains, the porosity, nw can be updated from n configuration to at n + 1 configuration using the Jacobian of solid grains, J s as follows: 1 − nw 1 n+1 = s w 1 − nn J

(3.42)

As discussed previously, the drainage condition in the soil is somewhere between the fully drained and the fully undrained condition. This condition is called the partially drained condition or the transient flow condition. For the transient flow condition, it may be presumed that the pore pressure is a function of the hydraulic conductivity and other parameters (stress-strain parameters). (This present study presents a relatively concise derivation of the coupled theory of mixtures: for a full derivation, see Prevost (1980)).

4 Coupling Yield Criteria and Micro-mechanics with the Theory of Mixtures

The behavior of geo-materials follows not only the theory of mixtures, but it also follows yield criteria. The yield behavior of various components (e.g. water, solid) of geo-materials differs. However, applying a complex paradigm of mechanics that deals with individual soil components is too complicated and time consuming at present. For this reason, the authors have restricted this chapter to macroscopic yield behavior, as shown in the coupling of theory of mixtures and yield criteria of geo-materials.

4.1 Soil Models There are many models for analyzing yield behavior of soils. One of the oldest soil models is Mohr-Coulomb yield criteria. In Mohr-Coulomb yield criteria, the soil behavior is expressed in terms of the friction angle and cohesion. And it is assumed that soils fail when the shear stress is greater than the shear strength of the soils. The shape of Mohr-Coulomb criteria in π-plane is shown in Fig. 4.1. It is a distorted hexagon because the compressive strength is greater than the extensional strength in soils. Mohr-Coulomb criteria are similar to Tresca criteria: in both sets of criteria, the materials fail at the highest shear stress (and the shape of yield criteria in π-plane is hexagon). Tresca criteria do not differentiate extension part from compression part as shown in Fig. 4.2 because Tresca criteria were initially developed for yield anlaysis of metals, and metals usually show identical tensile and compressive strength. Therefore, the shape of the Tresca yield criteria in π-plane is an undistorted hexagon. Considering the fact that soils exhibit different strength in extension and compression, Mohr–Coulomb criteria are more widely used for soils. Another difference between Mohr-Coulomb criteria and Tresca criteria is that MohrCoulomb can incorporate J1 (mean stress) dependency while Tresca cannot. Soils are essentially particulate materials; therefore, the J1 dependency is essential. Another widely used fundamental yield criterion is a von Mises yield criteria (Fig. 4.3). Von Mises criteria are also originally used for yield behavior analysis of metals. Von Mises criteria assume that materials fail when second stress invarient (J2 ) reaches its maximum state. Therefore, von Mises criteria

24

4 Coupling Yield Criteria and Micro-mechanics Mohr-Coulomb

(a)

(b)

Fig. 4.1. Mohr-Coulomb yield criteria (a) in stress space, (b) in π plane (Chen and Mizuno, 1990)

(a)

(b)

Fig. 4.2. Tresca yield criteria (a) in stress space, (b) in π-plane (Chen and Mizuno, 1990)

are very similar to Tresca, however these criteria yield a different shape in π-plane. The shape of Von Mises criteria in the π-plane is a circle with its center at the origin. Von Mises criteria do not incorporate the J1 dependency, either. For this reason von Mises is not used widely for the yield behavior analysis of soils. For metals, the true yield behavior is known to be somewhere in between von Mises and Tresca yield criteria. However, soils are different: readers should remember they are particulate materials, and models that are only J2 -dependent have limitations on their applicability.

4.1 Soil Models

25

Fig. 4.3. Von Mises yield criteria in the stress space (Chen and Mizuno, 1990)

The Drucker–Prager model, introduced relatively recently, compares with Mohr–Coulomb, Tresca, and von–Mises. The uniqueness of Drucker–Prager is the incorporation of J1 dependency in von–Mises model as shown in Fig. 4.4. The Drucker-Prager yield criteria is a circle in π-plane. The size (diameter) of the circle, however, varies depending on the mean principal stress (J1 ). When one takes away the J1 dependency from Drucker–Prager yield criteria, it is similar to von-Mises. Because the J1 dependency of Drucker–Prager yield criteria is a more realistic representation of the particulate characteristics of soils, it is one of the widely used soil models. Drucker–Prager assumes the same compressive and tensile behavior of the material and therefore its yield criteria are essentially the equation of a circle; the mathematics for Drucker– Prager is quite favorable. Compared to Drucker–Prager’s single equation of yield surface and its associated mathematical simplicity, the Mohr–Coulomb approach, however, is composed of six lines (equations) and the mathematics dealing with the manipulation of the equations is very complicated. For example, finding the normal direction in Mohr–Coulomb at one of the break points in Fig. 4.1, clearly involves exceptional mathematical difficulties. (Students and beginning engineers: note that the Mohr–Coulomb is the same Mohr–Coulomb in most soil mechanics textbooks. When one plots Mohr–Coulomb in the mean stress and deviatoric stress axis, one obtains a straight line that has slope φ [internal friction angle] and ordinate c [cohesion]).

26

4 Coupling Yield Criteria and Micro-mechanics

Fig. 4.4. Drucker–Prager yield criteria in stress space (Chen and Mizuno, 1990)

The above descriptions are discussions of classical soil models that are used for rigid plastic or elastic-perfect plastic behavior. The elastic-perfect plastic models assume that, after yielding, the material stays in yield (perfectly plastic) condition. However, field soils may not show this behavior. Field soils may show recovery of their strength, elastic rebound, and (sometimes) increased or decreased load carrying capacity even after yielding. This behavior is called hardening. Even during the loading process, a soil hardens (changes its strength); therefore, the incorporation of hardening process will be beneficial to formulating a realistic simulation of soil behavior. One of the pioneering soil models that can incorporate some of the above characteristics is the Cam Clay model, formulated by Roscoe, Schofield and Wroth (1957). One of the features of Cam Clay model is that it can incorporate the volume change (consolidation) phenomenon in the yield behavior of soils as shown in Fig. 4.5. By adding consolidation in the yield criteria, Cam Clay model yield criteria can simulate changes of J1 (mean normal stress) during shear. Therefore, the Cam Clay model yield criteria can expand during shear if J1 increases as shown in Fig. 4.6(a). If J1 decreases, the yield criteria do not shrink, but in this model, such a condition is considered an overconsolidated state. The Cam Clay model attracted great interest and showed good agreement with a vast range of test results, as shown in Fig. 4.6(b): The Cam Clay model assumes that the dissipated energy is carried out only through the plastic shear strain only. Later, this assumption changed,

4.1 Soil Models

27

Fig. 4.5. Cam Clay model in p, q and e (Chen and Mizuno, 1990)

(a)

(b)

Fig. 4.6. Cam Clay model (a) in stress space (Chen and Mizuno, 1990) (b) in p-q space (Wood, 1990)

28


q CSL

Yield Surface

Po Po/2

P

Fig. 4.7. Yield Locus of the modified Cam Clay model in p-q space

and it was assumed that the dissipated energy is carried out through both the plastic shear strain and plastic volumetric strain. This modified model is called the modified Cam Clay model. The advantage of the modified Cam Clay model is that it is more rational in fundamentals and mathematically more convenient than the original Cam Clay model: it uses only one equation for the yield surface while original Cam Clay model uses two equations in p vs. q space. This advantage solved the important mathematical problem of normality of plastic strain to the yield surface. The original Cam Clay model places the break point (point A and C in Fig. 4.6) between the Roscoe surface and the Hvoslev surface. This break point makes it mathematically difficult to find the normal vector to the yield surface as mentioned previously. In the modified Cam Clay model, the whole yield surface is a smooth ellipsoidal shape as shown in Figs. 4.7 and 4.8. Therefore, finding the normality of plastic strain with respect to the yield surface is not difficult at all. Although the modified Cam Clay model has the above-described advantages, it does not agree well with experimental data for over-consolidated soils. Typically, the modified Cam Clay model tends to over-estimate the strength of soils in over-consolidated state (Chen and Mizuno, 1990).

4.1 Soil Models

29

Space diagonal

Fig. 4.8. Yield criteria of modified Cam Clay Model in stress space

DiMaggio and Sandler (1971) and Baladi and Rohani (1979) proposed a mathematically advantageous yield criteria that also have good agreement with experimental data. This model is composed of two lines, one line for the over-consolidated state and another line for the normally consolidated state, as shown in Fig. 4.9. These two lines meet smoothly at the connection point (point B). At the connection point, the slopes of two lines are identical

Fig. 4.9. Cap Model (Desai and Siriwardane, 1984)

30


(the two lines are tangential to each other at point B). Therefore, the normality rule of plastic strains with respect to the yield surface does not cause mathematical problems. (For readers who wonder why the authors impose the normality rule, please note that the normality rule does not have to be imposed. However, employing the normality rule will simplify the formulation procedure.) Also, by taking flatter lines at the over-consolidated state, the above Cap model achieves a better agreement with experimental data than does the modified Cam Clay model. These models are some traditional models, the details of which can be found in Chen and Mizuno (1990), Desai and Siriwardane (1984) and Wood (1990). Readers should also note that although there are many soil models other than these traditional ones, this book restricts itself to a thorough discussion of the widely regarded and widely used soil model – the modified Cam Clay model.

4.2 Anisotropic Soil Model The above traditional models are isotropic models. Therefore, their application is of inherently limited use in addressing the anisotropic nature of soils. It is the same situation for hardening; even though soils may have anisotropic hardening characteristics, the previous models incorporate isotropic hardening only. Typically, the anisotropy of soils is divided into two kinds: inherent anisotropy and (stress)-induced anisotropy. Inherent anisotropy is caused by the natural texture of soils as shown in Fig. 4.10(a); (stress)-induced

(a)

(b)

Fig. 4.10. Anisotropic textures of clays (a) in natural deposits (b) in the shear plane (Mitchell, 1993)

4.2 Anisotropic Soil Model

31

anisotropy is caused by the re-orientation of the soil particles during shear as shown in Fig. 4.10(b). The shape of the modified Cam Clay model in π-plane is a circle with center at the origin. The overall shape in principal stress space is an egg-shaped ellipse, as shown in Fig. 4.8. With hardening, the size of the circle increases in π-plane, or the size of the egg-shape ellipse increases in the principal stress space. In order to incorporate the anisotropic characteristics of soils, the soil models need to have some additional features to the traditional modified Cam Clay model (that is, if one intends to use the modified Cam Clay model). Dafalias (1987), for example, proposed an anisotropic modified Cam Clay model that can incorporate the inherent anisotropy and the stress induced anisotropy. Dafalias incorporated the back stress concept in the modified Cam Clay model. Back stress may be interpreted as the internally embedded stress that changes the structure of the soil. When the structure of a soil changes, it becomes an essentially different soil. For this different soil, the yield surface is different in size and location, as shown in Fig. 4.11. In the π-plane, this behavior is represented as the translation of the yield criteria; in the principal stress space, this behavior is represented as the rotation of the yield criteria. Rotation in the principal stress space or translation in the π-plane represents the different material behavior depending on the three-dimensional stress conditions. In Fig. 4.11(b), the two dotted circles mean that the material is stronger to compression in σ1 and σ3 directions (if we say compression is positive). For σ2 direction, the soil has almost zero strength or some tensile strength. Figure 4.11 is a conceptual diagram to show that the required features for the anisotropic modified Cam Clay model. The following two sections address the essentials of (isotropic) modified Cam Clay model and formulate an anisotropic modified Cam Clay model.

Modified Cam Clay Model (Isotropic) From the equilibrium of work done and dissipated energy, (4.1) is obtained (Burland, 1965; Schofield and Wroth, 1968) as follows: pdεpv + qdεps = p{(dεpv )2 + M 2 (dεps )2 )1/2

(4.1)

where, the variables are defined as follows: p = mean principal stress = (σ1 + σ2 + σ3 )/3 = (σ1 + 2σ3 )/3 for triaxial condition q = deviatoric stress = (σ1 − σ2 )/2 εpv = plastic-volumetric strain εps = plastic-shear strain M = slope of critical state line in p vs. q space.

32


1

back stress

Ko-line

1

=

2

=

3

3 2

(a)

1

2

3

(b) Fig. 4.11. Concepts of anisotropic Cam Clay model (a) in stress space (after Banerjee and Yousif (1986)) (b) in π-plane

From (4.1), one can derive the well known yield criterion as follows: f = p2 − po p + (1/M 2 )q 2 = 0

(4.2)

In (4.2), po is the p at the hydrostatic condition (q = 0). Equation (4.2) can be rewritten as follows:


po 2

p− 2 po 2 2

+

q2 M p2o

2 = 1

33

(4.3)

Equation (4.3) represents the equation of ellipse with one axis length po and another axis length M po as shown in Fig. 4.7. When the stress condition is inside the ellipse, the material undergoes elastic behavior. When the stress condition is at the ellipse and load increment is positive, the material is in plastic deformation. The increase of effective stress due to consolidation during loading causes the increase of the mean principal stress po in (4.2). Increased mean principal stress causes the expansion of yield surface in Fig. 4.7. On unloading, the stress point lies inside the yield surface, and elasticity is incorporated. 4.2.1 Anisotropic Modified Cam Clay Model Equation (4.2) shows the evolution of the yield surface with an isotropic hardening behavior; po in (4.2) is the isotropic hardening parameter. However, the real soil is subjected to the anisotropic stress condition, and therefore the Anisotropic Modified Cam Clay Model (hereafter called AMCCM) is discussed here following the work of Dafalias (1987). The shape of AMCCM in the principal stress space is egg shaped as shown in Fig. 4.7. The cross section of the oval shape can be a circle, hexagon, or similar shape (not specifically known). However, one can guess that shape will be somewhere between the anisotropic von-Mises and Tresca type. When the yield locus in the principal stress space undergoes kinematic hardening, the yield surface will move around the π-plane. This behavior will appear as the rotation of the oval shape yield locus in the principal stress space. In the p vs. q plane, where p is the space diagonal and q is the deviatoric stress, the shape of the yield locus is the inclined cut of the three dimensional yield locus that appears in Fig. 4.12; that is, the yield locus of AMCCM is a tilted ellipse. Thus the kinematic hardening will result in the up and down rotation of the elliptical yield locus in the p vs. q space for which the origin does not change. We see, then, that the shape of the yield locus will be the distorted ellipse when a soil is subjected to anisotropic hardening. For ease of understanding, the AMCCM is first derived for the triaxial stress condition (σ2 = σ3 ). In (4.1), one can introduce the anisotropic hardening parameter α as shown in (4.4): pdεpv + qdεps = p

dεpv

2

1/2 2 1/2 + M 2 dεps + 2α dεpv dεps

(4.4)

In (4.4), α represents the so-called ‘back stress.’ One may use different expression for the inclusion of back stress in (4.4), such as α or 5α instead of 2α; however, the results will be the same. When one uses 2α, the integration becomes easier. Rearranging the terms in (4.4) yields the following relation:

34


q

M CSL C q

B A qo= Po Pc

O

Po P

Fig. 4.12. Kinematic hardening of AMCCM in p vs. q space (p is the mean principal stress, q is the deviatoric stress) (after Dafalias (1987))

dεpv 2αp2 − 2pq p = 2 dεs q − p2 M 2

(4.5)

Let η = q/p (η is different from M the slope of critical state line. η is the stress ratio at any condition, such as in elastic condition. At the critical state η and M become the same.) and making use of it in (4.5), one obtains the following relation: 2η − 2α dεps (4.6) p = dεv M 2 − η2 From the normality rule, one can obtain (4.7): dεps dp =− dεpv dq

(4.7)

Let the variable Ψ be defined as follows: Ψ=

dεpv M 2 − η2 p = dεs 2η − 2α

(4.8)

Making use of (4.7) in (4.8), one obtains dq M 2 − η2 = −Ψ = − dp 2η − 2α

(4.9)


35

From the relationship q = ηp and (4.9), one obtains (4.10) dq = pdη + ηdp = −Ψdp .

(4.10)

Rearranging the terms in (4.10) and making use of (4.7) into (4.10), one obtains the following expression: −

dη dp = 2 −η 2 p η+ M 2η−2α

(4.11)

Integrating (4.11) for p (pi to p) and η( η i to η) (where subscript i implies initial value), (4.12) is obtained as follows: p M 2 − α2 = 2 po M + η 2 − 2αη

(4.12)

Rearranging (4.12), one obtains f = p2 − ppo +

1 (q 2 − 2αpq + α2 ppo ) = 0 M2

(4.13)

Recalling the similar equation for the (isotropic) modified Cam Clay Model yields the following equation f = p2 − p o p +

1 2 q =0 M2

(4.14)

The same equations may be expressed as follows:

p − p2o po 2 2

2

q2

+

M p2o

2 = 1

(equation of ellipse)

(4.15)

One understands that the (4.13) is also the equation of ellipse with capability of rotating with α in p vs. q space. Equation (4.13) is identical to the one obtained by Dafalias (1987). Therefore (4.13) can have yield surface translation in the π-plane. For α in (4.13), Dafalias (1987) proposed the following evolution equation that is similar to that of the Armstrong and Frederick (1966):

1 + eo ∂f c ¯˙ (q − xαp) (4.16) α˙ = λ λ − κ ∂p po ¯˙ is the loading index, λ is the compreswhere, is the Macauley bracket, λ sion index from the e vs. ln p curve, κ is the recompression index from the e vs. ln p curve, eo is the initial void ratio, and c and x are constants. The ¯˙ expression λ(∂f /∂p) represents the plastic strain rate dε˙p . Making use of (4.6), (4.17) through (4.19) are obtained as shown below:

36


2η − 2α λ − κ dp + 2 dη 1 + eo p M + η 2 − 2αη 2η − 2α 2η − 2α λ − κ dp + 2 dη dεps = 1 + eo p M + η 2 − 2αη M 2 − η2

dεpv =

dεv = dεev + dεpv

dp λ κ 2η − 2α = + 1− dη 1 + eo p λ M 2 + η 2 − 2αη

(4.17) (4.18)

(4.19a) (4.19b)

Equations (4.13) through (4.19) are for the triaxial stress condition. For the generalized stress condition, some changes are made for q and α such as q = {(3/2)sij sij }1/2 and α = {(3/2)αij αij }1/2 . Equations (4.13) and (4.16) are now generalized to (4.20) and (4.21) as follows: 3 {(sij − pαij )(sij − pαij ) + (po − p)pαij αij } = 0 2M 2 (4.20)

∂f c 1 + e o ¯˙ (sij − xpαij ) tr = λ (4.21) λ − κ ∂σmn po

f = p2 − ppo +

α˙ ij

Equations (4.20) and (4.21) now represents an egg-shaped, three-dimensional yield surface in the principal stress space which can rotate anisotropically according to the magnitudes and directions of αij . Furthermore, note that the initial values of αij represent the (initial) inherent anisotropy due to its initial textures. Conversely, the evolution equation of the back stress αij represents the induced anisotropy due to shearing. Therefore (4.20) and (4.21) are expressions which can incorporate both the inherent anisotropy and induced anisotropy.

4.3 Elasto-Plastic Constitutive Relations Now one needs to incorporate the AMCCM into the constitutive relations to analyze the anisotropic behavior of soils. Assuming the normality rule (flow rule), (4.22) is obtained, ¯˙ ∂f = dλB ¯˙ ij dεpij = dλ ∂σij

(4.22)

where, f is the yield function, s is the current stress and εpij is the plastic strain. For the case of small elastic strains, one may make use of the additive decomposition of the incremental strain as follows: dεeij = dεij − dεpij

(4.23)

4.3 Elasto-Plastic Constitutive Relations

37

In order to obtain the constitutive relation, one uses the following relations dσij = Cijkl (dεkl − dεpkl ) = Cijkl dεkl − Cijkl dεpkl ¯˙ = C dε − C dλB ijkl

kl

ijkl

(4.24a) (4.24b) (4.24c)

kl

where Cijkl is the elastic stiffness matrix. Using equations (4.24), the incremental stress strain relation is expressed as follows: T Cmnrs + Cijpq ∂α∂fmn dαmn Bpq 13 dε−1 Cijkl Bmn Brs kl dσij = Cijkl − dεkl ∂f Bab Cabcd Bcd − ∂ε p Baa v dσij =

Cijkl −

+ Cijpq ∂α∂fmn Bpq Amnkl ∂f Bab Cabcd Bcd − ∂ε p Baa v

T Cmnrs Cijkl Bmn Brs

(4.25a) dεkl (4.25b)

Equations (4.25) express the well-known elasto-plastic stiffness equations similar to the form, [Dep ]−1 = [De ]−1 + [Dp ]−1 (where, [Dep ] represents the elasto-plastic stiffness, [De ] represents the elastic stiffness, and [Dp ] represents the plastic stiffness). The expression Amnkl in (4.25b) represents dαmn /dεkl , which essentially is back stress stiffness. From (4.25b), one sees that the back stress αij affects the plastic stiffness. This behavior is also physically correct because the different back stress means the different location of the stress point in the yield surface and the elasto-plastic behavior of the material should be different. In equations (4.25) Cijkl is the elastic stiffness matrix which is directly obtained from experiments. Bij is the gradient of the yield function with respect to the stress, as defined in (4.26). The terms in (4.26) are also expressed as follows: ∂f ∂f ∂p ∂f ∂sij = + ∂σij ∂p ∂σij ∂sij ∂σij 3 = 2p − po + {−2αij sij + po αij αij } 2M 2 1 = δij 3 3 = 2 (sij − pαij ) M 1 = δim δjn − δij δmn 3

Bij = ∂f ∂p ∂p ∂σij ∂f ∂sij ∂sij ∂σmn

(4.26) (4.27) (4.28) (4.29) (4.30)

Substituting equations (4.27) through (4.30) into (4.26), one obtains the following equation:

38


Bij =

(2p − po ) 1 δij + {(po αkl − 2skl )αkl }δij 3 2M2 3 1 + 2 (skl − pαkl ) δik δjl − δij δkl M 3

(4.31)

In (4.25b), ∂f /∂αij is the rate of change of the yield function with respect to the back stress. This rate of change is given in (4.32) by making use of (4.20); 3 ∂f = 2 {−p(sij − pαij ) + (po − p)pαij } ∂αij M 3 = 2 (ppo αij − psij ) M

(4.32a) (4.32b)

From Dafalias (1987) α˙ ij is defined by the relation given by (4.21). Integrating (4.21) with respect to time t, and expressing it in incremental form, one obtains 1 + eo p c |dεv | (sij − xpαij ) (4.33) dαij = 3 λ−κ po ∂f /∂εpv is obtained from the chain rule such that

1 + eo ∂f ∂f ∂po 3 ppo = = −1+ αij αij ∂εpv ∂po ∂εpv 2M 2 λ−κ

(4.34)

Substituting equations (4.31) through (4.34) into (4.25b), one obtains elastoplastic constitutive relationships for the anisotropic modified Cam Clay model. After this section, represents the usual plastic multiplier.

4.4 Micro-Mechanical Considerations/Bridging Different Length Scales Many soil models implicitly assume the uniform stresses or strains for both soil particles (micro-structure) and soil mass (macro-structure). However, it is known that micro-structural quantities may be quite different from those of macro-structural quantities. It is also known that micro-behaviors may affect macro-behaviors. The typical micro-behaviors that are known to affect macro behavior are the rotation of particles, the interaction of particles, the rate dependency, and the damage. Details of individual mechanisms are discussed in the following sections. Incorporating micro-behavior into macro-behavior of soil materials can be performed either by modifying some terms in the constitutive relations or by obtaining the macro-behavior from the average response of micro-behavior. Some micro- mechanisms are incorporated by the first method, and some other micro-mechanisms are incorporated by the second method. The second method is the more fundamental method for bridging two different scales,

4.4 Micro-Mechanical Considerations/Bridging Different Length Scales

39

and it is gaining greater attention in current research. This section describes details of modern RVE (Representative Volume Element) concepts and averaging schemes. 4.4.1 RVE Concept The RVE concept is based on the premise that the representative macro scale material properties are obtained by the averaging of the micro scale properties for the representative volume. According to this premise, at a material point, a non-local tensor, A, is expressed as the weighted average of its local counterpart A over a surrounding volume V at a small distance |s| ≤ l (l is an internal characteristic length, which is a material property) from x such that 1 A= h (s) A (x + s)dV (4.35) V V

where, h(s) is an empirical weighting function subject to the normalizing condition V h (s) dV = V . The normalizing condition ensures that A = A when A(x) is a constant. Baˇzant and Chang (1984) examined a number of weighting functions that are used in the integral equation of non-local theories. For simplicity, it is typically assumed that the weighting function is the identity tensor such that the non-local measure is simply the average of the local measure: 1 A= A (x + s)dV (4.36) V V

In order to develop the gradient dependent approximation of this averaging equation, the local tensor, A must be first approximated by a Taylor series expansion at s = 0 such that A (x + s) ≈ A + ∇A · s +

1 (2) ∇ A·s⊗s+ ··· 2!

(4.37)

where ∇(i) denotes the i-th order gradient operator evaluated at the macroscale. For a general three-dimensional case, the surrounding volume can be assumed to be a sphere with a radius equal to the material characteristic length such that V = 43 πl3 . Furthermore, truncating the Taylor series after the second order gradient term, the following expression is used for the nonlocal tensor: 1 4πl5 2 ∇ A A=A+ (4.38) 2!15 V where ∇2 is the Laplacian operator and is defined as the trace of the second gradient. By substituting the volume over which the local variable is averaged, we obtain the form used in this work for the non-local measure: A = A + c∇2 A

(4.39)

40


Equations (4.35) through (4.39) are based on the constant length scale l and the volume of RVE determined from the l. As we shall see, the use of this constant length scale in this manner calls for additional discussion (Voyiadjis and Abu al-Rub, 2005). 4.4.2 Characteristic Length A constant length scale l is easy to use; however, this approach fails to capture the full behavior of the variation in material strength with sizes as shown by the experimental results of different size shear band formations in different grain sizes. Length scales also change with the degree of accumulated plastic strain in the material. An evolution law for the length scale parameter is needed to address the proper modification required for the full utilization of the current gradient plasticity theories in solving the size effect problem. Gradient approaches typically retain terms in the constitutive equations with specific order spatial gradients with coefficients that represent lengthscale measures of the deformed microstructures associated with the non-local continuum. Aifantis (1984) was one of the first to study the gradient regularization in solid mechanics. An extensive review of the recent developments in gradient theories can be found in Voyiadjis et al. (2003). The full utilization of the gradient-type theories hinges on the ability to determine the intrinsic material length that scales with strain gradients. Gracia (1994) approximated the evolution of the mean dislocation spacing Ls in metals with the following equation, Ls =

δd δ + dp1/m

(4.40)

where d is the diameter of the grain size. It is taken as the mean diameter (D50 ) of the grains. δ is a constant coefficient on the order of 1.0 µm (for metals), p is the effective plastic strain, and m is a material constant. In this work it is assumed that the dislocation spacing is directly related to the intrinsic material length-scale. Thus one can write l in terms of the size d and the effective plastic strain p as follows, l=

hδD50 δ + D50 p1/m

(4.41-a)

where h is a constant. The intrinsic material length-scale can then be assumed to decrease from an initial value lo at yield to a final value of l → 0 at the saturation point due to the plastic deformation (corresponding to the conventional plasticity limit) at a rate characterized by a constant coefficient k1 , such that (4.41-b) l = lo exp(−k1 p)


41

This two-parameter function is thought to be the simplest form that gives enough freedom for the evolution equation of the material intrinsic lengthscale. Equations (4.40) through (4.41) show the asymptotic variation of the length scale l. The asymptotic variation of the length scale with the effective plastic strain is physically more appropriate than a constant value of l . Equations (4.40) through (4.41) assume the same length scale Ls or l. However, a question may arise regarding the cogency of an isotropic expression for l, as discussed below. The three dimensional Taylor expansion of the strain function γ is expressed as follows: γ(x, y, z) = γ(xo , yo , zo ) ∂f (xo , yo , zo ) ∂f (xo , yo , zo ) (∆x) + (∆y) + ∂x ∂y 1 ∂ 2 f (xo , yo , zo ) ∂f (xo , yo , zo ) (∆z) + (∆x2 ) + ∂z 2 ∂x2 1 ∂ 2 f (xo , yo , zo ) 1 ∂ 2 f (xo , yo , zo ) + (∆y 2 ) + (∆z 2 ) 2 2 ∂y 2 ∂z 2 ∂ 2 f (xo , yo , zo ) ∂ 2 f (xo , yo , zo ) ∆x∆y + ∆y∆z + ∂x∂y ∂x∂z ∂ 2 f (xo , yo , zo ) ∆z∆x + · · · (4.42-a) + ∂z∂x When one assumes symmetric gradient distributions and neglecting higher order terms, the above expression reduces to 1 ∂ 2 f (xo , yo , zo ) (∆x2 ) 2 ∂x2 1 ∂ 2 f (xo , yo , zo ) 1 ∂ 2 f (xo , yo , zo ) + (∆y 2 ) + (∆z 2 ) (4.42-b) 2 2 ∂y 2 ∂z 2

γ(x, y, z) = γ(xo , yo , zo ) +

The above expression maybe expressed into the familiar gradient expression such that γ(x, y, z) = γ(xo , yo , zo ) 1 1 1 2 2 2 2 2 2 + (∆x) (∇ γx ) + (∆y) (∇ γy ) + (∆z) (∇ γz ) (4.43) 2 2 2 where the terms 12 (∆x)2 , 12 (∆y)2 , 12 (∆z)2 may be considered a mathematical based length scale and the whole of the second terms may be considered a mathematical gradient. Equation (4.37) is the one dimensional Taylor expansion and (4.43) is the three dimensional Taylor expansion. Using the three dimensional gradients is a more realistic approach to the problems involved in multi dimensional stress or strain field. Shear bands, for example, will

42


Large

Small Fig. 4.13. Shear Band and Strain Gradient

have enormous strain gradients across the shear plane; however, there will be minimal strain gradients along the shear plane, as shown in Fig. 4.13. Therefore, the use of different length scales for the three different directions will be more appropriate for incorporating multi-dimensional problems. The fact that there are three different length scales implies that one may have rectangular or elliptical RVE’s instead of spherical or square block RVE’s. This implication will be especially valid for anisotropic cohesive soils in the natural alluvial deposits. Incorporating the concept of physical gradients to (4.43), one obtains γ(x, y, z) = γ(xo , yo , zo ) + c1x lx2 (∇2 γx ) + c1y ly2 (∇2 γy ) + c1z lz2 (∇2 γz ) (4.44) where cx , cy , cz are material parameters and lx , ly , lz represent the length scales. The above equation may be expressed in a simplified form as follows: γij = γ(xo , yo , zo )ij +

3

ck lk2 (∇2 γk )

(4.45)

k=1

By comparing equations (4.45) and (4.37), one observes that relation (4.37) is a kind of the isotropic versions of (4.45). An isotropic length scale may be justified for an isotropic material. By applying (4.45) to isotropic quantities such as the effective plastic strain, one obtains p = p(xo , yo , zo ) + c1 l12 (∇2 p)

(4.46)

Comparing equations (4.45), and (4.46) and using the typical strain gradient expression in relation (4.47), one concludes that (4.47) is an undesirable expression of (4.45). γij = γ(xo , yo , zo )ij + c1 l12 ∇2 γij

(4.47)


43

Equation (4.47) is only approximately correct. If one simplifies equation (4.45) for computational purposes, that is strictly for the computational purpose not for the realistic analysis. Equation (4.45) may be further reduced to the following form by incorporating c1 into the length scale itself: γij = γ(xo , yo , zo )ij +

3

(ck lk )2 (∇2 γk )

k=1

= γ(xo , yo , zo )ij +

3

(lk )2 (∇2 γk )

(4.48)

k=1

Note that in the above equations, lk and ∇2 γk provide a mutually compen2 sating behavior because ∇2 γk is expressed as ∂∂lγ2k . A larger lk (this is really 2

k

∆lk ) will produce a smaller ∂∂lγ2k . This is why it can reduce the mesh size k dependency of the numerical results. Anisotorpic nature of lk , therefore, may not be meaningful if one uses an average number. The use of single length scale may be regarded an approximate length scale. lk in soils represents the physical distance range in which the soil strains will be affected. One may expect that lk depends on many physical properties of the material. The mean diameter, the angularity, and the aspect ratio of soils are the major factors that may govern this length scale. One should expect changes in the magnitude of the length scale with the increase/decrease in the magnitude of the above factors. In addition to the above inherent material properties, other physical conditons such as the the effective plastic strain will affect the length scale because the material becomes more localized as the strains become higher. At elevated strains, the localization will become stabilized, and the length scale should be constant. In addition to the physical material properties discussed above, secondary properties such as the texture, the mineral composition, OCR (over consolidation ratio) and the strength may be involved in the change of the length scale. However, the only four detrimental parameters that may be easily determined will be incorporated in the further formulation as follows: lk = f (D50 , R, As , p)

(4.49)

where, D50 is the mean grain diameter, R is the roughness, As is the aspect ratio, and p is the effective plastic strain. Roughness of grains with reasonable accuracy may be easily obtained by comparing the roughness and shape of the grains to the chart shown in Fig. 4.14 for standard roughness and shape of grains (Alshibli and Alsaleh, 2004). Alsaleh (2004) also shows how to calculate the rougness from the microscopic image of the grains. The range of the roughness is zero to one. In natural soil deposit, the range of roughness ratio is 0.1 (very angular) to 0.85 (well rounded) (Alshibli and Alshaleh, 2004). The

44


Fig. 4.14. Visual Comparison Chart for Estimating Roundness and Sphericity (Alshibli and Alsaleh, 2004)

aspect ratio of the grain is obtained from the shape of the grain. A perfect sphere has an aspect ratio of unity, while a thin plate has an aspect ratio close to zero. When one uses the mean grain diameter as the primary parameter for the initial length scale lo , and assumes the effective plastic strain as the primary factor in controlling the length scale, one can set up the following expression: l = lo exp(−k1 p) = a1 D50 [a2 R + a3 As ] exp(−k1 p)

(4.50)

Equation (4.50) is similar in form to (4.41), but additional arameters a2 and a3 are added to better reflect the physical peoperties of soils. Equation (4.50) is merely an expression for one direction; similar expressions for other directions may be obtained accordingly.


45

RVE1

RVE2

Fig. 4.15. Multiple averaging with multiple RVE’s (Voyiadjis and Dorgan, 2003)

4.4.3 Bridging Different Length Scales The RVE technique is aimed to obtain the macro-properties from the microproperties. This RVE technique, however, is not limited to integrate only once. Depending on the level of micro-mechanism, multiple RVE’s may used as required. The concept is illustrated in Fig. 4.15. The multiple RVE and multiple averaging scheme is widely used in Voyiadjis and Deliktas (2000), Voyiadjis and Dorgan (2003). Voyiadjis and Deliktas (2000) derived a multi-scale gradient theory based on consistent thermodynamic formulations for metals and metal matrix composites. In this book, similar equations are derived for the anisotropic modified Cam Clay model. The thermoelastic Helmholtz free energy is expressed in terms of the internal state variables such that (p) (p) (p) ˆ ˆ(p) ˆ 2 ˆ(p) A(k) , ∇ A(k) (4.51) Ψ = Ψ εe , T, A(k) , ∇A(k) , ∇2 A(k) , ∇ where εe is the elastic strain, T is the temperature, superscript p represents (p) plasticity, and A(k) is a macroscale internal state variable such as for the isotropic hardening or the kinematic hardening in plasticity. ∇ represents (p) the first order gradient, ∇2 represents the second order gradient, and Aˆ(k) represents the microscale internal state variable in a representative volume element. At this point, readers may not feel comfortable with the inclusion of gradient terms in (4.51). Details of gradient terms will be explained in later chapters. However, readers may refer to (4.41) through (4.48) to see why the gradient terms may be included in the free energy equation.

46


When one assumes symmetrical distribution of the gradients, the odd order gradient terms vanish, and (4.51) becomes (p) (p) ˆ 2 ˆ(p) (4.52) A(k) Ψ = Ψ εe , T, A(k) , ∇2 A(k) , ∇ The link between macroscale internal variables and microscale internal variables is obtained by averaging the microscale internal variables in the representative volume element (RVE), 1 (p) (p) Aˆ dVRVE (4.53) A(k) = VRVE VRVE (k) where VRVE is the volume of the RVE. The first and second order gradients of the macro and micro internal variables are defined as follows: 2

∂ Aij (p) ∇2 A(k) = (4.54) ∂Xk ∂Xk (p) ∂ 2 Aˆ(k) 1 (p) 2 ˆ ˆ ∇ A(k) = dVRVE (4.55) VRVE ∂Xk ∂Xk VRVE

This integration maybe performed over a sub-volume of the RVE. One can express the time derivative of (4.52) in terms of its higher order state variables as follows, .

∂Ψ ∂Ψ (p) ˆ 2 Aˆ(p) (4.56) ˙ = ∂Ψ : ε˙e + ∂Ψ : A˙ (p) + ∇ : ∇2 A˙ (k) + Ψ (k) (k) (p) (p) (p) 2 ∂εe 2 ˆ ˆ ∂A(k) ∂∇ A(k) ∂ ∇ A(k) where super dots represent the time derivative. By substituting (4.56) into the Clausius–Duhem inequality, one obtains ∂Ψ ∂Ψ σ − ρ e : ε˙e + σ : ε˙p − ρ (p) : ∂ε ∂A(k) (p) A˙ (k) − ρ

∂Ψ

(p) : ∇ A˙ (k) − ρ 2

(p)

∂∇2 A(k)

∂Ψ ˆ 2 Aˆ(p) ∂∇ (k)

.

ˆ 2 Aˆ(p) ≥ 0 :∇ (k)

(4.57)

(p)

where the mesoscale gradient terms ∇2 Aˆ(k) are dependent on the macroscale (p)

variables, A(k) . However, the random periodic boundary condition ensures that there is no net flux of mesoscale gradients across the RVE boundary. Such a constraint would effectively prevent coupling between macroscale and mesoscale gradient terms. The first term in (4.57) represents the thermoelastic law of thermodynamics, and the remaining terms represent the total dissipation process due


47

to plasticity. This relationship can be expressed as the sum of the plastic dissipation as follows, (p) (p) (p) (p) (p) ˙ (p) Π = σ : ε˙p − V(k) : A˙ (k) − W(k) : ∇2 A˙ (k) − X(k) : ∇2 Aˆ(k) (r)

(r)

(4.58)

(r)

where V(k) , W(k) , and X(k) are the thermodynamic force conjugates and are expressed as follows: (p)

V(k) = ρ

∂Ψ ∂Ψ

(p)

W(k) = ρ

(4.59b)

(p)

∂∇2 A(k) ∂Ψ

(p)

X(k) = ρ

(4.59a)

(p)

∂A(k)

(4.59c)

(p)

ˆ 2 Aˆ ∂∇ (k)

One can now express the analytical form of the Helmholtz free energy as the quadratic form of its internal state variables as follows: ρΨ =

1 e 1 (p) (p) 1 (p) (p) (p) (p) ε : E(φ) : εe + a(k) A(k) : A(k) + b(k) ∇2 A(k) : ∇2 A(k) 2 2 2 1 (p) (p) ˆ 2 Aˆ(p) + c(k) ∇2 Aˆ(k) : ∇ (4.60) (k) 2

Using (4.59) and (4.60), the following definitions can be obtained for the thermodynamic forces as follows; (p)

(p)

(p)

V(k) = a(k) A(k) (p) W(k) (p) Z(k)

(4.61a)

=

(p) (p) b(k) ∇2 A(k)

(4.61b)

=

(p) (p) c(k) ∇2 Aˆ(k)

(4.61c)

The value of the thermodynamic forces can be obtained through the evolution relations of the internal state variables. However, it should be pointed out that there are two classes of evolution equations that need to be developed, normally one at the macroscale and the other at the mesoscale level. The former can be obtained by assuming the physical existence of the dissipation potential at the macroscale. The latter can be obtained by a micromechanical or phenomenological approach. One may also consider that the evolution equations of the internal state (p) variables A(k) can be obtained by integrating the evolution equations of the (p) local internal state variables at the mesoscale; that is, Aˆ over the domain (k)

(p) of the RVE. However, integration of the Aˆ(k) is a cumbersome task because (p) at the mesoscale, Aˆ(k) is a function of many different aspects of the material

48


inhomogeneities such as interaction of defects, size of defects, spacing between them, and distribution of defects within the sub RVE. Therefore, in this work the evolution equations of the mesoscale internal state variables are obtained through the use of the generalized normality rule of thermodynamics. In this regard the macroscale dissipation potential is defined only in terms of the macroscale flux variables as follows: (p) (p) ˙ (p) (4.62) Θ = Θ ε˙p , A˙ (k) , ∇2 A˙ (k) , ∇2 Aˆ(k) By using the Legendre–Fenchel transformation of the dissipation potential (Θ), one can obtain complementary laws in the form of the evolution laws of flux variables as function of the dual variables as follows: (p) (p) ˆ (p) . ˙ B(k) , ∇2 B(k) , ∇2 B (4.63) Θ∗ = Θ∗ σ, (k) Equation (4.63) is also expressed as follows, N N ∗ Θ = F σ, ˙ p˙o,i , α˙ i i=1

(4.64)

i=1

where p˙ o,i and αi are the isotropic and kinematic hardening forces respectively. The evolution equations of parameters in (4.64) are obtained as following way: ∂Ψ (p) (p) B˙ (k) = C A˙ (k) = C (4.65) (p) ∂B(k) The evolution equations of po,i and αi in (4.64) are also obtained from the dissipation. When one assumes Perzyna type elastic/viscoplasticity (1963, 1966) and the normality rule, one obtains the following evolution equations, ∂F = γφtr ∂f (4.66a) ε˙pv = γφtr ∂σij ∂σij ∂F ∂f α˙ ij = C1 γφ = C1 γφ (4.66b) ∂αij ∂αij ˙ ∂f ˆ 2α ˆ ij = C2 γφ ∂F ∇ = C2 γφ (4.66c) ˙2 ˙ ˆ α ˆ 2α ˆ ij ˆ ij ∂∇ ∂∇ ˙ ˆ 2p ô= C3 γφ ∇

∂F ∂f = C3 γφ ˙2 ˙ ˆ p ˆ 2p ô ô ∂∇ ∂∇

(4.66d)

where αij represent the back stress, γ(φ) represents the viscoplastic multiplier, is the MaCauley bracket, and γ is the viscosity. When one uses the anisotropic modified Cam Clay model expressed by Voyiadjis and Song (2000), the following expression is obtained for the yield criterion,


f = p2 − ppo +

49

3 {(sij − pαij )(sij − pαij ) + (po − p)pαij αij } = 0 (4.67) 2M 2

where p is the mean principal stress, po is the initial mean principal stress, M is the slope of the critical state line (CSL) in p vs. q space, sij is the deviatoric stress, and αij is the rotation of the yield surface with respect to the p axis (the dimensionless back stress factor). When one incorporates the gradient to the isotropic hardening factor po and the anisotropic hardening factor αij , the following expressions are obtained, ˙ ˆ 2p p˙ o = p˙o − a1 ∇2 p˙o − b1 ∇ ô ˙ ˆ 2α ˆ ij α ˜˙ ij = α˙ ij − a2 ∇2 α˙ ij − b2 ∇

(4.68a) (4.68b)

where p˙o , p˙o are global and local mean principal stresses for the isotropic hardening factor, and α ˜˙ ij and αij are global and local dimensionless back stresses for the kinematic hardening factor, respectively. In (4.68a) and (4.68b), the parameters a1 , b1 , a2 , b2 , also take into account the dimensional discrepancies between a quantity and the corresponding second order gradient of that quantity. One obtains the evolution equation for α˙ ij from equations (4.66b) (note that α˙ ij = α˙ ij /p, p = mean principle stress). One also obtains the evolution equation for ∇2 α ˆ˙ ij ˆ˙ ij from (4.66c). In this study, the evolution equation of α 2˙ is used to obtain the evolution equation of ∇ α ˆ ij . The partial derivatives of the yield function with respect to αij and ∇2 α ˆ˙ ij are as follows, ∂f 3 ˆ 2α ˆ ij ) − sij } = 2 p{po (αij − a2 ∇2 αij − b2 ∇ ∂αij M ∂f 3 ˆ 2α ˆ ij )} = [−b2 p{sij − p(αij − a2 ∇2 αij − b2 ∇ 2 ζM 2 ˆ ˆ ij ζ∇ α

(4.69a)

ˆ 2 pô )pb2 (αij − a2 ∇2 αij − b2 ∇ ˆ 2α ˆ ij )}] (4.69b) − (po − a1 ∇2 po − b1 ∇ where ζ accounts for the dimensional discrepancy problem b2 in (4.68b) accounts for both the dimensional discrepancy problem and the material characteristics. One also obtains similarly the evolution equation for p˙ o from (4.68a) as follows: 1 + eo po (po + ∆p) p ε˙v = K ε˙pv p˙ o = − (4.70) λ−k ∆p In the same way as in (4.69b), the following equation is obtained, ∂f 1 3 ˆ 2α ˆ ij ) = pb1 (αij − a2 ∇2 αij − b2 ∇ pb1 + 2 ξ 2M 2 ˆ ξ ∇ pô

ˆ 2α ˆ ij )) × (αij − a2 ∇2 αij − b2 ∇ where ξ is a parameter similar to ζ in (4.69b).

(4.71)

50


These derivations are based on normality rule assumptions. If one introduces other dissipation functions, the evolution equations of state variables will be changed accordingly.

4.5 Micro-mechanisms Micro-mechanisms of soils are reported by many researchers (Aifantis 1980, 1984; Dafalias 1983, 1984, 1985; Zbib 1993, 1994; Voyiadjis and Song 2005a, 2005b; Song and Voyiadjis 2005a; and many others). Traditional micromechanisms are studied for individual mechanisms. Recent study of Voyiadjis and Song (2005a) showed the coupled effects of micro-mechanisms in soils. Coupled effects of micro-mechanisms on soil behavior were quite different from those expected from other continua such as steel or composite materials. Soils consist of solids and pore spaces. Saturated soils consist of solids and pore water. Some micro-mechanisms affect the solid phase and some other micro-mechanisms affect the pore phase; therefore, some micro-mechanisms behave so as to cancel each other’s effect, and some micro-mechanisms behave so as to enhance each other’s effect. Details of mico-mechanisms are dealt in the following sections. Micro-mechanisms are essentially the products of the internal behavior of soils; therefore, many micro-mechanisms are related to the internal (embedded) stress that is also called back stress in continuum mechanics. Therefore, Sect. 4.5 starts with the discussions of back stress. 4.5.1 Back Stress The meaning of the back stress is better expressed as “embedded stress” or “internal stress.” The internal stress is embedded in the grain and at the contact points between the grains and so forth. The embedded stress inside the grain is termed “short range back stress,” and the embedded stress at the contact points between the grains (or outside the grain boundaries) is termed “long range back stress” (Lowe and Miller 1984, 1986; Moosbrugger and McDowell 1988; Lamar 1989). They are differentiated because their characteristics are different. For typical continua such as metals, a linear hardening rule is used for the evolution equation of short range back stress, and a non-linear hardening rule is used for the evolution equation of long range back stress. The use of independent corotational rates for the short range and the long range back stresses is also suggested by Zbib and Aifantis (1988). Justification or verification for using multiple back stresses is more usually found in discussion of metals than in discussions of geo-materials. For that reason, we will introduce at this point the concept of back stress as it relates to metallurgy. The ratcheting behavior of metals has been examined by many researchers (Abdel-Karim and Ohno 2000; Bari and Hassan 2000; Barbe et al. 2001;

4.5 Micro-mechanisms

51

Fig. 4.16. Variation of multiple back stresses (Lowe and Miller, 1986)

Moosbrugger et al. 2000; Yoshida 2000), and it has been explained by the concept of short range back stress. Experimentally, Montheillet et al. (1984) reported the existence of multiple induced textures of metals (short range back stress) and independent evolution rules during shear. Mughrabi (1975) has reported the evidence of separate short range and long range internal stress fields in copper. Mughrabi measured the radii of curvature of free primary dislocations pinned by irradiation-induced point defects using transmission electron microscopy. Long wavelength (∼6 µm) and short wavelength (∼1 µm) fluctuations in dislocation curvature were observed, indicating short range and long range internal stresses. There are many other sources of evidence for multiple back stresses; the interested reader may refer to Hirsch and Mitchell (1966), and to Kressel and Brown (1968). The graphical representation of short range and long range back stresses as reported by Lowe and Miller (1986) is illustrated in Fig. 4.16. Figure 4.16 clearly shows the different nature of short range and long range back stresses, and thus one can see that independent treatment of multiple back stresses is justified. The combined short range and long range back stresses are expressed in (4.72) as follows, N αi (4.72) α = α1 + i=2

where α is the composite back stress, α1 is a short range back stress, and αi (i = 2, 3, 4 . . .) is a long range back stress. The physical interpretation of multiple back stresses in soils is illustrated in Figs. 4.17 and 4.18. Figure 4.17 shows the assembly of clay particles: clusters (aggregations) and conglomeration of clusters. The back stresses α1 , α2 , and α3 for these clay structures are for different structural scales and conceptually understood from Fig. 4.17. Please note that Collins and McGown (1974) did not refer to the back stresses.

52


3

(inter cluster level)

2

(intra cluster inter granular level)

1

(intra granular level)

Fig. 4.17. Overall microfabrics in Tucson Silty Clay (freshwater alluvial deposit) (Collins and McGown, 1974)

3

1

2

(a) Fabric and particle orientation in Portsea Beach Sand (Lefeber

(b) Idealization of load carrying scheme of Sand (Petrakis and Dobry, 1986)

Fig. 4.18. Structure of granular materials


53

However, one can see the different nature of back stresses. Blenkinsop (2000) introduced 33 different microstructures for geo-materials, and ambitious readers may want to include all these 33 microstructures with 33 back stresses. This book, however, deals with only 2 to three back stresses for simplicity. Figure 4.18a shows the structure of sandy soils. It is observed that some sand grains have continuous contacts and contribute to the load carrying chain, and other sand grains do not have continuous contacts and do not contribute to the load carrying chain. However, sandy soils are believed to constitute hidden clustering structures as shown in Fig. 4.18b and are frequently modeled by a group of spherical grains through which the stresses are carried out (Petrakis, 1986). In Fig. 4.18b, the rectangular chains show the stress transferring mechanism. When the rectangular chains are continuous, the load carrying chain is effective; otherwise, the load carrying chain ends at the particle and the grain does not contribute to the load carrying chain. These grains are orphans. The applied stresses are not carried out by orphans but by the load carrying chains. In Fig. 4.18b, one can see that many particles are involved in more than one load carrying chain. That means these particles are subjected to more than one back stress. The number of back stresses are many, but this present study incorporate only two back stresses because of the lack of information about α3 and higher-order back stresses. The graphical representation of the evolution of multiple back stresses is shown in Fig. 4.19. The threshold shear strain is taken as 10−2 % for α1 because the typical threshold strain for the degradation of the shear modulus is experimentally reported to be approximately 10−2 % for most clayey soils (Song, 1986). Figure 4.19 shows the independent evolution of the two different back stresses by Yoshida (2000). The evolution equation of α1 is completely independent from the evolution of α2 . Similarly, adjusting parameters in the α2 curve does not give the properties of the α1 curve. Therefore, independent evolution equations are to be used for a more rational consideration of the embedded stress. One may assume that α1 is the back stress within the soil particles (short range), and α2 is the combination of the back stresses outside the soil particles (long range) as shown in Fig. 4.19. Equation (4.72) is now expressed as follows: (4.73) α = α1 + α2 The quantification of back stress should be incorporated with the fabric, inter-particle charges, and so forth. However, those relations are not available presently, and the authors applied the phenomenological approach using the existing evolution equations of back stresses. When one uses Prager’s linear evolution equation (See Chaboche (1991) and Zbib and Aifantis (1988)) for α1 , and Dafalias’ non-linear evolution equation for α2 (Dafalias, 1987), one obtains the following evolution equations for the back stresses,

54


Back Stress

(overall)

2 (long

range)

1 (short

range)

Strain Range Fig. 4.19. Schematics of evolution of short range back stress (α1 ), long range back stress (α2 ) and composite back stress (α)

2 C1 ε˙ p 3 ˙ 2 = C2 |ε˙ pv |(s − X2 pα2 ) α

˙1 = α

(4.74) (4.75)

where C1 is a material parameter for the linear kinematic hardening rule, ε˙ p is the plastic strain rate, ε˙ pv is the volumetric strain rate, s is the deviatoric stress, and p is the mean principal stress. C2 and X2 are similar to those given by Dafalias (1987). Also one may find that (4.74) and (4.75) show the same evolution relationship to the lines in Fig. 4.19. Each corotation of the different back stresses will follow its own corotational rate. Therefore, one should use the following corotational equations, ˙ i − Wis α ˙ i Wis ˙i+α αi = α

(no sum)

(4.76)

˙ i is the time where αi represents the corotational rate of the back stress, α rate of back stress, and W si is the modified spin for the back stress. Zbib and Aifantis (1988) suggested the following expressions for the modified spin for the short range back stress and the long range back stress: W s1 = W s − ξ 1 (α1 d − d α1 ) W s2 = W s − ξ 2 (α2 d − d α2 )

(short range) (long range)

(4.77) (4.78)

In (4.77), W s1 is the modified spin for short range back stress, W s is the spin, ξ 1 is a material constant, and d is the plastic strain rate. In (4.78), W s2 is the modified spin for long range back stress, and ξ 2 is a material constant. Equations (4.77) and (4.78) are essentially derived for metals. However, test results for soils (Song, 1986) have shown the consistent axial strain development during torsional cyclic loading by the resonant column test. (This


55

strain is not caused by void ratio change.) The authors have assumed a similar mechanism for soils as that used by Zbib and Aifantis (1988) in metals for the double back stresses and corresponding plastic spin in soils. More details of the plastic spin will be introduced in the next section. We will use the combination of back stresses α1 and α2 to incorporate back stresses into the yield function. When one uses the anisotropic modified Cam Clay model as shown in (4.79), one may substitute α1 + α2 into α: 3 {(sij −pαij )(s ij −pαij )+(p o −p)pαij αij } = 0 (4.79) 2M 2 However, for the plastic spin, the back stress α1 may not have to be included in conjunction with the spin tensor because intra-granular rotation stiffness will be much higher than inter-granular rotation stiffness (and therefore, intra-granular rotation may be neglected compared to inter-granular rotation.) f = p 2 = pp o +

4.5.2 Rotation of Particles Dramatically different behavior of the material at large strain was first reported by Truesdell (1955), who noted that an oscillatory stress solution is obtained when a standard linearly hypoelastic material is subject to a large deformation as shown in Fig. 4.20. Figure 4.20 shows the behavior of large strain models with or without considering the rotation of grains. The model that does not consider the rotation of grains (W, α = 0) shows unrealistic oscillation of stresses at the higher strain level. Voyiadjis and Kattan (1999) also reported the similar behavior.

Fig. 4.20. Oscillatory behavior for large strains (after Zbib and Aifantis, 1988)

56


a) Plastic Spin The plastic spin tensor modifies the constitutive equation that follows from the change in the substructure of the material. To clearly understand the physical concept of plastic spin, the stress co-rotation concept is addressed as follows. The stress transformation between two coordinate systems can be easily expressed as (4.80) T ∗ = QTQ T where T ∗ is the stress tensor transformed in the new coordinate system, T is the stress tensor in the original coordinate system, and Q is the transformation matrix. However, when one uses a stress rate in the above constitutive expression, the coordinate transformation is not as easy as (4.80). The time derivative of (4.80) is given below, ∗ T ˙ ˙ T + QT Q ˙T + Q TQ T˙ = QTQ

(4.81)

where the upper dot represents the time rate. However, one cannot assume that the first and third terms are zero in (4.81); therefore, one must introduce the spin tensor W to compensate for these non-zero terms as shown in (4.82), ∗ T˙ = T˙ − WT + TW

(4.82)

where, W is called the “spin tensor”. The spin tensor W corrects the constitutive relations for the rigid body rotation. Note also that even with the incorporation of the spin tensor W , the stress-strain response of the material is unstable at high strains in applications such as the simple shear problem (Voyiadjis and Kattan 1991; Dafalias 1983; Lee et al. 1983) apparently because of the absence of the micro-mechanical spin of the material in the constitutive model. The micro-mechanical spin having been incorporated, it becomes clear that some internal stress change is incorporated. Thus the spin tensor must be modified to reflect such a micro-mechanical change. The “modified spin tensor” is commonly expressed as W ∗ or Ω and is given as follows, ∗ T˙ = T˙ − W ∗ T + T W ∗ = T˙ − ΩT + T Ω

W∗ = Ω = W −WP

(4.83) (4.84)

where, W p is the plastic spin. One observes that the plastic spin is the term that incorporates the spin tensor W and reflects micro-mechanical changes of the material. The micro-mechanical changes may be a single-source phenomenon or may be a multi-source phenomenon. From intuition and the results of Anandarajah (1995), one may deduce that the plastic spin may be related to the plastic strain (or stress) and back stress. Dafalias (1983) quantified this concept in the following equation,


W p = η(αT − σT ) = η(ασ − σα)

57

(4.85)

where η is a constant, T is a tensor of physical quantities such as stresses or strains, and α is a constant. Equation (4.85) may be expressed as follows, W p = ξ(αd − d α)

(4.86)

where d represents the plastic strain rate and ξ is a function of η and the flow rule. (We will introduce more details of η and ξ later.) Equation (4.86) is the expression of plastic spin, and it can be incorporated into the conventional constitutive relationships. b) The Microplane Model The microplane model was proposed by Zienkiewicz and Pande (1977), Bazant (1984), Bazant and Kim (1986), and Prat and Bazant (1989). This model is quite different from the relative spin or plastic spin model. While the relative spin or plastic spin model is a plasticity-based model, the microplane model is a microscopic material characteristics-based model that is a kind of the classical slip theory of plasticity. The microplane method was first proposed by Taylor (1938), who suggested that the stress-strain relation be specified independently on planes of various orientations in the material. He also assumed that either the stresses on that plane (now called the microplane) are the resolved components of the macroscopic stress tensor (static constraint), or the strains on the plane are the resolved components of the macroscopic strain tensor (kinematic constraint). The responses on the planes of various orientations are then related to the macroscopic response simply by super position or, as has been done in recent works (Bazant, 1984; Carol et al., 1990), by means of the principle of virtual work. In the initial application to metals, beginning with Batdorf and Budiansky (1949), only the static constraint was considered. So it was the early applications to soils (Zienkiewicz and Pande, 1977; Pande and Sharma, 1980, 1983; Pande and Xiong, 1982) that successfully described some basic aspects of soil behavior other than strain softening. It appeared, however, that the microplane system under a static constraint becomes unstable when strain softening takes place (Bazant and Oh, 1983, 1985; Bazant and Gambarova, 1984). To cope with these problems, Prat and Bazant (1989) improved the microplane model for dynamic constraint. In the application of this model to clays, the microplanes may be imagined to represent the slip on the contact planes between clay platelets or the planes normal to the platelets on which slip is manifested by normal strain as shown in Fig. 4.21. Although the correlation to the microstructural mechanism of inelastic deformations is largely intuitive, the microplane model has the advantage of being able to distinguish among the intensities of inelastic strains at various orientations and describe how they are mutually

58


Clay Particles

1

2

Fig. 4.21. Microplane in Cohesive Soil: 1 = micro-plane as slip plane between clay platelets; 2 = micro-plane as normal plane to clay platelets (after Prat and Bazant, 1990)

constrained. Therefore, the microplane strains εN and εT in Fig. 4.22 may be imagined to represent the sum of the inelastic relative displacements on all the weak planes contained within a unit volume of the material plus the associated elastic deformations of all the particles. The equilibrium equation for constitutive law was set up by equating the macro level strain energy to the summation of the micro level strain energy caused by εN and εT as follows: ∆Vo (σ N δεN + σ T δεT )ΨN dΩ (4.87) ∆V o σ ij δεij = Ω

where, ∆Vo = unit volume, σεij = macroscopic stress tensor, δij = incre = normal stress in microplane, σT = ment of macroscopic strain tensor, σN tangential stress in microplane ΨN = αN vN , and dΩ = sin θdθdφ, with θ and φ = angular spherical coordinates. The prior advantage of the microplane model is that the constitutive law is written in terms of the current stresses and strains (not in terms of their increments), which allows the model to be explicit with all the numerical advantages shown by Carol et al. (1990). However, it is also pointed out by Prat and Bazant (1989) that it is impossible for the effective microplane


59

n

T

N

Fig. 4.22. Strain Components on microplane (after Prat and Bazant, 1990) stresses σN and σT to equilibrate the effective macrostress σij exactly since the microplanes are constrained kinematically.

c) Cosserat “Micropolar” Continuum Cosserat “Micropolar” continuum is a classical but one of the best continuum mechanics to account for the particle rotation (independently to the displacement). The Cosserat theory was first developed by the Cosserat brothers in 1859; 50 years later, researchers started revisiting and republishing Cosserat work until the 1960s, when Gunther marked the rebirth of the micromechanics in his papers. This section discusses the essentials of Cosserat “Micropolar” continuum. The Cosserat theory can separate the grain rotation from its translation, adding three other degrees of freedom to any point in the 3D continua. In classical continuum mechanics, one might have two different strain tensors, the Green–Lagrangian strain tensor and the Eulerian strain tensor. Either one of these tensors can be decomposed into a symmetric part (stretch tensor) and an antisymmetric part (spin tensor). Granular materials undergo high rotational and translational deformations at failure. However, the classical strain tensor fails to describe the real kinematics, such as micro-rotation in

60


granular materials, and other alternative tensors need to be used instead (Vardoulakis and Sulem, 1995; Oda and Iwishita, 1999). In geomechanics, interests in the Cosserat continuum began to show up in mid 1980’s, when links were made between Cosserat kinematics and strain localization phenomenon. In 1979, Kanatani used the micropolar theory to study the flow of granular materials; in this study, the grains were assumed to be homogenous rigid spheres with the same size. In his formulations, the velocity (vi ) and the rotation (ωij ) of the grains are considered two independent variables that describe the deformation of the continua. Making use of these independent kinematics in the macro conservation laws of mass, linear momentum, and angular momentum, Kanatani was able to propose equations of motion for the granular materials. However, the assumption that grains have uniform contact distribution because of the spherical shape of the particles is not correct: the non-uniformity of the shape must be accounted for. M¨ uhlhaus and Vardoulakis (1987) used the Cosserat kinematics for 2D space to investigate the thickness of shear bands in granular materials. Their work shows one of the strongest links between the Cosserat continuum and strain localization in granular materials. In their approach, the continuum has an overall rotation (ωij ) which is different from that of the grain or the c ). The deviation in the rotation would actually cause Cosserat rotation (ωij non-symmetry in the strain and stress tensors, and as a result, those tensors would be different from the classical ones. Assuming infinitesimal deformation in the pre-banding regime, the following kinematics are proposed for plane strain case: εij = e ij + ω ij − ω cij κi = w c3,i

(4.88) (4.89)

where e ij =

u i,j + u j,i u i,j − u j,i ; ω ij = 2 2

(4.90)

And ω cij = −e ij3 w c3

(4.91)

where eijk is the Ricci permutation tensor. The curvature or the rotation gradient ki is a measure of the relative rotation of a single grain with respect to the neighboring grain. If the rotation of the continuum coincides with the grain rotation, then (4.103) collapses into a classical strain tensor. The Cosserat continuum explicitly includes the rotation of grains, and therefore its equilibrium equations include the moment equilibrium as follows (refer to Fig. 4.23): σ 11,1 + σ 12,2 + f1 = 0 σ 21,1 + σ 22,2 + f2 = 0

(4.92) (4.93)

m1,1 + m2,2 + (σ 21 − σ 12 ) + Φ = 0.

(4.94)


61

Fig. 4.23. Equilibrium condition for Cosserat continuum

Also the equation of virtual work is expressed as the combination of the stretch and rotation as follows, (i) δW = (σ ij δ γ˙ ij + mi δki ) dV (4.95) V

where δ indicate the variation in the related quantity. In summary, the beauty of Cosserat continuum is that it considers grain rotations explicitly while many traditional continuum mechanics do not incorporate them. (For more details of Cosserat continuum, see Vardoulakis and Sulem (1995) and Oda and Iwishita (1999)).

62


d) Particulate Mechanics This approach may be said to be the advanced method since it deals with the mechanics of individual soil particles. By its nature, this method needs extended computation time. At the current state of knowledge, its application is primarily used for sandy soils. Particulate mechanics assumes that the behavior of sand aggregate is very similar to that of polycrystals because the individual grain packing within the sand could be considered in first approximation to behave like randomly oriented crystals (Voyiadjis et al., 1995, 1992). However, the main difference between sand particles and randomly ordered crystals is that the properties of these packed sand grains are pressure dependent, and the amount of slip affecting each of these packed grains, in contrast to the polycrystalline aggregate, depends on the mean stress. For example, a simple cubic array of equal spheres is a pressure dependent monocrystal with three sliding planes, with each plane containing two sliding directions (Voyiadjis and Foroozesh, 1990). Moreover, sand may experience dilation under shear that does not occur in polycrystalline aggregates. Finally, unlike metals, soils exhibit nonlinear inelastic stress strain behavior even at very small strains. The definition of “yielding” in granular media is critical because yielding in soils is likely to be a controversial topic. This yielding is a result of the nonlinear force-deformation behavior (Mindlin and Deresiewicz, 1953) at the interparticle contacts. This nonlinear force-deformation causes granular media to exhibit nonlinear inelastic stress-strain behavior at very small strain levels. Therefore, strictly speaking, cohesionless aggregates, unlike metals, do not have a clear “linear elastic region” defined by an initial yield surface. There are two distinct deformation mechanisms which operate during loading of a granular medium. At very small strains (γ < 10−2 %) there are no particles sliding and all macroscopic nonlinearity is the result of nonlinearities at the inter-granular contacts and of the redistribution of contact forces (one aspect of the material fabric) during loading. The normal component of the deformation at the contact is nonlinear elastic, while the tangential component is nonlinear inelastic as a result of the slip at the edges of the contact annulus between two spheres (Mindlin and Deresiewicz, 1953). In soils, hysteric behavior is observed during low level shear strain cycling in the resonant column device, but no permanent volumetric changes or pore pressure buildup accumulates. At larger strains (γ > 10−2 %), there is sliding between particles which move and rearrange themselves. Therefore, the geometric aspects of the fabric change as well. This change of geometric fabric manifests itself by irreversible volumetric changes if the loading takes place under drained conditions. The strain level at which this occurs has been experimentally determined (Dobry, 1985) to be on the order of 10−2 % for the level of mean stress used in soil testing (40–270 kPa) and it is called the threshold strain, γt . This sliding of particles is directly analogous to the “slip” in metals. The macroscopic


63

strain caused by grain slipping at the contact annulus is an order of magnitude smaller than the macroscopic strain caused by grain sliding and can be considered to be a second order effect. Therefore, one possible definition of yielding, in a manner directly analogous to yielding in metals, is the point in stress space at which the geometric fabric of the material changes irreversibly; that is when the first grain slides. Since the sliding of the first particle is very difficult to monitor in the laboratory, yielding could be defined as the locus of all points in stress space at which the value of the octahedral shear strain is equal to, or less than, 10−2 %. Following the above logic, the yield is defined as the locus of all points in stress space [τzθ (σzz − σrr )/2] that have the same value of total (elastic plus p c e c = γoct + γoct . The value of γoct should plastic) octahedral shear strain, γoct be as close to the threshold value as possible, given the restrictions posed by p will be close to zero. While a criterion the experimental device, so that γoct of an octahedral shear strain of 10−2 % does not necessarily imply that only one sphere has slipped, it is assumed that a small percentage of particles have slipped and that the yield loci obtained using this approach are homothetic to the true yield surface. Numerical simulations (Petrakis et al., 1991b) support this last hypothesis. In the experiments performed by Petrakis and Dobry (Petrakis et al., 1991a; Dobry et al., 1991), the “yield” criterion was set to 3×10−2 %. This value caused plastic strains to accumulate during the probing portion of the tests. The strain for yield is typically set at 1 × 10−2 % or less. With the proper constitutive model, this method is expected to include most of the micro-mechanical behavior and hardening characteristics of the yield surfaces of a granular medium under various levels of pre-strain. As mentioned previously, this model is one of the more advanced. However, its major difficulties are its associated computation costs and the fact that its application to clayey soils is not yet widely appreciated. 4.5.3 Grain Interaction As discussed in Sect. 4.5.2d), there is no substantial inter-grain motion when the shear strain is low. However, there may be substantial inter-grain motion when the shear strain is high. At finite strain, the severe grain interaction manifests itself and causes the stress transfer to the surrounding areas. Figure 4.24 shows the illustration of the grain interaction at the shear band (plane) by Di Prisco and Aifantis (1999). During shear, the shear strain in the shear band is greater than that in the surrounding area. In the case of large shear straining, the particles should be rearranged. Some particles may roll over neighboring particles, and some other particles may drop into the hollow space, thereby creating a change in volume. During volume expansion (dilatancy), the particles in the shear band push out the neighboring particles and increase the contact pressure; the ultimate result is higher shear strength in the shear band. When expansion is partly allowed, this behavior causes the stress to transfer from the shear

64


SHEAR BAND

Fig. 4.24. Illustration of grain interaction (Di Prisco and Aifantis, 1999)

band to the neighboring area. When the neighboring particles are actually pushed back, the stress in the shear band is redistributed and reduced. Note, however, that this mechanism should not be confused with that proposed by Vardoulakis and Aifantis (1989). According to Vardoulakis and Aifantis (1989), as volume increases, grain contact decreases, causing grains to move apart from each other. If, on the other hand, the volume decreases, then new contact is generated and the two grains move closer together. Dilatancy loosens grain-interlocking and reduces the stress or strain level in that region. For the volume contraction region, the quality of grain-interlocking is enhanced. This phenomenon alters stress or strain levels that are caused by the stress or strain gradients (Zbib 1994; Zbib and Aifantis 1988, 1989; Vardoulakis and Aifantis 1991). Although these two different mechanisms seem to conflict, they are essentially the same. Figure 4.25 illustrates the general stress-strain behavior of dense sand and loose sand specimens in drained condition. In Fig. 4.25, the dense specimen develops higher shear resistance at point 2 than at point 1. One the other hand, the void ratio at point 2 is greater than it is at point 1. This result shows that there is a dilatancy from point 1 to point 2, and the specimen shows higher strength. This is the mechanism of first particle interaction illustrated by Fig. 4.25 . The void ratio at point 2 is the maximum void (the loosest condition) that the specimen may experience during shear. Therefore, the specimen behaves similar to the loose specimen. The loose specimen, one the other hand, shows lower shear resistance at point 5 than at point 4. Because the void ratio at point 5 is greater than that at point 4, it is evident that the loose specimen is subject to void ratio reduction (contraction) during shear and gains strength as the number of contacts among the particles increases. This process is not different from the mechanism of second particle interaction illustrated by Vardoulakis and Aifantis (1989). Therefore, one must be cautious when considering the grain interactions in the constitutive relations because the positive and negative effects of grain interactions will be different


65

5 4 3

2

1

(b)

Fig. 4.25. Triaxial tests on “loose” and “dense” specimens of a typical sand: (a) stress-strain curves; (b) void ratio changes during shear (after Hirschfeld, 1963 and Holtz and Kovacs, 1981)

66


depending on the initial condition of the soils. The criterion by which soils may be judged “dense” or “loose” is easily determined from Fig. 4.25b. When the initial void ratio of the specimen is higher than the critical void ratio, then the soil is loose; if the contrary is the case, the soil is dense. The critical void ratio is defined as the ultimate void ratio at which continuous deformation occurs with no change in principal stress difference. (For further explanation of the critical void ratio, see Casagrande (1936)). Mathematically, the gradient concept is expressed as follows, σ ij = σ ij − a1 ∇2 σ ij

(4.96)

where σ ij represents the stress considering stress transfer (a state that is also called “homogenized stress”), and σ ij represents the stress before considering stress transfer (a state that is also called “localized stress”). Equation (4.96) is the gradient equation for stresses; however, the similar equation may be used for other physical quantities. The above discussions may be summarized as follows: “A stress or strain gradient around the shear plane will cause a change in volume and, eventually, a redistribution of stresses or strains.” The next section will discuss the formulation of this gradient concept. So far the gradient is described in terms of physical meaning. Gradient theory; however, is known for contributing to the “well posedness” of numerical solutions. This effect may be illustrated from a first order Taylor expansion (Chapra and Canale, 1988) x)(x − x ˜) f (x) ∼ x) + f (˜ = f (˜

(4.97)

This relationship can be employed to estimate the relative error of f (x) as in e[f (x)] =

x) ∼ f (˜ x) f (˜ x) − f (˜ (x − x ˜) = f (x) f (˜ x)

(4.98)

The relative error of x is given by e(x) =

x−x ˜ x ˜

(4.99)

A condition number is defined as the ratio of these relative errors as follows: Conditional Number =

x ˜f (˜ x) e[f (x)] = e(x) f (˜ x)

(4.100)

The condition number provides a measure of convergence or divergence of a numerical error. When the condition number is greater than 1, the error diverges; when the condition number is less than 1, the error converges. When the condition number is greater than one, it is called “ill conditioned” or “ill posed.” In the following example, one can see that the numerical solution can be ill posed in the shear plane while it is well posed in other areas.


67

Assume a function f (x) = tan x for x ˜ = π/2+0.1(π/2); then the condition number is computed as ˜) x ˜(1/ cos2 x tan x ˜ For x ˜ = π/2 + 0.1(π/2), 1.7279(4086) = −11.2 the condition number = −6.314 Condition number =

(4.101)

Thus, the condition is ill conditioned. Equation (4.101) shows that the major cause of the ill conditioning appears to be the derivative. For the areas of sharp strain change (or sharp stress change), one can expect a high derivative and an ill posed condition. For the same solution at the regular area (where we do not have high stress change), the derivative is small and the condition is well posed. When one includes the gradient it actually enhances the well posedness of a numerical solution as in the following second Taylor expansions: ˜ f (x) 2 (x − x ˜) . ˜(x − x ˜) + f (x) ∼ x) + f x = f (˜ 2

(4.102)

Of course a second order Taylor expansion is more accurate than a first order Taylor expansion, and the error should be smaller in the second order Taylor expansion. A smaller error will make a smaller condition number and improve the well posedness of a numerical solution. Note also that the third term on the right side of the equation is nothing but the second order gradient term. Therefore, the addition of higher order gradient term actually improves the stability of a numerical solution. The same example shows the effects of the gradient term on the condition number. For (4.102) the condition number is expressed as follows: ˜)] x ˜[f (˜ x) + 12 f (x − x e[f (x)] = e(x) f (˜ x) = 0.0968 for x ˜ = π/2 + 0.2(π/2).

Condition Number =

(4.103)

Voila! Equation (4.103) becomes a very well posed numerical condition just by the addition of the second order gradient term. An even higher order gradient may be added, but one can see from (4.103) that adding the second order gradient yields sufficiently stable results. It is therefore clear that the gradient theory has numerical justification as well as physical justification. In addition, note that the expression in (4.103) suggests that a constant x)2 . However, that conclusion is not a1 in (4.96) may be the same as (x−˜ 2 exactly true (The mathematical gradient and the physical gradient may not be the same.). The gradient constant in real life may be the whole amount x)2 and may be smaller or larger. For that reason, researchers use a of (x−˜ 2 constant that is a material parameter to generalize it.

68


Fig. 4.26. Effects of strain rate on normalized undrained shear (Kulhawy and Mayne, 1990)

4.5.4 Rate Dependency The viscous property of soils affects the rate dependency of materials. In localized shearing zones such as the shear band, the strain rate is much higher than it is outside the shear band. Therefore, materials within the shear band responds differently from the same materials outside the shear band. Traditionally, researchers have believed that a higher strain rate results in a higher strength of materials as shown in Fig. 4.26. The trend shown in Fig. 4.26 is also accepted in other material science areas. This trend is easy to understand; one may feel very little resistance when one is gently moving one’s hand in water, however, one will feel strong resistance when one is moving one’s hand quickly. Generally, the shearing resistance of a fluid is approximately proportional to the velocity of shearing mechanism. This is one of the reasons why a man can dive into the water while an airplane cannot. The rate dependency in soils, however, may not follow the general trend explained above. It is addressed below. Viscosity Dependent Rate Dependency The following discussion first describes the traditional viscosity related rate dependency; second, it describes non-traditional rate dependency. The additive decomposition of total strain rate into elastic, visco-plastic and damage parts gives the following relation,

4.5 Micro-mechanisms d ε˙ij = ε˙eij + ε˙vp ij + ε˙ij

69

(4.104)

where ε˙ij is the total strain rate, ε˙eij is the elastic strain rate, ε˙vp ij is the viscoplastic strain rate, and ε˙dij is the damage strain rate. The yield function f needs to be defined in terms of the effective stresses and strains. For modeling the rate dependency of soils, this study uses a Perzyna (1963) type viscoplasticity as follows, ∂fd (4.105) ε˙vp ij = γΦ(F ) ∂σij where · is the viscosity, is the McCauly Brackets, fd is a dynamic yield function incorporated with rate dependency, and Φ(F ) is a visco-plastic multiplier that is a function of F , which is defined as follows, F =

fd − ks ks

(4.106)

where ks is a hardening factor in static loading. Oka (1981) obtained from experimental results the expression of Φ(F ) as follows, põ (4.107) Φ(F ) = C exp m po where C and m are the material constants, po is an initial principal stress for the static condition, and põ is an initial principal stress for the dynamic condition. Equation (4.105) with Oka’s (1981)’s elaborated approach presents a workable solution for the rate dependency of soils. Later, Oka et al. (1999) developed a more sophisticated rate dependency. However, researchers are familiar with the flow rule that has expressions as follows: ε˙pij = dλ

∂f ∂σij

(4.108)

When one uses visco-plasticity, the yield function always expands or shrinks depending on the strain rate as shown in Fig. 4.27; therefore, researchers call this condition the dynamic yield surface. Since it is “dynamic”, it does not satisfy the consistency condition (df = 0) for the general flow rule from which the flow scalar dλ is determined. This was the major technical inconvenience in using Perzyna (1963) type visco-plasticity for continuum mechanics. Recent research results obtained by a Louisiana State University research group and Perzyna overcame the above difficulties and presented an easier way of incorporating visco-plasticity in the form of (4.22). For a modified Cam-Clay type with isotropic and kinematic hardening, the static yield function is defined as follows: fs = p2 − ppo +

3 {(sij − pαij )(sij − pαij ) + (po − p)pαij αij } = 0. (4.109) 2M 2

70


Ko

line

Fd

Fs

Fig. 4.27. Dynamic yield surface for rate dependent modified Cam Clay model

By mathematical arrangement of (4.109), the isotropic hardening parameter po is separated as follows: 1 1.5 sij sij − 2sij αij = po fs = 2 (4.110) pM 2 + M − 1.5αmn αmn p The extension of (4.110) to include the rate-dependent plasticity (viscoplasticity) implies that the stress state is no longer constrained to remain on the yield surface but can have fs ≥ 0. Here, the homogenization effect by rate dependency for anisotropic modified Cam Clay model is derived in line with a similar formulation for the von Mises material by Voyiadjis et al. (2003). We define the overstress as σ v = fs − R

(4.111)

where σ v is the viscous stress in the effective configuration (or the overstress; that is, the difference between the dynamic stress and its static counterpart), R is the isotropic hardening function, and denotes the MacAuley brackets defined by x = (x + |x|)/2. σ v is the common notion of visco-plasticity (Perzyna, 1966), which implies that an inelastic process can only take place if σ v is positive. In that case, fs ≥ 0. Therefore, we define the dynamic yield surface, f , as follows, ˙ ≤0 f = fs − R(p) − σ v (p, p)

(4.112)

where p and p˙ are the effective accumulative visco-plastic strain and its rate. ˙ is defined by The effective rate of the accumulative visco-plastic strain, p,


p˙ =

2 ˙ vp ˙ vp ε ε 3 ij ij

71

(4.113)

vp

where ε˙ is the visco-plastic strain rate in the effective configuration. One can write the evolution equation for the visco-plastic strain in the effective vp configuration, ε˙ , as follows: ∂f vp ε˙ ij = λ˙ vp ∂σ ij

(4.114)

It can be easily shown that p˙ expressed by (4.113) is related to λ˙ vp by ˙ ˙λvp = 3 √p (4.115) 2 Bii In classical visco-plastic models of the Perzyna-type (Perzyna, 1963, 1966), which are considered as penalty regularization of rate-dependent plasticity (visco-plasticity), the consistency parameter λ˙ vp in the effective configuration can be replaced by an increasing function of the overstress, as in the following example, ml σv 1 λ˙ vp = vp (4.116) η σ γp + po where m1 is the viscoplastic rate sensitivity parameter and η vp is the viscosity or fluidity parameter, which is referred to as the relaxation time according to the notation given by Perzyna (1988). By making use of (4.111) and (4.116), one can write an expression for the overstress function σ v as follows: σv = η

vp

3 p˙ √ 2 Bii

l/ml [σ yp + p o ]

(4.117)

Substituting σ v into (4.112) gives the following expression for the dynamic yield surface f in the effective configuration: 1/ml 3 p˙ √ f = fs − po 1 + ηvp ≡0 (4.118) 2 Bii This function is a generalization of the anisotropic modified Cam Clay model for rate-dependent materials. The rate-independent condition can be simply recovered by imposing η vp = 0 (no viscosity effect), so that one has the plasticity case f = fs ≤ 0. In the elastic domain, both fs and f are equivalent since, in that case, p˙ = 0 . Therefore, the admissible stress states are constrained to remain on or within the elastic domain so that one obtains a similar formulation to the rate-independent plasticity f ≤ 0. However, during the unloading process in rate dependent behavior, f < 0 and for a particular

72


strain-rate does not imply that the material is in the elastic domain; it may also be in a visco-plastic state with a smaller strain-rate. Moreover, from the relation in (4.26), it can be seen that as the viscosity parameter η vp goes to zero (rate-independent case), the consistency parameter λ˙ vp remains finite and positive (though indeterminate) because σ v also goes to zero. The extended criterion given by (4.118) will play a crucial rule in the dynamic finite element formulation described hereafter. It also allows a generalization of the standard Kuhn-Tucker loading/unloading conditions: (4.119) f ≤ 0, λ˙ vp ≥ 0, λ˙ vp f = 0 Thus, f still satisfies the constraint equation (f˙ = 0 ). For this reason, the dynamic yield surface can expand and shrink not only by softening or hardening effects, but also due to softening/hardening rate effects. A Gradient Dependent Formulation for a Rate Dependent Anisotropic Modified Cam Clay Model When one incorporates gradients in the dynamic yield surface equation expressed in (4.120), the following relations are obtained, ˙ ∇2 p˙ ≤ 0 f = fs − R p, ∇2 p − σ v p, ∇2 , p, p, (4.120) 1 1.5 s˜ij s˜ij − 2˜ sij α ˜ ij = põ (4.121) fs = 2 p˜M 2 + M − 1.5˜ αmn α ˜ mn p where upper “∼ ” represents a gradient implemented non-local quantity. In the above equation, s˜ij is determined from the strain gradient. When one incorporates the gradient into the isotropic hardening factor po and the anisotropic hardening factor αij , the following expressions are obtained, (4.122) p˜˙o = p˙o − a1 ∇2 p˙o α ˜˙ ij = α˙ ij − a2 ∇2 α˙ ij

(4.123)

where p˜˙o , p˙o are global and local mean principal stresses for the isotropic hardening effect and a ˜˙ ij , a˙ ij are global and local dimensionless back stresses for the kinematic hardening effect, respectively. In (4.122) and (4.123), the parameters a1 and a2 are length scales that account for both material properties and dimensional consistencies in the equations. One obtains the following ˙ evolution equation for α˙ ij , ε˙vp v and β ij assuming negligible micro level gradients (note that a˙ ij = α˙ ij /p, p = mean principle stress), ˙ vp tr ∂F = λ˙ vp tr ∂f (4.124a) ε˙vp = λ v ∂σij ∂σij ∂F ∂f α˙ ij = C1 λ˙ vp = C1 λ˙ vp (4.124b) ∂αij ∂αij ∂F ∂f β˙ ij = C4 λ˙ vp = C4 λ˙ vp (4.124c) ∂β ij ∂β ij


73

where αij represent the back stress for plasticity, β ij represent the back stress for damage, and λvp is the viscoplastic multiplier. Effects of Ground Water Flow on Rate Dependency of Saturated Soils So far, we have discussed the traditional viscosity related rate dependency. Saturated soils, however, are composed of two different phases: solids and pore water. Pore water flows in soils during the dynamic condition. (Please note that the “dynamic” in this section does not denote “vibration”; it just means a non-steady state.) When pore water flows in the soil, it changes the excess pore pressure and, ultimately, changes the effective stress. Pore water flow is a time-dependent behavior; therefore, it causes another kind of rate dependency. For the present, let us disregard the viscosity-related rate dependency. We know that the slower loading rate will generate smaller excess pore pressure (because there is ample time for pore pressure dissipation), larger effective stress, and ultimately greater shear strength. This result is a phenomenon that is exactly opposite to the viscosity- related rate dependency. Therefore, one should note that the rate dependency caused by the pore water flow is opposite to that caused by the viscosity . For the correct evaluation of rate dependency of saturated soils, one should use a coupled rate dependency equation that is nothing other than the coupling of (4.118) and (4.125). Detailed derivation of (4.125) was addressed in Sect. 3.2 (v w − v s ) = −

1 ws K (grad(P w ) − ρw b) γw

(4.125)

4.5.5 Damage of Solid Grains Damage of solid grains may be caused by the micro-failure of contact points or by micro-cracks. Clearly, such damage causes the reduction of modulus and affects the behavior of soils. Typically, sandy soils show more pronounced change in behavior from damage (perhaps the primary loading carrying capacity of sands comes from the stresses at the grain contacts.), and most research for soil damage is focused on sandy soils. Damage is incorporated with the gradient-implemented anisotropic modified Cam Clay model with visco-plasticity by a rather simple approach. Damage strain rate ε˙dij is expressed similarly to the plastic strain as follows: ∂g ε˙dij = λ˙ d ∂σij

(4.126)

The Damage potential g may be defined as follows: ˜ − Γ) ˜ :P ˜ : (Y ˜ − Γ) ˜ −1≤0 g = (Y

(4.127)

74


˜ is the damage conjugate force and Γ ˜ is similar to the back stress The term Y ˜ is the gradient dependent kinematic hardening of damage. Using the αij . Γ above damage function and from Voyiadjis and Deliktas (2000), the damage multiplier for damage λ˙ d is expressed as follows; λ˙ d =

∂g ∂g ˜ ∂ Y˜ ∂φ

+

∂g ∂σ ∂g d ∂K C

: σ˙ +

∂g ∂Γ

: B d : Ad

.

