Soil Mechanics E-Book

Roberto Nova

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 447

Veröffentlichungsjahr: 2012

Beschreibung

Knowledge of the behavior of soil mechanics is essential for forecasting the internal displacements and actions of any construction. This book, although theoretical at first glance, also offers a more practical scope, giving readers adequate tools to plan geotechnical projects correctly.

Leseprobe

Table of Contents

Preface

Chapter 1. Introduction: Basic Concepts

1.1. Soils and rocks

1.2. Engineering properties of soils

1.3. Soils as an aggregation of particles

1.4. Interaction with pore water

1.5. Transmission of the stress state in granular soil

1.6. Transmission of the stress state in the presence of a fluid

1.7. From discrete to continuum

1.8. Stress and strain tensors

1.9. Bibliography

Chapter 2. Field Equations for a Porous Medium

2.1. Equilibrium equations

2.2. Compatibility equations

2.3. Constitutive laws

2.4. Geostatic stress state and over-consolidation

2.5. Continuity equation and Darcy’s law

2.6. Particular cases

2.7. Bibliography

Chapter 3. Seepage: Stationary Conditions

3.1. Introduction

3.2. The finite difference method

3.3. Flow net

3.4. Excess pore pressure

3.5. Instability due to piping

3.6. Safety factor against piping

3.7. Anisotropic permeability

3.8. Transition between soils characterized by different permeability coefficients

3.9. Free surface problems

3.10. In situ methods for the permeability coefficient determination

3.11. Bibliography

Chapter 4. Seepage: Transient Conditions

4.1. One-dimensional consolidation equation

4.2. Excess pore pressure isochrones

4.3. Consolidation settlement

4.4. Consolidation settlement: approximated solution

4.5. Consolidation under different initial or boundary conditions

4.6. Load linearly increasing over time: under consolidation

4.7. Consolidation under axial symmetric conditions

4.8. Multidimensional consolidation: the Mandel-Cryer effect

4.9. Oedometer test and measure of cv

4.10. Influence of the skeleton viscosity

4.11. Bibliography

Chapter 5. The Constitutive Relationship: Tests and Experimental Results

5.1. Introduction

5.2. Fundamental requirements of testing apparatus

5.3. Principal testing apparatus

5.4. The stress path concept

5.5. Experimental results for isotropic tests on virgin soils

5.6. Experimental results for radial tests on virgin soils: stress, dilatancy relationship

5.7. Oedometric tests on virgin soil as a particular case of the radial test: earth pressure coefficient at rest

5.8. Drained triaxial tests on loose sands: Mohr-Coulomb failure criterion

5.9. Undrained triaxial tests on loose sands: instability line and static liquefaction

5.10. Drained tests on dense and medium dense sand: dilatancy and critical state

5.11. Strain localization: shear band formation

5.12. Undrained tests on dense and medium dense sands: phase transformation line

5.13. Sand behavior in tests in which the three principal stresses are independently controled: failure in the deviatoric plane

5.14. Normally consolidated and over-consolidated clays: oedometric tests with loading unloading cycles – extension failure

5.15. Drained and undrained triaxial tests on normally consolidated clays: normalization of the mechanical behavior

5.16. Over-consolidated clays

5.17. The critical state. Plasticity index

5.18. Natural soils: apparent over-consolidation – yielding surface

5.19. Soil behavior under cyclic loading: cyclic mobility and strength degradation

5.20. Bibliography

Chapter 6. The Constitutive Relationship: Mathematical Modeling of the Experimental Behavior