(4.128)

Equation (4.128) is a thermodynamics-based damage formulation. However, note that another popular approach for the damage for geo-materials called the “disturbed state” concept by Desai and Zhang (1998) and Desai et al. (1996). A simpler version of “disturbed state” damage formulation by Katti et al.(1999, 2000) is adopted in this study. The damage function g is assumed to be the same as the yield function f , and we further assumed the additive decomposition of d λ = d λp + d λd . Katti and Yazdani (2001) also found that the damage of sand particles depends primarily on the confining pressure. They suggested the following function for the scalar form of d λd : d λd = (1 − A(p)) dλ A2 A = 1 − e −A1 ∗p

(4.129) (4.130)

where A1 and A2 are material parameters. Combining the above equations with d λ = d λvp + d λd for viscoplastic material, one obtains the following equation: d λvp (4.131) dλ = A(p) The above equation assumes isotropic damage and does not differentiate between the damage of micro-structure and macro-structure. This equation is very easy to use. There is perhaps some doubt that the analysis of damage for sand can be applied to clays; however, we will assume that the fundamental mechanism of damage for sands and clays is similar. 4.5.6 Summary of Micro-mechanisms When a material is subjected to large deformation, substructure changes may take place. For this reason, substructure change needs to be taken into account for a more accurate analysis of the behavior of geo-materials. This substructure change is minimal in small strain problems such as linear elastic problems. However, it becomes very important for finite strains. The substructure change is caused by the external work energy, and thus it should be taken into account for the correct evaluation of the behavior of materials. For geo-materials, this behavior is more prominent because the bonding forces between the particles are relatively weak compared to other materials


75

such as steel. Also, the strain range for some geo technical problems such as post-failure problems, cone penetration tests, and so forth, are several tens to several hundreds percent in magnitude. Thus, severe substructure changes can take place. Another main characteristic of geo-materials is the anisotropy. As a consequence of the nature of the deposition of granular materials, inherent anisotropy exists in geo-materials. During deformation, the material undergoes an induced anisotropy because of the substructure change of the material. This change means that the initial anisotrophy of geo-materials evolves with the strain. Thus a valid assessment of the anisotropy of the geo-materials should also incorporate substructure changes. Comprehensive understanding of this substructure change and anisotropy is not yet well known. However, intensive experimental and theoretical studies have been performed by many researchers (see Masad et al. 1998; Anandarajah et al. 1996; Anandarajah 1995; Anandarajah and Kuganenthira 1994, 1995). Masad et al. (1998) showed a clear relationship between the microstructures and induced anisotropy, and presented an internal plastic energy dissipation formulation to account for fabric re-arrangement. Anandarajah and Kuganenthira (1995, 1994) presented experimental results that indicate the change of anisotropy during shear is accompanied by the microstructural change. This microstructural change may be illustrated by the rotation or realignment of the soil particles. The microstructural change is inevitably accompanied by the new arrangement of the inter-particle attraction and repulsion forces as pointed by Anandarajah (1995). This change of interparticle forces is caused by the so called “embedded stress,” and quite often this stress is referred to as the back stress. Thus, one may conclude that the change of anisotropy is related to the back stress. The embedded residual stress energy is part of the applied energy, and thus the constitutive equation must consider these terms for the correct equilibrium conditions. Application of this concept to constitutive relationship has been performed by many researchers (for example, Lee et al. 1983; Prat and Bazant 1990). This book presents a theoretical basis for the consideration of the substructure change and anisotropy. It also presents applications of this model by employing the anisotropic modified Cam Clay model and plastic spin. To deal with substructure change and its related anisotropy, one must incorporate the substructure level (micro-mechanical behavior of the geo-materials). There are several approaches for incorporating the micro-mechanical behavior of soils into the macro-mechanical modeling of soils, such as the micorplane approach (Prat and Bazant 1990), double slip approach (Zbib 1993), modified spin tensor (Dafalias 1998; Lee et al. 1983), micro-mechanical models (Dobry et al. 1991a, 1991b), and so forth. Even though the atomic or molecular level approach is the ultimate goal for implementing the micro-behavior

76


Yo, Yn, Yn+1 Vn+1 n+1 configuration Vn n configuration Vo o configuration

Xo, Xn, Xn+1 Zo, Zn, Zn+1 Fig. 4.28. Updated Lagrangian reference frame

of the material, the previously mentioned methods are computationally feasible at this time (computation time, numerical error accumulation).

4.6 Equation of Equilibrium of the External and Internal Forces in an Updated Lagrangian Reference Frame To avoid the large numerical errors caused by the large distortion and rotation, this book uses an updated Lagrangian reference frame. (Beginning geotechnical engineers should note that the difference between the engineering stress and true stress is significant when the deformation is large. To overcome this numerical dilemma, one can calculate the stresses for a small incremental deformation and repeat the calculation to the final deformation level. This kind of calculation is called an incremental scheme, and when one uses it together with the traditional coordinate system, this approach is called the updated Lagrangian frame.) A schematic diagram of the updated Lagrangian reference frame is shown in Fig. 4.28. As shown in Fig. 4.28, the incremental scheme is used and the configuration is updated at every calculation step. In this chapter, the basic coupled equilibrium equations for finite strain with plastic spin is derived. The equilibrium equations addressing other micro-mechanisms are discussed in individual section in Chap. 6. Following Bathe (1996), the principle of virtual work in an updated Lagrangian reference frame is obtained by (4.132),

4.6 Equation of Equilibrium of the External and Internal Forces

n+1

n+1 S AB δ n+1 εAB n n

R=

dn V

77

(4.132)

nV

where n V is the volume of the element at the nth configuration, n+1 n S AB is the second Piola–Kirchoff stress from nth to n + 1th configuration, δnn+1 εAB is the increment of Green–Lagrangian strain from nth to (n+1)th configuration, and n+1 R is the external force at the (n + 1)th configuration. Equation (4.132) can be now expressed as follows, n+1 R= (n σAB + ∆n SAB )δ(n eAB + n ηAB )dn V nV

(n σAB δ(n eAB + n ηAB )dn V

= nV

∆n SAB δ(n eAB + n ηAB )dn V

+

(4.133)

nV

where ∆n SAB is the increment of the second Piola–Kirchoff stress at the nth configuration, n eAB is the linear strain at the nth configuration, and n ηAB is the non-linear strain at the nth configuration. In (4.133), ∆n SAB can be expressed as t+∆t S˙ AB dt (4.134) ∆n SAB = t

where S˙ AB is the time rate of second Piola–Kirchoff stress. From Voyiadjis and Abu-Farsakh (1997), Voyiadjis (1988), and Voyiadjis and Kattan (1989), S˙ AB can be expressed as follows, considering the effective stress and the pore water pressure, and assuming the plastic spin to be zero: ∗ s s S˙ AB = DABCD P˙w δab ε˙CD + J s XA,a XB,b

(4.135)

The superscript “s” is used to distinguish the stress and strains of soils from ∗ is expressed as follows: those of water. In (4.135), DABCD s s s ∗ = Dabcd − σ cb δad − σ ac δbd + σ ab δcd + Pw δab δbd DABCD s s s s s −2P w δac δbd J XA,a XB,b XC,c XD,d (4.136) The symbol “ ” is used to describe the effective stress. In (4.135) and ∗ s s is the modified elasto-plastic modulus, XA,a = ∂ n+1 XA / (4.136), DABCD n s ˙ ∂ Xa , ε˙CD , is the strain rate, Pw is the pore water pressure, P w is the pore pressure rate, Dabcd is the elasto-plastic modulus, and J is the Jacobian. In this chapter, kinematic hardening is coupled with anisotropic modified Cam Clay model, and plastic spin is incorporated. Accounting for the plastic spin, (4.135) is modified such that

78


s s s s s S˙ AB = [Dabcd − σ cb δad − σ ac δbd + σ ab δcd + Pw δab δbd − 2P w δac δbd ]J XA,a

s s s s s s s XB,b XC,c XD,d ε˙CD + [σ mb Wma + σ aj Wjb ]J XA,a XB,b + J s X s X s P˙w δab s

A,a

s

s

B,b

(4.137)

Equation (4.137) is expressed in a simpler form such that ∗ ∗∗ s s S˙ AB = DABCD P˙w δab ε˙CD + DABCD ε˙CD + J s XA,a XB,b

(4.138)

∗ where DABCD is defined by (4.135). ∗∗ DABCD is explained as follows. The relation between the plastic spin tensor and the backstress tensor as given by Dafalias (1983, 1985), Lee et al. (1983), Paulun and Pecherski (1985), Voyiadjis and Kattan (1989,1990,1991) is shown below, ¯ ¯ s − d s α) (4.139) W s = ξ(αd

where, ξ is a function of the plastic strain. Paulun and Pecherski (1987) expressed ξ as follows, 3 3ε2eq ξ= ε˙eq (4.140) 2 1 + 3ε2eq where, εeq and ε˙eq are the equivalent plastic strain and its rate, respectively. The strain rate d s represents the strain rate of solid grains; it can be decomposed as shown below, (4.141) dsmn = dsmn + dsmn

where, dsmn is the total strain rate, dsmn is the elastic strain rate, and dsmn is the plastic strain rate. dsmn is the same as ε˙smn ; however, it is denoted as dsmn for consistency with most notations implying strain rates for finite deformations. Lower case subscripts indicate the spatial coordinate system while upper case subscripts indicate the material coordinate system. The plastic component of the strain rate can be expressed as follows,

dsij =

L nij H

(4.142)

where L is σijs nij (the back stress is already reflected in σijs ), H is the hardening modulus, and nij is the normal to the yield surface. For the elastic behavior, one obtains s dsmn (4.143) σkls = Eklmn

s where Eklmn is the modulus of elasticity corresponding to the soil skeleton, as expressed in

E klmn = Λs δkl δmn + Gs (δkm δln + δkn δlm ) s

(4.144)

where Λs and Gs are Lame’s constants for the soil skeleton. Using equations (4.141), (4.142), and (4.143), one obtains (4.145):


s σkls = Eklmn dsmn −

L n mn H

79

(4.145)

Taking the inner product of the corotational stress tensor with the normal to the yield surface σkls nkl , one obtains

s σkls nkl = Eklmn dsmn nkl −

s Lnmn nkl Eklmn H

(4.146)

However, σkls nkl = L, thus (4.144) is rewritten into the following form:

s s nkl nmn ) = H Eklmn dsmn nkl L(H + Eklmn

(4.147)

One can now solve for L:

s dmn nkl H Eklmn s H + Eabcd nab ncd

L=

(4.148a)

s = cEklmn dsmn nkl

(4.148b)

In (4.148b), c is given as follows: c=

H s H + Eabcd nab ncd

(4.149)

Substituting (4.148b) into (4.142), one obtains

dsij =

s dsmn nkl cEklmn nij H

(4.150)

Equation (4.150) may be rewritten in a simpler form as follows,

dsij = Mijmn dsmn

(4.151)

s where M = cEklmn nkl nij /H . Substituting (4.151) into (4.149) gives the following relation,

W s = ξ(¯ αam dsmb − dsan α ¯ nb )

(4.152a)

− Mancd dscd α ¯ nb ) s Mancd α ¯ nb )dcd

ξ(¯ αam Mmbcd dscd

= = ξ(¯ αam Mmbcd − = ξNabcd dscd

(4.152b) (4.152c) (4.153)

s s where Nabcd is defined as (αam Mmbcd −Mancd αnb ). The expression [σmb Wma + s s σaj Wjb ] in (4.137) is obtained as follows:

s s s Wma + σajs Wjb σmb

s ξNmacd dscd + σajs ξNjbcd dscd (4.154a) = σmb

s Nmacd + σajs Njbcd ]dscd = ξ[σmb

(4.154b)

80


By making use of the relation dscd = XC,c XD,d ε˙sCD and multiplying both sides of the equation by J XA,a XB,b , expression equation (4.154b) may be rewritten in the following form,

s s s [σmb Wma + σajs Wjb ]JXA,a XB,b

s = ξ[σmb Nmacd + σajs Njbcd ](JXA,a XB,b XC,c XD,d )ε˙sCD ∗∗ = DABCD ε˙sCD

(4.155a) (4.155b)

∗∗ where DABCD is defined as

∗∗ s DABCD = ξ[σmb Nmacd + σajs Njbcd ](JXA,a XB,b XC,c XD,d )

(4.156)

From (4.156) and (4.137), one obtains (4.157), ∗∗∗ s s S˙ AB = DABCD P˙w δab ε˙sAB + J s XA,a XB,b

(4.157)

∗∗∗ ∗ ∗∗ where DABCD is defined as DABCD + DABCD . Substituting (4.157) into (4.134), one obtains the new expression for the second Piola–Kirchoff stress:

∗∗∗ ∆n SAB = DABCD

t+∆t

t+∆t s s P˙w δab dt (4.158a) J s XA,a XB,b

ε˙sAB dt + t

t

∗∗∗ s s = DABCD ∆εsCD + J s XA,a XB,b ∆Pw δab

(4.158b)

Substituting (4.158b) into (4.134) and neglecting the ∆η · η term (because it is very small), one obtains (4.1.61): n+1 s R= (n σAB + n Pw δAB )δ n eAB dn V nV

s (n σAB + n Pw δAB )δ n ηAB dn V

+ nV

+

∗∗∗ DABCD (∆n eAB + ∆n ηAB )δ n eAB dn V

nV

+

∗∗∗ DABCD (∆n eAB )δ n ηAB dn V

nV

+

s s J s XA,a XB,b δ ∆Pw (n eAB + n ηAB )dn V w ab

(4.1.61 = 4.6.28)

nV

Equation (4.6.28) is the equation of equilibrium of the external and internal forces in an updated Lagrangian reference frame expressed in terms of the effective stress and pore water pressure.


81

So far, the relationship between the pore water pressure and the hydraulic conductivity has not been shown. The Prevost (1980) coupled theory of mixtures that was derived in Sect. 3.2 is used here as follows, nw ws s ∂Pw ∂ s−1 s−1 s s ε˙ij − JCij Cij XD,a × KAB Xa,A − ρw BB =0 J s Cij ∂Xp ρw ∂XB (4.159) s s s s where Cij = XK,I XK,I , ε˙ij , is the strain rate tensor, Xa,A = ∂ n+1 Xa /∂ n XA , s Bb = bb /X b,B , and J is the Jacobian. Using (4.6.28) and (4.159), one obtains the coupling of the stress, the deformation, the pore water pressure, and the hydraulic conductivity. In a matrix form, the coupling of (4.6.28) and (4.159) is expressed as shown below: − nΩ ∆U nK nΦ (4.160) = ∆W −n Ωt n Ψ δt nΠ

In equations (4.160), n K is the stiffness matrix, n Ω is the coupling matrix, is the flow vector, ∆U is the incremental nodal displacement, ∆W is the incremental pore water pressure, and t is the incremental time. More details of (4.160) are given below: nΨ

nK nK L

= n K L + n K nL + n K TnL + n K s = n B TL D ∗∗∗ n B L dn V

(4.161) (4.162)

nv

n K nL

=

B TL D ∗∗∗ n B N L dn V

(4.163)

∗T n ∗ n n B N L σb B N L d V

(4.164)

nv

nK

s

= nv

B ∗N L is the geometric nonlinear strain displacement matrix ∆U = incremental nodal displacement Ω = J s X sA,a X sB,b (n B TL + n B Tnl )N ab dn V (4.165) n nV

N = mN , m T = {1, 1, 0} ∆W = incremental nodal pore water pressure

(4.166)

82


nΦ =

n+1

R−

s−1 ws n J s (nw /ρw )C s−1 ij C ij n K AB N ,A N,B d V

= nV

nΠ

(4.167)

nV

nΨ

Tn n n B L σd V

(4.168)

= δtG − δtΨW n +

G=−

qn P w dn V

(4.169)

nS

ws s−1 n J s n w C s−1 ij Cij n K AB N ,A N ,B d V

(4.170)

nV

Thus, by solving (4.160), one can predict the behavior of the soil using an anisotropic modified Cam Clay model that is coupled with the large strain elasto-plastic constitutive equations in an updated Lagrangian reference frame that is again coupled with the theory of mixtures. Full derivations of (4.160) are shown in Sect. 5.1.

5 Finite Element Formulations

In solving finite strain problems, typical numerical issues are related to the difference between the initial configuration and the final configuration. The formulations that are based on the initial configuration are no longer valid. Typical example is the difference between the “engineering stress” and “true stress”. For infinitesimal strain problems, the engineering stress and true stress are reasonably the same. However, for finite strain problems, these two stresses may be quite different. To overcome these difficulties, two reference frames are used. One is an updated Lagrangian reference frame and another is an Eulerian reference frame. In geotechnical engineering, both formulations are used. In this book, an updated Lagrangian reference frame is used for the convenience of formulation as discussed in Sect. 4.6.

5.1 Updated Lagrangian Reference Sheme The finite element discretization is used here for the displacement u and the pore water pressure Pw as follows, u = h·U Pw = N · W

(4.1) (4.2)

where h is the displacement shape function, N is the pore water pressure shape functions, U is the nodal displacement, and W is the nodal pore water pressure. The linear and nonlinear strains may be expressed as follows: e = BL · U 1 η = BN L · U 2

(5.3a) (5.3b)

The linear and nonlinear strain variations are given as follows: δe = BL · δU 1 δη = BN L · δU 2 The pore water pressure gradient is given such that

(5.4a) (5.4b)

84


∂Pw = N ,B W ∂XB

(5.4c)

In (5.4), BL and BNL are the linear and nonlinear strain-displacement matrices. Substitution of (4.2) and (5.4c) into (4.159), yields δU T (n K L +n K N L +n K TN L +n K s )∆U − δU T

n Ω∆W

= δU T n Φ (5.5)

The above equation is valid for any δU T ; therefore one obtains (n K L +n K N L +n K TN L +n K s )∆U −n Ω∆W =n Φ

(5.6)

Making use of the following expression, nK

= (n K L +n K N L +n K TN L +n K s )

Equation (5.6) is expressed as follows: n K∆U

−n Ω∆W =n Φ

(5.7)

The components of n K are expressed as follows, T ∗∗∗ nB L dn V (linear stiffness matrix) (5.8) nK L = nBL D nV T ∗∗∗ nB N L dn V (non-linear stiffness matrix) (5.9) nK N L = nBL D nV s n nK = n C N L d V (non-linear geometric stiffness matrix) (5.10) nV

where CNL is defined as; nC N L

n =n B ∗T N L σn B N L

in (5.6) [See also (4.165)] is expressed as s s T ¯ n J s XA,a XB,b (n BLT + n BN nΩ = L )Nab d V

(5.11)

nΩ

(coupling matrix) (5.12)

nV

where ¯ = mN N

(5.13)

and m = {1 1 0} for two dimensions, and m dimensions such that σ = σ + mPw T

nΦ

T

= {1 1 1 0 0 0} for three (5.14)

in (5.7) is expressed as follows: n+1 R− nΦ =

Tn nBL nV

σ dn V

(5.15)

5.1 Updated Lagrangian Reference Sheme

85

From (3.41) and using Galerkin’s weighted residual method requires that s −1 s −1 s −1 ε˙ij − J s Cij Cij J s Cij nV w n ∂Pw ∂ WS P¯w dn V = 0 × KAB − ρw BB (5.16) ∂XA γw ∂XB The weak form of above equation is obtained by applying Green’s theory (Zienkiewicz and Pande 1977) as follows, nw s s−1 s−1 W S ∂Pw s−1 ε˙ij P¯w dn V − J CRS CRS KAB − ρw BB J s Cij ∂XB nV nV γw ¯ ∂ Pw n × d V − qn P¯w dn A = 0 (5.17) ∂XA nS where P¯w = N w ¯

(5.18)

is the weighted residual (virtual pore pressure). The pressure gradient is given as follows: ∂ P¯w = N,A w ¯ (5.19) ∂XA qn is the seepage velocity normal to the boundary surface. Substituting (5.1.4) and (5.19) into (5.17), one obtains −n Ω T u˙ + n ΨW = G − qn P¯w dn A (5.20) nS nw s s −1 s −1 W S Ψ = J CRS CRS KAB N,A N,B BB dn V (5.21) n nV γw Equation (5.20) can be solved by various processes of time stepping as follows, un+1 = un + δtn u˙ n+β

(5.22)

un+β = (1 − β)u˙ n + β u˙ n+1

(5.23)

where such that a particular value of β corresponds to a particular integration rule. For example, β = 0 corresponds to a forward difference integration, β = 1/2 corresponds to a linear variation and to the trapezoidal integration, and β = 1 corresponds to a backward difference integration. Considering the stability of the numerical time integration scheme, it reveals that for stability, β ≥ 1/2 is required (Prevost, 1981). Here the backward difference scheme is adapted with β = 1. Therefore (5.22) becomes un+1 = un + δtn u˙ n+1

(5.24)

86


Equation (5.20) maybe expressed as follows: −n Ω δtu˙ T

n+1

+ n Ψ δtn W

n+1

qn P¯w dn A

= δtn G − δt

(5.25)

nS

Utilizing the following relationships, ∆U = δtn u˙ n+1

(5.26)

n+1

(5.27)

W

n

= W + ∆W

Equation (5.25) can be rewitten as −n Ω T ∆U + n Ψ δt∆W = n Π where n Π is defined as nΠ

(5.28)

qn P¯w dn A

= δtn G − δtn ΨW n −

(5.29)

nS

Assembling (5.7) and (5.28), one obtains the following matrix form of coupled equations for the two-phase media: −n Ω ∆U nK nΦ (5.30) = ∆W −n Ω t δtn Ψ nΠ

5.2 Finite Element Implementation The proposed coupled system of equations derived earlier is implemented into the finite element program CS-Soil. This program is used to solve a large strain non-linear behavior of soils. In this process, element matrices are derived for 8-noded isoparametric plane strain element Q8P4. The use of the isoparametric elements has the advantage of describing the curved boundaries in the deformed configuraions. The shape functions N for the 8-noded isoparametric element are given as follows: h 1 0 h2 0 h3 0 h4 0 h5 0 h6 0 h7 0 h8 0 (5.31) h= 0 h1 0 h 2 0 h 3 0 h 4 0 h 5 0 h 6 0 h 7 0 h 8 The element displacements are related to the nodal displacement using the shape functions h as follows: ui =

8

(k)

hk ui

(5.32)

k=1

The linear and nonlinear incremental strains are related to the displacement derivatives as follows:

5.2 Finite Element Implementation n eAB n η AB

1 (n uA,B + n uB,A ) 2 1 = (n uK,A − n uK,B ) 2 =

87

(5.33) (5.34)

Because the shape functions h are expressed in terms of local coordinates r and s, a chain rule is applied in order to refer the displacement derivatives in terms of the global coordinates. The chain rule implies the following,     ∂h  ∂h        ∂nX     ∂r 1 = [J] (5.35)   ∂h  ∂h          ∂nX 2 ∂s where [J ] =

J11

J12

J21

J22



∂ n X1  ∂r =  ∂ n X1 ∂s

 ∂ n X2 ∂r   n ∂ X2  ∂s

The inverse of (5.35) gives the following relations:     ∂h  ∂h        ∂nX   ∂r   1 −1 = [J ]   ∂h  ∂h          n ∂ X2 ∂s

(5.36)

(5.37)

The displacement derivatives with respect to the global coordinates are given as follows, 8 ∂ui ∂hk (k) = U (i = 1, 2; j = 1, 2) (5.38) n ∂ Xj ∂ n Xj k=1

where

∂hk ∂hk ∂hk + J −1 = J −1 j1 j2 ∂ n Xj ∂r ∂s

(5.39)

By using the previous expressions and the nodal displacements, the deformation gradient matrix F for the increment displacement can be obtained as follows:   ∂u ∂u   n 1 n 1 

  ∂ X1 ∂ X2  ∂ui (5.40) = F = ∂u ∂u   ∂ n Xj    n 2 n 2  ∂ X1 ∂ X2 Once the deformation gradient matrix is obtained, the linear and nonlinear incremental strains can be computed. Using the definition of (5.33), the linear strain displacement matrix BL is expressed as follows:

88


n  [B L ] =  

h1,1 n

0 n

n

0

h2,1

h1,2

n

0 n

0

h3,1 . . . n h7,1

h2,2

0

...

n

0

n

0

h8,1

h7,2

0

 h82   (5.41) n h8,1 n

0

h1,2 n h1,1 n h2,2 n h2,1 n h3,2 . . . n h7,2 n h7,1

n



h82

In accordance with (5.41), the nonlinear strain-displacement matrix BNL may be expressed as follows, (5.42) BN L = G · Q · H where

n

 [G] =  

h1,1 0

n

n

0 n

h1,1

h1,2

n

0 n

0

h2,1

h1,2

0

. . . n h8,1

0 n

h2,2 . . .

n

0 n

0

h8,1

h8,2

h1,2 n h1,1 n h1,2 n h1,1 n h2,2 n h2,1 . . . n h8,2 n h8,1



 h82   n h8,1 n

0 n

0

h82

(5.43) n

u11 0

  0  [Q]T =   0  0

n

0

0

n 2 u1

0

0

0 . . . n u81 0

n 1 u1

0

0

0

n 2 u1

0

0

0 ...

0

n 1 u2

0

0

0

n 2 u2

0

0

0

0

0

n 1 u2

0

0

0

n 2 u2

0

0

h1,1

0

n

h2,1

0

. . . n h7,1

0

n  h1,2 0 n h2,2 0 . . . n h7,2 0  [H] =   0 n h1,1 0 n h2,1 . . . 0 n h7,1  n

0

h1,2

n

0

h2,2 . . .

n

0

n

0



 0    n 8 . . . u2 0   0 . . . n u82

n 8 u1

0

h8,1

0

n

h7,2

0

h82 0 0

(5.44)



    n h8,1   n h8,2 0

(5.45)

The nonlinear matrix n CNL is given as follows: [n C N L ] = [n B ∗

∗ T N L ] [n σ][n B N L ]

(5.46)

The geometric nonlinear strain displacement matrix [B∗NL ] may be arranged in the following form,  n h1,1 0 n h2,1 0 . . . n h7,1 0 n h8,1 0  n  h1,2 0 n h2,2 0 . . . n h7,2 0 n h82 0    (5.47) [B ∗N L ] =    0 n h1,1 0 n h2,1 . . . 0 n h7,1 0 n h8,1    0

n

h1,2

0

and the stress [n σ] is given by

n

h2,2 . . .

0

n

h7,2

0

n

h8,2

5.3 Remeshing and Return to Yield Surface

n

σ11 n  σ21  [n σ] =   0  0

n

σ21

0

n

σ22

0

0

n

σ11

0

n

σ21

0

89



 0    n σ21   n σ22

(5.48)

5.3 Remeshing and Return to Yield Surface In order to solve the system of nonlinear equations that arises from the mathematical formulation of the coupled equations, a finite strain problem is processed incrementally and iterations are performed within each incremental penetration. The full Newton-Raphson iterative method is used in order to obtain convergency. An abridged algorithm procedure is shown in this section. Full details of the algorithm procedure can be referred to Abu-Farsakh (1997) and Kiousis (1985). Incremental Loop The total penetration length is divided into smaller increments. At the beginning of each penetration increment, the incremental displacement ∆Uappl is computed by dividing the total penetration length by the number of increments. The Newton-Raphson iterative procedure is carried out within each increment in order to solve for the incremental load Rinc and excess pore pressures ∆Pw . Iterative Loop The applied iterative incremental load, Riter, for the first iteration is given by: Riter = Rinc The Newton–Rapson iteration loop is then carried out as described by the following steps: (1)

(2) (3) (4) (5)

Convert the iterative applied loads, Riter, to account for the skew boundaries, such that the degrees of freedom at the skew boundary nodes are normal and tangential to the skew boundary. Loop over the whole elements. Assemble the global stiffness matrix. Use a linear solver to solve the nonlinear equations for the iterative incremental scheme. Rotate back the iterative incremental displacements and loads at the skew boundaries to the original coordinate system.

90

(6) (7)

(8)

(9) (10)


Add the iterative incremental displacement to the previously computed quantities. Compute the Lagrangian iterative incremental strains ∆εi with respect to the previous configuration from the iterative incremental displacements ∆Ui . Compute the iterative incremental stresses ∆σ i using the subincrementation technique and applying certain corrections due to crossing the yield surface and the return to the yield surface [See Abu-Farsakh (1997) and Kiousis (1985) for details]. Update the constitutive matrix D and calculate the equilibrating forces for the element stresses (Requil ). Calculate the out-of-balance (the corrected) load vector from the accumulated applied load vector Rappl and the equilibrium load vector Reqt as follows: Rcor = Rappl − Requil

(11) Check the convergence of the solution using an appropriate convergence criterion. In this book, the displacement criterion is adapted as follows: ) ) uiter uiter ≤ tol uinc uinc where tol is the tolerence. If convergence does not meet, repeat the iterative steps 1 through 11. If convergence meets the criteria, then proceed to step 12. (12) Update the nodal coordinates by adding the incremental nodal displacements. (13) Move to the next load increment until the total load is applied.

6 Applications

6.1 Piezocone Penetration Test (PCPT) 6.1.1 Introduction A cone penetrometer is a device that is pushed into the ground at a constant rate to measure ground responses continuously. Typical measurements are made for tip resistance, friction resistance, and pore pressure response. From these measured quantities, the properties of ground are determined directly or indirectly. Crude types of penetrometers were introduced as early as the Roman era; at that time, the number of slaves required to push the rod into the ground was counted and used to quantify the strength of the ground. With the advent of modern science, this method was diversified and several different penetrometers were developed, such as the cone penetrometer, the standard penetrometer, the Swedish penetrometer, and others. Among these penetrometers, the cone penetrometer becames one of the widely used penetrometers because of its superior performance (repeatability, convenience, economy, and so forth). A modern cone penetrometer, the mechanical cone penetrometer, was first introduced in the early 20th century in Europe. The mechanical cone penetrometer pushed the cone into the ground by a mechanical driving system (chain + gear system), and measured the end resistance by probing rings. Later, a cone penetrometer that could simultaneously measure both end bearing and friction resistance appeared, and it was called the mantle cone penetrometer. Holland, which contributed much to the development of the modern cone penetrometer (including its unofficial but internationally known standard name, “Dutch Cone”) perhaps used the cone penetrometer most widely in earlier days. The early “Dutch Cone” was not equipped with modern sensors and automatic driving systems, but it presented more consistent and reliable results with less cost compared to its strong cousin, the SPT (Standard Penetration Test); thus, it continued to evolve with continued use. Traditional mechanical cone penetrometers use the double rod system – inner rods and outer rods. Measurement was carried out by pushing the inner rods for resistance measurements, and subsequently by pushing the outer rods for the

92

6 Applications

advancement of the whole system. Thus, the test procedure was not continuous; it was a “stop and go” process. With the advent of the modern sensors and electric technology, electric cone penetrometers were introduced in the 1970’s (Torstensson, 1975), a development that resulted in the greatly increased productivity and overall performance. The electric cone penetrometer used load cells and a motorized driving system instead of probing rings and a manual driving system; thus, readings could be recorded electronically without stopping the penetration process. The penetration speed also could be controlled more accurately. The piezocone is a type of electric cone penetrometer which, by virture of having pore pressure monitoring capability, can enhance the assessment of engineering parameters, especially the hydraulic properties of soils. Measurement of the pore water pressure generated while advancing a cone tip into the ground (and its subsequent dissipation when the penetration stops) was first made in Sweden in the early 1970’s (Wissa et al., 1975; Torstensson 1975). Early types of piezocone penetrometers did not have the ability to measure both cone resistance (end bearing and/or friction) and pore pressure simultaneously. Subsequent developments in transducer technology during the early 1980’s involved the incorporation of piezometric elements into the standard electric cone penetrometers. This development made it possible to measure pore pressure, cone resistance, and skin friction simultaneously. Tumay et al. (1981) is known as the first group who utilized the simultaneous measurement of cone resistance and pore pressure (Zuidberg et al., 1982). Later, many researchers contributed the valuable application of the simultaneous measurement of cone resistance and pore pressure (Baligh et al., 1981; Campanella and Robertson, 1981; Muromachi, 1981; de Ruiter, 1981; Zuidberg et al., 1982; Smiths, 1982; Lunne et al., 1997) , and opened the era of the fully equipped piezocone penetrometer. At present, many other sensors can be attatched to the cone probe simultaneously in order to obtain the various soil properties. Accelerometers can be attatched to detect the seismic response of soils (Campanella, 1994). Electric resistivity sensors, thermal sensors or infrared sensors can be attatched to detect ground contamination. Microphones may also be attatched to detect the sonic response of the ground, while the radioactive sensors can be attatched to detect the radioactive materials in the ground (Muromachi, 1981; Lunne et al., 1997). Recent developments include attaching a video camera and high intensity light to physically see the underground conditions (Envi, 1996). Virtually any kind of sensor can be incorporated with the modern cone penetrometers. For geotechnical purposes, the most widely used combination is two load cells for measuring the end bearing and the side friction, one piezometer for pore pressure response, and one inclination sensor for the inclination check. All of these sensors are electronic sensors, and an industrial-computer based electronic readout is adopted, so most of the measured response is recorded and analyzed on a real-time basis.

6.1 Piezocone Penetration Test (PCPT)

93

Fig. 6.1. Typical cone penetrometer tips (Courtesy of Dr. M. Tumay)

Recently, there have been several efforts to increase the efficiency of the piezocone penetration test (PCPT) even further by mechanical improvement. Tumay et al. (1998) and Tumay and Kurup (1999) devised a spiral rod penetrometer system for non-inturrupted continuous advancing of the cone tip, a development that eliminates time for connecting push rods. This same technique was used for underwater ground exploration at the sea bottom. Envi. Corp. (1996) developed the wireless piezocone penetrometer, which does not require the hassle of wiring during PCPT. Mayne and Rix (1996) developed the combined system of pressure meter and cone penetrometer for the simultaneous testing of the pressure meter and cone penetrometer. Typical cone penetrometer tips are shown in Fig. 6.1. Measuring sensors are built into the cone body. The white rings shown in Fig. 6.1 are the porous elements that protect piezometers. To improve the mobility of the cone penetrometer system, the system is typically mounted on all-wheel driven trucks or crawler-equipped vehicles. For off-shore application, the systems are mounted on the jack-up pontoons or barges. These vehicles or barges have their own power supply and data logging system for real time data processing. Figure 6.2 shows a typical truck-mounted driving system and data-collecting system that are currently used in LTRC (Louisiana Transportation Research Center). 6.1.2 Current Practice of Determining Hydraulic Properties from the Piezocone Penetrometer Determination or estimation of ground properties from the piezocone penetration tests is made by using empirical equations or theoretical analyses.

94

6 Applications

Fig. 6.2. Inside view of a truck-mounted driving and data collection system (Courtesy of Dr. M. Tumay, 1999)

Stress strain parameters are typically obtained from cone tip resistance and friction resistance. Hydraulic properties are typically obtained from the pore pressure dissipation response. Measurements of pore water pressures generated while advancing a probe into the ground and their subsequent dissipation were first made in the early 1970’s. The determination of the coefficient of consolidation and hydraulic conductivity from PCPT utilizes the pore pressure dissipation test data obtained during the stopping period of the Piezocone (Acar et al., 1982; Tumay and Acar, 1985). However, the above method has two major explicit drawbacks which make the required time for dissipation test very long, therefore making the continuous hydraulic conductivity profile virtually impossible. Typically, it takes several hours (including idling time) for one dissipation test. Without the dissipation test, the whole procedure for one piezocone penetration test takes 1.5 to 2 hours for a 30 m penetration. For this reason, the dissipation test greatly decreases the efficiency of the piezocone penetration test. Moreover, because of the prolonged waiting time for the dissipation test, only a limited number of dissipation tests can be carried out. As a result, continuous hydraulic conductivity profiles cannot be obtained even though other profiles (cone resistance, friction resistance, etc.) are practically continuous. This chapter presents a new method of determining the hydraulic conductivity of soils that utilizes the coupled theory of mixtures but does not require dissipation tests; hence, not only does it not require additional test time, but it potentially provides a continuous hydraulic conductivity profile.


95

It has been argued that the piezocone obtained hydraulic property is for the disturbed condition, not for the intact condition (Lunne et al., 1985). It is believed that the change of soil structures may affect the hydraulic conductivity. Previous research (Baligh and Levadoux, 1986; Robertson et al., 1992), however, had shown reasonable agreement between the laboratory test results and the piezocone dissipation test results. In general, it is believed that the hydraulic properties obtained from the piezocone penetration test are enough close to the field values. Another aspect of this chapter is the development of analytical tools for piezocone penetration tests. Along with the significant evolution of mechanical aspects of PCPT, there were also great achievements in the analysis method for the PCPT results. However, many of these works concentrated on the interpretation of stress-strain properties of soils. From the point of utilization of the penetration pore pressure response, most of the efforts were focused on the correction of end bearing and side friction, or on soil classification. Relatively fewer efforts were focused on the direct interpretation of penetration pore pressure response itself. This chapter focuses on the transient pore pressure response during the penetration, thereby extracting valuable information that has been overlooked in past research. 6.1.3 New Approaches Two new approaches for estimating hydraulic properties are presented in this section. The first method utilizes only one pore pressure data and it is termed the one-point method (OPM). The second method utilizes two pore pressure data and it is termed the two-point method (TPM). The one-point method is based on the idea that a measured pore pressure from PCPT is directly a function of hydraulic conductivity as follows, u = f (σ, k, . . . )

(6.1)

where, u is a pore pressure, σ is a stress tensor, and k is a hydraulic condutivity. Equation (6.1) is nothing but a conceptual expression of Biot’s coupled of theory of mixtures, and (6.2) is a working equation of this concept: −n Ω ∆U nK nΦ (6.2) = ∆W −n Ω t n Ψ δt nΠ The two-point method utilizes the pore pressure measurement results at two different locations and time so that the pore pressure dissipation effect is computed. The hydraulic conductivity is computed from the pore pressure dissipation response.

a) The One-Point Method (OPM) The deformation and consolidation of clayey soils around the advancing cone tip is essentially a large strain problem with partial drainage condi-

96

6 Applications

tions. Advanced consolidation theories incorporating the above condition were developed by Gibson et al. (1981); Schiffman (1980); Prevost (1980), and Voyiadjis and Abu-Farsakh (1997). While other formulations are based on an Eulerian reference frame, Voyiadjis and Abu-Farsakh (1997) adopted the updated Lagrangian reference frame to reduce numerical errors. The study in this book adopts the Voyiadjis and Abu-Farsakh (1997) updated Lagrangian reference frame approach. Traditional methods for estimating the hydraulic conductivity of soils from piezocone penetration tests are based on pore pressure dissipation test data (Torstensson, 1977; Baligh et al., 1980; Jamiolkowski et al., 1985; Campanella et al., 1985; Carter et al., 1979; Houlsby and Teh, 1988). The essentials of these conventional methods are suggested by Torstensson (1977). These methods are similar to solving the heat diffusion equation. The pore pressure dissipation test assumes a certain dissipation pattern (like a cylindrical or spherical heat sink in a homogeneous media in some ideal boundary conditions). Thus, the conventional method inherently has some limitations depending on the real soil conditions. The limitation of the conventional method is discussed below. The partial differential equations, boundary conditions and initial conditions for the conventional method are expressed respectively: Partial Differential Equation: ∂σ /∂t = (∂σ/∂t) − (∂u/∂t) = ch [(∂ 2 u/∂r2 + (1/R)(∂u/∂r)] + cz [(∂u/∂z2 ] (6.3) Boundary Conditions: u = 0 at r = ∞, z = ∞, and z = −∞ u = k at r = R, z = 0 (where k is a constant) Initial Condition: F = g (stress strain functions such as those obtained from the cavity expansion theory) In (6.3), σ is the effective stress, σ is the total stress, u is the excess pore pressure, t is the elapsed time, ch is the horizontal consolidation coefficient, cz is the vertical consolidation coefficient, R is the radius of the cone, r is the radial axis, and z is the vertical axis of the cylindrical coordinate system. The solution of (6.3) is not straightforward, but it can be solved using a special numerical technique such as the Crank – Nicholson technique. Torstensson (1975, 1977) simplified (6.3) by assuming negligible vertical drainage (assume the effect of cz and related term is minor) and a constant total stress (assume (∂σ/∂t) =0). Torstensson (1975, 1977) presented a convenient graphical


97

solution that is similar to that of Terzaghi’s (1943) one-dimensional consolidation solution. However, inaccurate results can also result if field conditions are not close to the assumptions given above. Gupta and Davidson (1986) used (6.3) instead of Torstensson’s simplified version (1975, 1977) with the assumed boundary conditions (given that (6.3) cannot be solved with boundary conditions such as u = 0 at r = ∞.) Although Gupta and Davidson (1986) obtained better results, they did not eliminate the fundamental drawbacks of (6.3) and its simplified solution by Torstensson (1975, 1977). The advantage of the conventional method is that because it does not require the coupled theory of mixtures, it is convenient to interprete. Torstensson (1977) also did not fully consider the different dissipation mechanisms at the cone tip, shaft and face. Levadoux and Baligh (1986) and Teh and Houlsby (1991) further improved the Torstensson (1977) method by adopting different initial pore pressure magnitudes and dissipation conditions at the cone tip, shaft, and face. Essentials of Pore Pressure Response The pore pressure response from the piezocone penetration test is not known in detail yet. Intuitively, however, we may expect that the response will follow the curves shown in Fig. 6.3. This figure shows a conceptual and a slightly exaggerated excess pore pressure response of a soil element that is located at the projected center-line of the piezocone penetrometer’s travel route. Initially, the piezocone tip is located far above this soil element, and there is no excess pore pressure. As

Fig. 6.3. Illustration of pore pressure response for a soil element during piezocone penetration (Song, 1999)

98

6 Applications

time passes, the penetrating cone tip comes closer to the soil element, and the induced stress by the penetrating cone tip gradually builds up over this soil element. This increase in the induced stress results in a build up of excess pore water pressure (an increase that may be either linear or non-linear). As the penetrating cone tip passes through this soil element, severe disturbance occurs and high excess pore pressure occurs. When the penetrating cone tip stops at this soil element, there is an immediate drop in the excess pore pressure as a result of the reduced axial force. In the mean time, the interaction of pore pressure between the near field and far field takes place (near field: location radially close to the cone tip; far field: location radially far from the cone tip), resulting in the small increase or decrease in the pore pressure. As observed previously by Voyiadjis and Abu-Farsakh (1997), the pore pressure during penetration is at a maximum at the cone face (typically known as a u1 position). Thus, if one has a porous element at the cone shoulder location (typically known as a u2 position), the measured pore pressure at during penetration will be smaller than the pore pressure at the cone face. Therefore, there will be a tendency toward a small increase of pore pressure at u2 position because of pore water inflow from the u1 position (high pore pressure position). (This explanation applies to the normally consolidated soils. The pore pressure response of an overconsolidated soil is not the same as that of a normally consolidated soil, and it is beyond the scope of this book.) We may conclude from the above discussion that the drainage condition during the cone penetration test is that of a partially drained condition and not that of the fully drained condition or fully undrained condition. Figure 6.3, also shows the difference between the assumed dissipation curve and the real dissipation curve. Comparing these two curves shows that the magnitude and initial timing of the excess pore pressure are different. Therefore, it is clear that the dissipation test is usually analyzed with an incorrect initial time and magnitude of initial pore pressure. As the drainage condition deviates more and more from the assumption of the fully undrained condition, the reliability of the assumed dissipation curve progressively decreases. Senneset et al. (1988) indirectly pointed out this aspect by questioning the validity of the magnitude of the initial pore pressure when Bq is less than 0.4 where Bq is the ratio of excess pore pressure to net cone resistance. The value Bq = 0.4 corresponds roughly to clayey silt. A small value of Bq implies a higher hydraulic conductivity. For soils with small Bq values, one can expect a substantial drop of ∆uo (initial pore pressure) as a result of pore pressure dissipation during penetration and pore water pressure interaction between the far field and the near field. Silty soils that have negative Bq values and high hydraulic conductivity exhibit negative excess pore pressure. Moreover, the validity of ∆uo for these soils is questionable. In other words, for materials of relatively high permeability, the penetration process of CPT is a partially drained condition and unlikely the fully undrained condition


99

to be valid. This deviance from the fully undrained condition may cause a substantial difference between the peaks of the two curves in Fig. 6.3. Elsworth (1993) showed that Bq varies with the coefficient of consolidation when Bq is less than 0.5. Elsworth (1993) also showed that Bq is almost constant when the coefficient of consolidation is less than a certain number. These results show that practically speaking, the undrained condition is obtained only when the coefficient of consolidation is less than a certain number. This phenomenon may again be caused by the fact that for higher permeable soils (Bq < 0.5), the validity of initial excess pore pressure is inaccurate because of the fact that in reality, the drainage condition is not that of a fully undrained condition. The results of Senneset et al. (1988) and Elsworth (1993) showed that the conventional method for determination of the hydraulic conductivity is desirable when the hydraulic conductivity is low enough and the interaction of pore pressure between the far field and near field is minimal. Another aspect of the piezocone-induced excess pore water pressure regime is that the measured penetration pore pressure is the result of simultaneous generation and dissipation. Thus, the measured pore pressures at the piezometer tip are the sum of the early-generated and dissipated pore pressures and the newly generated pore pressures. This casts some doubt on the the accuracy of the measurement of the magnitude of the initial pore pressure (Kurup and Tumay, 1997). Kurup and Tumay (1997) have shown that the dissipation of pore pressure during piezocone penetration is unavoidable and that the phenomenon results in the interference of the spatial distribution of pore pressures. Therefore, the pore pressure dissipation curve may result in a more gentle dissipation slope than a theoretically predicted one. Typically, the curvature represents the pore pressure dissipation rate. Therefore, the computed hydraulic conductivity or coefficient of consolidation may not be correct. The error induced from the above difficulties may be negligible, or it may be significant – depending on the soil condition. Therefore, an improved method for estimating the hydraulic conductivity from piezocone data is needed. New approaches to estimating the hydraulic conductivity from CPTinduced steady state excess pore pressure were proposed by Elsworth (1993, 1998), Manassero (1994) and House et al. (2001). Elsworth (1993, 1998) estimated the hydraulic properties of soils with a linear elastic soil model incorporating the dislocation method. The linear elastic soil model may show not be a perfect fit for an elastic-plastic model for the given strain. Therefore, higher or lower excess pore pressures may be obtained from the analysis. Typically, the prediction of the hydraulic conductivity based on the linear elastic soil model underpredicts the hydraulic conductivity of the soil. Elsworth (1993)’s comparison with experimental data also showed a lower predicted hydraulic conductivity than that obtained from the experimental data. Manassero (1994) correlated Bq and the hydraulic conductivity by semi-empiricism. Manassero’s correlation (1994) is obtained from a linear correlation between the given hydraulic conductivity data and the Bq coefficient.

100

6 Applications

Manassero’s method may therefore be used when one has existing data base for the hydraulic conductivity and the Bq parameter. House et al. (2001) experimentally incorporated the variable penetration speed of the piezocone penetrometer and accompanying excess pore pressure response and correlated them to the hydraulic conductivity. From the view of cost, the linear elastic model and the semi-empirical approach are quite efficient. However, considering the non-linear behavior of soils at the vicinity of the cone tip and the insufficient data base available for the empirical relationship between the hydraulic conductivity and the Bq or penetration speed, one sees the necessity of incorporating an elasto-plastic large strain approach in the analysis of the piezocone penetration test. Essentials of OPM The main objective of the OPM is to develop a method for the determination of the hydraulic conductivity that can overcome the drawbacks of the conventional dissipation test method and provide a more realistic theoretical framework. The method proposed in this study is based on the analysis of the steady state pore pressure during the piezocone penetration test so that the full interaction between the piezocone and the soil is considered. As a result, the problems related to the initial time and initial magnitude of the pore pressure discussed in the previous section are naturally resolved. The formulation of the coupled field equations for soils, using the theory of mixtures in an updated Lagrangian reference frame based on the principle of virtual work, is used in this work. The authors make use of an axi-symmetric finite element program developed for simulating the behavior of soils with the advancement of the piezocone tip. Finally, the results obtained using of the proposed method is compared with well-documented field test data as well as experimental results obtained using the Louisiana State University calibration chamber system. The piezocone penetrometer typically penetrates into the ground with the speed of 2 cm/sec, thereby inducing complete failure of soils around the cone tip. Researchers have shown that the strain at the vicinity of the cone tip ranges between ten percent to more than a hundred percent (Levadoux and Baligh 1986; Kiousis et al. 1986; Voyiadjis and Abu-Farsakh 1997). Thus the penetration of the piezocone is essentially a time-dependent large strain problem. Considering the complexity of the piezocone penetration test and the soil characterization, both the large strain theory and the visco-plasticity are desirable features for the analysis of the problem. However, in this section only the isotropic, elasto-plastic large strain approach with time dependent loading condition is adopted for the sake of simplicity, and viscositydependent time rate effects are disregarded. Also, the modified Cam clay model is used in this study to describe the plastic behavior of soils. The influence of soil compressibility on the estimation of the hydraulic conductivity is incorporated in the modified Cam clay model by using the compression index of the soil.


101

The drainage condition around the penetrating cone tip is again neither fully drained nor fully undrained. This condition is the partially drained condition or the transient flow condition. For the transient flow condition, it can be presumed that the pore pressure is a function of one or several of various factors such as hydraulic conductivity, material stiffness, and so forth. Hence, the soil must be looked upon as a multiphase material whose state is to be described by the stresses and displacements within each individual phase. In this work, Prevost (1980)’s theory of mixtures for two phase materials (saturated soil) is coupled with Terzaghi’s (1943) effective stress theory as shown by Kiousis et al. (1988) and Voyiadjis and Abu-Farsakh (1997). Voyiadjis and Abu-Farsakh (1997) implemented the coupled theory of soil-water mixtures to the effective stress and pore water pressure and derived the coupled equations of mixture in an updated Lagrangian reference frame. In this chapter, the coupling of the stress, deformation, pore water pressure, and hydraulic conductivity is obtained by using the work outlined in Voyiadjis and Abu-Farsakh (1997). (The details of the coupling equation are shown in Chap. 4.) Numerical Simulation From the work of Voyiadjis and Abu-Farsakh (1997), the coupled equations in the updated Lagrangian reference frame are obtained as shown in (6.3). The numerical simulation is conducted using an axi-symmetric finite element analysis code with the mesh shown in Fig. 6.4. The excess pore pressure at the cone face (typically known as u1 ) is used in this chapter. The piezocone penetrometer is assumed to be infinitely stiff and therefore, tensile stresses are not allowed to develop along the centerline boundaries. This assumption is reasonable because the test is assumed to take place at a depth great enough to assure that the initial stresses in the ground prevent the development of tensile stress. The continuous penetration of the piezocone is numerically simulated by applying an incremental vertical penetration rate of the cone of 2 cm/sec. This is the same rate as the piezocone penetration, and it allows the partial pore pressure dissipation during the penetration. A simple constraint approach at the nodal level of the interface is adapted in this study to account for the soil-penetrometer interface friction. An angle of friction, δ = 14◦ , is assumed between the soil and the piezocone face. During the piezocone penetration, three interface states can be identified: the fixed state, the slip state, and the free state. Solution for the new nodal interface state is determined iteratively. Based on the previous interface state (fixed, slip or free) and the loading criterion, a new interface state is assumed and solved in order to obtain a trial solution. The trial solution is then used to check if the assumed trial state is correct. If not, a new state is assumed that is more likely to be correct (Voyiadjis et al., 1998). (The input parameters used in this study are obtained from typical oedometer tests and triaxial compression tests as shown in Table 6.1.)

102

6 Applications

Fig. 6.4. Finite element mesh for the piezocone penetration analysis

At the beginning of penetration, all the nodes along the inclined conical surface and along the piezocone shaft are prevented from sliding along the surface and are forced to move vertically with the cone boundary incremental movement until the sliding potential occurs. The sliding potential is reached when the tangent frictional force (Ft ) of the node along the boundary surface reaches the allowable friction force (Ft > Fs ) given by (Voyiadjis et al. 1998) as follows: Table 6.1. Input parameters for the finite element analysis Parameter

Value

Units

Compression index, λ Recompression index, κ Initial void ratio, eo Poisson’s ratio, v Earth pressure coefficient (Ko ) Slope of critical line, M Unit Wt. of soil, γt Depth (from ground surface) Unit Wt. of water, γw

0.11 0.024 1.0 0.3 0.5 1.16 1.8 20 1.0

dimensionless dimensionless dimensionless dimensionless dimensionless dimensionless ton/m3 m ton/m3


Fs = Fka + Fkn tan δ

103

(6.4)

In (6.4), Fka is the soil-piezocone adhesion force at the load increment k, Fkn is the normal effective force at the load increment k, and δ is the angle of friction between the soil and the piezocone surface. These nodes are afterwards allowed to slide along the skew boundary surface and along the cone shaft surface. During the incremental penetration, the nodes along the boundary are continuously checked for appropriate boundary condition adjustments. Based on the previous boundary conditions at load increment (k-1), a new boundary condition state is assumed for the load increment (k), depending on the load and displacement criteria. The validity of the assumed trial boundary condition is tested before proceeding to the next loading increment. (Mesh sensitivity of this problem was tested by Abu-Farsakh et al. (1998), who demonstrated that the finite element mesh used in this present study is adequate. More details of the numerical simulation may also be found in Voyiadjis et al. (1998).)

Hydraulic Conductivity Estimation Procedure In (6.2), ∆W is the incremental excess pore pressure matrix, and n Ψ and n Π are functions of the hydraulic conductivity matrix. Thus one can solve the equations if the hydraulic conductivity is known. Computing the hydraulic conductivity matrix is possible if the ∆W matrix is known, but the ∆W matrix represents the distribution of the incremental excess pore pressures around the cone tip, a distribution that is not known. The accumulated excess pore pressure at the piezometric-element location is the only known (measured) quantity. Thus, a straightforward procedure is not possible. It is possible, by assuming the hydraulic conductivity matrix, to proceed using a trial and error method. With the assumed hydraulic conductivity matrix, the excess pore pressure at the piezometric-element location can be computed. This computed value is compared to the measured value. If these two values are within 10% of each other, the assumed hydraulic conductivity is considered a good estimation of the hydraulic conductivity of the soil. This general trial and error method is time consuming, and without a good initial hydraulic conductivity matrix, this process may take a substantially long time. Beside the numerical simulation of the piezocone penetration test for the soil properties given in Table 6.1, additional numerical simulations were carried out to check the sensitivity of the material parameters. The effect of the slope M of the critical state line in the p-q space was investigated for the range of M from 1 to 1.2. The effect of the void ratio was investigated for the range of initial void ratio from 0.8 to 1.16, where the initial void ratio (H) implies the void ratio at ln(p)= 0 in the e-ln p plot. The variability in

104

6 Applications 700

70

M=1.2, H=1.16 M=1.2, H=1.0 M=1.2, H=0.8 M=1.0, H=0.95 M=1.0, H=0.8

60

Ex. PWP, kPa

50

500

40

400

30

300

20 10

M : slope of critical state line H : Initial void ratio when ln(p)= 0

200 100 0

0 10

600

-1 8

10

-1 4

10

-1 0

10

-6

10 -2

Hydraulic conductivity, (m/sec) Fig. 6.5. Numerically predicted results for excess pore pressure and hydraulic conductivity (σv = 200 kPa)

the results with elastic stiffness is expected. However, this effect is not investigated here since the experimental data are normalized with respect to the undrained shear strength (which also can be said to be normalized with the undrained modulus). Figure 6.5 shows the effect of the variability of the material parameters on the excess pore pressure and hydraulic conductivity. These effects are negligible for the hydraulic conductivity higher than 10−9 m/sec., a finding that concurs with the results of a study by House et al. (2001). Figure 6.5 also indicates that there are clear and predominant relationships between the hydraulic conductivity and the piezocone-induced excess pore pressures considering that the effect of other material parameters such as stiffness is also included. These relationships are clearer when the hydraulic conductivity is in the range of 10−9 m/sec to 10−6 m/sec. The soils in this range of hydraulic conductivity are clayey silt to fine sand. Typically, soils out of this range of hydraulic conductivity are sands or very plastic clays. These results indicate that the coupled theory of mixtures has strong potential for making acceptable predictions of the hydraulic conductivity for these soils. Comparison with Test Results Test results were collected from well-documented piezocone penetration tests. Three piezocone penetration tests were also conducted at the LSU calibration chamber for a K-33 specimen (a mixture of 33% kaolinite and 67% sand) for evaluation of this approach. The average of the results of these three calibration chamber tests is shown in Fig. 6.6. Test data from the literature were


105

Excess Pore Pressure (kPa) -100

0

100

200

300

400

0

u4

u1 u2

10

u3

u3 u4

Penetration Depth (cm)

u2 20

u1

30

40

50

60

Fig. 6.6. Test set up and results from the LSU calibration chamber specimen

normalized to an undrained shear strength 60 kPa, which is the shear strength of the specimen tested in the LSU calibration chamber in this study. Three penetration tests were carried out in the soil specimens. The hydraulic system used for the cone penetration features dual-piston, double-acting hydraulic jacks on a collapsible frame. The frame is mounted on top of the upper lid of the chamber and allows for penetration of the sample in a single stroke of 640 mm or less. Such a single-stroke, continuous penetration is desirable especially in saturated cohesive specimens where stress relaxation and pore pressure dissipation can occur during a pause between strokes. The penetration depth is measured using an electronic analog to the digital converter depth decoding system. All tests are conducted at the standard penetration rate of 2 cm/sec. A total of three penetration tests are performed. Tests 1 and 2 are performed for the two different piezo-element configurations, the u1 , u3 , u4 configuration (see Fig. 6.6 for piezo-element configuration) and the u2 , u3 , u4 configuration, respectively. Test 3 has the same configuration as Test 2. The main purpose of Test 3 is to check the repeatability of the tests. (The discussions of these u2 , u3 , u4 will be published in a forthcoming paper; in this book, only the results of u1 are further discussed.) To include the scale effects in comparing the experimental data with the theoretical results, the following dimensionless excess pore water pressure

106

6 Applications

and dimensionless hydraulic conductivity will be used. ∆u/σv is used for the r r ) is used for the horizontal axis. (( kv )( rref ) is similar vertical axis, and ( kv )( rref to the dimensionless term used by House et al. (2001).) ∆u is the excess pore water pressure, σv is the effective vertical overburden, k is the hydraulic conductivity, v is the penetration speed of the piezocone penetrometer, r is the radius of the piezcone penetrometer’s head, and rref (1.784 cm) is the radius of the reference piezocone penetrometer’s head. The rationale for the normalization of the field data in Table 6.2 is based on the fact that the induced excess pore pressure is proportional to the undrained shear strength from the Cavity Expansion Theory (Vesic, 1972). Through this normalization, the dependency of the pore water pressure to the undrained shear strength of the material is reduced. To evaluate the effects of some factors (such as the penetration speed and effective overburden), the authors subjected performed several numerical analyses. Figure 6.7 shows the effect of the penetration speed on the excess pore pressure response. This effect is not quite linear; however, one can approximately assume a linear function, and in this study we assume that the effects of penetration speed are linear. Figure 6.8 shows the effects of the effective overburden on the excess pore pressure response. This effect is also not quite linear, but once again, one can reasonably assume a linear function, and we assume in this study that the effects of confining pressure are linear. As discussed previously, test results from the literature shown in Table 6.2 are plotted in Fig. 6.9 and compared with the theoretical results. Figure 6.9 shows reasonably good agreement between the test data and the theoretically predicted results. This agreement is good, considering that these data are obtained from different locations for different soils with different properties such as stiffness, and so forth. In Fig. 6.9, the test data covers a wide range of soils with dimensionless hydraulic conductivities ranging from 10−7 . to 10−4 . One can see from Fig. 6.7 that the change in the excess pore pressure is very small when the hydraulic conductivity is less than 10−7 or larger than 10−4 . Therefore, the drainage condition is practically that of a fully undrained condition for the dimensionless hydraulic conductivity of lower than 10−7 , and that of a fully drained condition for the dimensionless hydraulic conductivity of higher than 10−4 . A simpler theoretical approach that uses the drained cavity expansion or undrained cavity expansion may be applicable. However, for soils between these boundaries, the partial drainage effect is not negligible. Also, such a simpler approach may not provide reliable results. As we have noted above, the work of Senneset et al. (1988) and Elsworth (1993) showed that the partially drained condition is obtained when Bq is less than 0.4 or 0.5. The work presented here gives similar results by showing that the excess pore pressure during the piezocone penetration test is affected when the dimensionless hydraulic conductivity is lower than 10−10 .


107

Table 6.2. Test data of cone penetration induced excess pore pressure at u1 position and the hydraulic conductivity (σ v = 200 kPa)

No. Soil

Su (kPa)

k, (m/sec) Test Method

∆u (kPa)

∆unorm (kPa)

1

Stjørdal Silty Clay,

83

(0.7 − 5.2)×10−8 Oedometer

338

2

Glava Stjørdal Silty Clay

90

(3.6–6.3) × 10−10 Oedometer

3

Bakklandet Trondheim Silty Clay

100

4

Valøya Trondheim Silty Clay

5

OCR

Reference

244 = 338 × 60 ÷ 83

N.A.

Senn set et al. (1988)

800

533

3–4

Sandven (1990)

(0.2–1.1) × 10−9 Oedometer

800

480

1.7

Sandven (1990)

125

(0.2–1.0) × 10−9 Oedometer

1250

600

2–3

Sandven (1990)

Halsen Stjørdal Silty Clay

83

(3–9) × 10−8 Oedometer

300–400 216–289

N.A.

Sandven (1990)

6

Norco Silty Clay

50–60

(0.30–5) × 10−10 Oedometer

500–550 545–600

1–1.5

Tumay and Acar (1985)

7

Amherst Clayey Silt

71.8

(1–2) × 10−9 Lab. Perm. Test

450–500 376–417

1.3–3

Baligh and Levadoux (1986)

8

Pentre Clay

62.5

(2–8) × 10−9 Oedometer

600

1.2–1.8

Powell and Quarterman (1997)

9

Bothkenner Silty Clay

40–75

(1.4–3) × 10−9 830–870 664–696 Oedometer

1.0–1.5

Lunne et al. (1997)

10

LSU Calibration Chamber Artificial

60

8 × 10−9 Oedometer

560–624 560–624

1

Kurup (1993), K−50 (50% Kaolinite + 50% Sand)

11

Soil

85

2.1 × 10−8 Lab. Perm. Test

350–370 250–260

1.5

This study, K−33 (33% Kaolinite + 67% Sand)

12

Quiou Sand

N.A.

3 × 10−4 Hazen’s Law

10–15

10–15

1

Almeida et al. (1991)

13

Glass Bead

N.A.

9 × 10−3 Hazen’s Law

0

0

1

Peterson (1991)

576

108

6 Applications

Fig. 6.7. Effects of confirning pressure to excess pore pressure response of the piezocone penetrometer 700

Ex. P.W. P. (k Pa)

600 500 400

k=5x10-12 m /sec k=5x10-10 m /sec k=5x10-9 m /sec k=5x10-8 m /sec

300 200 100 0 0

1

2

3

4

Penetration Speed (cm/sec) Fig. 6.8. Effects of penetration speed to excess pore pressure response of the piezocone penetrometer


109

Fig. 6.9. Comparison of actual test data and predicted results of the excess pore pressure and hydraulic conductivity (for M = 1.2, H = 1.16)

Summary of OPM A new theoretical interpretation and experimental verification of the cone penetration induced excess pore pressures is presented in this study. In addition, it adopts the large strain coupled theory of mixtures formulation using an updated Lagrangian reference frame. Using this theory and the presented numerical simulation technique, the cone penetration-induced excess pore pressure is predicted and compared with existing test data. The test data agree well with the theoretically predicted results. Therefore, the potential exists for use of this method to interpret the continuous pore pressure measurements, and it may be possible to use this approach for the real time analysis for the hydraulic conductivity of saturated soils. Two threshold dimensionless hydraulic conductivities are obtained as 10−7 and 10−4 for the undrained condition and the free drainage condition, respectively. The coupled theory of mixtures should be used to predict the behavior of soils within the range of these threshold values. b) Two Points Method (TPM) OPM utilizes the coupled theory of mixtures to estimate the hydraulic conductivitis of soils. TPM, however, utilizes traditional (uncoupled) consolidation theory to estimate the hydraulic conductivities of soils. This method utilizes the difference of pore water pressure at u2 and u3 locations (See Fig. 6.6 for u2 and u3 locations). This method is valid when

110

6 Applications

Fig. 6.10. Excess pore pressure distribution around the cone tip (after Whittle and Aubeny, 1991)

there is a measurable difference between the pore pressures measured at u2 and u3 . The test data and analytical results show that there is a clear difference between u2 and u3 locations. Whittle and Aubeny (1991) analytically showed that there is a clear difference in pore pressure at u2 and u3 locations (See Fig. 6.10). Robertson et al. [1986] and Juran and Tumay [1989] showed this behavior experimentally. Also, the typical data showed the clear difference in pore pressures at u2 and u3 locations (See Fig. 6.11). Undoubtedly, there is a clear difference between the pore pressures at u2 and u3 locations. However, one can predict that the difference between the pore pressures at u2 and u3 locations for the fully undrained condition (very low hydraulic conductivity) and the fully drained condition (very high hydraulic conductivity) will be very small. The conceptual response of pore pressure difference that one can expect is shown in Fig. 6.12. From Fig. 6.12, one can expect that two hydraulic conductivities may be obtained from the one value of the pore pressure difference between the u2 and u3 locations. Figure 6.12 shows that the direct relationship between the amount of pore pressure difference at u2 and u3 locations and the hydraulic conductivity may not be an easy way for the quantification. A second method is the consolidation approach. The pore pressure at u2 and u3 locations is


111

Pore pressure difference, ( u2 - u3)

Fig. 6.11. Field measured pore pressure from PCPT at Pentre, U.K. (after Powell and Quarterman, 1997)

Hydraulic conductivity Fig. 6.12. Conceptual relation between pore pressure difference and hydraulic conductivity

different because of both consolidation and the stress conditions. One can see the possibility that the pore pressure at the u3 location is the dissipated pore pressure of the u2 location. At the steady state penetration, one can reasonably assume that the shearing stresses at u2 and u3 locations are identical. Then the pore pressure difference between u2 and u3 locations is due to the normal stress difference and pore pressure dissipation. Thus, if one can separate the shear stress-induced pore pressure and normal stress-induced

112

6 Applications

pore pressure, one can compute the amount of the pore pressure dissipation between the u2 and u3 locations. This computation would give the hydraulic property of the soil the u2 that is directly measured during the piezocone penetration test. In conclusion, the basic idea is that the pore pressure distribution at u2 and u3 represents the dissipation curve of the normal stress-induced excess pore pressure. Thus, by starting the virtual consolidation at the u2 location for the normal stress-induced pore pressure with the assumed hydraulic conductivity, one can obtain the pore pressure at u3 location. For the consolidation, u2 is not taken as the initial pore pressure without a proper justification. The simultaneous generation and dissipation for the pore water is taken into account from the beginning of the PCPT throughout the virtual consolidation. Thus, both the implicit and the explicit drawbacks of the conventional method are inherently removed. Formulation of the Uncoupled Consolidation For comparison with the result of coupled consolidation, approximate formulations for the uncoupled consolidation are derived. Using the notation of Fig. 6.6, one can show that the measured pore pressures at u2 and u3 locations are as follows: (6.5) u2 = us + ∆u2 u3 = us + ∆u3

(6.6)

In (6.5) and (6.6), us is the hydrostatic pressure, ∆u2 is the excess pore pressure at u2 location, and ∆u3 is the excess pore pressure at u3 location. The increment between u2 and u3 locations in (6.5) and (6.6) is expressed as (6.7); (6.7) u2 − u3 = (us + ∆u2 ) − (us + ∆u3 ) = (∆u2 − ∆u3 ) Also, by noting that the excess pore pressure is the function of both the normal stress change and the shear stress change, one obtains the following equation, ∆u = ∆un + ∆us = ∆uoct + 3a∆τoct = ∆uoct + 1.4142∆τoct

√ (Assume A = 1, a = ( 2/3)A)

(6.8)

where, ∆uoct is the excess pore pressure induced by octahedral normal stress, 3a∆τoct is the excess pore pressure induced by octahedral shear stress, “a” is Henkel’s pore pressure coefficient, and “A” is Skempton’s pore pressure coefficient. Consequently, (6.7) reduces to (6.9): ∆u2 − ∆u3 = (∆uoct + 1.4142∆τoct )2 − (∆uoct + 1.4142∆τoct )3

(6.9)


113

As mentioned previously, when one assumes that the shear stress at u2 and u3 location is identical, (6.9) reduces (6.10), ∆u2 − ∆u3 = ∆uoct,2 − ∆uoct,3

(6.10)

where, ∆uoct,2 is the octahedral normal stress induced excess pore pressure at the u2 location and ∆uoct,3 is the octahedral normal stress induced excess pore pressure at u3 . Also, one can see that ∆u2 is not exactly ∆uo because of the preceding dissipation: ∆u2 is smaller than ∆uo in nature (∆u2 < ∆uo ). Thus, one can assume that ∆uo,oct = ∆uo − 3a∆τoct ⇒ ∆uo,oct < ∆uo . ⇒ ∆u2 ≈ ∆uo ,oct (taking into account the preceding dissipation). Then the degree of consolidation is calculated as follows: U = (∆uo− ∆ut )/∆uo = (∆u2− ∆u3 )/∆uo,oct ≈ (∆u2− ∆u3 )/∆u2

(6.11)

From (6.11) and Torstensson’s (1975, 1977) graphical solution, one can calculate the radial coefficient of consolidation cr or permeability k (k = cr γ w /M, where k is the permeability, M is the constraint modulus, and γ w, is the unit weight of water) from U. Because of the inherent assumption of (6.11), the results are interpreted to have approximate values. Comparison with Test Results The collected field test results in Table 6.2 and Table 6.3 are plotted in Fig. 6.8 together with the theoretically predicted results. Because the collected field test results are for various soils, a normalization of the field test results is performed for the reference undrained shear strength of 60 kPa. This normalization technique is based on the fact that the induced excess pore pressure is proportional to the undrained shear strength as shown in (6.12) (Vesic, 1972). (6.12) ∆u = su [0.817αf + 2 ln(Rp /r)] In (6.12) su is the undrained shear strength, αf is Henkel’s pore pressure parameter, Rp is the radius of the plastic zone, and r is the distance to the center of cavity. To evaluate the relationships between the PCPT-induced pore pressures and the permeability using the coupled theory of mixtures in an updated Lagrangian reference frame with an incremental elasto-plastic constitutive model, the analyses are carried out utilizing the finite element program CS-S (Coupled System – Soils) as described in OPM. This evaluation is performed for various permeabilities and strength parameters. In Fig. 6.13, the parameter (∆u2 − ∆u3 )/∆u2 is used as the reference parameter for the quantification of the relationship between the excess pore pressure and permeability. (Note that both ∆u2 and ∆u3 are normalized.) This parameter is used because the direct increment of pore pressure (∆u2 − ∆u3 ) is not a function of a single permeability value, as previously discussed.

114

6 Applications 1 0.9 0.8

(u2-u3)/(u2-us)

0.7 0.6 M=1.2, H=0.9

0.5

M=1.0, H=1.0

0.4 0.3 0.2 0.1 0

-12

-10

-8

-6

-4

-2

0

Log of Permeability (M/SEC.) Fig. 6.13. Change of pore pressure ratio [(∆u2− ∆u3 )/∆u2 ] with hydraulic conductivity [In the figure, (u2 –u3 )/(u2 –us ) is the same as (∆u2 –∆u3 )/∆u2 , rectangles represent the field test data of Table 6.2, and solid dots represent the predicted values of permeability for the sites in Table 6.2 from the uncoupled consolidation]

In Fig. 6.13, the two solid lines represent the change of (∆u2 − ∆u3 )/∆u2 with hydraulic conuctivity (k) for different values of M and H, respectively (where M and H are the properties of Cam-Clay model). From the theoretically predicted lines in Fig. 6.13, one can see that there is a clear relationship between the (∆u2− ∆u3 )/∆u2 and the hydraulic conductivity in the range from 10−10 to 10−6 m/sec. Soils with hydraulic conductivity smaller than 10−10 m/sec are clayey soils and the soils with very low hydraulic conductivity higher than 10−6 m/sec are sandy soils. Considering the typical permeability criteria for the clay liners for sanitary land fill is 10−9 m/sec and that for the free drainage materials for vertical drains and (or) horizontal drains is 10−5 m/sec, the curves in Fig. 6.13 show the new possibility of estimating hydraulic conductivities for most field soils. In Fig. 6.13, the test results are indicated with rectangles instead of points. The rectangles reflect the transient variational nature of the measured pore pressures during the PCPT. The center of the rectangles is the same as the values in Table 6.2. In Table 6.3, ∆u2 and ∆u3 values are taken from the mean ∆u2 and ∆u3 of the field-measured pore pressure values. It seems that the agreement between the test results and the theoretically predicted results is not very excellent. However, considering that the representative field permeability values are typically larger than that of the laboratory values, as reported by Baligh and Levadoux (1980, 1986), Song et al. (1992) and Funeki (1976), one may recognize the tendency that the


115

Table 6.3. Cases of the cone penetration induced excess pore pressure and the permeability

Site

Cohesion (kPa)

Normalized Excess Pore Pressure for Excess Pore Cohesion = Pressure, ∆u 60 kPa, (kPa) ∆u (kPa)

Hydraulic conductivity, k, (m/sec)

∆u2

∆u3

∆u2

∆u3

OCR

Reference

10−9

550

250

330

150

1.7

Sandven [53]

1.1 ×

Bakklandet Trondheim (Norway)

100

Pentre (U.K)

62.5

(2–8) × 10−9

400

170

384

163

1.2–1.8 Powell and Quarterman [46]

40–75

(1.4–3) × 10−9

500

340

400

272

1.0–1.5 Lunne et al. [1]

Glava Stjørdal (Norway)

90

6.3 × 10−10

600

420

400

280

3–4

Sandven [53]

LSU Calibration Chamber

60

7.4 × 10−9

330

200

330

200

1.5

This study (K-33)

Bothkenner (U.K)

rectangles of the test results in Fig. 6.13 should move to the right. With this tendency having been recognized, one can obtain better agreement between the two. The arrows in Fig. 6.13 represent this tendency. The solid and thick dots in Fig. 6.13 represent analytical results by the uncoupled consolidation. The results from the uncoupled consolidation show a remarkably good agreement with the curves obtained from the coupled consolidation. Considering the computational simplicity and implicit assumptions, these results are unexpected. However, considering that the uncoupled consolidation theory does not take into account the pore pressure interactions around the cone penetrometer, the results of the uncoupled consolidation may deviate substantially in certain conditions: thus the uncoupled theory should be used very cautiously. 6.1.4 Evaluation The one-point method is applicable for soils with permeabilities in the range 10−9 m/sec to 10−6 m/sec. The two-point method shows a wider range of application. The OPM has its own advantages in that it can be used without

116

6 Applications Table 6.4. Computational Results Hydraulic Conductivity, k (m/sec)

Sites Bakklandet Trondheim (Norway) Pentre (U.K) Bothkenner (U.K) Glava Stjørdal (Norway) LSU Calibration Chamber ∗

Coupled Consolidation∗

Uncoupled Consolidation

Lab. Test (vertical)

2 × 10−8

2.3 × 10−8

.11 × 10−8

3 × 10−8

1.2 × 10−8

(.2 − .8) ×10−8

2 × 10−8

6 × 10−9

(1.4 − 3)×10−9

1.5 × 10−8

1.5 × 10−8

.063 × 10−8

1.3 × 10−8

1.7 × 10−8

.74 × 10−8

Hydraulic conductivity is back calculated from data in Table 6.3 using CS-S.

the modification of the existing piezocone penetrometer which has a piezoelement at the u1 position. However, the TPM has the advantage that it can be applied for a wider range of hydraulic conductivities. Thus, one can see these two methods are compensating each other’s disadvantages and constitute new methods when they are used together. In conclusion, the discussions in this study show the ability of the coupled theory of mixtures to predict the hydraulic conductivity of soils utilizing the penetrating pore pressure of PCPT. This study shows that the agreement with the test data is quite reasonable. Thus, one can see the possibility of obtaining the continuous permeability profile, an objective that was not possible in the past. Also, with the incorporation of the high-speed processor, the continuous permeability profile can be obtained with real time basis (“on the fly”).

6.2 (Shield) Tunneling 6.2.1 Introduction Rapid growth in urban development has called for continual upgrading and expansion of the existing infrastructure to meet all the demands of urban life, not least of which is the demand for space. Space must be found for transportation systems and various other kinds of networks, including communication systems, utilities, water supply, and sewage disposal pipelines. Space must also be found for storage of many kinds of materials and for facilities (shelters) that offer protection against natural and man-made disasters.

6.2 (Shield) Tunneling

117

Because underground structures can accommodate new demands for space as well as various other demands for new infrastructure, and because such structures have minimal impact on surface features, tunneling will play an increasingly important role in meeting these demands. The development of shield tunnel methods throughout the history has been concentrated on how to stabilize and support the cutting face during excavation. An unstable cutting face during excavation can be supported in different ways, such as by mechanical means, by compressed air, by fluid, and by the excavated soil itself. Mechanical support is not suitable and is very risky in soft ground, especially bellow groundwater level (Babenderede, 1991). The compressed air shield was the first approach used in soft ground. The next approach was the slurry shield, in which the face is supported by a fluid (usually water and additives such as bentonite). In 1974, the Earth Pressure Balance (EPB) shield system was introduced in Japan, in which the excavated soil material itself supports the cutting face. The pressure applied to the tunnel face counter-balances, in theory, the existing overburden and hydrostatic pressures. The EPB shield was first used in the USA in 1981 to drive a 3.7 m diameter, 915 m long tunnel for the San Francisco clean water project. Despite these advances, construction of tunnels in soft grounds, especially in urban regions, continues to pose a unique challenge to engineers. Design Engineers are facing an increasing challenge, with more responsibilities to build tunnels under different ground conditions and in congested town regions for various needs and purposes that meet both natural and legislated environmental restrictions. In urban areas, consideration must be given to protect pre-existing structures and underground conduits from damage during shield tunneling. Recent advances in tunneling technology reduce construction time with consequent decrease in cost. Unfortunately, theoretical advances have not kept pace with the recent advances in the tunneling technology. Construction of tunnels in soft ground, especially in urban regions, poses a unique challenge to engineers, and careful consideration must be given to the magnitude and distribution of settlements. At present, there is no generally valid method for predicting, prior to tunneling, the subsidence caused by tunneling. Up-to-date empirical procedures (e.g. Peck, 1969) have been widely used to asses potential ground deformation owing to tunneling, but empirical formulas have limitations in their applicability to different tunnel geometries, different construction techniques, and different soil conditions (Lee et al., 1992). It has been reported that deformation caused by tunneling (Ng et al., 1986) and the subsequent potential damage to adjacent and overlying services and structures depends on various factors: ground and groundwater condition, tunnel depth and geometry, and construction procedures, the last being the most important factor. Recently, the prediction of ground deformation and stress patterns during shield tunneling have been carried out by numerical

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6 Applications

analysis based on the finite element method. Due to simplicity and cost effectiveness, in many cases, researchers (e.g. Rowe et al., 1983; Rowe and Kack, 1983; Ng et al., 1986; Finno and Clough, 1985; Rowe and Lee, 1993) have adapted the two dimensional plane strain or axi-symmetrical approach for the tunnel transverse or longitudinal section. However, field results and theoretical analyses show that the general stress and displacement patterns around the tunnel are three dimensional and very different from that of the plane strain transverse section (Lee and Rowe, 1990a, 1990b). During the shield advance, before the face of the tunnel shield reaches the section, the soil is subjected to small settlement or heave movement. As the tunnel shield passes the section, rapid downward settlement of the soil occurs immediately after the tailpiece cleara the section invading the space (gap) between the tunnel boundary and the lining. At this stage, the lined tunnel section approaches the plain strain condition. The distance required for ground displacement to reach the plane strain condition depends on the amount of plasticity developed around the tunnel. Under idealized construction, when the tunneling machine is kept hard against the face, minimizing the stress changes and deformation into the face, the tunneling advances under perfect alignment and can be treated as a plane strain case. Under less conservative construction procedures, three-dimensional movement ahead of the tunnel face may be significant. Construction difficulties such as steering and alignment problems can cause over-excavation and remolding of adjacent soils. Usually, during tunneling, a significant zone of plastic behavior is induced around the tunnel. Several attempts have been made to use 3-D finite elements model (Lee and Rowe, 1989a,b, 1990a,b; Akagi, 1994) to simulate the tunneling process, especially for open face tunneling shields. Akagi (1994) used excavated elements ahead of the shield face to simulate the advance of the shield in the 3-D model. Finno and Clough (1985) introduced a 2-D model that is based on the combination of both the transverse and longitudinal plane sections. The longitudinal section analysis is used to provide information on the nature of the pressure distribution in order to simulate the heaving process in the transverse section. In this chapter, a two-dimensional computational model is developed and used to simulate the continuous advance of the Earth Pressure Balance (EPB) shield during the tunneling process in cohesive soils. The model is based on the plane strain “transverse-longitudinal” sections that are capable of simulating the continuous advances of the shield and incorporating the 3-D deformation of the soil around and ahead of the shield face. An elasto-plastic coupled system of equations (as described in Chap. 4) is used here in order to describe the time-dependent deformation of the saturated cohesive soil. The Remeshing technique is used in the longitudinal section in order to rearrange the finite element mesh ahead of the shield face do that the size and dimension of the excavated elements match the geometric shape and size of the shield advance. The computational model is used to analyze the N-2 tunnel project


119

excavated in 1981 in San Francisco using the EPB shield tunnel machine. The results of this analysis are compared with the in-situ field measurements of the N-2 tunnel project. 6.2.2 Method of Analysis Empirical formulas have been widely used to assess potential ground movements owing to tunneling (e.g., Peck, 1969; Mair et al., 1993). These formulas are based on the assumption that the transverse surface settlement (S) profile follows a normal probability (Gaussian distribution) curve as given below, 2

S = Smax e e(−x

/2i)

(6.13)

where Smax is the maximum settlement that occurs above the tunnel centerline; x is the distance from tunnel centerline; and i is the distance from the tunnel centerline to the point of inflection (as shown in Fig. 6.14). The total half width of the settlement profile is given by 2.5 times i. Nevertheless, empirical formulas have limitations in applicability to different tunnel geometries, ground conditions, and construction procedures (Lee et al., 1992). In addition, these formulas do not take into consideration the redistribution of the stress state and the development of excess pore pressure resulting from the advancing of the tunnel shield. For these reasons, an alternate method is used to calculate the stress distribution around the tunnel opening for the lining design. In recent years, several researchers have used numerical analysis based on the finite element technique in order to predict the ground deformation and stress patterns caused by the tunneling process. In many cases, numerical modeling has been treated as a two-dimensional plane

Fig. 6.14. Gaussian distribution of surface and subsurface settlement profiles (after Mair et al., 1993)

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6 Applications

strain or an axi-symmetric problem of the tunnel transverse or longitudinal sections. However, field studies and theoretical analysis show that the displacements around the tunnel are three-dimensional (Lee and Rowe, 1990a,b). Limited theoretical research has been conducted on the three-dimensional behavior of the ground movement ahead the tunnel face, particularly for soft soils, where nonlinear elasto-plastic conditions exist (Clough and Leca, 1993). For the sake of simplicity and cost effectiveness, most of the previous workers (e.g. Rowe and Kack, 1983; Finno and Clough, 1985; and Ng et al., 1986) adapted the two-dimensional, finite element approach. In this approach the tunnel process is modeled using a “transverse section” plane strain model. The transverse section cuts perpendicularly to the tunnel axis so that the under formed tunnel appears as a circle in the finite element mesh. There are two approaches to incorporating the effect of the threedimensional movement into the two-dimensional “transverse section” analysis. The first approach, adapted by Finno and Clough (1985), makes use of a second plane strain “longitudinal section” in order to represent a section cut parallel to the tunnel axis for the case of EPB shield tunneling. The “longitudinal section” is used to simulate the shield advance by applying incremental rigid translation, causing the soil to displace away from the tunnel face. The induced out-of-plane stresses are used to estimate the pressure distribution, which is used to simulate the initial heaving process in the “transverse section.” The plane strain “transverse section” is then used to simulate the closure of the tail gap and the long time-dependent deformations. The second approach has been used by Rowe et al. (1983), Rowe and Kack (1983), Ng et al. (1986), Rowe and Lee, (1992), and Lee et al. (1992) suggest the use of what is called the “gap parameter” to represent quantitatively the three-dimensional ground loss resulting from three-dimensional movement for an open face shield tunneling. The gap parameter (GAP) (Fig. 6.15) can be expressed (Rowe and Lee, 1992; Lee et al., 1992) as GAP = Gp + U∗3D + ω

(6.14)

where Gp = 2∆ + δ = the physical gap representing the geometric clearance between the outer skin of the shield and the lining, the gap being comprised of the thickness of the tailpiece (∆) and the clearance required for erection of ∗ the lining (∂); U 3D represents the equivalent three-dimensional elasto-plastic deformation at the tunnel face, which can be calculated from formulas given by Lee et al. (1992) and Rowe and Lee (1992), and ω takes into account the quality of workmanship. These movements can be approximately incorporated into the twodimensional plane strain model by assuming a larger excavated tunnel diameter with an additional volume corresponding to the volume of ground loss ahead and over the shield. Excavation of tunnel is then simulated by removing the excavated elements and replacing them with tractions around the tunnel opening. Tractions are then removed incrementally from around the tunnel periphery.


121

Fig. 6.15. Two-dimensional ground loss simulation (after Lee et al., 1992)

For the closed-face shield (such as the EPB) Yi et al. (1993) suggest applying a peripheral pressure at the tunnel opening until the observed heave is achieved in order to simulate the heaving caused by EPB. Tail gap closure is then simulated, allowing the soil to move inward until the closure of the theoretical gap. Lee and Rowe (1990a, 1990b, and 1991) used a three-dimensional finite element model to simulate the advance of an open-face tunneling shield. The continuous advance of the tunnel face was simulated by a two-stage analysis. At the beginning, the axial pressure ahead of the tunnel and the radial pressure around the periphery is released gradually, allowing the soil to move freely into the zone to be excavated until the total radical convergence of the soil at the tunnel crown and invert corresponds to the tunnel clearance soil in front of the heading moves both radially and axially toward the face. The volume between the final cut surface and the original position of the soil rep-

122

6 Applications

resents the ground loss due to the three-dimensional movement (U∗3D ). Once U∗ 3D is known, the total gap can be calculated. A second three-dimensional computation is then performed on the new dimensions. Pelli et al. (1991) used a three-dimensional finite elements analysis to simulate tunneling process in rocks. The excavation process was modeled by eliminating the “excavated” elements from the stiffness matrix. Akagi (1994) introduced excavated elements ahead of the shield face to simulate the advance of the shield in the three-dimensional model. 6.2.3 Finite Element Numerical Simulation The continuous advance of the tunneling process is simulated here by using the two-dimensional finite element analysis (Voyiadjis and Abu-Farsakh, 1998; Abu-Farsakh and Voyiadjis, 1999) that is based on the combination of both the plane strain “longitudinal and transverse” sections. The plane strain longitudinal section will be used to simulate the continuous advance of the shield and to study the short-term soil deformations, stress redistribution, and excess pore pressure ahead and around the shield, as well as the resulting initial heave or settlement caused from the shield advance. The analysis of the longitudinal section will provide important information – such as the initial surface heave or settlement and the distribution of the excess pore pressure around the tunnel opening – that is needed in the transverse analysis. a) Two-dimensional Longitudinal Section A 2-D Longitudinal section will be used to simulate the advancement of the EPB shield and the associated ground deformation and stresses redistribution resulting from the shield advancement. The advancement of the tunneling machine involves a change in geometry and removal of excavated soil, so a step-by-step incremental excavation procedure will be used in this simulation. The EPB shield machine can be operated in such a way that the rate of excavation of the soil is more less or equal to the rate of advancing the shield. This flexibility allows over-, under-, or perfect excavation simultaneously by adjusting the applied earth pressure at the shield face. In most cases, the EPB shield is operated so that the rate of excavation is less than the machine’s rate of advancement, a discrepancy that forces the soil away from the machine’s face in such a way as to cause small initial heave. This initial heave will reduce the amount of final settlement. The magnitude of the heave is directly proportional to the applied earth pressure. However, the proposed model will be capable of handling the three mentioned cases, and for both open- and closed-face shields. For the over excavation case, during each incremental advance, part of the soil will move inside the shield, causing the incremental excavation to be higher than the incremental advance. In the case of under-


123

Fig. 6.16. Stages of two-dimensional longitudinal section (Abu-Farsakh, 1996)

excavation, during each incremental advance, part of the soil ahead of the face will be excavated and part will be displayed away from the shield face. In the case of perfect excavation, the soil is neither displaced away nor moved inside. In the longitudinal plane strain analysis, the simulation of the advancement of the EPB shield tunneling will be accomplished in the following three stages as described in Fig. 6.16. Stage 1: Removal of the part of the soil ahead of the shield that has to be excavated, replacing it with the equivalent traction acting around the tunnel, and moving the face of the shield just ahead of the excavated part. The equivalent nodal tractions, Req, are calculated by integrating the total stresses over the excavated elements as given below:

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6 Applications

BT σdA

Req =

(6.15)

A

Stage 2: Under excavation: Apply incremental rigid translations of the nodes representing the face and the body of the EPB shield until the average earth pressure at the face reaches the specified controlled value (based on the machine operation). This will simulate the initial heaving process caused by displacing the soil away from the face. Over Excavation: Gradually release the nodes representing the face of the EPB shield until the average earth pressure at the face reaches the specified controlled value. This process will simulate the initial settlement process caused by inward movement of the soil into the shield chamber. Perfect Excavation: disregard this stage. Stage 3: Remeshing: The finite element mesh will be rearranged in the portion ahead of the shield face so that the size and dimensions of the excavated elements. For the next incremental step, match the geometric shape and size of the incremental shield advance (see Fig. 6.17). Stages 1–3 will be repeated for other excavation increments until a plane strain condition is reached just behind the shield tail or until a specified excavation distance is achieved. During the tunneling process, a region of disturbed (remolded) zone is created ahead of the shield face. The strength and the Poison’s ratio for the elements immediately ahead of the shield face are reduced in order to represent this disturbed zone.

Fig. 6.17. Remeshing of the finite element ahead of the tunnel boring machine (Voyiadjis and Abu-Farsakh, 1998)


125

Iso-parametric interface elements are used between the soil and the shield body in order to model the soil-shield interface friction. The Mohr-Coulomb elestoplastic yield function as given by Matsu and San (1989) is used in this work in order to describe the incremental stress-strain behavior of the interface element. This constitutive model is capable of describing the restrained dilatancy of the soil at the interface. More details of the soil-shield interface modeling are discussed later in Sect. 6.2.6. b) Two-dimensional Transverse Section In order to incorporate the three-dimensional deformation into the plane strain transverse section analysis, one must know the magnitude and distribution of the radial heave/settlement pressure that needs to be applied in the transverse section. Results of the longitudinal section analysis provide information about the amount of the initial surface heave/settlement (Spl ), the distribution of the excess pore pressure, and the stress change. The actual three-dimensional surface heave/settlement (S3D ) is expected to be less than the plane strain value (S3D < Spl or S3D = A.Spl , where A < 1). In this analysis the distribution of the heave/settlement pressure is obtained by applying a rigid translation of the nodes representing the shield body and face in a simple 3-D model as shown in Fig. 6.18. This pressure will be applied incrementally in the transverse section analysis until the excess pore pressure at the spring line reaches the value obtained from the longitudinal section. In future work, the author will simulate the 3-D model in order to obtain a good correlation between the 3-D model and the plane strain longitudinal section. In this way, enough information will be obtained from the longitudinal section analysis to feed the transverse section analysis (i.e. the three-dimensional initial heave/settlement S3D ). Simulation in transverse plane strain section will be accomplished in the following five stages as described in Fig. 6.19 (Voyiadjis and Abu-Farsakh, 1998, Abu-Farsakh and Voyiadjis, 1999). Stage 1: Apply (or unload for initial settlement) incrementally the heave or settlement pressure until the specified excess pore pressure at the spring line is achieved. (For perfect excavation, disregard this stage.) In this stage the peripheral nodal displacement can be estimated. Stage 2: Once the peripheral nodal displacement is known, repeat stage 1 with a anew mesh using a different tunnel opening so that at the end of the heaving stage, the actual opening is reached. Stage 3: Apply incremental unloading pressure around the tunnel periphery until the tail gap is closed (in case of grouting, unload until the grouting pressure is achieved). In this operation, nodes around the peripheral of the tunnel are allowed to move inward. The tail gap is considered closed when the relative movement

126

6 Applications

Fig. 6.18. Three dimensional mesh

of the node at the crown and at the invert is equal to the theoretical size of the tail gap. It is better to choose the distribution of the unloading pressure in such a way as to insure the closure of the gap at the spring line and at both the crown and the invert simultaneously. Reasonable results can be obtained by using a uniform unloading pressure distribution. Stage 4: Once the soil comes into contact with the lining, the soil-lining interaction will be activated by changing the material properties of the inner element (must have the same size of the lining).in order to represent the actual lining material. In addition, the weight of the erected lining and the remaining of the peripheral pressure will be applied incrementally to the soil elements around the tunnel. Stage 5: The time-dependent long term deformation resulting from the dissipation of the developed excess pore pressure is simulated here. 6.2.4 Remeshing The advancement of the tunneling machine during the excavation process involves removal of the excavated soil and change in geometry of the soil ahead and around the shield. Therefore, the finite element mesh is rearranged in the portion ahead and around the shield body so that the size and dimension of the excavated elements for the next incremental step match the geometric shape and size of the incremental shield advance as illustrated in Fig. 6.20.


127

Fig. 6.19. Two dimensional transverse section simulation

Adaptive methods for remeshing can be classified into three main types: the r-method, where the total number of nodes and elements are kept constant but the nodes are relocated (or moved) to new location with new coordinates; the h-method, where the elements of the initial mesh are refined into smaller elements or redefined into larger elements so as the error is distributed uniformly over the finite elements of the specified domain; and the p-method, which calls for applying higher (or lower) order interpolation polynomial shape functions used for element interpolation while keeping the total number of elements and nodes constant. A fourth type–the hp-method–may also be used, which is a combination of h and p method. Implementation of the p-method is more complicated than the h-method because extensive modification of the analysis is required. However, the convergence of the pmethod is better than the h-method. The r-method is used in this work to rearrange the finite element mesh in the portion ahead and around the shield body so that the geometric shapes

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6 Applications

Fig. 6.20. Rearrangement of the finite element mesh ((Voyiadjis and Abu-Farsakh, 1998; Abu-Farsakh and Voyiadjis, 1999)

and sizes of the finite elements in front of the shield match the next incremental shield advance during the tunneling process. The idea is to relocate the node coordinated of the mesh in order to serve the afore-mentioned goal. Since the number of nodes is unchanged during this adoption, the computational efficiency of the r-method makes it the best among the other methods (Tezuka, 1992). Nonetheless, though the r-method does not increase the number of elements, if the shape of the domain is complicated, some elements might be highly distorted and may lead to unacceptable results (Tezuka, 1992). 6.2.5 Mapping Variables In general, the mapping process mainly consists of three essential steps (Lee and Bathe 1994): (1) Element Identification (EI), to identify the element Eo that contains the node n. (2) Iso-Parametric inversion (IIN), to determine the iso-parametric local coordinates (ξ n , η n ) of the node n in the element Eo .


129

(3) Data transfer (DTR), to transfer data (values and variables) from element Eo to the node n with local coordinates, (ξ n , η n ). a) Mapping of Nodal Variables (Displacements and Pore Pressures) Referring to Fig. 6.20, the old mesh ID denoted by (α), while the new mesh is denoted by (αRM ) element. One needs first to identify the elements in the old mesh(α) that contain the node and thus determine the corresponding isoparametric local coordinates of that node using the modifies Newton-Raphson iterative method. If Nk is the interpolation function corresponding to the node k in the element, then the nodal coordinates (Xn , Yn ) of the nodes n in the new element are given by m Nk (ξ n , η n )X k (6.15a) Xn = k=1

Yn =

m

Nk (ξ n , η n )Y k

(6.15b)

k=1

where m is the number of nodes in the old element that contains node n, and (Xk , Yk ) are the coordinates of node k. To obtain the iso-parametric local coordinates ξ n , η n of the node n, one needs to solve (6.15a). To ensure that the node n lies inside the element Eo ., one must satisfy the following constraints: −1 < ξ n < 1 −1 < η n < 1

(6.16a) (6.16b)

In the case of the r-method adaptivity with no change of element node connectivity, step1 and 2 are done simultaneously. That is, the element identification (EI) and the iso-parametric inversion (IIN) are done at the same time for each node as follows: • Loop over the whole nodes of the new element in (αRM ) mesh. • Loop over the whole elements in the old mesh (α). For Each node, one needs to solve (6.16a) in order to obtain the iso-parametric local coordinates. The modified Newton-Raphson iterative method is used to solve (4.4) as follows: Let m Nk (ξn , η n )X k = 0 (6.17a) f1 (ξn , η n ) = X n − k=1

f2 (ξn , η n ) = Y n −

m k=1

Nk (ξn , η n )Y k = 0

(6.17b)

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6 Applications

Using the above equations, the ith iterative local coordinates are given as follows; f1 (ξin , ηin ) n ξi+1 = ξin − (6.18a) ∂f1 ∂ξ n n (ξi+1 ,ηi )

n ηi+1 = ηin −

n , ηin ) f1 (ξi+1 ∂f1 ∂ξ n n

(6.18b)

(ξi+1 ,ηi )

The iterations will continue until the square difference between two sub* sequent values is within acceptable tolerance (tol), such as (error)2 ≤ tol • Once the iso-parametric local coordinates (ξ n , η n ) are determined, one must check the conditions of (6.17a). If they are satisfied, then the node is identified inside that element; otherwise, one moves to other elements in the (α) domain. In some cases, the node lies along the border of two or more elements (ξ or η ∼ = 1) and is identified in more than one element. In this case the nodal variables are averaged smoothed. The method of obtaining the isoparametric local coordinates ξ n , η n of node n is referred as the “Inverse Iso-parametric Mapping Technique”. Once the iso-parametric local coordinates (ξ n , η n ) are known, then the nodal displacements and pore pressure can be interpolated from the Eonodes to node n using the interpolation functions Nk (may be taken to be different interpolation functions than those used in the finite element analysis) as follows:

Un =

m

Nk (ξ n , η n )U k

(6.19a)

n n k Nk (ξ , η )P

(6.19b)

k=1

Pn =

mp k=1

b) Mapping of Stresses and Strains at Gauss Points Since the mapping of stresses at the Gaussian elements interpolation points (I.P’s) is done after each increment, the new I.P’s of the (αRM ) element will remain inside the α element, the least square smoothing will be used for this purpose. The first step before smoothing is to obtain the iso-parametric local coordinates of the new I.P’s with respect to the old element “Iso-Parametric Inversion”. The modified Newton- Raphson iterative method is also used in order to obtain the local coordinates of the new I.P’s with respect to the α elements as follows: Let


f1 ξ np , η np = X np −

mip

Nk (ξ np , η np )X k = 0

131

(6.20a)

k=1

f2 (ξ np , η np ) = Y np −

mip

Nk (ξ np , η np )Y k = 0

(6.20b)

k=1

where mip is the number of integration points; (ξ np , η np ) is the total coordinate of the new integration points (I.P), np, of the (αRM ) elements with respect to α elements; (ξ np , η np ). And (X k , Y k ) are the global coordinates of the new integration point, np, and the previous integration point, k, respectively; and Nk is the interpolation function. Then the ith iterative coordinates of the np integration points will be given as follows (similar to the nodal local coordinates): np = ξinp − ξi+1

f1 (ξinp , ηinp ) ∂f1 ∂f2 np np

(6.21a)

np , ηinp ) f1 (ξi+1 ∂f1 ∂ξ np np

(6.21b)

(ξi ,ηi )

np = ηinp − ηi+1

(ξi+1 ,ηi )

* The iteration stops once a certain accuracy is achieved, (i.e (error)2 ≤ tol.). Once the iso-parametric local coordinates (ξ np , η np ) are obtained for all the new I.P’s in the αRM (with respect to the αRM elements), the least square smoothing of stresses and strains can be carried out over the whole finite elements domain, until “global smoothing” is attained. The problem is then to find the set of smoothed stresses from the unsmoothed stresses that minimize the squares of errors using smoothing shape function (may be taken to be different interpolation functions than those in the finite element analysis). c) Least Square Smoothing The least square smoothing procedure may be carried out over the whole finite element domain global smoothing, or performed separately over each individual element, a process called “local smoothing” (Hinton and Campbell, 1974). The local smoothing of the finite elements is adapted in this work. There are two types of local smoothing: 1) Local functional smoothing, in which the smoothed function g(ξ np , η np ) is assumed to be the least squares to the unsmoothed function σ(ξ np , η np ). 2) Local distance smoothing, in which the smoothed function g(ξ np , η np ) is assumed an exact least squares fit to selected values of σ(ξ np , η np ) (at the Gaussian integration points).

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6 Applications

d) Functional Smoothing Let the smoothed function of stresses (for example) at any point, n, within an element in the finite element formulation be given by the following expression, nip ∼ ∼ Nk σ k (6.22) g(ξ np , η np ) = k=1 ∼ Nk

are the smoothed shape functions, which may be taken as being where of different order from the interpolation shape functions used in the finite ∼ element analysis, and σk are the smoothed IP stresses. If σ(ξ, η) are the unsmoothed stresses at any point within the element, then the error between the smoothed and unsmoothed stresses at any point within the element is given by e(ξ, η) = σ(ξ, η) − g(ξ, η) (6.23) The problem now becomes finding the smoothed stresses, σk , which minimize the functional as follows; E= e(ξ, η)2 dx dy (6.24) A

For the error E to be minimized, ∂E =0 ∂σi

for I = 1, nip

(6.25)

To find the smoothed stresses, one needs to solve the following set of equations, e ∼ (6.26) [A] {σ} = {F }e where, the elements-smoothing matrix will be ∼ ∼ ∼ N σ det JDξdη . . . N det Jdξη N 1 A [A]e = A 1 ∼ ∼n ∼ N σ det JDξdη. . . . det Jdξη N n1 1 A A N nN n and the elements force vector will be ∼ σ det Jdξdη. N 1 A [F]e = ∼

σ det Jdξdη. N 1 A e) Discrete Smoothing: In discrete smoothing the problem is to minimize the following:

(6.27)

(6.28)


E=

nip

e(ξ, η)2

133

(6.29)

k=1

In order to minimize the error E, the following expression is obtained, which satisfies the condition of (6.23), ∼‘

[AD ]e { σ } = {FD }e

(6.30)

where the discrete element smoothing matrix will be as follows,   nip nip + ∼ k k ∼ k k + ∼‘ k k ∼ k k N1 (ξ , η ) N1 (ξ , η ) . . . N1 (ξ , η ) N1 (ξ , η )     k=1 [AD ]e =  k=1  nip  nip + ∼ k k ∼ k k + ∼‘ k k ∼ k k  N1 (ξ , η ) N1 (ξ , η ) . . . N1 (ξ , η ) N1 (ξ , η ) k=1

(6.31)

k=1

and the discrete element force vector will be as follows:   nip + ∼ k k      N1 (ξ , η )σk    k=1 [FD ]e = nip   + ∼ k k    N1 (ξ , η )σk   

(6.32)

k=1

f ) Determination of the Yield Stress at the New Gauss Points In the elastoplastic analysis, one needs to transfer the model’s yield surface from the old I.P’s of the α element to the new I.P’s of the αRM element. One way to do this transfer is by interpolating or by calculating the least square smoothing of the yield sureface from old I.P’s to the new I.P’s. This transfer can lead to values that are self-consistent (Lee and Bathe 1994); that is to say, the effective stress σ may be greater than the yield stress σy ). Lee and Bathe (1994) suggested that the yield stress (σy ) at the new I.P’s should be obtained from the mapped equivalent plastic strain and the strain hardening function as (6.33) σy = f (εpeq ) In the modified Cam Clay model, the evolution of the hardening parameter dPc is given as a function of the incremental volumetric plastic strain, dεpv as dPc =

(1 + e) Pc dεpv (λ − k)

(6.34)

Taking the integration of both sides of (6.33) over the increment, one obtains pc pco

p

(1 + eav ) dP c = Pc (λ − k)

ev

dεpv εp vo

(6.35)

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6 Applications

Therefore, ln(Pc ) − ln(Pco ) =

(1 + eav ) p (ε − εpvo ) (λ − k) v

(6.36)

where Pco and εpvo are the reference hardening parameter and volumetric plastic strain respectively. Because the hardening parameter is a function of the plastic volumetric strain εpv , the author proposes to map the equivalent plastic strain by the least square smoothing from the old IP’s in the α element to the new IP’s in the αRM element. The next step is to find the hardening parameter Pc at the new IP’s. Because remeshing is done after each load increment, the new IP’s are expected to be very close to the old IP’s. Voyiadjis and Abu-Farsakh (1998) propose to relate the hardening parameter at the new IP’s in the αRM element to the hardening parameter of the old closest IP’s in the α element by the formula (1 + enew ) εP ln(P c)new − ln(P co) vnew = ln(P c)old − ln(P co) (1 + eold ) εpvold

(6.37)

The equivalent plastic strain can be mapped by applying the same method used to map stresses and strains. Another option may be used by direct mapping of ln(Pc) by the least square smoothing technique similar to the stress and strain mapping. In mapping the hardening parameter, one needs to check the consistency between the stress state and tehe new mapped yield surface in the new IP’s such that the stress state can not be outside the yield surface, and the following condition must be satisfied, Py ≤ 1.0 Pc

(6.38)

where Py is the hardening parameter corresponding to the current state. P In the case Pyc > 1.0, correction of the yield surface will be carried out by returning the stress state to the yield surface. 6.2.6 Modeling of Interface Friction The engineering literature contains a variety of interface elements formulations. Goodman et al. (1968) developed a simple rectangular, two-dimensional element with eight degrees of freedom. The element formulation is derived on the basis of relative nodal displacements of the solid elements adjacent to the interface element. The thickness of the element is assumed to be zero. Zienkiewicz et al., (1970) proposed the use of continuous solid elements as interface elements with a simple nonlinear material property for shear and normal stresses, assuming uni-form strain in the thickness direction. However, not many critical and systematic studies and implementations of the concept are available in the published literature (Desai et al., 1984). Numerical difficulties may arise for zero-thick elements, from ill conditioning of the stiffness


135

Fig. 6.21. Thin-layer interface element

matrix due to very large off-diagonal or small diagonal terms (Ghaboussi et al., 1973). In recent years, a number of investigators have considered the use of thin-layer interface elements (i.e Desai et al., 1984; Sharma and Desai, 1992). In this present study, the 6-noded thin-layer iso-parametric element is used to model the soil-shield interface friction. a) Interface Element Consider a 6-noded nonlinear iso –parametric thin layer interface element as shown in Fig. 6.21. This element is used to model the soil–shield interface friction. Using the local coordinates, (ξ, η), the shape functions, Ni , are defined as follows: N1 = −(ξ/4)(1 − ξ)(1 − η) N2 = (ξ/4)(1 + ξ)(1 − η) N3 = (ξ/4)(1 + ξ)(1 + η) N4 = −(ξ/4)(1 − ξ)(1 + η) N5 = (1/2)(1 − ξ 2 )(1 − η) N6 = (1/2)((1 − ξ 2 )(1 + η) One defines the nodal displacement vector, {d}, as follows: {d} = {u1 v1 u2 v2 u3 v3 u4 v4 u6 v6 } The displacement at any arbitrary point can be released to the nodal displacement vector using the shape function, Ni, as

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6 Applications

u

v

where [N] =

= [N ]{d}

N 1 0 N2 0 N3 0 N4 0 N5 0 N6 0 0 N1 0 N2 0 N3 0 N4 0 N5 0 N6

(6.39) (6.40)

b) The Elasto-Plastic Stress-Strain Relation An elastoplastic constitutive model of the Coulomb yield function and its associated flow rule (elastic-perfectly plastic model) is used in this study as proposed by Matsui and Sam (1989) and as shown in Fig. 6.22. This model can represent the restrained dilatancy. The shear stress (τ ) is given by τ = Ca + σn · tan δ

(6.41)

where Ca is the adhesion, σn is the effective normal to the surface stress, and δ is the angle of friction between the soil and the shield body. The yield function, f, can be defined (Matsui and San, 1989) as F = τ 2 − (Ca + σn · tan δ)2 = 0

(6.42)

Assuming the associated plastic flow rule, the plastic spatial strain rate, d p , is given by ∂f dp = λ = 0 (6.43) ∂σ

Fig. 6.22. Coulomb yield function for interfaces (Matsui and San, 1989)


137

The consistency condition implies df =

∂f dσ = 0 ∂σ

(6.44)

Substitution of (6.44) into (6.42) leads to df = 2τ dτ − 2(Ca + σn · tan δ) tan δ · dσn = 0

(6.45)

The total spatial strain rate, d, can be decomposed into elastic, de , and plastic, d p , spatial strains rate components as d = de + dp

(6.46)

de = d − dp

(6.47)

The plastic spatial strain is then becomes   0  ∂f Dp = λ = λ −2S   ∂σ 2τ

(6.48)

where S = (Ca + σn · tan δ) tan δ. The co-rotational stress rate, σ , is related to the spatial strain rate as {σ } = [De ]{de } = [De ]({d} − {dp })        ds  σs  0  = [De ]  dn − λ −2S  σn       2τ dγ τ where the elastic stiffness matrix, [De ], is given by   C1 C2 0 [De ] =  C2 C1 0  0 0 G in which C1 =

E(1 − v) ; (1 + v)(1 − 2v)

C1 =

Ev (1 + v)(1 − 2v)

(6.49)

(6.50)

(6.51)

where E is the elastic (young’s) modulus, v is the poison’s ratio, and G is the shear modulus. By substituting (6.47) into (6.43) one obtains the scalar parameter, λ, such that λ=

τ Gdγs − SEdεn 2τ 2 G + 2S 2 E

Substituting (6.52) into (6.50) leads to

(6.52)

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6 Applications

       0 0 0  ds   ds   σs  1  0 S 2 E −τ SG  dn σn = [Dep ] dn − 2 (6.53)     τ G + S2E   τs dγ dγ 0 −τ SG τ 2 G or

     ds   σs  σn = [Dep ] dn     τs dγ

(6.54)

[Dep ] = [De ] − [Dp ]

(6.55)

where The plastic stiffness matrix is given by   0 0 0 [Dp ] =  0 D11 D12  0 D21 D22

(6.56)

where D11 =

S2E2 τ 2 G2 −τ SEG ; D ; and D12 = D21 = 2 = 22 2 2 2 2 S E+τ G S E+τ G S E + τ 2G

The element stiffness matrix is then given by [K]ξ,η = [B]T [Dep ][B]detJdξdη

(6.57)

A

Transforming the stiffness matrix from local (ξ, η) to global (x,y) coordinates axis, one obtains (6.58) [K]xy = [T ][K]ξ,η [T ] where [T] is the transformation matrix containing the direction cosines as given below,   sn2 cn.sn cn2 cn −cn.sn  (6.59) [T ] =  sn2 −2cn.sn 2cn.sn cn2 − sn2 where cn and sn are the direction cosines between the local and global coordinates. 6.2.7 Case Study of N-2 Tunnel a) Subsurface Condition The proposed Computational model is used here in order to analyze the N-2 tunnel project located in San Francisco and constructed in 1981 (Abu-Farsakh and Voyiadjis, 1999). The soil condition of the N-2 tunnel site consists of an average of 6.1 m of rubble fill underlain by 9.1 m of soft sediment, known locally as Recent Bay Mud as illustrated in Fig. 6.23.


139

Fig. 6.23. Subsurface profile

A stratum of colluviums and residual sandy clay is encountered below the Bay Mud (compete description of the subsurface condition can be found in Clough et al., 1983). A tunnel of 3.7 m diameter was advanced within the Recent Bay Mud stratum. The fill is classified as loose to medium dense. The recent Bay Mud consists mainly of silt and lean clay that is normally consolidated. The ground water table is located at 3 m below ground surface. b) Description of the Earth Pressure Balance (EPB) Shield The EPB tunneling machine, as shown in Fig. 6.24, usually is a cylindrical shape within which the excavation takes place and the liner is erected. It advances itself by jacking against the in-place linear segments. After the shield is pushed forward by a full stroke of propulsive jacks, the jacks are withdrawn and the next ring of liner segment is erected within the “tail” of the shield. As the shield advances, soil is excavated through slots in a rotating cutter head and deposited into a soil-retaining area located between the cutter head and the bulkhead. Two rows of cutting bits are usually set between the open slots in the cutter head to cut the soil at the face of the shield. The soil is removed from the enclosed spoil retaining area via a screw

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6 Applications

Fig. 6.24. Earth pressure balance (EPB) shield section (Abu-Farsakh, 1996)

auger which takes it through the bulkhead and deposits it onto a conveyer belt. The rates of soil excavation and removal are controlled by the operation of the shield. The fundamental idea of the earth pressure balance shield is to allow for the control of soil removed from the spoil-retaining area as the shield advances. The total earth pressures are measured inside the spoil retaining area as the shield advances. This pressure is applied to the tunnel face to counter-balance, in theory, the existing earth and hydrostatic pressure. If this pressure is too large, it produces an upheaval at the surface, and if it is too small, it leads to excessive settlements at the surface. c) Finite Elements Analysis Figure 6.25 represents the finite element mesh used in the longitudinal section analysis, while Fig. 6.26 represent the finite element mesh used in the transverse section analysis. The eight-nodded iso-parametric finite element, Q8P4, is used to represent the cohesion soil of the Recent Bay Mud. The Q8 element is used to represent the cohesion-less soil of the fill and colluviums. The pre pressures are kept fixed at the bottom of the rubble fill and at the top of the colluviums. Six-nodded, thin-layer iso-parametric interface slip elements (Q6P4) are arranged between the shield machine and the soil in order to model the soil-shield interface friction. Desai et al. (1984) carried out parametric study on the effect of the thickness (t) to length (l) ratio of the interface element. They reported that the


141

Fig. 6.25. Finite element mesh for the longitudinal section analysis

use of t/l ratio in the range of 0.01 to 0.1 gives satisfactory results for interface behavior. In this study, the ratio t/l is taken as 0.1. The modified Cam Clay is used in this study in order to describe the plastic behavior of normally consolidated cohesive soil. (The modified Cam-Clay model is discussed in Chap. 4.) A nonlinear hyperbola model is used to describe the cohesionless soil response (fill and colluviums). This model has been used by several investigators. Detailed Derivations of the hyperbolic formulas for Et and ν t can be found in various publications (i.e. Clough and Duncan, 1972; Desai, 1971; and Chang and Duncan, 1971). The final equations are given below, Et = Ei (1 − λt )2 σ A1 −B 1 log Pa3 vt = (σ −σ3 )G [1− Ei1(1−λ ] t)

(6.60)

(6.61)

where the initial tangent modulus, Ei , is related to the confining pressure, σ3 , through the following empirical formula (Janbu 1963), n σ3 (6.62) Ei = K P a Pa and λt =

Rf (1 − Sinφ)(σ1 − σ3 ) 2c Cosφ + 2σ3 Sinφ

(6.63a)

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6 Applications

Fig. 6.26. Finite element mesh for the transverse section analysis

Rf =

(σ1 − σ3 )f (σ1 − σ3 )ult

(6.63b)

where c and φ the cohesion and international friction angle as in the MohrCoulomb shear strength; Rf is the failure ratio; (σ1 − σ3 ) is the stress difference; Pa is the atmospheric pressure expressed in appropriate units; and the soil parameter A1 , B1 ,G, K and n can be determined from conventional laboratory tests such as the tri-axial tests. For unloading-reloading of the soil, the unloading-reloading modulus, Eur , has been found to be related to the confining pressure, σ3 , in the same manner as shown in (6.62) for the initial tangent modulus (Chang and Duncan, 1971), n σ3 Eur = Kur Pa (6.64) Pa where the unloading-reloading modulus number, Kur , can be determined from cyclic unloading-reloading laboratory tests. An elasto-plastic constitutive frictional model of the Mohr-Coulomb type is used to describe the response of the interface elements as described earlier. The interface friction angle, δ, can be taken as δ = (2/3)φ, in which φ is the angle of friction of the soil. The soil parameters used in this analysis are taken from Finno and Clough (1985). Table 6.5 describes the soil parameters from the nonlinear hyperbolic model used for the fill and colluviums stratas chosen from the test results for similar materials. The modified Cam-clay soil parameters used for the recent Bay Mud presented in Table 6.6 are based on iso-tropically consolidated undrained triaxial compression (CIU) tests. For the interface elements, the soil parameters are the following: The Young’s modulus E = 14,700 kPa; Poisson’s ratio v = 0.3; adhesion Ca = 0; and shield-soil interface


143

Table 6.5. Soil Parameters for the Hyperbolic Model (after Finno and Clough, 1985) Parameter

Fill

Colluvium

v, Poisson’s ratio Rf , Failure ratio φ, friction angle C, Cohesion, in kPa n, modulus exponent K, Primary loading modulus Kur , unloading-reloading modulus Ko , Lateral earth pressure co-efficient γt , Total unit weight, in kN/M3

0.3 0.9 30◦ 14.36 0.5 400 600 0.5 15.75

0.35 0.9 20◦ 47.88 0.4 945 1400 0.8 19.68

Table 6.6. Soil Parameters for Cam-Clay Model (after Finno and Clough, 1985) Parameter

Recent Bay Mud

λ, Slope of isotropic compression line k , Slop of isotropic unloading-reloading line M, Slope of the critical state line Void ratio at critical state and unit pressure Ratio of Shear modulus of effective overburden pressure Kv; kh = 5 kv, coefficient of permeability, in m/sec γt , Total unit weight, in kN/m3

0.326 0.043 1.2 3.72 40 10−8 16.53

angle δ = 20◦ . In the longitudinal analysis, the average applied face pressure is kept at 74.25 kPa. The EPB shield is successfully advanced a distance equal to 1.5 times the shield length as illustrated in Fig. 6.25 by the dashed element. It has been accomplished by six subsequent excavation steps. At each step of excavation, a rigid translation is applied to the nodes representing the shield face and body until the average applied earth pressure reaches the specified value (74.25 kPa). The strength of the soil is reduced by 20% for the elements just ahead of the shield face in order to represent the disturbed (remolded) zone created ahead of the tunnel face. The distribution of the heaving pressure in the transverse section is determined from the analysis of a simple 3-D with rigid translation of the shield face and body. An elliptical pressure distribution of the ratio 1:5:1 for the crown, spring line, and invert respectively is adapted in this analysis. The heaving process is completed when a certain criterion is reached. The criterion is taken by many workers to be based on the field measurements. This criterion can be taken as either when the observed (measured) field

144

6 Applications

heave is reached (Rowe et al., 1990a) or when the lateral observed displacement is achieved (Finno and Clough, 1985). These two versions of the one criterion assume that one knows the field measurements before the analysis. The aim in this study is to predict the soil response before the event. At the present state of study, the heaving process is considered completed when the predicted excess pore pressure at the spring line during heaving in the transverse section reaches the predicted excess pore pressure from the longitudinal analysis. This assumption seems reasonable in this study. The authors suggest that in the future, the initial heave/settlement be obtained from the longitudinal plane strain analysis. The distribution of the unloading pressure during the closure of the gap of 15 cm is taken as 1:1:5:1 for the crown, spring line, and invert (respectively) in order to ensure that the closure of the gap at the spring line, invert, and crown occurs simultaneously. If a uniform unloading pressure distribution is adapted, consideration must be given to the fact that closure of the gap around the tunnel opening may not occur simultaneously for crown, spring line, and invert. Proper considereation of this possibility can be achieved, for example, by monitoring each of the peripheral nodes separately. Once any node closes the corresponding gap, that node will be prevented from further movements. This monitoring will be continued until the all of the peripheral nodes close the gap. d) Results Deformation of the Longitudinal Section The predicted longitudinal displacements ahead of the shield that are obtained from the longitudinal section analysis of the shield advance are compared with field data measured at 1.2 m and 5.5 m in front of the shield as shown in Fig. 6.27: Most of the longitudinal displacement occurs in the Recent Bay Mud layer. Figure 6.28 compares the predicted and observed surface heaving. Good agreement can be seen between the predicted and observed longitudinal displacements at 1.2 m. However, the predicted displacements at 5.5 m and the predicted surface heaving are higher than the measured ones. This result is expected because the soil is restricted from moving laterally, and therefore the plane train displacements are expected to be higher than those obtained from the three-dimensional analysis. A 3-D computational model needs to be developed in the future that will help to obtain a correlation between the actual 3-D displacements and the longitudinal plane strain displacements. This correlation will help analysizing the transverse section, especially with regard to the value of the initial surface heave/settlement.


145

Fig. 6.27. Longitudinal displacement ahead of the shield; (a) at 1.2 m ahead of the face; (b) at 5.5 m ahead of the face

Excess Pore Pressure and Stress Changes in the Longitudinal Section The contour of the predicted excess pore pressure in the Recent Bay Mud Layer resulting from the shield-advance in the longitudinal analysis is presented in Fig. 6.29. The time required to advance the shield 7.5 m is estimated to be 0.82 days based on the average daily advancement (9.1 m/day). This time allows partial dissipation of the soil. The predicted excess pore pressure at the end of the shield advancement is 32.5 kPa immediately in front of the

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6 Applications

Fig. 6.28. Predicted and observed surface initial heaving at the longitudinal section

Fig. 6.29. Contours of Excess pore pressure (kPa)

shield face and decreases rapidly with increasing distance from the shield face. Figures 6.30 and 6.31 represent the contour of axial stress change and shear stress respectively. The maximum predicted axial stress change of


147

Fig. 6.30. Contours of axial stress change (kPa) at the longitudinal section

Fig. 6.31. Contour of normal stress (kPa) at the longitudinal section

21.8 kPa occurs around the bottom corner of the shield face. The predicted shear stress ranges from −16 kPa along the bottom of the shield body to −6 kPa along the top of the shield body. This variation is due to the fact that the effective normal stress is higher near the bottom of the shield.

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6 Applications

Fig. 6.32. Lateral displacement at maximum heave; (a) at 2.05 m from centerline; (b) at 5.5 m from centerline

Fig. 6.33. Predicted and observed lateral displacement distribution at maximum heave


149

Deformations of the Transverse Section The predicted lateral deformation at the time of maximum heave as compared with the lateral measurements at a distance of 2.05 m and 5, 5 m from the tunnel centerline is shown in Fig. 6.32. Good agreement id found between the predicted and observed lateral deformation. Both the observed and predicted deformations indicate that the maximum lateral deformation occurs slightly above the spring line and that most of the lateral deformation occurs in the Recent Bay Mud layer. The predicted and observed lateral displacement distributions obtained at the time of the maximum heave near the crown, spring line, and invert are shown in Fig. 6.33. A good agreement with the field data can be seen for the predicted deformations that are less than 5 m from the centerline. Figure 6.33 shows that the lateral displacement decreases rapidly with distance from the tunnel centerline. The development of the surface initial heave and final settlement (after consolidation) profiles are drawn in Fig. 6.34. The surface heaves up to

Fig. 6.34. Predicted and observed distribution of surface displacements

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6 Applications

0.88 cm initially above the tunnel vertical center axis and ultimately settles down to 2.89 cm. The observed final settlements are quite close to the predicted values.

6.3 Estimation of Hydraulic Conductivity using Acoustic Technique 6.3.1 Introduction Hydraulic conductivity of geo-materials is one of the major geotechnical parameters used for designing and analyzing geotechnical structures. However, the procedure for determining the hydraulic conductivity (sometimes termed permeability with some controversy) of these materials is complicated. Although considerable theoretical and experimental work has been done on developing accurate methods for determining the hydraulic conductivity of soils, existing methods are still time consuming and expensive. Conventional tests, both in situ and in the laboratory, tend to change the texture of soils in the process of drilling and sampling. This chapter will discuss the development of a new method for estimating the hydraulic conductivity of soils using an acoustic technique. Because this method does not disturb soil texture, it may deliver an accurate assessment of the hydraulic conductivity. The theoretical background of this study is based on the established fact that there are two compression waves for saturated soils: (1) the wave that has in- phase motion of fluids and solids, and (2) the wave that has out of phase motion of fluids and solids. The first wave, called the fast P-wave, has much less attenuation and propagates faster. The second wave, called the slow P-wave, has significant attenuation compared to the first one and propagates slower. The different behavior of the fast P-wave and the slow P-wave was first predicted by Biot (1956a, b, c, d) theoretically, and experimentally confirmed by others later (Berryman 1981). The measurement of slow P-wave velocity involves complex procedures, including the capture of a weak signal associated with the high attenuation. Yamamoto, (2003), Xiang and Sabatier, (2003), Bouzidi and Schmitt (2002), and Batzle et al. (2001) have recently reported the measurement of the slow P-wave velocity by implementing a spread spectrum technology called Pseudo Random Noise, and the measurements of the slow P-wave are now considered well documented (Yamamoto, 2003; Xiang and Sabatier, 2003). Recently Song and Kim (2005) reported an easier acoustic method of estimating soil hydraulic conductivity, which is based on the attenuation of fast P-wave with different frequencies. This chapter discusses essentials of acoustic methods for soil permeability estimation.

6.3 Estimation of Hydraulic Conductivity using Acoustic Technique

151

6.3.2 Basics of Wave Propagation in Saturated Media Seismic wave is one of the most conveniently used ground investigation tools for profiling geotechnical conditions. Cross Hole technique is used for an accurate and direct measurement of seismic wave velocities, while SASW (Spectral Analysis of Surface Wave) technique is used for convenient and quick determination of seismic wave velocities. There are also other methods such as Down Hole technique, Up Hole technique, and many others. Seismic wave velocities obtained by such methods are used for the determination of modulus, for the evaluation of liquefaction potential, for the profiling of ground layers, and for many other purposes. The essentials of seismic waves and acoustic waves are not very different. Acoustic wave (a compression wave) is an extension of seismic wave, but the frequency range is that of audio wave. Typically the seismic frequency is between 1 and 100 Hz, but the acoustic frequency is between 20 and 20,000 Hz. Seismic wave is easy to feel but acoustic wave is easy to hear. Typically the field-measured P-wave velocity from the seismic technique is the first P-wave velocity because the magnitude of slow P-wave velocity is very weak and is typically hidden in the noise. However, the slow P-wave has much interaction with pore properties that it can be used to detect the pore properties. Theoretical background for the relationship between the slow P-wave velocity and the permeability can be found in Biot (1956a,b,c,d). Biot (1956a,b,c,d) developed macroscopic equations for the propagation of elastic waves in poro-elastic media. In addition to the existence of a fast compressional P-wave and a shear wave, Biot’s theory predicted the existence of a third wave mode referred to as the slow compressional P-wave. The essentials of Biot (1956a,b,c,d) are shown in this chapter. Assuming a conservative physical system which was statistically isotropic, Biot derived the following stress-strain relations containing four distinct elastic constants: σ x = 2N ex + Ae + Qε σ y = 2N ey + Ae + Qε σ z = 2N ez + Ae + Qε

(6.65) (6.66) (6.67)

τ x = N γx τ y = N γy

(6.68) (6.69)

τ z = N γz s = Qe + Rε

(6.70) (6.71)

where, N is the shear modulus, A is the Lame’s constant, Q and R are coupling moduli, e is the strain of solids, ε is the strain of the pore fluid, and s is the pore pressure.

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6 Applications

The dynamic equilibrium relations assuming no dissipation are expressed as follows: ∂σ x ∂τ z ∂τ y ∂2 + + = 2 (ρ11 u x + ρ12 U x ) (6.72) ∂x ∂y ∂z ∂t ∂2 ∂s = 2 (ρ12 u x + ρ22 U x ), etc (6.73) ∂x ∂t where ρ11 is the volume fraction of solids, ρ12 is the coupled volume fraction of solids and pore fluid, and ρ22 is the volume fraction of pore fluid. The equations for wave propagation are obtained by substituting expressions (6.65) through (6.71) into (6.72) and (6.73) and applying divergence operation: ∂2 (ρ11 e + ρ12 ε) ∂t2 ∂2 ∇2 (Qe + Rε) = 2 (ρ12 e + ρ22 ε), with definition P = A + 2N ∂t

∇2 (P e + Qε) =

(6.74) (6.75)

With additional mathematical arrangements, the above equations are expressed thus: 1 ∂2 (γ11 e + γ12 ε) Vc2 ∂t2 1 ∂2 ∇2 (σ 12 e + σ 22 ε) = 2 2 (γ12 e + γ22 ε) Vc ∂t ∇2 (σ 11 e + σ 12 ε) =

(6.76) (6.77)

Solutions of these equations are written in the form, e = C1 exp[i(lx + αt)]

(6.78)

ε = C2 exp[i(lx + αt)]

(6.79)

And the velocity V of these waves is V =

α l

This velocity is determined by substituting expressions (6.78) and (6.79) into (6.76) and (6.77). Putting z = Vc2 /V 2 and by applying mathematical arrangements and orthogonality relations, one obtains: zi =

(i)2

+ 2γ12 C1 + γ22 C2

(i)2

+ 2σ12 C1 + σ22 C2

γ11 C1 σ11 C1

(i)

(i)2

(i)

(i)2

(6.80)

In the above equation, there are two z’s for two phase materials (i = 1, 2), and they are: V12 = Vc2 /z1

(6.81)

V22

(6.82)

=

Vc2 /z2


153

Equation (6.80) is obtained using orthogonality relations, an expression that means that one solution represents the fast P-wave and another solution represents the slow P-wave. This orthogonal relation also means that one solution is for in-phase motion, and another one should be for out-of-phase motion. Equations (6.65) though (6.82) are shown just to indicate the origin of two different wave velocities. (For details of derivation, see Biot (1956a)). When one adds dissipation to the above equations, the following equations are obtained: 2 (σ11 σ22 − σ12 )2 1 f 1 V1 ζ = 1− ζ ζ + ζ + ζ ζ (6.83) 1 2 1 2 1 2 Vc 2 fc (γ12 + γ22 )2 2 2 2 f Lc 1 σ11 σ22 − σ12 = |ζ1 ζ2 | (6.84) x1 2 γ12 + γ22 fc 1/2 2 V2 ) f (σ11 σ22 − σ12 = 2 (6.85) Vc fc (γ12 + γ22 ) 1/2 2 1 f σ11 σ22 − σ12 Lc = . (6.86) x2 2 fc γ12 + γ22 Where, f is the frequency, fc is the characteristic frequency, ζ1 = z1 − 1, ζ2 = z2 − 1, Lxic represent attenuation of the first P-wave and the second P-wave. Above results are for the low frequency range where the input frequency is less than the characteristic frequency. The Biot characteristic frequency is the frequency near which propagation and diffusion effects have approximately equal contributions so that the material shows the maximum attenuation. One can also derive similar relations for the high frequency range. When one plots relations (6.73) through (6.86) for low and high frequency ranges, one obtains Figs. 6.35 and 6.36.

Fig. 6.35. Sweep Curve for Velocity

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6 Applications

Fig. 6.36. Sweep Curve for attenuation

Figures 6.35 and 6.36 do not appear exactly same for the fast P-wave and the slow P wave; however, the overall trend is the same. Figures 6.35 and 6.36 show that there is a major change at the characteristic frequency. Biot (1956a) relates the characteristic to the viscosity and permeability as follows: ωc =

ηφ Kρ

(6.87)

where, ωc is the characteristic frequency (angular frequency), η is the viscosity, φ is the porosity, K is the intrinsic permeability (Darcy unit), and ρ is the fluid density. When ωc is determined from the experiment, the permeability K can be estimated from (6.87). Modification of (6.87) for familiar quantities in soil mechanics presents (6.88), fc =

φ·g 2π · k

(6.88)

where g is the gravity, f is the frequency in hertz, and k is the hydraulic conductivity in m/sec unit. Equations (6.87) and (6.88) assume Darcy flow. Similar equations are available for different flow conditions. For squirt flow, (6.89) is suggested as follows by Batzle et al. (2001), ωc =

Kα3 η

(6.89)

where, K is frame modulus and α is crack aspect ratio. Based on the Kozeny equation, Kelder (1998) also suggested the following equation, 2 · η · φ · ρb Vpf ko = (6.90) Vps · f (w) · [Kf · (Kb + 4G 3 )] where ko is the hydraulic conductivity, Vpf is the fast P-wave velocity, Vps is the slow P-wave velocity, ρb is the bulk density, Kf and Kb are respectively the fluid and bulk moduli, G is the shear modulus, and f(w) is a function of frequency that approaches one for w equal to zero. Recently, Yamamoto


155

(2003) reported a “super-k regime” for a more reliable imaging of permeability profile of highly permeable carbonate formations. The studies cited above show that a promising theoretical base is already in place. The main problem yet to be worked out is how to catch the characteristic frequency. The experimental program that follows this section presents a suitable methodology for solving this problem. 6.3.3 Applicability The applicability of the above equations has some limitations. These equations are based one the fact that the characteristic frequency or wave velocities are reliably obtained. Batzle et al. (2001) reported that for low permeability materials (k < 200 mD), the characteristic frequency could not be captured since the characteristic frequency is lower than the measurable range. However, they reported that the estimation of permeability for highly permeable materials such as sand would be easier to ascertain. Converted into a conventional geotechnical hydraulic conductivity, the expression k = 200 mD is somewhere near 10−6 m/s. This is a typical permeability range for silt to sandy soils. Therefore, application of the proposed method will be feasible for some soils. The quick calculation of Biot’s characteristic frequency by (6.89) gives the following results: Table 6.7. Approximate range of Biot’s characteristic frequency Permeability (m/s) −6

10 10−5 10−4 10−3

fc (kHz)

Remarks

500 50 5 0.5

ultrasonic ultrasonic sonic sonic

Researchers (Batzle et al. 2001, Yamato, 2003) presented the measured slow P-wave velocity up to 106 Hz or a higher frequency range. Of course these results are for rock specimens. The received signals for soils (from the same intensity of source signal) will be weaker than these same signals for rocks because of a higher attenuation. Fortunately, the widespread availability of spread spectrum signal processing technique and precision sensors make capturing this weak signal more effective than ever. Yamamoto (2003) and Xiang and Sabatier (2003) recently reported the implementation of spread spectrum technology called PSRN(Pseudo Random Noise). Yamamoto reported that he could achieve S/N ratio higher than 40% by using PSRN encoded signal. Also using the singular value decomposition method developed by Bregmann et al. (1989a,b), Yamanoto (2003) calculated the attenuation of the slow Pwave reliably.

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6 Applications

wave

Fig. 6.37. Equipment set up

The widely used acoustic experimental program is very similar to that of conventional surface wave technique. Figure 6.37 shows equipment set ups. This procedure requires a digital oscilloscope with one source and one receiver and triggering capability. Measurement technique is similar to that for the conventional seismic wave measurement. The oscilloscope records the source signal and received signal. Two signals will be shown in the oscilloscope screen, and the arrival time of slow P-wave is calculated. So far, the measurement scheme is about the same as that for a typical surface reflection test. For this purpose, a frequency controlled source is used instead of random frequency shock wave because the velocity of slow P-wave is also a function of frequency. Sweeping from low frequency to high frequency presents the variation of slow P-wave velocity or attenuation as shown in Figs. 6.35 and 6.36. From this sweep curve reveals a characteristic frequency. Once the characteristic frequency is found, the characteristic slow P-wave velocity is found. From this characteristic slow P-wave velocity or characteristic frequency, one can estimate the permeability with proper equations. Recently Song and Kim (2005) reported the application of the similar method to sandy soils as shown in Fig. 6.38. Song and Kim (2005) used the fast P-wave to capture the characteristic frequency. Figure 6.38 clearly shows that the characteristic frequency can be captured and may be used for the hydraulic conductivity computation. The hydraulic conductivity as measured in the laboratory by Song and Kim (2005) is 0.0000542 m/sec., while the predicted by the Biot equation is 0.000167 m/sec. This discrepancy, however, is not substantial considering the nature of the hydraulic conductivity. Moreover, the predicted hydraulic conductivity by Biot equation may not

6.4 Evaluation

157

damping

Saturated soil

frequency (Hz)

Fig. 6.38. Frequency response of damping characteristic of mortar sand (Song and Kim, 2005)

necessarily be exactly the same as that measured in the laboratory as pointed out by Berryman (2005). As the more experimental data are collected, the better relationship between the static (laboratory) hydraulic conductivity and dynamic hydraulic conductivity may be established.

6.4 Evaluation There are many methods to evaluate the hydraulic conductivity of soils. Typical laboratory tests include falling head tests and constant head tests. Both tests may be performed for disturbed and intact specimens. However, laboratory tests for intact specimens are very difficult to conduct; therefore, disturbed specimens are commonly used. Because of this unavoidable sample disturbance, laboratory test results may not be very reliable even though they are performed under well controlled experimental conditions. Typical field tests include in situ falling head tests and in situ constant head tests. Field tests are less prone to the sample disturbance; however, field tests are not performed in a highly controlled environment. For example, typical field tests are performed in drilled bore holes, and one does not have much control over the wall condition of bore holes. The bore hole diameter may be larger than it should be, or the bore hole wall may be coated with drilling fluid. Many times engineers do not have any other option but to use the bore hole as is. Another disadvantage of the field hydraulic conductivity tests is that it is time-consuming. Typically, such a test requires several days. Therefore the amount of test data is limited, and a detailed hydraulic conductivity profile can seldom be provided.

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6 Applications

The BAT is used to minimize the sample disturbance problem. It operates in a bore hole bottom. BAT uses a hypodermic needle to inject pressurized water into the ground and monitors the dissipation of this injected water with an associated piezometer. By analyzing the dissipation of injected pressurized water, it obtains the hydraulic conductivity of the soil at the bore-hole bottom. This method is one of the best methods invented so far; however, bore hole bottoms are usually contaminated with the drilling fluid and sludge. When the bore hole bottom is not clean (sometimes, thorough cleaning of bore hole bottom is impossible.), the BAT results may bring out the hydraulic conductivity of the drilling sludge, not that of the soils. Several methods to evaluate the hydraulic conductivity using the piezocone penetrometer have recently been developed. Piezocone is a device that penetrates into the ground and measures the pore pressure and penetration resistance. Rust et al. (1995) estimated the coefficient of consolidation from the pore pressure dissipation curve obtained during the resting time (that is required to reconnect the driving rod) Robertson et al. (2000) applied the same analytical method as the one used for the analysis of conventional dissipation curve for piezocone holding tests. Song et al. (1999) tried to estimate the hydraulic conductivity of soil from the pore pressure difference between u2 (the measured pore pressure at the shoulder of the conventional piezocone penetrometer tip) and u3 (the measured pore pressure at the cone shaft that is approximately 14 cm apart from u2 location of the piezocone penetration test). Comparison of the pore pressures at u2 and u3 (u3 is measured seven seconds after u2 ), shows that the pore pressure at u3 is usually smaller than that at u2 . The difference is caused by the dissipation of excess pore pressure in seven seconds, because the piezocone penetrometer moves with 2 cm/sec between u2 and u3 . Using this dissipation data, one can estimate the hydraulic conductivity of the soil. House et al. (2001) tried to estimate the pore pressure responses of soils by changing the penetration speed of the piezocone penetration test. They found that the soils with high hydraulic conductivity show no significant difference between measured pore pressures for two different penetration speed, and the soils with a low hydraulic conductivity show significant difference between measured pore pressures for two different penetration speed. When one has the difference of those two pore pressures, one can estimate the hydraulic conductivity of the soil. Voyiadjis and Song (2003) reported the calculation of the hydraulic conductivity based on back analysis of pore pressure measured during the penetration of piezocone. They used the coupled theory mixtures and soil plasticity to estimate the hydraulic conductivity. Using Voyiadjis and Song’s (2003) methods, one can obtain virtually continuous hydraulic conductivity profiles of the ground without additional time-consuming experiments. These recent efforts have contributed significantly to the easy evaluation of hydraulic

6.4 Evaluation

159

conductivity. They also show how desperate the engineers are to find more effective and reliable ways to evaluate hydraulic conductivities. Although the acoustic method is in line with these recent studies, it is a completely different method. This method does not use any penetrometer that may disturb grounds; it uses non destructive wave techniques, which do not disturb the structure of soils. Another benefit of this method is that it may provide the complete spectrum of the permeability of the ground when it is combined with the inversion technique. Used in conjunction with the inversion technique, the acoustic method can provide the tomographical view of hydraulic conductivities of a job site. Continuous and three-dimensional hydraulic conductivity information will enhance the current practice of geotechnical engineering substantially (e.g. design and analysis of soft soil problems).

7 Advanced Topics

Traditionally, coupled theory for soils was mainly focused on micromechanisms. However, modern theories of molecular and quantum mechanics combined with formidable computation power extends the coupled theory to the nano level. This chapter discusses some topics of Nano mechanics for geo-materials (7.1). Also, the coupled behavior of micro-mechanisms is different in saturated soils from the coupled behavior of other continua such as composite materials or metals. This chapter also discusses topics of coupled behavior of micro-mechanisms (7.2).

7.1 Nano-mechanics for Geotechnical Engineering 7.1.1 Introduction The properties of soils vary extensively. Naturally, the prediction of soil behavior is definitely challenging even with modern geotechnical engineering. For accurate predictions of soil behavior, sophisticated constitutive relations are currently used. However, the prediction of detailed behavior (e.g. fluidclay interactions) is extremely difficult; these sophisticated constitutive relations work with new input parameters that are not easy to determine. Some parameters, especially parameters for micro-mechanics, are known to exist, but testing methods are not well developed at present. Many researchers (including these authors) are striving to find these parameters. A new method that provides these material parameters rationally and predicts the behavior of soils will provide a quantum leap for the whole geotechnical community. In this section the authors present a method for implementing a Nano-mechanics approach into geotechnical Engineering to overcome the associated difficulties and to provide a new method for predicting and analyzing soil behaviors. Nano-mechanics is an emerging technology that deals with Nano (10−9 m) level particle size, that is the size of a cluster of atoms and molecules. Thus, Nano-mechanics is essentially based on quantum and molecular mechanics, and naturally it deals with the equilibrium states of atomic and molecular forces. One advantage of using Nano-mechanics is that its input parameters are well established physical quantities, such as electron charge, dihedral angle, electron permittivity, and so on. Another advantage of using Nanomechanics is that the detailed physical properties (or input parameters for

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continuum mechanics) of materials including soils are obtained from the statistical average behavior of atoms and molecules. The principal method in this concept is that one starts from Nano-level material behavior and continue to integrate to obtain the real (macro) scale material behavior without resorting to difficult input parameters. The authors have addressed many applications in nanocomposites (Voyiadjis et al., 2003; Meng and Voyiadjis, 2003; Srinivasan et al., 2005). The great challenge to this averaging and integrating process is the substantial amount of computation time required. However, promising timesaving averaging and integrating techniques have recently been developed, and some researchers (including one of the co-authors of this book) successfully predicted the behavior of materials such as carbon-nano-fibers within a reasonable computation time (Srinivasan et al., 2005). Soils of course are different from carbon-nano-fibers in many ways; however, both materials are made up with nothing but atoms and molecules, and the operating mechanics are fundamentally the same. Eventually, the behavior of soils can be understood and predicted in great detail using Nano-mechanics. The applications of this method are many. One of the applications may be to the NASA space program, such as the prediction of soils behavior in the Moon and Mars. 7.1.2 Brief History of Nano-mechanics Richard P. Feynman provoked the scientific community by predicting an emerging nanotechnology on December 29th, 1959 at the annual meeting of the American Physical Society at the California Institute of Technology in his speech “There’s Plenty of Room at the Bottom.” Regis (1995) comments that “Some in the audience (Many of them were physicists, after all, and they’d earned all their hundreds of advanced degrees by reading books, studying books, going to the library for books, big damned fat square volumes with page after page of dense text)” suspected Feynman’s nano-concept when he talked about writing a full volume of Encyclopedia Britannica on a pin head. Some of them took it as a typical dinner table joke.” However, nanotechnology emerged just as Feynman predicted, and is now a reality in this modern scientific world. The discovery of carbon nanotubes by Iijima in 1991 has inspired a number of scientific investigators to explore their unique properties and potential applications. In 1992, the eminent scientist K. Eric Drexler testified at the Senate Committee on Commerce, Science, and Transportation, Subcommittee on Science, Technology, and Space chaired by the ex vice-president Al Gore. He convinced the country’s leaders of the strategic potential of nanotechnology (Regis, 1995; US Congress, 1992). Since then, nanotechnology has found its way to broader and more profound applications because of its ability to make things smaller, faster, or stronger. Consequently, new materials with exceptional properties have emerged, allowing for the design of machines that will lead to new manufacturing paradigms. (CMP Cientifica,

7.1 Nano-mechanics for Geotechnical Engineering

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2002). So far, nanotechnology has been successfully focused on the fabrication of nano-materials. Subsequently, analytical works by nano-mechanics are required to better understand the underlying principles and mechanisms at atomic and molecular level. Nano-mechanics is a combination of molecular mechanics and quantum mechanics; because these are the fundamental building blocks of every mechanics, the same technique can be used for geo-continuum mechanics. In spite of advancement of nanotechnology and nano-mechanics in many scientific areas, it has had little influence and use in geotechnical engineering. This chapter addresses the application of nano-mechanics to the geotechnical area, its purpose being to improve the current state of practice in geotechnical engineering. 7.1.3 Nano-mechanics as a General Platform for Studying Detailed Behavior of Geo-materials Because of nano-level length scales and also pico to femto seconds time scales, nano-mechanics provides extremely detailed material behavior. Modern continuum mechanics (Cui et al. 1996; Jasiuk and Ostoja-Starzewski 2003; Kaliakin et al. 2000; Ling et al. 2002, 2003; Manzari 2004; Song and Voyiadjis 2005a; Voyiadjis and Song 2005b, Voyiadjis et al. 1999) and discrete mechanics (Anandarajah, 2004, 1994; Peters, 2004) approaches are very capable of providing macro-scale material behavior but their ability to provide detailed material behavior is limited. The contrasts of different scales are well depicted in Fig. 7.1. Some of the advanced continuum mechanics are capable of

Fig. 7.1. Comparison of length scales and time scales of different mechanics (Cygan, 2001)

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7 Advanced Topics

providing detailed material behavior to some extent; however, the determination of input parameters is still a challenging task. To analyze the detailed behavior of geo-materials such as clay-fluid interaction, nano-mechanics is the tool of choice at present (Smith, 1998; Shroll and Smith, 1999). Some of the merits of nano-mechanics are its extraordinarily well established theories and simple input parameters. One of the theories could be a potential energy equation shown in (7.1); some of the input parameters are traditional physical and chemical constants, such as permittivity of a vacuum and angle bend force constant which are already well defined in Table 7.1. ET otal = ECoul + EV DW + EBond

Stretch

+ EAngle

Bend

+ ET orsion

(7.1)

where ECoul , the Coulombic energy, and EV DW , the van der Waals energy, representing so-called non-bonded energy components, and the final three terms represent the explicit bonded energy components associated with bond stretching, angle bending, and torsion dihedral, respectively. Important features of energy components are summarized in Table 7.1. Using (7.1) and from the optimized energy or force field, one can compute material properties easily. One can compare the potential energy along a certain direction and equalize it to the strain energy in the same direction and compute spring constant or modulus. This is the key concept of AndersonParrinello-Rahmen theory (Ray and Rahman, 1984; Ray, 1988; Parrinello and Rahmen, 1981). A preliminary result by the authors using the above procedure is presented in Fig. 7.2. The major challenges to this process are the difficulties associated with the theory of molecular mechanics or quantum mechanics and the great amount of computation time. Yet this challenge is very much attenutated nowadays, since the concepts for molecular mechanics or quantum mechanics are well established. Ever since the establishment of the time-independent Schrödinger equation and the Born-Oppenheimer approximation that effectively decouples nuclear and electronic motions, molecular interactions are easily handled and computed. This challenge is also partly overcome by the advent of accessible molecular mechanics software such as LAMMPS, NAMD, BioCoRE, and Materials Studio. With these accessible software programs and the basic knowledge of Physics and Chemistry, one can even perform quantum level analyses satisfactorily. Dynamic conditions occur when one has interaction of two different stable molecular systems. When the two materials are mixed (allowed to interact), molecular interaction takes place and finally achieves a new equilibrium with new optimum (minimum) potential energy. The chemical procedure of obtaining the new equilibrium strictly follows molecular dynamics rules (Katti et al., 2004). Therefore, the interaction of two different materials is conveniently analyzed by molecular dynamics. When the chemical reaction is completed, one can use molecular mechanics to predict the physical properties. A part of computer outputs in this respect obtained by the Song and


165

Table 7.1. Important features of potential energy components Name Coulombic

Governing Equations + qi qj e2 ECoul = 4πε where qi rij o i=j

van der Waals (2nd term of Lennard-Jones 12-6 potential energy) Bond Stretch

Angle Bend

Torsion

Remarks

and qj represent the charge of the two interacting atoms (ions), e is the electron charge, and εo is the permittivity (dielectric constant) of a vacuum. Lennard-Jones Potential Energy EV DW 6 12 + = Do rRijo − 2 rRijo i=j

where Do and Ro represent empirical parameters. EBond Stretch = k1 (r − ro )2 where r is the separation distance for the bonded atoms, ro is the equilibrium bond distance, and k1 is an empirical force constant. EAngle Bend = k2 (θ − θo )2 where θ is the measured bond angle for the configuration, θo is the equilibrium bond angle and k2 is the angle bend force constant. ET orsion = k3 (1 + cos 3ϕ) where k3 is an empirical force constant and ϕ is the dihedral angle.

Remark The Coulombic energy is based on the classical description of charged particle interactions and varies inversely with the distances rij . The second term is the van der Waals energy representing the attractive molecular interactions.

The bond stretch term can be represented as a simple quadratic (harmonic) expression.

The energy equation for angle bend for a bonded system is typically expressed using a harmonic potential. The torsional dihedral interactions expressed with the dihedral angle ϕ is defined as the angle formed by the terminal bonds of quartet of sequentially bonded atoms as viewed along the axis of the intermediate bond.

Additional terms may be added to the total potential energy expression of (7.1), such as an out-of-plane stretch term for systems that have a planar equilibrium structure.

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7 Advanced Topics

Fig. 7.2. A part of computer outputs showing calculated material properties from molecular mechanics (Software: Material Studio)

Tentative Results

Polymer molecules are placed around CNT. (t =0 s.)

Olem iss Civil Engineering

System aft er m olecular dynamics simulation (15 ps.) This simulation w ould help us understanding interfacial mechanical properties.

Fig. 7.3. A part of computer outputs showing molecular dynamics (The left figure shows the initial condition when the polymer is introduced around the carbon nano tube. The right figure shows the condition after 15 pico seconds)

Al-Ostaz (2005) for a carbon nano-tube is shown in Fig. 7.3. It is believed that the same concept may be used to evaluate the interaction between clay minerals (e.g. Yazoo clay in Mississippi) and pore water. However, the process of applying nano-mechanics to geo-materials is not without difficlties. Although equations and minimization techniques for


167

potential energies are known to be correct, each equation or technique has its own intrinsic assumption(s) and approximation(s) which may not be quite reasonable for geo-materials (see Belytschko, 2005). A typical myth is “Molecular mechanics does not assume anything and solves the problem very accurately”. In reality, substantial research is required to evaluate and select the proper optimization technique(s) for geo-materials. 7.1.4 Nano-mechanics as a Tool to Study Macro-level Material Properties Through Continuumization The previous section described the capability of nano-mechanics to evaluate detailed properties of geo-materials. On the other hand, nano-mechanics is not as capable as continuum mechanics in providing and predicting macroscale material behavior because of its extensive computation time. Therefore one needs a carrier that bridges nano-mechanics and continuum mechanics and also provides nano-scale material behavior in detail as well as continuum scale macro-material behavior. With the carrier, one can make use of nano-mechanics in conjunction with the well established modern continuum mechanics described in Table 7.2. Essentially, nano-mechanics will supply the fundamental properties that continuum mechanics needs (e.g. continuum mechanics parameters shown in Table 7.2, and continuum mechanics will work out the micro and macro scale behavior of geo-materials. In this way, one can enjoy the full features of modern continuum mechanics without expending exhaustive efforts to evaluate input parameters. One of the promising techniques in continuum mechanics called RVE (Representative Volume Element) casts helpful light on the feasibility of combining nano-mechanics and continuum mechanics so that the detailed material behavior as well as macro-scale material behavior is obtained without consuming excessive computational costs. The underlying idea of RVE concept is based on the concept that the macro scale behavior is nothing but an averaged lower scale behavior. Upon implanting nano-mechanics in the lowest RVE of continuum mechanics, one may obtain detailed properties with minimum input parameters. Another technique called “Equivalent Beam or Truss Element Method” is also useful. These approaches represent the lattice structure of molecules into joints and beams in traditional numerical analyses. Elongation stiffness and torsional stiffness are obtained from the equalization of a bond stretch potential and a twisting potential in molecular mechanics to the elongation strain energy and the torsional strain energy in traditional mechanics. In this way the molecular level simulations are effectively performed in popular commercial software such as ANSYS. This method turned out to be quite effective for continua (Srinivasan et al., 2005).

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7 Advanced Topics Table 7.2. Micro-mechanical behavior in continuum mechanics

Micromechanical Behavior Rotation of particles

Interaction of particles

Damage of particles

Viscosity of pore fluid

Flow characteristics of pore fluid

Form of Governing Equations*

Remark

¯ s − ds α) ¯ W s = ξ(αd ¯ and ds are the where, W s ,ξ, α, plastic spin tensor, constant, back stress, and deformation, respectively.

Dafalias (1998), Cosserat (1909), Song and Voyiadjis (1999 to 2005), Voyiadjis and Kattan (1990, 1991) Di Prisco and Aifantis (1999), Zbib (1994), Zbib and Aifantis (1988), Voyiadjis and Song (2005), Song and Voyiadjis (2005a) Voyiadjis and Song (2005a)

ε˜˙vp = ε˙vp −k∇2 ε˙vp vp where, ε˜˙ ε˙vp v , and k are the homogenized strain rate, visco-plastic volumetric strain rate and a constant, respectively. ∂g , where, ε˙dij , ε˙dij = λ˙ d ∂σ ij λ˙ d , g and σ are the damage strain, damage multiplier, damage potential, and stress, respectively. )

f = fs − po [1 + ηvp

3 2

√p¯˙ )1/m1 ] ≡ 0 Bii

where, f, fs ,po, ηvp ,p¯˙ , Bii, and m 1 are the dynamic yield surface, static yield surface, mean principal stress, viscosity, time rate of the mean stress, the differentiation of f with stress, and a constant, respectively, −div [(nw /ρw ) K ws (grad P w − ρw b + ρw a w )] + div v s = 0 where, n, ρs , ρw aw K ws v s Pw and b are the porosity, mass density of the soil, mass density of the water, acceleration of water, permeability tensor, solid velocity, pore water pressure, and body force vector.

Perzyna (1963, 1966, 1988), Song and Voyiadjis (2005a), Voyiadjis and Song (2005)

Coupled theory of mixtures Biot (1955, 1978), Prevost (1980, 1982), Muraleetharan et al. (1994), Schrefler et al. (1990), Wei and Muraleetharan (2002) Song and Voyiadjis (1999 to 2005) Voyiadjis and Song (2000 to 2005)

• Equations may vary for different researchers. These equations are generally accepted forms.

Detailed concepts and comparisons of these two methods are shown in Table 7.3. Using this continuumization technique, researchers, including one of the co-authors of this study (Srinivasan et al., 2005), have successfully predicted the behavior of materials such as carbon nano-fibers based on ad-


169

Table 7.3. Features of averaging techniques Method RVE

Governing Rules* A = V1 AdV v

Where, A is the property at upper scale, V is the volume of RVE, A is the property at lower scale.

Remark Similar to homogenization technique in continuum mechanics. Full capability of incorporating molecular interactions. Belytschko and Xiao (2003) Huang and Jiang (2005) Liu et al. (2005) Belytschko (2005) Song and Al-Ostaz (2005)

Equivalent beam or truss element

Ku = f where, u = [uxi , uyi , uzi , θxi , θyi , θzi , uxj , uyj , uzj , θxj , θyj , θzj ]T f = [fxi , f yi , f zi , mxi , myi , mzi , fxj , f yj , f zj , mxj , myj , mzj ]T Kii Kij K= Kij Kjj

This is the continuum expression of molecular mechanics. Input parameters for stiffness, such as Young’s modulus, was obtained by comparing the potential energy for stretching to the equivalent strain energy as follows: EBondStretch = k1 (r − ro )2 = Eaxialstretch = 12 EA (∆L)2 L This method is not very effective for simulating dynamic procedure such as interaction of minerals and pore fluid. Li and Chou (2003) Odegard et al. (2001) Ostoja-Starzewski, (2002) Srinivasan et al. (2005) Wang et al. (2005)

• Equations may vary for different researchers. These equations are generally accepted forms.

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vanced cotinuumization algorithms (Belytschko and Xiao 2003; Belytschko 2005; Li and Chou 2003; Odegard et al. 2001; Ostoja-Starzewski, 2002). Soils, however, have both continuum properties and particulate properties. For example, sands are continua inside the grain boundary; they are discrete media outside the grain boundary. Therefore, nano-mechanics provides fundamental properties to discrete element, and the continuumization process between discrete element and continuum (Cosserat brothers 1909; Anandarajah 2004; Tordesillas et al. 2004; Peters, 2004) will provide macro-behavior. The Nano-mechanics approach for soils is quite challenging and must be different from that for other continua. The rational procedure shall be as follows: 1) Implement nano-mechanics for grains and grains-liquid interaction. 2) Obtain the material properties for discrete particles and surrounding liquid 3) Use a continuumization technique to expand the discrete properties to continuum properties (Maiti et al., 2004) presented a different method for averaging called “DPD (Dissipative Particle Dynamics)” that is quite frequently used in Chemistry to analyze the interaction of two different fluids. The DPD approach utilizes the positions and momenta of “fluid droplets” illustrated in Fig. 7.4 rather than individual atoms for dynamic analysis. Therefore, DPD technique solves molecular dynamics in terms of RVE, and it may be used to compute the equilibrium condition for long length and time scales. Maiti et al., 2004 revised DPD theory to incorporate the interactions between carbon nano-tubes and fluid, and they obtained the equilibrium morphology that can be used as an input to traditional finite element codes. This method is partly similar to the RVE concept in continuum mechanics; but it is also different because it is still at molecular level; it is commonly called “coarse grained” molecular dynamics. The revised DPD (Maiti et al., 2004) therefore is equivalent to a molecular level RVE, and it requires no extra averaging scheme. As a result, DPD technique seems very cost effective to study interactions between clay particles and the pore fluid. The material properties from nano-mechanics will be transferred to the continuum properties by RVE and accompanying integration technique. The link between nano-scale molecular mechanics and macro-scale continuum mechanics, therefore, may be summarized as follows: • Clayey Soils: Molecular mechanics with DPD (or similar mechanism) – Lowest RVE for continuum mechanics • Sandy Soils: Molecular mechanics to evaluate grain properties – DEM to evaluate grain interactions – Lowest RVE for continuum mechanics The linking procedure is not easy; nevertheless, the preliminary research has shown quite convincing possibilities.


171

Fig. 7.4. Fluid droplets in DPD (Maiti et al., 2004)

This method also has some challenges. Traditional continuumization techniques are primarily good for the continua but not for particulate media. DPD technique is promising for clayey soils; however, further extensive research is necessary before it becomes fully feasible. 7.1.5 Space Science Application Nano-mechanics predicts material properties based on the chemical, physical and electrical interactions of the constituents. Its advantage in space exploration is evident: whereas performing physical or chemical experiments would be extremely difficult, a nano-mechanics approach can reasonably predict the properties and behavior of new materials in the outer space (insofar as one knows the chemical composition of the materials). One of the great dreams of human beings is to create habitats on the Moon or Mars. The behavior of materials will be different on the Moon from their behavior on the Earth. The properties of muscovite and montmorillonite in wet condition are very different from each other, but they are similar in dry conditions. The surface of the Moon and Mars are completely dry, and properties of materials there will be very different from their properties on Earth. It is predicted that the behavior of these materials in outer space is very different from that on Earth; geotechnical engineering based in nano-mechanics cast the new possibility in this aspect for it can predict the material behaviors in such extreme conditions. One example of a material that calls for nano-mechanics is beta quartz formed at high temperature and pressure – a material that does not exist in normal conditions at thte earth’s surface. Nano-mechnics makes the analysis of these special materials possible without much difficulty.

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7 Advanced Topics

Nano-mechanics for geo-materials is an area of science that is still growing and which stands in need of much more research. However, it is quite sure that modern nano-mechanics has a strong potential for providing a new paradigm for geotechnical engineering.

7.2 Coupled Behavior of Micro-Mechanisms 7.2.1 Introduction The macro-behavior of geo-materials is the result of the complex interaction of the coupled micro-behaviors of many constituent elements. Micromechanical behavior includes the rotation of grains, grain interactions such as grain interlocking, the viscosity of the pore fluid, and the integrity of grains (Voyiadjis and Song, 2002). The rotation of grains affects the equilibrium equations because part of the applied energy is used for the rotation of the grains. There are many ways to consider the rotation of grains; this study accounts for it by employing the concept of plastic spin. Grain interaction affects the stress-strain redistribution and wellposedness of numerical methods. When soils with lower than the critical void ratio are subject to shear, the dilation (volume expansion) occurs primarily along the shear band; the soil particles along the shear band are more highly stressed than those in the surrounding area; and the higher stress along the shear plane is transferred to the surrounding area followed by the volume expansion. This mechanism reduces the stress in the shear band, and thus strain is reduced accordingly. The reduced strain also contributes to the wellposedness of the numerical methods as discussed in Sect. 4.5.3. Due to the stress (or strain) redistribution (or homogenization), this phenomenon is also termed (almost officially) the homogenization mechanism. The viscous property of the soil affects the rate dependency of materials. In localized shearing zones such as the shear band region, the strain rate is much higher than that outside the shear band. Therefore, the material within the shear band responds more strongly than the material outside the shear band. This phenomenon tends to expand the shear band to the neighboring area. Rate dependency, therefore, brings out another kind of localization or homogenization mechanism. The integrity of grains may be degraded during the loading and unloading process because of particle crushing, micro-cracks or wear-out of sharp edges. The degradation of the friction angle due to particle crushing is well documented by de Beer (1963), Lee (1965), and Lee and Seed (1967). This phenomenon will cause the reduction of the modulus. Desai and Zhang (1998) and Desai et al. (1997) discussed the reduction of modulus by “disturbed state”. Ultimately all damage mechanisms will cause the reduction of the macroscopic stiffness of the soils. For the localized shearing zone such as the

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shear band, the reduction in the stiffness of the material will cause the softening of the material with localized high strains. This mechanism enhances the strain localization. From the above discussion, one can see that the micro-mechanical behavior is coupled. Some mechanisms enhance localization while some mechanisms enhance homogenization. Saturated soils are essentially a two-phase material (e.g. solid grains and pore water). Interaction between the two components of two-phase material produces effective stresses and pore pressure. The effective stress and pore pressure compete with each other and reduce each other’s magnitude. Therefore the effects of the above micro-mechanical mechanisms are expected to be even more complicated in soils. Saturated soils are also pervious, which in addition causes the response of the soils to be affected by the hydraulic conductivity. This section investigates the coupled effects of the above mentioned micromechanical mechanisms on the behavior of saturated soils around a penetrating object. Several factors – the rotation of particles, the interaction of particles, the rate dependency, the soundness of the particles, and the coupling of particles with pore fluid – are all incorporated in the anisotropic modified Cam Clay model through the plastic spin, the gradient theory, the viscoplasticity, the damage theory, and the coupled theory mixtures, respectively. Material parameters are not easy to define for the above micro-mechanical behavior; therefore, a parametric study is performed to estimate the proper material properties. 7.2.2 Simplification of Equations Incorporating the Physical Behavior of Soils The rate-dependent anisotropic modified Cam Clay model with strain gradient and plastic spin are incorporated with the coupled theory of mixtures as presented in Sect. 4.5. The formulations presented in the above chapter are further simplified, reflecting the physical behavior of soils. In this way, many parameters introduced in above sections are simplified for practical purposes. Viscosity-dependent rate dependency is reduced to (7.2) as shown by Horsky (2002); m p¯˙ ≡0 (7.2) f = fs − po 1 + η where fs is the static yield surface, η is the viscosity parameter that has sec−1 unit, and m is a dimensionless parameter. In this work m is used instead of 1/m1 to reduce numerical instabilities. Equation (7.2) also assumes a constant Bii . A non-local accumulative effective visco-plastic strain rate is introduced as follows; (7.3) ε˜˙vp = ε˙vp −k∇2 ε˙vp

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Fig. 7.5. Finite element mesh for the piezocone penetration analysis

Considering that the effective visco-plastic strain gradient is related to the volume change of the material (Di Prisco and Aifantis, 1999), (7.3) is further simplified as follows; ε˜˙vp = ε˙vp −K ε˙vvp

(7.4)

where, ε˙vp v is the visco-plastic volumetric strain rate. Equation (7.4) naturally accommodates strain reduction for dilative behavior and strain increase for the contractive behavior depending on the sign of K. Numerical Simulation of the Cone Penetration Test To investigate the performance of the proposed formulations, numerical simulations of the cone penetration tests are performed and compared with the experimental results. For the numerical simulations, the cone penetration tests are modeled as shown in Fig. 7.5. The detailed procedure of numerical simulation is also shown in Sect. 6.1.


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Fig. 7.6. Gradation of K-33 specimen

Model Calibration To calibrate the fundamental model parameters, anisotropic triaxial test results for artificially manufactured K-33 soil (33% Kaolinite, 67% Sand) carried out by Kim (1999) are used. The fundamental properties of K-33 soil are shown in Table 7.3 and Fig. 7.6. Determination of material properties or parametric studies for rate dependency, gradient, and damage are explained in each relevant section. 7.2.3 Rate Dependency Rate dependency of the cone penetration test was analyzed for 7 different penetration velocities, e.g. 0.02 cm/s, 0.2 cm/s, 2 cm/s, 5 cm/s, 10 cm/s, 20 cm/s, and 200 cm/s. To determine the material properties η and m of (7.2), a series of parametric study was carried out. Trial η was changed from 0.1 to 100,000 (η is infinity for rate independent materials with the assumption of positive m). Trial m was changed from 0.1 to 1. The responses of the cone penetration test turned out to be most reasonable to these parameters when η is approximately 10 and m is 1. These numbers are used for the analyses. The pore pressure responses of the cone penetration tests for these 7 velocities are shown in Fig. 7.7a. Figure 7.7a shows the magnitude of the excess pore pressure is low at the high permeability, and it is high at the low permeability. This trend agrees with the expected behavior. Figure 7.7a also shows the two-stage rate dependency. The rate dependency is minimal when the penetration velocity is lower than 10 cm/s. The rate dependency is substantial when the penetration velocity is higher than 10 cm/s. The rate dependency is subdued again when

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7 Advanced Topics Table 7.4. Basic material properties Parameter

Value

Units

Compression index, λ Recom pression index, κ Initial void ratio, eo Poisson’s ratio, ν Slope of critical line, M Unit Wt. of soil, γt Depth (from ground surface) Unit Wt. of water, γw Atterberg Limit Specific Gravity, Gs Water Content, w Undrained Shear Strength, su

0.11 0.024 1.0 0.3 1.16 1.8 20 1.0 LL=20, PL=14 2.67 18.56 80

dimensionless dimensionless dimensionless dimensionless dimensionless ton/m3 m ton/m3 % dimensionless % kPa

Fig. 7.7a. Excess pore pressure response of cone penetration test for different penetration velocities and different hydraulic conductivities

the penetration velocity is higher than 20 cm/s. Analysis for the penetration velocity higher than 200 cm/s. was not performed in this analysis because the additional dynamic effects need to be considered for the analysis of such a fast penetration. This variation of excess pore pressure with the penetration velocity is somewhat unexpected because continuously increasing excess pore pressure with increasing penetration velocity was expected. To analyze this strange behavior, Figs. 7.7b and 7.7c are plotted below. Figure 7.7b shows the magnitude of the effective stress at the cone tip is higher for high permeability soils (silty soils), while it is lower for the low permeability soils (fat clays). This is because one has high excess pore pressure in low conductivity soils. This trend is a generally expectable behavior.


177

Total Vertical Stress (kPa)

Fig. 7.7b. Effective stress response at the cone tip for different penetration velocities and different hydraulic conductivities

715 k=10-15 m/s

710

k=19-9 m/s

705

k=10-8 m/s k=10-7 m/s

700 695 690 685 680 675 0.01

0.1

1

10

100

1000

Penetration Velocity (cm/S)

Fig. 7.7c. Total stress response at the cone tip for different penetration velocities and different hydraulic conductivities

Figure 7.7b also shows that the variation of the effective stress at the cone tip decreases with increasing penetration velocity. But it does not show substantial difference for different hydraulic conductivities when the penetration velocity exceeds 2 cm/s. In other words, the magnitude of the effective stress at the cone tip changes with the permeability only when the penetration velocities are lower than 2 cm/s. The reasons for this behavior are not confirmed here due the absence of existing experimental data to study such phenomena. Therefore Fig. 7.7c is plotted for a better understanding of this behavior.

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Fig. 7.8. Influence of rate of penetration on cone resistance in a varved clay (reprinted form Roy et al., 1982)

Figure 7.7c shows the total stress at the cone tip for different penetration velocities and different hydraulic conductivities. It shows essentially rate independent total vertical stress at the low penetration velocity (when velocity is less than 0.2 cm/s), reduced total vertical stress at the intermediate penetration velocity (when velocity is less than 10 cm/s and larger than 2 cm/s), sharp increase of total vertical stress at the critical penetration velocity (when velocity is between 10 cm/s and 20 cm), and subdued response at the higher penetration velocity (when velocity is higher than 20 cm/s). This behavior is quite interesting and unprecedented. However, comparison of Fig. 7.7c with Roy et al. (1982) and Bemben and Myers (1974) shows that the soil behavior exhibited in Fig. 7.7c is the expected behavior for saturated soils. A copy of a figure from Roy et al. (1982) is presented in Fig. 7.8 for completeness. Figure 7.8 shows variation of the normalized cone resistance qc with respect to the preconsolidation pressure. To compare Fig. 7.7c and Fig. 7.8, one needs to understand that the total vertical stress in Fig. 7.7c multiplied by the cross sectional area of the cone penetrometer is approximately the same as the qc in Fig. 7.8. Therefore, Figs. 7.7c and 7.8 should have linear relationships, and the trend should be the same. These two figures show exactly the same trend. Bemben and Myers (1974) attributed the shape of the curve to a combination of viscosity and pore pressure drainage effects. They further elaborately predicted the mechanism of this behavior such as “up to about 0.05 mm/s drained conditions apply and above 5 cm/s undrained conditions apply”. Based on the results obtained from Bemben and Myers (1974) and Roy et al. (1982), Lunne et al. (1997) speculated that the background mechanisms are as follows:


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For very slow rates of penetration qc is predominantly of a drained nature. As the rate of penetration increases, qc decreases due to the decrease in effective stress and reduction in strength. As the penetration rate increases, the viscous forces offset the strength reduction and the curve will pass through a minimum. Then viscous forces will tend to dominate the process and qc will increase.

This mechanism was the hypothesis predicted by Lunne et al. (1997) when Bemben and Myers (1974) and Roy et al. (1982) reported the test results. At this time, however, one can confirm that this mechanism is a fact, and the mechanisms that are displayed in Figs. 7.7a to 7.7c are generally true. 7.2.4 (Strain) Gradient Gradient dependency of the cone penetration test was simulated with two different gradient conditions. Gradient parameter K of (7.4) was determined from a series of parametric studies. Parameter K was increased from 0 with 0.1 increments. K equals zero implies zero gradient effect. Bigger K means a more pronounced effect of the gradient. From the parametric study, it turned out that K cannot be larger than 0.3. At that point, the singularity condition was observed for a certain element. In (7.4) one can observe that a very big value of K will change the sign of the strain. This implies that a very big value of K will change dilative behavior to contractive behavior or contractive behavior to dilative behavior. These results do not represent the correct behavior of the gradient. Therefore, two gradient conditions (e.g. K = 0.1 and K = 0.2) are used for further study. Gradient causes the homogenization of the stress and strain, and therefore, one can expect the reduced maximum pore pressure and maximum effective vertical stress. Figure 7.9a shows these behaviors. Figure 7.9a demonstrates the noticeable effects of the gradient on the excess pore pressure. One observes that the flat curve of the excess pore pressure is extended up to a penetration velocity of 20 cm/s when the gradient is incorporated. The effects of the gradient coefficient K are not especially large; however, bigger values of K produce smaller excess pore pressure. Figure 7.9b shows the effects of gradient on the effective vertical stress. The overall trend is similar to Fig. 7.9b. The effects of gradients on the magnitude of the effective vertical stress are not pronounced. The authors believe that slightly reduced excess pore pressure in Fig. 7.9a causes slightly high effective stress in Fig. 7.9b. However, that effect is minimized when the penetration velocity is higher than 2cm/s. Therefore, the total vertical stress is not significantly affected by the gradient. Figure 7.9c shows the variation of total stress at the cone tip: the response is similar to Fig. 7.9c. However, the flat part of the total stress is extended again to a penetration velocity of 20 cm/s. when the gradient is considered.

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Excess Pore Pressure (kPa)

590 580 570

k=10-15 m/s. K=0 k=10-15 m/s. K=0.1 k=10-15 m/s. K=0.2 k=10-9 m/s. K=0 k=10-9 m/s. K=0.1 k=10-9 m/s. K=0.2

560 550 540 530 0.01

0.1

1

10

100

1000

Penetration Velocity (cm/s) Fig. 7.9a. Excess pore pressure response at the cone tip for different penetration velocities, hydraulic conductivities, and gradient constants

Effective Vertical Stress (kPa)

160 150 140 130 120 110 100 0.01

k=10-15 m/s. K=0 k=10-15 m/s. K=0.1 k=10-15 m/s. K=0.2 k=10-9 m/s. K=0 k=10-9 m/s. k=10-9 m/ s. K=0.2 K=0.1 k=10-9 m/ s. K=0.2 k=-9,kd02

0.1

1

10

100

1000

Penetration Velocity (cm/s) Fig. 7.9b. Effective stress response at the cone tip for different penetration velocities, hydraulic conductivities, and gradient constants

7.2.5 Damage Damage dependency of the cone penetration test was simulated with two different damage magnitudes. Damage parameters A1 and A2 of (4.77) were determined from a series of parametric studies. Considering that the skeleton


181

Total Vertical Stress (kPa)

715 710 705 700

k=10-15 m/s. K=0 k=10-15 m/s. K=0.1 k=10-15 m/s. K=0.2 k=10-9 m/s. K=0 k=10-9 m/s. K=0.2 k=10-9 m/ s. K=0.1 k=-9,kd02 k= 10-9 m/s. K= 0.2

695 690 685 680 675 0.01

0.1

1

10

100

1000

Penetration Velocity (cm/s) Fig. 7.9c. Total stress response at the cone tip for different penetration velocities, hydraulic conductivities, and gradient constants

stiffness of soils is much smaller than that of the soil particle, a smaller magnitude of damage is assumed. One or two percent of damage is assumed from a 200 kPa to a 1000 kPa confining pressure. These assumptions correspond to A1 = 24.04, A2 = −0.23535 and A1 = 50, A2 = −0.35366, respectively. Figure 7.10a shows the variation of excess pore pressures with different damage magnitudes. For higher damage, lower excess pore pressures are observed. However, sharp changes of excess pore pressures at the critical penetration velocity are not observed for the damaged condition in Fig. 7.10a. This result is believed to be effected by the reduced stiffness of particles caused by damage at high penetration velocity. Figure 7.10b shows the variation of the effective stress at the cone tip with different damage magnitudes. It shows higher effective stresses at higher damage magnitudes. This phenomenon is somewhat unexpected since damage causes a reduction in the physical quantity. The authors speculate this phenomenon is due to the interaction of grain damage and excess pore pressure. Excess pore pressure in Fig. 7.10a is decreased for higher damage. The decrease in excess pore pressure causes the increase in the effective stress although damage might decrease the effective stress itself. However, at the present the authors do not have experimental data to support this hypothesis. Total stress variation at the cone tip is shown in Fig. 7.10c. The overall shape is similar to Fig. 7.9c. The curves with damage show smaller total stresses.

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590 570 550 530 510 490

D=0% D=0.1% D=0.2%

470 450 0.01

0.1

1

10

100

1000

Penetration Velocity(cm/s) Fig. 7.10a. Excess pore pressure response at the cone tip for different penetration velocities, different hydraulic conductivities, and different damages (gradient constant = 0.2)

Effective Stress (kPa)

220 D=0% D=0.1% D=0.2%

200 180 160 140 120 100 0.01

0.1

1

10

100

1000

Penetration Velocity (cm/s)

Fig. 7.10b. Effective stress response at the cone tip for different penetration velocities, hydraulic conductivities, and damages (gradient constant = 0.2)

7.2.6 Permeability Effects of permeability on the cone penetration tests are reported by Voyiadjis and Song (2003). Figure 7.11 is replotted from Voyiadjis and Song (2003). Figure 7.11 shows that the Piezocone penetration pore pressure is sensitive to the permeability only within a certain range of the permeability. However, the penetration pore pressure is not very sensitive to the permeability at very high or very low permeability ranges. From Figs. 7.7b to 7.10b one can also see that the effective stress at the cone tip shows little change for different hydraulic


183

710 D=0% D=0.1% D=0.2%

Total Stress (kPa)

700 690 680 670 660 650 640 0.01

0.1

1 10 100 Pe ne tration Ve locity (cm/s)

1000

Fig. 7.10c. Total stress response at the cone tip for different penetration velocities, hydraulic conductivities, and damages (gradient constant = 0.2)

Fig. 7.11. Variation of excess pore pressure with hydraulic conductivity (Voyiadjis and Song, 2003)

conductivities, except at very slow penetration velocities. These results show that the permeability affects the excess pore pressure response rather than the effective stress response. These responses are more pronounced within a certain range of permeability (10−9 m/sec ≤ k ≤ 10−6 m/sec).