6.1. Introduction

6.2. Nonlinear elasticity

6.3. Perfect elastic-plasticity

6.4. Yielding of metals

6.5. Taylor and Quinney experiments: the normality postulate

6.6. Generalized variables of stress and strain

6.7. Plastic strains for a material behaving as described by the Mohr-Coulomb criterion

6.8. Drucker-Prager and Matsuoka-Nakai failure criteria

6.9. Dilatancy: non-associated flow rule

6.10. Formulation of an elastic-perfectly plastic law

6.11. Cam clay model

6.12. Reformulation of the Cam clay model as an elastic-plastic hardening model

6.13. Comparison between experimental behavior and mathematical modeling for normally consolidated clays

6.14. Lightly over-consolidated clays

6.15. Heavily over-consolidated clays

6.16. Subsequent developments and applications

6.17. Non-associated flow rule: the Nova-Wood model

6.18. Sinfonietta classica: a model for soils and soft rocks

6.19. Models for soils subjected to cyclic loading

6.20. Conceptual use of constitutive soil behavior models

6.21. Bibliography

Chapter 7. Numerical Solution to Boundary Value Problems

7.1. Introduction

7.2. The finite element method for plane strain problems

7.3. Earth pressures on retaining structures

7.4. Settlements and bearing capacity of shallow foundations

7.5. Numerical solution of boundary value problems for fully saturated soil

7.6. Undrained conditions: short-term bearing capacity of a footing

7.7. Short- and long-term stability of an excavation

7.8. Bibliography

Postscript: From Soil Mechanics to Geotechnical Engineering

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. The translation of this book has been funded by SEPS (Segretariato Europeo per le Pubblicazioni Scientifiche) Via Val d’Aposa 7, 40123 Bologna, Italy, [email protected] – www.seps.it

Originally published in Italian under the title: Fondamenti di meccanica delle terre ISBN 88-386-0894-6 © The McGraw-Hill Compagnies, S. r. l. Milano, 2002. Permission for this edition was arranged through The McGraw-Hill Compagnies, Srl. – Publishing Group Italia.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd 27–37 St George’s Road London SW19 4EU UK

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John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

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© ISTE Ltd 2010

The rights of Roberto Nova to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Nova, Roberto.

  [Fondamenti di meccanica delle terre. English]

  Soil mechanics / Roberto Nova ; translated by Laura Gabrieli.

      p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-102-5

1. Soil mechanics. I. Title.

TA710.N58713 2009

324.1'5136--dc22

2009034935

British Library Cataloguing-in-Publication Data

Preface

Atque neque, uti docui, solido cum corpore mundi naturast, quoniam admixtumst in rebus inane …

Titus Lucretius Caro De Rerum Natura

According to the engineering nomenclature, soil mechanics is concerned with the behavior of clastic rocks, or “soils”, under different loading conditions: external loading, such as that transmitted by the foundations of any structure, or generated by the seepage of water, and also by its own weight as a consequence of geometric changes, induced for instance by excavation or tunneling.

Knowledge of soil mechanical behavior is, in fact, an essential element for the prediction of the displacements and internal actions of a structure founded on or interacting with it. Soil Mechanics is, therefore, the fundamental subject of geotechnical engineering, the branch of civil engineering concerned with soil and with the interacting soil structure, dealing with the design and the construction of civil and industrial structures and environment defense works against geological hazards.

Aristotle said “Φαντασία δέ πᾶσα ᾕ λογιστική ᾕ αίσθητική”: any prediction is based either on a rational calculation or on intuitive perception. Although the latter has been for a long time the starting-point of any construction and still plays a relevant role in design, it is the former that allows the definition of the structure’s dimensions and safety assessment. In fact, it allows rational prediction of the structure’s behavior in the different construction phases and during its life.

This calculation must be based on a mathematical model of the structure and the soil. This should schematize the geometry of the problem, the mechanical behavior both of materials and structures, as well as the loading. The definition of an overall mathematical model of the structure and the soil is a very complex problem that is beyond the scope of this book. In the following, only the bases upon which a mathematical model of soil behavior can be formulated will be outlined.

Though limited in scope, soil modeling is rather complex and requires different levels of abstract thinking. First, it is necessary to pass from the physical nature of soil, composed of a discrete and innumerable number of solid mineral particles and voids, into which fluids such as air, water or mineral oils can seep, to its representation as a continuum. In fact, this allows a much more feasible mathematical formulation. In order to achieve this goal, it is necessary to assume the soil to be a special medium obtained by “overlapping” two continua: a solid continuum, modeling the skeleton composed of the mineral particles, the “solid skeleton”, and a fluid continuum, modeling the fluid, or the fluid mixture, seeping through the voids.

The most relevant aspect lies in the fact that both these continua completely occupy the same region of space. They interact by parting the stress state in a way that directly derives from the conditions of conservation of energy and mass, and that is a function of how the behavior of the solid continuum under loading and the fluid seepage in the soil are independently modeled. Hence, it is necessary to mathematically formulate models for the description of the mechanical behavior of the solid skeleton (stress-strain relationship) and a conceptually equivalent law ruling the motion of fluid with respect to the solid skeleton.