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7.2.7 Anisotropy and Plastic Spin In this section a comparison is made to see the effect of anisotropic and isotropic models. Figure 7.12 shows the excess pore pressure contours obtained from the finite element analysis. Figures 7.12a and 7.12b show the results of anisotropic modified Cam Clay model (AMCCM) with the plastic spin and without the plastic spin, respectively. Figure 7.12c shows the results of isotropic modified Cam Clay model (IMCCM) with a mean principle stress that is the same as the vertical effective stress (hydrostatic stress condition). Figure 7.12d shows the IMCCM with a mean principal stress that is the same as Fig. 7.12a. The results of IMCCM with initial anisotropic stress condition are shown in Fig. 7.12e. The initial confining condition is the same as in Fig. 7.12a. Figure 7.12f shows the results of AMCCM with initial isotropic stress condition at which the mean principle stress is the same as the vertical effective stress. In Figs. 7.12a and 7.12b one can see that the spatial distributions of the excess pore pressures in both figures look alike whether the plastic spin is incorporated or not. However, different maximum excess pore pressures are obtained in each case. This is due to the high concentration of microstructural changes at the vicinity of the cone tip where the strains are extremely large. This behavior is shown indirectly in Fig. 7.13 by the tensor component (N2133 ) of the plastic spin. In the region of the cone face adjacent to the shoulder, one notes that most of the plastic spin activity occurs. This is also the region of maximum strain (Voyiadjis and Abu-Farsakh, 1997). Comparison of Fig. 7.12a with Fig. 7.12c, Fig. 7.12d, and Fig. 7.12e shows the difference between the AMCCM and the IMCCM constitutive relations. Figure 7.12a, Fig. 7.12d, and Fig. 7.12e essentially have the same mean principle stress. The difference is the initial confining condition and the applied soil model. From Fig. 7.12a and Fig. 7.12d, one can see that the pore pressure response of AMCCM is substantially higher than that of IMCCM (for the same mean principle stress). However, Fig. 7.12c shows that when one uses a confining pressure that is the same as the vertical effective stress in IMCCM, one obtains almost similar results with the AMCCM and the plastic spin. However, it should be noted that this agreement is mainly due to the higher mean confining pressure. Figure 7.12e shows that even with an anisotropic initial confining condition for IMCCM, the response is not much different when the mean principle stress is the same. Figure 7.12f shows that the combination of high initial mean principle stress (same as vertical effective stress) with AMCCM results in the highest pore pressure response. From this discussion, one can see that the pore pressure response of cone penetration tests is substantially affected by the applied model and applied initial confining condition. However, one should acknowledge that the combination of anisotropic initial stress condition and AMCCM with plastic spin is the most realistic condition.

(b) AMCCM without plastic spin

Fig. 7.12. Excess pore pressure contours of AMCCM and IMCCM

(a) AMCCM with plastic spin

7.2 Coupled Behavior of Micro-Mechanisms 185

Fig. 7.12 (Continued)

stress same as vertical stress

(c) IMCCM for mean principle

(d) IMCCM for the same mean principle stress as (a)

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Fig. 7.12 (Continued)

(e) IMCCM for same initial anisotropic stress condition as (a)

(f) AMCCM with initial isotropic stress condition as (b)

7.2 Coupled Behavior of Micro-Mechanisms 187

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Fig. 7.13. Distribution of plastic spin around a cone tip

Figure 7.14 shows the variation of the cone-penetrometer-induced pore pressure with different permeability. It is noted that Fig. 7.14 is for the case of ground depth 20 m (saturated) and the stiffness similar to the LSU (Louisiana State University) calibration chamber test specimen (recompression index κ = 0.024, Poisson’s ratio ν = 0.3). For the different soils such as stiffer soils, the back-bone curves in Fig. 7.14 may be shifted up especially at lower permeability range. From Fig. 7.14, one can know that the pore pressure response from the AMCCM (the one which considers both initial anisotropy and induced anisotropy) is closer to the experimental data than that from IMCCM. This condition (the one which considers both initial anisotropy and induced anisotropy) is closer to true field conditions; thus the results must agree well with the field test results. This behavior is quite rational considering the realistic behavior of soils always reflect the anisotropy and some degree of texture changes during shear.

7.2.8 Experimental Verification of Coupled Pore Pressure Around a Cone Tip This chapter experimentally investigates the distribution of excess pore pressures along the surface of a penetrating object during the penetration. A penetrometer embodied with three pore pressure measuring units is used, and tests are performed in the calibration chamber facility at Louisiana State University. A large strain elasto-plastic coupled theory of mixtures that incorporates the modified Cam-Clay model is used for analyzing the problem. To incorporate the effects of micro-mechanical grain rotations at the close vicinity of the penetrating object, the back stress and plastic spin are included


189


800 Size of rectangles shows the range of test data Solid circles represent the LSU/CALCHAS test results

600 500 400 300 200

All Isotropic Initial Aniso + Induced Aniso

100

Initial Iso + Induced Aniso

1.0E-16

700

1.0E-14

1.0E-12

1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

0 1.0E+00

Hydraulic Conductivity (m/sec)

Fig. 7.14. FEM results of pore pressure prediction of PCPT and experimental results

in the numerical model. The results show that the maximum excess pore pressure is observed at the tip (face) of the penetrating object while smaller excess pore pressure is observed at the shaft of the penetrating object. A similar trend is expected for other researches (Torstensson, 1977; Houlsby and Teh, 1988; Teh and Houlsby, 1991), and the same behavior is confirmed in a highly controlled test environment in this study. It is also shown that the interaction and accompanying dissipation of the shear-induced pore pressure and compression-induced pore pressure play a major role for the excess pore pressure response at the shaft of the penetrating object. The Three-Piezo-element Miniature Penetrometer The three-piezo-element miniature penetrometer used to obtain test results was specially designed and fabricated especially. A photograph of the penetrometer is shown in Fig. 7.15. It has a projected penetrometer area of 2 cm2 and an apex angle of 60◦ . The maximum pore pressure capacity is 700 kPa. The penetrometer tip is detachable so that u1 and u2 porous elements are exchangeable. The u1 , u2 , u3 numbering system follows the typical numbering system for the piezocone penetrometer. The u4 is numbered last because it is behind u3 . The penetrometer can have two set-ups depending on the combination of porous elements. The possible configurations are pore pressure measurements at u1 , u3 , and u4 or at u2 , u3 , and u4 . The pore pressure transducers are Precision Measurement Miniature Pressure Transducer Model 150F full bridge electric resistor type strain gauges.

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u4 (with porous tip attached)

15 (adjustable)

Diameter = 1.61 cm Area = 2 cm2

0.61 1

6

For u1, u3, u4 configuration the current penetrometer tip is removed and the new penetrometer tip with u1 is attached.

Sleeve is exchangeable for adjusting u3 – u4 distance

u3 (with porous tip removed) Circular dot is the sensing membrane of piezometer

u2 (with porous tip removed ) Small dots are the leader holes for pore pressure transmission

Penetrometer tip, u1 is attached here for u1, u3, u4 configuration

Fig. 7.15. The three-piezo-element miniature penetrometer (showing u2 , u3 , u4 configuration. Units are in cm.)

For the u1 or u2 location, conventional leader holes are used to transmit the water pressure to the sensor (see Fig. 7.15). During the saturation process, a hypodermic needle is used to inject water into the leader hole. However, for the u3 and u4 locations, because the sensing membrane is directly placed just next to the porous protective cover (see u3 in Fig. 7.15), the leader hole


191

is not needed. With this configuration, the saturation process for u3 and u4 piezometers is substantially simplified and the volumetric displacement during the pore pressure measurement is minimized. Measurements for friction resistance or end resistance are not performed for this penetrometer. Only three pore pressure transducers are used because of the limited space in the penetrometer body. For the end resistance and friction data, independent penetration tests are carried out with another miniature penetromenter by Lim (1999). This calibration chamber test system and sample preparation procedure are essentially the same as those used by Lim (1999) and Kim (1999). The essentials of the test system and sample preparation procedure are outlined below. The details are given in the works of Lim (1999) and Kim (1999). The specimen is essentially a combination of sand, kaolin and deionized water at a water content of twice the liquid limit. This water content is found to be adequate to minimize the segregation of soil grains and air entrapment in the slurry. The grain size distribution of the kaolin and fine sand is shown in Fig. 7.6. Mixing is done in two large 40 gallon polyethylene tanks using a specially designed, hand-held heavy duty agitator. A mixture of 33% kaolin and 67% Edgar fine sand by weight is used to prepare the K-33 specimens. The Atterberg limits of the soil mixture are shown in Table 7.4. Slurry is placed very carefully inside the consolidometer (H×D = 1624 mm × 525 mm) with a large spoon. A vertical consolidation pressure of 138 kPa is gradually applied to the slurry by an ENERPAC hydraulic pump. The vertical pressure is selected in order to obtain an initial soil specimen of minimum strength to withstand its own weight. Pore pressures are monitored at two different elevations and at various radial distances. During the slurry consolidation, a Ko condition is maintained due to the rigid wall structure of the slurry consolidometer. The drainage is allowed at the top and bottom of the specimen through the filter papers, but the drainage is not allowed at the side wall. At the end of the first stage of slurry consolidation, the specimen enclosed in the membrane is transferred to the calibration chamber (H × D = 945 mm × 525 mm) where it is subjected to a second stage of consolidation at higher stresses. The calibration chamber has similar boundary conditions as Houlsby and Teh (1988): flexible walls and rigid top and bottom ends. The drainage condition of the calibration chamber is the same as the slurry consolidometer: drainage through the top and bottom plates, no drainage through the side wall. The chamber consolidation is performed with the initial back pressure (uo ) of 138 kPa while the effective stress is kept intact to ensure saturation. After checking the B parameter for saturation, the confining pressure is adjusted for the desired pressure condition. In this work, the vertical pressure is increased to σv = 273 kPa and adjusted to σv = 182 kPa (vertical) to make a lightly overconsolidated specimen. The horizontal pressure is increased to σh = 112 kPa and reduced to σh = 75 kPa (horizontal). In this way a lightly

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overconsolidated (OCR = 1.5) clayey specimen is obtained that is very similar to specimens obtained in typical field conditions. The vertical stress is applied through a vertical loading jack. Horizontal stress is applied through the cell pressure. Test Procedure Using Three-Piezo-Element Miniature Penetrometer Three penetration tests are carried out in the soil specimens. Dissipation tests are performed at the end of the penetration tests. The hydraulic system used for the penetration consists of dual piston, double-acting hydraulic jacks on a collapsible frame. The frame is mounted on the top lid of the chamber and allows for penetration of the sample up to 640 mm or less in a single stroke. Such a single stroke continuous penetration is desirable, especially in saturated cohesive specimens where stress relaxation and pore pressure dissipation can occur during a pause between strokes. Data acquisition is carried out both manually and automatically. The pore pressure response is recorded with digital voltmeters, which are hooked up to the computer’s data acquisition system. The readings of the digital voltmeters are also recorded by a video camera in case of data acquisition system failure. Data is taken every one second. During the dissipation test, the sampling frequency is lowered to reduce the data file size. The test equipment set up is shown in Fig. 7.16. Validity of the penetration rate is carefully investigated because of the scale difference of the three-piezo-element miniature penetrometer and other penetrating objects. This penetrometer has cross sectional area 2 cm2 , and this area is much smaller than that of the typical international reference cone penetrometer or piles. Regarding this scale difference, an adjustment of penetration speed for the miniature penetrometer was considered. Comprehensive comparison of penetration speed effects for the cone penetrometer by Lunne et al. (1997) showed an inconsistent trend. Also, the results of Roy et al. (1982) showed a non-significant difference in excess pore pressure change for the penetration speed range 3 cm/sec – 240 cm/sec. The test results by Lim (1999) also showed a non-significant difference in excess pore pressure response between a reference penetrometer and a miniature penetrometer with 1 cm2 cross sectional area as shown in Fig. 7.17. Voyiadjis and Song (2003) investigated the scale effect of the miniature cone penetrometer on the pore pressure response extensively, and they showed that there is no significant scale effects for a penetration speed of around 2 cm/sec. Consequently, no predominant effect of the strain rate is considered, and 2 cm/sec penetration speed is adopted in this test. Therefore, the effects of penetration speed are not considered in this study.


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3 Channel Signal Amplifier 3 Digital Voltmeters

Penetrometer

Data Logger (PC)

Fig. 7.16. Test set up

Results The test results are shown in Fig. 7.18. The combined penetration results for the u1 , u2 , u3 , and u4 pore pressure measuring locations are indicated in the figure. The pore pressure response for the u1 location shows a similar trend to a typical pore pressure response obtained from the penetration test. The steady state pore pressure for u1 is obtained for approximately two seconds after the initiation of penetration. Also, u2 shows a similar overall response to u1 . The magnitude of the pore pressure for u2 is a little smaller than that of u1 and is similar to the results determined by Robertson et al. (1992) and Powell and Quartermann (1988). The results of the pore pressure for u3 and u4 in Fig. 7.18 show a substantial initial fluctuation and an increase up to the steady state condition. This is an unexpected behavior. However, one should note that the distances from the penetrometer tip to the u3 and u4 locations are substantially large. They are 7 cm and 22 cm, respectively. These results imply that the initial equilibrium condition and “on-the-penetration” equilibrium condition for u3 and u4 may be very different from those for u1 and u2 . Therefore the discussion of pore pressure response is provided with respect to the equilibrium condition as follows: Figure 7.18 shows that the steady state condition for u1 is obtained in a relatively short time (about 2 seconds). This response is similar to the

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Fig. 7.17. Comparison of penetration results for the reference cone (10 cm2 ) and miniature cone (1 cm2 ) penetrometer (Lim, 1999)


195

Excess Pore Pressure (kPa) -100

0

100

200

300

400

0

u4


10

u3

u1 u2 u3 u4

u2 20

u1

30

40

50

60

Fig. 7.18. Combined penetration test results for u1 , u2 , u3 , and u4 locations

u1 response of other researchers (Kurup et al. 1994; Lunne et al. 1997). The rapid achievement of the steady state represents the rapid achievement of the new (on-the-penetration) equilibrium condition. The steady state includes constant equilibrium conditions such as stabilized pore water flow and stressstrain conditions. u1 is initially under the Ko condition without any flow. After initiation of penetration, the Ko condition and the zero-flow condition are no longer valid. Once “on-the-penetration” new equilibrium conditions are developed and stabilized for u1 , they reach the steady state pore pressure condition. The pore pressure response for u2 is slightly different from that for u1 . The u2 response shows a slight initial hesitation and that is followed by a trend similar to that of u1 . Regarding the initial hesitation, it is important to note the pore pressure response for u3 because the initial hesitation is more clearly observed in the response for u3 . In u3 , this initial hesitation period is much longer than that for u2 . The enlarged version of the pore pressure response for u3 is shown in Fig. 7.19. Note that u3 shows a two-stage, steady-state condition.

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E xcess P ore P ressure (kP a) -50

0

50

100

150

200

0 5 10


15 20 25

Ko Condition Maximum ShearInduced Pore Pressure Reduced Shear-Induced Pore pressure (note: OCR=1.5) Entering into old u2 location Entering into the passage of u1 location

30 35 40 45 50

Fig. 7.19. Details of pore pressure at u3

Initially u3 is subjected to the Ko stress condition, and with no flow condition. With the initiation of penetration, u3 undergoes shear motion. Therefore u3 shows the shear induced pore pressure, which is maximum at a 2 cm penetration. As penetration exceeds 2 cm, the shear induced pore pressure decreases. The reason for this decrease should be attributed to the overconsolidation of the test specimen. The overconsolidation ratio (OCR) of the test specimen is 1.5, which implies a lightly overconsolidated condition. However, it seems that this much OCR is enough to cause the dilative behavior of the specimen at the cone shaft. During this dilative behavior, the pore pressure for u3 achieves a “stage one” steady condition even though that is only for a very short time period. (Note that u4 shows a longer and clearer “stage one” steady state condition). The reduction of pore pressure by dilative behavior stops when the u3 element enters into the old cone tip location (initial location of u2 and u1 ). The initial location of u2 and u1 has some residual pore


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pressure that was generated by the cone tip penetration. Therefore, when u3 enters into the old cone tip location, the coupling of shear-generated negative pore pressure and the residual from the compression-generated positive pore pressure is triggered. After some time, the coupling process achieves the equilibrium that is the “stage two” steady state condition. Note that u2 is located at the penetrometer shaft; however, it is very close to u1 . The authors could not capture the initial fluctuation of the pore pressure for u2 . However, the essentials of initial fluctuation for u2 should be same as that of u3 . Independently from this study, Elsworth (1993) predicted the time to reach the steady state penetration pore pressure based on the dislocation scheme. For the case of a penetration speed of 2 cm/sec and a coefficient of consolidation 28.3 × 10−3 m2 /sec, the time required to reach 95% of steady state penetration pore pressure is about 0.5 seconds for the penetrometer tip. That number is 10 seconds for the penetrometer shaft at a location ten times the radius of the penetrometer behind the face. One notes Elsworth (1993) that the minimum penetration depth for achieving the steady state condition is quite short for the penetrometer tip. However, that distance is increased substantially for the penetrometer shaft. Elsworth’s 1993 findings do not exactly agree with the experimental results in this study, but they show a conceptual agreement with our results. From this discussion, one can see that for a better analysis of the penetration test, a substantial amount of penetration is required, especially for the analysis of the friction on the penetrometer shaft. Figure 7.19 also shows the steady state pore pressure distribution which is high at the penetrometer tip and low (gradual decrease) along the penetrometer shaft. This trend agrees well with Levadoux and Baligh’s experimental results (1986), and with Whittle and Aubeny’s (1991) and Elsworth’s (1993) theoretical results. Figure 7.20 shows the dissipation test results carried out at the end of the penetration for all four piezometer locations. Note that u1 shows a typical dissipation curve. The computed permeability from the u1 dissipation curve is about 2 × 10−8 m/sec (t50 method of Robertson et al., 1992). Considering the fact that the measured permeability value in the laboratory (constant head permeability tests for the triaxial specimen with a confining pressure having the same magnitude as the effective vertical overburden pressure of the specimen) is 2.1 × 10−8 m/sec, one concludes that the computed permeability is in good agreement with the laboratory test results. However, u2 shows some deviation from the standard back-bone shape of the dissipation curve. Note also that u3 and u4 show even a larger deviation from the standard dissipation curve. The following observations are made with reference to Fig. 7.19 with respect to the mechanism of the unstable pore pressure dissipation curve. For the u1 location, a faster pore pressure dissipation is observed, which may indicate a spherical dissipation. The spherical dissipation is proportional to

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Fig. 7.20. Dissipation curves of u1 , u2 , u3 , and u4 locations

r3 , while the cylindrical dissipation is proportional to r2 (where r is the radius of the penetrometer). For the u2 , u3 , and u4 locations, the dissipation condition may be assumed to be of a cylindrical drainage. However, the initial pore pressure is very unstable and is far from the standard back-bone shape. Through the previously discussed two stage equilibrium, one observes that small (negative) pore pressure is generated at the penetrometer shaft, and it approaches the “stage one” steady state condition until it enters a new equilibrium phase. After the penetration of the penetrometer stops, an interaction starts between the penetrometer tip generated pore pressure and the penetrometer shaft generated pore pressure (possibly negative pressure) in order to attain pore pressure equalization. The high pore pressure from the far field flows into the near field of the penetrometer shaft. Therefore, it shows an initial increase in the pore pressure response for the u2 , u3 and u4 as indicated in Fig. 7.20. The above discussion explains the non-standard back bone shape of the pore pressure dissipation curves for u1 , u2 , u3 and u4.


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Therefore one can say that pore pressure dissipation curves for u2 , u3 , and u4 are adequate to use for the evaluation of the coefficient of consolidation or the permeability. Comparison with Numerical Results The calibration chamber test results are compared with numerical simulation results. Numerical simulation is carried out based on the Prevost (1980) coupled theory of mixtures and an anisotropic modified Cam Clay model (Song and Voyiadjis 2000). An updated Lagrangian reference frame is used for the large strain elasto-plasticity model. Numerical simulation results are shown in Fig. 7.21, which shows a high excess pore pressure at the face (u1 position) and smaller excess pore pressure at the penetrometer shoulder (u2 position). Figure 7.21 also shows that the magnitude of u1 is very close to the experimental result. In addition, u2 shows smaller pore pressure than u1 , a result that also agrees fairly well with the experimental result. Therefore, one can say that Fig. 7.21 and Fig. 7.18 agree fairly well for the pore pressures of u1 and u2 . The background theory for Fig. 7.21 is the coupled theory of mixtures. The coupled theory of mixtures takes into account the interaction

Fig. 7.21. Analytical prediction of the excess pore pressure distribution around a penetrating object

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of the cone shaft generated pore pressure and the cone face (tip) generated pore pressure. The micro-level rotation of the soil grains is incorporated in the numerical simulation through the plastic spin, which essentially produces more accurate pore pressures. (Song and Voyiadjis 2000) showed that the effect of the plastic spin is especially large at the u1 location. The Possible interaction of the near field pore pressure and the far field pore pressure can be also predicted from Fig. 7.21. In Fig. 7.21, one can expect the dissipation of the pore water pressure from the face (shown by arrows) to the cone shoulder direction. As the penetration proceeds, u2 will go into the high excess pore pressure region, an effect that accounts for the observed gradual increase. Fig. 7.21 does not show the pore pressure response for large distances from the tip such as those at u3 and u4 because the numerical simulation was performed for a penetration distance of 2.5 cm, and the required penetration length for steady state is more than 10 cm and 35 cm for u3 and u4 , respectively (see Fig. 7.18). Numerical simulation for this long penetration is not possible at this time due to the accumulated numerical errors. Considering that the test cone used in this study is a miniature cone that has 2 cm2 cross sectional area compared to the reference cone that has 10 cm2 cross sectional area, one can see that the numerical simulation of the cone shaft behavior for the reference cone requires even a larger penetration length. The numerical simulation of several centimeters of penetration for the shaft friction will not present reliable results. Evaluation This study evaluates a new concept for the experimental investigation of the penetration induced excess pore pressure. Our analysis is based on the large strain coupled theory of mixtures formulation using the modified Cam Clay model and incorporating micro-mechanics through the plastic spin in an updated Lagrangian reference frame. To carry out our experiment, we used a three-piezo-element penetrometer that was designed and tested in the Louisiana State University Calibration Chamber. We offer the following conclusions: • The generated pore pressure at the tip of the penetrating object is mainly due to the compressive stress, which is usually positive. • The generated pore pressure at the shaft of the penetrating object is mainly due to the shear stress. • The measured pore pressure at the shaft of the penetrating object is a combination of shear induced pore pressure, compression induced pore pressure, and dissipation of early generated pore pressure. • Sometimes, the shear induced pore pressure is negative, and the combined pore pressure can be negative or very small.


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• Due to this interaction of shear induced pore pressure and compression induced pore pressure at the shoulder of the penetrating object, the dissipation of pore pressure does not show the standard back-bone shape when the pore pressure measurement position is not at the tip. • The effect of this interaction is ever more severe as the distance from the cone tip increases, and the steady state pore pressure magnitude is very different from that of the tip or face of the penetrating object. • The steady state for the shaft of the penetrating object is obtained in two stages because of the change in equilibrium conditions. In the case of the penetrating object tip, the steady state is obtained in one stage because of the constant equilibrium conditions. For the pore pressure at the cone shaft, the pore water dissipation curve (especially u3 , and u4 ) deviates largely from the standard back-bone shape curve due to the different mechanism in achieving the steady state. 7.2.9 Back Stress Multiple back stresses are discussed in Sect. 4.5.1, which investigates the effects of multiple back stresses. Figure 7.22 shows the comparison of excess pore pressure around the cone tip for the case of single back-stress and for the case of dual back-stress. As shown in Fig. 7.22, the difference between the two back-stress models is not substantial in close proximity to the penetrating cone tip. However, the difference is observed at a relatively far distance from the cone tip (see the area encircled by the dotted line.). This behavior is contrary to what was expected. It is typically known that the activity of the back-stress is pronounced in the high strain region (close vicinity of the cone tip). However, recalling the basic concept of the short range back-stress, one can explain the above behavior. The material at close proximity to the cone tip undergoes very large strains, and effect of short range back-stress is smudged. However, the behavior of the material at a far distance from the cone tip usually undergoes small strains; the behavior will be affected primarily by the short range back-stress. This behavior is predicted by the careful observation of Fig. 7.22. In Fig. 7.22, the short range back-stress is mobilized rapidly and its amount is larger than that of the long range back-stress at relatively low strains. The long range back-stress, however, is mobilized relatively slowly and the initial magnitude is smaller than that of the short range back-stress. Therefore its effect is not substantial at low strain range. Figure 7.22 shows the excess pore pressure response of the penetrating cone tip predicted by the isotropic modified Cam Clay model (without any back-stress). Comparing Figs. 7.22(b) and 7.16, one can see that the effect of the long range back-stress is mostly pronounced at close vicinity of the cone tip. It may be assumed that the effect of the short range back-stress for steel is substantial, especially for repeated cyclic loading (Montheillet et al.. 1984).

____ Umax=357.3 kPa

(a) Short Range + Long Range Back-Stress Condition

Cone Shaft

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Cone Shaft

202

____ Umax=357.1 kPa

(b) Long Range Only Back-Stress Condition

Fig. 7.22. Comparison of excess pore pressure contours for dual back-stress and single back-stress condition

One of the most widely known behaviors for steel which is related to the short range back-stress is “ratcheting behavior”. Ratcheting behavior in metals is easily observed by the length change of the test specimen subjected to the cyclic torsional loading. Similar behavior for soil (sand) is reported by Song (1986). Measurement of the height change of resonant column test specimens (ASTM C-109 Ottawa Sand) shows a


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consistent change of the height of the test specimen (Song, 1986). This change is distinctly different from the void ratio reduction which is typically caused by dynamic loading. This behavior is similar to the ratcheting behavior for metals, and it is believed to be caused by the short range back-stress. The research for the ratcheting behavior of clay is not reported yet. However, the authors believe that the ratcheting behavior of the clay may not be as significant as ratcheting behavior in sand because the magnitude of the short range back-stress relative to the long range back-stress is believed to depend on the stiffness of the grain relative to the stiffness of the skeleton of the structure. Because the stiffness of the grain is larger than that of the material skeleton, the internal stress will cause the relative movement of the grains rather than cause the deformation of the grains. This mechanism addresses that the soil grains dos not store much energy. In metals, the stiffness of the macroscopic skeleton structure is much greater that that of the sand or clay; however, the macroscopic stiffness is similar or lower. Therefore, relatively larger short range back-stresses can be stored in the metal microstructure. In sands, the stiffness of the sand grains is larger than that of the soil skeleton. It follows that the probability for short range back-stress storage is relatively lower for sand than for steel. In clays, the stiffness of the clay particles is much larger than that of the clay skeleton, and the short range back-stress is hardly stored in the clay particles. This result shows the relatively insignificant effect of the short range backstress on the very large strain region. However, it does not mean that the short range back-stress in not important for geo-materials. As discussed previously, the short range back-stress is more important for materials that have higher stiffness, such as sands or over-consolidated clays. It is noted that the effects of α3 are not taken into account here. It is beyond the scope of this study, since the material parameters for α3 are not available at this moment. Evaluation In this section the effect of the micro-structural change for soils is investigated through the dual back-stress approach – the so called short range back-stress and long range back-stress approach. The anisotropic modified Cam Clay model is used, which incorporates the plastic spin as an internal state variable. This soil model is used to solve the problem of the cone penetrometer test that is a typical example for large strain problems in soils. This study shows that the effect of the short range back-stress is not significant in the large strain region. However, its effect is shown to be significant in the low strain region. This is primarily because the ultimate magnitude of the short range backstress is much smaller than that of the long range back-stress. The magnitude of the short range back-stress is greater than that of the long range backstress at the earlier stage (low strain stage). This relationship holds for the

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soil condition used in this study. However, the effect of the short range backstress may be larger for some other soils such as sands or stiffer skeleton materials because the stiffness of the clay particles is much greater than that of the clay skeleton, and it is much easier for the internal energy to cause the relative movement of the clay particles than to store the back-stress in the clay particles. The authors also expect that this short range back-stress may play a substantial role in soils that have higher skeleton stiffness, such as sands. 7.2.10 Evaluation In our discussion of the individual micro-mechanical mechanisms, we have noted that each individual micro-mechanical mechanism has its own effect. The penetration rate effect, however, is not very strong for the typical penetration velocity range. The mechanism is such that when the penetration is initiated, the response of the soil is governed by two different rate dependency mechanisms: the viscosity related rate dependency and the pore fluid flow dependent rate dependency. When the effect of the viscosity related rate dependency is dominant (as is the case for the relatively high penetration speed), the penetration resistance is increased as the penetration speed is increased. When the effective stress is dominant (as is the case for the relatively low penetration speed), the penetration resistance is increased as the penetration speed is decreased. In the typical penetration velocity range, the viscosity related rate dependency and the porous fluid flow related rate dependency are balanced, and no pronounced rate effects are observed. The effects of gradients are expected to cause the reduction of the peak effective stress and the peak excess pore pressure. In this section, the reduction of the excess pore pressure due to the gradient appears to be bigger than that of the effective stress. This behavior causes the overall reduction of the total stress at the cone tip. Negligible changes in the effective stress are believed to be caused by the combination of the reduced effective stress by gradient mechanism and the increased effective stress by reduction of the pore pressure. The damage effect causes a substantial reduction in the excess pore pressure. It is reasonable to anticipate that the effective stress at the cone tip will be reduced. However, the effective stress at the cone tip is actually increased because of the substantial decrease of the excess pore pressure. The authors conclude that the effects of the individual micro-mechanical mechanisms are inter-linked with each other. The saturated clayey soil is essentially a two-phase material composed of the pore fluid and soil grains. For the multi-phase materials, the coupled effects of these micro-mechanical mechanisms are quite different from what is to be expected from the uncoupled soil models. The primary reason is the interaction of the pore fluid and the soil grains, which is essentially the interaction of the pore pressure and the


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effective stress. Pore pressure and effective stress nullify each other; therefore, the effects of the micro-mechanical mechanism are not always consistent. The summary of the coupled micro-mechanisms may be outlined as follows: • Homogenization mechanisms such as gradient, damage, rate dependency impact both the effective stress and pore pressure. • Viscosity governs the rate dependency of the Piezocone tip resistance only at higher penetration speeds such as higher than 10 cm/sec. • Pore fluid flow governs the rate dependency of the Piezocone tip resistance only at lower penetration speeds (i.e., lower than 0.2 cm/sec). • Viscosity related rate dependency and pore pressure related rate dependency are balanced at the intermediate penetration speed range, and the corresponding Piezocone tip resistance is not sensitive to the penetration speed. • Gradient theory reduces the peak pore pressure. However, the effective stress is not affected substantially. • This behavior is believed to be due to the increase of effective stress by the reduced pore pressure and the decrease of the effective stress by the gradient. • Damage reduced the pore pressure around the Piezocone tip. The reduced pore pressure increases the effective stress even with damage of grains.

Appendix: Fortran Codes of CS-S

c PROGRAM CS-S.f C---------------------------------------------------------------------C THIS PROGRAM ANALYSE GENERAL FINITE DEFORMATION, FINITE STRAIN C INELASTIC TIME INDEPENDENT PROBLEMS USING UPDATED LAGRANGIAN C REFERENCE FRAME. C---------------------------------------------------------------------C Back up is cone8.f C ORIGINALLY CODED BY S. M. Sivakumar C Modifid by MURAD ABU-FARSAKH for CPT with modified Cam Clay model C MODIFIED BY CHUNG R. SONG (1/26/98) for anisotrophy & plastic spin C Modified by Chung R. Song (4/27/00) for multiple back stress C Last modified by Chung R. Song (8/20/03) C Last accessed by Chung R. Song (8/12/03) C C C C

NOTE BY CHUNG R. SONG (4/27/00) FINITE STRAIN : consider PLASTIC SPIN. COUPLED THEORY OF MIXTURE : PREVOST (1980, 1981) TIME INDEPENDENT : ACTUALLY SEMI TIME DEPENDENT BY CONSIDERING C THE DISSIPATION OF PORE WATER DURING C EACH PENETRATION STEP. -- NOT VISCO ELASTO-PLASTIC C UPDATED LAGRANGIAN : INCREMENTAL APPROACH FOR LARGE STRAIN PROBLEM C SOIL MODEL : Anisotropic - MODIFIED CAM CLAY MODEL WITH GENERAL HARDENING AND YIELD C SURFACE CORRECTION C This program also considers the micro-mechanical behavior of soil by c plastic spin. C Rate Dependency 08/07/03 c Damage 08/08/03 c Gradient 08/12/03 C

----- INITIAL SETUP IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ CHARACTER*60

208


INFILE,OUTFILE1,OUTFILE2,OUTFILE3,OUTFILE4,OUTFILE5 COMMON/DEVICE/LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON/PARS/PYI,ASMVL,ZERO COMMON/NSIZE/NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW DIMENSION XYZ(3,MNODES),NCONN(NTPE,MEL),LTYP(MEL),MAT(MEL) DIMENSION NQ(MNODES),NW(MNODES+1),NLST(NTPE) C

----- OPEN INPUT & OUTPUT FILE

C C

WRITE(*,*)’ENTER INPUT FILE NAME>’ READ(*,’(A)’)INFILE INFILE=’Chung21.TXT’ C WRITE(*,*)’ENTER MAIN OUTPUT FILE NAME>’ C READ(*,’(A)’)OUTFILE1 OUTFILE1=’SONG2’ C WRITE(*,*)’ENTER EQUIL OUTPUT FILE NAME>’ C READ(*,’(A)’)OUTFILE2 OUTFILE2=’SONG3’ C WRITE(*,*)’ENTER STRESS OUTPUT FILE NAME>’ C READ(*,’(A)’)OUTFILE3 OUTFILE3=’SONG4’ C WRITE(*,*)’ENTER CAM-PAR OUTPUT FILE NAME>’ c READ(*,’(A)’)OUTFILE4 OUTFILE4=’SONG5’ OUTFILE5=’SONG6’ OPEN(1,FILE=INFILE,FORM=’FORMATTED’,STATUS=’UNKNOWN’) OPEN(2,FILE=OUTFILE1,FORM=’FORMATTED’,STATUS=’UNKNOWN’) OPEN(3,FILE=OUTFILE2,FORM=’FORMATTED’,STATUS=’UNKNOWN’) OPEN(4,FILE=OUTFILE3,FORM=’FORMATTED’,STATUS=’UNKNOWN’) OPEN(5,FILE=OUTFILE4,FORM=’FORMATTED’,STATUS=’UNKNOWN’) OPEN(10,FILE=’CONSOLV’,FORM=’UNFORMATTED’,STATUS=’UNKNOWN’) c OPEN(8,FILE=OUTFILE5,FORM=’FORMATTED’,STATUS=’UNKNOWN’)

c

C C C

LINP=1 LOUT1=2 LOUT2=3 LOUT3=4 LOUT4=5 LOUT5=8 LSOLV=10

----- CALCULATION SEQUENCY INPUT NODE COORDINATE & ELEMENT DATA CALL INPUT(XYZ,NCONN,MAT,LTYP,NLST)

Appendix: Fortran Codes of CS-S C C C C

209

CALCULATE NO OF D.O.F FOR EACH NODE CALL MAKENZ(NEL,NNODES,NCONN,LTYP,NQ,INXL) GENERATE GLOPAL NUMBERS FOR ALL D.O.F CALL CALDOF(NNODES,NNOD1,NDF,NW,NQ) PRINT OUT ARRAYS CALL GPOUT(LOUT1,NEL,NNODES,NDF,NCONN,MAT,LTYP,NLST) CALL THE MAIN SUMROUTINE CALL MINT(XYZ,NCONN,MAT,LTYP,NW)

C C --- MAJOR SUBROUTINES ARE ATTATCHED BY ITS ORDER RIGHT AFTER THE CONTROL PROGRAM. C --- MINOR SUBROUTINES ARE ATTATCHED BY ITS ORDER AFTER THE MAJOR SUBROUTINES STOP END C ----- END OF CONTROL PROGRAM

C

----- MAJOR SUBROUTINE -----

C********************************************************************** SUBROUTINE INPUT(XYZ,NCONN,MAT,LTYP,NLST) C********************************************************************** C THIS SUBPROGRAM GETS ALL THE VALUES NEEDED AS INPUT FOR THE PROGRAM IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ CHARACTER*80 TITLE CHARACTER*80 HEADER1 CHARACTER*80 HEADER2 CHARACTER*80 HEADER3 CHARACTER*80 HEADER4 COMMON/DEVICE/LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON/PARS/PYI,ASMVL,ZERO COMMON/NSIZE/NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON/ELINF/LINFO(50,15) COMMON/SKBC/ISPB(20),DIRCOS(20,3) DIMENSION DUMMY(6),NLST(NTPE),M(20) DIMENSION XYZ(3,MNODES),NCONN(NTPE,MEL),MAT(MEL), 1 LTYP(MEL),TEMP(3)

210


C

----- INITIALIZE ARRAYS CALL ZEROR2(XYZ,3,MNODES) CALL ZEROR2(DIRCOS,20,3) CALL ZEROI2(NCONN,NTPE,MEL) CALL ZEROI1(LTYP,MEL) CALL ZEROI1(MAT,MEL) CALL ZEROI1(NLST,NTPE) CALL ZEROI1(ISPB,20)

C

----- SET SOME CONSTANTS PYI=4.0D0*ATAN(1.0D0) ASMVL=1.0D-20 ZERO=0.0D0 ASMVL=ASSIGNED MINIMUM VALUE

C C

----- READ THE TITLE READ(LINP,101)TITLE WRITE(LOUT1,101)TITLE 101 FORMAT(A80)

C

----- READ PL. STR-AXI. SYM INDEX NPLAX & THE LINEARITY INDIX LINR READ(LINP,101)HEADER1 READ(LINP,*)NPLAX,LINR,NDIM WRITE(LOUT1,*)’LINEARITY=’,LINR

C C C C C C C C C

----- SET SOME VARIABLES INXL - INDEX TO NO. OF D.O.F OF FIRST NODE OF ELEMENT NSP - ONE DIMENSIONAL INTEGRATION NUMBER OF SAMPLING POINTS NS - SIZE OF D-MATRIX NPT - MAXM NUMBER OF DISPLACEMENT NODES ALONG ELEMENT EDGE MXEN,MXLD - SIZE OF ARRAYS IN COMMON BLOCKS PRSLD,PRLDI MXLD - MAXIMUM NUMBER OF ELEMENT EDGES WITH PRESSURE LOADING MXEN - MAXIMUM NUMBER OF DISPLACEMENT NODES ALONG AN EDGE X 2 MXFXT - MAXIMUM NUMBER OF FIXITIES (SIZE OF ARRAYS MF, NFIX,DXYT)

C

----INXL=20 NSP=5 IF(NDIM.EQ.2)NS=4 IF(NDIM.EQ.3)NS=6 NPT=LV MXEN=10 MXLD=100 MXFXT=200

Appendix: Fortran Codes of CS-S C

310

----- READ AND GENERATE THE NODAL COORDINATES I=0 READ(LINP,*)NNODES,NSKEW WRITE(LOUT1,*)’NUMBER OF NODES =’,NNODES READ(LINP,101)HEADER2 NNOD1=NNODES+1 READ(LINP , *) K,(DUMMY( IDIR ) , IDIR = 1 ,NDIM),INCR DO IDIR=1,NDIM XYZ(IDIR,K)= DUMMY(IDIR) ENDDO I=I+1

C

----- Interpolation of the nodal coordinate IF (INCR.NE.0)THEN N=(K-K1)/INCR DX=(XYZ(1,K)-XYZ(1,K1))/N DY=(XYZ(2,K)-XYZ(2,K1))/N DZ=(XYZ(3,K)-XYZ(3,K1))/N K2=K-INCR DO J=K1,K2,INCR N1=(J-K1)/INCR XYZ(1,J)=XYZ(1,K1)+N1*DX XYZ(2,J)=XYZ(2,K1)+N1*DY XYZ(3,J)=XYZ(3,K1)+N1*DZ I=I+1 ENDDO I=I-1 ENDIF K1=K IF(I.LT.NNODES) GO TO 310

C WRITE(LOUT1, 20) DO K1 = 1, NNODES WRITE(LOUT1, 10)K1,(XYZ(IDIR,K1),IDIR=1,NDIM) ENDDO C

C

C

---- READ SKEW BOUNDARY NODES & ITS DIRECTION COSINES READ(LINP,101)HEADER3 IF(NSKEW.GT.0) THEN DO K=1,NSKEW READ(LINP,*)INODE ,(TEMP(IDIR),IDIR=1,2) TEMP(1)=5.640D0 TEMP(2)=9.780D0 --- Above values are for 60 degree cone tip. ISPB(K)=INODE TEMP3=SQRT(TEMP(1)**2+TEMP(2)**2)

211

212

Appendix: Fortran Codes of CS-S DIRCOS(K,1)=TEMP(2)/TEMP3 DIRCOS(K,2)=TEMP(1)/TEMP3 IF(NDIM.EQ.3)DIRCOS(K,3)=TEMP(3) ENDDO END IF

C

----- READ, WRITE AND GENERATE THE ELEMENTS READ(LINP,101)HEADER4 I = 0 READ(LINP,*)NEL,NELNOD WRITE(LOUT1,*)’NUMBER OF ELEMENTS =’,NEL 410

READ(LINP, *)K,ITYP,IMAT,INCR,(NLST(IK),IK=1,NELNOD)

C NDN=LINFO(1,ITYP) LTYP(K)=ITYP MAT(K)=IMAT C DO IK=1,NDN NUM=NLST(IK) NCONN(IK,K)=NUM ENDDO C I = I + 1 IF(INCR.EQ.0) THEN K1 = K C ELSE K2 = (K - K1)/INCR DO NODE = 1, NDN M( NODE ) = (NCONN(NODE,K)-NCONN(NODE,K1))/K2 ENDDO C DO IELEM = K1+INCR, K-INCR, INCR LTYP(IELEM)=LTYP(K) MAT(IELEM)=MAT(K) I = I + 1 IELEM1 = IELEM - INCR DO NODE = 1,NDN NCONN(NODE,IELEM) = NCONN(NODE,IELEM1) + M( NODE ) ENDDO ENDDO END IF IF(I.LT.NEL) GO TO 410 C DO I=1,NEL


C

213

LT=LTYP(I) NDN=LINFO(1,LT) WRITE(LOUT1,*)I,(NCONN(J,I),J=1,NDN) ENDDO RETURN 10 FORMAT(I5,4X,3F12.4) 20 FORMAT(/,12X,’COORDINATES OF THE NODES’/,45(1H-),/,3X,’NODE’, 1 11X,’X’,11X,’Y’,11X,’Z’,/,45(1H-)) END

C********************************************************************** BLOCK DATA C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) C---------------------------------------------------------------------C DATA PRESENTED BY LIN (FIRST INDEX) C 1 - TOTAL NUMBER OF NODES (DISPLACEMENT + POREPRESSURE)......NDPT C 2 - TOTAL NUMBER OF VERTEX NODES..............................NVN C 3 - TOTAL NUMBER OF ELEMENT EDGES............................NEDG C 4 - TOTAL NUMER OF ELEMENT FACES (3D)........................NFAC C 5 - TOTAL NUMBER OF DISPLACEMENT NODES........................NDN C 6 - TOTAL NUMBER OF POREPRESSURE NODES........................NPN C 7 - NO. OF DISPLACEMENT NODES PER EDGE (EXCLUDING END NODES).NDSD C 8 - NO. OF POREPRESSURE NODES PER EDGE (EXCLUDING END NODES).NPSD C 9 - NUMBER OF INNER DISPLACEMENT NODES.......................NIND C 10 - NUMBER OF INNER POREPRESSURE NODES.......................NINP C 11 - NUMBER OF INTEGRATION POINTS (=GAUSS POINT)...............NGP C 12 - INDEX TO WEIGHTS AND INTEGRATION POINT COORDINATES.......INDX C 13 - INDEX TO VERTEX NODES OF ELEMENTS (ARRAY NFC).............INX C 14 - INDEX TO NODES ALONG EDGE (ARRAYS NP1, NP2).............INDED C 15 - NUMBER OF LOCAL OR AREA COORDINATES........................NL C 16 - TOTAL NUMBER OF DEGREES OF FREEDOM (D.O.F.) IN ELEMENT...MDFE C 17 - CENTROID INTEGRATION POINT NUMBER........................NCGP C 21 - ONWARDS THE NUMBER OF D.O.F. OF EACH NODE OF ELEMENT.....NDFN C C C C C C C C C

ELEMENT TYPES (SECOND INDEX) 1 - 3-NODED BAR ....................(2-D) 2 - 6-NODED LST TRIANGLE............(2-D) 3 - 6-NODED LST TRIANGLE............(2-D CONSOLIDATION) 4 - 8-NODED QUADRILATERAL...........(2-D) 5 - 8-NODED QUADRILATERAL...........(2-D CONSOLIDATION) 6 - 15-NODED CUST TRIANGLE..........(2-D) 7 - 22-NODED CUST TRIANGLE..........(2-D CONSOLIDATION)

**

214


C C C C C

8 9 10 11

C

-

20-NODED 20-NODED 10-NODED 10-NODED

**

BRICK..................(3-D) BRICK..................(3-D CONSOLIDATION) TETRA-HEDRA............(3-D) TETRA-HEDRA............(3-D CONSOLIDATION)

** **

ELEMENT TYPES NOT IMPLEMENTED IN THIS VERSION

C********************************************************************** COMMON /ELINF/ LIN(50,15) COMMON /DATL / SL(4,100) COMMON /DATW / W(100) COMMON /SAMP / POSSP(5),WEIGP(5) DATA LIN(1,1),LIN(2,1),LIN(3,1),LIN(4,1),LIN(5,1),LIN(6,1), 1 LIN(7,1),LIN(8,1),LIN(9,1),LIN(10,1),LIN(11,1),LIN(12,1), 2 LIN(13,1),LIN(14,1),LIN(15,1),LIN(16,1),LIN(17,1), 3 LIN(21,1),LIN(22,1),LIN(23,1)/ 3 3,2,1,1,3,0,1,0,0,0,5,0,0,0,1,6,3,2,2,2/ DATA LIN(1,2),LIN(2,2),LIN(3,2),LIN(4,2),LIN(5,2),LIN(6,2), 1 LIN(7,2),LIN(8,2),LIN(9,2),LIN(10,2),LIN(11,2),LIN(12,2), 2 LIN(13,2),LIN(14,2),LIN(15,2),LIN(16,2),LIN(17,2), 3 LIN(21,2),LIN(22,2),LIN(23,2),LIN(24,2),LIN(25,2),LIN(26,2)/ 4 6,3,3,1,6,0,1,0,0,0,7,5,0,0,3,12,7,2,2,2,2,2,2/ DATA LIN(1,3),LIN(2,3),LIN(3,3),LIN(4,3),LIN(5,3),LIN(6,3), 1 LIN(7,3),LIN(8,3),LIN(9,3),LIN(10,3),LIN(11,3),LIN(12,3), 2 LIN(13,3),LIN(14,3),LIN(15,3),LIN(16,3),LIN(17,3),LIN(21,3), 3 LIN(22,3),LIN(23,3),LIN(24,3),LIN(25,3),LIN(26,3)/ 4 6,3,3,1,6,3,1,0,0,0,7,5,0,0,3,15,7,3,3,3,2,2,2/ DATA LIN(1,4),LIN(2,4),LIN(3,4),LIN(4,4),LIN(5,4),LIN(6,4), 1 LIN(7,4),LIN(8,4),LIN(9,4),LIN(10,4),LIN(11,4),LIN(12,4), 2 LIN(13,4),LIN(14,4),LIN(15,4),LIN(16,4),LIN(17,4), 3 LIN(21,4),LIN(22,4),LIN(23,4),LIN(24,4),LIN(25,4), 4 LIN(26,4),LIN(27,4),LIN(28,4)/ 4 8,4,4,1,8,0,1,0,0,0,4,12,4,3,2,16,9,2,2,2,2,2,2,2,2/ DATA LIN(1,5),LIN(2,5),LIN(3,5),LIN(4,5),LIN(5,5),LIN(6,5), 1 LIN(7,5),LIN(8,5),LIN(9,5),LIN(10,5),LIN(11,5),LIN(12,5), 2 LIN(13,5),LIN(14,5),LIN(15,5),LIN(16,5),LIN(17,5), 3 LIN(21,5),LIN(22,5),LIN(23,5),LIN(24,5),LIN(25,5), 4 LIN(26,5),LIN(27,5),LIN(28,5)/ 4 8,4,4,1,8,4,1,0,0,0,4,12,4,3,2,20,9,3,3,3,3,2,2,2,2/ DATA LIN(1,6),LIN(2,6),LIN(3,6),LIN(4,6),LIN(5,6),LIN(6,6), 1 LIN(7,6),LIN(8,6),LIN(9,6),LIN(10,6),LIN(11,6),LIN(12,6), 2 LIN(13,6),LIN(14,6),LIN(15,6),LIN(16,6),LIN(17,6), 3 LIN(21,6),LIN(22,6),LIN(23,6), 4 LIN(24,6),LIN(25,6),LIN(26,6),LIN(27,6),LIN(28,6),LIN(29,6), 5 LIN(30,6),LIN(31,6),LIN(32,6),LIN(33,6),LIN(34,6),LIN(35,6)/ 5 15,3,3,1,15,0,3,0,3,0,16,21,0,0,3,30,16, 6 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2/


215

DATA LIN(1,7),LIN(2,7),LIN(3,7),LIN(4,7),LIN(5,7), LIN(6,7),LIN(7,7),LIN(8,7),LIN(9,7),LIN(10,7),LIN(11,7), LIN(12,7),LIN(13,7),LIN(14,7),LIN(15,7),LIN(16,7),LIN(17,7), LIN(21,7),LIN(22,7),LIN(23,7),LIN(24,7),LIN(25,7), LIN(26,7),LIN(27,7),LIN(28,7),LIN(29,7),LIN(30,7), LIN(31,7),LIN(32,7),LIN(33,7),LIN(34,7),LIN(35,7),LIN(36,7), LIN(37,7),LIN(38,7),LIN(39,7),LIN(40,7),LIN(41,7),LIN(42,7)/ 22,3,3,1,15,10,3,2,3,1,16,21,0,0,3,40,16,3,3,3,2,2,2,2,2,2,2,2,2, 2,2,2,1,1,1,1,1,1,1/ DATA LIN(1,8),LIN(2,8),LIN(3,8),LIN(4,8),LIN(5,8),LIN(6,8), 1 LIN(7,8),LIN(8,8),LIN(9,8),LIN(10,8),LIN(11,8),LIN(12,8), 2 LIN(13,8),LIN(14,8),LIN(15,8),LIN(16,8),LIN(17,8), 3 LIN(21,8),LIN(22,8),LIN(23,8),LIN(24,8),LIN(25,8), 4 LIN(26,8),LIN(27,8),LIN(28,8), 4 LIN(29,8),LIN(30,8),LIN(31,8),LIN(32,8),LIN(33,8),LIN(34,8), 5 LIN(35,8),LIN(36,8),LIN(37,8),LIN(38,8),LIN(39,8),LIN(40,8)/ 6 20,8,12,6,20,0,1,0,0,0,8,37,4,3,3,60,27,3,3,3,3,3,3,3,3,3,3,3, 7 3,3,3,3,3,3,3,3,3/ DATA LIN(1,9),LIN(2,9),LIN(3,9),LIN(4,9),LIN(5,9), 1 LIN(6,9),LIN(7,9),LIN(8,9),LIN(9,9),LIN(10,9), 2 LIN(11,9),LIN(12,9),LIN(13,9),LIN(14,9),LIN(15,9),LIN(16,9), 3 LIN(17,9),LIN(21,9),LIN(22,9),LIN(23,9),LIN(24,9),LIN(25,9), 4 LIN(26,9),LIN(27,9),LIN(28,9),LIN(29,9),LIN(30,9),LIN(31,9), 5 LIN(32,9),LIN(33,9),LIN(34,9),LIN(35,9),LIN(36,9),LIN(37,9), 6 LIN(38,9),LIN(39,9),LIN(40,9)/ 7 20,8,12,6,20,8,1,0,0,0,8,37,4,3,3,68,27,4,4,4,4,4,4,4,4, 8 3,3,3,3,3,3,3,3,3,3,3,3/ DATA LIN(1,10),LIN(2,10),LIN(3,10),LIN(4,10),LIN(5,10), 1 LIN(6,10),LIN(7,10),LIN(8,10),LIN(9,10),LIN(10,10), 2 LIN(11,10),LIN(12,10),LIN(13,10),LIN(14,10),LIN(15,10), 3 LIN(16,10),LIN(17,10),LIN(21,10),LIN(22,10),LIN(23,10), 4 LIN(24,10),LIN(25,10),LIN(26,10),LIN(27,10),LIN(28,10), 5 LIN(29,10),LIN(30,10)/ 5 10,4,6,4,10,0,1,0,0,0,4,64,28,15,4,30,0,3,3,3,3,3,3,3,3,3,3/ DATA LIN(1,11),LIN(2,11),LIN(3,11),LIN(4,11),LIN(5,11), 1 LIN(6,11),LIN(7,11),LIN(8,11),LIN(9,11),LIN(10,11), 2 LIN(11,11),LIN(12,11),LIN(13,11),LIN(14,11),LIN(15,11), 3 LIN(16,11),LIN(17,11),LIN(21,11),LIN(22,11),LIN(23,11), 4 LIN(24,11),LIN(25,11),LIN(26,11),LIN(27,11),LIN(28,11), 5 LIN(29,11),LIN(30,11)/ 5 10,4,6,4,10,4,1,0,0,0,4,64,28,15,4,34,0,4,4,4,4,3,3,3,3,3,3/ C---------------------------------------------------------------------C AREA COORDINATES - LINEAR STRAIN TRIANGLE - ELEMENT TYPE 2,3 C---------------------------------------------------------------------DATA SL(1,6),SL(2,6),SL(3,6),SL(1,7),SL(2,7),SL(3,7),SL(1,8), 1 SL(2,8),SL(3,8),SL(1,9),SL(2,9),SL(3,9),SL(1,10),SL(2,10), 1 SL(3,10),SL(1,11),SL(2,11),SL(3,11),SL(1,12),SL(2,12),SL(3,12)/ 1 .797426985353087245,.101286507323456343,.101286507323456343 1 2 3 4 5 6 7 8

216


1,.101286507323456343,.797426985353087245,.101286507323456343 1,.101286507323456343,.101286507323456343,.797426985353087245 1,.597158717897698279E-01,.470142064105115082,.470142064105115082 1,.470142064105115082,.597158717897698279E-01,.470142064105115082 1,.470142064105115082,.470142064105115082,.597158717897698279E-01 1,.333333333333333329,.333333333333333329,.333333333333333329/ C---------------------------------------------------------------------C LOCAL COORDINATES - LINEAR STRAIN QUADRILATERAL - ELEM TYPE 4, 5 C---------------------------------------------------------------------DATA SL(1,13),SL(2,13),SL(1,14),SL(2,14),SL(1,15),SL(2,15), 1 SL(1,16),SL(2,16),SL(1,17),SL(2,17),SL(1,18),SL(2,18), 1 SL(1,19),SL(2,19),SL(1,20),SL(2,20),SL(1,21),SL(2,21)/ 1 -0.577350269189626,-0.577350269189626, 1 0.577350269189626,-0.577350269189626, 1 0.577350269189626, 0.577350269189626, 1 -0.577350269189626, 0.577350269189626, 1 0.0D0,0.0D0, 1 0.0D0,0.0D0, 1 0.0D0,0.0D0, 1 0.0D0,0.0D0, 1 0.0D0,0.0D0/ C---------------------------------------------------------------------C AREA COORDINATES - CUBIC STRAIN TRIANGLE - ELEMENT TYPE 6,7 C---------------------------------------------------------------------DATA SL(1,22),SL(2,22),SL(3,22),SL(1,23),SL(2,23),SL(3,23), 1 SL(1,24),SL(2,24),SL(3,24),SL(1,25),SL(2,25),SL(3,25), 1 SL(1,26),SL(2,26),SL(3,26),SL(1,27),SL(2,27),SL(3,27), 1 SL(1,28),SL(2,28),SL(3,28),SL(1,29),SL(2,29),SL(3,29)/ 1 0.898905543365938,0.050547228317031,0.050547228317031, 1 0.050547228317031,0.898905543365938,0.050547228317031, 1 0.050547228317031,0.050547228317031,0.898905543365938, 1 0.658861384496478,0.170569307751761,0.170569307751761, 1 0.170569307751761,0.658861384496478,0.170569307751761, 1 0.170569307751761,0.170569307751761,0.658861384496478, 1 0.081414823414554,0.459292588292723,0.459292588292723, 1 0.459292588292723,0.081414823414554,0.459292588292723/ DATA SL(1,30),SL(2,30),SL(3,30),SL(1,31),SL(2,31),SL(3,31), 1 SL(1,32),SL(2,32),SL(3,32),SL(1,33),SL(2,33),SL(3,33), 1 SL(1,34),SL(2,34),SL(3,34),SL(1,35),SL(2,35),SL(3,35), 1 SL(1,36),SL(2,36),SL(3,36),SL(1,37),SL(2,37),SL(3,37)/ 1 0.459292588292723,0.459292588292723,0.081414823414554, 1 0.008394777409958,0.728492392955404,0.263112829634638, 1 0.008394777409958,0.263112829634638,0.728492392955404, 1 0.263112829634638,0.008394777409958,0.728492392955404, 1 0.728492392955404,0.008394777409958,0.263112829634638, 1 0.728492392955404,0.263112829634638,0.008394777409958, 1 0.263112829634638,0.728492392955404,0.008394777409958, 1 0.333333333333333,0.333333333333333,0.333333333333333/


217

C---------------------------------------------------------------------C LOCAL COORDINATES - 20-NODED BRICK - ELEM TYPE 8, 9 C---------------------------------------------------------------------DATA SL(1,38),SL(2,38),SL(3,38),SL(1,39),SL(2,39),SL(3,39), 1 SL(1,40),SL(2,40),SL(3,40),SL(1,41),SL(2,41),SL(3,41), 1 SL(1,42),SL(2,42),SL(3,42),SL(1,43),SL(2,43),SL(3,43), 1 SL(1,44),SL(2,44),SL(3,44),SL(1,45),SL(2,45),SL(3,45)/ 1 -0.577350269189626,-0.577350269189626, 0.577350269189626, 1 0.577350269189626,-0.577350269189626, 0.577350269189626, 1 0.577350269189626, 0.577350269189626, 0.577350269189626, 1 -0.577350269189626, 0.577350269189626, 0.577350269189626, 1 -0.577350269189626,-0.577350269189626,-0.577350269189626, 1 0.577350269189626,-0.577350269189626,-0.577350269189626, 1 0.577350269189626, 0.577350269189626,-0.577350269189626, 1 -0.577350269189626, 0.577350269189626,-0.577350269189626/ C 1 SL(1,46),SL(2,46),SL(3,46),SL(1,47),SL(2,47),SL(3,47), C 1 SL(1,48),SL(2,48),SL(3,48),SL(1,49),SL(2,49),SL(3,49), C 1 SL(1,50),SL(2,50),SL(3,50),SL(1,51),SL(2,51),SL(3,51), C 1 SL(1,52),SL(2,52),SL(3,52),SL(1,53),SL(2,53),SL(3,53), C 1 SL(1,54),SL(2,54),SL(3,54),SL(1,55),SL(2,55),SL(3,55), C 1 SL(1,56),SL(2,56),SL(3,56),SL(1,57),SL(2,57),SL(3,57), C 1 SL(1,58),SL(2,58),SL(3,58),SL(1,59),SL(2,59),SL(3,59), C 1 SL(1,60),SL(2,60),SL(3,60),SL(1,61),SL(2,61),SL(3,61), C 1 SL(1,62),SL(2,62),SL(3,62),SL(1,63),SL(2,63),SL(3,63), C 1 SL(1,64),SL(2,64),SL(3,64)/ C 1 0.0D0,0.0D0/ C---------------------------------------------------------------------C WEIGHTS - LINEAR STRAIN TRIANGLE - ELEMENT TYPE 2,3 C---------------------------------------------------------------------DATA W(6),W(7),W(8),W(9),W(10),W(11),W(12)/ 1 .062969590272413570,.062969590272413570,.062969590272413570, 1 .066197076394253089,.066197076394253089,.066197076394253089, 1 .112499999999999996/ C---------------------------------------------------------------------C WEIGHTS - LINEAR STRAIN QUADRILATERAL - ELEMENT TYPE 4,5 C---------------------------------------------------------------------DATA W(13),W(14),W(15),W(16),W(17),W(18),W(19),W(20),W(21)/ 1 1.00000000000000,1.00000000000000, 1 1.00000000000000,1.00000000000000, 1 0.0D0,0.0D0, 1 0.0D0,0.0D0, 1 0.0D0/ C---------------------------------------------------------------------C WEIGHTS - CUBIC STRAIN TRIANGLE - ELEMENT TYPE 6,7 C---------------------------------------------------------------------DATA W(22),W(23),W(24),W(25),W(26),W(27),W(28),W(29), 1 W(30),W(31),W(32),W(33),W(34),W(35),W(36),W(37)/

218


1 .016229248811599,.016229248811599,.016229248811599, 1 .051608685267359,.051608685267359,.051608685267359, 1 .047545817133642,.047545817133642,.047545817133642, 1 .013615157087217,.013615157087217,.013615157087217, 1 .013615157087217,.013615157087217,.013615157087217, 1 .072157803838893/ C---------------------------------------------------------------------C WEIGHTS - 20-NODED BRICK - ELEMENT TYPE 8,9 C---------------------------------------------------------------------DATA W(38),W(39),W(40),W(41),W(42),W(43),W(44),W(45)/ 1 1.000000000000000,1.000000000000000, 1 1.000000000000000,1.000000000000000, 1 1.000000000000000,1.000000000000000, 1 1.000000000000000,1.000000000000000/ C---------------------------------------------------------------------C ONE-DIMENSIONAL INTEGRATION C---------------------------------------------------------------------DATA POSSP(1),POSSP(2),POSSP(3),POSSP(4),POSSP(5)/ 1 -0.906179845938664,-0.538469310105683,0.0D0, 1 0.538469310105683,0.906179845938664/ DATA WEIGP(1),WEIGP(2),WEIGP(3),WEIGP(4),WEIGP(5)/ 1 0.236926885056189,0.478628670499366,0.568888888888889, 1 0.478628670499366,0.236926885056189/ END C********************************************************************** SUBROUTINE MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL) C********************************************************************** C SETS UP THE NQ ARRAY WHICH CONTAINS THE NUMBER C OF DEGREES OF FREEDOM ASSOCIATED WITH EACH NODE C FOR ELEMENTS IN THIS ASSEMBLY. C 1 CALLED BY INSITU. IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),LTYP(MEL),NQ(MNODES) COMMON /ELINF/ LINFO(50,15) C---------------------------------------------------------------------C INXL - INDEX TO NO. OF DEGREES OF FREEDOM OF FIRST NODE OF ELEMENT C (SEE BLOCK DATA ROUTINES BDATA1, MAIN2) C---------------------------------------------------------------------DO 8 J=1,NN 8 NQ(J)=0 C DO 20 J=1,NEL IF(LTYP(J).LT.0) GOTO 20 LT=LTYP(J)


219

NDPT=LINFO(1,LT) C DO 10 I=1,NDPT NDFN=LINFO(I+INXL,LT) NOD=NCONN(I,J) IF(NDFN.GT.NQ(NOD)) NQ(NOD)=NDFN 10 CONTINUE 20 CONTINUE C RETURN END C********************************************************************** SUBROUTINE CALDOF(NN,NN1,NDF,NW,NQ) C********************************************************************** C ROUTINE TO CALCULATE GLOBAL NUMBER FOR D.O.F. INCLUDE ’PARM.FOR’ DIMENSION NW(MNODES+1),NQ(MNODES) C NC=1 NW(1)=1 C DO 10 I=1,NN NC=NC+NQ(I) 10 NW(I+1)=NC C NDF=NW(NN1)-1 C RETURN END

C C********************************************************************** SUBROUTINE GPOUT(LOUT1,NEL,NN,NDF,NCONN, 1 MAT,LTYP,NLST) C********************************************************************** C ROUTINE TO PRINTOUT ARRAYS SET-UP IN GEOMETRY PART OF PROGRAM IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NLST(NTPE) COMMON /ELINF / LINFO(50,15) C WRITE(LOUT1,902)

220


C DO 20 JU=1,NEL IF(NEL.EQ.0)GOTO 20 MPR=JU LT=LTYP(MPR) NDPT=LINFO(1,LT) C DO 10 IN=1,NDPT NP=NCONN(IN,MPR) 10 NLST(IN)=NP C WRITE(LOUT1,906)JU,LT,MAT(MPR),(NLST(IN),IN=1,NDPT) 20 CONTINUE C C C C C

WRITE(LOUT1,908)(NQ(IN),IN=1,NN) WRITE(LOUT1,910)(NW(IN),IN=1,NN1) WRITE(LOUT1,911)NN WRITE(LOUT1,912)NDF

C RETURN 902 FORMAT(//10X,30H ELEMENT MATERIAL TYPE AND, 1 15H NODE NUMBERS//1X,7HELEMENT,1X,4HTYPE,2X,3HMAT, 2 19H 1 2 3 4, 3 55H 5 6 7 8 9 10 11 12 13 14 4 35H 16 17 18 19 20 21 22/) 906 FORMAT(I5,2I6,22I5) 911 FORMAT(//25H TOTAL NUMBER OF NODES =,I8) 912 FORMAT(/40H TOTAL DEGREES OF FREEDOM IN SOLUTION =,I8) END

15,

C********************************************************************** SUBROUTINE MINT(XYZ,NCONN,MAT,LTYP,NW) C********************************************************************** C This is the main calculation subroutine IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED COMMON /PRLDI / PRSLDI(10,100),LEDI(100),NDI1(100),NDI2(100),ILOD


221

C DIMENSION NW(MNODES+1),NP1(NPL),NP2(NPL) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES) DIMENSION PR(NPR,NMT),NTY(NMT),PCONI(MDOF),PEQT(MDOF) c write(*,*)pr(1,1) C

C

C

C

----- CALL SUBROUTINE SETUP NP CALL SETNP(NP1,NP2,NPL) ----- CALCULATE INITIAL INSITU STRESSES LINK1=1 CALL INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW, 1 NP1,NP2,PR,NTY,INCS,INCF) ----- DEFINE INITIAL STRESS STATUS FOR THE MODEL CALL INMODST(STRESS,MAT,PR,NTY,LTYP,NEL,NS,NDIM) ----- Major calculation CALL TOTSOL(XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,PR,NTY, 1 NP1,NP2,INCS,INCF)

C RETURN END C----- FROM HERE, MINOR SUBROUTINES BEGIN C C C----- MINOR SUBROUTINES FOR SUBROUTINE INPUT(...) C SUBROUTINE ZEROR2(V,L1,L2) C********************************************************************** C ROUTINE TO INITIALISE A 2-DIMENSIONAL REAL ARRAY C********************************************************************** c 1 CALLED BY INPUT C 2 CALLED BY TOTSOL C 3 CALLED BY TOTSOL IMPLICIT REAL*8(A-H,O-Z) DIMENSION V(L1,L2) C DO 10 I=1,L1 DO 10 J=1,L2 10 V(I,J)=0.0D0 RETURN END

222


SUBROUTINE ZEROI2(N,L1,L2) C********************************************************************** C ROUTINE TO INITIALISE A 2-DIMENSIONAL INTEGER ARRAY C********************************************************************** C 1 CALLED BY INPUT C 2 CALLED BY INMODST C 3 CALLED BY TOTSOL DIMENSION N(L1,L2) C DO 10 J=1,L2 DO 10 I=1,L1 10 N(I,J)=0 RETURN END

SUBROUTINE ZEROI1(N,LN) C********************************************************************** C ROUTINE TO INITIALISE A 1-DIMENSIONAL INTEGER ARRAY C********************************************************************** C 1 CALLED BY INPUT DIMENSION N(LN) C DO 10 I=1,LN 10 N(I)=0 RETURN END

C----- MINOR SUBROUTINES FOR SUBROUTINE MINT(...) C SUBROUTINE SETNP(NP1,NP2,NPL) C********************************************************************** C SET UP ARRAYS NP1 AND NP2 WHICH GIVE THE INDEX TO ARRAY C NCONN FOR NODES AT EITHER END OF EACH ELEMENT EDGE C********************************************************************** C 1 CALLED BY MINT SETNP(NP1,NP2,NPL) DIMENSION NPL1(21),NPL2(21),NP1(NPL),NP2(NPL) C---------------------------------------------------------------------C INDEXES OF ARRAYS NPL1,NPL2,NP1,NP2 C INDEX ELEMENT TYPE C 1 - 3 1, 2, 3, 6, 7 C 4 - 7 4, 5 C 4 - 15 8, 9 C 16 - 21 10,11 C----------------------------------------------------------------------


223

DATA NPL1(1),NPL1(2),NPL1(3),NPL1(4),NPL1(5),NPL1(6),NPL1(7), 1 NPL1(8),NPL1(9),NPL1(10),NPL1(11),NPL1(12),NPL1(13),NPL1(14), 2 NPL1(15),NPL1(16),NPL1(17),NPL1(18),NPL1(19),NPL1(20),NPL1(21)/ 3 1,2,3,1,2,3,4,5,6,7,8,1,2,3,4,1,2,3,1,2,3/ DATA NPL2(1),NPL2(2),NPL2(3),NPL2(4),NPL2(5),NPL2(6),NPL2(7), 1 NPL2(8),NPL2(9),NPL2(10),NPL2(11),NPL2(12),NPL2(13),NPL2(14), 2 NPL2(15),NPL2(16),NPL2(17),NPL2(18),NPL2(19),NPL2(20),NPL2(21)/ 3 2,3,1,2,3,4,1,6,7,8,5,5,6,7,8,2,3,1,4,4,4/ C DO 10 I=1,NPL NP1(I)=NPL1(I) 10 NP2(I)=NPL2(I) C RETURN END

SUBROUTINE INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW, + NP1,NP2,PR,NTY,INCS,INCF) C********************************************************************** C MAIN CONTROLLING ROUTINE - INSITU STRESSES C********************************************************************** C 1 Called by MINT INITIAL(LINK1,XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,NP1,NP2,PR,NTY) IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ CHARACTER*80 HEADER5 DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),P(MDOF), 1 PT(MDOF),PCOR(MDOF),PEQT(MDOF),XYFT(MDOF),PCONI(MDOF) DIMENSION NW(MNODES+1),IDFX(MDOF),NP1(NPL),NP2(NPL) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION NTY(NMT),PR(NPR,NMT) COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON COMMON COMMON COMMON COMMON

/DATL /DATW /ELINF /FIX /PRSLD

/ / / / /

SL(4,100) W(100) LINFO(50,15) DXYT(4,200),MF(200),NFIX(4,200),NF PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED

224

Appendix: Fortran Codes of CS-S COMMON /PRLDI / PRSLDI(10,100),LEDI(100),NDI1(100),NDI2(100),ILOD COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, + ICOR ,IDCHK,IOUT ,INCT ,IWL , + NOIB ,JS ,JINCB,NLOD ,NLDS

C

----LINK2=1 TTIME=ZERO TGRAV=ZERO READ(LINP,101)HEADER5 READ(LINP,*)IDCHK WRITE(LOUT1,922)IDCHK IF(IDCHK.EQ.0)WRITE(LOUT1,930) IF(IDCHK.EQ.1)WRITE(LOUT1,935) IF(IDCHK.EQ.2)WRITE(LOUT1,940) -----IF ONLY TO TEST GEOMETRY DATA STOP HERE IF(IDCHK.EQ.1)STOP IF(LINK1.EQ.LINK2) GO TO 1 WRITE(LOUT1,904)LINK1,LINK2 STOP

C

C

----- ROUTINE TO READ CONTROL OPTIONS AND MATERIAL PROPERTIES 1

CALL RDPROP(NPLAX,NDIM,NMAT,NOIB,INCS,INCF,INCT, 1 ICOR,PR,NTY)

C

----- READ & SETUP IN-SITU STRESSES AND CHECK FOR EQUILIBRIUM IF(INCS.EQ.1)CALL INSITU(XYZ,STRESS,P,PT,PCOR,PEQT, + XYFT,PCONI,NCONN,MAT,LTYP,NW,IDFX,NP1,NP2,PR,NTY) RETURN 101 904 922 930 935 940

FORMAT(A80) FORMAT(//10X,32HERROR ---- LINK CODE MISMATCH,2I5) FORMAT(/1X,20HDATA CHECK OPTION =,I5/) FORMAT(1X,32HCOMPLETE ANALYSIS IS CARRIED OUT/) FORMAT(1X,30HONLY GEOMETRY DATA ARE CHECKED/) FORMAT(1X,42HGEOMETRY DATA AND IN-SITU STRESSES CHECKED/) END

SUBROUTINE INMODST(STRESS,MAT,PR,NTY,LTYP,NEL,NS,NDIM) C********************************************************************** C This subroutine define initial stress status for the model. C********************************************************************** C 1 CALLED BY MINT IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’

Appendix: Fortran Codes of CS-S COMMON/ELINF /LINFO(50,15) COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) DIMENSION STRESS(NVRS,NIP,MEL),MAT(MEL),PR(NPR,NMT),NTY(NMT), 1 LTYP(MEL),TEMP(6) C

----- Initializing CALL ZEROI2(MCODE,NIP,MEL) CALL ZEROR3(PQMOD,NIP,MEL,2) CALL ZEROR1(TEMP,6)

C DO MR=1,NEL KM=MAT(MR) KGO=NTY(KM) IF(KGO.NE.3.AND.KGO.NE.4)GO TO 100 PRM=PR(4,KM) LT=LTYP(MR) NGP=LINFO(11,LT) DO IP=1,NGP PC=STRESS(NS+3,IP,MR) PQMOD(IP,MR,1)=PC P=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0 DO IS=1,NS TEMP(IS)=STRESS(IS,IP,MR) ENDDO QE=Q(TEMP,NS,NDIM) PQMOD(IP,MR,2)=QE/P C IF(KGO.EQ.3) THEN PCS=PC/2.0D0 PY=P+QE*QE/(P*PRM*PRM) ELSEIF(KGO.EQ.4) THEN PCS=PC/EXP(1.0d0) PY=P*EXP(QE/(PRM*P)) ENDIF C IF(P.GE.PCS) THEN !NC IF(PY.GE.0.9950D0*PC)THEN MCODE(IP,MR)=2 !Iitial point at Pc ELSE MCODE(IP,MR)=1 !Roscoe surface & inside ENDIF ELSEIF(P.LT.PCS) THEN !OC IF(PY.GE.0.9950D0*PC)THEN MCODE(IP,MR)=4 !OC+hardening(expansion) ELSE MCODE(IP,MR)=3 !OC, Hvoslev surface & inside ENDIF

225

226

Appendix: Fortran Codes of CS-S ENDIF

C ENDDO 100 CONTINUE ENDDO RETURN END

SUBROUTINE TOTSOL(XYZ,STRESS,PCONI,PEQT,NCONN,MAT,LTYP,NW,PR,NTY, 1 NP1,NP2,INCS,INCF) C********************************************************************** C This is a main controlling routine. C********************************************************************** C 1 Called by MINT. IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /DATL / SL(4,100) COMMON /DATW / W(100) COMMON /ELINF / LINFO(50,15) COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /PARS / PYI,ASMVL,ZERO COMMON /LOADS / FB(2,15) COMMON /OUT / IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED COMMON /PRLDI / PRSLDI(10,100),LEDI(100),NDI1(100),NDI2(100),ILOD COMMON /PTLOAD/PT1(MDOF),INDPT COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, + ICOR ,IDCHK,IOUT ,INCT ,IWL , + NOIB ,JS ,JINCB ,NLOD ,NLDS COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !back stress variable C DIMENSION NQ(MNODES),NW(MNODES+1),IDFX(MDOF) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES) DIMENSION PR(NPR,NMT),NTY(NMT),NP1(NPL),NP2(NPL) DIMENSION PINC(MDOF),PCOR(MDOF),PREV(MDOF),PT(MDOF),PEQT(MDOF), 1 PCONI(MDOF),PITER(MDOF) DIMENSION DINC(MDOF),DITER(MDOF),DIPR(MDOF),DA(MDOF) DIMENSION PDISLD(3,LV),PRES(3,LV) DIMENSION STRAIN(NVRN,NIP,MEL) DIMENSION


227

XYFIB(MDOF),PIB(MDOF),PEXIB(MDOF),PEXI(MDOF),XYFT(MDOF) DIMENSION JEL(MEL),FXYZ(3),DXYT1(4,200) DIMENSION RINCC(2500),DTM(2500),IOPT(2500) DIMENSION DIMENSION DIMENSION DIMENSION DIMENSION C

VARC(9,NIP,MEL),MCS(MEL),MNGP(MEL),NELCM(MEL) LCS(NIP,MEL),LNGP(NIP,MEL) NCHAIN(100,2),NUMD(MDOF,2),DD(4,200),DD1(4,200) PNOD(MDOF),DP(MDOF),DP1(MDOF) FRICT(MDOF),FRICT1(MDOF),FRICTPR(MDOF),XMUFR(2)

----- MAXIMUM NUMBER OF INCREMENTS IN A INCREMENT BLOCK INCZ=2500 ILINR=0 IBCAL=0 FRMAX0=0.0D0 FRMAX3=0.0D0 TOTPEN=0.0D0 NDIM1=NDIM+1 IF(IDCHK.EQ.0)GOTO 10 WRITE(LOUT1,907) STOP

C

C

!=0 for me

----- INITIALIZE SOME ARRAYS 10 CALL ZEROR3(STRAIN,NVRN,NIP,MEL) CALL ZEROR2(DD,4,200) CALL ZEROI2(NUMD,MDOF,2) CALL ZEROR1(XMUFR,2) DO IDF=1,MDOF XYFT(IDF)=0.0D0 PEXI(IDF)=0.0D0 PCOR(IDF)=0.0D0 DP(IDF)=0.0D0 DP1(IDF)=0.0D0 PNOD(IDF)=0.0D0 FRICT(IDF)=0.0D0 FRICT1(IDF)=0.0D0 FRICTPR(IDF)=0.0D0 DA(IDF)=0.0D0 ENDDO ----- START OF INCREMENT BLOCK LOOP INDPT=0 CALL ZEROR1(PT1,MDOF) ! Initializing

DO 250 JINCB=1,NOIB INCR1=0

! Start of big Do loop, NOIB=1 for me.

228

Appendix: Fortran Codes of CS-S WRITE(LOUT1,908) WRITE(LOUT2,908) WRITE(LOUT3,908) WRITE(LOUT4,908)

C

JINCB JINCB JINCB JINCB

----- INITIALISE LOAD VECTORS DO IDF=1,MDOF XYFIB(IDF)=0.0D0 PIB(IDF)=0.0D0 PEXIB(IDF)=0.0D0 PINC(IDF)=0.0D0 PITER(IDF)=0.0D0 DINC(IDF)=0.0D0 DIPR(IDF)=0.0D0 ENDDO CALL ZEROR2(DXYT1,4,200) ! Initializing CALL ZEROR2(PRSLDI,10,100) ! Initializing

C ILOD=0 CALL ZEROI1(JEL,MEL) CALL ZEROI1(IOPT,2500) CALL ZEROR1(DTM,2500) CALL ZEROR1(RINCC,2500) FRACT=0 C

! ! ! !

Initializing Initializing Initializing Initializing

----- READ INCREMENT CONTROL OPTIONS READ(LINP,101)HEADER15 READ(LINP,*)IBNO,NLOD,ILDF,NFXEL,NFXNOD,IOUTS, 1IOCD,DTIME,ITMF,DGRAV INC1=INCS INC2=INCF IDUMMY=1 IF(IBNO.EQ.2)INC2=2

C WRITE(*,*)IBNO,INC1,INC2,IDUMMY,NLOD,ILDF,NFXEL,NFXNOD,IOUTS, C 2 IOCD,DTIME,ITMF !,DGRAV

WRITE(LOUT1,912)IBNO,INC1,INC2,IDUMMY,NLOD,ILDF,NFXEL,NFXNOD,IOUTS, 2IOCD !,! DTIME,ITMF !,DGRAV NOINC=INC2+1-INC1 IF(NOINC.LE.INCZ)GOTO 70 !NOINC=10 INCZ=2500 -- MAXIMUM VALUE OF NOINC WRITE(LOUT1,950)NOINC STOP 70 IF(IBNO.EQ.JINCB) GO TO 76 !IBNO=1 JINCB=1 WRITE(LOUT1,913) IBNO,JINCB


229

STOP 101 FORMAT(A80) C ----- CALCULATE BODY FORCE LOAD VECTOR C ----- FOR SELF-WEIGHT LOADING AND GRAVITY LOADING 76 CALL SEL1(LOUT1,NDIM,NEL,XYZ,PIB,NCONN,MAT, 1 LTYP,NW,PR,DGRAV)

C

C C

----- READ LOAD FACTORS, TIME FACTORS AND OUTPUT OPTIONS CALL FACTOR(LINP,LOUT1,NOINC,ILDF,IOCD,ITMF,IOUTS, 1RINCC,DTM,IOPT,DTIME) IF(NLOD.EQ.0)GO TO 95 !NLOD=0 IF(NLOD.GT.0)GO TO 82 ----- Skip from here to 95 ----- PRESSURE LOADING ALONG ELEMENT EDGE WRITE(LOUT1,1000) NLDS=IABS(NLOD) IF(NDIM.EQ.2)GOTO 78 WRITE(LOUT1,955) 955 FORMAT(/1X,’NO OPTION TO CALCULATE NODAL LOADS’,1X, 1 ’FROM PRESSURE LOADING IN 3-D PROBLEM (ROUTINE TOTSOL)’) STOP

C 78 DO 80 KLOD=1,NLDS ! NLDS = No. of loaded element. (guess) READ(LINP,*)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,NDIM),IV=1,NPT) WRITE(LOUT1,1002)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,NDIM),IV=1,NPT) C CALL ZEROR2(FB,2,15) DO 100 IV=1,NPT DO 100 ID=1,NDIM IDR=NDIM+1-ID 100 PRES(ID,IV)=PDISLD(IDR,IV) C DO 110 IV=1,NPT DO 110 ID=1,NDIM 110 PDISLD(ID,IV)=PRES(ID,IV) C CALL EDGLD(LOUT1,NDIM,NCONN,LTYP,LNE,ND1,ND2,NP1,NP2, 1PDISLD,PRES,KLOD,NPT,0,MXLD) C CALL DISTLD(LOUT1,XYZ,PIB,NCONN,LTYP,NW,NP1, 1 NP2,PRES,LNE,ND1,ND2,1,1,1.0D0) 80 CONTINUE GO TO 95 C

----- READ INCREMENTAL POINT LOADS

230

Appendix: Fortran Codes of CS-S 82 WRITE(LOUT1,916)

C DO 90 JJ=1,NLOD READ(LINP,*)KK,(FXYZ(ID),ID=1,NDIM) WRITE(LOUT1,940)KK,(FXYZ(ID),ID=1,NDIM) C ----- NO PROVISION FOR PORE PRESSURE TERMS IN ’APPLIED’ NODAL LOADS FTT=ZERO KJ=KK N1=NW(KJ)-1 IDF=NW(KJ+1)-NW(KJ) IF(IDF.EQ.1)GO TO 84 C DO 83 ID=1,NDIM 83 XYFIB(N1+ID)=FXYZ(ID) IF(IDF.EQ.NDIM1)XYFIB(N1+NDIM1)=FTT GO TO 90 84 XYFIB(N1+1)=FTT 90 CONTINUE C 95 IF(NFXEL.EQ.0.AND.NFXNOD.EQ.0) GO TO 137 !NFXEL=8, NFXNOD=0 C

----- READ CHANGE WRITE(LOUT1,931)

TO

NODAL

FIXITIES

C IF(NDIM.EQ.2)THEN READ(LINP,101)HEADER16 CALL FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFXEL, 1 NFXNOD) ENDIF IF(NDIM.EQ.3)THEN READ(LINP,101)HEADER16 CALL FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFXEL,NFXNOD) ENDIF 137 CONTINUE C

----- FIND {NQ} CALL MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL)

C

----- READ DATA FOR PENETRATION PROBLEM READ(LINP,101)HEADER17 READ(LINP,*)NPEN,IPEN IF(NPEN.GE.1)THEN ! IF(IPEN.EQ.1)CALL INPENT(LINP,NPEN,XREF,YREF,DYREF,DXPEN, !IPEN=1 1 XCOS,XSIN,XMUFR,NCHAIN,KSLID0,KSLID3)


231

IF(KSLID3.EQ.0)THEN NSKEW1=NSKEW NSKEW=0 ENDIF ENDIF C

----- Initializing incremental variables for back stress by Song 2/22/98 PREP=1 PREQ=0 PREETA=0 PREALPH=0

C

----- START OF INCREMENT LOOP INCT=0 ! ADDED BY SONG 9/2/98 DO 200 JS=INC1,INC2 !*** INCREMENTAL LOOP STARTS*** INCT=INCT+1 INCR1=INCR1+1 IF(JS.EQ.INCT)GO TO 138 WRITE(LOUT1,933)JS,INCT STOP 138 JC=JS+1-INC1 FRACLD=RINCC(JC) !FRACLD=RINCC=FSTD=1/No. of Inc C FRACLD=Fragment !Song FRACT=FRACT+FRACLD !AccumulatED total displ for unit pen. leng. DTIMEI1=DTM(JC) TTIME=TTIME+DTIMEI1 DTIMEI=DTIMEI1 DGRAVI=FRACLD*DGRAV !DGRAV = increment in gravity field =0 TGRAV=TGRAV+DGRAVI IOUT=IOPT(JC) C ----- ASSIGN THE APPROPRIATE POINTERS TO THE DOF AND STIFFNESS MATRIX CALL LOCINIT(LTYP,NCONN,NQ,NW,NDF,NDIM) C IWL=0 IF(JINCB.EQ.NOIB.AND.JS.EQ.INC2)IWL=1 C---------------------------------------------------------------------C BOUNDARY CONDITIONS (LOADS AND DISPLACEMENTS) ARE PRINTED C EVERY IBC INCREMENTS C IBC = 0 NOT PRINTED IN ANY INCREMENT C IBC = 1 PRINTED IN EACH INCREMENT C IBC = N PRINTED IN EVERY NTH INCREMENT C---------------------------------------------------------------------IOUTP=0 IF(IBC.EQ.0)GOTO 130 NJS=IBC*(JS/IBC)

232


IF(NJS.EQ.JS)IOUTP=1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC IF(INCT.LE.5.OR.INCT.EQ.8)IOUTP=1 IF(INCT.EQ.10.OR.INCT.EQ.15)IOUTP=1 IF(INCT.EQ.20.OR.INCT.EQ.30)IOUTP=1 IF(INCT.EQ.40.OR.INCT.EQ.50)IOUTP=1 IF(INCT.EQ.75.OR.INCT.EQ.100)IOUTP=1 IF(INCT.EQ.150)IOUTP=1 IF(IBNO.GE.2)IOUTP=1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC WRITE(*,*)’INCR=’,INCT IF(IOUTP.EQ.1)THEN WRITE(LOUT1,919)DTIMEI,TTIME WRITE(LOUT2,919)DTIMEI,TTIME WRITE(LOUT3,919)DTIMEI,TTIME WRITE(LOUT4,919)DTIMEI,TTIME ENDIF C 130 DO 140 IM=1,NDF C XYFT(IM)=XYFT(IM)+XYFIB(IM)*FRACLD+FRICT(IM) C 140 PINC(IM)=FRACLD*PIB(IM)+FRACLD*XYFIB(IM)+DP(IM)+FRICT(IM) XYFT(IM)=XYFT(IM)+XYFIB(IM)*FRACLD+DP1(IM)+FRICT(IM)+FRICT1(IM) 140 PINC(IM)=FRACLD*PIB(IM)+FRACLD*XYFIB(IM)+DP(IM)+DP1(IM)+FRICT1(IM) CALL ZEROR1(FRICT,MDOF) CALL ZEROR1(FRICT1,MDOF) C DO 145 IM=1,NDF 145 PEXI(IM)=(1.0D0-FRACT)*PEXIB(IM)+PNOD(IM) C

----- UPDATE LIST OF PRESSURE LOADING ALONG ELEMENT EDGES IF(NLOD.GE.0)GO TO 162

C

C ***

150 160 162 C

DO 160 ISD=1,NLDS LNE=LEDI(ISD) ND1=NDI1(ISD) ND2=NDI2(ISD) ICT=0 N2D = 2 FOR TWO DIMENSIONAL PROBLEMS N2D=2 DO 150 IK=1,NPT DO 150 IJ=1,N2D ICT=ICT+1 PRES(IJ,IK)=FRACLD*PRSLDI(ICT,ISD) CALL LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,0,MXLD) CONTINUE CONTINUE ----- INITIALISE INCREMENTAL DISPLACEMENTS


233

ICONV=0 ITER=1 MAXITER=10 TOL=0.080D0 C DO I10=1,NDF PITER(I10)=PINC(I10)+PCOR(I10) c PITER(I10)=PINC(I10) ENDDO DO IDF=1,MDOF DINC(IDF) =0.0D0 DIPR(IDF) =0.0D0 DITER(IDF)=0.0D0 PREV(IDF) =0.0D0 PINC(IDF) =0.0D0 ENDDO C---------------------------------------------------------------------c IF(INCR1.EQ.1.AND.LINR.GE.1)THEN c ILINR=LINR c LINR =0 c ELSEIF(INCR1.GT.1.AND.ILINR.GE.1)THEN c LINR =ILINR c ILINR=0 c ENDIF c write(2,*)’linr=’,linr C---------------------------------------------------------------------C

----- START OF ITERATION LOOP ******ITERATION STARTS****** DO WHILE (ICONV.EQ.0.AND.ITER.LE.MAXITER)

C

----- CONVERT PITER TO COUNT FOR SKEW BOUNDARIES IF(NSKEW.GT.0)CALL ROTBC(PITER,NW,NDIM,NSKEW,1)

C

C

----- SOLVE EQUATIONS USING SKY LINE CALL SKSOLV(XYZ,DA,DITER,DIPR,DD,STRESS,PITER,NQ,NW,LTYP, 1 NTY,MAT,NCONN,PR,IOUTP,ITER) !js REMOVED TENTATIVELY IF(ITER.EQ.1) THEN DO IJ=1,NF DO ID=1,NDIM DXYT1(ID,IJ)=DXYT(ID,IJ) DXYT(ID,IJ)=0.0D0 DD1(ID,IJ)=DD(ID,IJ) DD(ID,IJ)=0.0D0 ENDDO ENDDO ENDIF ----- ROTATE BACK DISPL & LOADS AT SKEW BOUNDARIES TO ORIGINAL

234


COORD. IF(NSKEW.GT.0) THEN CALL ROTBC(DITER,NW,NDIM,NSKEW,-1) CALL ROTBC(PITER,NW,NDIM,NSKEW,-1) END IF C

----- FIND THE OUT-OF-BALANCE LOAD PCOR CALL EQUIBLOD(XYZ,NCONN,MAT,LTYP,NQ,NW,NP1,NP2,PR,NTY,DITER, 1 DIPR,STRESS,STRAIN,PEXI,IDFX,PITER,PT,PCOR,PEQT,XYFT,PCONI, 2 LCS,LNGP,NELCM,MCS,MNGP,NCAM,ITER,IOUTP,JS)

C

----- ROTATE BACK PCOR AT SKEW BOUNDARIES TO ORIGINAL COORD. IF(NSKEW.GT.0)CALL ROTBC(PCOR,NW,NDIM,NSKEW,-1)

C DO I10=1,NDF DINC(I10)=DINC(I10)+DITER(I10) ENDDO C

----- CHECK CONVERGENCE CALL CONVCH(NDF,NN,NDIM,TOL,ICONV,2,NW,DITER,DINC,PREV,PCOR, 1 ITER,RNRM1)

C DO I10=1,NDF PINC(I10)=PINC(I10)+PITER(I10) PITER(I10)=PCOR(I10) PREV(I10)=PCOR(I10) DIPR(I10)=DITER(I10) ENDDO CALL ZEROR1(DITER,MDOF) DTIMEI=0.0001D0 DTIMEI=DTIMEI1/100.0D0 WRITE(*,*)" Iter. No.",ITER ITER=ITER+1 c

ENDDO

!***** END OF ITERATION LOOP (iteration until converge)

C NITER=ITER-1 IF(IOUTP.EQ.1)WRITE(LOUT1,*)’ICONV=’,ICONV, 1 ’NO. OF ITERATIONS=’,NITER C DO IJ=1,NF DO ID=1,NDIM DXYT(ID,IJ)=DXYT1(ID,IJ) DXYT1(ID,IJ)=0.0D0


235

DD(ID,IJ)=DD1(ID,IJ) DD1(ID,IJ)=0.0D0 ENDDO ENDDO C

C

----- UPDATE AND OUTPUT CALCULATIONS CALL UPOUT(XYZ,DINC,DA,STRESS,STRAIN,PINC,PT,PCOR,PEQT,NCONN, 1 MAT,NTY,PR,LTYP,NW,NQ,IDFX,VARC,YREF,IOUTP) ----- ADJUST B.C FOR THE PENETRATION PROBLEM IF(NPEN.GE.1)THEN YREF=YREF-DYREF*FRACLD !FRACLD=FSTD=RINCC=1/NOINC TOTPEN=TOTPEN+DYREF*FRACLD IF(KSLID3.NE.0) THEN

c CALL BCADJST(NPEN,NDIM,NSKEW,XREF,YREF,DXPEN,XSIN,XCOS, NCHAIN,XYZ,NW,PEQT,PT,FRICT1,FRICTPR,XMUFR,PNOD,DD,DP,DP1, INCT,NUMD,IOUTP,TOTPEN,IBCAL,FRMAX0,FRMAX3)

1 2 c

ENDIF IF(KSLID0.EQ.0.OR.KSLID3.EQ.0)THEN CALL CHSLIDE(NPEN,NSKEW,NSKEW1,DXPEN,KSLID0,KSLID3,XSIN,XCOS, 1 XMUFR,NCHAIN,NW,PEQT,PT,FRICT,FRICTPR,INCT,FRMAX0,FRMAX3) ENDIF ENDIF 200 CONTINUE C

!*** END OF INCREMENTAL LOOP (for incremental penet.)