Once the model is defined, in order to mathematically reproduce with the best approximation possible the experimental results obtained by elementary tests, the parameters describing the soil (or the different soil layers) behavior have to be specified for the case under examination.

Finally, a further step in modeling is necessary to transform the system of differential equations and boundary conditions ruling any soil mechanics problem in the light of continuum mechanics into a system of algebraic equations that can be solved by means of a computer.

This book will be developed in logical sequence according to what has been previously outlined.

Chapter 1 presents some elementary concepts necessary to pass from the discrete nature of soil to its continuum representation. Differential and boundary equations for a generic soil mechanics problem will then be presented in Chapter 2. Special cases will be analyzed, such as stationary seepage conditions (Chapter 3), “rapid” loading conditions (undrained conditions), and transient seepage conditions (Chapter 4). In this last case, under constant loading, the stress state is transferred from the water to the solid skeleton, inducing soil deformations and structure assessments over time (consolidation). For the sake of simplicity, in this case soil will be assumed to be characterized by an incrementally linear behavior.

Nevertheless, the mechanical behavior of the solid skeleton is much more complex. In fact, it is nonlinear, irreversible, and highly influenced by the average pressure to which it is subjected. These aspects will be detailed in Chapter 5, which is dedicated to the study of the response of elementary soil samples in laboratory tests. In Chapter 6, mathematical models of increasing complexity describing the behavior outlined in the previous chapter will be formulated. Finally, in Chapter 7, methods of discretizing the continuum and integration procedures will be mentioned. A few examples, referring to archetypes of geotechnical problems (foundations, sheet piles, slopes), will illustrate the results that can be obtained in this way.

This book is not intended to be exhaustive on all the geotechnical issues or to give “practical” suggestions. For these purposes several good and topical books already exist and there is no reason to write another. On the contrary, the goal of this work is to tackle the fundamental aspects of a very complex subject at a deeper level than current works. These aspects can have remarkable consequences on the choices that the engineer has to make in order to build the geotechnical model of the soil that is appropriate for the particular case under examination (geometry of the problem, type of model to describe soil behavior, parameters to be assumed, type of numerical solution) and thus, as a consequence, on the design.

Having worked in the field of soil mechanics for many years, I know that there is some confusion concerning the fundamental principles which this subject is based on. Frequently, even people working in the geotechnical engineering field do not completely understand the formulae that they use, especially the computer methods, whose bases they do not have knowledge of. The dialog between the several actors involved in a geotechnical project (civil and environmental engineers, geologists, architects) risks becoming a dialog between deaf people, in which not even the specific role of each of them is clear.

As any good geotechnical engineer knows, a safe structure has to be based on solid foundations. The book is therefore intended to give, to those who will have the patience to read it, the bases necessary to understand the fundamentals of soil mechanics. It is my firm belief that only through the thorough understanding of such fundamentals can appropriate geotechnical characterization and soil modeling be carried out. Though the main point of this book is undoubtedly theoretical, its final goal is very practical: to give adequate means for a correct framing of geotechnical design.

In writing this book, I was privileged to collaborate with some young colleagues: Claudio di Prisco, Roberta Matiotti, Silvia Imposimato, Riccardo Castellanza, Francesco Calvetti, Cristina Jommi, Rocco Lagioia, Claudio Tamagnini, Stefano Utili, Giuseppe Buscarnera, Matteo Oryem Ciantia, Giuseppe Dattola and Federico Pisanò. They helped me to clarify the text (in addition to taking care of the graphics). To them and to all those who have been so kind as to highlight mistakes and omissions or simply been willing to discuss the non-traditional approach followed in this book, my most sincere thanks.

This book is dedicated to “my” Maddalena, Tommaso and Tobia, who patiently bore the consequences of its writing.

Chapter 1

Introduction: Basic Concepts

1.1. Soils and rocks

The term “soil” is used in civil engineering to describe a material composed of a natural accumulation of mineral particles, whose sizes range between specified limits, according to a conventional classification system.

Soil is the result of the chemical-physical alteration of rocks due to atmospheric agents (weathering), rocks being the primary element that constitutes the Earth’s crust. Soil particles can be completely uncemented or weakly cemented, depending on the degree of alteration of the parent rock. On the other hand, soil that is exposed to atmospheric agents for a long period of time undergoes chemical reactions that cement the particles, so that deposits that were originally composed of uncemented particles are gradually transformed into sedimentary rocks (diagenesis).