----- ZERO ALL NON-ZERO PRESCRIBED VALUES IF(NF.EQ.0)GOTO 240

C DO JJ=1,MXFXT DO II=1,4 DD(II,JJ)=0.0d0 DXYT(II,JJ)=0.0d0 ENDDO ENDDO C DO JJ=1,NDF FRICT1(JJ)=0.0D0 DP1(JJ)=0.0D0 PNOD(JJ)=PNOD(JJ)+DP(JJ) DP(JJ)=0.0D0 ENDDO 240 CONTINUE C 250 CONTINUE !*** END OF INCREMENTAL LOOP (for the incremental blocknumber)

236


907 FORMAT(/1X,24HANALYSIS NOT CARRIED OUT/) 908 FORMAT(//120(1H=)// 1 1X,43HSTART OF LOAD INCREMENT BLOCK NUMBER ,I5/1X,48(1H-)) 912 FORMAT(/ 11X,23HINCR BLOCK NUMBER.....=,I5,4X,23HSTARTING INCR NUMBER..=,I8/ 21X,23HFINISHING INCR NUMBER.=,I5,4X,23HNO. OF ELEMENT CHANGES=,I8/ 31X,23HNUMBER OF LOADS.......=,I5,4X,23HLOAD RATIO OPTION.....=,I8/ 41X,23HNUM OF ELEM FIXITIES..=,I5,4X,23HNUM OF NODE FIXITIES..=,I5/ 51X,23HSTD OUTPUT CODE.......=,I8/ !i8 61X,23HOUTPUT OPTION.........=,I8/) C 74X,23HTIME INCREMENT........=,F10.1/ C 71X,23HTIME RATIO OPTION.....=,I5/) C 84X,23HINCR IN GRAVITY FIELD.=,F10.1/) !temp kill 913 FORMAT(//1X,26HERROR IN INCR BLOCK NUMBER,2I6) 914 FORMAT(//28H LIST OF ELEMENT ALTERATIONS/1X,27(1H-)/) 916 FORMAT(//,3X,32H LIST OF INCREMENTAL NODAL LOADS/3X,33(1H-),/, 1 3X,4HNODE,6X,1HX,9X,1HY,9X,1HZ,/,3X,33(1H-),/) 920 FORMAT(1X,10I8) 931 FORMAT(/1X,29HPRESCRIBED BOUNDARY CONDITONS/1X,29(1H-)/) 933 FORMAT(//1X,’ERROR IN INCREMENT NUMBER’,2I6,2X,’(ROUTINE TOTSOL)’) 940 FORMAT(1X,I5,3F10.3) 950 FORMAT(/1X,46HINCREASE SIZE OF ARRAYS RINCC, DTM AND IOPT TO, 1 I5,2X,’ALSO SET INCZ IN ROUTINE TOTSOL’) 1000 FORMAT(39H SPECIFIED NODAL VALUES OF SHEAR/NORMAL, 1 36H STRESSES AND EQUIVALENT NODAL LOADS/1X,74(1H-)/5H0ELEM, 2 1X,4HNDE1,2X,4HNDE2,2X,4HSHR1,8X,4HNOR1,8X,4HSHR2,8X,4HNOR2, 3 8X,4HSHR3,8X,4HNOR3,8X,4HSHR4,8X,4HNOR4,8X,4HSHR5,8X,4HNOR5/ 1 1X,16H(LOAD DIRECTION),2X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y), 2 9X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y),9X,3H(X),9X,3H(Y)/) 1002 FORMAT(1X,3I4,10E12.4) C 915 FORMAT(//120(1H=)// C 1 1X,32HSTART OF LOAD INCREMENT NUMBER ,I5, C 2 4X,22HINCREMENT BLOCK NUMBER,I5,4X,13HLOAD RATIO =,F5.2/ C 3 1X,90(1H-)) C 917 FORMAT(/22H INCR GRAVITY LEVEL =,E12.4, C 1 24H TOTAL GRAVITY LEVEL =,E12.4) 919 FORMAT(/80(1H=),/18H TIME INCREMENT =,G12.4,/, 1 14H TOTAL TIME =,G12.4) RETURN END

C----- MINOR OF MINOR SUBROUTINES C C SUBROUTINE


237

RDPROP(NPLAX,NDIM,NMAT,NOIB,INCS,INCF,INCT,ICOR,PR,NTY) C********************************************************************** C READ CONTROL OPTIONS AND MATERIAL PROPERTIES C********************************************************************** C 1 CALLED BY INITIAL IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ CHARACTER*80 HEADER6 DIMENSION PR(NPR,NMT),NTY(NMT) COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /OUT / IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF COMMON/MATPROP/C,X C---------------------------------------------------------------------C ICOR - OPTION TO APPLY OUT-OF-BALANCE LOADS AS CORRECTING C LOADS IN THE NEXT INCREMENT C ICOR = 0 - CORRECTING LOADS ARE NOT APPLIED C ICOR = 1 - CORRECTING LOADS ARE APPLIED C---------------------------------------------------------------------ICOR=0 C READ(LINP,101)HEADER6 READ(LINP,*)NMAT,NOIB,INCS,INCF WRITE(LOUT1,922)NMAT,NOIB,INCS,INCF NOINC=INCF-INCS+1 IF(NOINC.GT.0)GOTO 5 WRITE(LOUT1,925)NOINC,INCS,INCF STOP C 5 CONTINUE C---------------------------------------------------------------------C INCT - COUNTER OF INCREMENT NUMBER C---------------------------------------------------------------------INCT=INCS-1 IF(NDIM.NE.3)GOTO 8 WRITE(LOUT1,928) GOTO 10 8 IF(NPLAX.EQ.0)WRITE(LOUT1,930) IF(NPLAX.EQ.1)WRITE(LOUT1,931) 10 CONTINUE C

----- READ OUTPUT REDUCING OPTIONS. THIS OPERATES ON RE READ(LINP,101)HEADER7 READ(LINP,*)IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF WRITE(LOUT1,945)IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF

C

----- READ MATERIAL PROPERTIES

238

Appendix: Fortran Codes of CS-S CALL ZEROR2(PR,NPR,NMT) WRITE(LOUT1,932) READ(LINP,101)HEADER8 DO 20 I=1,NMAT READ(LINP,*)II,NTY(II),(PR(JJ,II),JJ=1,NPR) WRITE(LOUT1,936)II,NTY(II),(PR(JJ,II),JJ=1,NPR) 20 CONTINUE

READ(LINP,101)HEADER81 !back stress parameter READ(LINP,*)pr(6,2),pr(7,2) !C,X read(LINP,101)HEADER82 !VISCOSITY HEADER read(linp,*)pr(1,2),pr(2,2) !VISCOSITY DATA read(linp,101)header83 !Damage header read(linp,*)pr(3,2),pr(4,2) !A1 & A2 read(linp,101)header84 !Gradient Header read(linp,*)pr(5,2) !gradcon c write(*,*)pr(5,2)

C C C C

RETURN 101 FORMAT(A80) 922 FORMAT(/ 1 10X,46HNUMBER OF MATERIALS..........................=,I5/ 2 10X,46HNUMBER OF INCREMENT BLOCKS...................=,I5/ 3 10X,46HSTARTING INCR NUMBER OF ANALYSIS.............=,I5/ 4 10X,46HFINISHING INCR NUMBER OF ANALYSIS............=,I5/ 5 /120(1H*)/) 6 10X,46HNUMBER OF PRIMARY ELEMENT CHANGES............=,I5/ 7 10X,46HOPTION TO UPDATE COORDINATES.................=,I5/ 8 10X,46HOPTION TO STOP/RESTART ANALYSIS..............=,I5/ 9 /120(1H*)/) 925 FORMAT(/1X,29HERROR IN NO. OF INCREMENTS =,I5, 1 4X,7HINCS =,I5,4X,7HINCF =,I5,2X,16H(ROUTINE RDPROP)) 928 FORMAT(//1X,22H3-DIMENSIONAL ANALYSIS) 930 FORMAT(//1X,21HPLANE STRAIN ANALYSIS) 931 FORMAT(//1X,22HAXI-SYMMETRIC ANALYSIS) 932 FORMAT(//24H MATERIAL PROPERTY TABLE 1 /1X,23(1H-) 2 //2X,8HMAT TYPE,7X,1H1,11X,1H2,11X,1H3,11X,1H4,11X,1H5, 3 11X,1H6,11X,1H7,11X,1H8,11X,1H9,11X,2H10/) 936 FORMAT(1X,2I5,(10E12.4/)) 945 FORMAT(//120(1H*)/ 1 10X,46HOPTION TO PRINT BOUNDARY CONDITIONS..........=,I5/ 2 10X,46HOPTION TO PRINT REACTIONS....................=,I5/ 3 10X,46HSTARTING VERTEX NODE NUMBER FOR OUTPUT.......=,I5/ 4 10X,46HFINISHING VERTEX NODE NUMBER FOR OUTPUT......=,I5/ 5 10X,46HSTARTING MIDSIDE NODE NUMBER FOR OUTPUT......=,I5/ 6 10X,46HFINISHING MIDSIDE NODE NUMBER FOR OUTPUT.....=,I5/ 7 10X,46HSTARTING ELEMENT NUMBER FOR OUTPUT...........=,I5/


239

8 10X,46HFINISHING ELEMENT NUMBER FOR OUTPUT..........=,I5/ 9 /120(1H*)/) END

SUBROUTINE LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,ILST,MXLD) C********************************************************************** C ROUTINE TO STORE CUMULATIVE LIST OF APPLIED C PRESSURE LOADING ALONG ELEMENT EDGES C********************************************************************** C 1 Called by EDGLD IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION PRES(3,LV) COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED C

----- SKIP IF NEW LIST IF(NLED.EQ.0.OR.ILST.EQ.1)GO TO 22

C

----- SEARCH FOR LNE IN EXISTING LIST DO 20 J=1,NLED IF(LNE.NE.LEDG(J))GO TO 20 N1=NDE1(J) N2=NDE2(J) IF(N1.EQ.ND1.AND.N2.EQ.ND2)GO TO 25 20 CONTINUE

C

C

----- ADD NEW EDGE TO THE LIST 22 NLED=NLED+1 IF(NLED.LE.MXLD)GO TO 23 WRITE(LOUT1,900) 900 FORMAT(/27H INCREASE SIZE OF ARRAYS IN, 1 51H COMMON BLOCK PRSLD ALSO SET MXLD IN ROUTINE MAXVAL/ 2 25X,16H(ROUTINE LODLST)) STOP 23 JE=NLED GO TO 30 ----- UPDATE EXISTING LIST 25 JE=J GO TO 35

C 30 LEDG(JE)=LNE NDE1(JE)=ND1 NDE2(JE)=ND2 C 35 IC=0

240

Appendix: Fortran Codes of CS-S DO 40 IPT=1,NPT DO 40 IK=1,NDIM IC=IC+1 40 PRESLD(IC,JE)=PRESLD(IC,JE)+PRES(IK,IPT) RETURN END

SUBROUTINE ZEROR1(A,LA) C********************************************************************** C ROUTINE TO INITIALISE A 1-DIMENSIONAL REAL ARRAY C********************************************************************** C 1 CALLED BY INMODST C 2 CALLED BY TOTSOL IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(LA) C DO 10 I=1,LA 10 A(I)=0.0D0 RETURN END SUBROUTINE ZEROR3(V,L1,L2,L3) C********************************************************************** C ROUTINE TO INITIALISE A 3-DIMENSIONAL REAL ARRAY C********************************************************************** C 1 CALLED BY INMODST C 2 CALLED BY TOTSOL IMPLICIT REAL*8(A-H,O-Z) DIMENSION V(L1,L2,L3) C DO 10 I=1,L1 DO 10 J=1,L2 DO 10 K=1,L3 10 V(I,J,K)=0.0D0 RETURN END SUBROUTINE DETJCB(LOUT1,NDIM,DJACB,NDN,ELCOD,DS,IP,MUS,KSTGE) C********************************************************************** C CALCULATES DETERMINANT OF JACOBIAN MATRIX * C********************************************************************** C 1 CALLED BY SELF IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO


C C

241

DIMENSION ELCOD(3,NDMX),DS(3,20),XJAC(3,3) ----- NXJ - SIZE OF ARRAY XJAC NXJ=3 CALL ZEROR2(XJAC,3,3)

C DO 10 ID=1,NDIM DO 10 JD=1,NDIM DO 10 IN=1,NDN 10 XJAC(ID,JD)=XJAC(ID,JD)+DS(ID,IN)*ELCOD(JD,IN) C IF(NDIM.NE.2)GOTO 20 DJACB=XJAC(1,1)*XJAC(2,2)-XJAC(1,2)*XJAC(2,1) GOTO 50 C 20 DJACB=XJAC(1,1)*(XJAC(2,2)*XJAC(3,3)-XJAC(2,3)*XJAC(3,2)) DJACB=DJACB-XJAC(1,2)*(XJAC(2,1)*XJAC(3,3)-XJAC(2,3)*XJAC(3,1)) DJACB=DJACB+XJAC(1,3)*(XJAC(2,1)*XJAC(3,2)-XJAC(2,2)*XJAC(3,1)) C 50 IF(DJACB.GT.ZERO)GO TO 60 WRITE(LOUT1,900)DJACB,MUS,IP 900 FORMAT(1X,10H JACOBIAN,E16.5,3X,11HIS NEGATIVE,2X, 1 7HELEMENT,I5,2X,10HINT. POINT,I5,2X,16H(ROUTINE DETJCB)) C WRITE(LOUT1,910)KSTGE 910 FORMAT(/1X,’CODE TO INDICATE STAGE OF ANALYSIS =’,I5// 1 4X,’CODE’,20X,’STAGE OF THE ANALYSIS’,// 1 6X,’1 - CALLED BY INSITU/EQLOD/SELF CALCULATION OF’, 2 1X,’INSITU SELF WEIGHT LOADS’,/6X,’2 - CALLED BY’, 3 1X,’TOTSOL/CHANGE/SELF LOADS DUE TO ELEMENT CHANGES’,/ 4 6X,’3 - CALLED BY TOTSOL/SEL1/SELF INCREMENTAL SELF’, 5 1X,’WEIGHT LOADS/6X,25H4 - CALLED BY UPOUT/EQLOD’, 6 ’/SELF SELF WEIGHT LOADS FOR EQUILIBRIUM CHECK’) STOP 60 RETURN END

SUBROUTINE DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,JL,IP,ISTGE) C********************************************************************** C CALCULATES DETERMINANT AND INVERSE OF A SQUARE 3X3 MATRIX C********************************************************************** C 1 Called by FORMB2 C 2 Called by ELMSTIF C 3 Called by EQUIBLOD C 4 Called by DMCAM C 5 Called by UPOUT IMPLICIT REAL*8(A-H,O-Z)

242

Appendix: Fortran Codes of CS-S INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO DIMENSION XJACM(3,3),XJACI(3,3)

C IF(NDIM.EQ.3)GOTO 20 DJACB=XJACM(1,1)*XJACM(2,2)-XJACM(1,2)*XJACM(2,1) IF(DJACB.GT.ZERO)GOTO 15 GOTO 60 C 15 XJACI(1,1)= XJACM(2,2)/DJACB XJACI(2,2)= XJACM(1,1)/DJACB XJACI(1,2)=-XJACM(1,2)/DJACB XJACI(2,1)=-XJACM(2,1)/DJACB RETURN C 20 XJACI(1,1)= (XJACM(2,2)*XJACM(3,3)-XJACM(2,3)*XJACM(3,2)) XJACI(1,2)=-(XJACM(1,2)*XJACM(3,3)-XJACM(1,3)*XJACM(3,2)) XJACI(1,3)= (XJACM(1,2)*XJACM(2,3)-XJACM(1,3)*XJACM(2,2)) C XJACI(2,1)=-(XJACM(2,1)*XJACM(3,3)-XJACM(2,3)*XJACM(3,1)) XJACI(2,2)= (XJACM(1,1)*XJACM(3,3)-XJACM(1,3)*XJACM(3,1)) XJACI(2,3)=-(XJACM(1,1)*XJACM(2,3)-XJACM(1,3)*XJACM(2,1)) C XJACI(3,1)= (XJACM(2,1)*XJACM(3,2)-XJACM(2,2)*XJACM(3,1)) XJACI(3,2)=-(XJACM(1,1)*XJACM(3,2)-XJACM(1,2)*XJACM(3,1)) XJACI(3,3)= (XJACM(1,1)*XJACM(2,2)-XJACM(2,1)*XJACM(1,2)) C 1

DJACB=XJACM(1,1)*XJACI(1,1)+XJACM(1,2)*XJACI(2,1)+ XJACM(1,3)*XJACI(3,1) IF(DJACB.GT.ZERO)GOTO 32 GOTO 60

C 32 DJACBI=1.0D0/DJACB C DO 35 ID=1,NDIM DO 35 JD=1,NDIM 35 XJACI(ID,JD)=XJACI(ID,JD)*DJACBI RETURN

60 WRITE(LOUT1,900)DJACB,JL,IP 900 FORMAT(/1X,9HJACOBIAN ,E16.5,3X,10HOF ELEMENT,I6,3X, 1 17HINTEGRATION POINT,I5,3X,29HIS NEGATIVE (ROUTINE DETMIN)) WRITE(LOUT1,910)ISTGE 910 FORMAT(/1X,36HCODE TO INDICATE STAGE OF ANALYSIS =,I5// 1 4X,4HCODE,20X,21HSTAGE OF THE ANALYSIS// 2 6X,49H1 - CALLED BY RDSTRS/EQLIB/FORMB2 LOAD EQUIVALENT, 3 19H TO INSITU STRESSES/6X,33H2 - CALLED BY CHANGE/EQLIB/FORMB2,


243

4 5 6 7 8

32H CALCULATION OF IMPLIED LOADINGS/6X, 35H3 - CALLED BY SKSOLV/ELMSTIF/FORMB2, 32H CALCULATION OF STIFFNESS MATRIX/ 6X,38H4 - CALLED BY UPOUT/FORMB2 CALCULATION, 1X,24HOF STRAINS. OUTPUT STAGE) STOP END

SUBROUTINE DETMIN1(LOUT1,NDIM,XJACM,XJACI,DJACB,JL,IP,ISTGE) C********************************************************************** C CALCULATES DETERMINANT AND INVERSE OF A SQUARE 3X3 MATRIX C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO DIMENSION XJACM(3,3),XJACI(3,3) 20 XJACI(1,1)= (XJACM(2,2)*XJACM(3,3)-XJACM(2,3)*XJACM(3,2)) XJACI(1,2)=-(XJACM(1,2)*XJACM(3,3)-XJACM(1,3)*XJACM(3,2)) XJACI(1,3)= (XJACM(1,2)*XJACM(2,3)-XJACM(1,3)*XJACM(2,2)) C XJACI(2,1)=-(XJACM(2,1)*XJACM(3,3)-XJACM(2,3)*XJACM(3,1)) XJACI(2,2)= (XJACM(1,1)*XJACM(3,3)-XJACM(1,3)*XJACM(3,1)) XJACI(2,3)=-(XJACM(1,1)*XJACM(2,3)-XJACM(1,3)*XJACM(2,1)) C XJACI(3,1)= (XJACM(2,1)*XJACM(3,2)-XJACM(2,2)*XJACM(3,1)) XJACI(3,2)=-(XJACM(1,1)*XJACM(3,2)-XJACM(1,2)*XJACM(3,1)) XJACI(3,3)= (XJACM(1,1)*XJACM(2,2)-XJACM(2,1)*XJACM(1,2)) C DJACB=XJACM(1,1)*XJACI(1,1)+XJACM(2,1)*XJACI(1,2)+ XJACM(3,1)*XJACI(1,3)

1 C

32 DJACBI=1.0D0/DJACB C DO 35 ID=1,NDIM DO 35 JD=1,NDIM 35 XJACI(ID,JD)=XJACI(ID,JD)*DJACBI RETURN END SUBROUTINE DISTLD(LOUT1,XYZ,RHS,NCONN,LTYP, 1 NW,NP1,NP2,PRES,LNE,ND1,ND2,IPRINT,IST,FC) C********************************************************************** C ROUTINE TO CALCULATE EQUIVALENT NODAL LOADS FOR SPECIFIED *

244


C C C C

PRESSURE LOADING ALONG ELEMENT EDGES USING 5 POINT (NSP) * INTEGRATION RULE. INTEGRATES POLYNOMIAL OF ORDER NINE OR LESS * EXACTLY. ARRAYS ILOC,PRES,PEQLD,ELCD,SHF,DERIV ARE * TO CATER FOR A MAXIMUM OF FIVE NODES (NPT) ALONG AN ELEMENT EDGE * C (ALL 2-D ELEMENTS UP TO ORDER FIVE). * C********************************************************************** C 1 Called by EQLOD IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /ELINF / LINFO(50,15) COMMON /SAMP / POSSP(5),WEIGP(5) COMMON /LOADS / FB(2,15) DIMENSION NCONN(NTPE,MEL),LTYP(MEL), 1 NW(MNODES+1),NP1(NPL),NP2(NPL) DIMENSION RHS(MDOF),XYZ(3,MNODES),PRES(3,LV) DIMENSION ILOC(5),PSP(2),DSP(2),PEQLD(3,5),ELCD(2,5) DIMENSION SHF(5),DERIV(5),PCOM(3) NP=5 TPI=2.0D0*PYI C NE=MREL(LNE) NE=LNE LI1=ND1 C LI1=NREL(ND1) LT=LTYP(NE) IF(IST.EQ.1)GOTO 5 LT=IABS(LT) 5 IF(LT.GT.0)GOTO 10 WRITE(LOUT1,900)LNE 900 FORMAT(/1X,44H**** ERROR : YOU HAVE PUT A PRESSURE LOAD ON, 1 8H ELEMENT,I5,2X,28HWHICH IS NOT PRESENT IN MESH, 2 17H (ROUTINE DISTLD)/) RETURN 10 NVN=LINFO(2,LT) NEDG=LINFO(3,LT) NDSD=LINFO(7,LT) NTSD=NDSD+2 INDED=LINFO(14,LT) C DO 20 K1=1,NEDG J1=NP1(K1+INDED) J2=NP2(K1+INDED) I1=NCONN(J1,NE)

Appendix: Fortran Codes of CS-S IF(LI1.EQ.I1)GOTO 25 20 CONTINUE WRITE(LOUT1,903)LNE,ND1,ND2 903 FORMAT(/21H **** ERROR : ELEMENT,I5, 1 2X,22H DOES NOT HAVE NODES :,2I5, 2 3X,16H(ROUTINE DISTLD)) RETURN C

----- STORE LOCATIONS OF NODE (IN NCONN) IN ARRAY ILOC 25 LC1=NVN+(J1-1)*NDSD ILOC(1)=J1 ILOC(NTSD)=J2 IF(NDSD.EQ.0)GOTO 31

C DO 30 JP=1,NDSD 30 ILOC(JP+1)=LC1+JP C

----- SET UP LOCAL ARRAY FOR CO-ORDINATES IN ELCD 31 DO 32 KC=1,NTSD ILC=ILOC(KC) NDE=NCONN(ILC,NE)

C

C

DO 32 ID=1,NDIM 32 ELCD(ID,KC)=XYZ(ID,NDE) INITIALISE PEQLD CALL ZEROR2(PEQLD,3,5)

C

----- LOOP FOR NUMERICAL INTEGRATION DO 60 ISP=1,NSP XI=POSSP(ISP)

C

----- EVALUATE SHAPE FUNCTION FOR SAMPLING POINT CALL SFR1(LOUT1,XI,SHF,DERIV,NTSD,LNE,LT)

C ----- CALCULATE COMPONENTS OF THE EQUIVALENT NODAL LOADS PEQLD DO 40 IDOF=1,NDIM PSP(IDOF)=ZERO DSP(IDOF)=ZERO C DO 40 IEDG=1,NTSD PSP(IDOF)=PSP(IDOF)+PRES(IDOF,IEDG)*SHF(IEDG) 40 DSP(IDOF)=DSP(IDOF)+ELCD(IDOF,IEDG)*DERIV(IEDG) C DV=WEIGP(ISP) IF(NPLAX.EQ.0)GOTO 48 RAD=0.0D0 C

245

246

Appendix: Fortran Codes of CS-S DO 45 IEDG=1,NTSD 45 RAD=RAD+ELCD(1,IEDG)*SHF(IEDG) DV=DV*TPI*RAD 48 PCOM(1)=DSP(1)*PSP(2)-DSP(2)*PSP(1) PCOM(2)=DSP(1)*PSP(1)+DSP(2)*PSP(2)

C DO 50 IEDG=1,NTSD DO 50 ID=1,NDIM 50 PEQLD(ID,IEDG)=PEQLD(ID,IEDG)+PCOM(ID)*SHF(IEDG)*DV C 60 CONTINUE IF(IPRINT.EQ.1)WRITE(LOUT1,905)LNE,ND1,ND2, 1 ((PEQLD(ID,IP),ID=1,2),IP=1,NTSD) 905 FORMAT(1X,3I4,10E12.4/) C

----- SLOT LOADS INTO ARRAY RHS DO 80 IJ=1,NTSD JL=ILOC(IJ) IF(JL.GT.15)WRITE(LOUT1,*)’WARNING..JL > 15 (S. DISTLD)’ NDE=NCONN(JL,NE) N1=NW(NDE)-1

C DO 80 ID=1,NDIM FB(ID,JL)=FB(ID,JL)+PEQLD(ID,IJ) 80 RHS(N1+ID)=RHS(N1+ID)+PEQLD(ID,IJ)*FC RETURN END

SUBROUTINE EDGLD(LOUT1,NDIM,NCONN,LTYP, 1 LNE,ND1,ND2,NP1,NP2,PDISLD,PRES,KLOD,NPT,KINS,MXLD) C********************************************************************** C ROUTINE TO ALIGN NODES ALONG LOADED EDGE IN THE ANTI-CLOCKWISE * C ORDER AND TO STORE THE INFORMATION * C THE PRESSURES AT THE BEGINNING OF AN INCREMENT BLOCK ARE STORED * C IN A TEMPORARY ARRAY COMMON BLOCK PRLDI * C THE RATIOS OF THESE LOADING ARE ADDED TO THE CUMULATIVE LIST * C (COMMON BLOCK PRSLD) * C OF PRESSURE LOADS AT THE BEGINNING OF EACH INCREMENT * C********************************************************************** C Called by INSITU IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /ELINF / LINFO(50,15) COMMON /PRLDI / PRSLDI(10,100),LEDI(100),NDI1(100),NDI2(100),ILOD DIMENSION NCONN(NTPE,MEL),LTYP(MEL),NP1(NPL),NP2(NPL) DIMENSION PDISLD(3,LV),PRES(3,LV) C


247

CALL ZEROR2(PRES,3,LV) NE=LNE LI1=ND1 LI2=ND2 LT=LTYP(NE) IF(LT.GT.0)GOTO 15 WRITE(LOUT1,901)NE 901 FORMAT(1X,7HELEMENT,I6,2X,27HNOT PRESENT IN CURRENT MESH, 1 1X,16H(ROUTINE EDGLD)) RETURN 15 NEDG=LINFO(3,LT) NDSD=LINFO(7,LT) NTSD=NDSD+2 INDED=LINFO(14,LT) C DO 20 K1=1,NEDG J1=NP1(K1+INDED) J2=NP2(K1+INDED) I1=NCONN(J1,NE) I2=NCONN(J2,NE) IF(LI1.EQ.I1.AND.LI2.EQ.I2)GO TO 25 IF(LI1.EQ.I2.AND.LI2.EQ.I1)GO TO 21 20 CONTINUE WRITE(LOUT1,903)KLOD,LNE,ND1,ND2 903 FORMAT(/13H **** ERROR :,I5,17H TH LOAD. ELEMENT,I5, 1 2X,25H DOES NOT CONTAIN NODES :,2I5, 2 2X,15H(ROUTINE EDGLD)) STOP C

----- ALIGN NODES IN SEQUENCE 21 LIT=LI1 LI1=LI2 LI2=LIT NT=ND1 ND1=ND2 ND2=NT

C

----- PRES - CONTAINS THE PRESSURE COMPONENTS ALIGNED IN SEQUENCE DO 24 J=1,NTSD JBACK=NTSD+1-J DO 24 I=1,2 24 PRES(I,J)=PDISLD(I,JBACK) GO TO 35 25 DO 30 J=1,NTSD DO 30 I=1,2 30 PRES(I,J)=PDISLD(I,J)

C

----- UPDATE OR READ IN A NEW LIST

248

C C C C

Appendix: Fortran Codes of CS-S 35 IF(KINS.EQ.0)GO TO 40 ----- PRESSURE LOADS IN EQUILIBRIUM WITH IN-SITU STRESSES NEW LIST - READ DIRECTLY INTO COMMON PRSLD ----- Call LODLST ! Store cumulative list of applied pressure loading along element edges. CALL LODLST(LOUT1,NDIM,LNE,ND1,ND2,PRES,NPT,1,MXLD) GO TO 55

C ----- PRESSURE LOADS FOR NEW INCREMENT BLOCK READ INTO COMMON PRSLDI 40 ILOD=KLOD LEDI(ILOD)=LNE NDI1(ILOD)=ND1 NDI2(ILOD)=ND2 IC=0 DO 50 IV=1,NTSD DO 50 IJ=1,2 IC=IC+1 50 PRSLDI(IC,ILOD)=PRES(IJ,IV) 55 CONTINUE RETURN END

SUBROUTINE EQLBM(LOUT2,NDIM,IEQL,NN,NDF,NW,NQ,IDFX,P,PT,PCOR, 1 PEQT,IEQOP,ICOR,IRAC,IOUTP) C********************************************************************** C CARRIES OUT AN EQUILIBRIUM CHECK C CALCULATE AND PRINTOUT UNBALANCED NODAL LOADS C********************************************************************** C 1 CALLED BY EQLOD IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO DIMENSION NW(MNODES+1),NQ(MNODES),IDFX(MDOF) DIMENSION P(MDOF),PT(MDOF),PCOR(MDOF),PEQT(MDOF) DIMENSION PAR(6),RMAX(6),TER(3) C

C

----- MP - ARRAY SIZE OF PAR, RMAX MP=6 NDIM1=NDIM+1 NDIM2=2*NDIM IF(IRAC.EQ.1)CALL REACT(LOUT2,NDIM,NN,NW,NQ,IDFX,PEQT,PT,IOUTP) --- Above subroutine calculate the reaction at the restrained


249

points. C ----- INCLUDE ALL PORE-PRESSURE TERMS IN THE LIST OF FIXED D.O.F. C ALL EXCESS PORE PRESSURE D.O.F. ARE CONSIDERED TO BE FIXED DO 2 NI=1,NN NQL=NQ(NI) IF(NQL.NE.1.AND.NQL.NE.NDIM1)GO TO 2 ILC=NW(NI)+NQL-1 IDFX(ILC)=1 2 CONTINUE C

C

c

C C

----- CALCULATE OUT-OF-BALANCE LOADS FOR ALL FREE D.O.F. DO 5 IK=1,NDF IF(IDFX(IK).EQ.1) GO TO 3 PCOR(IK)=PT(IK)-PEQT(IK) GO TO 5 3 PCOR(IK)=ZERO 5 CONTINUE IF(IEQL.NE.1.OR.IOUTP.NE.1) RETURN ----- OUTPUT EQUILIBRIUM, OUT-OF-BALANCE AND APPLIED NODAL LOADS IF(IEQOP.EQ.0)GOTO 25 IF(NDIM.EQ.3) GOTO 22 WRITE(LOUT2,900) WRITE(LOUT2,904) GOTO 23 22 WRITE(LOUT2,930) WRITE(LOUT2,934) 23 CONTINUE DO 20 JR=1,NN J=JR NQL=NQ(J) IF(NQL.LE.1)GOTO 20 IF(IEQOP.EQ.1.AND.JR.GT.NDZ)GOTO 20 N1=NW(J) N2=N1+NDIM-1 IF(NDIM.EQ.2)WRITE(LOUT2,901)JR,(P(JJ),JJ=N1,N2), 1 (PT(JJ),JJ=N1,N2),(PEQT(JJ),JJ=N1,N2),(PCOR(JJ),JJ=N1,N2) IF(NDIM.EQ.3)WRITE(LOUT2,931)JR,(P(JJ),JJ=N1,N2), 1 (PT(JJ),JJ=N1,N2),(PEQT(JJ),JJ=N1,N2),(PCOR(JJ),JJ=N1,N2) 20 CONTINUE 25 CALL ZEROR1(RMAX,6) ----- CALCULATE MAXIMUM OF APPLIED AND OUT-OF-BALANCE ----- LOADS IN ALL DIRECTIONS DO 50 IK=1,NN NQL=NQ(IK) IF(NQL.LE.1)GOTO 50

250

Appendix: Fortran Codes of CS-S N1=NW(IK) N2=N1+NDIM-1 IC=0

C DO 35 KN=N1,N2 IC=IC+1 PAR(IC)=PT(KN) 35 PAR(IC+NDIM)=PCOR(KN) C DO 40 IC=1,NDIM2 RV=PAR(IC) IF(ABS(RV).LT.ASMVL)GOTO 40 IF(ABS(RV).GT.RMAX(IC))RMAX(IC)=ABS(RV) 40 CONTINUE 50 CONTINUE C ----- OUTPUT MAXIMUM OF (1) APPLIED LOADS (2) OUT-OF-BALANCE LOADS C ----- IN ALL DIRECTIONS WRITE(LOUT2,902) C IWARN=0 PMAXT=RMAX(1) DO 55 ID=2,NDIM 55 IF(RMAX(ID).GT.PMAXT)PMAXT=RMAX(ID) IF(PMAXT.LT.ASMVL) GOTO 132 DO 130 ID=1,NDIM 130 TER(ID)=100.0D0*RMAX(ID+NDIM)/PMAXT GOTO 125 132 IWARN=1 DO 135 ID=1,NDIM 135 TER(ID)=ZERO C 125 IF(NDIM.EQ.3) GOTO 122 WRITE(LOUT2,903) WRITE(LOUT2,905) WRITE(LOUT2,907)(RMAX(JQ),JQ=1,NDIM2),(TER(ID),ID=1,NDIM) GOTO 123 122 WRITE(LOUT2,933) WRITE(LOUT2,935) WRITE(LOUT2,937)(RMAX(JQ),JQ=1,NDIM2),(TER(ID),ID=1,NDIM) 123 CONTINUE IF(IWARN.EQ.1)WRITE(LOUT2,910) C INCR

----- ZERO PCOR IF NO CORRECTING LOADS ARE TO BE APPLIED IN NEXT IF(ICOR.NE.0)RETURN

C


251

DO 140 IK=1,NDF 140 PCOR(IK)=ZERO RETURN 900 FORMAT(//59X,19HLOADS EQUIVALENT TO/9X, 1 24HINCREMENTAL APPLIED LOAD,3X,18HTOTAL APPLIED LOAD, 1 6X,16HELEMENT STRESSES,7X,19HOUT-OF-BALANCE LOAD/ 2 9X,24(1H-),3X,18(1H-),6X,16(1H-),7X,19(1H-)) 901 FORMAT(1X,I5,2X,8E12.4) 902 FORMAT(//1X,17HEQUILIBRIUM CHECK/1X,17(1H-)) 903 FORMAT(/3X,20HMAXIMUM APPLIED LOAD,2X, 1 24HMAXM OUT-OF-BALANCE LOAD,3X, 2 31HPERCENTAGE ERROR IN EQUILIBRIUM/ 3 3X,20(1H-),2X,24(1H-),3X,31(1H-)/) 904 FORMAT(/1X,5H NODE,8X,1HX,12X,1HY,11X,1HX,11X,1HY,11X,1HX, 1 11X,1HY,11X,1HX,11X,1HY//) 905 FORMAT(8X,1HX,11X,1HY,11X,1HX,10X,1HY,15X,1HX,15X,1HY/) 907 FORMAT(1X,4E12.4,4X,F10.4,5X,F10.4) 910 FORMAT(/40H WARNING **** NO APPLIED LOADING - CHECK, 1 1X,49HWHETHER ALL BOUNDARY CONDITIONS ARE DISPLACEMENTS, 2 2X,15H(ROUTINE EQLBM)) 930 FORMAT(/15X,24HINCREMENTAL APPLIED LOAD,5X, 1 38HLOADS EQUIVALENT TO TOTAL APPLIED LOAD, 1 8X,16HELEMENT STRESSES,20X,19HOUT-OF-BALANCE LOAD/ 2 15X,24(1H-),5X,38(1H-),8X,16(1H-),20X,19(1H-)) 931 FORMAT(1X,I5,2X,12E12.4) 933 FORMAT(/10X,20HMAXIMUM APPLIED LOAD,13X, 1 24HMAXM OUT-OF-BALANCE LOAD,11X, 2 31HPERCENTAGE ERROR IN EQUILIBRIUM/ 3 10X,20(1H-),13X,24(1H-),11X,31(1H-)/) 934 FORMAT(/2X,5H NODE,8X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ, 1 11X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ//) 935 FORMAT(8X,1HX,11X,1HY,11X,1HZ,11X,1HX,11X,1HY,11X,1HZ, 1 13X,1HX,11X,1HY,11X,1HZ/) 937 FORMAT(1X,6E12.4,3F12.4) END

SUBROUTINE EQLIB(JJ,LT,NGP,INDX,NDN,NAC,XYZ,STRESS,F,NCONN,ISTGE) C********************************************************************** C ROUTINE TO CALCULATE FORCES EQUILIBRATING C ELEMENTAL STRESSES C********************************************************************** C 1 CALLED BY RDSTRS IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /PARS / PYI,ASMVL,ZERO

252

Appendix: Fortran Codes of CS-S COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /DATW / W(100) COMMON /DATL / SL(4,100) COMMON /JACB / XJACI(3,3),DJACB

1

DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),ELCOD(3,NDMX), 1 DS(3,20),SHFN(20),CARTD(3,NDMX),B(6,NB), 2 F(3,NDMX),SLL(4),NCONN(NTPE,MEL) CR=1.0D0 IF(NPLAX.EQ.1)CR=2.0D0*PYI CALL ZEROR2(F,3,NDMX)

!Initializing

DO 20 KN=1,NDN NDE=NCONN(KN,JJ) DO 20 ID=1,NDIM 20 ELCOD(ID,KN)=XYZ(ID,NDE) DO 60 IP=1,NGP IPA=IP+INDX C DO 30 IL=1,NAC 30 SLL(IL)=SL(IL,IPA) C

----- Call FORMB2 - Forms B matrix CALL FORMB2(JJ,R,RI,NDN,NAC,ELCOD,DS,SHFN,CARTD,B,SLL,LT,IP,ISTGE) F9=CR*DJACB*W(IPA) IF(NPLAX.EQ.1)F9=F9*R

C U=STRESS(NS+1,IP,JJ) SIGXT=STRESS(1,IP,JJ)+U SIGYT=STRESS(2,IP,JJ)+U SIGZT=STRESS(3,IP,JJ)+U TXY=STRESS(4,IP,JJ) IF(NDIM.EQ.2)GOTO 35 C TYZ=STRESS(5,IP,JJ) TZX=STRESS(6,IP,JJ) C DO 50 IN=1,NDN F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+CARTD(2,IN)*TXY 1 +CARTD(3,IN)*TZX)*F9 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY 1 +CARTD(3,IN)*TYZ)*F9 F(3,IN)=F(3,IN)+(CARTD(3,IN)*SIGZT+CARTD(2,IN)*TYZ

Appendix: Fortran Codes of CS-S 1 50 CONTINUE GOTO 60

253

+CARTD(1,IN)*TZX)*F9

C 35 DO 40 IN=1,NDN F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+SHFN(IN)*SIGZT*RI 1 +CARTD(2,IN)*TXY)*F9 40 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY)*F9 60 CONTINUE RETURN END

SUBROUTINE EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ, P,PT, 1 PCOR,PEQT,XYFT,PCONI,PR,IEQOP,ICOR,TGRAV,IRAC,KSTGE,IEQL,IOUTP) C---------------------------------------------------------------------C ROUTINE TO CALCULATE EQUIVALENT NODAL LOADS FOR C APPLIED LOADING TO CARRY OUT AN EQUILIBRIUM CHECK C********************************************************************** C 1 Called by INSITU IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL), 1 NW(MNODES+1),NQ(MNODES),IDFX(MDOF),NP1(NPL),NP2(NPL) DIMENSION XYZ(3,MNODES),P(MDOF),PT(MDOF),PCOR(MDOF),PEQT(MDOF), 1 XYFT(MDOF),PCONI(MDOF),F(3,NDMX),PR(NPR,NMT),PRES(3,LV) COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON COMMON COMMON COMMON

/PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED /PTLOAD/PT1(MDOF),INDPT /ELINF / LINFO(50,15) /PARS / PYI,ASMVL,ZERO

C CALL ZEROR1(PT,MDOF) C

----- (1) PRESSURE LOADING ALONG ELEMENT EDGE IF(NLED.EQ.0.AND.TGRAV.LT.ASMVL)GO TO 62 IF(NLED.EQ.0.OR.INDPT.NE.0)GO TO 32

C DO 30 KE=1,NLED LNE=LEDG(KE) NE=LNE LT=LTYP(NE)

254


IF(LT.GT.0)GOTO 10 IF(KSTGE.EQ.4)GOTO 30 WRITE(LOUT1,900)LNE 900 FORMAT(/1X,45H *** ERROR : IN SITU PRESSURE LOAD APPLIED TO,1X, 1 7HELEMENT,I5,2X,28HWHICH IS NOT PRESENT IN MESH,1X, 2 15H(ROUTINE EQLOD)/) GOTO 30 10 ND1=NDE1(KE) ND2=NDE2(KE) C *** N2D = 2 FOR TWO DIMENSIONAL PROBLEMS N2D=2 ICT=0 DO 20 IV=1,NPT DO 20 ID=1,N2D ICT=ICT+1 20 PRES(ID,IV)=PRESLD(ICT,KE) C C ----- Call DISTLD CALL DISTLD(LOUT1,XYZ,PT1,NCONN,LTYP,NW,NP1, 1 NP2,PRES,LNE,ND1,ND2,0,1,1.0D0) 30 CONTINUE INDPT=1 32 DO IDF=1,NDF PT(IDF)=PT1(IDF) ENDDO C

----- (2) SELF WEIGHT LOADING IF(TGRAV.LT.ASMVL) GO TO 62 DO 60 KL=1,NEL LT=LTYP(KL) IF(LT.LT.0)GO TO 60 JK=KL NDN=LINFO(5,LT) INDX=LINFO(12,LT) NAC=LINFO(15,LT) KM=MAT(KL)

C C

----- FIND IF ELEMENT HAS BEEN ADDED IN THIS INCREMENT BLOCK ----- THEN USE LOAD RATIO FRACT ON GRAVITY LOADING

C C C C C C

DO 40 IM=1,NEL MUS=JEL(IM) IF(MUS.EQ.0)GO TO 42 MPR=MREL(MUS) IF(KL.EQ.MPR)GO TO 44 40 CONTINUE FA=1.0D0


C

255

GO TO 45 44 FA=FRACT 45 DENS=PR(8,KM)*TGRAV*FA

C C

----- Call Self CALL SELF(LOUT1,KL,NDN,NAC,XYZ,F,NCONN,MAT,LT,INDX,DENS,JK,KSTGE) C DO 55 KK=1,NDN NCOR=NCONN(KK,KL) KKK=NW(NCOR)-1 C DO 55 ID=1,NDIM 55 PT(KKK+ID)=PT(KKK+ID)+F(ID,KK) 60 CONTINUE 62 CONTINUE C

C C C C

C

----- ADD CONTRIBUTIONS FROM POINT LOADS DO 70 J=1,NDF 70 PT(J)=PT(J)+XYFT(J)+PCONI(J) ----- CONVERT PT TO COUNT FOR SKEW BOUNDARIES IF(NSKEW.GT.0) CALL ROTBC(PT,NW,NDIM,NSKEW,1) ! Nskew=8 for me --- This subroutine turn back the cal. displ. to original coord. system --- for skew boundary nodes. ----- FIND DOF WHICH ARE RESTRAINED CALL RESTRN(NDIM,NW,IDFX) ----- EQUILIBRIUM CHECK CALL EQLBM(LOUT2,NDIM,IEQL,NN,NDF,NW,NQ,IDFX, 1 P,PT,PCOR,PEQT,IEQOP,ICOR,IRAC,IOUTP) RETURN END

C********************************************************************** SUBROUTINE FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFX,NFXNOD) C---------------------------------------------------------------------C ROUTINE TO MAINTAIN A LIST OF NODAL FIXITIES. INTERPRETS C FIXITIES ALONG ELEMENT EDGES INTO NODAL FIXITIES C---------------------------------------------------------------------C 1 Called by INSITU C 2 Called by TOTSOL

256 c

Appendix: Fortran Codes of CS-S NFX=NFXEL IN THE INPUT HEADER 11 IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),LTYP(MEL) DIMENSION NP1(NPL),NP2(NPL),IND(5),FV(5),V(LV) ! LV=3 from PARM.FOR COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /ELINF / LINFO(50,15)

C NFZ=200 NDIM1=NDIM+1 IF(NFX.EQ.0)RETURN CALL ZEROI1(IND,5) CALL ZEROR1(FV,5) WRITE(LOUT1,900) C---------------------------------------------------------------------C LOOP ON ALL FIXED EDGES I.E. EDGES WITH PRESCRIBED C DISPLACEMENT/EXCESS PORE PRESSURES C---------------------------------------------------------------------C Read in fixity data C WRITE(LOUT1,101)NFX C 101 FORMAT(//I5) IF(NFX.EQ.0) GO TO 201 DO 200 JX=1,NFX !NFX=NFXEL READ(LINP,*)ML,ND1,ND2,IVAR,IFX,V ! V is an array of size (3). WRITE(LOUT1,902)JX,ML,ND1,ND2,IVAR,IFX,V NE=ML LI1=ND1 LI2=ND2 LT=LTYP(NE) LT=IABS(LT) NVN=LINFO(2,LT) NEDG=LINFO(3,LT) NDSD=LINFO(7,LT) IF(IVAR.EQ.NDIM1)NDSD=LINFO(8,LT) NTSD=NDSD+2 INDED=LINFO(14,LT) C DO 20 K1=1,NEDG J1=NP1(K1+INDED) J2=NP2(K1+INDED) I1=NCONN(J1,NE) I2=NCONN(J2,NE) IF(LI1.EQ.I1.AND.LI2.EQ.I2)GO TO 25 IF(LI1.EQ.I2.AND.LI2.EQ.I1)GO TO 21


257

20 CONTINUE WRITE(LOUT1,903)JX,ML,ND1,ND2 GOTO 200 C---------------------------------------------------------------------C ALIGN END NODES OF EDGE IN CORRECT SEQUENCE. (ANTICLOCKWSIE C OREDER ABOUT ELEMENT CENTRE) C---------------------------------------------------------------------21 LIT=LI1 LI1=LI2 LI2=LIT NT=ND1 ND1=ND2 ND2=NT C DO 24 J=1,NTSD JBACK=NTSD+1-J 24 FV(J)=V(JBACK) GO TO 35 C 25 DO 30 J=1,NTSD 30 FV(J)=V(J) C---------------------------------------------------------------------C IND - LIST OF NODES ALONG EDGE. START WITH END NODES C---------------------------------------------------------------------35 IND(1)=LI1 IND(NTSD)=LI2 IF(NTSD.EQ.2)GO TO 42 LC1=NVN+(K1-1)*NDSD IF(IVAR.EQ.NDIM1)LC1=LINFO(5,LT)+(K1-1)*NDSD C---------------------------------------------------------------------C INTERMEDIATE NODES (IF NTSD=2 NO INTERMEDIATE NODES) C---------------------------------------------------------------------DO 40 JP=1,NDSD ILC=LC1+JP 40 IND(JP+1)=NCONN(ILC,NE) C---------------------------------------------------------------------C LOOP ON ALL NODES ALONG EDGE C---------------------------------------------------------------------42 DO 100 KND=1,NTSD I=IND(KND) IF(NF.EQ.0)GO TO 58 C DO 50 J=1,NF IF(I.EQ.MF(J))GO TO 55 50 CONTINUE C GO TO 58

258


C---------------------------------------------------------------------C UPDATE EXISTING VALUES C---------------------------------------------------------------------55 JF=J GO TO 60 C 58 NF=NF+1 IF(NF.LE.NFZ)GO TO 59 WRITE(LOUT1,904) STOP 59 JF=NF 60 MF(JF)=I NFIX(IVAR,JF)=IFX DXYT(IVAR,JF)=FV(KND) 100 CONTINUE 200 CONTINUE 201 IF(NFXNOD.EQ.0) RETURN WRITE(LOUT1,905)

C

! NFXNOD=0 for me.

----- This subroutine is not used now. 1/26/98 DO 202 INODE=1,NFXNOD READ(LINP,*)NODE,IVAR,IFX,VALUE WRITE(LOUT1,906)NODE,IVAR,IFX,VALUE DO J=1,NF KNODE=MF(J) IF(KNODE.EQ.NODE) THEN NFIX(IVAR,J)=IFX DXYT(IVAR,J)=VALUE GO TO 202 ENDIF ENDDO NF=NF+1 MF(NF)=NODE NFIX(IVAR,NF)=IFX DXYT(IVAR,NF)=VALUE 202 CONTINUE RETURN 900 FORMAT(/1X,4HSIDE,4X,7HELEMENT,3X,5HNODE1,3X,5HNODE2, 1 3X,3HDOF,4X,11HFIXITY CODE,8X,4HVAL1,6X,4HVAL2,6X,4HVAL3, 2 6X,4HVAL4,6X,4HVAL5/) 902 FORMAT(1X,I3,4X,I5,5X,I4,4X,I4,5X,I2,12X,I3,3X,5F9.4) 903 FORMAT(/13H **** ERROR :,I5,19H TH FIXITY. ELEMENT, 1 I5,25H DOES NOT CONTAIN NODES :,2I5,2X,14H(ROUTINE FIXX)) 904 FORMAT(/42H INCREASE SIZE OF ARRAYS MF, NFIX AND DXYT/ 1 1X,34HIN COMMON BLOCK FIX (ROUTINE FIXX))


259

905 FORMAT(/3X,4HNODE,3X,3HDOF,6X,11HFIXITY CODE,6X,3HVAL/) 906 FORMAT(1X,I4,5X,I2,10X,I3,5X,F12.6) END

C********************************************************************** SUBROUTINE FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFX,NFXNOD) C********************************************************************** C ROUTINE TO MAINTAIN A LIST OF NODAL FIXITIES. INTERPRETS C FIXITIES ALONG (3-D) ELEMENT FACE INTO NODAL FIXITIES. C AT PRESENT TO CATER FOR THE 3-D BRICK ELEMENTS ONLY. C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),LTYP(MEL) DIMENSION IND(8),FV(8) DIMENSION KX(48),NDU(8),NDP(8),NXC(4),NXM(4),KNL(8) COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /ELINF / LINFO(50,15) C---------------------------------------------------------------------C ARRAY KX(48) GIVES THE INDEX TO ARRAY NCONN FOR THE FOUR C CORNER NODES OF EACH FACE OF THE ELEMENT FOLLOWED BY THE C MIDSIDE NODES. C---------------------------------------------------------------------DATA KX(1),KX(2),KX(3),KX(4),KX(5),KX(6),KX(7),KX(8),KX(9), 1 KX(10),KX(11),KX(12),KX(13),KX(14),KX(15),KX(16),KX(17), 1 KX(18),KX(19),KX(20),KX(21),KX(22),KX(23),KX(24),KX(25), 1 KX(26),KX(27),KX(28),KX(29),KX(30),KX(31),KX(32),KX(33), 1 KX(34),KX(35),KX(36),KX(37),KX(38),KX(39),KX(40),KX(41), 1 KX(42),KX(43),KX(44),KX(45),KX(46),KX(47),KX(48)/ 1 1,2,3,4,9,10,11,12,6,5,8,7,13,16,15,14,1,5,6,2,17,13,18,9, 1 2,6,7,3,18,14,19,10,4,3,7,8,11,19,15,20,5,1,4,8,17,12,20,16/ C DO 5 IU=1,8 KNL(IU)=0 NDU(IU)=0 NDP(IU)=0 5 CONTINUE C NFZ=200 NDIM1=NDIM+1 IF(NFX.EQ.0) GO TO 201 WRITE(LOUT1,900) C---------------------------------------------------------------------C IF NEW 3-D ELEMENT TYPES ARE ADDED THEN NC, NFCD C AND LVL (=NFCD) SHOULD BE OBTAINED FROM ARRAY LINFO

260


C IN ORDER TO MAKE THE ROUTINE GENERAL. C NC - NUMBER OF VERTEX NODES ON ELEMENT FACE C NFCD- TOTAL NUMBER OF DISPLACEMENT NODES ON FACE C---------------------------------------------------------------------NC=4 NFCD=8 C---------------------------------------------------------------------C LOOP ON ALL FACES WITH FIXITIES I.E FACES WITH PRESCRIBED C DISPLACEMENT/EXCESS PORE PRESSURES. C---------------------------------------------------------------------LVL=NFCD DO 200 JX=1,NFX READ(LINP,*)ML,(NDU(J),J=1,NC),IVAR,IFX,(FV(K),K=1,LVL) WRITE(LOUT1,910)JX,ML,(NDU(J),J=1,NC),IVAR,IFX,(FV(K),K=1,LVL) NE=ML C DO 30 IN=1,NC ND=NDU(IN) 30 NDP(IN)=ND C LT=LTYP(NE) LT=IABS(LT) NFAC=LINFO(4,LT) C---------------------------------------------------------------------C LOOP ON ALL FACES OF ELEMENT TO IDENTIFY THE FACES OF THE C ELEMENT WITH PRESCRIBED VALUES C---------------------------------------------------------------------DO 90 IFAC=1,NFAC ISX=NFCD*(IFAC-1) C GET INDEXES OF NODES TO NCONN DO 40 IN=1,NC NXC(IN)=KX(ISX+IN) C IF NOT PORE-PRESSURE D.O.F, ADDITIONAL NODES ALONG C EDGE ARE PRESENT IF(IVAR.NE.NDIM1)NXM(IN)=KX(ISX+NC+IN) 40 CONTINUE C GET VERTEX NODES OF FACE FROM NCONN DO 50 IN=1,NC IP=NXC(IN) 50 KNL(IN)=NCONN(IP,NE) C---------------------------------------------------------------------C LOOP ON ALL STARTING NODES C TRY TO MATCH THE NODES SPECIFIED BY THE USER C WITH THE NODES OF EACH FACE. EACH NODE IN TURN C IS CONSIDERED AS A STARTING NODE. C---------------------------------------------------------------------DO 80 IS=1,NC ISV=IS


261

TRY MATCHING THE NODES DO 60 IN=1,NC IF(NDP(IN).NE.KNL(IN))GOTO 65 60 CONTINUE GOTO 95 C START WITH NEXT NODE. THE SEQUENCES OF THE NODES ARE STILL THE SAME 65 CALL ALTER(LOUT1,KNL,NC) 80 CONTINUE 90 CONTINUE C FACE NOT FOUND WRITE(LOUT1,930)JX,ML,(NDU(J),J=1,NC) C GOTO 200 C 95 IF(ISV.EQ.1)GOTO 105 IS1=ISV-1 C SORT THE INDEXES TO MATCH WITH NODE SEQUENCE KNL DO 100 IM=1,IS1 CALL ALTER(LOUT1,NXC,NC) IF(IVAR.NE.NDIM1)CALL ALTER(LOUT1,NXM,NC) 100 CONTINUE C IF PORE-PRESSURE FIXITY 105 CONTINUE IF(IVAR.NE.NDIM1)GOTO 125 C DO 120 IL=1,NC IP=NXC(IL) 120 IND(IL)=NCONN(IP,NE) NSDN=NC GOTO 132 C IF DISPLACEMENT FIXITY 125 DO 130 IL=1,NC IM=NXC(IL) IN=NXM(IL) IND(2*IL-1)=NCONN(IM,NE) 130 IND(2*IL)=NCONN(IN,NE) NSDN=NFCD 132 CONTINUE C---------------------------------------------------------------------C LOOP ON ALL NODES ALONG FACE C---------------------------------------------------------------------142 DO 180 KND=1,NSDN I=IND(KND) IF(NF.EQ.0)GOTO 158 C DO 150 J=1,NF IF(I.EQ.MF(J))GOTO 155

262


150 CONTINUE C GOTO 158 C---------------------------------------------------------------------C UPDATE EXISTING VALUES C---------------------------------------------------------------------155 JF=J GOTO 160 C 158 NF=NF+1 IF(NF.LE.NFZ)GOTO 159 WRITE(LOUT1,940) STOP 159 JF=NF 160 MF(JF)=I NFIX(IVAR,JF)=IFX DXYT(IVAR,JF)=FV(KND) 180 CONTINUE 200 CONTINUE C 201 IF(NFXNOD.EQ.0) RETURN WRITE(LOUT1,905) DO 202 INODE=1,NFXNOD READ(LINP,*)NODE,IVAR,IFX,VALUE WRITE(LOUT1,906)NODE,IVAR,IFX,VALUE DO J=1,NF KNODE=MF(J) IF(KNODE.EQ.NODE) THEN NFIX(IVAR,J)=IFX DXYT(IVAR,J)=VALUE GO TO 202 ENDIF ENDDO NF=NF+1 MF(NF)=NODE NFIX(IVAR,NF)=IFX DXYT(IVAR,NF)=VALUE 202 CONTINUE RETURN C 900 FORMAT(/19X,16H......NODES.....,8X,6HFIXITY// 1 1X,4HFACE,4X,7HELEMENT,3X,16H1 2 3 4, 2 3X,3HDOF,3X,4HCODE,7X,4HVAL1,5X,4HVAL2,5X,4HVAL3, 3 5X,4HVAL4,5X,4HVAL5,5X,4HVAL6,5X,4HVAL7,5X,4HVAL8//) 910 FORMAT(1X,I3,4X,I5,3X,I4,1X,I4,1X,I4,1X,I4,4X,I2,3X,I3,3X,8F8.4) 930 FORMAT(/1X,20H***** ERROR : FIXITY,I4,2X,8HIN LIST.,3X, 1 7HELEMENT,I5,2X,29HDOES NOT HAVE FACE WITH NODES,4I5)


263

940 FORMAT(/42H INCREASE SIZE OF ARRAYS MF, NFIX AND DXYT/ 1 1X,35HIN COMMON BLOCK FIX (ROUTINE FIXX3)) 905 FORMAT(/1X,4HNODE,3X,3HDOF,3X,11HFIXITY CODE,3X,3HVAL) 906 FORMAT(1X,I4,5X,I2,10X,I3,5X,F10.3) END

C********************************************************************** SUBROUTINE ALTER(LOUT1,IM,N) C********************************************************************** C ROUTINE TO SHIFT ARRAY FORWARD BY ONE PLACE C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) DIMENSION IM(N) C IF(N.LE.1)GOTO 100 NM1=N-1 IMT=IM(1) C DO 10 K=1,NM1 10 IM(K)=IM(K+1) IM(N)=IMT RETURN 100 WRITE(LOUT1,900)N 900 FORMAT(/1X,45HERROR * ARRAY CONTAINS LESS THAN OR EQUAL TO,I5,2X, 1 40HMEMBERS (ROUTINE ALTER) CALLED BY FIXX3) RETURN END

SUBROUTINE FORMB2(J,R,RI,NDN,NAC,ELCOD,DS,SHFN, 1 CARTD,B,SLL,LT,IP,ISTGE) C********************************************************************** C FORMS B MATRIX FROM AREA/LOCAL COORDS SLL(NAC) C IN ELEMENT J FOR INTEGRATION POINT IP C********************************************************************** C 1 CALLED BY EQLIB IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20), 1 CARTD(3,NDMX),B(6,NB),SLL(4),XJACM(3,3) COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /JACB / XJACI(3,3),DJACB

264 C

Appendix: Fortran Codes of CS-S ----- INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS) CALL ZEROR2(DS,3,20) CALL ZEROR1(SHFN,20) CALL ZEROR2(B,6,NB)

C CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,J) CALL ZEROR2(XJACM,3,3) C NDN2=2*NDN C DO 15 IDIM=1,NDIM DO 15 JDIM=1,NDIM SUM=ZERO C DO 12 IN=1,NDN 12 SUM=SUM+DS(IDIM,IN)*ELCOD(JDIM,IN) 15 XJACM(IDIM,JDIM)=SUM C

----- Call DETMIN CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,J,IP,ISTGE)

C

----- CALCULATE RADIUS R=ZERO RI=ZERO IF(NPLAX.EQ.0)GOTO 28

FOR

C DO 25 IN=1,NDN 25 R=R+ELCOD(1,IN)*SHFN(IN) RI=-1.0D0/R C 28 DO 35 IN=1,NDN DO 35 ID=1,NDIM SUM=ZERO C DO 30 JD=1,NDIM 30 SUM=SUM-DS(JD,IN)*XJACI(ID,JD) 35 CARTD(ID,IN)=SUM C IF(NDIM.NE.2)GOTO 52 C

----- 2 - D ELEMENT DO 50 IN=1,NDN B(1,IN)=CARTD(1,IN) B(2,NDN+IN)=CARTD(2,IN) IF(NPLAX.EQ.0)GOTO 45 B(3,IN)=SHFN(IN)*RI 45 B(4,NDN+IN)=B(1,IN) 50 B(4,IN)=B(2,NDN+IN)

AXI-SYM

B

MATRIX


265

C 52 IF(NDIM.NE.3)GOTO 62 C

----- 3 - D ELEMENT DO 60 IN=1,NDN B(1,IN)=CARTD(1,IN) B(2,NDN+IN)=CARTD(2,IN) B(3,NDN2+IN)=CARTD(3,IN) B(4,IN)=CARTD(2,IN) B(4,NDN+IN)=CARTD(1,IN) B(5,NDN+IN)=CARTD(3,IN) B(5,NDN2+IN)=CARTD(2,IN) B(6,IN)=CARTD(3,IN) B(6,NDN2+IN)=CARTD(1,IN) 60 CONTINUE

C 62 CONTINUE RETURN END

SUBROUTINE INSITU(XYZ,STRESS,P,PT,PCOR,PEQT,XYFT,PCONI, 3 NCONN,MAT,LTYP,NW,IDFX,NP1,NP2,PR,NTY) C================================================================ ======= C SETUP INSITU STRESSES AND CHECK FOR EQUILIBRIUM C================================================================ ======= C 1 CALLED BY RDPROP IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED COMMON /PARS

/ PYI,ASMVL,ZERO

COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW

1

COMMON /OUT

/ IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF

COMMON /ANLYS/ TTIME,DTIMEI,TGRAVI,DGRAVI,FRACT,FRACLD, ICOR ,IDCHK,IOUT ,INCT ,IWL , NOIB ,JS ,JINCB,NLOD ,NLDS

+ +

266

Appendix: Fortran Codes of CS-S DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),P(MDOF), 1 PT(MDOF),PCOR(MDOF),PEQT(MDOF),XYFT(MDOF),PCONI(MDOF) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL), 1 NW(MNODES+1),NQ(MNODES),IDFX(MDOF),NP1(NPL),NP2(NPL) DIMENSION PR(NPR,NMT),PDISLD(3,LV),PRES(3,LV),NTY(NMT),TEMP(3)

C

----- CODE TO INDICATE STAGE OF THE ANALYSIS KSTGE=1

C

----- INITIALISE PRESSURE LOADS NDIM1=NDIM+1 CALL ZEROR1(PCONI,MDOF) CALL ZEROR1(TEMP,3) CALL ZEROR2(PRESLD,10,100) CALL ZEROI1(LEDG,100) CALL ZEROI1(NDE1,100) CALL ZEROI1(NDE2,100) CALL ZEROI1(MF,200) CALL ZEROI2(NFIX,4,200) CALL ZEROR2(DXYT,4,200)

C

----- SET UP IN-SITU STRESS SYSTEM READ(LINP,101)HEADER9 READ(LINP,*)KT,NI WRITE(LOUT1,926)KT,NI IF(NI.EQ.0)NI=1 ! If NI = 0 use a value of 1 to avoid array size of 0. IF(NI.LE.100)GOTO 40 WRITE(LOUT1,910)NI 101 FORMAT(A80) 910 FORMAT(/1X,’INCREASE SIZE OF ARRAYS 1 ’IN ROUTINE INSITU’) STOP 40 CONTINUE

C

NLI AND NHI TO’,I6,2X,

----- Read in the initial stress data CALL RDSTRS(KT,XYZ,STRESS,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,NI)

C INITIALISE FIXED LOADS, TOTAL POINT LOADS AND TOTAL DISPLACEMENTS C NF - NUMBER OF FIXITIES c ----- Initializing NF=0 CALL ZEROR1(PCOR,MDOF) CALL ZEROR1(XYFT,MDOF) CALL ZEROR1(P,MDOF)


----- READ LOADS IN EQUILIBRIUM WITH IN-SITU STRESSES NLED=0 TGRAVI=ZERO IF(KT.EQ.0)GO TO 62

C READ(LINP,101)HEADER11 READ(LINP,*)NLODI,NLDNOD,NFXEL,NFXNOD,TGRAVI WRITE(LOUT1,952)NLODI,NLDNOD,NFXEL,NFXNOD,TGRAVI C IF(NLODI.EQ.0)GO TO 52 WRITE(LOUT1,960) C READ(LINP,101)HEADER12 DO 50 KL=1,NLODI READ(LINP,*)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,2),IV=1,NPT) WRITE(LOUT1,964)LNE,ND1,ND2,((PDISLD(ID,IV),ID=1,2),IV=1,NPT) C DO 100 IV=1,NPT DO 100 ID=1,NDIM IDR=NDIM+1-ID 100 PRES(ID,IV)=PDISLD(IDR,IV) C DO 110 IV=1,NPT DO 110 ID=1,NDIM 110 PDISLD(ID,IV)=PRES(ID,IV) C C

C

----- Call EDGLD ! Allign nodes along the loaded edge CALL EDGLD(LOUT1,NDIM,NCONN,LTYP,LNE,ND1,ND2,NP1,NP2,PDISLD, 1 PRES,KL,NPT,1,MXLD) 50 CONTINUE 52 IF(NLDNOD.EQ.0) GO TO 53 ----- Read point load data WRITE(LOUT1,965) READ(LINP,101)HEADER13 DO KL=1,NLDNOD READ(LINP,*)NODE,(TEMP(IDIR),IDIR=1,NDIM) WRITE(LOUT1,966)NODE,(TEMP(IDIR),IDIR=1,NDIM) DO IDIR=1,NDIM N1=NW(NODE)-1 NID=N1+IDIR PCONI(NID)=TEMP(IDIR) ENDDO ENDDO

C 53 IF(NFXEL.EQ.0.AND.NFXNOD.EQ.0)GO TO 62 C

----- IN-SITU BOUNDARY CONDITIONS

267

268

C

Appendix: Fortran Codes of CS-S WRITE(LOUT1,930) READ(LINP,101)HEADER14 IF(NDIM.EQ.2) THEN READ(LINP,101)HEADER14 CALL FIXX2(LINP,LOUT1,NDIM,NCONN,LTYP,NP1,NP2,NFXEL, 1 NFXNOD) !Interpret fixities along element edges into nodal fixities. ENDIF IF(NDIM.EQ.3) THEN READ(LINP,101)HEADER14 CALL FIXX3(LINP,LOUT1,NDIM,NCONN,LTYP,NFXEL,NFXNOD) ENDIF

C

----- Call Make NZ CALL MAKENZ(NEL,NN,NCONN,LTYP,NQ,INXL) TTGRV=1.0D0

C

----- Call EQLOD ! Convert the applied load to equal nodal force. CALL EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ,P, 1PT,PCOR,PEQT,XYFT,PCONI,PR,2,0,TTGRV,IRAC,KSTGE,1,1)

C 62 RETURN 907 FORMAT(//1X,38HLIST OF REMOVED ELEMENTS TO FORM, 1 14H PRIMARY MESH/1X,52(1H-)/) 920 FORMAT(20I6/) 926 FORMAT(//10X,30HIN-SITU STRESS OPTION........=,I10 1 /10X,30HNUMBER OF IN-SITU NODES......=,I10/) 930 FORMAT(/1X,27HIN-SITU BOUNDARY CONDITIONS/1X,27(1H-)/) 952 FORMAT(/ 1 10X,46HNUMBER OF EDGES WITH PRESSURE LOAD...........=,I5/ 2 10X,46HNUMBER OF POINT LOAD NODES...................=,I5/ 3 10X,46HNUMBER OF EDGES RESTRAINED...................=,I5/ 4 10X,46HNUMBER OF INDIVIDUAL RESTRAINED NODES........=,I5/ 5 10X,46HIN-SITU GRAVITY ACCELERATION FIELD...........=,F8.1,2X, 6 1HG//) 960 FORMAT(/1X,38HSPECIFIED NODAL VALUES OF SHEAR/NORMAL, 1 19H STRESSES (IN-SITU)/1X,57(1H-)/1X,4HELEM, 2 1X,4HNDE1,2X,4HNDE2,2X,4HSHR1,8X,4HNOR1,8X,4HSHR2,8X,4HNOR2, 3 8X,4HSHR3,8X,4HNOR3,8X,4HSHR4,8X,4HNOR4,8X,4HSHR5,8X,4HNOR5/) 964 FORMAT(1X,3I4,10E12.4) 965 FORMAT(/5X,’SPECIFIED POINT LOAD VALUES’,/5X,27(1H-),/1X,’NODE’, 1 5X,’X’,6X,’Y’,6X,’Z’/) 966 FORMAT(1X,I4,3E12.4) END


269

SUBROUTINE INSTRS(LOUT1,NEL,NDIM,NS,XYZ,STRESS,NCONN,MAT,LTYP,NTY) C********************************************************************** C ROUTINE TO PRINT OUT INITIAL IN-SITU STRESSES C********************************************************************** C 1 Called by RDSTRS IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20),TEMP(6), 1 CIP(3),SLL(4),NTY(NMT) COMMON /ELINF / LINFO(50,15) COMMON /DATL / SL(4,100) COMMON /PARS / PYI,ASMVL,ZERO C CALL ZEROR1(TEMP,6) ! Initializing NS1=NS+1 WRITE(LOUT1,900) 900 FORMAT(/1X,34HINTEGRATION POINT IN-SITU STRESSES/ 1 1X,34(1H-)/) IF(NDIM.EQ.2)WRITE(LOUT1,901) IF(NDIM.EQ.3)WRITE(LOUT1,931) C DO 60 MR=1,NEL IF(MR.EQ.0)GO TO 60 J=MR LT=LTYP(J) IF(LTYP(J).LT.0)GO TO 60 NDN=LINFO(5,LT) NGP=LINFO(11,LT) INDX=LINFO(12,LT) NAC=LINFO(15,LT) KM=MAT(J) KGO=NTY(KM) GO TO(11,11,12,60,60),KGO WRITE(LOUT1,910)MR,KGO GOTO 60 11 ICAM=0 GO TO 14 12 ICAM=1 14 CONTINUE C DO 18 KN=1,NDN NDE=NCONN(KN,J) DO 18 ID=1,NDIM 18 ELCOD(ID,KN)=XYZ(ID,NDE)

270


C DO 40 IP=1,NGP IPA=IP+INDX C DO 25 IL=1,NAC 25 SLL(IL)=SL(IL,IPA) C

----- Call shape function CALL SHAPE (LOUT1,SLL,NAC,DS,SHFN,LT,1,MR)

C DO 35 ID=1,NDIM SUM=ZERO DO 30 I=1,NDN 30 SUM=SUM+SHFN(I)*ELCOD(ID,I) 35 CIP(ID)=SUM C DO IS=1,NS TEMP(IS)=STRESS(IS,IP,MR) ENDDO IF(ICAM.NE.1)GO TO 38 EI=STRESS(NS+2,IP,J) PCI=STRESS(NS+3,IP,J) PE=(STRESS(1,IP,J)+STRESS(2,IP,J)+STRESS(3,IP,J))/3.0D0 QE=Q(TEMP,NS,NDIM) IF(NDIM.EQ.2)WRITE(LOUT1,903)J,IP,(CIP(ID),ID=1,NDIM), 1 (STRESS(IK,IP,J),IK=1,NS1),PE,QE,PCI,EI IF(NDIM.EQ.3)WRITE(LOUT1,933)J,IP,(CIP(ID),ID=1,NDIM), 1 (STRESS(IK,IP,J),IK=1,NS1),PE,QE,PCI,EI GO TO 40 38 WRITE(LOUT1,903)J,IP,(CIP(ID),ID=1,NDIM), 1 (STRESS(IK,IP,J),IK=1,NS1) 40 CONTINUE 60 CONTINUE RETURN 901 FORMAT(1X,7H ELM-IP,4X,1HX,6X,1HY,9X,2HSX,10X, 1 2HSY,10X,2HSZ,10X,3HTXY,9X,1HU,10X,2HPE, 2 11X,1HQ,10X,2HPC,7X,4HVOID/) 903 FORMAT(1X,I3,I2,2F8.4,8E12.4,F7.4) 910 FORMAT(1X,7HELEMENT,I5,2X,27HIS OF UNKNOWN MATERIAL TYPE,I5, 1 2X,16H(ROUTINE INSTRS)) 931 FORMAT(1X,7H ELM-IP,4X,1HX,7X,1HY,7X,1HZ,8X,2HSX,10X, 1 2HSY,10X,2HSZ,9X,3HTXY,9X,3HTYZ,9X,3HTZX,11X,1HU,10X,2HPE, 2 11X,1HQ,10X,2HPC,6X,4HVOID/) 933 FORMAT(1X,I3,I2,3F8.4,10E12.4,F7.4) END


271

SUBROUTINE RDSTRS(KT,XYZ,STRESS,PEQT,NCONN,MAT,LTYP,NW,PR,NTY,NI) C********************************************************************** C SET UP IN-SITU STRESSES C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION XYZ(3,MNODES),STRESS(NVRS,NIP,MEL),PEQT(MDOF) DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20), 1 FI(3,NDMX) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION NW(MNODES+1) DIMENSION YI(100),VAR(NVRS,100),NLI(100),NHI(100) DIMENSION CIP(3),SLL(4),PR(NPR,NMT),NTY(NMT) COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /DATL / SL(4,100) COMMON /ELINF / LINFO(50,15) C

----- ISTGE - CODE TO INDICATE STAGE OF THE ANALYSIS ISTGE=1

C

----- INITIALISE STRESS - INTEGRATION CALL ZEROR3(STRESS,NVRS,NIP,MEL)

C C

C

POINT

VARIABLES

----- INITIALISE PEQT - CONTRIBUTION OF FORCES DUE TO ELEMENT IN-SITU STRESSES CALL ZEROR1(PEQT,MDOF) IF(KT.EQ.0) WRITE(LOUT1,904) IF(KT-1) 200,8,82 8 IF(NDIM.EQ.2)WRITE(LOUT1,906) IF(NDIM.EQ.3)WRITE(LOUT1,926)

----- Read in Node Y coordinate and initial stress data READ(LINP,101)HEADER10 101 FORMAT(A80) DO 10 J=1,NI NDAT=NS+3 ! NS=4 for 2 D problems READ(LINP,*)YI(J),(VAR(JJ,J),JJ=1,NDAT) IF(NDIM.EQ.2)WRITE(LOUT1,910) J,YI(J),(VAR(JJ,J),JJ=1,NDAT) IF(NDIM.EQ.3)WRITE(LOUT1,930) J,YI(J),(VAR(JJ,J),JJ=1,NDAT) 10 CONTINUE C MI=NI-1

272

Appendix: Fortran Codes of CS-S DO 20 IN=1,MI N1=IN N2=IN+1 Y1=YI(N1) Y2=YI(N2)

C IF(Y1.LT.Y2) THEN NMIN=N1 NMAX=N2 ELSE NMAX=N1 NMIN=N2 END IF NLI(IN)=NMIN NHI(IN)=NMAX 20 CONTINUE C

----- LOOP ON ALL GEOMETRY MESH ELEMENTS DO 80 J=1,NEL LT=LTYP(J) IF(LT.LT.0)GOTO 80 CC LT=IABS(LT) JUS=J GO TO(80,22,22,22,22,22,22,22,22,22,22,80,80,80,80),LT WRITE(LOUT1,915)JUS,LT GOTO 80 22 KM=MAT(J) NGP=LINFO(11,LT) NDN=LINFO(5,LT) INDX=LINFO(12,LT) NAC=LINFO(15,LT) C DO 30 KN=1,NDN NDE=NCONN(KN,J) DO 30 ID=1,NDIM ELCOD(ID,KN) = XYZ(ID,NDE) 30 CONTINUE C

C

C

----- LOOP ON DO 60 IP=1,NGP

ALL

INTEGRATION

----- CALCULATE INTEGRATION IPA=IP+INDX DO 35 IL=1,NAC 35 SLL(IL)=SL(IL,IPA)

POINTS

POINT

COORDINATES

----- Call Shape functin CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,1,JUS)

Appendix: Fortran Codes of CS-S C DO 40 ID=1,NDIM SUM=ZERO DO 38 I=1,NDN 38 SUM=SUM+SHFN(I)*ELCOD(ID,I) 40 CIP(ID)=SUM YY=CIP(2) C

----- SEARCH FOR DO 45 JJJ=1,MI NSM=NLI(JJJ) NLA=NHI(JJJ) YMIN=YI(NSM) YMAX=YI(NLA)

RELEVANT

IN-SITU

LAYER

C IF(YY.LT.YMIN.OR.YY.GT.YMAX)GO TO 45 GO TO 48 C 45 CONTINUE WRITE(LOUT1,950)JUS,IP GO TO 60 C

----- DIRECT INTERPOLATION 48 DY=YI(JJJ)-YI(JJJ+1) YR=(YY-YMIN)/DY

FROM

IN-SITU

MESH

NODES

C DO 50 I=1,NVRS 50 STRESS(I,IP,J)=VAR(I,NSM)+(VAR(I,JJJ)-VAR(I,JJJ+1))*YR KGO=NTY(KM) GO TO(60,60,52,60,60),KGO C C

C C C

----- CALCULATE MEAN EFFECTIVE STRESS P’ 52 P=(STRESS(1,IP,J)+STRESS(2,IP,J)+STRESS(3,IP,J))/3.0D0 CALCULATE PC’ (PC) AND CRITICAL STATE VALUE OF P’(PU) PC=STRESS(NS+3,IP,J) IF(KGO.NE.3)GO TO 54 PU=0.50D0*PC GO TO 55 54 PU=PC/EXP(1.0d0) ----- CALCULATE VOID RATIO 55 STRESS(NS+2,IP,J)=PR(3,KM)-PR(1,KM)*ALOG(P)1(PR(2,KM)-PR(1,KM))*ALOG(PU) 55 STRESS(NS+2,IP,J)=PR(3,KM)-PR(1,KM)*LOG(P)1(PR(2,KM)-PR(1,KM))*LOG(PU) 60 CONTINUE 80 CONTINUE

273

274

Appendix: Fortran Codes of CS-S GOTO 92

C

C

C

----- DIRECT SPECIFICATION OF IN-SITU STRESSESS 82 IF(KT.NE.2)GO TO 92 ! KT=1 for me -- skip WRITE(LOUT1,955) *** READ FOR ALL INTEGRATION POINTS DO 90 IM=1,NEL READ(LINP,*)MUS IL=MREL(MUS) IL=MUS LT=LTYP(IL) NGP=LINFO(11,LT)

C DO 85 IP=1,NGP READ(LINP,*)(STRESS(JJJ,IP,IL),JJJ=1,NVRS) 85 WRITE(LOUT1,960)(STRESS(JJJ,IP,IL),JJJ=1,NVRS) 90 CONTINUE C C

----- CALCULATE EQUILIBRIUM LOADS FOR INSITU STRESSES ----- ASSEMBLE ELEMENT CONTRIBUTION (FI) INTO PEQT 92 CR=1.0D0 IF(NPLAX.EQ.1)CR=2.0D0*PYI DO 100 J=1,NEL LT=LTYP(J) IF(LT.LE.0)GO TO 100 NDN=LINFO(5,LT) NGP=LINFO(11,LT) INDX=LINFO(12,LT) NAC=LINFO(15,LT)

C

----- Call EQLIB

(Calculate equilibrium loads for in situ stresses) CALL EQLIB(J,LT,NGP,INDX,NDN,NAC,XYZ,STRESS,FI,NCONN,ISTGE)

C

----- SLOT EQUILIBRIUM LOADS INTO PEQT DO IK=1,NDN NCOR=NCONN(IK,J) N1=NW(NCOR)-1

C DO 95 ID=1,NDIM NID=N1+ID 95 PEQT(NID)=PEQT(NID)+FI(ID,IK) END DO


275

100 CONTINUE C C

----- COUNT FOR SKEW BOUNDARIES ----- Turn back the displacement to the original coordinate. (Global to local) IF(NSKEW.GT.0) CALL ROTBC(PEQT,NW,NDIM,NSKEW,1) !NSKEW>0 for my case

C C C

----- PRINT OUT INITIAL IN-SITU STRESSES CALL INSTRS(LOUT1,NEL,NDIM,NS,XYZ,STRESS,NCONN,MAT,LTYP,NTY)

200 CONTINUE RETURN 904 FORMAT(//1X,36HIN-SITU STRESSES ALL SET TO ZERO/1X,36(1H-)) 906 FORMAT(//1X,19HIN-SITU MESH DATA/1X,19(1H-)/ 1 /3X,4HNODE,8X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ, 2 9X,3HTXY,10X,1HU,22X,2HPC/) 910 FORMAT(1X,I5,10F12.3) 915 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5) 926 FORMAT(//1X,19HIN-SITU MESH DATA/1X,19(1H-)/ 1 /3X,4HNODE,8X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ, 2 9X,3HTXY,10X,3HTYZ,9X,3HTZX,9X,1HU,22X,2HPC/) 930 FORMAT(1X,I5,12F12.3) 950 FORMAT(1X,46HWARNING --- POINT OUTSIDE IN-SITU STRESS SPACE, 1 2X,9HELEMENT =,I5,2X,4HIP =,I5,2X,16H(ROUTINE RDSTRS)) CC951 FORMAT(2I4,7E14.4) 955 FORMAT(//1X,40HDIRECT SPECIFICATION OF IN-SITU STRESSES 1 /1X,39(1H-)) 960 FORMAT(1X,10E12.5) C 985 FORMAT(/1X,37HEQUILIBRIUM LOADS FOR INSITU STRESSES/ C 1 1X,37(1H-)//(10E12.4)) END

SUBROUTINE REACT(LOUT2,NDIM,NN,NW,NQ,IDFX,PEQT,PT,IOUTP) C********************************************************************** C CALCULATES REACTION TO EARTH AT RESTRAINED NODES C********************************************************************** C 1 CALLED BY EQLBM IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION PEQT(MDOF),PT(MDOF),NW(MNODES+1),NQ(MNODES),IDFX(MDOF) DIMENSION R(500),NDENO(500),NDIR(500) C

----- NCT - SIZE OF ARRAYS R, NDENO AND NDIR NCT=500

276 C

Appendix: Fortran Codes of CS-S ----- ICT - COUNTER OF TOTAL NO. OF REACTIONS ICT=0