Since the processes of weathering and diagenesis are gradual, the distinction between soil and rock is to a certain extent arbitrary. To the geotechnical engineer soil is any accumulation of mineral particles with weak chemical bonds, such that the stress levels typical of civil engineering applications can easily exceed their strength. On the other hand, rock is defined as a material with strong chemical bonds. The deformation and failure of rock masses are governed by the mechanical behavior of the pre-existing geometric discontinuities (faults or joints) rather than by the intrinsic characteristics of the rock itself.

Several geological materials (e.g. tuff, clay stone, marble, limestone, etc.) have an intermediate behavior. These materials behave as rocks if subjected to relatively low stresses, and as soil if subjected to stresses high enough to break the chemical bonds cementing the particles.

Soil grains are mainly composed of silica minerals (e.g. silicon dioxide and other silica-based minerals), which are more resistant to chemical-physical attack by weathering than other minerals. Quartz (SiO2) is almost insoluble in water, is relatively acid proof, and is a very stable mineral. It is primarily composed of rounded or prismatic particles of the order of a millimeter or less and is the main mineral of silica sands, followed by feldspars.

Feldspars are chemically altered by water, oxygen and carbon dioxide. The gradual breakdown of feldspar crystals forms microcolloidal particles of kaolin. Similarly, phillosillicates, existing in large quantities in igneous rocks, delaminate along their basal plane, due to their mineralogical foil structure, and form illite and smectite. Kaolin, illite and smectite are the primary minerals appearing in clay; they are characterised by plate-like particles with length and width in the order of a micron.

Soil particles are also composed of calcite and gypsum, as well as of minerals of volcanic origin (pyroclasts). Particles formed by these types of minerals are usually weaker than those formed by silica minerals; therefore, they have a greater influence on the strain behavior of these materials.

The shape of the particles and their structural arrangement depends on the materials that compose them and on their geological history.

For example, on the one hand, rounded shape sand grains with faces and angles bevelled by abrasion are typical of sand deposits formed after wind or water transportation. On the other hand, sand grains that remain in their original location, where weathering of the parent rock took place, are angular and have an irregular shape.

The chemical environment in which the particles are deposited has a significant influence on the structure of clay that can aggregate in different ways. If clay particles align in the same direction (face-to-face orientation, Figure 1.1a) it is referred to as dispersed structure, while a structure similar to a card house (edge-toface or edge-to-edge orientation) is referred to as flocculated structure (Figure 1.1b) and is much more unstable than the former. With the change in the deposit chemical conditions, the structure can pass from dispersed to flocculated and vice versa.

Figure 1.1.a) Dispersed clay structure; b) flocculated clay structure (“cardhouse”)

1.2. Engineering properties of soils

As seen in the previous section, several types of minerals compose a soil, its “solid skeleton”, and its fabric are influenced by its geological history and by the chemical environment. However, for the majority of engineering aims, different types of soil can be initially classified according to the size of the constituent particles. The classification of the different types of soils is somewhat arbitrary. Examples of classifications adopted by British Standards (BS), Italian Geotechnical Association (AGI) and American Association of State Highway Officials (AASHO) are listed in Table 1.1.

Table 1.1.Classification of different types of soils. Sizes are in mm

Note that in the proposed classifications there is no direct reference to the grain chemical composition, to the type of parent rock or to the formation process of the deposit (for transport or in situ alteration). This type of classification has two advantages. Firstly, it is a quantitative classification, and hence it is almost free from the subjectivity of the operator. Secondly, it allows for the direct identification of a property that has a fundamental influence on the soil mechanical behavior. The range of possible particle sizes is enormous. Soil particle sizes range from sub microscopic clay particles, discernible only by a scanning electron microscope, to rounded sand grains with a diameter a thousand times larger, to cobbles with a diameter a hundred times larger.

On a single particle, both body forces (weight) and surface forces (electrostatic forces) have effect. The former, are proportional to the volume of the particle, while the latter are proportional to the external surface. An initial difference between fine and coarse particles consists of the different role interplayed by the electrostatic forces on their surface. An indicator of the relative role played by the two types of forces is the specific surface, Ss, defined as the ratio of the area of the surface of the particle to the mass of the particle, ρV:

[1.1]

where ρ is the density and V is the volume of the particle.