C DO 25 JR=1,NN IF(JR.EQ.0)GOTO 25 J=JR NQL=NQ(J) C

----- SKIP IF NODE HAS PORE PRESSURE D.O.F. ONLY IF(NQL.LE.1)GOTO 25 N1=NW(J) N2=N1+NDIM-1 IDF=0

C DO 20 KN=N1,N2 IDF=IDF+1 IF(IDFX(KN).NE.1)GOTO 20 ICT=ICT+1 IF(ICT.GT.NCT)GOTO 30 R(ICT)=-(PEQT(KN)-PT(KN)) NDENO(ICT)=JR NDIR(ICT)=IDF 20 CONTINUE 25 CONTINUE C IF(IOUTP.NE.1) RETURN WRITE(LOUT2,901) WRITE(LOUT2,903)(NDENO(JCT),NDIR(JCT),R(JCT),JCT=1,ICT) RETURN 30 WRITE(LOUT2,906) STOP 901 FORMAT(//1X,18H LIST OF REACTIONS/2X,17(1H-)/ 1 2X,3(4HNODE,4X,9HDIRECTION,7X,8HREACTION,11X)/) 903 FORMAT(3(1X,I5,5X,I4,5X,E14.4,10X)) 906 FORMAT(/1X,35HINCREASE ARRAY SIZE OF R,NDENO,NDIR, 1 1X,16HIN ROUTINE REACT) END

SUBROUTINE RESTRN(NDIM,NW,IDFX) C********************************************************************** C ROUTINE TO IDENTIFY ALL DISPLACEMENT BOUNDARY CONDITIONS WHICH C ARE SPECIFIED. (SET IDFX = 1 FOR ALL DOF WHICH ARE RESTRAINED) C********************************************************************** C 1 CALLED BY EQLOD IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’


277

DIMENSION NW(MNODES+1),IDFX(MDOF) COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF C C C C

----- LOOP ON ALL NODES WITH ONE OR MORE FIXITIES CALL ZEROI1(IDFX,MDOF) DO 10 J=1,NDF 10 IDFX(J)=0 IF(NF.EQ.0)RETURN DO 40 JN=1,NF NDE=MF(JN) NFS=NW(NDE)-1

C

----- BY-PASS IF NODE HAS ONLY PORE-PRESSURE DOF JP=NW(NDE+1)-NW(NDE) IF(JP.EQ.1)GO TO 40

C DO 20 JF=1,NDIM NCDE=NFIX(JF,JN) IF(NCDE.EQ.0)GO TO 20 IDFX(NFS+JF)=1 20 CONTINUE 40 CONTINUE RETURN END

SUBROUTINE SELF(LOUT1,I,NDN,NAC,XYZ,F,NCONN, 1 MAT,LT,INDX,DENS,MUS,KSTGE) C********************************************************************** C CALCULATES SELF WEIGHT LOADS C********************************************************************** C 1 CALLED BY EQLOD C 2 Called by SEL1 IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),MAT(MEL) DIMENSION XYZ(3,MNODES),ELCOD(3,NDMX),DS(3,20), 1 SHFN(20),F(3,NDMX),SLL(4),GCOM(3) COMMON /ELINF / LINFO(50,15) COMMON /DATL / SL(4,100) COMMON /DATW / W(100) COMMON /PARS / PYI,ASMVL,ZERO COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW

278


C TPI=2.0D0*PYI NGP=LINFO(11,LT) K=MAT(I) C

----- INITIALISE ARRAY F CALL ZEROR2(F,3,NDMX)

C IF(DENS.LE.ASMVL)GO TO 100 GCOM(1)= ZERO GCOM(2)=-DENS GCOM(3)= ZERO C

----- SET UP LOCAL ARRAY FOR CO-ORDINATES DO 10 KC=1,NDN NDE=NCONN(KC,I)

C DO 10 ID=1,NDIM 10 ELCOD(ID,KC)=XYZ(ID,NDE) C

----- LOOP FOR NUMERICAL INTEGRATION DO 60 IP=1,NGP IPA=IP+INDX

C DO 35 IL=1,NAC 35 SLL(IL)=SL(IL,IPA) C C

----- EVALUATE SHAPE FUNCTION FOR INTEGRATION POINT ----- Call shape & detjcb CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MUS) CALL DETJCB(LOUT1,NDIM,DJACB,NDN,ELCOD,DS,IP,MUS,KSTGE) !Determinant of J. DV=DJACB*W(IPA) IF(NPLAX.EQ.0)GO TO 45 C RAD=ZERO C DO 40 IN=1,NDN 40 RAD=RAD+ELCOD(1,IN)*SHFN(IN) DV=DV*TPI*RAD C 45 DO 50 IN=1,NDN DO 50 ID=1,NDIM 50 F(ID,IN)=F(ID,IN)+GCOM(ID)*SHFN(IN)*DV 60 CONTINUE 100 CONTINUE RETURN END


279

SUBROUTINE SFR1(LOUT1,S,SHF,DERIV,NSD,LNE,LT) C********************************************************************** C SHAPE FUNCTIONS AND DERIVATIVES FOR ONE-DIMENSIONAL C GAUSSIAN INTEGRATION ALONG ELEMENT EDGE C********************************************************************** C 1 Called by DISTLD IMPLICIT REAL*8(A-H,O-Z) DIMENSION SHF(5),DERIV(5) C

----- INITIALISE CALL ZEROR1(SHF,5) CALL ZEROR1(DERIV,5)

C GO TO(80,21,31,41,51),NSD WRITE(LOUT1,900)LNE,LT 900 FORMAT(1X,7HELEMENT,I5,2X,7HOF TYPE,I5,2X, 1 22HUNKNOWN (ROUTINE SFR1)) STOP C

----- 2 NODES ALONG EDGE 21 CONTINUE WRITE(LOUT1,910)LT 910 FORMAT(/1X,12HELEMENT TYPE,I5,2X, 1 30HNOT IMPLEMENTED (ROUTINE SFR1)) GO TO 80

C

----- 3 NODES ALONG EDGE 31 CONTINUE SHF(1)=0.50D0*S*(S-1.0D0) SHF(2)=(1.0D0-S)*(1.0D0+S) SHF(3)=0.50D0*S*(S+1.0D0) DERIV(1)=S-0.50D0 DERIV(2)=-2.0D0*S DERIV(3)=S+0.50D0 GO TO 80

C

----- 4 NODES ALONG EDGE 41 CONTINUE WRITE(LOUT1,910)LT GO TO 80

C

----- 5 NODES ALONG EDGE 51 S0=S S1=S+0.50D0 S2=S-0.50D0 S3=S+1.0D0 S4=S-1.0D0

280

Appendix: Fortran Codes of CS-S C1=2.0D0/3.0D0 C2=8.0D0/3.0D0 C3=4.0D0 SHF(1)= C1*S0*S1*S2*S4 SHF(2)=-C2*S0*S2*S3*S4 SHF(3)= C3*S1*S2*S3*S4 SHF(4)=-C2*S0*S1*S3*S4 SHF(5)= C1*S0*S1*S2*S3 DERIV(1)= C1*(S2*S4*(S1+S0)+S0*S1*(S2+S4)) DERIV(2)=-C2*(S2*S4*(S3+S0)+S0*S3*(S2+S4)) DERIV(3)= C3*(S3*S4*(S1+S2)+S1*S2*(S3+S4)) DERIV(4)=-C2*(S3*S4*(S1+S0)+S1*S0*(S3+S4)) DERIV(5)= C1*(S2*S3*(S1+S0)+S1*S0*(S2+S3)) 80 CONTINUE RETURN END

SUBROUTINE SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,ICODE,MUS) C================================================================ ======= C SHAPE FUNCTIONS AND DERIVATIVES FOR DIFFERENT ELEMENT TYPES C================================================================ ======= C 1 CALLED BY RDSTRS C 2 CALLED BY INSTRS C 3 CALLED BY SELF INCLUDE ’PARM.FOR’ IMPLICIT REAL*8(A-H,O-Z) DIMENSION SLL(4),SHFN(20),DS(3,20) C AC1=SLL(1) AC2=SLL(2) IF(NAC.LT.3)GOTO 10 AC3=SLL(3) IF(NAC.LT.4)GOTO 10 AC4=SLL(4) C 10 GOTO(11,13,13,14,14,15,15,17,17,18,18),LT WRITE(LOUT1,910)MUS,LT STOP C

C

----- SHAPE FUNCTIONS AND DERIVATIVES FOR BAR ELEMENT 11 CONTINUE WRITE(LOUT1,910)MUS,LT GOTO 80 ----- SHAPE FUNCTIONS AND DERIVATIVES FOR LST


C C

13 SHFN(1)=AC1*(2.0D0*AC1-1.0D0) SHFN(2)=AC2*(2.0D0*AC2-1.0D0) SHFN(3)=AC3*(2.0D0*AC3-1.0D0) SHFN(4)=4.0D0*AC1*AC2 SHFN(5)=4.0D0*AC2*AC3 SHFN(6)=4.0D0*AC1*AC3 IF(ICODE.EQ.1)GOTO 80 !icode=1 means initial stress condition otherwise icode=2 DS(1,1)=4.0D0*AC1-1.0D0 DS(1,2)=0.0D0 DS(1,3)=-(4.0D0*AC3-1.0D0) DS(1,4)=4.0D0*AC2 DS(1,5)=-4.0D0*AC2 DS(1,6)=4.0D0*(AC3-AC1)

C DS(2,1)=0.0D0 DS(2,2)=4.0D0*AC2-1.0D0 DS(2,3)=-(4.0D0*AC3-1.0D0) DS(2,4)=4.0D0*AC1 DS(2,5)=4.0D0*(AC3-AC2) DS(2,6)=-4.0D0*AC1 GO TO 80 C

C

C

----- SHAPE FUNCTIONS AND DERIVATIVES FOR QUADRILATERALS 14 SHFN(1)=-0.250D0*(AC1-1.0D0)*(AC2-1.0D0)*(1.0D0+AC1+AC2) SHFN(2)=0.250D0*(AC1+1.0D0)*(AC2-1.0D0)*(1.0D0-AC1+AC2) SHFN(3)=0.250D0*(AC1+1.0D0)*(AC2+1.0D0)*(AC1+AC2-1.0D0) SHFN(4)=-0.250D0*(AC1-1.0D0)*(AC2+1.0D0)*(AC2-AC1-1.0D0) SHFN(5)=0.50D0*(AC1**2-1.0D0)*(AC2-1.0D0) SHFN(6)=-0.50D0*(AC1+1.0D0)*(AC2**2-1.0D0) SHFN(7)=-0.50D0*(AC1**2-1.0D0)*(AC2+1.0D0) SHFN(8)=0.50D0*(AC1-1.0D0)*(AC2**2-1.0D0) IF(ICODE.EQ.1)GOTO 80 derivatives for ac1 DS(1,1)=-0.250D0*(AC2-1.0D0)*(2.0D0*AC1+AC2) DS(1,2)=0.250D0*(AC2-1.0D0)*(-2.0D0*AC1+AC2) DS(1,3)=0.250D0*(AC2+1.0D0)*(2.0D0*AC1+AC2) DS(1,4)=-0.250D0*(AC2+1.0D0)*(-2.0D0*AC1+AC2) DS(1,5)=AC1*(AC2-1.0D0) DS(1,6)=-0.50D0*(AC2**2-1.0D0) DS(1,7)=-AC1*(AC2+1.0D0) DS(1,8)=0.50D0*(AC2**2-1.0D0) derivatives for ac2 DS(2,1)=-0.250D0*(AC1-1.0D0)*(2.0D0*AC2+AC1) DS(2,2)=0.250D0*(AC1+1.0D0)*(2.0D0*AC2-AC1) DS(2,3)=0.250D0*(AC1+1.0D0)*(2.0D0*AC2+AC1) DS(2,4)=-0.250D0*(AC1-1.0D0)*(2.0D0*AC2-AC1)

281

282

Appendix: Fortran Codes of CS-S DS(2,5)=0.50D0*(AC1**2-1.0D0) DS(2,6)=-AC2*(AC1+1.0D0) DS(2,7)=-0.50D0*(AC1**2-1.0D0) DS(2,8)=AC2*(AC1-1.0D0)

c GOTO 80 C

C

----- SHAPE FUNCTIONS AND DERIVATIVES FOR CUBIC STRAIN TRIANGLE 15 CONTINUE C1=32.0D0/3.0D0 C2=64.0D0 C3=128.0D0/3.0D0 C4=128.0D0 T11=AC1-0.250D0 T12=AC1-0.50D0 T13=AC1-0.750D0 T21=AC2-0.250D0 T22=AC2-0.50D0 T23=AC2-0.750D0 T31=AC3-0.250D0 T32=AC3-0.50D0 T33=AC3-0.750D0 ----- SHAPE FUNCTIONS SHFN(1) =C1*AC1*T11*T12*T13 SHFN(2) =C1*AC2*T21*T22*T23 SHFN(3) =C1*AC3*T31*T32*T33 SHFN(4) =C3*AC1*AC2*T11*T12 SHFN(5) =C2*AC1*AC2*T11*T21 SHFN(6) =C3*AC1*AC2*T21*T22 SHFN(7) =C3*AC2*AC3*T21*T22 SHFN(8) =C2*AC2*AC3*T21*T31 SHFN(9) =C3*AC2*AC3*T31*T32 SHFN(10)=C3*AC1*AC3*T31*T32 SHFN(11)=C2*AC1*AC3*T11*T31 SHFN(12)=C3*AC1*AC3*T11*T12 SHFN(13)=C4*AC1*AC2*AC3*T11 SHFN(14)=C4*AC1*AC2*AC3*T21 SHFN(15)=C4*AC1*AC2*AC3*T31 IF(ICODE.EQ.1)GOTO 80

C DS(1,1)=C1*(T12*T13*(T11+AC1)+AC1*T11*(T13+T12)) DS(1,2)= 0.0D0 DS(1,3)=-C1*(T32*T33*(AC3+T31)+AC3*T31*(T32+T33)) DS(1,4)= C3*AC2*(T11*T12+AC1*(T11+T12)) DS(1,5)= C2*AC2*T21*(AC1+T11) DS(1,6)= C3*AC2*T21*T22

Appendix: Fortran Codes of CS-S DS(1,7)=-C3*AC2*T21*T22 DS(1,8)=-C2*AC2*T21*(AC3+T31) DS(1,9)=-C3*AC2*(T31*T32+AC3*(T31+T32)) DS(1,10)=-C3*(AC1*AC3*(T31+T32)-T31*T32*(AC3-AC1)) DS(1,11)= C2*(AC1*AC3*(T31-T11)+T31*T11*(AC3-AC1)) DS(1,12)= C3*(AC1*AC3*(T11+T12)+T11*T12*(AC3-AC1)) DS(1,13)= C4*AC2*(AC1*AC3+T11*(AC3-AC1)) DS(1,14)= C4*AC2*T21*(AC3-AC1) DS(1,15)=-C4*AC2*(AC1*AC3+T31*(AC1-AC3)) C DS(2,1) = 0.0D0 DS(2,2) = C1*(T22*T23*(AC2+T21)+AC2*T21*(T22+T23)) DS(2,3) =-C1*(T32*T33*(AC3+T31)+AC3*T31*(T32+T33)) DS(2,4) = C3*AC1*T11*T12 DS(2,5) = C2*AC1*T11*(AC2+T21) DS(2,6) = C3*AC1*(T21*T22+AC2*(T21+T22)) DS(2,7) = C3*(AC2*AC3*(T21+T22)+T21*T22*(AC3-AC2)) DS(2,8) = C2*(AC2*AC3*(T31-T21)+T21*T31*(AC3-AC2)) DS(2,9) =-C3*(AC2*AC3*(T31+T32)+T31*T32*(AC2-AC3)) DS(2,10)=-C3*AC1*(T31*T32+AC3*(T31+T32)) DS(2,11)=-C2*AC1*T11*(AC3+T31) DS(2,12)=-C3*AC1*T11*T12 DS(2,13)= C4*AC1*T11*(AC3-AC2) DS(2,14)= C4*AC1*(AC2*AC3+T21*(AC3-AC2)) DS(2,15)=-C4*AC1*(AC2*AC3+T31*(AC2-AC3)) GO TO 80 C C

----- SHAPE FUNCTIONS AND DERIVATIVES FOR BRICK ELEMENT ----- SHAPE FUNCTIONS 17 SHFN(1)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0+AC3) 1 *(-AC1-AC2+AC3-2.0D0) SHFN(2)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0+AC3) 2 *(AC1-AC2+AC3-2.0D0) SHFN(3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0+AC3) 3 *(AC1+AC2+AC3-2.0D0) SHFN(4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0+AC3) 4 *(-AC1+AC2+AC3-2.0D0) SHFN(5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0-AC3) 5 *(-AC1-AC2-AC3-2.0D0) SHFN(6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0-AC3) 6 *(AC1-AC2-AC3-2.0D0) SHFN(7)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0-AC3) 7 *(AC1+AC2-AC3-2.0D0) SHFN(8)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0-AC3) 8 *(-AC1+AC2-AC3-2.0D0) SHFN(9) =0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)*(1.0D0+AC3) SHFN(10)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)*(1.0D0+AC3) SHFN(11)=0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)*(1.0D0+AC3)

283

284

C

Appendix: Fortran Codes of CS-S SHFN(12)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)*(1.0D0+AC3) SHFN(13)=0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2)*(1.0D0-AC3) SHFN(14)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2)*(1.0D0-AC3) SHFN(15)=0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2)*(1.0D0-AC3) SHFN(16)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2)*(1.0D0-AC3) SHFN(17)=0.250D0*(1.0D0-AC1)*(1.0D0-AC2)*(1.0D0-AC3**2) SHFN(18)=0.250D0*(1.0D0+AC1)*(1.0D0-AC2)*(1.0D0-AC3**2) SHFN(19)=0.250D0*(1.0D0+AC1)*(1.0D0+AC2)*(1.0D0-AC3**2) SHFN(20)=0.250D0*(1.0D0-AC1)*(1.0D0+AC2)*(1.0D0-AC3**2) IF(ICODE.EQ.1)GOTO 80 DERIVATINES DS(1,1)=0.1250D0*(1.0D0-AC2)*(1.0D0+AC3)*(2.0D0*AC1+AC2-AC3+1.0D0) DS(1,2)=0.1250D0*(1.0D0-AC2)*(1.0D0+AC3)*(2.0D0*AC1-AC2+AC3-1.0D0) DS(1,3)=0.1250D0*(1.0D0+AC2)*(1.0D0+AC3)*(2.0D0*AC1+AC2+AC3-1.0D0) DS(1,4)=0.1250D0*(1.0D0+AC2)*(1.0D0+AC3)*(2.0D0*AC1-AC2-AC3+1.0D0) DS(1,5)=0.1250D0*(1.0D0-AC2)*(1.0D0-AC3)*(2.0D0*AC1+AC2+AC3+1.0D0) DS(1,6)=0.1250D0*(1.0D0-AC2)*(1.0D0-AC3)*(2.0D0*AC1-AC2-AC3-1.0D0) DS(1,7)=0.1250D0*(1.0D0+AC2)*(1.0D0-AC3)*(2.0D0*AC1+AC2-AC3-1.0D0) DS(1,8)=0.1250D0*(1.0D0+AC2)*(1.0D0-AC3)*(2.0D0*AC1-AC2+AC3+1.0D0) DS(1,9) =-0.50D0*AC1*(1.0D0-AC2)*(1.0D0+AC3) DS(1,10)= 0.250D0*(1.0D0-AC2**2)*(1.0D0+AC3) DS(1,11)=-0.50D0*AC1*(1.0D0+AC2)*(1.0D0+AC3) DS(1,12)=-0.250D0*(1.0D0-AC2**2)*(1.0D0+AC3) DS(1,13)=-0.50D0*AC1*(1.0D0-AC2)*(1.0D0-AC3) DS(1,14)= 0.250D0*(1.0D0-AC2**2)*(1.0D0-AC3) DS(1,15)=-0.50D0*AC1*(1.0D0+AC2)*(1.0D0-AC3) DS(1,16)=-0.250D0*(1.0D0-AC2**2)*(1.0D0-AC3) DS(1,17)=-0.250D0*(1.0D0-AC2)*(1.0D0-AC3**2) DS(1,18)= 0.250D0*(1.0D0-AC2)*(1.0D0-AC3**2) DS(1,19)= 0.250D0*(1.0D0+AC2)*(1.0D0-AC3**2) DS(1,20)=-0.250D0*(1.0D0+AC2)*(1.0D0-AC3**2)

C DS(2,1)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC3)*(AC1+2.0D0*AC2-AC3+1.0D0) DS(2,2)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC3)* 1 (-AC1+2.0D0*AC2-AC3+1.0D0) DS(2,3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC3)*(AC1+2.0D0*AC2+AC3-1.0D0) DS(2,4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC3)* 1 (-AC1+2.0D0*AC2+AC3-1.0D0) DS(2,5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC3)*(AC1+2.0D0*AC2+AC3+1.0D0) DS(2,6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC3)* 1 (-AC1+2.0D0*AC2+AC3+1.0D0) DS(2,7)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC3)*(AC1+2.0D0*AC2-AC3-1.0D0) DS(2,8)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC3)* 1 (-AC1+2.0D0*AC2-AC3-1.0D0) DS(2,9) =-0.250D0*(1.0D0-AC1**2)*(1.0D0+AC3) DS(2,10)=-0.50D0*(1.0D0+AC1)*AC2*(1.0D0+AC3) DS(2,11)= 0.250D0*(1.0D0-AC1**2)*(1.0D0+AC3) DS(2,12)=-0.50D0*(1.0D0-AC1)*AC2*(1.0D0+AC3)


285

DS(2,13)=-0.250D0*(1.0D0-AC1**2)*(1.0D0-AC3) DS(2,14)=-0.50D0*(1.0D0+AC1)*AC2*(1.0D0-AC3) DS(2,15)= 0.250D0*(1.0D0-AC1**2)*(1.0D0-AC3) DS(2,16)=-0.50D0*(1.0D0-AC1)*AC2*(1.0D0-AC3) DS(2,17)=-0.250D0*(1.0D0-AC1)*(1.0D0-AC3**2) DS(2,18)=-0.250D0*(1.0D0+AC1)*(1.0D0-AC3**2) DS(2,19)= 0.250D0*(1.0D0+AC1)*(1.0D0-AC3**2) DS(2,20)= 0.250D0*(1.0D0-AC1)*(1.0D0-AC3**2) C DS(3,1)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)* (-AC1-AC2+2.0D0*AC3-1.0D0) DS(3,2)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)*(AC1-AC2+2.0D0*AC3-1.0D0) DS(3,3)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)*(AC1+AC2+2.0D0*AC3-1.0D0) DS(3,4)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)* 1 (-AC1+AC2+2.0D0*AC3-1.0D0) DS(3,5)=0.1250D0*(1.0D0-AC1)*(1.0D0-AC2)*(AC1+AC2+2.0D0*AC3+1.0D0) DS(3,6)=0.1250D0*(1.0D0+AC1)*(1.0D0-AC2)* 1 (-AC1+AC2+2.0D0*AC3+1.0D0) DS(3,7)=0.1250D0*(1.0D0+AC1)*(1.0D0+AC2)* 1 (-AC1-AC2+2.0D0*AC3+1.0D0) DS(3,8)=0.1250D0*(1.0D0-AC1)*(1.0D0+AC2)*(AC1-AC2+2.0D0*AC3+1.0D0) DS(3,9) = 0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2) DS(3,10)= 0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2) DS(3,11)= 0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2) DS(3,12)= 0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2) DS(3,13)=-0.250D0*(1.0D0-AC1**2)*(1.0D0-AC2) DS(3,14)=-0.250D0*(1.0D0+AC1)*(1.0D0-AC2**2) DS(3,15)=-0.250D0*(1.0D0-AC1**2)*(1.0D0+AC2) DS(3,16)=-0.250D0*(1.0D0-AC1)*(1.0D0-AC2**2) DS(3,17)=-0.50D0*(1.0D0-AC1)*(1.0D0-AC2)*AC3 DS(3,18)=-0.50D0*(1.0D0+AC1)*(1.0D0-AC2)*AC3 DS(3,19)=-0.50D0*(1.0D0+AC1)*(1.0D0+AC2)*AC3 DS(3,20)=-0.50D0*(1.0D0-AC1)*(1.0D0+AC2)*AC3 GOTO 80 1

C

----- SHAPE FUNCTIONS AND DERIVATIVES FOR TETRA-HEDRA 18 CONTINUE WRITE(LOUT1,910)MUS,LT 910 FORMAT(/1X,7HELEMENT,I5,2X,14HIS OF TYPE ***,I5,2X, 1 31HNOT IMPLEMENTED (ROUTINE SHAPE)) 80 CONTINUE RETURN END

286


SUBROUTINE FACTOR(LINP,LOUT1,NOINC,ILDF,IOCD,ITMF,IOUTS, 1 RINCC,DTM,IOPT,DTIME) C********************************************************************** C LOAD RATIOS, TIME RATIOS (CONSOLIDATION ANALYSIS) AND OUTPUT C OPTIONS FOR ALL INCREMENTS IN THE BLOCK C********************************************************************** C 1 CALLED BY TOTSOL IMPLICIT REAL*8(A-H,O-Z) DIMENSION RINCC(2500),DTM(2500),IOPT(2500) COMMON /PARS / PYI,ASMVL,ZERO C

----- READ LOAD RATIOS FOR INCREMENTS FSTD=1.0D0/FLOAT(NOINC) !FSTD=factor of increment for unit pen. length IF(ILDF.EQ.0)GO TO 98 !ILDF=0 and skip --Song WRITE(LOUT1,948) READ(LINP,*)(RINCC(IN),IN=1,NOINC) WRITE(LOUT1,954)(RINCC(IN),IN=1,NOINC) GO TO 122 98 DO 100 IK=1,NOINC 100 RINCC(IK)=FSTD C

----- READ OUTPUT OPTIONS 122 IF(IOCD.EQ.0)GO TO 127 !IOCD=0 WRITE(LOUT1,960) READ(LINP,*)(IOPT(IN),IN=1,NOINC) WRITE(LOUT1,964)(IOPT(IN),IN=1,NOINC) GO TO 131

C 127 DO 130 IK=1,NOINC 130 IOPT(IK)=IOUTS C

----- READ TIME RATIOS FOR INCREMENTS 131 IF(DTIME.LT.ASMVL.OR.ITMF.EQ.0)GO TO 132 WRITE(LOUT1,965) READ(LINP,*)(DTM(IN),IN=1,NOINC) WRITE(LOUT1,968)(DTM(IN),IN=1,NOINC) GO TO 136

!ITMF=0

C 132 DO 135 IK=1,NOINC 135 DTM(IK)=FSTD*DTIME 136 CONTINUE RETURN 948 FORMAT(/1X,34HLIST OF LOAD RATIOS FOR INCREMENTS/1X,34(1H-)/) 954 FORMAT(1X,10F8.1) 960 FORMAT(/1X,35HLIST OF OUTPUT CODES FOR INCREMENTS/1X,35(1H-)/) 964 FORMAT(1X,10I6)


287

965 FORMAT(/1X,33HLIST OF TIME STEPS FOR INCREMENTS/1X,33(1H-)/) 968 FORMAT(1X,8G10.1) END

C********************************************************************** SUBROUTINE SKSOLV(XYZ,DA,DITER,DIPR,DD,STRESS,PITER,NQ,NW, 1 LTYP,NTY,MAT,NCONN,PR,IOUTP,ITER) C********************************************************************** C C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION XYZ(3,MNODES),DA(MDOF),DITER(MDOF),DAPPL(MDOF), 1 DIPR(MDOF),DSOLVD(MDOF),STRESS(NVRS,NIP,MEL),PITER(MDOF), 2 P(MDOF),GSTIF(MKSIZ) DIMENSION NQ(MNODES),NW(MNODES+1) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION PR(NPR,NMT),NTY(NMT),DD(4,200) COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW

C

C

COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, + ICOR ,IDCHK,IOUT ,INCT ,IWL , + NOIB ,JS ,JINCB ,NLOD ,NLDS INITIALIZE THE GLOBAL STIFFNESS MATRIX DO I=1,MKSIZ GSTIF(I)=0.0D0 ENDDO INITIALIZE DAPPL,P CALL ZEROR1(DAPPL,MDOF) CALL ZEROR1(P,MDOF) CALL ZEROR1(PDIS,MDOF) CALL ZEROR1(DITER,MDOF) NDIM1=NDIM+1

C IF(ITER.EQ.1) THEN DO INODE=1,NF NODE=MF(INODE) IDF=NW(NODE)-1 KDF=NQ(NODE) IF(KDF.EQ.1)GO TO 10

288

Appendix: Fortran Codes of CS-S DO ID=1,KDF IF(ID.EQ.NDIM1)GO TO 10 DAPPL(IDF+ID)=DXYT(ID,INODE)*FRACLD+DD(ID,INODE) DITER(IDF+ID)=DAPPL(IDF+ID) ENDDO GO TO 20 10 NTTI=NFIX(NDIM1,INODE) IF(NTTI.EQ.2) THEN DAPPL(IDF+KDF)=DXYT(NDIM1,INODE)-DA(IDF+KDF) DITER(IDF+KDF)=DAPPL(IDF+KDF) NFIX(NDIM1,INODE)=1 DXYT(NDIM1,INODE)=0.0D0 ENDIF 20 CONTINUE ENDDO ENDIF

c

THE LOAD MATRIX IS REARRANGED SUCH THAT BCS ARE TAKEN AWAY. DO INODE=1,NNODES KDF=NQ(INODE) NWDF=NW(INODE)-1 DO IDF=1,KDF LOC=NWDF+IDF LOCGLB=LOCDOF(LOC) IF(LOCGLB.GT.0)P(LOCGLB)=PITER(LOC) enddo enddo

C CALL GLASSEM(GSTIF,XYZ,DA,DAPPL,DIPR,STRESS,P,NQ,NW,LTYP, 1 NTY,MAT,NCONN,PR,DTIMEI,ITER,JS) C DO I=1,NEQTNS DSOLVD(I)=P(I) ENDDO C CALL SOLSYM(GSTIF,DSOLVD) c c

DO I=1,NDF DITER(I)=0.0D0 LOC=LOCDOF(I) IF(LOC.GT.0) THEN DITER(I)=DSOLVD(LOC)


289

ENDIF ENDDO C RETURN END C********************************************************************** SUBROUTINE GLASSEM(GSTIF,XYZ,DA,DAPPL,DIPR,STRESS,P,NQ,NW, 1 LTYP,NTY,MAT,NCONN,PR,DTIMEI,ITER,JS) C********************************************************************** C THIS SUBROUTINE ASSEMBLES THE GLOBAL STIFFNESS MATRIX AND CALCULATE C THE EQUIVALENT LOAD FOR THE APPLIED DISPLACEMENT C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION XYZ(3,MNODES),DA(MDOF),DAPPL(MDOF),DIPR(MDOF), 1 STRESS(NVRS,NIP,MEL),P(MDOF),GSTIF(MKSIZ),ESTIF(MDFE,MDFE), 2 EDINC(NB) DIMENSION NQ(MNODES),NW(MNODES+1),LOCELM(MDFE) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL) DIMENSION PR(NPR,NMT),NTY(NMT) COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS COMMON/ELINF /LINFO(50,15) COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW C

C

C C

LOOP OVER EACH ELREMENT DO IELEM=1,NEL ITYPE=LTYP(IELEM) NELN=LINFO(1,ITYPE) NEDOF=LINFO(16,ITYPE) GET THE ELEMENT DISPL FROM GLOBAL DISPL DO JDN=1,NELN NIN=NDIM*(JDN-1) JN=ABS(NCONN(JDN,IELEM)) JL=NW(JN)-1 DO ID=1,NDIM EDINC(NIN+ID)=DIPR(JL+ID) ENDDO ENDDO GET THE CORRESPONDANCE BETWEEN ELEMENT AND GLOBAL DOFS CALL DOFL2G(LOCELM,NW,NQ,NCONN,NELN,IELEM) CALCULATE ELEMENT STIFNESS MATRIX

290

Appendix: Fortran Codes of CS-S CALL ELMSTIF(IELEM,ESTIF,DTIMEI,ITYPE,XYZ,DA,EDINC,STRESS,P, 2 NCONN,MAT,NW,PR,NTY,IOUTP,ITER,JS)

C C C C C C C C

IF(IELEM.EQ.1) THEN WRITE(2,*)’ESTIF’ DO I1=1,20 WRITE(2,*)(ESTIF(I1,J1),J1=1,20) ENDDO ENDIF DO I=1,NEDOF IGDF=LOCELM(I) IDOF=LOCDOF(LOCELM(I)) IF(IDOF.GT.0) THEN LOCD=JDIAG(IDOF) DO J=1,NEDOF JGDF=LOCELM(J) JDOF=LOCDOF(LOCELM(J))

IF(IDOF.GE.JDOF.AND.JDOF.GT.0) THEN LOCA=LOCD+IDOF-JDOF GSTIF(LOCA)=GSTIF(LOCA)+ESTIF(I,J) C WRITE(*,*)"STIFF",GSTIF(LOCA) ENDIF C C C C

IF(JDOF.GT.0) THEN P(JDOF)=P(JDOF)+ESTIF(I,J)*DAPPL(IGDF) PDIS(JGDF)=PDIS(JGDF)+ESTIF(I,J)*DAPPL(IGDF) ENDIF ENDDO ELSE IF(IDOF.EQ.0) THEN DO J=1,NEDOF JDOF=LOCDOF(LOCELM(J)) IF(JDOF.GT.0) THEN P(JDOF)=P(JDOF)-ESTIF(I,J)*DAPPL(IGDF) ENDIF ENDDO ENDIF ENDDO

C ENDDO RETURN END


291

C********************************************************************** SUBROUTINE SOLSYM(A,R) C********************************************************************** C THIS SUBROUTINE SOLVES FOR THE DISPLACEMENTS BY SOLVING THE C SYMMETRIC LINEAR SET OF EQUATIONS C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS DIMENSION A(MKSIZ),R(MDOF)

c

perform l*d*l factorization of the stiffness matrix do n=1,NEQTNS kn = jdiag( n ) kl = kn + 1 ku = jdiag(n+1) - 1 kh = ku - kl if (kh.gt.0)then k = n - kh ic = 0 klt = ku do j = 1, kh ic = ic + 1 klt = klt - 1 ki = jdiag( k ) nd = jdiag( k + 1 ) - ki - 1 if (nd.gt.0)then kk = min0(ic,nd) c = 0. do l = 1, kk c = c + a(ki + l)*a(klt + l) enddo a( klt ) = a( klt ) - c endif k = k + 1 enddo endif if(kh.ge.0)then k = n b = 0.d0 do kk = kl, ku k = k - 1 ki = jdiag( k ) c = a( kk )/ a( ki )

292

C

c

Appendix: Fortran Codes of CS-S b = b + c*a( kk ) a( kk ) = c enddo a( kn ) = a( kn ) - b endif if (a(kn).le.0)then if (a(kn).EQ.0)then write( *, 2000) n, a( kn ) stop endif enddo reduce the right-hand-side load vector do n = 1, NEQTNS kl = jdiag( n ) + 1 ku = jdiag( n + 1) - 1 if(ku-kl.ge.0)then k = n c = 0.d0 do kk = kl, ku k = k - 1 c = c + a( kk )*r( k ) enddo r( n ) = r( n ) - c endif enddo

C c C

back-substitute do n = 1, NEQTNS k = jdiag( n ) r( n ) = r( n )/ a( k ) enddo if (NEQTNS.eq.1) return n = NEQTNS do l = 2, NEQTNS kl = jdiag( n ) + 1 ku = jdiag( n + 1 ) - 1 if (ku-kl.ge.0)then k = n do kk = kl, ku k = k - 1 r( k ) = r( k ) - a( kk )*r( n ) enddo endif n = n - 1


293

enddo return 2000 format(//1x,’stop - stiffness matrix not positive definite ’// 1 1x,’ ZERO PIVOT FOR EQUATION ’,i4//1x,’pivot = ’,e20.12) end

C********************************************************************** SUBROUTINE LOCINIT(LTYP,NCONN,NQ,NW,NDF,NDIM) C********************************************************************** C THS SUBROUTINE REASSIGN THE APPROPRIATE POINTERS TO THE DOF MATRIX C AND STIFFNESS MATRIX FOR STORAGE ALLOCATION AND DEPENDNG ON THE WAY C THE LINEAR SYSTEM IS SOLVED. C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NQ(MNODES),NW(MNODES+1) DIMENSION NCONN(NTPE,MEL),LTYP(MEL) COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS C ELIMINATE THE DOFS THAT ARE CONSTRAINED BY ASSIGNING ZERO TO LOCDOF C CORRESPOND TO THAT DOF M=0 NDIM1=NDIM+1 CALL ZEROI1(LOCDOF,MDOF) CALL ZEROI1(JDIAG,MDOF) C C WRITE(2,*)’NODE NFIX LOCDOF ’ DO INODE=1,NF NODE=MF(INODE) IDF=NW(NODE)-1 KDF=NQ(NODE) IF(KDF.EQ.1) THEN LOCDOF(IDF+KDF)=NFIX(NDIM1,INODE) ELSE DO ID=1,KDF LOCDOF(IDF+ID)=NFIX(ID,INODE) ENDDO ENDIF ENDDO C C

WRITE(2,*)’ LOCDOF 1’ WRITE(2,*)(LOCDOF(IDF),IDF=1,NDF)

294


C DO IDOF=1,NDF IF(LOCDOF(IDOF).EQ.0) THEN M=M+1 LOCDOF(IDOF)=M ELSE IF(LOCDOF(IDOF).GT.0) THEN LOCDOF(IDOF)=0 ENDIF ENDDO NEQTNS=M C C

WRITE(2,*)’ LOCDOF 2’ WRITE(2,*)(LOCDOF(IDF),IDF=1,NDF) CALL DIAG(LTYP,NCONN,NQ,NW) RETURN END

C********************************************************************** SUBROUTINE DOFL2G(LOCGLB,NW,NQ,NCONN,NELN,IELEM) C********************************************************************** C THIS SUBROUTINE FINDS THE CORRESPONDENCE BETWEEN ELEMENT AND GLOBAL C DOFS C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NQ(MNODES),NW(MNODES+1) DIMENSION NCONN(NTPE,MEL),LOCGLB(MDFE) C IESTRT=0 NWQ=0 DO IELN=1,NELN NODE=NCONN(IELN,IELEM) NODDOF=NQ(NODE) IESTRT=IESTRT+NWQ IGSTRT=NW(NODE)-1 DO IDOF=1,NODDOF IELOC=IESTRT+IDOF LOCGLB(IELOC)=IGSTRT+IDOF ENDDO NWQ=NODDOF ENDDO RETURN END


295

C********************************************************************** SUBROUTINE DIAG(LTYP,NCONN,NQ,NW) C********************************************************************** C THIS SUBROUTINE FINDS THE DIAGONAL LOCATION OF THE ELEMENT IN THE C STIFFNESS MATRIX (JDIAG) C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NQ(MNODES),NW(MNODES+1) DIMENSION NCONN(NTPE,MEL),LTYP(MEL) COMMON/ELINF /LINFO(50,15) COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS COMMON/NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW C

C C

FIND THE COLUMN HEIGHTS DO IELEM=1,NEL ITYPE=LTYP(IELEM) NELN=LINFO(1,ITYPE) MAXDOF=0 MINDOF=1000000 DO IELN=1,NELN NODE=NCONN(IELN,IELEM) NODDOF=NQ(NODE) DO IDOF=1,NODDOF K=NW(NODE)-1+IDOF IF(LOCDOF(K).GT.0) THEN IF(LOCDOF(K).GT.MAXDOF)MAXDOF=LOCDOF(K) IF(LOCDOF(K).LT.MINDOF)MINDOF=LOCDOF(K) MAXDOF=MAX0(MAXDOF,LOCDOF(K)) MINDOF=MIN0(MINDOF,LOCDOF(K)) ENDIF ENDDO ENDDO

C DO IELN=1,NELN NODE=NCONN(IELN,IELEM) NODDOF=NQ(NODE) DO IDOF=1,NODDOF K=NW(NODE)-1+IDOF IF(LOCDOF(K).GT.0) THEN ID=LOCDOF(K) MHT=ID-MINDOF+1 IF(MHT.GT.JDIAG(ID)) JDIAG(ID)=MHT ENDIF

296

Appendix: Fortran Codes of CS-S ENDDO ENDDO ENDDO

C MHT=1 ID=0 DO K=1,NEQTNS+1 ID=ID+MHT MHT=JDIAG(K) JDIAG(K)=ID ENDDO NKSIZ=JDIAG(NEQTNS+1)-JDIAG(1) IF(NKSIZ.GT.MKSIZ)THEN WRITE(2,*)’NKSIZ=’,NKSIZ WRITE(2,*)’NKSIZ GREATER THAN MKSIZ’ STOP ENDIF RETURN END

C C

SUBROUTINE SEL1(LOUT1,NDIM,NEL,XYZ,P,NCONN,MAT, 1 LTYP,NW,PR,DGRAV) -----------------------------------------------------------------1 CALLED BY TOTSOL IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION XYZ(3,MNODES),P(MDOF),F(3,NDMX) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NW(MNODES+1) DIMENSION PR(NPR,NMT) COMMON /ELINF / LINFO(50,15) COMMON /PARS / PYI,ASMVL,ZERO

C

-----CODE TO INDICATE STAGE OF THE ANALYSIS KSTGE=3

C

----- ITERATE DO 50 J=1,NEL JK=J

FOR

C

----- BY-PASS LT=LTYP(J)

ADDITION

ALL

ELEMENTS

IF

ELEMENT

NOT

IN

CURRENT

MESH


297

IF(LT.LT.0)GO TO 50 GOTO(50,22,22,22,22,22,22,22,22,22,22),LT WRITE(LOUT1,900)JK,LT 900 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5, 1 14H(ROUTINE SEL1)) 22 INDX=LINFO(12,LT) NDN=LINFO(5,LT) NAC=LINFO(15,LT) K=MAT(J) DENS=DGRAV*PR(8,K) IF(DENS.LE.ASMVL)GO TO 50 C

----- Call Self CALL SELF(LOUT1,J,NDN,NAC,XYZ,F,NCONN,MAT,LT,INDX,DENS,JK,KSTGE) C DO 30 JJ=1,NDN JN=NCONN(JJ,J) JL=NW(JN)-1 C DO 30 ID=1,NDIM 30 P(JL+ID)=P(JL+ID)+F(ID,JJ) 50 CONTINUE RETURN END C********************************************************************** SUBROUTINE ELMSTIF(K,STIFF,DTIME,LT,XYZ,DA,EDISPINC,STRESS,P, 2 NCONN,MAT,NW,PR,NTY,IOUTP,ITER,JS) C********************************************************************** C CALCULATION AND ASSEMBLY OF STIFFNESS MATRIX C********************************************************************** C This subroutine calculate the stiffness of each element. C It is combined with global stiffness in GLobalASSEMbly. Song 3/31/99 C Called by Glassem C K=ILEM=element number IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION PERM(3) DIMENSION SG(KES),XYZ(3,MNODES),DA(MDOF),DAB(3,3), 1 STRESS(NVRS,NIP,MEL),P(MDOF),D(6,6),ELCOD(3,NDMX),DS(3,20), 2 EDISPINC(NB),SHFN(20),CARTD(3,NDMX),BL(6,NB), 3 BNL(6,NB),BNLS(9,NB),BL1(9,NB),DB(6,NB), 3 EKSTIF(NB,NB),ELCODP(3,NPMX),XJACM(3,3), 4 E(3,NPMX),RN(NB),AA(NPMX),EFLOW(NPMX,NPMX), 5 ECOUPT(NB,NPMX),SPK(9,9),S(6),F(3,3),FINV(3,3)

298


dimension ETE(NPMX,NPMX),RLT(NB,NPMX) DIMENSION NCONN(NTPE,MEL),MAT(MEL),NW(MNODES+1), 1 NWL(NPMX),SLL(4),PR(NPR,NMT),NTY(NMT) DIMENSION STIFF(MDFE,MDFE) DIMENSION GDT(NPMX),BOD(3),PE(3,NPMX) DIMENSION SALPHA(4),A(6) !A added just for matching COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /DATW / W(100) COMMON /DATL / SL(4,100) COMMON /PARS / PYI,ASMVL,ZERO COMMON /ELINF / LINFO(50,15) COMMON /JACB / XJACI(3,3),DJACB COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) COMMON /NSIZE / NNODES,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW C---------------------------------------------------------------------CR=1.0D0 IF(NPLAX.EQ.1)CR=2.0D0*PYI C----------INITIALISE EKSTIF,ECOUPT AND EFLOW CALL ZEROR2(EKSTIF,NB,NB) CALL ZEROR2(ECOUPT,NB,NPMX) CALL ZEROR2(RLT,NB,NPMX) CALL ZEROR2(EFLOW,NPMX,NPMX) CALL ZEROR2(ETE,NPMX,NPMX) CALL ZEROR1(GDT,NPMX) CALL ZEROR1(BOD,3) C WRITE(*,*)"ELMSTIF" C NDN=LINFO(5,LT) NPN=LINFO(6,LT) NGP=LINFO(11,LT) INDX=LINFO(12,LT) NAC=LINFO(15,LT) NDV=NDIM*NDN NDPT=LINFO(1,LT) GOTO(1,1,2,1,2,1,2,1,2,1,2),LT WRITE(LOUT1,910)K,LT 910 FORMAT(1X,7HELEMENT,I5,2X,18HIS OF UNKNOWN TYPE,I5, 1 2X,17H(ROUTINE ELMSTIF)) STOP C 1 ICPL=0 IBLK=1 NPN=0 GOTO 14 2 ICPL=1 IBLK=0


299

C---------------------------------------------------------------------C SETUP LOCAL ARRAY OF NW AS NWL GIVING THE INDEX TO C PORE-PRESSURE VARIABLES C---------------------------------------------------------------------IPP=0 C----------INXL - INDEX TO NODAL D.O.F. (SEE BLOCK DATA) DO 12 IV=1,NDPT IQ=LINFO(IV+INXL,LT) IF(IQ.EQ.NDIM)GO TO 12 IPP=IPP+1 NDE=NCONN(IV,K) NDE=IABS(NDE) C----------COORDINATES OF POREPRESSURE NODES OF ELEMENT DO 10 ID=1,NDIM 10 ELCODP(ID,IPP)=XYZ(ID,NDE) NWL(IPP)=NW(NDE)+IQ-1 12 CONTINUE C 14 KM=MAT(K) C---------------------------------------------------------------------C LOCAL ARRAY OF COORDINATES OF DISPLACEMENT NODES OF ELEMENT C---------------------------------------------------------------------DO 20 KN=1,NDN NDE=NCONN(KN,K) NDE=IABS(NDE) C DO 20 ID=1,NDIM 20 ELCOD(ID,KN)=XYZ(ID,NDE) C IF(NTY(KM)-2)26,28,28 C----------CONSTANT ELASTICITY D MATRIX 26 CALL DCON(K,IBLK,MAT,PR,D,NDIM,BK) C---------------------------------------------------------------------C ITERATE FOR ALL INTEGRATION POINTS C---------------------------------------------------------------------28 DO 80 IP=1,NGP !NGP = No. of iteration point = No. of Gauss Point IPA=IP+INDX ICOD=MCODE(IP,K) !icod=1,2,3,4,5 depend on the stress point C DO 30 IL=1,NAC 30 SLL(IL)=SL(IL,IPA) ISTGE=3 C---------------------------------------------------------------------C INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS) C----------------------------------------------------------------------

300


CALL ZEROR2(DS,3,20) CALL ZEROR1(SHFN,20) C---------------------------------------------------------------------C CALCULATE SHAPE FUNCTIONS AND DERIVATIVES W.R.T LOCAL COOR. C---------------------------------------------------------------------CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,K) CALL ZEROR2(XJACM,3,3) C---------------------------------------------------------------------C CALCULATE [JAC], [JAC]-1 & DETJAC C---------------------------------------------------------------------DO 15 IDIM=1,NDIM DO 15 JDIM=1,NDIM SUM=ZERO C DO 112 IN=1,NDN 112 SUM=SUM+DS(IDIM,IN)*ELCOD(JDIM,IN) 15 XJACM(IDIM,JDIM)=SUM C CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,K,IP,ISTGE) C---------------------------------------------------------------------C CALCULATE RADIUS FOR AXI-SYM B MATRIX C---------------------------------------------------------------------R=ZERO RI=ZERO IF(NDIM.EQ.3.OR.NPLAX.EQ.0)GOTO 38 C DO 25 IN=1,NDN 25 R=R+ELCOD(1,IN)*SHFN(IN) RI=-1.0D0/R C---------------------------------------------------------------------C CALCULATE CARTESIAN DERIVATIVES OF SHAPE FUNCTIONS C---------------------------------------------------------------------38 DO 35 IN=1,NDN DO 35 ID=1,NDIM SUM=ZERO C DO 130 JD=1,NDIM 130 SUM=SUM-DS(JD,IN)*XJACI(ID,JD) 35 CARTD(ID,IN)=SUM C---------------------------------------------------------------------C CALCULATE THE LINEAR STRAIN-DISPL MATRIX [BL] C---------------------------------------------------------------------CALL BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL) F9=CR*DJACB*W(IPA) IF(NDIM.EQ.2.AND.NPLAX.EQ.1)F9=F9*R C IF(LINR.GT.0) THEN


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C---------------------------------------------------------------------C CALCULATE THE GEOMETRIC NONLINEAR MATRIX B\_NLS C---------------------------------------------------------------------CALL BNLNRS(SHFN,CARTD,RI,BNLS,NDIM,NDN,NPLAX) C---------------------------------------------------------------------C RETRIEVE THE SECOND PIOLA-KIRCHHOF STRESS VECTOR C---------------------------------------------------------------------DO IS=1,NS S(IS)=STRESS(IS,IP,K) ENDDO IF(NDIM.NE.3.AND.NPLAX.NE.1) THEN S(3)=S(4) S(4)=0.0D0 END IF C---------------------------------------------------------------------C CONVERT S VECTOR TO SPK MATRIX C---------------------------------------------------------------------CALL CONVERT(S,SPK,NDIM,NPLAX) C IF(LINR.GT.1) THEN C---------------------------------------------------------------------C CALCULATE THE NONLINEAR STRAIN-DISPL MATRIX BNL C---------------------------------------------------------------------CALL BNLNR(EDISPINC,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX) C CALL ADDBMAT(BL,BNL) END IF C---------------------------------------------------------------------C COMPUTE [BNLS]T[S][BNLS] AND ADD TO THE EKSTIF C---------------------------------------------------------------------NDD=NDIM*NDIM IF(NDIM.EQ.2.AND.NPLAX.EQ.1) NDD=NDD+1 CALL BTDB(BNLS,SPK,EKSTIF,F9,NDD) C END IF C---------------------------------------------------------------------C CALCULATE THE LINEAR DISPL-DISPL MATRIX BL1 C---------------------------------------------------------------------CALL BLNR1(CARTD,BL1,NDIM,NDN) C---------------------------------------------------------------------C CALCULATE THE DEFORMATION GRADIENT MATRIX [F],[FINV] & DETF C---------------------------------------------------------------------CALL DEFGRAD(BL1,F,FINV,DETF,EDISPINC,NDIM,LINR) C---------------------------------------------------------------------C CALCULATE PORE PRESSURE SHAPE FUNCTIONS & DERIVATIVES C---------------------------------------------------------------------IF(ICPL.EQ.1)CALL JPC(K,NDIM,NPN,NAC,NPLAX, 1 DS,CARTD,BL,E,RN,AA,SLL,LT)

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C KGO=NTY(KM) C---------------------------------------------------------------------C COMPUTE [D] MATRIX C---------------------------------------------------------------------GO TO(39,32,33),KGO 32 CALL DLIN(K,IBLK,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK) GO TO 39 33 CALL DMCAM(IP,K,IBLK,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB, +SALPHA,A,H) GO TO 39 39 CONTINUE

C---------------------------------------------------------------------C CALCULATE EKSTIF MATRIX C---------------------------------------------------------------------CALL LSTIFA(EKSTIF,BL,D,DB,F9,NS) C---------------------------------------------------------------------C BYPASS IF NOT COUPLED CONSOLIDATION C---------------------------------------------------------------------IF(ICPL.EQ.0)GO TO 80 C---------------------------------------------------------------------C FORM PERM*E C---------------------------------------------------------------------PERM(1)=PR(9,KM) PERM(2)=PR(10,KM) PERM(3)=PERM(1) GAMMAW=PR(7,KM) XGAM=GAMMAW C KGO=NTY(KM) GO TO(60,60,52,60,60),KGO C---------------------------------------------------------------------C CALCULATE VOID RATIO C---------------------------------------------------------------------52 EI=STRESS(NS+2,IP,K) POR=EI/(1.0D0+EI) c POR=POR*DETF STRESS(NS+2,IP,K)=POR/(1.0D0-POR) XGAM=GAMMAW/POR C XGAM=GAMMAW 60 CONTINUE C---------------------------------------------------------------------C CALCULATE ELEMENT FLOW MATRIX C---------------------------------------------------------------------c WRITE(*,*)"2222" CALL ELFLOW(NDIM,NPN,FINV,DETF,E,F9,EFLOW,PERM,XGAM,DTIME)


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C DO 40 JJ=1,NPN DO 40 IM=1,NDIM PE(IM,JJ)=PERM(IM)*E(IM,JJ) 40 CONTINUE C---------------------------------------------------------------------c FORM ET*PERM*E C---------------------------------------------------------------------DO 41 II=1,NPN DO 41 JJ=1,NPN DO 41 KK=1,NDIM 41 ETE(II,JJ)=ETE(II,JJ)+E(KK,II)*PE(KK,JJ)*DTIME*F9/GAMMAW C---------------------------------------------------------------------C FORM LT C---------------------------------------------------------------------DO 42 II=1,NDV DO 42 JJ=1,NPN 42 RLT(II,JJ)=RLT(II,JJ)+RN(II)*AA(JJ)*F9 C---------------------------------------------------------------------C FORM G.DT DO 50 II=1,NPN DO 50 KK=1,NDIM 50 GDT(II)=GDT(II)+PE(KK,II)*BOD(KK)*DTIME*F9*POR C---------------------------------------------------------------------C CALCULATE THE ELEMENT COUPLING MATRIX [ECOUP]T C---------------------------------------------------------------------CALL ELCPT(AA,BL,FINV,DETF,ECOUPT,NPN,NDV,NS,F9) C---------------------------------------------------------------------C END OF INTEGRATION POINT LOOP C---------------------------------------------------------------------80 CONTINUE

C TRANSFORM THE EKSTIF & ECOUPT MATRIX TO COUNT FOR SKEW BOUNDARIES IF(NSKEW.GT.0) CALL STFTRN(K,NDIM,NSKEW,EKSTIF,ECOUPT,NCONN,LT) C C C C C C C C C C

WRITE(LOUT1,*)’EKSTIF MATRIX OF’,K DO I1=1,NB WRITE(LOUT1,1104)(EKSTIF(I1,J1),J1=1,20) ENDDO WRITE(LOUT1,*)’EFLOW MATRIX’ DO I=1,NPN WRITE(LOUT1,1103)(EFLOW(I,J),J=1,NPN) ENDDO WRITE(LOUT1,*)’ECOUPT MATRIX’

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C C C

DO I3=1,NB WRITE(LOUT1,1103)(ECOUPT(I3,J3),J3=1,NPN) ENDDO

C---------------------------------------------------------------------C FORM STIFFNESS MATRIX SG FROM EKSTIF, ECOUPT AND EFLOW C---------------------------------------------------------------------CALL ASSEMPLE(SG,DA,GDT,P,EKSTIF,EFLOW,ECOUPT,NWL, 1 NPN,NDIM,NDN,LT,ICPL,ITER) II=0 DO J1=1,20 DO I1=1,J1 II=II+1 STIFF(I1,J1)=SG(II) STIFF(J1,I1)=STIFF(I1,J1) ENDDO ENDDO c WRITE(*,*)"4444" C C c IF(IOUTP.EQ.1.AND.K.EQ.5) then c WRITE(2,*)’EKSTIF MATRIX’ c DO I1=1,10 c WRITE(2,1104)(EKSTIF(I1,J1),J1=1,10) c ENDDO c C WRITE(2,*)’EFLOW MATRIX’ C DO I3=1,NPN C WRITE(2,1103)(EFLOW(I3,J3),J3=1,NPN),(ETE(I3,J3),J3=1,NPN) C ENDDO C C C C

WRITE(2,*)’ECOUPT MATRIX’ DO I3=1,16 WRITE(2,1103)(ECOUPT(I3,J2),J2=1,NPN),(RLT(I3,J3),J3=1,NPN) ENDDO

c

ENDIF 1101 FORMAT(6E10.3) 1103 FORMAT(8E10.3,/) 1104 FORMAT(20E10.3,/)

C C C C C C 1101

WRITE(LOUT1,*)’ELEMENT NO=’,K WRITE(LOUT1,*)’[D] MATRIX’ DO I=1,NS WRITE(LOUT1,1101)(D(I,J),J=1,NS) ENDDO FORMAT(6E10.3,/)


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C WRITE(LOUT1,*)’BLNR MATRIX’ C DO I=1,NS C WRITE(LOUT1,1102)(BL(I,J),J=1,40) C ENDDO C 1102 FORMAT(40E10.3,/) C WRITE(LOUT1,*)’FULL ELEMENT STIFFNESS MATRIX’ C DO I=1,68 C WRITE(LOUT1,1102)(STIFF(I,J),J=1,20) C ENDDO C 1102 FORMAT(20E10.3,/) C RETURN END C********************************************************************** SUBROUTINE FORMP(J,NDIM,NPN,NAC,DS,SFP,CARTD,SLL,LT) C********************************************************************** C FORMS CARTD MATRIX FOR AREA COORDS SLL(NAC) C IN TRIANGLE J FOR INTEGRATION POINT IP C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION SLL(4) DIMENSION DS(3,10),SFP(10),CARTD(3,NPMX) COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /PARS / PYI,ASMVL,ZERO COMMON /JACB / XJACI(3,3),DJACB C---------------------------------------------------------------------C CALCULATE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS) C---------------------------------------------------------------------CALL SHFNPP(LOUT1,SLL,NAC,DS,SFP,LT,1,J) C DO 35 IN=1,NPN DO 35 ID=1,NDIM SUM=ZERO C DO 30 JD=1,NDIM 30 SUM=SUM-DS(JD,IN)*XJACI(ID,JD) 35 CARTD(ID,IN)=SUM RETURN END C********************************************************************** SUBROUTINE JPC(J,NDIM,NPN,NAC,NPLAX,DS,CARTD,B,E,RN,AA,SLL,LT) C---------------------------------------------------------------------C CALCULATES SHAPE FUNCTIONS AND DERIVATIVES C FOR EXCESS PORE PRESSURE VARIATION C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’

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Appendix: Fortran Codes of CS-S DIMENSION DS(3,10),CARTD(3,NPMX),B(6,NB), 1 E(3,NPMX),RN(NB),AA(NPMX),SLL(4) COMMON /PARS / PYI,ASMVL,ZERO

C CALL FORMP(J,NDIM,NPN,NAC,DS,AA,CARTD,SLL,LT) C---------------------------------------------------------------------C FORM RN C---------------------------------------------------------------------NCOM=NDIM IF(NPLAX.EQ.1.AND.NCOM.EQ.2)NCOM=NDIM+1 C DO 30 IB=1,NB SUM=ZERO C DO 20 ID=1,NCOM 20 SUM=SUM+B(ID,IB) 30 RN(IB)=SUM C---------------------------------------------------------------------C FORM E C---------------------------------------------------------------------DO 50 IN=1,NPN DO 50 ID=1,NDIM 50 E(ID,IN)=CARTD(ID,IN) RETURN END C********************************************************************** SUBROUTINE ASSEMPLE(SG,DA,GDT,P,EKSTIF,EFLOW,ECOUPT,NWL,NPN, 1 NDIM,NDN,LT,ICPL,ITER) C********************************************************************** C FORM ELEMENT STIFFNESS MATRIX SG FROM EKSTIF, ECOUPT AND EFLOW C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION KP(29),KD(94),NXP(15),NXD(15) DIMENSION SG(KES),DA(MDOF),P(MDOF),EKSTIF(NB,NB), 1 EFLOW(NPMX,NPMX),ECOUPT(NB,NPMX),NWL(NPMX),GDT(NPMX) COMMON /PARS /PYI,ASMVL,ZERO COMMON/GLBSTF/LOCDOF(MDOF),JDIAG(MDOF),NKSIZ,NEQTNS C---------------------------------------------------------------------C INDEX TO ROWS/COLUMNS OF SG FOR ROWS/COLUMNS OF ETE C INDEX TO COLUMNS OF SG FOE COLUMNS OF ECOUPT (FOR CONSOLIDATION) C---------------------------------------------------------------------C----------ELEMENT TYPE 3 - LST---------------------------------------DATA KP(1),KP(2),KP(3)/ 1 3,6,9/


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C----------ELEMENT TYPE 5 - QUADRILATERAL-----------------------------DATA KP(4),KP(5),KP(6),KP(7)/ 1 3,6,9,12/ C----------ELEMENT TYPE 7 - CUST--------------------------------------DATA KP(8),KP(9),KP(10),KP(11),KP(12),KP(13),KP(14),KP(15), 1 KP(16),KP(17)/ 2 3,6,9,34,35,36,37,38,39,40/ C----------ELEMENT TYPE 9 - BRICK-------------------------------------DATA KP(18),KP(19),KP(20),KP(21),KP(22),KP(23),KP(24),KP(25)/ 2 4,8,12,16,20,24,28,32/ C----------ELEMENT TYPE 11 - TETRA-HEDRA------------------------------DATA KP(26),KP(27),KP(28),KP(29)/ 1 4,8,12,16/ C---------------------------------------------------------------------C INDEX TO FIRST DISPLACEMENT VARIABLE OF EACH NODE IN SG C INDEX TO ROWS/COLUMNS OF SG FROM ROWS/COLUMNS OF SS C INDEX TO ROWS OF SG FOR ROWS OF ECOUPT (FOR CONSOLIDATION ELEMENT) C---------------------------------------------------------------------C----------ELEMENT TYPE 1(2), 2(6), 4(8), 6(15)-----------------------DATA KD(1),KD(2),KD(3),KD(4),KD(5),KD(6),KD(7),KD(8),KD(9),KD(10), 2 KD(11),KD(12),KD(13),KD(14),KD(15)/ 3 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29/ C----------ELEMENT TYPE 8(20), 10(10)---------------------------------DATA KD(16),KD(17),KD(18),KD(19),KD(20),KD(21),KD(22),KD(23), 1 KD(24),KD(25),KD(26),KD(27),KD(28),KD(29),KD(30),KD(31), 2 KD(32),KD(33),KD(34),KD(35)/ 3 1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58/ C----------ELEMENT TYPE 3(6)------------------------------------------DATA KD(36),KD(37),KD(38),KD(39),KD(40),KD(41)/ 1 1,4,7,10,12,14/ C----------ELEMENT TYPE 5(8)------------------------------------------DATA KD(42),KD(43),KD(44),KD(45),KD(46),KD(47),KD(48),KD(49)/ 1 1,4,7,10,13,15,17,19/ C----------ELEMENT TYPE 7(15)-----------------------------------------DATA KD(50),KD(51),KD(52),KD(53),KD(54),KD(55),KD(56),KD(57), 1 KD(58),KD(59),KD(60),KD(61),KD(62),KD(63),KD(64)/ 2 1,4,7,10,12,14,16,18,20,22,24,26,28,30,32/ C----------ELEMENT TYPE 9(20)-----------------------------------------DATA KD(65),KD(66),KD(67),KD(68),KD(69),KD(70),KD(71),KD(72), 1 KD(73),KD(74),KD(75),KD(76),KD(77),KD(78),KD(79),KD(80), 2 KD(81),KD(82),KD(83),KD(84)/ 3 1,5,9,13,17,21,25,29,33,36,39,42,45,48,51,54,57,60,63,66/ C----------ELEMENT TYPE 11(10)----------------------------------------DATA KD(85),KD(86),KD(87),KD(88),KD(89), 1 KD(90),KD(91),KD(92),KD(93),KD(94)/ 2 1,5,9,13,17,20,23,26,29,32/ C----------------------------------------------------------------------

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C NXP AND NXD GIVE STARTING INDEX TO ARRAYS KP AND KD C RESPECTIVELY FOR DIFFERENT ELEMENT TYPES C---------------------------------------------------------------------DATA NXP(1),NXP(2),NXP(3),NXP(4),NXP(5),NXP(6),NXP(7), 1 NXP(8),NXP(9),NXP(10),NXP(11)/ 2 0,0,0,0,3,0,7,0,17,0,25/ DATA NXD(1),NXD(2),NXD(3),NXD(4),NXD(5),NXD(6),NXD(7), 1 NXD(8),NXD(9),NXD(10),NXD(11)/ 2 0,0,35,0,41,0,49,15,64,15,84/ C---------------------------------------------------------------------C---------- SIZE OF ARRAYS KP AND KD----------------------------------NKP=29 NKD=94 C---------------------------------------------------------------------INXD=NXD(LT) C----------BYPASS IF NOT COUPLED CONSOLIDATION IF(ICPL.EQ.0)GOTO 96 C---------------------------------------------------------------------C COUPLED CONSOLIDATION C---------------------------------------------------------------------INXP=NXP(LT) C----------CALCULATE RIGHT HAND SIDE FOR PORE PRESSURES DO 94 II=1,NPN N1=NWL(II) SUM=ZERO C DO 92 JJ=1,NPN N2=NWL(JJ) IF(ITER.EQ.1) THEN SUM=SUM+EFLOW(II,JJ)*DA(N2) ELSE SUM=0.0D0 ENDIF 92 CONTINUE LOCGLB=LOCDOF(N1) 94 P(LOCGLB)=P(LOCGLB)+SUM+GDT(II) C---------------------------------------------------------------------C FORM SG FROM EKSTIF C---------------------------------------------------------------------96 DO 150 J=1,NDN NJ=KD(J+INXD)-1 C DO 150 JD=1,NDIM NJA=NJ+JD JA=JD+(J-1)*NDIM NCN=NJA*(NJA-1)/2 C DO 150 I=1,NDN


309

NI=KD(I+INXD)-1 C DO 140 ID=1,NDIM NIA=NI+ID IA=ID+(I-1)*NDIM IF(NIA.GT.NJA)GOTO 140 LOC=NCN+NIA SG(LOC)=EKSTIF(IA,JA) 140 CONTINUE 150 CONTINUE C IF(ICPL.EQ.0)GOTO 200 C---------------------------------------------------------------------C SLOT ECOUPT C---------------------------------------------------------------------DO 160 JA=1,NPN NJA=KP(JA+INXP) NCN=NJA*(NJA-1)/2 C DO 160 I=1,NDN NI=KD(I+INXD)-1 C DO 160 ID=1,NDIM NIA=NI+ID IA=ID+(I-1)*NDIM LOC=NIA+NCN IF(NIA.GT.NJA)LOC=NIA*(NIA-1)/2+NJA 160 SG(LOC)=ECOUPT(IA,JA) C---------------------------------------------------------------------C SLOT EFLOW C---------------------------------------------------------------------DO 180 JE=1,NPN NJ=KP(JE+INXP) NCN=NJ*(NJ-1)/2 C DO 180 IE=1,JE NI=KP(IE+INXP) 180 SG(NI+NCN)=-EFLOW(IE,JE) 200 CONTINUE RETURN END C********************************************************************** SUBROUTINE LSTIFA(EKSTIF,B,D,DB,F9,NS) C********************************************************************** C ROUTINE TO CALCULATE D*B AND BT*D*B C FOR EACH INTEGRATION POINT C**********************************************************************

310


IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION EKSTIF(NB,NB),D(6,6),DB(NVRN,NB),B(6,NB) C---------------------------------------------------------------------C FORM D*B C---------------------------------------------------------------------CALL ZEROR2(DB,NVRN,NB) C DO 20 JJ=1,NB DO 20 II=1,NS DO 20 KK=1,NS 20 DB(II,JJ)=DB(II,JJ)+D(II,KK)*B(KK,JJ) C---------------------------------------------------------------------C FORM BT*D*B C---------------------------------------------------------------------DO 30 JJ=1,NB DO 30 II=1,JJ DO 30 KK=1,NS 30 EKSTIF(II,JJ)=EKSTIF(II,JJ)+DB(KK,JJ)*B(KK,II)*F9 DO 40 JJ=2,NB JJM1=JJ-1 DO 40 II=1,JJM1 40 EKSTIF(JJ,II)=EKSTIF(II,JJ) RETURN END C********************************************************************** SUBROUTINE SHFNPP(LOUT1,SLL,NAC,DS,SFP,LT,IFL,MUS) C********************************************************************** C SHAPE FUNCTIONS AND DERIVATIVES FOR PORE PRESSURE VARIATION C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION SFP(10),DS(3,10),SLL(4) C RL1=SLL(1) RL2=SLL(2) IF(NAC.LT.3)GOTO 10 RL3=SLL(3) IF(NAC.LT.4)GOTO 10 RL4=SLL(4) C 10 GOTO(80,80,13,80,25,80,37,80,49,80,71),LT WRITE(LOUT1,900)MUS,LT 900 FORMAT(/1X,7HELEMENT,I5,2X,22HIS OF UNKNOWN TYPE ***,I5,2X, 1 16H(ROUTINE SHFNPP)) STOP C----------------------------------------------------------------------


311

C LINEAR STRAIN TRIANGLE C---------------------------------------------------------------------13 IF(IFL.EQ.0)GO TO 23 DS(1,1)=1.0D0 DS(1,2)=0.0D0 DS(1,3)=-1.0D0 DS(2,1)=0.0D0 DS(2,2)=1.0D0 DS(2,3)=-1.0D0 C 23 SFP(1)=RL1 SFP(2)=RL2 SFP(3)=RL3 RETURN C---------------------------------------------------------------------C QUADRILATERAL ELEMENT C---------------------------------------------------------------------25 IF(IFL.EQ.0)GO TO 35 DS(1,1)=0.250D0*(RL2-1.0D0) DS(1,2)=-0.250D0*(RL2-1.0D0) DS(1,3)=0.250D0*(RL2+1.0D0) DS(1,4)=-0.250D0*(RL2+1.0D0) DS(2,1)=0.250D0*(RL1-1.0D0) DS(2,2)=-0.250D0*(RL1+1.0D0) DS(2,3)=0.250D0*(RL1+1.0D0) DS(2,4)=-0.250D0*(RL1-1.0D0) C 35 SFP(1)=0.250D0*(RL1-1.0D0)*(RL2-1.0D0) SFP(2)=-0.250D0*(RL1+1.0D0)*(RL2-1.0D0) SFP(3)=0.250D0*(RL1+1.0D0)*(RL2+1.0D0) SFP(4)=-0.250D0*(RL1-1.0D0)*(RL2+1.0D0) RETURN C---------------------------------------------------------------------C CUBIC VARIATION IN PORE-PRESSURE C---------------------------------------------------------------------37 C1=9.0D0/2.0D0 C2=27.0D0/2.0D0 C3=27.0D0 T11=RL1-1.0D0/3.0D0 T12=RL1-2.0D0/3.0D0 T21=RL2-1.0D0/3.0D0 T22=RL2-2.0D0/3.0D0 T31=RL3-1.0D0/3.0D0 T32=RL3-2.0D0/3.0D0 IF(IFL.EQ.0)GO TO 40 C DS(1,1)=C1*(T11*T12+RL1*(T11+T12)) DS(1,2)=0.0D0

312

Appendix: Fortran Codes of CS-S DS(1,3)=-C1*(T31*T32+RL3*(T31+T32)) DS(1,4)=C2*RL2*(RL1+T11) DS(1,5)=C2*RL2*T21 DS(1,6)=-C2*RL2*T21 DS(1,7)=-C2*RL2*(RL3+T31) DS(1,8)=C2*RL3*T31-C2*RL1*(RL3+T31) DS(1,9)=C2*RL3*(RL1+T11)-C2*RL1*T11 DS(1,10)=C3*RL2*RL3-C3*RL2*RL1

C DS(2,1)=0.0D0 DS(2,2)=C1*(T21*T22+RL2*(T21+T22)) DS(2,3)=-C1*(T31*T32+RL3*(T31+T32)) DS(2,4)=C2*RL1*T11 DS(2,5)=C2*RL1*(RL2+T21) DS(2,6)=C2*RL3*(RL2+T21)-C2*RL2*T21 DS(2,7)=C2*RL3*T31-C2*RL2*(RL3+T31) DS(2,8)=-C2*RL1*(RL3+T31) DS(2,9)=-C2*RL1*T11 DS(2,10)=C3*RL1*RL3-C3*RL1*RL2 C 40 SFP(1) =C1*RL1*T11*T12 SFP(2) =C1*RL2*T21*T22 SFP(3) =C1*RL3*T31*T32 SFP(4) =C2*RL1*RL2*T11 SFP(5) =C2*RL1*RL2*T21 SFP(6) =C2*RL2*RL3*T21 SFP(7) =C2*RL2*RL3*T31 SFP(8) =C2*RL1*RL3*T31 SFP(9) =C2*RL1*RL3*T11 SFP(10)=C3*RL1*RL2*RL3 RETURN C---------------------------------------------------------------------C PORE-PRESSURE SHAPE FUNCTIONS AND DERIVATIVES FOR BRICK ELEMENT C---------------------------------------------------------------------49 IF(IFL.EQ.0)GO TO 45 C DERIVATIVES DS(1,1)=-0.1250D0*(1.0D0-RL2)*(1.0D0+RL3) DS(1,2)= 0.1250D0*(1.0D0-RL2)*(1.0D0+RL3) DS(1,3)= 0.1250D0*(1.0D0+RL2)*(1.0D0+RL3) DS(1,4)=-0.1250D0*(1.0D0+RL2)*(1.0D0+RL3) DS(1,5)=-0.1250D0*(1.0D0-RL2)*(1.0D0-RL3) DS(1,6)= 0.1250D0*(1.0D0-RL2)*(1.0D0-RL3) DS(1,7)= 0.1250D0*(1.0D0+RL2)*(1.0D0-RL3) DS(1,8)=-0.1250D0*(1.0D0+RL2)*(1.0D0-RL3) C DS(2,1)=-0.1250D0*(1.0D0-RL1)*(1.0D0+RL3) DS(2,2)=-0.1250D0*(1.0D0+RL1)*(1.0D0+RL3) DS(2,3)= 0.1250D0*(1.0D0+RL1)*(1.0D0+RL3)


313

DS(2,4)= 0.1250D0*(1.0D0-RL1)*(1.0D0+RL3) DS(2,5)=-0.1250D0*(1.0D0-RL1)*(1.0D0-RL3) DS(2,6)=-0.1250D0*(1.0D0+RL1)*(1.0D0-RL3) DS(2,7)= 0.1250D0*(1.0D0+RL1)*(1.0D0-RL3) DS(2,8)= 0.1250D0*(1.0D0-RL1)*(1.0D0-RL3) C DS(3,1)= 0.1250D0*(1.0D0-RL1)*(1.0D0-RL2) DS(3,2)= 0.1250D0*(1.0D0+RL1)*(1.0D0-RL2) DS(3,3)= 0.1250D0*(1.0D0+RL1)*(1.0D0+RL2) DS(3,4)= 0.1250D0*(1.0D0-RL1)*(1.0D0+RL2) DS(3,5)=-0.1250D0*(1.0D0-RL1)*(1.0D0-RL2) DS(3,6)=-0.1250D0*(1.0D0+RL1)*(1.0D0-RL2) DS(3,7)=-0.1250D0*(1.0D0+RL1)*(1.0D0+RL2) DS(3,8)=-0.1250D0*(1.0D0-RL1)*(1.0D0+RL2) C SHAPE FUNCTIONS 45 SFP(1)=0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)*(1.0D0+RL3) SFP(2)=0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)*(1.0D0+RL3) SFP(3)=0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)*(1.0D0+RL3) SFP(4)=0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)*(1.0D0+RL3) SFP(5)=0.1250D0*(1.0D0-RL1)*(1.0D0-RL2)*(1.0D0-RL3) SFP(6)=0.1250D0*(1.0D0+RL1)*(1.0D0-RL2)*(1.0D0-RL3) SFP(7)=0.1250D0*(1.0D0+RL1)*(1.0D0+RL2)*(1.0D0-RL3) SFP(8)=0.1250D0*(1.0D0-RL1)*(1.0D0+RL2)*(1.0D0-RL3) RETURN C---------------------------------------------------------------------C TETRA-HEDRA ELEMENT C---------------------------------------------------------------------71 CONTINUE WRITE(LOUT1,910)MUS,LT 910 FORMAT(/1X,7HELEMENT,I5,2X,14HIS OF TYPE ***,I5,2X, 1 31HNOT IMPLEMENTED (ROUTINE SHAPE)) 80 RETURN END C********************************************************************** SUBROUTINE ELFLOW(NDIM,NPN,FINV,DETF,E,F9,EFLOW, 1 PERM,GAMMAW,DTIME) C********************************************************************** C THIS SUBPROGRAM CALCULATES THE ELEMENT FLOW MATRIX C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION FINV(3,3),E(3,NPMX),EFLOW(NPMX,NPMX),PERM(3) DIMENSION PE(3,NPMX),TEMP(3,3) C CALL ZEROR2(TEMP,3,3) CALL ZEROR2(PE,3,NPMX)

314


C CALCULATE THE FACTOR [CIJ]-1 * [CIJ]-1 DO I=1,NDIM DO J=1,NDIM TEM = 0.0D0 DO K=1,NDIM TEM = TEM + FINV(I,K)*FINV(J,K) END DO TEMP(I,J)=TEM END DO END DO C TEMP1=0.0D0 DO I=1,NDIM DO J=1,NDIM TEMP1=TEMP1+TEMP(I,J)*TEMP(I,J)/NDIM ENDDO ENDDO CNST=F9*DETF*TEMP1 C C

FORM

PERM*E

DO 10 JJ=1,NPN DO 10 IM=1,NDIM 10 PE(IM,JJ)=PERM(IM)*E(IM,JJ) C-------------------------------------------------------------------C FORM [EFLOW] C-------------------------------------------------------------------DO 20 II=1,NPN DO 20 JJ=1,NPN DO 20 KK=1,NDIM 20 EFLOW(II,JJ)=EFLOW(II,JJ)+E(KK,II)*PE(KK,JJ)*DTIME*CNST/GAMMAW RETURN END C ********************************************************************* SUBROUTINE ELCPT(AA,BL,FINV,DETF,ECOUPT,NPN,NDV,NS,F9) C ********************************************************************* C THIS SUBPROGRAM CALCULATES THE ELEMENT COUPLING MATRIX C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION FINV(3,3),AA(NPMX),TEMP(3,3),BL(6,NB) DIMENSION ECOUPT(NB,NPMX),TEMP1(NVRN),RN(NB) C CALL ZEROR1(TEMP1,NVRN)


315

CALL ZEROR2(TEMP,3,3) CALL ZEROR1(RN,NB) C CALCULATE [FINV]*[FINV]*KRONICOR DELTA =[TEMP] DO 10 I=1,3 DO 10 J=1,3 TEM=0.0D0 DO 10 K=1,3 TEM=TEM+FINV(I,K)*FINV(J,K) 10 TEMP(I,J)=TEM C CONVERT [TEMP] INTO VECTOR, MULTIPLY BY BL, PUT IN RN TEMP1(1)=TEMP(1,1) TEMP1(2)=TEMP(2,2) TEMP1(3)=TEMP(3,3) TEMP1(4)=TEMP(1,2) C

C C

DO 20 IB=1,NB SUM=0.0D0 DO 30 ID=1,NS 30 SUM=SUM+BL(ID,IB)*TEMP1(ID) 20 RN(IB)=SUM CALCULATE THE ELEMENT COUPLING MATRIX [ECOUP]T=RN*AA*DETF*F9 DO 60 II=1,NDV DO 60 JJ=1,NPN 60 ECOUPT(II,JJ)=ECOUPT(II,JJ)+RN(II)*AA(JJ)*F9*DETF RETURN END

C ********************************************************************* SUBROUTINE BLNR1(CARTD,BL1,NDIM,NDN) C ********************************************************************* C THIS SUBPROGRAM CALCULATES THE B\_L* MATRIX C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION CARTD(3,NDMX),BL1(9,NB) C INITIALIZE B\_L MATRIX TO ZERO CALL ZEROR2(BL1,9,NB) C ASSIGN THE APPROPRIATE VALUES TO THE APPROPRIATE ELEMENTS OF B\_L C IF(NDIM.NE.2) GO TO 10 C

316


C

2-DIMENSIONAL ELEMENT DO IELN=1,NDN IDOF1=2*IELN-1 IDOF2=IDOF1+1 BL1(1,IDOF1)=CARTD(1,IELN) BL1(2,IDOF1)=CARTD(2,IELN) BL1(3,IDOF2)=CARTD(1,IELN) BL1(4,IDOF2)=CARTD(2,IELN) ENDDO 10

C

IF(NDIM.NE.3) GO TO 50 3-DIMENSIONAL ELEMENT

DO IELN=1,NDN IDOF1=3*IELN-2 IDOF2=IDOF1+1 IDOF3=IDOF2+1 BL1(1,IDOF1)=CARTD(1,IELN) BL1(2,IDOF1)=CARTD(2,IELN) BL1(3,IDOF1)=CARTD(3,IELN) BL1(4,IDOF2)=CARTD(1,IELN) BL1(5,IDOF2)=CARTD(2,IELN) BL1(6,IDOF2)=CARTD(3,IELN) BL1(7,IDOF3)=CARTD(1,IELN) BL1(8,IDOF3)=CARTD(2,IELN) BL1(9,IDOF3)=CARTD(3,IELN) ENDDO 50 CONTINUE RETURN END C ********************************************************************* SUBROUTINE BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL) C ********************************************************************* C THIS SUBPROGRAM IN CALCULATES THE B\_L MATRIX C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION SHFN(20),BL(6,NB),CARTD(3,NDMX) C INITIALIZE B\_L MATRIX TO ZERO CALL ZEROR2(BL,6,NB) C IF(NDIM.NE.2) GO TO 10 C---------------------------------------------------------------------C 2-DIMENSIONAL ELEMENT C---------------------------------------------------------------------DO IELN=1,NDN


20

317

IDOF1=2*IELN-1 IDOF2=IDOF1+1 BL(1,IDOF1)=CARTD(1,IELN) BL(2,IDOF2)=CARTD(2,IELN) IF(NPLAX.NE.1) GO TO 20 BL(3,IDOF1)=SHFN(IELN)*RI BL(4,IDOF1)=CARTD(2,IELN) BL(4,IDOF2)=CARTD(1,IELN) ENDDO

10 IF(NDIM.NE.3) GO TO 50 C---------------------------------------------------------------------C 3-DIMENSIONAL ELEMENT C---------------------------------------------------------------------DO IELN=1,NDN IDOF1=IELN*3-2 IDOF2=IDOF1+1 IDOF3=IDOF2+1 BL(1,IDOF1)=CARTD(1,IELN) BL(2,IDOF2)=CARTD(2,IELN) BL(3,IDOF3)=CARTD(3,IELN) BL(4,IDOF1)=CARTD(2,IELN) BL(4,IDOF2)=CARTD(1,IELN) BL(5,IDOF2)=CARTD(3,IELN) BL(5,IDOF3)=CARTD(2,IELN) BL(6,IDOF1)=CARTD(3,IELN) BL(6,IDOF3)=CARTD(1,IELN) ENDDO 50 CONTINUE RETURN END C ********************************************************************* SUBROUTINE BNLNRS(SHFN,CARTD,RI,BNLS,NDIM,NDN,NPLAX) C ********************************************************************* C THIS SUBPROGRAM CALCULATES GEOMETRIC NONLIN STRESS-DISP MATRIX B\_NLS. C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION SHFN(20),CARTD(3,NDMX), . BNLS(9,NB)

C

INITIALIZE B\_NL MATRIX CALL ZEROR2(BNLS,9,NB)

318 C C C

Appendix: Fortran Codes of CS-S ASSIGN THE APPROPRIATE VALUES TO THE SPECIFIC ELEMENTS OF B\_NLS IF(NDIM.NE.2) GO TO 10 2-DIMENSIONAL ELEMENT DO IELN=1,NDN IDOF1=2*IELN-1 IDOF2=IDOF1+1 BNLS(1,IDOF1)=CARTD(1,IELN) BNLS(2,IDOF1)=CARTD(2,IELN) IF(NPLAX.NE.1) GO TO 20 BNLS(3,IDOF1)=SHFN(IELN)*RI BNLS(4,IDOF2)=CARTD(1,IELN) BNLS(5,IDOF2)=CARTD(2,IELN) GO TO 30 20 BNLS(3,IDOF2)=CARTD(1,IELN) BNLS(4,IDOF2)=CARTD(2,IELN) 30 CONTINUE ENDDO

C C

C C C C C

10 IF(NDIM.NE.3) GO TO 50 3-DIMENSIONAL ELEMENT DO IELN=1,NDN IDOF1=IELN*3-2 IDOF2=IDOF1+1 IDOF3=IDOF2+1 BNLS(1,IDOF1)=CARTD(1,IELN) BNLS(2,IDOF1)=CARTD(2,IELN) BNLS(3,IDOF1)=CARTD(3,IELN) BNLS(4,IDOF2)=CARTD(1,IELN) BNLS(5,IDOF2)=CARTD(2,IELN) BNLS(6,IDOF2)=CARTD(3,IELN) BNLS(7,IDOF3)=CARTD(1,IELN) BNLS(8,IDOF3)=CARTD(2,IELN) BNLS(9,IDOF3)=CARTD(3,IELN) END DO 50 CONTINUE RETURN END ********************************************************************* SUBROUTINE CONVERT(S,SPK,NDIM,NPLAX) ********************************************************************* THIS SUBPROGRAM CONVERTS THE S VECTOR CONSISTING OF 4 COMPONENTS INTO SPK MATRIX OF 5X5 -------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION S(6),SPK(9,9)

Appendix: Fortran Codes of CS-S C INITIALIZE SPK TO ZERO DO I=1,9 DO J=1,9 SPK(I,J)=0.0D0 ENDDO ENDDO

C ASSIGN THE ELEMENTS OF S APPROPRIATELY IN SPK IF(NDIM.NE.2) GO TO 10 C 2-DIMENSIONAL ELEMENT SPK(1,1)=S(1) SPK(2,2)=S(2) IF(NPLAX.NE.1) GO TO 20 SPK(3,3)=S(3) SPK(4,4)=S(1) SPK(5,5)=S(2) SPK(4,5)=S(4) SPK(5,4)=S(4) SPK(1,2)=S(4) SPK(2,1)=S(4) GO TO 10 20 SPK(3,3)=S(1) SPK(4,4)=S(2) SPK(1,2)=S(3) SPK(2,1)=S(3) SPK(3,4)=S(3) SPK(4,3)=S(3) 10 IF(NDIM.NE.3) GO TO 40 DO K=1,NDIM K1=3*K-2 K2=K1+1 K3=K1+2 SPK(K1,K1)=S(1) SPK(K2,K2)=S(2) SPK(K3,K3)=S(3) SPK(K1,K2)=S(4) SPK(K2,K1)=S(4) SPK(K2,K3)=S(5) SPK(K3,K2)=S(5) SPK(K1,K3)=S(6) SPK(K3,K1)=S(6) ENDDO 40 CONTINUE RETURN END

319

320


C ********************************************************************* SUBROUTINE BTDB(B,D,ESTIF,CNSTIP,NS) C ********************************************************************* C THIS SUBPROGRAM CALCULATE THE MATRIX PRODUCT B\^T D B GIVEN B AND D C MATRICES. C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION B(9,NB),D(9,9),ESTIF(NB,NB),TEMP(9,NB) DO I=1,NS DO J=1,NB TEM=0.0D0 DO K=1,NS TEM=TEM+D(I,K)*B(K,J) ENDDO TEMP(I,J)=TEM ENDDO ENDDO DO I=1,NB DO J=1,NB TEM=0.0D0 DO K=1,NS TEM=TEM+B(K,I)*TEMP(K,J) ENDDO ESTIF(I,J)=ESTIF(I,J)+TEM*CNSTIP ENDDO ENDDO RETURN END

C ********************************************************************* SUBROUTINE BNLNR(EDISP,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX) C ********************************************************************* C THIS SUBPROGRAM CALCULATES THE NONLIN STRESS-DISP. MATRIX B\_NL C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION SHFN(20),CARTD(3,NDMX),EDISP(NB),BNL(6,NB) DIMENSION BB(NVRN+1,NB),C(NVRN+1) NS1=NS+1 C INITIALIZE MATRICES CALL ZEROR2(BB,NVRN+1,NB) CALL ZEROR2(BNL,6,NB) IF(NDIM.NE.2) GO TO 10


321

C---------------------------------------------------------------------C 2-DIMENSIONAL ELEMENT C---------------------------------------------------------------------C FIND THE INTERMEDIATE MATRIX BB DO IELN=1,NDN IDOF1=IELN*2-1 IDOF2=IDOF1+1 BB(1,IDOF1)=CARTD(1,IELN) BB(2,IDOF2)=CARTD(2,IELN) IF(NPLAX.NE.1) GO TO 20 BB(3,IDOF1)=SHFN(IELN)*RI 20 BB(4,IDOF1)=CARTD(2,IELN) BB(5,IDOF2)=CARTD(1,IELN) ENDDO C FIND [C]=[BB][U] DO 1 I=1,NS1 C(I)=0.0D0 DO 1 J=1,NB 1 C(I)=C(I)+BB(I,J)*EDISP(J) C CALCULATE BNL MARIX DO IELN=1,NDN IDOF1=IELN*2-1 IDOF2=IDOF1+1 BNL(1,IDOF1)=C(1)*CARTD(1,IELN) BNL(1,IDOF2)=C(5)*CARTD(1,IELN) BNL(2,IDOF1)=C(4)*CARTD(2,IELN) BNL(2,IDOF2)=C(2)*CARTD(2,IELN) IF(NPLAX.NE.1) GO TO 15 BNL(3,IDOF1)=C(3)*SHFN(IELN)*RI 15 BNL(4,IDOF1)=C(4)*CARTD(1,IELN)+C(1)*CARTD(2,IELN) BNL(4,IDOF2)=C(2)*CARTD(1,IELN)+C(5)*CARTD(2,IELN) ENDDO C 10 IF(NDIM.NE.3) GO TO 100 C---------------------------------------------------------------------C 3-DIMENSIONAL ELEMENT C---------------------------------------------------------------------C

FIND THE INTERMEDIATE MATRIX BB DO IELN=1,NDN IDOF1=IELN*3-2 IDOF2=IDOF1+1 IDOF3=IDOF2+1

322


BB(1,IDOF1)=CARTD(1,IELN) BB(2,IDOF2)=CARTD(2,IELN) BB(3,IDOF3)=CARTD(3,IELN) ENDDO 100 CONTINUE RETURN END C********************************************************************** SUBROUTINE DEFGRAD(BL1,F,FINV,DETF,EDISPINC,NDIM,LINR) C********************************************************************** C THIS SUBPROGRAM CALCULATES THE DEFORMATION GRADIENT MATRIX [F], C [FINV] AND DET[F] C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION EDISPINC(NB),BL1(9,NB) DIMENSION UDRV(9),F(3,3),FINV(3,3),ff(3,3) NDD=NDIM*NDIM CALL ZEROR1(UDRV,9) CALL ZEROR2(F,3,3) CALL ZEROR2(Ff,3,3) CALL ZEROR2(FINV,3,3) C---------------------------------------------------------------------C FIND THE DERIVATIVES OF DISPLACEMENTS {U1,1 U1,2 U2,1 U2,2} FOR 2-D C AND {U1,1 U1,2 U1,3 U2,1 U2,2 U2,3 U3,1 U3,2 U3,3 } FOR 3-D C---------------------------------------------------------------------DO I=1,NDD !ndd=ndim*ndim TEMP=0.0D0 DO J=1,NB !NB=60 from parm.for TEMP=TEMP+BL1(I,J)*EDISPINC(J) !BL1=linear strain-displ matrix !edispinc=element incremental displacement=EDINC. !temp=accumulated strain???? Wrong logic? ENDDO UDRV(I)=TEMP ENDDO C

FIND THE DEFORMATION GRADIENT F DO I=1,3 DO J=1,3 F(I,J)=0.0D0 FINV(I,J)=0.0D0 IF(I.EQ.J)THEN F(I,J)=1.0D0 FINV(I,J)=1.0D0

Appendix: Fortran Codes of CS-S ENDIF ENDDO ENDDO IF(LINR.GT.0) THEN IF(NDIM.NE.2) GO TO 10 2-DIMENSIONAL ELEMENT F(1,1)=1.0D0+UDRV(1) F(2,2)=1.0D0+UDRV(4) F(3,3)=1.0D0 F(1,2)=UDRV(2) F(2,1)=UDRV(3) GO TO 101

C

C 10

3-DIMENSIONAL ELEMENT F(1,1)=1.0D0+UDRV(1) F(1,2)=UDRV(2) F(1,3)=UDRV(3) F(2,1)=UDRV(4) F(2,2)=1.0D0+UDRV(5) F(2,3)=UDRV(6) F(3,1)=UDRV(7) F(3,2)=UDRV(8) F(3,3)=1.0D0+UDRV(9)

101 CONTINUE END IF

C C

FIND THE INVERSE OF THE DEFORMATION GRADIENT FINV IF(NDIM.NE.2) GO TO 15 2-DIMENSIONAL ELEMENT DETF=F(1,1)*F(2,2)-F(1,2)*F(2,1) FINV(1,1)=F(2,2)/DETF FINV(2,2)=F(1,1)/DETF FINV(3,3)=1.0D0 FINV(1,2)=-F(1,2)/DETF FINV(2,1)=-F(2,1)/DETF GO TO 100

C 15

3-DIMENSIONAL ELEMENT FINV(1,1)= (F(2,2)*F(3,3)-F(2,3)*F(3,2)) FINV(1,2)=-(F(1,2)*F(3,3)-F(1,3)*F(3,2)) FINV(1,3)= (F(1,2)*F(2,3)-F(1,3)*F(2,2))

C FINV(2,1)=-(F(2,1)*F(3,3)-F(2,3)*F(3,1)) FINV(2,2)= (F(1,1)*F(3,3)-F(1,3)*F(3,1)) FINV(2,3)=-(F(1,1)*F(2,3)-F(1,3)*F(2,1))

!DET(F) IS J

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C FINV(3,1)= (F(2,1)*F(3,2)-F(2,2)*F(3,1)) FINV(3,2)=-(F(1,1)*F(3,2)-F(1,2)*F(3,1)) FINV(3,3)= (F(1,1)*F(2,2)-F(2,1)*F(1,2)) C DETF=F(1,1)*FINV(1,1)+F(1,2)*FINV(2,1)+ F(1,3)*FINV(3,1)

1 C

DO 35 ID=1,NDIM DO 35 JD=1,NDIM 35 FINV(ID,JD)=FINV(ID,JD)/DETF 100 CONTINUE c c

DETF=Finv(1,1)*Finv(2,2)-Finv(1,2)*Finv(2,1) DETF=1.0d0 !DET(F) IS J

!DET(F) IS J

RETURN END C ********************************************************************* SUBROUTINE ADDBMAT(BL,BNL) C ********************************************************************* IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION BL(6,NB),BNL(6,NB) DO I=1,6 DO J=1,NB BL(I,J)=BL(I,J)+BNL(I,J) ENDDO ENDDO RETURN END C********************************************************************** SUBROUTINE STFTRN(IELEM,NDIM,NSKEW,EKSTIF,ECOUPT,NCONN,LT) C********************************************************************** C THIS SUBRPROGRAM TRANFORMS THE STIFFNESS MATRIX & COUPLING MATRIX C TO ACCOUNT FOR THE ROTATION OF DOFS IN THE SKEW BOUNDARIES C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /ELINF / LINFO(50,15) COMMON/SKBC /ISPB(20),DIRCOS(20,3) DIMENSION EKSTIF(NB,NB),ECOUPT(NB,NPMX),ROTMAT(3,3) DIMENSION NCONN(NTPE,MEL),TEMP(3),TEMP1(3)

Appendix: Fortran Codes of CS-S C NDPT =LINFO(1,LT) NEDOF=LINFO(16,LT) NPN=LINFO(6,LT) C DO 100 I=1,NDPT NODE=ABS(NCONN(I,IELEM)) DO ISKEW=1,NSKEW INODE=ISPB(ISKEW) IF(INODE.EQ.NODE) THEN K1=NDIM*(I-1) CALL ROTM(DIRCOS,ROTMAT,NDIM,ISKEW) DO J = 1,NEDOF DO K = 1,NDIM TEMP( K ) = 0.0D0 DO IDOF = 1,NDIM ID = K1 + IDOF TEMP(K) = TEMP(K)+EKSTIF(J,ID)*ROTMAT(IDOF,K) ENDDO ENDDO DO K = 1,NDIM ID = K1 + K EKSTIF(J,ID) = TEMP( K ) ENDDO ENDDO C DO J = 1,NEDOF DO K = 1,NDIM TEMP(K) = 0.0D0 DO IDOF = 1,NDIM ID = K1 + IDOF TEMP(K) = TEMP(K)+EKSTIF(ID,J)*ROTMAT(IDOF,K) ENDDO ENDDO DO K = 1, NDIM ID = K1 + K EKSTIF(ID,J) =TEMP(K) ENDDO ENDDO C DO J = 1,NPN DO K = 1,NDIM TEMP1(K)= 0.0D0 DO IDOF = 1,NDIM

325

326

Appendix: Fortran Codes of CS-S ID = K1 + IDOF TEMP1(K)= TEMP1(K)+ECOUPT(ID,J)*ROTMAT(IDOF,K) ENDDO ENDDO DO K = 1, NDIM ID = K1 + K ECOUPT(ID,J)=TEMP1(K) ENDDO ENDDO

C ENDIF ENDDO 100 CONTINUE RETURN END SUBROUTINE ROTBC(R,NW,NDIM,NSKEW,NUM) C********************************************************************* C THIS SUBPROGRAM ROTATE BACK THE OBTAINED CURRENT DISPLACEMENTS TO C THE ORIGINAL COORD. SYSTEM AT THE SKEW BOUNDARY. C---------------------------------------------------------------------C 1 CALLED BY RDSTRS IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON/SKBC /ISPB(20),DIRCOS(20,3) DIMENSION R(MDOF),NW(MNODES+1) DIMENSION TEMP(6),ROTMAT(3,3) C DO ISKEW=1,NSKEW INODE=ISPB(ISKEW) ICODE=NW(INODE)-1 C C

----- Call ROTM ! Cal Rotation Matrix CALL ROTM(DIRCOS,ROTMAT,NDIM,ISKEW) DO I=1,NDIM TEMP1=0.0D0 DO J=1,NDIM LOC=ICODE+J IF(NUM.GT.0) TEMP1=TEMP1+R(LOC)*ROTMAT(J,I) IF(NUM.LT.0) TEMP1=TEMP1+R(LOC)*ROTMAT(I,J)


327

ENDDO TEMP(I)=TEMP1 ENDDO C DO I=1,NDIM LOC=ICODE+I R(LOC)=TEMP(I) ENDDO ENDDO RETURN END

SUBROUTINE ROTM(DIRCOS,ROTMAT,NDIM,ISKEW) C********************************************************************** C THIS SUBPROGRAM FINDS THE ROTATION MATRIX NEEDED FOR ROTATING C THE DOFS W.R.T. THAT NODE C---------------------------------------------------------------------C 1 CALLED BY ROTBC C 2 CALLED BY ROTBC BY EQLOD IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION DIRCOS(20,3),ROTMAT(3,3)

C

IF(NDIM.EQ.3) GO TO 10 ASSIGN COS(TH\_X) & COS(TH\_Y) FOR TWO DIMENSION ROTMAT(1,1)= DIRCOS(ISKEW,1) ROTMAT(2,1)=-DIRCOS(ISKEW,2) ROTMAT(1,2)= DIRCOS(ISKEW,2) ROTMAT(2,2)= DIRCOS(ISKEW,1) 10 RETURN END

C********************************************************************** SUBROUTINE DCON(I,IET,MAT,PR,D,NDIM,BK) C********************************************************************** C CALCULATES STRESS-STRAIN MATRIX FOR ANISOTROPIC ELASTICITY C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION MAT(MEL),D(6,6),PR(NPR,NMT)

328


C KM=MAT(I) AN=PR(1,KM)/PR(2,KM) A=PR(2,KM)/((1.0D0+PR(3,KM))*(1.0D0-PR(3,KM)-2.0D0*AN*PR(4,KM)* 1 PR(4,KM))) D(1,1)=A*AN*(1.0D0-AN*PR(4,KM)*PR(4,KM)) D(1,2)=A*AN*PR(4,KM)*(1.0D0+PR(3,KM)) D(1,3)=A*AN*(PR(3,KM)+AN*PR(4,KM)*PR(4,KM)) D(2,1)=D(1,2) D(2,2)=A*(1.0D0-PR(3,KM)*PR(3,KM)) D(2,3)=D(1,2) D(3,1)=D(1,3) D(3,2)=D(2,3) D(3,3)=D(1,1) D(4,4)=PR(5,KM) BK=(D(2,2)+2.0D0*D(2,1))/3.0D0 IF(NDIM.EQ.2)GOTO 5 D(5,5)=PR(5,KM) D(6,6)=PR(5,KM) 5 IF(IET.EQ.0) GO TO 20 C DO 10 J=1,3 DO 10 JJ=1,3 10 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK 20 RETURN END C********************************************************************** SUBROUTINE DLIN(I,IET,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK) C********************************************************************** C CALCULATES STRESS-STRAIN MATRIX FOR LINEAR ELASTIC C BEHAVIOUR WHEN ELASTIC PROPERTIES VARY LINEARLY WITH DEPTH C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION ELCOD(3,NDMX),SHFN(20),D(6,6) DIMENSION MAT(MEL),PR(NPR,NMT) COMMON /PARS / PYI,ASMVL,ZERO C KM=MAT(I) CC IPA=IP+INDX YY=ZERO DO 5 IN=1,NDN 5 YY=YY+SHFN(IN)*ELCOD(2,IN) E=PR(1,KM)+PR(3,KM)*(PR(2,KM)-YY) G=E/(2.0D0*(1.0D0+PR(4,KM))) A=E/((1.0D0+PR(4,KM))*(1.0D0-2.0D0*PR(4,KM)))


329

BK=E/(3.0D0*(1.0D0-2.0D0*PR(4,KM))) D(1,1)=A*(1.0D0-PR(4,KM)) D(1,2)=A*PR(4,KM) D(1,3)=D(1,2) D(2,1)=D(1,2) D(2,2)=D(1,1) D(2,3)=D(1,3) D(3,1)=D(1,3) D(3,2)=D(2,3) D(3,3)=D(1,1) D(4,4)=G IF(NDIM.EQ.2)GOTO 8 D(5,5)=G D(6,6)=G 8 IF(IET.EQ.0)RETURN DO 10 J=1,3 DO 10 JJ=1,3 10 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK RETURN END C********************************************************************** SUBROUTINE DMCAM(IP,I,IET,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB, +SALPHA,A,H) C********************************************************************** C CALCULATES STRESS-STRAIN MATRIX [Dep] FOR MODIFIED CAM-CLAY C********************************************************************** C IP=Gauss point, I=element number, IET=iteration number C ITER=iteration number, JS=increment number IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ C I=ELEMENT NUMBER COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !back stress variable COMMON/NORMFACT/XI DIMENSION STRESS(NVRS,NIP,MEL),D(6,6),MAT(MEL) DIMENSION S(6),A(6),B(6),PR(NPR,NMT) DIMENSION DUM4(4),DUM5(4),HPA(4,4),DEPSI(4),DAB(3,3),DABI(3,3) DIMENSION SALPHA(4),SALPH22(4),SALPH11(4),BB21(4),BB31(4) DIMENSION DUM1(4),DELALPH(4) dimension epsiequi(2,170,4),epsiaccu(2,170,4,3,3),graddab(3,3) C I=ELEMENT NUMBER C KM=MAT(I) PRM=PR(4,KM) ICOD=MCODE(IP,I)

330


SX=STRESS(1,IP,I) !new sigma x SY=STRESS(2,IP,I) !new sigma y and so and so SZ=STRESS(3,IP,I) TXY=STRESS(4,IP,I) E=STRESS(NS+2,IP,I) !Void ratio PC=ABS(STRESS(NS+3,IP,I)) c Correction of stress for rate dependency ------- 08/13/03 call rate(epsiequi,epsiaccu,i,ip) c call rate1(dabequi,dab) viscos=pr(1,2) pm1=pr(2,2) !pm1=m1 just for making real number SX=STRESS(1,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1) ! SY=STRESS(2,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1) SZ=STRESS(3,IP,I)*(1+(epsiequi(1,i,ip)/viscos)**pm1) delsx=STRESS(1,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1) delsy=STRESS(2,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1) delsz=STRESS(3,IP,I)*((epsiequi(1,i,ip)/viscos)**pm1) TXY=STRESS(4,IP,I)+0.5*(delsx-delsy) !! correct? c Correction of py for rate dependency ----------- 08/07/03 c viscos=pr(1,2) c pm1=pr(2,2) !pm1=m1 just for making real number c pc=(1+(epsiequi(1,i,ip)/viscos)**pm1)*pc !i=ien in sub-rate c write(*,*)pc c pause P=(SX+SY+SZ)/3.0D0 Q2=SX*(SX-SY)+SY*(SY-SZ)+SZ*(SZ-SX)+3.0D0*TXY*TXY !general exp. IF(NDIM.EQ.2)GOTO 10 C TYZ=STRESS(5,IP,I) TZX=STRESS(6,IP,I) Q2=Q2+3.0D0*TYZ*TYZ+3.0D0*TZX*TZX 10 Q=SQRT(Q2) ETA=Q/P c write(*,*)pr(4,km) PY=P+Q*Q/(P*PR(4,KM)*PR(4,KM)) c Correction of py for rate dependency ----------- 08/07/03 c viscos=pr(1,2) c pm1=pr(2,2) !pm1=m1 just for making real number c py=(1+(epsiequi(1,i,ip)/viscos)**pm1)*py !i=ien in sub-rate c if(epsiequi(1,i,ip).gt.0.0)write(*,*)"999",py,pc

BK=(1.0D0+E)*P/PR(1,KM) !K constrained modulus C---------------------------------------------------------------------C CALCULATE ELASTIC STRESS-STRAIN MATRIX C---------------------------------------------------------------------G=PR(5,KM) !G=Poisson’s ratio


331

IF(G.LT.1.0D0) G=BK*1.50D0*(1.0D0-2.0D0*PR(5,KM))/(1.0D0+PR(5,KM)) AL=(3.0D0*BK+4.0D0*G)/3.0D0 DL=(3.0D0*BK-2.0D0*G)/3.0D0 C CALL ZEROR2(D,6,6) D(1,1)=AL D(2,1)=DL D(3,1)=DL D(1,2)=DL D(2,2)=AL D(3,2)=DL D(1,3)=DL D(2,3)=DL D(3,3)=AL D(4,4)=G IF(NDIM.EQ.2)GOTO 12 D(5,5)=G D(6,6)=G C 12 IF(PY.LT.0.9950d0*PC) GO TO 50 !PY=Po !bypass C---------------------------------------------------------------------C CALCULATE PLASTIC STRESS-STRAIN MATRIX IF CURRENT C POINT ON YIELD LOCUS AND SET PC NEGATIVE C---------------------------------------------------------------------PCS=.50D0*PC PB=P/PCS S(1)=SX-P S(2)=SY-P S(3)=SZ-P S(4)=2.0D0*TXY IF(NDIM.EQ.2)GOTO 16 S(5)=2.0D0*TYZ S(6)=2.0D0*TZX C -----------------------------

16 c

CONTINUE

Adjustment of Dab for gradient gradcon=pr(5,2) volepsi=dab(1,1)+dab(2,2)-dab(1,1)*dab(2,2) c write(*,*)"vol",volepsi,dab(1,1) do ii=1,3 do jj=1,3 c if(volepsi.ge.0)goto 4 !consider gradient when volume expand c subtraction for volume exp. addition for volume contr.

332


graddab(ii,jj)=dab(ii,jj)-gradcon*volepsi enddo enddo c Turn back gradient dab to dab for further calculation do ii=1,3 do jj=1,3 dab(ii,jj)=graddab(ii,jj) enddo enddo 4

C Correction to avoid tension failure C DO II=1,3 C IF(S(II).LE.0.0)S(II)=0.0 C ENDDO IF(STRESS(1,IP,I).LE.10.0)GOTO 200 IF(STRESS(2,IP,I).LE.10.0)GOTO 200 IF(STRESS(3,IP,I).LE.10.0)GOTO 200 c Calculate the back stress. 17 CALL BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS,IP,DELALPH,I,S) c i=ien=element number watch out!!!!! CALL BKSTRS1(P,Q,ETA,PRM,PR,ITER,JS,IP,I,SALPH11,DAB, 1epsiaccu) c write(*,*) "song" c write(*,*) epsiaccu(1,1,1,1,1) SALPHA(1)=SALPH22(1)+SALPH11(1) SALPHA(2)=SALPH22(2)+SALPH11(2) SALPHA(3)=SALPH22(3)+SALPH11(3) SALPHA(4)=SALPH22(4)+SALPH11(4) c write(*,*)"sapha",salpha(1),SALPH11(1),SALPH22(1) GOTO 210 C

-----------------------------

C---BB=(df/dp)(dp/dsig) without Kronecker dij 3x3 tensor C--C=(df/dq)(dq/dsig) without Sij C--{A}={B}ij C--{S}=Sij 200 do ii=1,4 salpha(ii)=0 ! 11,22,33,13 delalph(ii)=0 enddo C---Calculate Bij 210 BB11=(2.0d0*P-PC)*PRM**2.0d0 BB11=BB11/3 ! really x delta ij

Appendix: Fortran Codes of CS-S BB12=0D0 DO II=1,4 BB12=BB12+1.5D0*(PC*SALPHA(II)*SALPHA(II)-2D0*SALPHA(II) 1*S(II)) c !above term is scalar quantity. ENDDO BB12=BB12/3 BB21(1)=(2D0/3D0)*S(1)-(1D0/3D0)*S(2)-(1D0/3D0)*S(3) BB21(1)=3*BB21(1) BB21(2)=-(1D0/3D0)*S(1)+(2D0/3D0)*S(2)-(1D0/3D0)*S(3) BB21(2)=3*BB21(2) BB21(3)=-(1D0/3D0)*S(1)-(1D0/3D0)*S(2)+(2D0/3D0)*S(3) BB21(3)=3*BB21(3) BB21(4)=(2D0/3D0)*S(4) BB21(4)=3*BB21(4) BB31(1)=3D0*P*((2D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)6(1D0/3D0)*SALPHA(3)) BB31(2)=3D0*P*(-(1D0/3D0)*SALPHA(1)+(2D0/3D0)*SALPHA(2)1(1D0/3D0)*SALPHA(3)) BB31(3)=3D0*P*(-(1D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)+ 1(2D0/3D0)*SALPHA(3)) BB31(4)=3D0*P*(2D0/3D0)*SALPHA(4) DO II=1,3 A(II)=BB11+BB12+BB21(II)-BB31(II) ENDDO A(4)=0+0+BB21(4)-BB31(4) BII=A(1)+A(2)+A(3) !Bii tr(df/dsigma) c WRITE(*,*)"111" C

Complete Bij

!{A}={B}ij matrix

18 DO 20 J=1,3 B(J)=0.0D0 DO 20 JJ=1,3 20 B(J)=B(J)+D(J,JJ)*A(JJ) ! {B}=[C]ijkl[B]kl = (De)(df/dsig) B(4)=D(4,4)*A(4) IF(NDIM.EQ.2)GOTO 25 B(5)=D(5,5)*A(5) B(6)=D(6,6)*A(6) c WRITE(*,*)"222" C---df/dev 25 XI=(1.0D0+E)/(PR(2,KM)-PR(1,KM)) !XI=(1+e)/(lambda-kappa) ALPHA=0D0 !ALPHA ij Alpha ij DO II=1,4 ALPHA=ALPHA+SALPHA(II)*SALPHA(II) !ALPHA=scalar, aijxaij sum

333

334


ENDDO AA1=XI*P*PC*(-2D0*PRM**2D0+3D0*ALPHA) !SONG AA2=BII AA=AA1*AA2 !AA=[Hp] currently negative H=AA !To transfer H to the next routine C WRITE(*,*)"AA1,AA2",AA1,AA2,P,PC,ALPHA

df/d epsilon vp

C---Compute the additional change of stiffness matrix by back stress. C by Chung R. Song 3/08/99 DUM2=0 DO II=1,4 DUM1(II)=3D0*(P*PC*SALPHA(II)-P*S(II)) !df/dalpha DELALPH(II)=DELALPH(II) DUM2=DUM2+DUM1(II)*DELALPH(II) ! (df/dalpha)(dalpha) SCALAR !magic ENDDO C WRITE(*,*)"333" C----C C C

Inversion of depsilon DAB=d edpsilon, DABI=inverse of d epsilon DEPS=DAB This is the vectorial form Dab Song DAB is depsi !3/23/99 Song

C WRITE(*,*)"DAB",DAB(1,1),DAB(1,2) C WRITE(*,*)"DAB",DAB(2,1),DAB(2,2),ITER DJACOB1=DAB(1,1)*(DAB(2,2)*DAB(3,3)-DAB(3,2)*DAB(2,3)) DJACOB2=DAB(2,1)*(DAB(1,2)*DAB(3,3)-DAB(3,2)*DAB(1,3)) DJACOB3=DAB(3,1)*(DAB(1,2)*DAB(2,3)-DAB(2,2)*DAB(1,3)) DJACOB=DJACOB1-DJACOB2+DJACOB3 C DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1) !radial strain = 0 IF(DJACOB.EQ.0)GOTO 27 !skip singular condition CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE) !special c IF(ITER.EQ.1)GOTO 27 C CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE) C DABI(1,1)=DAB(2,2)/DJACOB C DABI(2,2)=DAB(1,1)/DJACOB C DABI(1,2)=-DAB(2,1)/DJACOB C DABI(2,1)=-DAB(1,2)/DJACOB GOTO 75 27 DO 70 JJJ1=1,4 70 DEPSI(JJJ1)=0D0 GOTO 77


335

C WRITE(*,*)"444" C-----Change the depsilon\_1 tensor to depsilon\_1 vector 75 DEPSI(1)=DABI(1,1) DEPSI(2)=DABI(2,2) DEPSI(3)=0.0 !DABI(3,3) DEPSI(4)=DABI(1,2) !Axi-symmetric C-----Substitute depsilon inverse, into [Hpa] equation 77 CONTINUE DO JS1=1,4 DUM5(JS1)=0.0D0 !clear ENDDO C WRITE(*,*)"555" DO 90 JS1=1,4 !2-D case, Axisymmetric DUM4(JS1)=DUM2*A(JS1) ! (df/dalpha)(dalpha)[B] vectorized tensor DO 91 JS2=1,4 !2-D case 91 DUM5(JS1)=DUM5(JS1)+D(JS1,JS2)*DUM4(JS2) !2nd oderized 4th order,sum tensor DO 90 JS2=1,4 90 HPA(JS1,JS2)=(1D0/3D0)*DUM5(JS1)*DEPSI(JS2) ![HPA]=(1/3)[C](df/dalpha)(dalpha)[B][depsil]

C

IF(ETA.LT.PRM.AND.AA.LT.0.0D0)AA=0000000000.0 !AA=infinitive IF(ETA.GT.PRM.AND.AA.GT.0.0D0)AA=0.0D0 IF(ICOD.EQ.5)AA=0.0D0 AB=0.0D0 !clear AB DO 30 J=1,NS 30 AB=AB+A(J)*B(J) !AB=[He]

(df/dsig)(De)(df/dsig)

C c WRITE(*,*)"666" BETA=-AA+AB ![Hp]+[He] BETAA=10.0D17 IF(BETA.GT.BETAA)BETA=BETAA !prevent overflow DO 40 J=1,NS DO 40 JJ=1,NS C WRITE(*,*)"H, D",BETA Call Damage(pr,npr,nmt,pc,ap) !Damage consideration 40 D(JJ,J)=D(JJ,J)-(B(JJ)*B(J)+HPA(JJ,J))/(BETA*ap)

C

! [Cijkl]-{[C]ijkl[B]kl[C]ijkl[B]kl+[HPP]}/{[Hp]+[He]} 40 WRITE(*,*)"Dep",D(JJ,J)

336


c 50 CONTINUE C IF(IET.EQ.0) GOTO 80 C DO 60 J=1,3 DO 60 JJ=1,3 60 D(JJ,J)=D(JJ,J)+PR(7,KM)*BK !Bulk Modulus 80 CONTINUE c WRITE(*,*)"777" c rate dependency c rate depencency -------------------------call rate(epsiequi,epsiaccu,i,ip) c write(*,*)epsiequi(1,1,1) RETURN END

C ********************************************* Subroutine damage(pr,npr,nmt,pc,ap) c This subroutine cnsiders the damage. c Called from DMCAM c Chung R. Song 08/09/2003 c ********************************************* IMPLICIT REAL*8(A-H,O-Z) dimension PR(NPR,NMT) c write(*,*)"999" A1=pr(3,2) A2=pr(4,2) dum=A1*pc**A2 c write(*,*)dum c write(*,*)"9999",pc Ap=1-2.71828**(-dum) if(ap.LE.0.95)ap=0.95 c write(*,*)"99999" c beta=beta*Ap !!!! Cancelled. ap is used directly in dmcam if(ap.le.0.5)write(*,*)"Too much damage!!!!" return end

C ********************************************* subroutine rate(epsiequi,epsiaccu,i,ip) c This subroutine change the yield function based on Song and Voyiadjis (2003) c Called from DMCAM c Chung R. Song 08/06/2003 c *********************************************


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IMPLICIT REAL*8(A-H,O-Z) COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, + ICOR ,IDCHK,IOUT ,INCT ,IWL , + NOIB ,JS ,JINCB ,NLOD ,NLDS DIMENSION epsiequi(2,170,4),epsiaccu(2,170,4,3,3) ien=i EPSIEQUI(1,ien,ip)=epsiaccu(2,ien,ip,1,1) 1*epsiaccu(2,ien,ip,1,1) 2+epsiaccu(2,ien,ip,2,2)*epsiaccu(2,ien,ip,2,2) 3+epsiaccu(2,ien,ip,3,3)*epsiaccu(2,ien,ip,3,3) EPSIEQUI(1,ien,ip)=epsiequi(1,ien,ip)*sqrt(0.666667) c write(*,*) ttime epsiequi(1,ien,ip)=epsiequi(1,ien,ip)/ttime !ttime=elapsed time c write(*,*) epsiequi(1,ien,ip) c Adjustment of pc is made in the beginning part of DMCAM return end C ********************************************* subroutine rate1(dabequi,dab) c This subroutine change the yield function based on Song and Voyiadjis (2003) c Called from DMCAM c Chung R. Song 08/13/2003 c********************************************* IMPLICIT REAL*8(A-H,O-Z) COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, + ICOR ,IDCHK,IOUT ,INCT ,IWL , + NOIB ,JS ,JINCB ,NLOD ,NLDS DIMENSION dab(3,3) ien=i dabEQUI=dab(1,1)*dab(1,1)+dab(2,2)*dab(2,2)+dab(3,3)*dab(3,3) dabEQUI=dabequi*sqrt(0.666667) c write(*,*) ttime dabequi=dabequi/ttime c Adjustment of po is made by adjusting sigmaxx, sigmayy, sigmazz. return end C********************************************************************** subroutine bkstrs2(p,q,eta,prm,salph22,pr,iter,js,ip,delalph 1,ien,s) c SUBROUTINE BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS, c 1IP,DELALPH,IEN,S) c CALL BKSTRS2(P,Q,ETA,PRM,SALPH22,PR,ITER,JS,IP,DELALPH,I,S) C**********************************************************************

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C 1. CALLED FROM EQUIBLOD, DMCAM C 2. This subroutine calculate the back stress and supply it to yield f. C 3. Chung R. Song, 2/25/99. C 4. Modified for multiple back stress C****** Long range back stress, ALPHA 2 ********** IMPLICIT REAL*8(A-H,O-Z) COMMON/BSTRESS/ PREP,PREQ,PREETA,PREALPH !Back stress variable COMMON/MATPROP/C,X DIMENSION PREP2(170,2,4,8),PREQ2(170,2,4,8),PREETA2(170,2,4,8) DIMENSION DALPHA(170,2,4,8,4),SSALPHA(170,4,4),DELS(4), 1SALPH22(4),PR(10,10) !PR added just for matching DIMENSION PRE\_S(170,2,4,8),S(6) !6 just for matching with Dmcam DIMENSION DELALPH(4) C elem no, Incr no, gauss pt, iteration, direction c (IEN,JS,IP,ITER,J) C Js was reduced to 2 by magic. (repeated use) c WRITE(*,*)"BKSTRS",IEN C---Clear DO J=1,4 SALPH22(J)=0.0 DELS(J)=0.0 ENDDO C---Iteration for direction ij (11,22,33,12) DO J=1,4 !Beginning of loop c--- Prevent numerical noise C--- Do not remove this part. This part makes the program more stable. IF(P.LE.0.0000000001)P=0 IF(Q.LE.0.0000000001)Q=0 c IF(S(J).LE.0.0000000001)S(J)=0 IF(ETA.LE.0.0000000001)ETA=0 IF(SALPH22(J).LE.0.0000000001)SALPH22(J)=0 DO I=1,2 DO II=1,8 IF(PREP2(IEN,I,IP,II).LE.0.0000000001)PREP2(IEN,I,IP,II)=0.0 IF(PREQ2(IEN,I,IP,II).LE.0.0000000001)PREQ2(IEN,I,IP,II)=0.0 IF(PRE\_S(IEN,I,IP,II).LE.0.0000000001)PRE\_S(IEN,I,IP,II)=0.0 IF(PREETA2(IEN,I,IP,II).LE.0.0000000001)PREETA2(IEN,I,IP,II)= 10.0 IF(DALPHA(IEN,I,IP,II,J).LE.0.0000000001) 1DALPHA(IEN,I,IP,II,J)=0.0 ENDDO ENDDO

Appendix: Fortran Codes of CS-S IF(JS.EQ.1.AND.ITER.EQ.1) THEN !Very Beginning DELP=0.0D0 !P-PREP DELQ=0.0D0 !Q-PREQ DELS(J)=0.0D0 DELETA=0.0D0 !ETA-PREETA DELALPH(I)=0.D0 !SALPHA-PREALPH ENDIF IF(JS.EQ.1.AND.ITER.EQ.2) THEN ! Beginning DELP=P-PREP2(IEN,JS,IP,ITER-1) DELQ=Q-PREQ2(IEN,JS,IP,ITER-1) DELS(J)=S(J)-PRE\_S(IEN,JS,IP,ITER-1) DELETA=ETA-PREETA2(IEN,JS,IP,ITER-1) DELALPH(I)=DALPHA(IEN,JS,IP,1,J) ENDIF IF(JS.EQ.1.AND.ITER.GE.3) THEN ! Beginning DELP=P-PREP2(IEN,JS,IP,ITER-1) DELQ=Q-PREQ2(IEN,JS,IP,ITER-1) DELS(J)=S(J)-PRE\_S(IEN,JS,IP,ITER-1) DELETA=ETA-PREETA2(IEN,JS,IP,ITER-1) DELALPH(I)=DALPHA(IEN,1,IP,ITER-1,J)-DALPHA(IEN,1,IP,ITER-2,J) ENDIF IF(JS.EQ.2.AND.ITER.EQ.1) THEN DELP=P-PREP2(IEN,1,IP,2) !JS=increm. penet, IP=Gauss point No. DELQ=Q-PREQ2(IEN,1,IP,2) DELS(J)=S(J)-PRE\_S(IEN,1,IP,2) DELETA=ETA-PREETA2(IEN,1,IP,2) DELALPH(I)=DALPHA(IEN,1,IP,2,J) !Previous value SALPHAPREALP2(JS-1,IP) ENDIF IF(JS.GT.2.AND.ITER.EQ.1) THEN !Intermediate Beginning DELP=P-PREP2(IEN,2,IP,2) !JS=increm. penet, IP=Gauss point No. DELQ=Q-PREQ2(IEN,2,IP,2) DELS(J)=S(J)-PRE\_S(IEN,2,IP,2) DELETA=ETA-PREETA2(IEN,2,IP,2) DELALPH(I)=DALPHA(IEN,2,IP,2,J) !Previous value SALPHAPREALP2(JS-1,IP) ENDIF IF(JS.GT.1.AND.ITER.EQ.2) THEN !Intermediate 2 DELP=P-PREP2(IEN,2,IP,1) DELQ=Q-PREQ2(IEN,2,IP,1) DELS(J)=S(J)-PRE\_S(IEN,2,IP,1) DELETA=ETA-PREETA2(IEN,2,IP,1) DELALPH(I)=DALPHA(IEN,2,IP,1,J)-DALPHA(IEN,1,IP,2,J)

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ENDIF IF(JS.GT.1.AND.ITER.GE.3) THEN !Intermediate 3 and others DELP=P-PREP2(IEN,2,IP,ITER-1) DELQ=Q-PREQ2(IEN,2,IP,ITER-1) DELS(J)=S(J)-PRE\_S(IEN,2,IP,ITER-1) DELETA=ETA-PREETA2(IEN,2,IP,ITER-1) DELALPH(I)=DALPHA(IEN,2,IP,ITER-1,J)-DALPHA(IEN,2,IP,ITER-2,J) ENDIF C=PR(6,2) X=PR(7,2) ALP=ALP !SSALPHA(IEN,IP,3)-( 1) ! Use alpha1-alpha3 from previous calculation c WRITE(*,*)"P,PRM,ETA,ALP",P,PRM,ETA,ALP SALPHA1=DELP/P+(2D0*ETA2D0*ALP)*DELETA/(PRM**2D0+ETA**2D012D0*ALP*ETA) SALPHA2=(C/P)*(DELS(J)-X*DELP*SALPH22(J)) C WRITE(*,*)"DELP",ITER,IEN,IP,J,ETA SALPH22(J)=SALPH22(J)+3*(SALPHA1*SALPHA2) C WRITE(*,*)"FDDF",J,JS,SALPHA1,SALPHA2,SALPHA(J),DELS(J) C WRITE(*,*)"DFD", DELP,P,ETA,ALP,DELETA,PRM C IF(ABS(SALPHA(J)).GT.PRM)THEN C SALPHA(J)=PRM C ENDIF C SALPHA(J)=0 !MAKE ISOTROPIC C ---------- Save current P, Q, ETA, SDELS, SALPHA for next iteration IF(JS.EQ.1)THEN !JS=incremental number PREP2(IEN,1,IP,ITER)=P PREQ2(IEN,1,IP,ITER)=Q PRE\_S(IEN,1,IP,ITER)=S(J) PREETA2(IEN,1,IP,ITER)=ETA DALPHA(IEN,1,IP,ITER,J)=SALPH22(J) SSALPHA(IEN,IP,J)=SSALPHA(IEN,IP,J)+DALPHA(IEN,1,IP,ITER,J) 1 -DALPHA(IEN,1,IP,ITER-1,J) SALPH22(J)=SSALPHA(IEN,IP,J) ALP=SALPH22(1)-SALPH22(3) ENDIF


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IF(JS.EQ.2)THEN PREP2(IEN,2,IP,ITER)=P PREQ2(IEN,2,IP,ITER)=Q PRE\_S(IEN,2,IP,ITER)=S(J) PREETA2(IEN,2,IP,ITER)=ETA DALPHA(IEN,2,IP,ITER,J)=SALPH22(J) SSALPHA(IEN,IP,J)=SSALPHA(IEN,IP,J)+DALPHA(IEN,2,IP,ITER,J) 1 -DALPHA(IEN,2,IP,ITER-1,J) SALPH22(J)=SSALPHA(IEN,IP,J) ALP=SALPH22(1)-SALPH22(3) ENDIF IF(JS.GE.3)THEN PREP2(IEN,1,IP,ITER)=PREP2(IEN,2,IP,ITER) PREP2(IEN,2,IP,ITER)=P PREQ2(IEN,1,IP,ITER)=PREQ2(IEN,2,IP,ITER) PREQ2(IEN,2,IP,ITER)=Q PRE\_S(IEN,1,IP,ITER)=PRE\_S(IEN,2,IP,ITER) PRE\_S(IEN,2,IP,ITER)=S(J) PREETA2(IEN,1,IP,ITER)=PREETA2(IEN,2,IP,ITER) PREETA2(IEN,2,IP,ITER)=ETA DALPHA(IEN,1,IP,ITER,J)=DALPHA(IEN,2,IP,ITER,J) DALPHA(IEN,2,IP,ITER,J)=SALPH22(J) SSALPHA(IEN,IP,J)=SSALPHA(IEN,IP,J)+DALPHA(IEN,2,IP,ITER,J) 1 -DALPHA(IEN,2,IP,ITER-1,J) SALPH22(J)=SSALPHA(IEN,IP,J) ALP=SALPH22(1)-SALPH22(3) ENDIF C ---------ENDDO !End of big do loop RETURN END

C********************************************************************** SUBROUTINE BKSTRS1(P,Q,ETA,PRM,PR,ITER,JS,IP,IEN,SALPH11 1,DAB,epsiaccu) C********************************************************************** C CALLED FROM Dmcam C This subroutine compute the short range back stress by Prager’s C linear hardening rule. 5/30/00 C alpha1(dot)=(2/3)C1*epsilonp(dot)---------

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IMPLICIT REAL*8(A-H,O-Z) DIMENSION EPSIACCU(2,170,4,3,3),DAB(3,3) !elem,gaus,i,j DIMENSION ALP1DOT(170,4,3,3),SALPH11(4),PR(10,10) !PR added for matching C elem no, Incr no, gauss pt, iteration, direction c (IEN,JS,IP,ITER,J) c accumulation for incremental steps only (JS) c for each element, for each Gauss point, for each direction i,j !!! c

----- Clear ALP1DOT do i=1,3 do j=1,3 ALP1DOT(IEN,IP,I,J)=0.0 enddo enddo c DAB(i,j) is the strain increment. inew=1 iold=2 c ----- Initializing EPSIACCU if(js.EQ.1)EPSIACCU(iold,ien,ip,i,j)=0.0d0 do i=1,3 do j=1,3 C do ien=1,150 !already in iterating loop c do ip=1,8 !already in iterating loop c ien=element no. ip=gauss no. EPSIACCU(inew,ien,ip,I,J)=EPSIACCU(iold,ien,ip,I,J)+DAB(i,j) !EPSIACCU=Accumulated Plastic Strain C ----- STABILIZE alpha1 after THRESHOLD strain limit if (epsiaccu(inew,ien,ip,i,j).GT.0.001)GOTO 100 C1=.0 !temperal ALP1DOT(ien,ip,I,J)=(2/3.0)*C1*EPSIACCU(inew,ien,ip,I,J) !ip added 5/10/00 c ----- Vectorize ALP1dot(ien,ip,i,j), indecies in salpha( ) is same as J c ----- in previous Bkstrs subroutine SALPH11(1)=ALP1DOT(IEN,IP,1,1) SALPH11(2)=ALP1DOT(IEN,IP,2,2) SALPH11(3)=ALP1DOT(IEN,IP,3,3) SALPH11(4)=ALP1DOT(IEN,IP,1,2) !REST = 0 c c

ENDDO ENDDO


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ENDDO ENDDO do i=1,3 do j=1,3 EPSIACCU(iold,ien,ip,i,j)=EPSIACCU(inew,ien,ip,i,j) enddo enddo 100

RETURN END

C********************************************************************** SUBROUTINE STRSCOR(IP,I,NDIM,NS,STRESS,VAR,MAT,PR,FYLD,DSIG,PC, 1ITER,JS,DAB,SALPHA,ICOD,H,A) C********************************************************************** C CALLED FROM EQUIBLOD C This subroutine correct the stress which cross the yield surface. 2/25/99. C Corrected again at 5/1/99 by C.R. Song IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION STRESS(NVRS,NIP,MEL),D(6,6),MAT(MEL) DIMENSION DSIG(6),VAR(6),S(6),A(6),B(6),PR(NPR,NMT) DIMENSION DAB(3,3),DABI(3,3),HPA(4,4),DEPSI(4),DUM4(4),DUM5(4) DIMENSION SALPHA(4),BB21(4),BB31(4),DUM1(4),DELALPH(4) C Subroutine modified by Song for default KGO=3 4/30/99 C C WRITE(*,*)"STRSCOR" KM=MAT(I) PRM=PR(4,KM) SX=VAR(1) SY=VAR(2) SZ=VAR(3) TXY=VAR(4) C WRITE(*,*)"VAR",VAR(1),VAR(2),VAR(3) E=STRESS(NS+2,IP,I) P=(SX+SY+SZ)/3.0D0 Q2=SX*(SX-SY)+SY*(SY-SZ)+SZ*(SZ-SX)+3.0D0*TXY*TXY IF(NDIM.EQ.2)GOTO 10 C TYZ=VAR(5) TZX=VAR(6) Q2=Q2+3.0D0*TYZ*TYZ+3.0D0*TZX*TZX 10 Q=SQRT(Q2) ETA=Q/P PY=P+Q*Q/(P*PR(4,KM)*PR(4,KM)) PCS=PC/2.0D0

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Appendix: Fortran Codes of CS-S PB=P/PCS BK=(1.0D0+E)*P/PR(1,KM)

C---

CALCULATE ELASTIC STRESS-STRAIN MATRIX G=PR(5,KM) IF(G.LT.1.0D0)G=BK*1.50D0*(1.0D0-2.0D0*PR(5,KM))/(1.0D0+PR(5,KM)) AL=(3.0D0*BK+4.0D0*G)/3.0D0 DL=(3.0D0*BK-2.0D0*G)/3.0D0

C CALL ZEROR2(D,6,6) D(1,1)=AL D(2,1)=DL D(3,1)=DL D(1,2)=DL D(2,2)=AL D(3,2)=DL D(1,3)=DL D(2,3)=DL D(3,3)=AL D(4,4)=G IF(NDIM.EQ.2)GOTO 12 D(5,5)=G D(6,6)=G C---

CALCULATE

{A}={DF/DSIG} YIELD FUNCTION DERIVATIVES

12 S(1)=SX-P S(2)=SY-P S(3)=SZ-P S(4)=2.0D0*TXY IF(NDIM.EQ.2)GOTO 16 S(5)=2.0D0*TYZ S(6)=2.0D0*TZX 16 CONTINUE C C C C

Correction to avoid tension failure by C.R. Song DO II=1,3 IF(S(II).LE.0.0)S(II)=0.0 ENDDO

IF(VAR(1).LE.10.0)GOTO 200 IF(VAR(2).LE.10.0)GOTO 200 IF(VAR(3).LE.10.0)GOTO 200 C--- Calculate the back stress. C 17 CALL BKSTRS2(P,Q,ETA,PRM,SALPHA,PR,ITER,JS,IP,DELALPH,I,S) GOTO 210

Appendix: Fortran Codes of CS-S C WRITE(*,*)"BSTRSED IN STRSCOR" C ----------------------------c 16 BB=-2.0D0*(1.0D0-PB)/(3.0D0*PCS) c C=3.0D0/(PCS*PCS*PR(4,KM)*PR(4,KM)) C BB=PRM**2.0D0*(2.0d0*P-PC)/3.0D0 !TEMP. BY SONG C C=3.0D0 !TEMP. BY SONG C SJJ2 IS THE SECOND STRESS DETERMINANT J2 !SONG C WRITE(*,*)"ALPHA",SALPHA C SJJ2=(Q**2D0)/(3.0D0) !SONG C---BB=(df/dp)(dp/dsig) without Kronecker dij C--C=(df/dq)(dq/dsig) without Sij C--{A}={B}ij C--{S}=Sij 200 do ii=1,4 salpha(ii)=0 delalph(ii)=0 enddo

C---Caculate Bij 210 BB11=(2.0d0*P-PC)*PRM**2.0d0 BB11=BB11/3 BB12=0D0 DO II=1,4 BB12=BB12+1.5D0*(PC*SALPHA(II)*SALPHA(II)-2D0*SALPHA(II) 1*S(II)) ENDDO BB12=BB12/3 BB21(1)=(2D0/3D0)*S(1)-(1D0/3D0)*S(2)-(1D0/3D0)*S(3) BB21(1)=3*BB21(1) BB21(2)=-(1D0/3D0)*S(1)+(2D0/3D0)*S(2)-(1D0/3D0)*S(3) BB21(2)=3*BB21(2) BB21(3)=-(1D0/3D0)*S(1)-(1D0/3D0)*S(2)+(2D0/3D0)*S(3) BB21(3)=3*BB21(3) BB21(4)=(2D0/3D0)*S(4) BB21(4)=3*BB21(4) BB31(1)=3D0*P*((2D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)6(1D0/3D0)*SALPHA(3)) BB31(2)=3D0*P*(-(1D0/3D0)*SALPHA(1)+(2D0/3D0)*SALPHA(2)1(1D0/3D0)*SALPHA(3)) BB31(3)=3D0*P*(-(1D0/3D0)*SALPHA(1)-(1D0/3D0)*SALPHA(2)+ 1(2D0/3D0)*SALPHA(3)) BB31(4)=3D0*P*(2D0/3D0)*SALPHA(4) DO II=1,3

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A(II)=BB11+BB12+BB21(II)-BB31(II) ENDDO A(4)=0+0+BB21(4)-BB31(4) BII=A(1)+A(2)+A(3) !Bii C

Complete Bij

!{A}={B}ij matrix

18 DO 20 J=1,3 B(J)=0.0D0 DO 20 JJ=1,3 20 B(J)=B(J)+D(J,JJ)*A(JJ) ! {B}=[C]ijkl[B]kl = (De)(df/dsig) B(4)=D(4,4)*A(4) IF(NDIM.EQ.2)GOTO 25 B(5)=D(5,5)*A(5) B(6)=D(6,6)*A(6) C---df/dev 25 XI=(1.0D0+E)/(PR(2,KM)-PR(1,KM)) !XI=(1+e)/(lambda-kappa) ALPHA=0D0 !ALPHA ij Alpha ij DO II=1,4 ALPHA=ALPHA+SALPHA(II)*SALPHA(II) !ALPHA=scalar ENDDO AA1=XI*P*PC*(-2D0*PRM**2D0+3D0*ALPHA) !SONG df/d epsilon vp AA2=BII AA=AA1*AA2 !AA=[Hp] H=AA !To transfer H to the next routine C WRITE(*,*)"H in STRscor",H,AA1,AA2 C--------- Compute the additional change of stiffness matrix by back stress. C--------- by Chung R. Song 3/08/99 DUM2=0 DO II=1,4 DUM1(II)=3D0*(P*PC*SALPHA(II)-P*S(II)) !df/dalpha DELALPH(II)=DELALPH(II) DUM2=DUM2+DUM1(II)*DELALPH(II) ! (df/dalpha)(dalpha) SCALAR !magic ENDDO C----- Inversion of depsilon C DAB=d edpsilon, DABI=inverse of d epsilon C DEPS=DAB This is the vectorial form Dab Song C DAB is depsi !3/23/99 Song C WRITE(*,*)"DAB",DAB(1,1),DAB(1,2) C WRITE(*,*)"DAB",DAB(2,1),DAB(2,2),ITER

Appendix: Fortran Codes of CS-S C C C C C C C C C C C C C

DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1) IF(DJACOB.EQ.0)GOTO 27 !skip singular condition IF(ITER.EQ.1)GOTO 27 CALL DETMIN(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE) DABI(1,1)=DAB(2,2)/DJACOB DABI(2,2)=DAB(1,1)/DJACOB DABI(1,2)=-DAB(2,1)/DJACOB DABI(2,1)=-DAB(1,2)/DJACOB GOTO 75 27 DO 70 JJJ1=1,4 70 DEPSI(JJJ1)=0D0 GOTO 77

C----- Inversion of depsilon C DAB=d edpsilon, DABI=inverse of d epsilon C DEPS=DAB This is the vectorial form Dab Song C DAB is depsi !3/23/99 Song C WRITE(*,*)"DAB",DAB(1,1),DAB(1,2) C WRITE(*,*)"DAB",DAB(2,1),DAB(2,2),ITER DJACOB1=DAB(1,1)*(DAB(2,2)*DAB(3,3)-DAB(3,2)*DAB(2,3)) DJACOB2=DAB(2,1)*(DAB(1,2)*DAB(3,3)-DAB(3,2)*DAB(1,3)) DJACOB3=DAB(3,1)*(DAB(1,2)*DAB(2,3)-DAB(2,2)*DAB(1,3)) DJACOB=DJACOB1-DJACOB2+DJACOB3 C DJACOB=DAB(1,1)*DAB(2,2)-DAB(1,2)*DAB(2,1) !radial strain = 0 IF(DJACOB.EQ.0)GOTO 27 !skip singular condition CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE) !special c IF(ITER.EQ.1)GOTO 27 C CALL DETMIN1(LOUT1,NDIM,DAB,DABI,DJACB,JL,IP,ISTGE) C DABI(1,1)=DAB(2,2)/DJACOB C DABI(2,2)=DAB(1,1)/DJACOB C DABI(1,2)=-DAB(2,1)/DJACOB C DABI(2,1)=-DAB(1,2)/DJACOB GOTO 75 27 DO 70 JJJ1=1,4 70 DEPSI(JJJ1)=0D0 GOTO 77 C WRITE(*,*)"444" C-----Change the depsilon\_1 tensor to depsilon\_1 vector 75 DEPSI(1)=DABI(1,1)

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DEPSI(2)=DABI(2,2) DEPSI(3)=0.0 !DABI(3,3) DEPSI(4)=DABI(1,2) !Axi-symmetric C-----Substitute depsilon inverse, into [Hp] equation 77 CONTINUE DO JS1=1,4 !clear DUM4(JS1)=0.0D0 ENDDO DO 90 JS1=1,4 !2-D case, Axisymmetric DUM4(JS1)=DUM2*A(JS1) ! (df/dalpha)(dalpha)[B] tensor DO 91 JS2=1,4 !2-D case 91 DUM5(JS1)=DUM5(JS1)+D(JS1,JS2)*DUM4(JS2) !2nd oderized 4th order tensor DO 90 JS2=1,4 90 HPA(JS1,JS2)=(1D0/3D0)*DUM5(JS1)*DEPSI(JS2) ![HPA]=(1/3)[C](df/dalpha)(dalpha)[B][depsil]

C

vectorized

IF(ETA.LT.PRM.AND.AA.LT.0.0D0)AA=0000000000.0 !AA=infinitive IF(ETA.GT.PRM.AND.AA.GT.0.0D0)AA=0.0D0 IF(ICOD.EQ.5)AA=0.0D0 AB=0.0D0

C CALCULATE ALPHA PARAMETER C--------------------------------------------------------------------DO 30 J=1,NS 30 AB=AB+A(J)*B(J) BETA=AA+AB ALPHA=FYLD/BETA C--------------------------------------------------------------------C CALCULATE DSIG = ALPHA*[D]*{A} C--------------------------------------------------------------------DO 40 J=1,NS 40 DSIG(J)=ALPHA*B(J) RETURN END

C********************************************************************** SUBROUTINE EQUIBLOD(XYZ,NCONN,MAT,LTYP,NQ,NW,NP1,NP2,PR,NTY,DI, 1 DIPR,STRESS,STRAIN,PEXI,IDFX,P,PT,PCOR,PEQT,XYFT,PCONI, + LCS,LNGP,NELCM,MCS,MNGP,NCAM,ITER,IOUTP,JS)


349

C---------------------------------------------------------------------C THIS SUBROUTINE COMPUTES THE EQUILIBRATING GLOBAL LOAD VECTOR C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES),STRAIN(NVRN,NIP,MEL) DIMENSION NCONN(NTPE,MEL),MAT(MEL),LTYP(MEL),NW(MNODES+1), 1 NQ(MNODES),NP1(NPL),NP2(NPL) DIMENSION BL(6,NB),BNL(6,NB),BL1(9,NB),FT(3,NDMX) DIMENSION SLL(4),NWL(NPMX),AA(NPMX) DIMENSION PR(NPR,NMT),NTY(NMT) DIMENSION D(6,6),ELCOD(3,NDMX),DS(3,20),SHFN(20),CARTD(3,NDMX) DIMENSION MCS(MEL),MNGP(MEL),ST(6),ST1(6) DIMENSION DI(MDOF),DIPR(MDOF),EDINC(NB),EDINCP(NB),ED(2), 1 XJACM(3,3) DIMENSION P(MDOF),PCOR(MDOF),PT(MDOF),XYFT(MDOF),PCONI(MDOF) DIMENSION PEXI(MDOF),PEQT(MDOF),IDFX(MDOF) DIMENSION LCS(NIP,MEL),LNGP(NIP,MEL),NELCM(MEL) COMMON /DATL / SL(4,100) COMMON /DATW / W(100) COMMON /COUNT / NCS,NNGP COMMON /ELINF / LINFO(50,15) COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /PRSLD / PRESLD(10,100),LEDG(100),NDE1(100),NDE2(100),NLED COMMON /DEVICE/ LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV COMMON /PARS / PYI,ASMVL,ZERO COMMON /OUT / IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF COMMON /JACB / XJACI(3,3),DJACB COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW C---------------------------------------------------------------------LED=2 IEL=0 NCAM=0 NDIM1=NDIM+1 CR=1.0D0 IF(NPLAX.EQ.1)CR=2.0D0*PYI C INITIALIZE THE EQUILIBRIATING LOAD VECTOR CALL ZEROR1(PEQT,MDOF) IF(ITER.EQ.1) CALL ZEROR1(PT,MDOF) C---------------------------------------------------------------------C INITIALISE C---------------------------------------------------------------------DO 18 IM=1,NEL MCS(IM)=0

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MNGP(IM)=0 NELCM(IM)=0 DO 18 IP=1,NIP LCS(IP,IM)=0 18 LNGP(IP,IM)=0 C---------------------------------------------------------------------IEL=0 NCAM=0 C GET THE ELEMENT DISPLACEMENTS FROM THE GLOBAL DISPLACEMENT VECTOR DO 200 MR=1,NEL ICAM=0 C LT=LTYP(MR) IF(LT.LT.0)GOTO 200 NDN=LINFO(5,LT) NGP=LINFO(11,LT) INDX=LINFO(12,LT) NPN=LINFO(6,LT) NDPT=LINFO(1,LT) NAC=LINFO(15,LT) C CALCULATE ELEMENT INCREMENTAL DISPLACEMENT EDINC CALL ZEROR1(EDINCP,NB) CALL ZEROR1(EDINC,NB) DO JDN=1,NDN NIN=NDIM*(JDN-1) JN=ABS(NCONN(JDN,MR)) JL=NW(JN)-1 DO ID=1,NDIM EDINCP(NIN+ID)=DIPR(JL+ID) EDINC(NIN+ID)=DI(JL+ID) ENDDO ENDDO C---------------------------------------------------------------------C SETUP LOCAL NODAL COORDINATES OF ELEMENT C---------------------------------------------------------------------DO 20 KN=1,NDN NDE=NCONN(KN,MR) DO 20 ID=1,NDIM 20 ELCOD(ID,KN)=XYZ(ID,NDE) C GOTO(25,25,23,25,23,25,23,25,23,25,23),LT C---------------------------------------------------------------------C SETUP LOCAL ARRAY OF NW AS NWL GIVING THE INDEX TO C PORE-PRESSURE VARIABLES C---------------------------------------------------------------------23 IPP=0 DO 24 IV=1,NDPT IQ=LINFO(IV+INXL,LT)


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IF(IQ.NE.NDIM1.AND.IQ.NE.1)GOTO 24 IPP=IPP+1 NDE=NCONN(IV,MR) NWL(IPP)=NW(NDE)+IQ-1 24 CONTINUE 25 KM=MAT(MR) KGO=NTY(KM) !Explanation of KGO IF(NTY(KM)-2)27,28,28 27 CALL DCON(MR,0,MAT,PR,D,NDIM,BK) 28 IEL=IEL+1 C INITIALIZE THE EQUILIBRIATING LOAD VECTOR FT CALL ZEROR2(FT,3,NDMX) C---------------------------------------------------------------------C LOOP ON INTEGRATION POINTS C---------------------------------------------------------------------DO 125 IP=1,NGP IPA=IP+INDX C DO 35 IL=1,NAC 35 SLL(IL)=SL(IL,IPA) C---------------------------------------------------------------------C INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS) C---------------------------------------------------------------------CALL ZEROR2(DS,3,20) CALL ZEROR1(SHFN,20) C CALCULATE SHAPE FUNCTIONS AND DERIVATIVES W.R.T LOCAL COOR. CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MR) CALL ZEROR2(XJACM,3,3) C CALCULATE [JAC], [JAC]-1 & DETJAC DO 15 IDIM=1,NDIM DO 15 JDIM=1,NDIM SUM=ZERO DO 112 IN=1,NDN 112 SUM=SUM+DS(IDIM,IN)*ELCOD(JDIM,IN) 15 XJACM(IDIM,JDIM)=SUM c C WRITE(*,*)"EQUIBLOD" CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,MR,IP,ISTGE) C---------------------------------------------------------------------C CALCULATE RADIUS FOR AXI-SYM B MATRIX C---------------------------------------------------------------------R=ZERO RI=ZERO IF(NPLAX.EQ.0)GOTO 38 C DO IN=1,NDN R=R+ELCOD(1,IN)*SHFN(IN)

352

C

Appendix: Fortran Codes of CS-S ENDDO RI=-1.0D0/R CALCULATE CARTESIAN DERIVATIVES OF SHAPE FUNCTIONS 38 DO IN1=1,NDN DO ID=1,NDIM SUM=ZERO

C DO 130 JD=1,NDIM 130 SUM=SUM-DS(JD,IN1)*XJACI(ID,JD) CARTD(ID,IN1)=SUM ENDDO ENDDO C

CALCULATE THE LINEAR STRAIN-DISPL MATRIX [BL] CALL BLNR(NDIM,NDN,NPLAX,RI,SHFN,CARTD,BL)

C C

IF(LINR.GT.1) THEN CALCULATE THE NONLINEAR STRAIN-DISPL MATRIX BNL CALL BNLNR(EDINCP,SHFN,CARTD,RI,NDIM,NDN,BNL,NS,NPLAX)

C

END IF CALCULATE THE LINEAR DISPL-DISPL MATRIX BL1 CALL BLNR1(CARTD,BL1,NDIM,NDN)

C CALL ZEROR1(ST,NS) C DO 44 II=1,NDN IN=NCONN(II,MR) N1=NW(IN) N2=N1+1 ST(1)=ST(1)+CARTD(1,II)*DI(N1) ST(2)=ST(2)+CARTD(2,II)*DI(N2) ST(3)=ST(3)+SHFN(II)*DI(N1)*RI ST(4)=ST(4)+CARTD(1,II)*DI(N2)+CARTD(2,II)*DI(N1) IF(NDIM.EQ.2)GOTO 44 N3=N1+2 ST(3)=ST(3)+CARTD(3,II)*DI(N3) ST(5)=ST(5)+CARTD(3,II)*DI(N2)+CARTD(2,II)*DI(N3) ST(6)=ST(6)+CARTD(3,II)*DI(N1)+CARTD(1,II)*DI(N3) 44 CONTINUE C DO IS=1,NS ST1(IS)=STRAIN(IS,IP,MR) ENDDO ED(1)=EDS(ST1,NS,NDIM) C GOTO(70,70,66,70,66,70,66,70,66,70,66),LT


353

66 CALL SHFNPP(LINP,SLL,NAC,DS,AA,LT,0,MR) SUM=0.0D0 DO 68 IC=1,NPN IVR=NWL(IC) 68 SUM=SUM+AA(IC)*DI(IVR) V=ST(1)+ST(2)+ST(3) UI=SUM GOTO 72 70 V=ST(1)+ST(2)+ST(3) UI=PR(7,KM)*V*BK 72 CONTINUE C CALL STRSTN(IP,MR,NDN,KGO,ELCOD,SHFN,MAT,PR,D,BK, 1 EDINC,BL,BNL,BL1,STRESS,UI,ST,ITER,JS) C DO 45 IS=1,NS 45 STRAIN(IS,IP,MR)=STRAIN(IS,IP,MR)+ST(IS) DO IS=1,NS ST1(IS)=STRAIN(IS,IP,MR) ENDDO ED(2)=EDS(ST1,NS,NDIM) C IF(KGO.NE.3.AND.KGO.NE.4)GOTO 85 C---------------------------------------------------------------------C CALCULATE EXTRA VARIABLES FOR CAM-CLAY ONLY C---------------------------------------------------------------------85 CALL STRSEQ(MR,IP,IPA,NDIM,NDN,NS,STRESS,SHFN,CARTD,FT,DJACB, 1 R,RI,CR,NPLAX) C

END OF INTEGRATION POINTS LOOP 125 CONTINUE

C---------------------------------------------------------------------C ASSEMBLE EQUILIBRATING NODAL FORCES INTO GLOBAL ARRAY - PEQT C---------------------------------------------------------------------DO 150 IK=1,NDN II=NCONN(IK,MR) N1=NW(II)-1 C DO 150 ID=1,NDIM 150 PEQT(N1+ID)=PEQT(N1+ID)+FT(ID,IK) C END OF ELEMENTS LOOP 200 CONTINUE

354


C---------------------------------------------------------------------C CALCULATE OUT-OF-BALANCE NODAL LOADS C---------------------------------------------------------------------DO 230 IM=1,NDF 230 PEQT(IM)=PEQT(IM)+PEXI(IM) C

COUNT FOR SKEW BOUNDARIES IF(NSKEW.GT.0) CALL ROTBC(PEQT,NW,NDIM,NSKEW,1)

C KSTGE=4 TTGRV=1.0D0 C IF(ITER.EQ.1) THEN CALL EQLOD(NCONN,MAT,LTYP,NW,NQ,IDFX,NP1,NP2,XYZ,P, 1 PT,PCOR,PEQT,XYFT,PCONI,PR,0,1,TTGRV,0,KSTGE,0,IOUTP) ELSE CALL RESTRN(NDIM,NW,IDFX) DO 5 IK=1,NDF IF(IDFX(IK).EQ.1) THEN PCOR(IK)=ZERO ELSE PCOR(IK)=PT(IK)-PEQT(IK) END IF 5 CONTINUE ENDIF C RETURN END C ********************************************************************* SUBROUTINE CONVCH(NDF,NNODES,NDIM,TOL,ICONV,ICRIT,NW,UITER,UINC, 1 REPRE,RE,ITER,RNRM1) C---------------------------------------------------------------------C THIS SUBPROGRAM FINDS OUT IF THE CONVERGENCE HAS OCCURED BY USING C THE PREVIOUS AND CURRENT UNBALANCED LOADS OR DISPLACEMENTS OR BY C ENERGY CRITERIA C --------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION UITER(MDOF),UINC(MDOF),REPRE(MDOF),RE(MDOF),NW(MNODES+1)

C

ICONV=0 INTERNAL ENERGY CRITERION: ICONV1=0

Appendix: Fortran Codes of CS-S IF(ICRIT.EQ.1)THEN ERGPREV=ERG ERG=0.0D0 C DO I=1,NNODES N1=NW(I)-1 DO JJ=1,NDIM ERG=ERG+UITER(N1+JJ)*(RE(N1+JJ)-REPRE(N1+JJ)) ENDDO ENDDO IF(ITER.EQ.1)THEN ERG1=ERG RETURN ELSE IF(ABS(ERG).LT.ABS(TOL*ERG1))ICONV=1 IF(ABS(ERG).GT.ABS(ERGPREV))ICONV1=1

C

ENDIF DISPLACEMENT CRITERION: ELSE IF(ICRIT.EQ.2)THEN TOTNRM=0.0D0 CURNRM=0.0D0

C DO I=1,NNODES N1=NW(I)-1 DO JJ=1,NDIM TOTNRM=TOTNRM+UINC(N1+JJ)*UINC(N1+JJ) CURNRM=CURNRM+UITER(N1+JJ)*UITER(N1+JJ) ENDDO ENDDO C TOTNRM=SQRT(TOTNRM) CURNRM=SQRT(CURNRM) IF(CURNRM.LT.TOL*TOTNRM)ICONV=1 C IF UNBALANCED FORCE CRITERION: ELSE IF(ICRIT.EQ.3)THEN DO I=1,NDF RTMPNRM=0.0D0 RTMPNRM=RTMPNRM+RE(I)*RE(I) ENDDO RNRM=SQRT(RTMPNRM)

355

356

Appendix: Fortran Codes of CS-S IF(ITER.EQ.1)THEN RNRM1=RNRM RETURN ELSE IF(RNRM.LT.TOL*RNRM1)ICONV=1 ENDIF ENDIF RETURN END

C********************************************************************** SUBROUTINE UPOUT(XYZ,DI,DA,STRESS,STRAIN,P,PT,PCOR,PEQT,NCONN,MAT, 1 NTY,PR,LTYP,NW,NQ,IDFX,VARC,YREF,IOUTP) C---------------------------------------------------------------------C UPDATE AND OUTPUT ROUTINE C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NQ(MNODES),NW(MNODES+1),IDFX(MDOF) DIMENSION NCONN(NTPE,MEL),MAT(MEL),NTY(NMT),LTYP(MEL),PR(NPR,NMT) DIMENSION STRESS(NVRS,NIP,MEL),XYZ(3,MNODES) DIMENSION P(MDOF),PCOR(MDOF),PT(MDOF),PEQT(MDOF) DIMENSION DI(MDOF),DA(MDOF) DIMENSION ELCOD(3,NDMX),DS(3,20),SHFN(20) DIMENSION CIP(3),SLL(4) DIMENSION VARC(9,NIP,MEL),STRAIN(NVRN,NIP,MEL) DIMENSION NELPR(MEL),NELUS(MEL) DIMENSION SPA(3) DIMENSION XJACM(3,3) COMMON COMMON COMMON COMMON COMMON COMMON COMMON

/DATL / /ELINF / /DEVICE/ /PARS / /COUNT / /OUT / /JACB /

SL(4,100) LINFO(50,15) LINP,LOUT1,LOUT2,LOUT3,LOUT4,LOUT5,LSOLV PYI,ASMVL,ZERO NCS,NNGP IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF XJACI(3,3),DJACB

1

COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW

+ +

COMMON /ANLYS/ TTIME,DTIMEI,TGRAV,DGRAVI,FRACT,FRACLD, ICOR ,IDCHK,IOUT ,INCT ,IWL , NOIB ,JS ,JINCB ,NLOD ,NLDS


357

C ISTGE=4 LED=2 NS1=NS+1 NDIM1=NDIM+1 C---------------------------------------------------------------------C BREAK OUTPUT CODE C---------------------------------------------------------------------IOUT4=IOUT/1000 IOUT3=(IOUT-1000*IOUT4)/100 IOUT2=(IOUT-1000*IOUT4-100*IOUT3)/10 IOUT1=(IOUT-1000*IOUT4-100*IOUT3-10*IOUT2) IF(IOUT1.LT.1)GOTO 66 LT1=LTYP(1) LT1=IABS(LT1) IF(IOUTP.NE.1) GO TO 66 C GOTO(1,1,2,1,2,1,2,3,4,1,2),LT1 C 1 WRITE(LOUT1,902) C GOTO 66 C 2 WRITE(LOUT1,901) C GOTO 66 C 3 WRITE(LOUT1,933) C GOTO 66 C 4 WRITE(LOUT1,934) 66 CONTINUE C---------------------------------------------------------------------C UPDATE ABSOLUTE DISPLACEMENTS C---------------------------------------------------------------------CR=1.0D0 IF(NPLAX.EQ.1)CR=2.0D0*PYI DO 5 KD=1,NDF 5 DA(KD)=DA(KD)+DI(KD) C DO 10 JR=1,NN IF(JR.EQ.0)GOTO 10 J=JR NQL=NQ(JR) IF(NQL.EQ.0) GOTO 10 N1=NW(JR) IF(IOUT1.EQ.0)GOTO 10 IF(IOUT1.EQ.1.AND.JR.GT.NN)GOTO 10 IF(JR.LT.NN)GOTO 6 IF(JR.LT.NMOS.OR.JR.GT.NMOF)GOTO 10 GOTO 8 6 CONTINUE IF(JR.LT.NVOS.OR.JR.GT.NVOF)GOTO 10 8 CONTINUE

358


C N2=N1+NQL-1 IF(IOUTP.NE.1) GO TO 14 C IF(NDIM.EQ.3) GOTO 9 C IF(NQL.EQ.3)WRITE(LOUT1,900)JR,(DI(JJ),JJ=N1,N2),(DA(JJ),JJ=N1,N2) C IF(NQL.EQ.2)WRITE(LOUT1,910)JR,(DI(JJ),JJ=N1,N2),(DA(JJ),JJ=N1,N2) C IF(NQL.EQ.1)WRITE(LOUT1,911)JR,DI(N1),DA(N1) C GOTO 10 C 9 CONTINUE C IF(NQL.EQ.4)WRITE(LOUT1,940)JR,(DI(JJ),JJ=N1,N2),(DA(JJ),JJ=N1,N2) C IF(NQL.EQ.3)WRITE(LOUT1,941)JR,(DI(JJ),JJ=N1,N2),(DA(JJ),JJ=N1,N2) 10 CONTINUE IF(NDIM.EQ.3) GOTO 12 IF(IOUT2.EQ.2)WRITE(LOUT3,904) IF(IOUT2.EQ.1)WRITE(LOUT3,906) GOTO 14 12 CONTINUE IF(IOUT2.EQ.2)WRITE(LOUT3,904) IF(IOUT2.EQ.1)WRITE(LOUT3,936) 14 CONTINUE C---------------------------------------------------------------------C INITIALISE C---------------------------------------------------------------------DO 18 IM=1,NEL NELPR(IM)=0 NELUS(IM)=0 18 CONTINUE C---------------------------------------------------------------------C UPDATE NODAL CO-ORDINATES C---------------------------------------------------------------------ND=NN DO 220 J=1,NN N1=NW(J)-1 DO 220 ID=1,NDIM 220 XYZ(ID,J)=XYZ(ID,J)+DI(N1+ID) C---------------------------------------------------------------------C OUTPUT NODAL COORDINATES AND PWP C---------------------------------------------------------------------IF(IOUTP.EQ.1)THEN IF(NDIM.EQ.2)WRITE(LOUT1,301) IF(NDIM.EQ.3)WRITE(LOUT1,331) DO I33=1,NN NWL=NW(I33)-1 NQL=NQ(I33) ND1=NDIM+1 NDL=NWL+ND1 XNODE=XYZ(1,I33) YNODE=XYZ(2,I33)


359

YLIM=YREF-0.040D0 IF(XNODE.LT.0.040D0.AND.YNODE.GE.YLIM)THEN XNORM=XNODE/0.005640D0 YNORM=(YNODE)/0.005640D0 IF(NQL.EQ.ND1) THEN c IF(NDIM.EQ.2)WRITE(LOUT1,303)I33,XNORM,YNORM, IF(NDIM.EQ.2)WRITE(LOUT1,303)I33,XNODE,YNODE, 1 DA(NDL) IF(NDIM.EQ.3)WRITE(LOUT1,333)I33,(XYZ(IDIM,I33),IDIM=1,NDIM), 1 DA(NDL) c ELSE c IF(NDIM.EQ.2)WRITE(LOUT1,313)I33,(XYZ(IDIM,I33),IDIM=1,NDIM) c IF(NDIM.EQ.3)WRITE(LOUT1,323)I33,(XYZ(IDIM,I33),IDIM=1,NDIM) ENDIF c ENDIF ENDDO ENDIF C---------------------------------------------------------------------C CALCULATE AND OUTPUT STRESSES AT THE NODES C---------------------------------------------------------------------IF(IOUTP.EQ.1)CALL NODSTRS(NEL,NN,NS,LTYP,XYZ,NCONN,STRESS, 1 YREF) C---------------------------------------------------------------------C CALCULATE AND OUTPUT STRAINS AT THE NODES C---------------------------------------------------------------------IF(IOUTP.EQ.1)CALL NODSTRN(NEL,NN,NS,LTYP,XYZ,NCONN,STRAIN, 1 YREF) C---------------------------------------------------------------------C LOOP OVER THE ELEMENTS C---------------------------------------------------------------------IEL=0 DO 200 MR=1,NEL

c

KMAT=MAT(MR) KGO=NTY(KMAT) J=MR C IF(J.EQ.0)GOTO 200 LT=LTYP(J) NDN=LINFO(5,LT) NGP=LINFO(11,LT) INDX=LINFO(12,LT) NPN=LINFO(6,LT) NDPT=LINFO(1,LT) NAC=LINFO(15,LT) C---------------------------------------------------------------------C SETUP LOCAL NODAL COORDINATES OF ELEMENT C---------------------------------------------------------------------DO 20 KN=1,NDN

360

Appendix: Fortran Codes of CS-S NDE=NCONN(KN,J) DO 20 ID=1,NDIM 20 ELCOD(ID,KN)=XYZ(ID,NDE)

C 25 IF(IOUT2.NE.2.OR.IOUTP.NE.1)GOTO 26 IF(MR.LT.NELOS.OR.MR.GT.NELOF)GOTO 26 WRITE(LOUT3,908)MR IF(NDIM.EQ.2)WRITE(LOUT3,914) IF(NDIM.EQ.3)WRITE(LOUT3,944) 26 CONTINUE IEL=IEL+1 NELUS(IEL)=MR NELPR(IEL)=J C---------------------------------------------------------------------C LOOP ON INTEGRATION POINTS C---------------------------------------------------------------------DO 125 IP=1,NGP IPA=IP+INDX C DO 35 IL=1,NAC 35 SLL(IL)=SL(IL,IPA) C---------------------------------------------------------------------C INITIALISE SHAPE FUNCTION AND DERIVATIVES (LOCAL COORDS) C---------------------------------------------------------------------CALL ZEROR2(DS,3,20) CALL ZEROR1(SHFN,20) C CALCULATE SHAPE FUNCTIONS AND DERIVATIVES W.R.T LOCAL COOR. CALL SHAPE(LOUT1,SLL,NAC,DS,SHFN,LT,2,MR) CALL ZEROR2(XJACM,3,3) C CALCULATE [JAC], [JAC]-1 & DETJAC DO 15 IDIM=1,NDIM DO 15 JDIM=1,NDIM SUM=ZERO DO 112 IN=1,NDN 112 SUM=SUM+DS(IDIM,IN)*ELCOD(JDIM,IN) 15 XJACM(IDIM,JDIM)=SUM C CALL DETMIN(LOUT1,NDIM,XJACM,XJACI,DJACB,MR,IP,ISTGE) C---------------------------------------------------------------------C OUTPUT ABSOLUTE STRESSES C---------------------------------------------------------------------CALL PRINC(STRESS(1,IP,J),STRESS(2,IP,J),STRESS(4,IP,J),SPA) IF(IOUT2.EQ.0)GOTO 175 IF(IOUT2.EQ.1)GOTO 120 IKM=IP GOTO 122 120 IF(IOUT2.NE.1.OR.IP.NE.NGP)GOTO 175


361

IKM=MR C 122 DO 124 ID=1,NDIM SUM=ZERO C DO 123 IN=1,NDN 123 SUM=SUM+SHFN(IN)*ELCOD(ID,IN) 124 CIP(ID)=SUM IF(IOUTP.NE.1) GO TO 175 IF(MR.LT.NELOS.OR.MR.GT.NELOF)GOTO 175 IF(NDIM.EQ.2)WRITE(LOUT3,916)IKM,(CIP(ID),ID=1,NDIM), 1 (STRESS(IK,IP,J),IK=1,NS1),(SPA(JL),JL=1,3) IF(NDIM.EQ.3)WRITE(LOUT3,946)IKM,(CIP(ID),ID=1,NDIM), 1 (STRESS(IK,IP,J),IK=1,NS1) C 175 IF(KGO.NE.3.AND.KGO.NE.4)GOTO 125 C---------------------------------------------------------------------C CALCULATE EXTRA VARIABLES FOR CAM-CLAY AND UPDATE MCODE C---------------------------------------------------------------------CALL UPDCAM(STRESS,IP,MR,KMAT,NS,NDIM,PR,KGO,VARC) 125 CONTINUE C 200 CONTINUE C---------------------------------------------------------------------C OUTPUT ADDITIONAL PARAMETERS FOR CAM-CLAYS C---------------------------------------------------------------------CALL CAMOUT(LOUT4,LTYP,MAT,NTY,IOUT3,NEL,VARC,IOUTP) C 225 CONTINUE KSTGE=4

CALL EQLBM(LOUT2,NDIM,1,NN,NDF,NW,NQ,IDFX,P,PT,PCOR,PEQT, IOUT4,1,IRAC,IOUTP)

1

C---------------------------------------------------------------------C WRITE RESULTS ON SAVE FILE C---------------------------------------------------------------------C IF(ISR.EQ.0)GOTO 250 C IF(ISR.EQ.2)GOTO 240 C IF(ISR.EQ.1.AND.IWL.EQ.1)GOTO 240 C GOTO 250 C 240 WRITE(IW2) TTIME,TGRAV,XYZ,STRESS,STRAIN,DA,XYFT,PCOR,PCONI,LTYP,NMOD C WRITE(IW2) NF,MF,NFIX,DXYT C WRITE(IW2) NLED,LEDG,NDE1,NDE2,PRESLD C C 250 CONTINUE

362


RETURN 301 FORMAT(//,14X,’UPDATED NODE COORDINATES’,/,4X,48(’-’),/, 1 5X,’NODE’,12X,’X’,12X,’Y’,12X,’PWP’,/,4X,48(’-’)) 303 FORMAT(4X,I4,4X,2F12.6,4X,F10.3) 313 FORMAT(4X,I4,4X,2F12.6) 331 FORMAT(//,20X,’UPDATED NODE COORDINATES’,/,4X,58(’-’), 1 /,5X,’NODE’,10X,’X’,11X,’Y’,11X,’Z’,12X,’PWP’,/,4X,58(’-’)) 333 FORMAT(4X,I4,4X,4F12.4) 323 FORMAT(4X,I4,4X,3F12.4) 900 FORMAT(1X,I5,6E12.4) 901 FORMAT(//46H NODAL DISPLACEMENTS AND EXCESS PORE PRESSURES/ 1 1X,45(1H-)//21X,11HINCREMENTAL,26X,8HABSOLUTE// 12X,4HNODE,7X,2HDX,10X,2HDY,10X,2HDU,10X,2HDX,10X,2HDY,10X,2HDU/) 902 FORMAT(//20H NODAL DISPLACEMENTS/1X,19(1H-)// 1 18X,11HINCREMENTAL,33X,8HABSOLUTE// 1 2X,4HNODE,7X,2HDX,13X,2HDY,28X,2HDX,13X,2HDY/) 904 FORMAT(//40H ABSOLUTE STRESSES AT INTEGRATION POINTS/1X,39 (1H-)/) 906 FORMAT(//30H STRESSES AT ELEMENT CENTROIDS/1X,29(1H-)//8H ELEMENT, 1 3X,1HX,13X,1HY,11X,2HSX,11X,2HSY,11X,2HSZ,10X,3HTXY,12X,1HU, 1 10X,5HSIG-1,8X,5HSIG-2,7X,5HTH-XY) 908 FORMAT(/15H ELEMENT NUMBER,I5/1X,19(1H-)) 910 FORMAT(1X,I5,2E12.4,12X,2E12.4) 911 FORMAT(1X,I5,30X,E12.4,30X,E12.4) 914 FORMAT(2X,2HIP,7X,1HX,9X,1HY,10X,2HSX,10X,2HSY,10X,2HSZ, 1 9X,3HTXY,9X,1HU,9X,5HSIG-1,7X,5HSIG-2,7X,5HTH-XY/) 916 FORMAT(1X,I3,2F10.4,7E12.4,F10.1) 933 FORMAT(//20H NODAL DISPLACEMENTS/1X,19(1H-)// 1 18X,11HINCREMENTAL,51X,8HABSOLUTE// 1 2X,4HNODE,7X,2HDX,13X,2HDY,13X,2HDZ,28X,2HDX,13X,2HDY,13X,2HDZ/) 934 FORMAT(//46H NODAL DISPLACEMENTS AND EXCESS PORE PRESSURES/ 1 1X,45(1H-)//21X,11HINCREMENTAL,37X,8HABSOLUTE// 1 4X,4HNODE,5X,2HDX,10X,2HDY,10X,2HDZ,10X,2HDU, 1 10X,2HDX,10X,2HDY,10X,2HDZ,10X,2HDU/) 936 FORMAT(//30H STRESSES AT ELEMENT CENTROIDS/1X,29(1H-)//8H ELEMENT, 1 3X,1HX,13X,1HY,12X,1HZ,11X,2HSX,11X,2HSY,11X,2HSZ,11X,3HTXY, 1 11X,3HTYZ,10X,3HTZX,11X,1HU/) 940 FORMAT(1X,I5,8E12.4) 941 FORMAT(1X,I5,3E12.4,12X,3E12.4) 944 FORMAT(2X,2HIP,7X,1HX,9X,1HY,9X,1HZ,8X,2HSX, 1 10X,2HSY,10X,2HSZ,10X,3HTXY,9X,3HTYZ,10X,3HTZX,9X,1HU/) 946 FORMAT(1X,I3,3F10.4,7E12.4) END


363

C********************************************************************** SUBROUTINE STRSTN(IP,MR,NDN,KGO,ELCOD,SHFN,MAT,PR,D, 1 BK,EDINC,BL,BNL,BL1,STRESS,UI,DEPS,ITER,JS) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) COMMON/NSIZE / NN,NEL,NDF,NNOD1,NS,NPT,NSP,INXL, 1 MXEN,MXLD,MXFXT,NPLAX,LINR,NDIM,NSKEW DIMENSION EDINC(NB),F(3,3),FINV(3,3),EDOT(3,3),TEMP2(3,3), 1 DAB(3,3),SPK(3,3),SIG(3,3),SIGP(6),DEPS(6),TEMP1(6),TEMP3(6), 2 STRESS(NVRS,NIP,MEL),BL(6,NB),BNL(6,NB),BL1(9,NB) DIMENSION ELCOD(3,NDMX),SHFN(20),MAT(MEL),PR(NPR,NMT),D(6,6), 1 DSIG(6),SIGDOT(3,3),SPKDOT(3,3),DSIGCOR(6),VAR(6) DIMENSION DD(6,6),DDD(6,6),A(6),SALPHA(4),D2(3,3,3,3),IJ1(6) C WRITE(*,*)"STRSTN" C C IK(M,N)=(M/N)*(N/M) C KM=MAT(MR) PRM=PR(4,KM) ICOD=MCODE(IP,MR) C

INITIALIZE ARRAYS DO I1=1,6 DSIGCOR(I1)=0.0D0 TEMP1(I1)=0.0D0 TEMP3(I1)=0.0D0 VAR(I1)=0.0D0 SIGP(I1)=0.0D0 DSIG(I1)=0.0D0 ENDDO DO I1=1,3 DO J1=1,3 TEMP2(I1,J1)=0.0D0 DAB(I1,J1)=0.0D0 EDOT(I1,J1)=0.0D0 SIG(I1,J1)=0.0D0 SIGDOT(I1,J1)=0.0D0 SPKDOT(I1,J1)=0.0D0 ENDDO ENDDO

C IF(LINR.GT.1)CALL ADDBMAT(BL,BNL)

364


C

C

DO II1=1,NS TEMP=0.0D0 DO II2=1,NB TEMP=TEMP+BL(II1,II2)*EDINC(II2) write(2,*)’TEMP=’,TEMP ENDDO TEMP1(II1)=TEMP ENDDO

C---------------------------------------------------------------------C CONVERT IT INTO MATRIX TEMP1----> EDOT(I,J) C---------------------------------------------------------------------DO II1=1,3 DO II2=1,3 EDOT(II1,II2)=0.0D0 ENDDO ENDDO EDOT(1,1)=TEMP1(1) EDOT(2,2)=TEMP1(2) EDOT(3,3)=TEMP1(3) EDOT(1,2)=TEMP1(4) EDOT(2,1)=EDOT(1,2) IF(NDIM.NE.3) GO TO 101 EDOT(2,3)=TEMP1(5) EDOT(3,2)=EDOT(2,3) EDOT(1,3)=TEMP1(6) EDOT(3,1)=EDOT(1,3) 101 CONTINUE C---------------------------------------------------------------------C CALCULATE THE DEFORMATION GRADIENT MATRIX [F],[FINV] & DETF C---------------------------------------------------------------------CALL DEFGRAD(BL1,F,FINV,DETF,EDINC,NDIM,LINR) C---------------------------------------------------------------------C TRANSFORM THE STRAIN [EDOT] TO [DAB] BY: [DAB]=[FINV]\^T [EDOT][FINV] C---------------------------------------------------------------------DO II=1,3 DO IJ=1,3 TEMP=0.0D0 DO K=1,3 TEMP=TEMP+EDOT(II,K)*FINV(K,IJ) ENDDO TEMP2(II,IJ)=TEMP ENDDO ENDDO DO II=1,3 DO IJ=1,3


365

TEMP=0.0D0 DO K=1,3 TEMP=TEMP+FINV(K,II)*TEMP2(K,IJ) ENDDO DAB(II,IJ)=TEMP ENDDO ENDDO C WRITE(*,*)"STRSTN" C WRITE(*,*)"INI",DAB(1,1),DAB(1,2) C WRITE(*,*)"INI",DAB(2,1),DAB(2,2) C c c c

CORRECTION FOR ANTI-TENSION FAILURE DO IJ=1,3 IF(STRESS(IJ,IP,MR).LE.10.0)STRESS(IJ,IP,MR)=10.0 ENDDO

C---------------------------------------------------------------------C CALCULATE 2-ND PIOLA-KIRCHHOFF STRESS TENSOR SPK(3,3) C---------------------------------------------------------------------CALL ZEROR2(SPK,3,3) SPK(1,1)=STRESS(1,IP,MR) SPK(2,2)=STRESS(2,IP,MR) SPK(3,3)=STRESS(3,IP,MR) SPK(1,2)=STRESS(4,IP,MR) SPK(2,1)=STRESS(4,IP,MR) IF(NDIM.EQ.3)THEN SPK(2,3)=STRESS(5,IP,MR) SPK(3,2)=STRESS(5,IP,MR) SPK(1,3)=STRESS(6,IP,MR) SPK(3,1)=STRESS(6,IP,MR) ENDIF C---------------------------------------------------------------------C TRANSFORM SPK INTO SIG (CAUCHY STRESS) C---------------------------------------------------------------------DO II=1,3 DO IJ=1,3 TEMP=0.0D0 DO K=1,3 TEMP=TEMP+SPK(II,K)*F(K,IJ) ENDDO TEMP2(II,IJ)=TEMP ENDDO ENDDO DO II=1,3 DO IJ=1,3 TEMP=0.0D0 DO K=1,3

366

Appendix: Fortran Codes of CS-S TEMP=TEMP+F(K,II)*TEMP2(K,IJ) ENDDO SIG(II,IJ)=TEMP/DETF ENDDO ENDDO

C---------------------------------------------------------------------C CHANGE THE TENSORS TO VECTORS (SIG AND DAB) C---------------------------------------------------------------------DEPS(1)=DAB(1,1) SIGP(1)=SIG(1,1) DEPS(2)=DAB(2,2) SIGP(2)=SIG(2,2) DEPS(3)=DAB(3,3) SIGP(3)=SIG(3,3) DEPS(4)=DAB(1,2) SIGP(4)=SIG(1,2) DEPS(5)=DAB(2,3) SIGP(5)=SIG(2,3) DEPS(6)=DAB(1,3) SIGP(6)=SIG(1,3) C STRESS(1,IP,MR)=SIG(1,1) STRESS(2,IP,MR)=SIG(2,2) STRESS(3,IP,MR)=SIG(3,3) STRESS(4,IP,MR)=SIG(1,2) IF(NDIM.EQ.3)THEN STRESS(5,IP,MR)=SIG(2,3) STRESS(6,IP,MR)=SIG(1,3) ENDIF C---------------------------------------------------------------------C CALCULATE F(P,PC,Q) C---------------------------------------------------------------------IF(KGO.EQ.3.OR.KGO.EQ.4) THEN P1=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0 PC1=STRESS(NS+3,IP,MR) Q1=Q(SIGP,NS,NDIM) ETA1=Q1/P1 C ENDIF C---------------------------------------------------------------------C CALL THE CONSTITUTIVE MODEL TO FIND [D] C---------------------------------------------------------------------C WRITE(*,*)"ip=",IP,’MR=’,mr C PAUSE izero=0 GOTO(1,2,3),KGO


367

2 CALL DLIN(IP,izero,NDIM,NDN,ELCOD,SHFN,MAT,D,PR,BK) GOTO 1 3 CALL DMCAM(IP,MR,izero,NDIM,NS,STRESS,MAT,D,PR,BK,ITER,JS,DAB, +SALPHA,A,H) C WRITE(*,*)"HH",salpha(1) GOTO 1 C---------------------------------------------------------------------C COMPUTE EFFECTIVE COROTATIONAL/CAUCHY STRESS RATE TENSOR C---------------------------------------------------------------------1 CONTINUE DO 60 II=1,NS DSIG(II)=0.0D0 DO 60 JJ=1,NS c WRITE(*,*)"DDD",DSIG(II),D(II,JJ),DEPS(JJ) 60 DSIG(II)=DSIG(II)+D(II,JJ)*DEPS(JJ) C---------------------------------------------------------------------C CONVERT DSIG VECTOR TO SIGDOT TENSOR C---------------------------------------------------------------------SIGDOT(1,1)=DSIG(1) SIGDOT(2,2)=DSIG(2) SIGDOT(3,3)=DSIG(3) SIGDOT(1,2)=DSIG(4) SIGDOT(2,1)=DSIG(4) SIGDOT(2,3)=DSIG(5) SIGDOT(3,2)=DSIG(5) SIGDOT(1,3)=DSIG(6) SIGDOT(3,1)=DSIG(6) C---------------------------------------------------------------------C FIND OUT EFFICTIVE SPK STRESS INCREMENT AS: C [SPKDOT]=DETF [FINV][FINV]\{[SIGDOT]-[DAB][SIG]-[DBC][SIG]+DCC[SIG] C---------------------------------------------------------------------DCC=DAB(1,1)+DAB(2,2)+DAB(3,3) DO I=1,3 DO J=1,3 TEMP=0.0D0 TEM=0.0D0 DO K=1,3 TEMP=TEMP+DAB(I,K)*SIG(K,J) TEM=TEM+SIG(I,K)*DAB(J,K) ENDDO c WRITE(*,*)"SIGDOT",DCC,SIG(I,J),TEMP,TEM,SIGDOT(I,J) C WRITE(*,*)"LINT",LINR IF(LINR.GT.1)THEN

368 C Song

Appendix: Fortran Codes of CS-S SPKDOT(I,J)=DCC*SIG(I,J)-TEMP-TEM+SIGDOT(I,J) !alive 5/30/99 by SPKDOT(I,J)=SIGDOT(I,J) ELSE SPKDOT(I,J)=SIGDOT(I,J) ENDIF ENDDO ENDDO DO I=1,3 DO J=1,3 TEMP=0.0D0 DO K=1,3 TEMP=TEMP+SPKDOT(I,K)*FINV(K,J) ! Sik Xkj ENDDO TEMP2(I,J)=TEMP ENDDO ENDDO DO I=1,3 DO J=1,3 TEMP=0.0D0 DO K=1,3 TEMP=TEMP+FINV(K,I)*TEMP2(K,J) ! Xki Sik Xkj ENDDO SPKDOT(I,J)=TEMP*DETF ! Xki Sik Xkj J ENDDO ENDDO

C---------------------------------------------------------------------C UPDATE PORE WATER PRESSURE U C---------------------------------------------------------------------TEMP=0.0D0 DO I=1,NDIM TEMP=TEMP+F(I,I)*F(I,I) ENDDO STRESS(NS+1,IP,MR)=STRESS(NS+1,IP,MR)*TEMP/(NDIM*DETF) PWP=STRESS(NS+1,IP,MR) !Change of variable for D* routine C---------------------------------------------------------------------C UPDATE EFFECTIVE STRESSES STRESS C---------------------------------------------------------------------VAR(1)=STRESS(1,IP,MR)+SPKDOT(1,1) VAR(2)=STRESS(2,IP,MR)+SPKDOT(2,2) VAR(3)=STRESS(3,IP,MR)+SPKDOT(3,3) VAR(4)=STRESS(4,IP,MR)+SPKDOT(1,2) IF(NDIM.EQ.3)THEN VAR(5)=STRESS(5,IP,MR)+SPKDOT(2,3) VAR(6)=STRESS(6,IP,MR)+SPKDOT(1,3) ENDIF

Appendix: Fortran Codes of CS-S c c c c c C

369

correction for anti-tension failure DO IJ=1,3 IF(STRESS(IJ,IP,MR).LE.10.0)STRESS(IJ,IP,MR)=10.0 IF(VAR(IJ).LE.10.0)VAR(IJ)=10.0 ENDDO

IF(KGO.EQ.3.OR.KGO.EQ.4) THEN C---------------------------------------------------------------------C CORRECT THE YIELD SURFACE C---------------------------------------------------------------------P2=(VAR(1)+VAR(2)+VAR(3))/3.0D0 Q2=Q(VAR,NS,NDIM) IF(KGO.EQ.3) THEN PY=P2+Q2*Q2/(P2*PRM*PRM) ELSEIF(KGO.EQ.4)THEN PY=P2*EXP(Q2/(PRM*P2)) ENDIF IF(PY.LT.PC1.AND.ICOD.NE.4)GOTO 10 C---------------------------------------------------------------------C UPDATE THE HARDENING PARAMETER PC C---------------------------------------------------------------------ETA2=Q2/P2 DP12=P2-P1 DQ12=Q2-Q1 P12=(P1+P2)/2.0D0 ETA12=(ETA1+ETA2)/2.0D0 DETA=DQ12/P12-DP12*ETA12/P12 DPC=PC1*(DP12/P12+(2.0D0*ETA12*DETA)/(ETA12*ETA12+PRM*PRM)) IF(ICOD.EQ.4.AND.DPC.GT.0.0D0)DPC=0.0D0 IF(ICOD.NE.4.AND.DPC.LT.0.0D0)DPC=0.0D0 IF(ICOD.EQ.5)DPC=0.0D0 PC2=PC1+DPC write(2,*)’P2=’,p2,’ Q2=’,q2 c WRITE(2,*)’ETA2=’,eta2 c write(2,*)’DPC=’,dpc,’ PC2=’,pc2 C------------------------------------------------------------------C CORRECT STRESSES DUE TO DRIFTING OF YIELD SURFACE C------------------------------------------------------------------FYB=FPQ(P2,PC2,Q2,PRM,KGO) c WRITE(2,*)’FYB2=’,FYB 20 CALL STRSCOR(IP,MR,NDIM,NS,STRESS,VAR,MAT,PR,FYB,DSIGCOR,PC2, 1 ITER,JS,DAB,SALPHA,ICOD,H,A) VAR(1)=VAR(1)-DSIGCOR(1)

370

Appendix: Fortran Codes of CS-S VAR(2)=VAR(2)-DSIGCOR(2) VAR(3)=VAR(3)-DSIGCOR(3) VAR(4)=VAR(4)-DSIGCOR(4) IF(NDIM.EQ.3)THEN VAR(5)=VAR(5)-DSIGCOR(5) VAR(6)=VAR(6)-DSIGCOR(6) ENDIF

C

C

P2=(VAR(1)+VAR(2)+VAR(3))/3.0D0 Q2=Q(VAR,NS,NDIM) FYC=FPQ(P2,PC2,Q2,PRM,KGO) IF(KGO.EQ.3) THEN PY=P2+Q2*Q2/(P2*PRM*PRM) ELSEIF(KGO.EQ.4)THEN PY=P2*EXP(Q2/(PRM*P2)) ENDIF STRESS(NS+3,IP,MR)=PY

c

WRITE(2,*)’FYC=’,FYC ENDIF

C-------------------------------------------------------------------C UPDATE STRESSES AND PORE PRESSURE C-------------------------------------------------------------------10 STRESS(1,IP,MR)=VAR(1) STRESS(2,IP,MR)=VAR(2) STRESS(3,IP,MR)=VAR(3) STRESS(4,IP,MR)=VAR(4) IF(NDIM.EQ.3)THEN STRESS(5,IP,MR)=VAR(5) STRESS(6,IP,MR)=VAR(6) ENDIF STRESS(NS+1,IP,MR)=STRESS(NS+1,IP,MR)+UI C

Update the constitutive element stiffness matrix D* call Dstar(D,DD,SIG,FINV,DETF,NDIM,NS,LINR,PWP,D2)

C Update the constitutive element stiffness matrix for plastic spin D***=D*+D** C C WRITE(*,*)"LKK",SALPHA(1) CALL Dstar2(D2,A,SALPHA,DDD,SIG,FINV,DETF,NDIM,NS,LINR,H,IJ1, 1P2,IP,MR) C Combination of D* and D**, D*** = D* + D**


C C C

371

WRITE(*,*),D(1,1) do i=1,6 do j=1,6 D(i,j)=DD(i,j)+DDD(i,j) enddo enddo WRITE(*,*)"3",D(1,1) Save the modified constitutive matrix for use in ELMSTIF routine write(pp,11)(DD(I,J),J=1,NS),I=1,NS) RETURN END

C ********************************************************************* SUBROUTINE Dstar(D1,DD,SIG,FINV,DETF,NDIM,NS,LINR,PWP,D2) C --------------------------------------------------------------------C COMPTUTE [DD]=(D* TENSOR)=JF-1F-1F-1(D-SIG*IK-SIG*IK....) C --------------------------------------------------------------------C This subroutine was initially called from nowhere. C Now it is corrected & called from ELMSTIF Song, 4/16/99 IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ INTEGER*2 DIMENSION DIMENSION DIMENSION C C C C C

C

A,B D1(6,6),DD(6,6),DD1(6,6),SIG(3,3),FINV(3,3) D2(3,3,3,3),D3(3,3,3,3),IJ(6),MN(33),IK(3,3) De(6,6)

D1=Dij, uncorrected D 6x6 D2=DABCD, uncorrected D 3x3x3x3 D3=D*ABCD DD=Dij, corrected D 6x6 DD1=Dij, uncorrected D 6x6

Definition of Kronecker delta DO M=1,3 DO N=1,3 IF(M.EQ.N) THEN IK(M,N)=1 ELSE IK(M,N)=0 ENDIF ENDDO ENDDO

372


C

IK(M,N)=(M/N)*(N/M) LOUT1=2 ! ?

C

Give the numbers to index IJ DO I=1,6 IJ(I)=0 ENDDO IJ(1)=1 IJ(2)=5 IJ(3)=9 IJ(4)=2 IF(NDIM.EQ.3) THEN IJ(5)=6 IJ(6)=3 ENDIF

C

Give the numbers to index MN DO I=1,33 MN(I)=0 ENDDO MN(11)=1 MN(22)=2 MN(33)=3 MN(21)=4 MN(12)=4 MN(23)=5 MN(32)=5 MN(13)=6 MN(31)=6

C Give the numbers to M and M1 / Convert 9x9 D tensor to 3x3x3x3 D tensor. C logic confirmed by Song, 4/14/99 C--- Set D14, D24,D34 7/16/99 C This is for the non-coaxiality of principal direction & x,y,z D1(1,4)=D1(1,1)/10.0 D1(2,4)=D1(1,1)/10.0 D1(3,4)=D1(1,1)/10.0 D1(4,1)=D1(1,4) D1(4,2)=D1(2,4) D1(4,3)=D1(3,4)

C

DO I=1,3 DO J=1,3 M=(I-1)*3+J M=J+10*I ! 11,12,13,21,22,23,31,32,33 M1=MN(M) ! 1,4,6,4,2,5,6,5,3


373

DO K=1,3 DO L=1,3 C N=(K-1)*3+L N=L+10*K ! 11,12,13,21,22,23,31,32,33 N1=MN(N) ! 1,4,6,4,2,5,6,5,3 D2(I,J,K,L)=D1(M1,N1) ! 11111,1112,1113,1121,1122,1123, 1131,1132,1133,... C ! 11,14,16,14,12,15,16,15,13,... c WRITE(*,*)"111",I,J,K,L,M1,N1 c WRITE(*,*)"222",D2(I,J,K,L),D1(M1,N1) c PAUSE C D2 is also transferred and used in Dstar2 instead of [E]klmn ENDDO ENDDO ENDDO ENDDO C

Compute D* (Dstar) [D3] represents D* IF(LINR.GT.1)THEN DO A=1,3 DO B=1,3 DO C=1,3 DO D=1,3 TEMP=0.0D0 DO I=1,3 DO J=1,3 DO K=1,3 DO L=1,3 TEMP1=D2(I,J,K,L)-SIG(K,J)*IK(I,L)-SIG(I,K)*IK(J,L) + +SIG(I,J)*IK(K,L)-PWP*IK(I,J)*IK(K,L)+2.0D0*PWP*IK(I,K)*IK(J,L) TEMP2=FINV(I,A)*FINV(J,B)*FINV(K,C)*FINV(L,D) IF(LINR.LT.2)TEMP2=1 TEMP=TEMP+TEMP1*TEMP2 ENDDO ENDDO ENDDO ENDDO D3(A,B,C,D)=TEMP*DETF ENDDO ENDDO ENDDO ENDDO ENDIF

C Return back 3x3x3x3 D3 matrix to 9x9 DD matrix C Logic confirmed by Song. 4/14/99 DO I=1,NS

374

Appendix: Fortran Codes of CS-S M=IJ(I) ! 1,5,9,2,6,3 II=(M-1)/3+1 ! 1,2,3,1,2,1 JJ=M-(II-1)*3 ! 1,2,3,2,3,3 DO J=1,NS N=IJ(J) ! 1,5,9,2,6,3 KK=(N-1)/3+1 ! 1,2,3,1,2,1 LL=N-(KK-1)*3 ! 1,2,3,2,3,3 IF(LINR.GT.1)THEN DD(I,J)=D3(II,JJ,KK,LL) !1111,1122,1133,1112,1123,1113,2211, 2222,.... ELSE DD(I,J)=D2(II,JJ,KK,LL) ENDIF DD1(I,J)=D2(II,JJ,KK,LL) ENDDO ENDDO

C

Convert DD to D DO I=1,NS DO J=1,NS D1(I,J)=DD(I,J) ENDDO ENDDO RETURN END

C ********************************************************************* SUBROUTINE Dstar2(D2,A,SALPHA,DDD,SIG,FINV,DETF,NDIM,NS,LINR,H,IJ, 1P2,IP,MR) C --------------------------------------------------------------------C COMPTUTE [D**]=n[sigmbNmacd + sigajNjbcd](JXA,aXB,b...) C --------------------------------------------------------------------C This subroutine compute the D** matrix which considers plastic spin. C

Song, 4/16/99 IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’

COMMON/PLSPIN/PSPIN(27,500) DIMENSION DD1(6,6),TM(3,3,3,3),TN(3,3,3,3),B(3,3),Eta(3,3) DIMENSION Alpha(3,3),D4(3,3,3,3),DDD(6,6),IK(3,3),A(6),IJ(6) DIMENSION D2(3,3,3,3),SIG(3,3),FINV(3,3),SALPHA(4),Dumm(3,3)

Appendix: Fortran Codes of CS-S c WRITE(*,*)"DSTAR2" C D2=DABCD, uncorrected D 3x3x3x3 C A=Bij from DMCAM C D3=D*ABCD C DDD=Dij, corrected D 6x6 C DDD=Dij, uncorrected D 6x6 C Definition of Kronecker delta !This routine is O.K. checked 4/26/99 DO M=1,3 DO N=1,3 IF(M.EQ.N) THEN IK(M,N)=1 ELSE IK(M,N)=0 ENDIF ENDDO ENDDO LOUT1=2

! ?

C---- Hardening Modulus H (roundf/roundepsilon v)Bii C Hardening Modulus H was taken from H at DMCAM C---- Elastic Stiffness E C E was taken from D2 of previous step n-1 because this is incremental scheme. C D2 is 3x3x3x3 matrix from initial C---- (Eta)kl = [B]ij/norm [B]ij C [B]ij is taken from [A]n of DMCAM C Change the vector [B] to tensor [B] C Clear Bij first do i=1,3 do j=1,3 B(i,j)=0 enddo enddo B(1,1)=A(1) B(2,2)=A(2) B(3,3)=A(3) B(1,2)=A(4) B(1,3)=A(5) B(2,3)=A(6) !Other numbers are zero Bnorm1 = B(1,1)**2+B(2,2)**2 Bnorm2= B(3,3)**2+2*B(1,2)**2 Bnorm3=2*B(1,3)**2

375

376

c

Appendix: Fortran Codes of CS-S Bnorm4=2*B(2,3)**2 !Scalar this time Bnorm=SQRT(Bnorm1+Bnorm2+Bnorm3+Bnorm4) WRITE(*,*)"BNORM",Bnorm do i=1,3 !Cal. Eta do j=1,3 Eta(i,j)=B(i,j)/Bnorm enddo enddo

C---- parameter c !Scalar this time C c=H/(H+Eabcd nab ncd) C Calculate E nab ncd first dum=0 ! Initialize dum Do k=1,3 Do l=1,3 Dumm(k,l)=0 !Clear enddo enddo Do k=1,3 Do l=1,3 Do i=1,3 Do j=1,3 Dumm(k,l)=Dumm(k,l)+D2(i,j,k,l)*Eta(i,j) enddo enddo enddo enddo Do kk=1,3 Do ll=1,3 dum=dum+Dumm(kk,ll)*Eta(kk,ll) enddo enddo c=H/(H+dum) C---- [M] tensor C [M]=(c E n n)/H C Clear Do i=1,3 Do j=1,3 Do k=1,3 Do l=1,3 TM(i,j,k,l)=0 enddo enddo

Appendix: Fortran Codes of CS-S enddo enddo c

WRITE(*,*)"H in Dstar2",H Do k=1,3 Do l=1,3 Dumm(k,l)=0 !Clear enddo enddo

Do m=1,3 Do n=1,3 Do k=1,3 Do l=1,3 Dumm(m,n)=Dumm(m,n)+D2(k,l,m,n)*Eta(k,l) c WRITE(*,*)"K",k,l,m,n,D2(k,l,m,n) c pause enddo enddo enddo enddo

c c c

Do i=1,3 Do j=1,3 Do m=1,3 Do n=1,3 TM(m,n,i,j)=c*Dumm(m,n)*Eta(i,j)/(-H) WRITE(*,*)"m,n,i,j",m,n,i,j WRITE(*,*)"M",TM(m,n,i,j),Dumm(m,n),Eta(i,j) pause enddo enddo enddo enddo

C---- Back stress alpha C Back stress was taken from DMCAM & BKSTRS C Alpha was taken as Salpha times identity matrix. do i=1,3 do j=1,3 Alpha(i,k)=0 enddo enddo Alpha(1,1)=P2*SALPHA(1) Alpha(2,2)=P2*SALPHA(2) Alpha(3,3)=P2*SALPHA(3) Alpha(1,2)=P2*SALPHA(4)

377

378


Alpha(2,1)=P2*SALPHA(4) c WRITE(*,*)"111",SALPHA(1),ALPHA(2,2),ALPHA(3,3),ALPHA(1,2)

C---- [N] matrix C [N]=alpha M - M alpha do ia=1,3 do ib=1,3 do ic=1,3 do id=1,3 TM1=0 TM2=0 do m=1,3 TM1=TM1+alpha(ia,m)*TM(m,ib,ic,id) enddo do m=1,3 TM2=TM2+TM(ia,m,ic,id)*alpha(m,ib) enddo TN(ia,ib,ic,id)=TM1-TM2 ielem=MR if(TN(ia,ib,ic,id).EQ.0.0D0)GOTO 10 c write(*,*),TN(ia,ib,ic,id),ia,ib,ic,id c pause PSPIN(IP,ielem)=TN(ia,ib,ic,id) 10 CONTINUE c write(6,100),TM1,TM2,TN(ia,ib,ic,id) c pause enddo enddo enddo enddo c 100 FORMAT(/,3x,F15.13,3x,F15.13,3x,F15.13) C---- Etadot ! Scalar function C Eta was assumed 0.001 to 0.1 as constant Etadot=0.001 C---- D** ! Final Product C D**=Eta*[sig N +sig N](J XA,a XB,b XC,c XD,d) do do do do

iaa=1,3 ibb=1,3 icc=1,3 idd=1,3

do ia=1,3


379

do ib=1,3 do ic=1,3 do id=1,3 Dum1=0 do m=1,3 Dum1=Dum1+SIG(m,ib)*TN(m,ia,ic,id)+SIG(ia,m)*TN(m,ib,ic,id) enddo Dum2=Dum2+Dum1*FINV(iaa,ia)*FINV(ibb,ib)*FINV(icc,ic)* 1FINV(idd,id) enddo enddo enddo enddo D4(iaa,ibb,icc,idd)=Etadot*Dum2*DETF enddo enddo enddo enddo C

Give the numbers to index IJ DO I=1,6 IJ(I)=0 ENDDO IJ(1)=1 IJ(2)=5 IJ(3)=9 IJ(4)=2 IF(NDIM.EQ.3) THEN IJ(5)=6 IJ(6)=3 ENDIF

DO I=1,NS M=IJ(I) ! 1,5,9,2,6,3 II=(M-1)/3+1 ! 1,2,3,1,2,1 JJ=M-(II-1)*3 ! 1,2,3,2,3,3 DO J=1,NS N=IJ(J) ! 1,5,9,2,6,3 KK=(N-1)/3+1 ! 1,2,3,1,2,1 LL=N-(KK-1)*3 ! 1,2,3,2,3,3 DDD(I,J)=D4(II,JJ,KK,LL) !1111,1122,1133,1112,1123,1113,2211, 2222,....

380 C

Appendix: Fortran Codes of CS-S WRITE(*,*)"DDD",DDD(1,1) ENDDO ENDDO RETURN END

C********************************************************************** SUBROUTINE UPDCAM(STRESS,IP,MR,KM,NS,NDIM,PR,KGO,VARC) C********************************************************************** C CALCULATE EXTRA STRESS PARAMETERS FOR CAM-CLAYS C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) DIMENSION STRESS(NVRS,NIP,MEL),PR(NPR,NMT) DIMENSION VARC(9,NIP,MEL),TEMP(6) C CALL ZEROR1(TEMP,6) PRM=PR(4,KM) U=STRESS(NS+1,IP,MR) C---------------------------------------------------------------------C CHECK IF MCODE STATUS FOR CAM MODEL NEEDS TO BE CHANGED C---------------------------------------------------------------------DO IS=1,NS TEMP(IS)=STRESS(IS,IP,MR) ENDDO PC1=PQMOD(IP,MR,1) ETA1=PQMOD(IP,MR,2) P2=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0 Q2=Q(TEMP,NS,NDIM) PC2=STRESS(NS+3,IP,MR) ETA2=Q2/P2

ICOD=MCODE(IP,MR)

IF(KGO.EQ.3) THEN PCS=PC2/2.0D0 PY=P2+Q2*Q2/(P2*PRM*PRM) ELSEIF(KGO.EQ.4)THEN PCS=PC2/EXP(1.0d0) PY=P2*EXP(Q2/(PRM*P2)) ENDIF


381

C Mcode is the index parameter for elastic, plastic, hardening condition. C Later it is changed to ICOD in DMCAM IF(ICOD.EQ.2) THEN !Initial Pc IF(PY.LT.0.9950D0*PC2)THEN !Pc2=new Pc IF(P2.LT.PCS)MCODE(IP,MR)=3 !OC - Hvoslev IF(P2.GE.PCS)MCODE(IP,MR)=1 !Roscoe Same as initial cond. ELSEIF(PY.GE.0.9950D0*PC2)THEN !Initial point Pc IF(ETA2.GE.PRM.AND.PC2.GT.PC1) THEN !OC above and need expansion YC C PC2=PC1+(PRM-ETA1)*(PY-PC1)/(ETA2-ETA1) MCODE(IP,MR)=5 C GO TO 100 ELSEIF(P2.LT.PCS.AND.PC2.LE.PC1)THEN !OC elastic MCODE(IP,MR)=4 !Condition stress adjusted to much ENDIF ENDIF GO TO 200

C C

C C

ELSEIF(ICOD.EQ.4) THEN ! IF(ETA2.LT.PRM.AND.PC2.LT.PC1) THEN !below CSL OC or NC PC2=PC1-(ETA1-PRM)*(PC1-PY)/(ETA1-ETA2) MCODE(IP,MR)=5 GO TO 100 ELSEIF(PC2.GE.PC1) THEN !Need hardening IF(P2.LT.0.9950D0*PY.AND.P2.LT.PCS)MCODE(IP,MR)=3 !+OC IF(P2.LT.0.9950D0*PY.AND.P2.GE.PCS)MCODE(IP,MR)=1 !+NC ENDIF GO TO 200 ELSEIF(ICOD.EQ.1) THEN IF(PY.LT.0.9950D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=3 IF(PY.GE.0.9950D0*PC2)THEN IF(ETA2.GT.PRM.AND.PC2.GT.PC1) THEN PC2=PC1+(PRM-ETA1)*(PY-PC1)/(ETA2-ETA1) MCODE(IP,MR)=5 GO TO 100 ELSEIF(ETA2.LT.PRM.AND.P2.GE.PCS)THEN MCODE(IP,MR)=2 ELSEIF(P2.LT.PCS.AND.PC2.LE.PC1)THEN MCODE(IP,MR)=4 ENDIF ENDIF GO TO 200 ELSEIF(ICOD.EQ.3) THEN IF(PY.LT.0.9950D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=1 IF(PY.GE.0.9950D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=2

382

Appendix: Fortran Codes of CS-S IF(PY.GE.0.9950D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=4 GO TO 200

C C C

C C C C C C C C C C C C C C C C C C C C C C C

ELSEIF(ICOD.EQ.5) THEN IF(PY.LT.0.990D0*PC2.AND.P2.GE.PCS)MCODE(IP,MR)=1 IF(PY.LT.0.990D0*PC2.AND.P2.LT.PCS)MCODE(IP,MR)=3 ENDIF GO TO 200 100 STRESS(NS+3,IP,MR)=PC MCODE(IP,MR)=5 FYD=FPQ(P2,PC,Q2,PRM,KGO) CALL STRSCOR(IP,MR,NDIM,NS,STRESS,MAT,PR,FYD,DSIGCOR,KGO) STRESS(1,IP,MR)=STRESS(1,IP,MR)-DSIGCOR(1) STRESS(2,IP,MR)=STRESS(2,IP,MR)-DSIGCOR(2) STRESS(3,IP,MR)=STRESS(3,IP,MR)-DSIGCOR(3) STRESS(4,IP,MR)=STRESS(4,IP,MR)-DSIGCOR(4) IF(NDIM.EQ.3)THEN STRESS(5,IP,MR)=STRESS(5,IP,MR)-DSIGCOR(5) STRESS(6,IP,MR)=STRESS(6,IP,MR)-DSIGCOR(6) ENDIF 200 CONTINUE DO IS=1,NS TEMP(IS)=STRESS(IS,IP,MR) ENDDO P2=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0 Q2=Q(TEMP,NS,NDIM) ETA2=Q2/P2 PC=STRESS(NS+3,IP,MR) FYD=FPQ(P2,PC,Q2,PRM,KGO 200 PQMOD(IP,MR,1)=PC2 PQMOD(IP,MR,2)=ETA2

c c C C C

ICOD2=MCODE(IP,MR) WRITE(2,*)’ICOD1=’,ICOD,’ICOD2=’,ICOD2 DO IS=1,NS TEMP(IS)=STRESS(IS,IP,MR) ENDDO

C C C C C

QT=Q(TEMP,NS,NDIM) PE=(STRESS(1,IP,MR)+STRESS(2,IP,MR)+STRESS(3,IP,MR))/3.0D0 EE=STRESS(NS+2,IP,MR) PC=STRESS(NS+3,IP,MR)

C

IF(KGO.EQ.3) THEN

Appendix: Fortran Codes of CS-S C C C C C c C C C

C

383

PY=PE+QT*QT/(PE*PRM*PRM) PCS=PC/2.0D0 ELSEIF(KGO.EQ.4) THEN PY=PE*EXP(QT/(PRM*PE)) PCS=PC/EXP(1.0d0) ENDIF FYC=FPQ(PE,PC,QT,PRM,KGO) IF(PCS.LT.0.0D0) THEN WRITE(*,*)’WARNING: PC LESS THAN ZERO FOR ELM-IP’,MR,IP STOP ENDIF EE=PR(3,KM)-PR(1,KM)*ALOG(P2)-(PR(2,KM)-PR(1,KM))*ALOG(PCS) IF(P2.GT.0.0D0)EE=PR(3,KM)-PR(1,KM)*LOG(P2)1(PR(2,KM)-PR(1,KM))*LOG(PCS)

C VARC(1,IP,MR)=P2 VARC(2,IP,MR)=Q2 VARC(3,IP,MR)=P2+U VARC(4,IP,MR)=PC2 VARC(5,IP,MR)=ETA2 VARC(6,IP,MR)=Q2/(P2*PRM) VARC(7,IP,MR)=PY/PC2 VARC(8,IP,MR)=EE STRESS(NS+2,IP,MR)=EE C CALL ANGTH(STRESS,IP,MR,THETA) VARC(9,IP,MR)=THETA C IF(NDIM.EQ.3)VARC(9,IP,MR)=0.0D0 RETURN END C********************************************************************** SUBROUTINE ANGTH(STRESS,IP,J,THETA) C********************************************************************** C ROUTINE TO CALCULATE ANGLE IN PI PLANE C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION STRESS(NVRS,NIP,MEL) COMMON /PARS / PYI,ASMVL,ZERO C ALAR=1.0D+25 SX=STRESS(1,IP,J)

384

Appendix: Fortran Codes of CS-S SY=STRESS(2,IP,J) SZ=STRESS(3,IP,J) TXY=STRESS(4,IP,J)

C

8 9

10 15

PIBY4=0.250D0*PYI SD=0.50D0*(SX-SY) SM=0.50D0*(SX+SY) RAD=SQRT(SD*SD+TXY*TXY) SIG1=SM+RAD SIG3=SM-RAD DY=SY-SM IF(ABS(TXY).LT.ASMVL.AND.ABS(DY).LT.ASMVL)GOTO 8 THXY2=ATAN2(TXY,DY) GOTO 9 THXY2=0.50D0*PYI THXY=0.50D0*THXY2 THXYD=THXY*180.0D0/PYI IF(ABS(THXY).LT.PIBY4)GOTO 10 PSIGX=SIG1 PSIGY=SIG3 GOTO 15 PSIGX=SIG3 PSIGY=SIG1 PSIGZ=SZ

C

CC

SIGX=(PSIGZ-PSIGY)/SQRT(2.0D0) SIGY=(2.0D0*PSIGX-PSIGY-PSIGZ)/SQRT(6.0D0) RADO=SQRT(SIGX*SIGX+SIGY*SIGY) IF(ABS(SIGX).LT.ASMVL.AND.ABS(SIGY).LT.ASMVL)GOTO 20

C THETA=ATAN2(SIGY,SIGX) IF(THETA.LT.ZERO)THETA=2.0D0*PYI+THETA THETA=THETA*180.0D0/PYI GOTO 25 C 20 THETA=ALAR 25 CONTINUE C RETURN END C********************************************************************** SUBROUTINE STRSEQ(JJ,IP,IPA,NDIM,NDN,NS,STRESS,SHFN,CARTD,F,DJACB, 1 R,RI,CR,NPLAX) C********************************************************************** C ROUTINE TO CALCULATE FORCES EQUILIBRATING C ELEMENTAL STRESSES (INTEGRATION POINT CONTRIBUTION) C**********************************************************************


385

IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION STRESS(NVRS,NIP,MEL),SHFN(20),CARTD(3,NDMX) DIMENSION F(3,NDMX) COMMON /DATW / W(100) C F9=CR*DJACB*W(IPA) IF(NPLAX.EQ.1)F9=F9*R C U=STRESS(NS+1,IP,JJ) C SIGXT=STRESS(1,IP,JJ)+U SIGYT=STRESS(2,IP,JJ)+U SIGZT=STRESS(3,IP,JJ)+U TXY=STRESS(4,IP,JJ) IF(NDIM.EQ.2)GOTO 35 C TYZ=STRESS(5,IP,JJ) TZX=STRESS(6,IP,JJ) C DO 30 IN=1,NDN F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+CARTD(2,IN)*TXY 1 +CARTD(3,IN)*TZX)*F9 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY 1 +CARTD(3,IN)*TYZ)*F9 F(3,IN)=F(3,IN)+(CARTD(3,IN)*SIGZT+CARTD(2,IN)*TYZ 1 +CARTD(1,IN)*TZX)*F9 30 CONTINUE GOTO 60 C 35 DO 40 IN=1,NDN F(1,IN)=F(1,IN)+(CARTD(1,IN)*SIGXT+SHFN(IN)*SIGZT*RI 1 +CARTD(2,IN)*TXY)*F9 40 F(2,IN)=F(2,IN)+(CARTD(2,IN)*SIGYT+CARTD(1,IN)*TXY)*F9 60 CONTINUE RETURN END C********************************************************************** SUBROUTINE PRINC(C,D,E,B) C********************************************************************** C CALCULATES PRINCIPAL STRESSES AND THEIR DIRECTIONS C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) DIMENSION B(3) COMMON /PARS / PYI,ASMVL,ZERO C AP=C+D

386

Appendix: Fortran Codes of CS-S AD=C-D S=SQRT(.250D0*AD*AD+E*E) B(1)=.50D0*AP+S B(2)=.50D0*AP-S B(3)=90.0D0 IF(ABS(AD).LT.ASMVL) GO TO 2 B(3)=28.64790D0*ATAN(2.0D0*E/AD) 2 RETURN END

C********************************************************************** SUBROUTINE NODSTRS(NEL,NNODES,NS,LTYP,XYZ,NCONN,STRESS,YREF) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),LTYP(MEL),XYZ(3,MNODES),SHAPE(4) DIMENSION STRESS(NVRS,NIP,MEL),SIG(MNODES,6),ICOUNT(MNODES) DIMENSION R(8),S(8),PRST(3),SPIN(537) COMMON /ELINF / LINFO(50,15) COMMON/PLSPIN/PSPIN(NIP,MEL) DATA R(1),R(2),R(3),R(4),R(5),R(6),R(7),R(8)/ 1 -1.73205080D0,1.73205080D0,1.73205080D0,-1.73205080D0, 2 0.0D0,1.73205080D0,0.0D0,-1.73205080D0/ DATA S(1),S(2),S(3),S(4),S(5),S(6),S(7),S(8)/ 1 -1.73205080D0,-1.73205080D0,1.73205080D0,1.73205080D0, 2 -1.73205080D0,0.0D0,1.73205080D0,0.0D0/ C---------------------------------------------------------------------C INITIALIZE C---------------------------------------------------------------------CALL ZEROI1(ICOUNT,MNODES) CALL ZEROR2(SIG,MNODES,6) CALL ZEROR1(SHAPE,4) C---------------------------------------------------------------------C LOOP OVER THE ELEMENTS C---------------------------------------------------------------------DO IELM=1,NEL LT=LTYP(IELM) NDN=LINFO(5,LT) NGP=LINFO(11,LT) C---------------------------------------------------------------------C RETURN IF THE ELEMENT DOES NOT HAVE 8-NODES,4 INTEGRATION POINT C----------------------------------------------------------------------


387

IF(NDN.NE.8.OR.NGP.NE.4)THEN WRITE(*,*)’CANNOT CALCULATE NODAL STRESSES FOR ELM=’,IELM,NDN,NGP RETURN ENDIF C---------------------------------------------------------------------C LOOP OVER THE ELEMENT NODES C---------------------------------------------------------------------DO INOD=1,NDN NODE=NCONN(INOD,IELM) ICOUNT(NODE)=ICOUNT(NODE)+1 C---------------------------------------------------------------------C SHAPE FUNCTIONS FOR 8-NODES,4 INTEGRATION POINT CASE C---------------------------------------------------------------------SHAPE(1)=0.250D0*(1.0D0-R(INOD))*(1.0D0-S(INOD)) SHAPE(2)=0.250D0*(1.0D0+R(INOD))*(1.0D0-S(INOD)) SHAPE(3)=0.250D0*(1.0D0+R(INOD))*(1.0D0+S(INOD)) SHAPE(4)=0.250D0*(1.0D0-R(INOD))*(1.0D0+S(INOD)) C---------------------------------------------------------------------C LOOP OVER THE INTEGRATION POINTS C---------------------------------------------------------------------DO IP=1,NGP DO IS=1,NS SIG(NODE,IS)=SIG(NODE,IS)+STRESS(IS,IP,IELM)*SHAPE(IP) ENDDO SPIN(NODE)=SPIN(NODE)+PSPIN(IP,IELM)*SHAPE(IP) !other spin = 0 ENDDO ENDDO ENDDO C WRITE(2,10) C---------------------------------------------------------------------C FIND AVERAGE STRSSS AT NODES C---------------------------------------------------------------------DO INOD=1,NNODES A=FLOAT(ICOUNT(INOD)) DO IS=1,NS SIG(INOD,IS)=SIG(INOD,IS)/A ENDDO C---------------------------------------------------------------------C CALCULATE PRINCIPLE STRESSES C---------------------------------------------------------------------CALL PRINC(SIG(INOD,1),SIG(INOD,2),SIG(INOD,4),PRST) C----------------------------------------------------------------------

388


C OUTPUT NODAL STRESSES C---------------------------------------------------------------------XNODE=XYZ(1,INOD) YNODE=XYZ(2,INOD) YLIM=YREF-0.040D0 c IF(XNODE.LT.0.040D0.AND.YNODE.GE.YLIM)THEN XNORM=XNODE/0.005640D0 YNORM=(YNODE)/0.005640D0 c WRITE(2,100)INOD,XNORM,YNORM,(SIG(INOD,J),J=1,NS), WRITE(2,100)INOD,XNODE,YNODE,(SIG(INOD,J),J=1,NS), 1 (PRST(K),K=1,3),SPIN(INOD) c ENDIF ENDDO

10 1 2 3 100

FORMAT(//,3X,’NODE’,3X,’NODE COORDINATES’,19X,’NODE STRESSES’, 18x,’PRINCIPAL STRESSES’,/,1X,120(’-’),/,11X,’X’,8X,’Y’, 12X,’SX’,10X, ’SY’,10X,’SZ’,9X,’SXY’,9X,’S1’,10X,’S2’, 10X,’TH’,10X,’SPIN’/,1X,120(’-’)) FORMAT(1X,I4,1X,2F9.5,2X,6E12.4,F9.2,2X,E12.4)

RETURN END

C********************************************************************** SUBROUTINE NODSTRN(NEL,NNODES,NS,LTYP,XYZ,NCONN,STRAIN,YREF) C********************************************************************** C 1This subroutine update the nodal strain C 2. Called by DMCAM IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCONN(NTPE,MEL),LTYP(MEL),XYZ(3,MNODES),SHAPE(4) DIMENSION STRAIN(NVRN,NIP,MEL),SIG(MNODES,6),ICOUNT(MNODES) DIMENSION R(8),S(8),PRST(3) COMMON /ELINF / LINFO(50,15) DATA R(1),R(2),R(3),R(4),R(5),R(6),R(7),R(8)/ 1 -1.73205080D0,1.73205080D0,1.73205080D0,-1.73205080D0, 2 0.0D0,1.73205080D0,0.0D0,-1.73205080D0/ DATA S(1),S(2),S(3),S(4),S(5),S(6),S(7),S(8)/ 1 -1.73205080D0,-1.73205080D0,1.73205080D0,1.73205080D0, 2 -1.73205080D0,0.0D0,1.73205080D0,0.0D0/


389

C---------------------------------------------------------------------C INITIALIZE C---------------------------------------------------------------------CALL ZEROI1(ICOUNT,MNODES) CALL ZEROR2(SIG,MNODES,6) CALL ZEROR1(SHAPE,4) C---------------------------------------------------------------------C LOOP OVER THE ELEMENTS C---------------------------------------------------------------------DO IELM=1,NEL LT=LTYP(IELM) NDN=LINFO(5,LT) NGP=LINFO(11,LT) C---------------------------------------------------------------------C RETURN IF THE ELEMENT DOES NOT HAVE 8-NODES,4 INTEGRATION POINT C---------------------------------------------------------------------IF(NDN.NE.8.OR.NGP.NE.4)THEN WRITE(*,*)’CANNOT CALCULATE NODAL STRAINS FOR ELM=’,IELM,NDN,NGP RETURN ENDIF C---------------------------------------------------------------------C LOOP OVER THE ELEMENT NODES C---------------------------------------------------------------------DO INOD=1,NDN NODE=NCONN(INOD,IELM) ICOUNT(NODE)=ICOUNT(NODE)+1 C---------------------------------------------------------------------C SHAPE FUNCTIONS FOR 8-NODES,4 INTEGRATION POINT CASE C---------------------------------------------------------------------SHAPE(1)=0.250D0*(1.0D0-R(INOD))*(1.0D0-S(INOD)) SHAPE(2)=0.250D0*(1.0D0+R(INOD))*(1.0D0-S(INOD)) SHAPE(3)=0.250D0*(1.0D0+R(INOD))*(1.0D0+S(INOD)) SHAPE(4)=0.250D0*(1.0D0-R(INOD))*(1.0D0+S(INOD)) C---------------------------------------------------------------------C LOOP OVER THE INTEGRATION POINTS C---------------------------------------------------------------------DO IP=1,NGP DO IS=1,NS !NS=size of D matrix=4 for 2-D,=6 for 3-D SIG(NODE,IS)=SIG(NODE,IS)+STRAIN(IS,IP,IELM)*SHAPE(IP) !Eventhough sig is used here that is really strain. !Just to save the memory.

390

Appendix: Fortran Codes of CS-S ENDDO ENDDO ENDDO ENDDO

C WRITE(2,10) C---------------------------------------------------------------------C FIND AVERAGE STRSSS AT NODES C---------------------------------------------------------------------DO INOD=1,NNODES A=FLOAT(ICOUNT(INOD)) DO IS=1,NS SIG(INOD,IS)=SIG(INOD,IS)/A ENDDO C---------------------------------------------------------------------C CALCULATE PRINCIPLE STRAINS C---------------------------------------------------------------------CALL PRINC(SIG(INOD,1),SIG(INOD,2),SIG(INOD,4),PRST) C---------------------------------------------------------------------C OUTPUT NODAL STRAINS C---------------------------------------------------------------------XNODE=XYZ(1,INOD) YNODE=XYZ(2,INOD) YLIM=YREF-0.040D0 c IF(XNODE.LT.0.040D0.AND.YNODE.GE.YLIM)THEN XNORM=XNODE/0.005640D0 YNORM=(YNODE)/0.005640D0 c WRITE(2,100)INOD,XNORM,YNORM,(SIG(INOD,J),J=1,NS), WRITE(2,100)INOD,XNODE,YNODE,(SIG(INOD,J),J=1,NS), 1 (PRST(K),K=1,3) c ENDIF ENDDO 10 1 1 1 100

FORMAT(//,3X,’NODE’,3X,’NODE COORDINATES’,19X,’NODE STRAINS’, 18x,’PRINCIPAL STRAINS’,/,1X,106(’-’),/,11X,’X’,8X,’Y’, 12X,’STX’,9X, ’STY’,9X,’STH’,9X,’STXY’,9X,’S1’,10X,’S2’, 10X,’TH’,/,1X,106(’-’)) FORMAT(1X,I4,1X,2F9.5,2X,6E12.4,F9.2)

RETURN END C********************************************************************** FUNCTION FPQ(P,PC,Q,PM,KGO) C********************************************************************** C THIS FUNCTION CALCULATES F(P,PC,Q)


391

C---------------------------------------------------------------------IMPLICIT REAL*8(A-H,O-Z) IF(KGO.EQ.3) THEN FPQ=(P*P)/(PC*PC)-(P/PC)+(Q*Q)/(PC*PC*PM*PM) ELSEIF(KGO.EQ.4)THEN FPQ=LOG(P)-LOG(PC)+Q/(P*PM) C FPQ=ALOG(P)-ALOG(PC)+Q/(P*PM) ENDIF RETURN END C********************************************************************** SUBROUTINE CAMOUT(LOUT4,LTYP,MAT,NTY,IOUT3,NEL,VARC,IOUTP) C********************************************************************** C *** OUTPUT ADDITIONAL PARAMETERS CAM-CLAYS C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /MODEL /PQMOD(NIP,MEL,2),MCODE(NIP,MEL) COMMON /ELINF / LINFO(50,15) COMMON /OUT / IBC,IRAC,NVOS,NVOF,NMOS,NMOF,NELOS,NELOF DIMENSION MAT(MEL),NTY(NMT),LTYP(MEL),VARC(9,NIP,MEL) C IF(IOUT3.EQ.0.OR.IOUTP.NE.1)GOTO 25 IF(IOUT3.EQ.1)WRITE(LOUT4,911) IF(IOUT3.EQ.1)WRITE(LOUT4,902) IF(IOUT3.EQ.2)WRITE(LOUT4,912) IF(IOUT3.EQ.2)WRITE(LOUT4,901) C DO 20 J=1,NEL KM=MAT(J) KGO=NTY(KM) LT=LTYP(J) NGP=LINFO(11,LT) IF(KGO.NE.3.AND.KGO.NE.4)GO TO 20 IF(IOUTP.NE.1) GO TO 20 IF(J.LT.NELOS.OR.J.GT.NELOF)GOTO 20 IF(IOUT3.EQ.1)GOTO 12 IF(IOUT3.EQ.2)WRITE(LOUT4,904)J C DO 10 IGP=1,NGP WRITE(LOUT4,905)IGP,(VARC(IK,IGP,J),IK=1,9),MCODE(IGP,J) 10 CONTINUE GOTO 20 12 WRITE(LOUT4,905)J,(VARC(IK,NGP,J),IK=1,9), 1 (MCODE(IP,J),IP=1,NGP)

392

Appendix: Fortran Codes of CS-S 20 CONTINUE WRITE(LOUT4,935)

C 25 CONTINUE RETURN 901 FORMAT(2X,6HELM-IP,6X,2HPE,11X,1HQ,11X,2HPT,11X, 1 2HPC,9X,3HETA,5X,5HETA/M,6X,2HYR,4X,6HE-VOID,3X, 2 4HTH-3,2X,3HCDE) 902 FORMAT(2X,6HELM-IP,6X,2HPE,11X,1HQ,11X,2HPT,11X, 1 2HPC,9X,3HETA,5X,5HETA/M,6X,2HYR,4X,6HE-VOID,3X, 2 4HTH-3,2X,14H 1 2 3 4 5 6 7) 904 FORMAT(I4) 905 FORMAT(2X,I4,4E13.5,2F9.3,3X,F6.3,F8.4,2X,F7.1, 1 2X,8I2/5X,9I2) 911 FORMAT(/33H CAM CLAY PARAMETERS AT CENTROIDS/ 1 1X,32(1H-)/) 912 FORMAT(/42H CAM CLAY PARAMETERS AT INTEGRATION POINTS/ 1 1X,41(1H-)/) 935 FORMAT(//) END C********************************************************************** FUNCTION Q(A,N,NDIM) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(N) Q2=0.50D0*((A(1)-A(2))*(A(1)-A(2))+(A(2)-A(3))*(A(2)-A(3)) 1 +(A(3)-A(1))*(A(3)-A(1)))+3.0D0*A(4)*A(4) IF(NDIM.EQ.2)GOTO 10 Q2=Q2+3.0D0*A(5)*A(5)+3.0D0*A(6)*A(6) 10 Q=SQRT(Q2) RETURN END C********************************************************************** FUNCTION EDS(A,N,NDIM) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(N) EDS2=0.50D0*((A(1)-A(2))*(A(1)-A(2))+(A(2)-A(3))*(A(2)-A(3)) 1 +(A(3)-A(1))*(A(3)-A(1)))+.750D0*A(4)*A(4) IF(NDIM.EQ.2)GOTO 10 EDS2=EDS2+0.750D0*A(5)*A(5)+0.750D0*A(6)*A(6) 10 EDS=2.0D0*SQRT(EDS2)/3.0D0 RETURN END


393

SUBROUTINE INPENT(LINP,NPEN,XREF,YREF,DYREF,DXPEN, 1 XCOS,XSIN,XMUFR,NCHAIN,KSLID0,KSLID3) C********************************************************************** C 1 Called by TOTSOL IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ DIMENSION NCHAIN(100,2),XMUFR(2) C

----- INITIALIZE NCHAIN MATRIX CALL ZEROI2(NCHAIN,100,2)

C

----- READ INPUT DATA FOR PENETRATION READ(LINP,101)HEADER18 101 FORMAT(A80) READ(LINP,*)XREF,YREF,DXPEN,XDIM,YDIM,KSLID0,KSLID3, 1 (XMUFR(I),I=1,2) TEMP=SQRT(XDIM**2+YDIM**2) XCOS=YDIM/TEMP XSIN=XDIM/TEMP C DYREF=DXPEN/XSIN !DYREF=incremental distance at cone face DYREF=DXPEN/XCOS !by Song to control it by vertical displ. C !Above manipulation should not make cal. diff. READ(LINP,101)HEADER19 DO I=1,NPEN READ(LINP,*)(NCHAIN(I,J),J=1,2) ENDDO C RETURN END

SUBROUTINE CHSLIDE(NPEN,NSKEW,NSKEW1,DXPEN,KSLID0,KSLID3,XSIN, 1 XCOS,XMUFR,NCHAIN,NW,PEQT,PT,FRICT,FRICTPR,INCR,FRMAX0,FRMAX3) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF DIMENSION NCHAIN(100,2),NW(MNODES+1),PEQT(MDOF), 1 PT(MDOF),XMUFR(2),FRICT(MDOF),FRICTPR(MDOF) C---------------------------------------------------------------------NODSLID0=0 NODSLID3=0 ICOUNT0=0 ICOUNT3=0

394

Appendix: Fortran Codes of CS-S TOTFR0=0.0D0 TOTFR3=0.0D0 TOTALFR0=0.0D0 TOTALFR3=0.0D0 FVNOD0=0.0D0 FMNOD0=0.0D0 FVNOD3=0.0D0 FMNOD3=0.0D0 ICV0=0 ICM0=0 ICV3=0 ICM3=0

C---------------------------------------------------------------------DO 100 I=1,NPEN NODE =NCHAIN(I,1) INDEX=NCHAIN(I,2) IVM=NODE-2*(NODE/2) N1=NW(NODE)-1 IF(INDEX.NE.0.AND.INDEX.NE.3) GO TO 100 C DO II=1,NF INODE=MF(II) IF(INODE.EQ.NODE) GO TO 1 ENDDO 1 CONTINUE C---------------------------------------------------------------------C CHECK NODES WITH INDEX=0 C---------------------------------------------------------------------IF(INDEX.EQ.0.AND.KSLID0.EQ.0) THEN ICOUNT0=ICOUNT0+1 RNORM = -PEQT(N1+1) FRICT0=-PT(N1+2)+PEQT(N1+2) TOTFR0=TOTFR0+FRICT0 ALLFR0=RNORM*XMUFR(1) IF(IVM.EQ.1)THEN FVNOD0=FVNOD0+ALLFR0 ICV0=ICV0+1 ELSEIF(IVM.EQ.0)THEN ICM0=ICM0+1 FMNOD0=FMNOD0+ALLFR0 ENDIF TOTALFR0=TOTALFR0+ALLFR0 IF(FRICT0.LT.ALLFR0) THEN NODSLID0=NODSLID0+1 ENDIF


395

C---------------------------------------------------------------------C CHECK NODES WITH INDEX=3 C---------------------------------------------------------------------ELSEIF(INDEX.EQ.3.AND.KSLID3.EQ.0)THEN ICOUNT3=ICOUNT3+1 RXX= -PEQT(N1+1) RYY= -PEQT(N1+2) RNORM =RXX*XCOS-RYY*XSIN FRICT3=-RXX*XSIN-RYY*XCOS TOTFR3=TOTFR3+FRICT3 ALLFR3=RNORM*XMUFR(2) IF(IVM.EQ.1)THEN ICV3=ICV3+1 FVNOD3=FVNOD3+ALLFR3 ELSEIF(IVM.EQ.0)THEN ICM3=ICM3+1 FMNOD3=FMNOD3+ALLFR3 ENDIF TOTALFR3=TOTALFR3+ALLFR3 IF(FRICT3.LT.ALLFR3) THEN NODSLID3=NODSLID3+1 ENDIF ENDIF 100 CONTINUE C---------------------------------------------------------------------C CHECK IF SLIDING OCCURS AT NODES WITH INDEX=0 C---------------------------------------------------------------------IF(NODSLID0.GE.(ICOUNT0))GO TO 2 IF(TOTFR0.LT.TOTALFR0)GO TO 2 GO TO 20 2 CONTINUE AVRM0=FMNOD0/ICM0 AVRV0=FVNOD0/ICV0 FRMAX0=TOTALFR0/ICOUNT0 KSLID0=1 DO 200 I=1,NPEN NODE =NCHAIN(I,1) INDEX=NCHAIN(I,2) INDFOR=NCHAIN(I+1,2) N1=NW(NODE)-1 IF(INDEX.NE.0)GO TO 200

396


C DO II=1,NF INODE=MF(II) IF(INODE.EQ.NODE) GO TO 3 ENDDO 3 CONTINUE IF(INDFOR.EQ.3.AND.KSLID3.EQ.0)GO TO 200 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0 FRICT(N1+2)=-PT(N1+2)+PEQT(N1+2) FRICTPR(N1+2)=-PT(N1+2)+PEQT(N1+2) 200 CONTINUE 20 CONTINUE C---------------------------------------------------------------------C CHECK IF SLIDING OCCURS AT NODES WITH INDEX=3 C---------------------------------------------------------------------IF(NODSLID3.GE.ICOUNT3)GO TO 4 IF(TOTFR3.LT.TOTALFR3)GO TO 4 GO TO 30 4 CONTINUE AVRM3=FMNOD3/ICM3 AVRV3=FVNOD3/ICV3 FRMAX3=TOTALFR3/ICOUNT3 KSLID3=1 NSKEW=NSKEW1 DO 300 I=1,NPEN NODE =NCHAIN(I,1) INDEX=NCHAIN(I,2) INDPR=NCHAIN(I-1,2) N1=NW(NODE)-1 IF(INDEX.NE.3)GO TO 300 C DO II=1,NF INODE=MF(II) IF(INODE.EQ.NODE) GO TO 5 ENDDO 5 CONTINUE IF(INDPR.EQ.0.AND.KSLID0.NE.0)THEN DO IIP=1,NF INODEP=MF(IIP) IF(INODEP.EQ.(NODE-1)) GO TO 11 ENDDO 11 N11=NW(NODE-1)-1 NFIX(2,IIP)=0 DXYT(1,IIP)=0.0D0 DXYT(2,IIP)=0.0D0


397

FRICT(N11+2)=-PT(N11+2)+PEQT(N11+2) FRICTPR(N11+2)=-PT(N11+2)+PEQT(N11+2) ENDIF NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)=DXPEN DXYT(2,II)=0.0D0 RXX= -PEQT(N1+1) RYY= -PEQT(N1+2) RYPAR=-RXX*XSIN-RYY*XCOS FRICT(N1+1)=RYPAR*XSIN FRICT(N1+2)=RYPAR*XCOS FRICTPR(N1+1)=RYPAR*XSIN FRICTPR(N1+2)=RYPAR*XCOS 300 CONTINUE 30 CONTINUE RETURN END C********************************************************************** SUBROUTINE BCADJST(NPEN,NDIM,NSKEW,XREF,YREF,DXPEN,XSIN, 1 XCOS,NCHAIN,XYZ,NW,PEQT,PT,FRICT,FRICTPR,XMUFR,PNOD,DD,DP,DP1, 2 INCR,NUMD,IOUTP,TOTPEN,IBCAL,FRMAX0,FRMAX3) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) INCLUDE ’PARM.FOR’ COMMON /FIX / DXYT(4,200),MF(200),NFIX(4,200),NF COMMON /SKBC /ISPB(20),DIRCOS(20,3) DIMENSION XYZ(3,MNODES),NCHAIN(100,2),NW(MNODES+1),PEQT(MDOF), 1 PT(MDOF),DD(4,200),DP(MDOF),DP1(MDOF),NUMD(MDOF,2),PNOD(MDOF) DIMENSION QC(MDOF),FRICT(MDOF),FRICTPR(MDOF),XMUFR(2) C IBCAL=IBCAL+1 FRMAX0=-2.0d-03 FRMAX3=-0.8082640d-03 c XMUFR(1)=0.250D0 c XMUFR(2)=0.250D0 C CALL ZEROR1(QC,MDOF) NUM=40 NUMB=40 I1=0 C DO 100 I=1,NPEN NODE =NCHAIN(I,1) INDEX=NCHAIN(I,2)

398

Appendix: Fortran Codes of CS-S XNODE=XYZ(1,NODE) YNODE=XYZ(2,NODE) N1=NW(NODE)-1

C DO II=1,NF INODE=MF(II) IF(INODE.EQ.NODE) GO TO 1 ENDDO 1 CONTINUE C---------------------------------------------------------------------C CHECK NODES WITH INDEX=0 IF TENSION OCCURS (0) C---------------------------------------------------------------------IF(INDEX.EQ.0) THEN QC(N1+2)=PT(N1+2)-PEQT(N1+2)

c

C C

C

c c C

REACTX=PT(N1+1)-PEQT(N1+1) INDFOR1=NCHAIN(I+1,2) INDFOR2=NCHAIN(I+2,2) IF(IBCAL.EQ.1.AND.INDFOR1.EQ.3)THEN DP1(N1+2)=-PEQT(N1+2)/10.0D0 NUMD(N1+2,1)=10 WRITE(2,*)’NODE=’,NODE,’DP1=’,DP1(N1+2) ENDIF IF(REACTX.GT.0.0D0.AND.INDFOR1.NE.0.AND.INDFOR2.NE.0) THEN IF(REACTX.GT.0.0D0) THEN CHANGE INDEX FROM 0 TO 4 AND FREE THE NODE NCHAIN(I,2)=4 NFIX(1,II)=0 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0 DD(1,II)=0.0D0 APPLY LOAD=REACT & THEN UNLOAD IT INCREMENTALLY NUMBER=INT(ABS(REACTX)/1.0D-04)+1 DP(N1+1) =REACTX/NUMBER PNOD(N1+1)=REACTX NUMD(N1+1,1)=NUMBER WRITE(2,*)’NODE=’,NODE,’NUM=’,NUMBER,’PNOD=’,PNOD(N1+1), 1 ’DP=’,DP(N1+1) CHECK FOR FRICTION ELSEIF(REACTX.LT.0.0D0)THEN

C ALLFRICT=REACTX*XMUFR(1) FPND=ALLFRICT+PNOD(N1+2) IF(FPND.GT.0.0D0)GO TO 10 APPLFR=FRICTPR(N1+2)


c c C

399

IF(ALLFRICT.LT.APPLFR.AND.APPLFR.GT.FRMAX0)THEN FRDIF=ALLFRICT-FRICTPR(N1+2) IF(FRDIF.GE.(-2.0D-05)) THEN FRICT(N1+2)=FRDIF ELSE FRICT(N1+2)=-2.0D-05 ENDIF FRICTPR(N1+2)=FRICTPR(N1+2)+FRICT(N1+2) WRITE(2,*)’NODE=’,NODE,’FRICT=’,FRICT(N1+2),’FRICTPR=’, 1 FRICTPR(N1+2),’RNORM=’,REACTX

ENDIF ENDIF GO TO 10 C---------------------------------------------------------------------C CHECK NODES WITH INDEX=4 IF XNODE < XREF (4) C---------------------------------------------------------------------ELSEIF(INDEX.EQ.4) THEN INDFOR1=NCHAIN(I+1,2) INDFOR2=NCHAIN(I+2,2) QC(N1+2)=PT(N1+2)-PEQT(N1+2)

C

DXNODE=XNODE-XREF INDPR=NCHAIN(I-1,2) IF(DXNODE.LT.0.0D0) THEN CONSTRAIN THE NODE IN THE X-DIRECTION & CHANGE INDEX FROM 4 TO 0 NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0 DP(N1+1)=0.0D0 PNOD(N1+1)=0.0D0 NUMD(N1+1,1)=0 NUMBER2=INT(ABS(DXNODE)/(1.0D-05))+1 NUMD(N1+1,2)=NUMBER2 DD(1,II)=-DXNODE/NUMBER2 NCHAIN(I,2)=0 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) GOTO 10

C ELSEIF(INDFOR1.EQ.0.AND.INDFOR2.EQ.0)THEN C

CONSTRAIN THE NODE IN THE X-DIRECTION & CHANGE INDEX FROM 4 TO 0 NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0

400

Appendix: Fortran Codes of CS-S DP(N1+1)=0.0D0 PNOD(N1+1)=0.0D0 NUMD(N1+1,1)=0 NUMBER2=INT(ABS(DXNODE)/(1.0D-05))+1 NUMD(N1+1,2)=NUMBER2 DD(1,II)=-DXNODE/NUMBER2 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) NCHAIN(I,2)=0 ENDIF GO TO 10

C---------------------------------------------------------------------C CHECK NODES WITH INDEX=3 (SKEW BOUNDARY) (3) C---------------------------------------------------------------------ELSE IF(INDEX.EQ.3) THEN INDPR=NCHAIN(I-1,2) RXX = PT(N1+1)-PEQT(N1+1) RYY = PT(N1+2)-PEQT(N1+2) QC(N1+2)=-(-PEQT(N1+1)*XSIN+PEQT(N1+2)*XCOS)

C

IF(XNODE.GT.XREF.AND.INDPR.NE.3) THEN FREE THE SKEW NODE AND CHANGE INDEX FROM 3 TO 4 NFIX(1,II)=0 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0 DD(1,II)=0.0D0 DD(2,II)=0.0D0 NCHAIN(I,2)=4

C

REMOVE THE NODE FROM SKEW BOUNDARY AND SHIFT THE ISPB ARRAY CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,-1)

C

ADD LOADS = REACTIONS AND THEN UNLOAD THEM RXX = PT(N1+1)-PEQT(N1+1) RYY = PT(N1+2)-PEQT(N1+2) ABRXX=ABS(RXX*XCOS) NUMBER=INT(ABRXX/1.0D-04)+1 DP(N1+1) = RXX*XCOS/NUMBER DP(N1+2) =-RXX*XSIN/NUMBER PNOD(N1+1)=RXX*XCOS PNOD(N1+2)=-RXX*XSIN NUMD(N1+1,1)=NUMBER NUMD(N1+2,1)=NUMBER NUMD(N1+1,2)=0


c c C

c

c c c c

401

NUMD(N1+2,2)=0 WRITE(2,*)’NODE=’,NODE,’NUM=’,NUMBER,’PNOD=’,PNOD(N1+1), 1 PNOD(N1+2),’DP=’,DP(N1+1),DP(N1+2) ELSE CHECK FOR FRICTION RXX1=RXX+PNOD(N1+1)*XCOS IF(RXX1.LT.0.0D0)THEN ALLFRICT=RXX1*XMUFR(2) APPLFR=FRICTPR(N1+1)*XSIN+FRICTPR(N1+2)*XCOS IF(ALLFRICT.LT.APPLFR.AND.APPLFR.GT.FRMAX3)THEN IF(ALLFRICT.LT.APPLFR)THEN IF(XNODE.LT.0.0010D0)GOTO 55 FRDIF=ALLFRICT-APPLFR IF(FRDIF.GE.(-2.0D-05)) THEN RYPAR=FRDIF ELSE RYPAR=-2.0D-05 ENDIF FRICT(N1+1)=RYPAR*XSIN FRICT(N1+2)=RYPAR*XCOS FRICTPR(N1+1)=FRICTPR(N1+1)+FRICT(N1+1) FRICTPR(N1+2)=FRICTPR(N1+2)+FRICT(N1+2) WRITE(2,*)’NODE=’,NODE,’FRICT=’,(FRICT(N1+IFR),IFR=1,2), 1 ’FRICTPR=’,applfr,’FRMAX3=’,frmax3 write(2,*)’pt=’,(pt(n1+ifr),ifr=1,2),’peqt=’, 1 (peqt(n1+ifr),ifr=1,2),’rxx=’,rxx,’ryy=’,ryy ENDIF 55 CONTINUE ENDIF

C C IF(XNODE.LT.0.0D0) THEN C APPLY DISPL TO MOVE THE NODE BACK ALONG THE POSITIVE SKEW BOUNDARY DD(1,II)=0.0D0 XNODE3=-XNODE/XSIN NUMBER2=INT(ABS(XNODE3)/(1.0D-05)) IF(NUMBER2.LT.1)NUMBER2=1 DD(2,II)=XNODE3/NUMBER2 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) NUMD(N1+1,2)=0 NUMD(N1+2,2)=NUMBER2 ENDIF ENDIF GO TO 10

402


C---------------------------------------------------------------------C CHECK THE NODE WITH INDEX = 2 (2) C---------------------------------------------------------------------ELSE IF(INDEX.EQ.2) THEN DYNODE=YNODE-(YREF+XNODE*XCOS/XSIN) INDPR=NCHAIN(I-1,2)

C

C

C

IF(XNODE.LT.0.0D0.AND.YNODE.LT.YREF) THEN RECONSTRAIN THE NODE & CHANGE INDEX FROM 2 TO 1 I1=I1+1 NCHAIN(I,2)=1 NFIX(1,II) =1 NUMBER2=INT(ABS(XNODE)/(1.0D-05)) IF(NUMBER2.LT.1)NUMBER2=1 DD(1,II) =-XNODE/NUMBER2 NUMD(N1+1,2)=NUMBER2 NUMD(N1+2,2)=0 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) PNOD(N1+1)=0.0D0 DP(N1+1)=0.0D0 GO TO 20 ELSEIF(XNODE.GT.0.0D0.AND.DYNODE.GE.0.0D0) THEN CHANGE INDEX FROM 2 TO 3 AND ADD THE NODE TO SKEW BOUNDARY NCHAIN(I,2)=3 CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,1) CHANGE THE BOUNDARY CONDITION OF THE NODE NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)=DXPEN DXYT(2,II)=0.0D0

C

C C C C C C

XNOD2=DYNODE*XSIN NUMBER2=INT(ABS(XNOD2)/(1.0D-05))+1 DD(1,II) = DYNODE*XSIN/NUMBER2 DD(2,II) = 0.0D0 PNOD(N1+2)=PNOD(N1+1)*XSIN DP(N1+2) =DP(N1+1)*XSIN PNOD(N1+1)=0.0D0 DP(N1+1) =0.0D0 NUMD(N1+2,1)=NUMD(N1+1,1) NUMD(N1+1,1)=0 NUMD(N1+1,2)=NUMBER2 NUMD(N1+2,2)=0 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2)


403

C GO TO 10

C

ELSEIF(XNODE.LE.0.0D0.AND.YNODE.GE.YREF) THEN CHANGE THE INDEX FROM 2 TO 3 AND ADD THE NODE TO SKEW BOUNDARY NCHAIN(I,2)=3 NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)= DXPEN DXYT(2,II)= 0.0D0 A=YNODE-YREF B=-XNODE*XSIN/XCOS IF(A.GE.B) THEN XNOD2=DYNODE*XSIN NUMBER2=INT(ABS(XNOD2)/(1.0D-05))+1 DD(1,II)=DYNODE*XSIN/NUMBER2 DD(2,II)=0.0D0 NUMD(N1+1,2)=NUMBER2 NUMD(N1+2,2)=0 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) ELSEIF(A.LT.B) THEN NUMBER2=INT(ABS(B)/(1.0D-05)) IF(NUMBER2.LT.1)NUMBER2=1 DD(1,II)=(-XNODE*XCOS+(YNODE-YREF)*XSIN)/NUMBER2 DD(2,II)=(-XNODE*XSIN-(YNODE-YREF)*XCOS)/NUMBER2 NUMD(N1+1,2)=NUMBER2 NUMD(N1+2,2)=NUMBER2 WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2) ENDIF CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,1)

C C C C C C C

PNOD(N1+2)=PNOD(N1+1)*XSIN DP(N1+2) =DP(N1+1)*XSIN PNOD(N1+1)=0.0D0 DP(N1+1) =0.0D0 NUMD(N1+2,1)=NUMD(N1+1,1) NUMD(N1+1,1)=0

ENDIF GO TO 10 C---------------------------------------------------------------------C CHECK THE FIRST NODE WITH INDEX=1 IF TENSION OCCURS (1) C----------------------------------------------------------------------

404

Appendix: Fortran Codes of CS-S ELSE IF(INDEX.EQ.1.AND.I1.EQ.0) THEN I1=I1+1 INDPR=NCHAIN(I-1,2) DYNODE=YNODE-YREF REACTX=PT(N1+1)-PEQT(N1+1)

C C

C

IF(YNODE.GE.YREF) THEN CHANGE INDEX FROM 1 TO 3 AND ADD THE NODE TO SKEW BOUNDARY I1=I1-1 NCHAIN(I,2)=3 CALL CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,1) CHANGE THE BOUNDARY CONDITION OF THE NODE NFIX(1,II)=1 NFIX(2,II)=0 DXYT(1,II)=DXPEN DXYT(2,II)=0.0D0

C

c c

NUMBER2=INT(ABS(DYNODE*XSIN)/(1.0D-05))+1 DD(1,II) = DYNODE*XSIN/NUMBER2 DD(2,II) = 0.0D0 PNOD(N1+1)=REACTX NUMBER=INT(ABS(REACTX)/1.0D-04)+1 DP(N1+1)=REACTX/NUMBER NUMD(N1+1,1)=NUMBER NUMD(N1+1,2)=NUMBER2 NUMD(N1+2,2)= 0 WRITE(2,*)’NODE=’,NODE,’NUM=’,NUMBER,’PNOD=’,PNOD(N1+1), 1 ’DP=’,DP(N1+1) WRITE(2,*)’NODE’,NODE,’N=’,NUMBER2,’dd=’,(dd(ifr,ii),IFR=1,2)

C C

C

c c

ELSEIF(YNODE.LT.YREF.AND.REACTX.GT.0.0D0.AND.INDPR.NE.2) THEN CHANGE INDEX FROM 1 TO 2 AND FREE THE NODE NCHAIN(I,2)=2 NFIX(1,II)=0 NFIX(2,II)=0 DXYT(1,II)=0.0D0 DXYT(2,II)=0.0D0 APPLY LOAD=REACT AND THEN UNLOAD IT INCREMENTALLY NUMBER=INT(ABS(REACTX)/1.0D-04)+1 DP(N1+1) =REACTX/NUMBER PNOD(N1+1)=REACTX NUMD(N1+1,1)=NUMBER WRITE(2,*)’NODE=’,NODE,’NUM=’,NUMBER,’PNOD=’,PNOD(N1+1), 1 ’DP=’,DP(N1+1) ENDIF GO TO 10 ENDIF

Appendix: Fortran Codes of CS-S 10 CONTINUE 100 CONTINUE 20 CONTINUE c c

IF(IOUTP.EQ.1) THEN DO I33=1,20

c c c c c c c

NODE =NCHAIN(I33,1) INDEX=NCHAIN(I33,2) XNODE=XYZ(1,NODE) YNODE=XYZ(2,NODE) ND1=NW(NODE) ENDDO ENDIF DO INODE=1,NF NODE=MF(INODE) IDF=NW(NODE)-1 DO ID=1,NDIM NUM1=NUMD(IDF+ID,1) NUM2=NUMD(IDF+ID,2) IF(NUM1.GT.0) THEN PNOD(IDF+ID)=PNOD(IDF+ID)*(NUM1-1)/NUM1 NUMD(IDF+ID,1)=NUM1-1 ELSEIF(NUM1.EQ.0) THEN PNOD(IDF+ID)=0.0D0 DP(IDF+ID)=0.0D0 DP1(IDF+ID)=0.0D0 ENDIF IF(NUM2.GT.0) THEN NUMD(IDF+ID,2)=NUM2-1 ELSEIF(NUM2.EQ.0) THEN DD(ID,INODE)=0.0D0 ENDIF ENDDO

c ENDDO C IF(IOUTP.EQ.1) THEN WRITE(2,30)INCR,YREF,TOTPEN WRITE(2,40) DO I=1,NPEN NODE =NCHAIN(I,1) N1=NW(NODE)-1 WRITE(2,50)(NCHAIN(I,J),J=1,2),(XYZ(J,NODE),J=1,2),QC(N1+2)

405

406

Appendix: Fortran Codes of CS-S ENDDO ENDIF 30 FORMAT(/,2X,’INCR=’,I5,4X,’YREF=’,F12.6,4X,’TOTPEN=’,F12.6) 40 FORMAT(/,6X,’NCHAIN’,9X,’COORDINATES’,15X,’QC’,/,55(’-’)) 50 FORMAT(2X,2I5,2F12.6,4X,E15.7)

c C C C C C C

WRITE(2,*)’ NODE+ DXYT+ NFIX’ DO II=1,20 WRITE(2,100)MF(II),(DXYT(J,II),J=1,2),(NFIX(J,II),J=1,2) ENDDO 100 FORMAT(1X,I5,2F12.6,2I5) 200 FORMAT(1X,I3,4I4,2X,6E12.4) RETURN END

C********************************************************************** SUBROUTINE CHSKEW(NSKEW,NDIM,NODE,XCOS,XSIN,IND) C********************************************************************** IMPLICIT REAL*8(A-H,O-Z) COMMON /SKBC /ISPB(20),DIRCOS(20,3)

C

C

IF(IND.EQ.1) THEN ADD THE NODE FROM SKEW BOUNDARY DO ISKEW=1,NSKEW KNODE=ISPB(ISKEW) IF(KNODE.GT.NODE) THEN DO KSKEW=ISKEW,NSKEW KK=NSKEW-KSKEW+1 ISPB(ISKEW+KK)=ISPB(ISKEW+KK-1) DIRCOS(ISKEW+KK,1)=DIRCOS(ISKEW+KK-1,1) DIRCOS(ISKEW+KK,2)=DIRCOS(ISKEW+KK-1,2) ENDDO ISPB(ISKEW)=NODE DIRCOS(ISKEW,1)=XCOS DIRCOS(ISKEW,2)=XSIN GO TO 25 ENDIF ENDDO ISPB(NSKEW+1)=NODE DIRCOS(NSKEW+1,1)=XCOS DIRCOS(NSKEW+1,2)=XSIN 25 NSKEW=NSKEW+1 ELSEIF(IND.EQ.-1) THEN REMOVE THE NODE FROM SKEW BOUNDARY AND SHIFT THE ISPB ARRAY DO ISKEW=1,NSKEW KNODE=ISPB(ISKEW)

Appendix: Fortran Codes of CS-S IF(KNODE.EQ.NODE) THEN DO IK=ISKEW,NSKEW-1 ISPB(IK)=ISPB(IK+1) DO ID=1,NDIM DIRCOS(IK,ID)=DIRCOS(IK+1,ID) ENDDO ENDDO ISPB(NSKEW)=0 DO ID=1,NDIM DIRCOS(NSKEW,ID)=0.0D0 ENDDO NSKEW=NSKEW-1 GO TO 30 ENDIF

30

ENDDO CONTINUE ENDIF RETURN END

C******************************************************************

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Index

Acoustic propagation 6 Acoustic technique 149, 151, 153, 155 Anderson-Parrinello-Rahmen theory 164 Anisotropy 6, 30, 31, 35, 75, 184, 188 Angle of internal friction 3 Aspect ratio 44 Atom 161 Attenuation 150–156 Back stress 21–38, 48–56, 72–78, 168, 188–203 BAT 158 Binary model 5 BioCoRE 164 Body force 8, 20, 168 (Non) bonded energy 164 Born-Oppenheimer approximation 164 Cam clay model 26-55, 70–83, 100, 114, 133–143, 184–207 Capillarity 5 Cap model 29, 30 Carbon-nano-fiber 162 Cartesian coordinate 16 Characteristic frequency 153–155 Characteristic length 39, 40 Clausius-Duham inequality 46 Clay 3, 15, 30, 41, 63, 74, 170 Clay-fluid interaction 164 Clay liner 114 Coefficient of consolidation 197, 199 Cohesion 24, 26, 125, 142 Cohesionless 62, 141 Compressibility 2, 21 100 Compression index 35, 100, 102, 176 Compression wave 150, 151

Condition number 66, 67 Cone face 98, 101, 184, 200 Cone penetrometer 91–106, 158, 168, 192, 203 Cone resistance 92, 94, 98, 178 Cone shoulder 98, 200 Conservation principle 2 Consolidation 6–17, 26–43, 94–115, 158, 178, 191 Consolidation coefficient 15, 96 Constitutive equation 2, 16, 40, 56, 75, 82 Constitutive law 8, 58, 63, 64 Constitutive model 2, 8, 10, 113, 125, 136, 184 Constitutive relation 16, 36–38, 56, 57, 75, 161 Constitutive theory 6, 9 Continuum 4–19, 40, 50, 59–69, 162–170 Continuum models 2, 16 Cosserat (continuum), 59–61 Coulombic energy 164 Coupled consolidation 112, 115 Coupled theory 10–21, 81, 94–116, 158–200 Coupled theory of mixtures 10, 12, 13–21, 81, 94, 97, 104, 109, 113, 116, 173, 188, 199, 200 Coupling modulus 152 Coupling matrix 84 Crank-Nicholson technique 96 Critical void ratio 65, 66, 172 Cross hole technique 151 Cyclic shear strain 7 Damage 11, 38, 68–74, 117, 168–205 Damping (ratio), 7, 157

436

Index

Darcy’s law 13 Densification 7, 8 Deviator stress 2, 1 Diffusion 4, 15, 16, 96, 97, 153 Diffusivity 16 Dihedral angle 161 Dilatancy 63 Dilational wave 6 Discrete mechanics 163 Displacement tensor 17 Dissipation potential 47, 48 Dissipation test 94, 96, 98, 100, 192, 197 Disturbed state 74 Divergence 19, 20, 66, 152 DPD (Dissipative Particle Dynamics), 170 Drucker-Prager 25, 26 Dutch cone 91 Dynamic yield function 69 Earthquake 1, 7 Effective stress 1, 5, 7–18, 33, 69–80, 96, 102, 133, 173–205 Elasticity 33, 78 Electric cone penetrometer 92 Electron charge 161 Electron permittivity 161 Embedded stress 31, 40, 43, 75 Endochronic (theory) 8 Entropy 6, 8 EPB shield 117 Equivalent beam or truss element method 167 Equivalent Linear Method (ELM) 7 Equivalent plastic strain 78, 133, 134 Excess pore water pressure 1, 13–17, 98, 99, 105–106 Failure 3, 59, 63, 65, 100, 142, 143, 192 Fast P-wave 150, 151, 153 Finite deformation 6 FEM (Finite Element Method) 2, 9–18, 72, 83, 86, 90–103, 113–132, 170–174 Finite strain 17–21, 63, 74, 76, 83, 89 Flow rule 36, 57, 69, 136 Fluid droplet 170

Fourier equation 4 Friction angle 23, 26, 142, 143, 172 Friction resistance 91, 94, 191 Fully drained 21, 98, 101, 106, 111 Fully undrained 21, 98, 101, 106, 111 Gauss point 133, 213 Gradient 2, 9, 13, 37, 39–49, 60–87, 173–207 Grain interaction 11, 63, 64, 170, 172 Grain rotation 11, 18, 59, 61, 188 Hardening 26, 30–34, 45–54, 64–78 134, 135 Heat diffusion 15, 16, 96, 97 Henkel’s pore pressure coefficient 112 Helmholtz free energy 46, 47 Hvoslev surface 28 Hydraulic conductivity 12–21, 94–116, 149–183 Hydraulic gradient 13 Hydrodynamic 4 Hydro-mechanical 18 Ill conditioned 66 Ill posedness 66, 67 Immiscible mixture 3 Incremental scheme 76, 89 Induced anisotropy 30, 31, 36, 75, 188 Inelastic 10, 57, 58, 62, 70 Inherent anisotropy 30, 31, 36, 76 In-phase motion 150 Interaction 3–11, 38, 38, 48, 63, 64, 98–200, 115, 151, 161–173, 181–204 Interface element 125, 134, 135, 142 Internal friction angle 26 Internal state variable 45, 47, 48, 203 Intrinsic permeability 154 Isoparametric (element) 86, 129, 130 Isothermal 6 Jacobian

21, 77, 81

Kinetic theory

4

Lagrangian reference frame 10, 76, 80–93, 96–113, 199, 200 Lame’s constant 152

Index Laplacian operator 39 Large strain 2, 10, 55, 82, 86, 95, 100, 109, 119, 188, 198–207 Length scale 38–45, 73, 163 Lennard-Jones 12–6 potential energy 165 Lime 3 Linear stiffness matrix 84 Liquefaction 1, 7, 8, 18, 151 Liquefaction potential 151 Long range back stress 50, 51, 54, 201–203 MaCauley bracket 35, 48, 70 Macroscale 39, 45–48, 167 Macroscopic yield 35 Mandel-Cryer effect 16, 17 Mantle cone 91 Material Studio 164 Mathematical gradient 41, 67 Mechanical cone 91 Mesh size dependency 43 Mesoscale 46–48 Micro-mechanical 9–18, 38–56, 63, 75, 168, 173, 188, 204–207 Microplane model 57, 58 Micropolar (continuum) 59, 60 Microscale 45, 46 Minimization 166 Mixture 1–23, 81, 94–104, 113 115 168 173 188–207 Modified spin tensor 56 Mohr-Coulomb 23–25, 125, 142 Molecular mechanics 163 Molecule 161 Mud 3 Multi phase 1 4 11 204 Multi-polar continuum 9 NAMD 164 Nanocomposite 162 Nano-mechanics 161–167, 170, 171 Nanotechnology 162 (Non) bonded energy 164 Nonlinear 2, 10, 62, 81–90, 120, 134, 135, 141, 142 Non-linear stiffness matrix 84 Octahedral normal stress 112 Octahedral shear stress 112

437

OPM (One Point Method) 95, 100 Orthogonality relation 152 Oscilloscope 156 Out-of-phase motion 150 Overstress 70, 71 Partial differential equation 15, 96 Partially drained 21, 98, 101, 106 Partially saturated 10 Permeability 6–20, 98–116, 143–199 Permeability tensor 20, 168 Physical gradient 42, 67 Piezocone penetrometer test 91–96, 100, 104, 115, 158 Piola-Kirchoff stress 77, 80 π-plane 23–25, 31–35, 58, 120, 165 Poisson’s ratio 16, 166 Pore water pressure 1, 7–17, 77–109, 168–200 Poro-elastic 8, 151 Porosity 18, 20, 154, 168 Porous media 1–10, 20 Preconsolidation pressure 178 PSRN (Pseudo Random Noise), 155 Quantum mechanics Quasi-static 6, 7 Quicksand 3

163

Radius of plastic zone 113 Rate dependency 11, 38, 68–74, 172, 175 Recompression index 35, 102, 188 Remeshing 89, 118, 124–127, 134 Resonant column test 54 Rigid plastic 26 Rigid porous solid 5 Roscoe surface 28 Rotational wave 6 Roughness 43 RVE 39, 40, 42–46, 167, 170 Sand

3–8, 18, 52, 53, 62, 64, 73, 74, 104, 139, 155, 170, 191, 203, 204 SASW 151 Seepage 13, 85 Senate Committee on Commerce Science and Transportation 162 Shear band 40–42, 63, 64, 68, 172

438

Index

Shear modulus 7, 16, 53, 137, 143, 152, 166 Shear wave 151 Shield tunnel 116, 117 Short range back stress 50, 51, 54, 201–203 Schr¨ odinger equation 164 Simple shear 7 Singular value decomposition method 155 Skeleton 1, 7, 18, 78, 203, 204 Skempton’s pore pressure coefficient 112 Sliding potential 102 Slip theory 57 Slow P-wave 150, 151, 156 Slurry consolidation 191 Small strain 2, 16, 62, 74, 201 S/N ratio 155 Source 155 Spin tensor 56 Spread spectrum 150 SPT 91 Steady flow 13 Stiffness 7, 37, 55, 81, 84, 89–106, 122, 134–138, 167–204 Stress invariant 23 Stress tensor 59 Swedish penetrometer 91

Total stress 1, 7, 12, 96, 178–183, 204 TPM (Two Point Method) 95, 109 Transient flow 13, 21, 101 Tresca 23–25, 33 Trigger 155 Two phase material 11, 101, 152, 173, 204

Taylor expansion 41, 66, 67 Thermodynamics 4, 15, 46, 48, 74 Three phase material 11 Threshold strain 62

Yazoo clay 166 Yield criteria 10, 23–40, 42–46, 58, 80 Yield surface 26–36, 49, 63, 64, 69–72, 78, 79, 89, 90, 133, 134, 168, 173

Uncoupled consolidation 112, 114, 115 Updated Lagrangian 18, 21, 76, 80–113, 199–200 van der Walls energy 164 Varved clay 178 Virtual consolidation 112 Virtual work 18, 57, 61, 76, 100 Visco-elasticity 33 Visco-plasticity 48, 69, 70 Viscosicity 48, 68–73, 100, 154, 168, 172, 173, 178, 204 Void 1, 11, 15. 35, 55, 66, 76, 102–104, 143, 157, 172, 203, 266 Volume fraction 2, 3, 5, 9, 152 Volumetric strain 7, 16–31, 54, 134, 168, 174 von Mises 23–25, 33, 70 Wave propagation 6, 9, 10, 151, 152 Well posedness 66, 67

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