In the case of a rounded particle of silica sand, the specific surface is inversely proportional to the diameter of the grain, dg.

[1.2]

Quartz density is equal to 2.65 g/cm3; hence, for a rounded particle of diameter 1 mm the specific surface will be 0.00226 m2/g. A clay particle, of plate-like shape, has instead a specific surface equal to:

[1.3]

where s is the thickness of the particle. The particle thickness largely depends on the type of clay. For kaolin it can be of the order of a tenth of a micron, while it can be of the order of only 10 Å (10-3 μm) for the smallest particles, this is typical of montmorillonite.

For kaolin the specific surface is of the order of 10 m2/g (more than 3,000 times the value of the sand considered). For montmorillonite the specific surface is of the order of 1,000 m2/g. Electrostatic forces are then negligible in sand, however, they become relevant when dealing with clay. In the presence of water, clay particles attract a layer of water molecules that can not be separated from the mineral particles by means of mechanical forces or processes. This layer is referred to as adsorbed water. The water forming this layer has very different mechanical properties in comparison with those of free water: for instance, adsorbed water is capable of transferring shear stresses. In practice, adsorbed water can be considered, as a first approximation, as an integral part of the mineral clay particle. Unless otherwise specified, the mineral particle is assumed to be coated by a layer of adsorbed water. From the mechanical point of view, interactions between clay particles coated by adsorbed water do not qualitatively differ from the ones that take place among sand grains.

Moving away from the surface of the particle, the attractive force decreases and progressively water starts behaving as free water, which can be gradually removed from a sample of soil; for example, by applying compression stresses.

From an engineering point of view, the most relevant aspect related to the particle size distribution is the ability of water or other fluids, such as oil, to seep through the soil pores.

Figure 1.2.A soil element as an aggregation of particles

A soil element can be visualized as an aggregation of solid particles, weakly cemented or uncemented, the void space between the particles containing one or more fluids, principally air and/or water (Figure 1.2). A fluid can seep through a soil more or less easily depending on the width of the flow channel section. The average velocity of a fluid in laminar flow is proportional to the square of the hydraulic radius, which is of the same order of magnitude of the soil particle size. The size of a clay particle is approximately a thousand times smaller than the one of a sand grain. Thereafter, water discharge velocity in a clay layer must be a million times lower than the one in a sand layer, all other conditions being equal. As will be observed in the following, this difference has relevant practical consequences.

A load applied on a sample of soil provokes the rearrangement of its structure. Since soil grains are principally composed of extremely resistant and rigid minerals, the deformability of a soil element is mainly associated with a change in the configuration among grains, which is related to a change in the volume occupied by voids. In fact, grain deformability is negligible; with the exception of soils composed of calcareous or pyroclastic grains or of soils that are extremely porous and crush under the action of limited loads. Water is also considered, under the stress levels typical of civil engineering applications, to be an incompressible fluid. If soil is fully saturated by water, a change in volume can take place only if water is free to drain throughout the soil. If soil is coarsely grained, drainage is instantaneous and the particles are free to change their configuration while loads are applied. On the contrary, in fine grained soils, water flow is subjected to a higher resistance. The time necessary for water to drain through the pores is of several orders of magnitude higher than the one required to complete the load process (e.g. the construction of a building, a road embankment or an excavation). In the initial phases of the load, referred to as “short term”, the possible configurations are only the ones that maintain the total volume constant, which means that the soil has an internal kinematic constraint. With time, water gradually drains through the soil and at “long term” also fine grained soils can freely change their configuration without any internal constraints.

The first and main difference between coarsely and fine grained soils is then apparent. Fine grained soils change their configuration after a change in load, even though initially without a change in volume. Coarsely grained soils change their configuration step by step with the change in load and complete their settlement at the end of the load process. The gradual expulsion of water from the pores implies also a change over time in the structural arrangement of the solid particles. Therefore, the strain process continues also after the stabilization of the load.

It is worth noting, however, that in relatively coarsely grained soils, such as fine sands, there can be kinematic constraints preventing changes in volume after rapid variations in load, as in the case of earthquakes. Moreover, a prevailingly sandy soil can rearrange its structure over time due to the presence of fine particles.

Another important difference between fine and coarsely grained soils is represented by capillary rise. Let Ts be the surface tension of water, α the angle between the tangent to the meniscus and the wall of the capillary tube, γw the unit weight of water and d the diameter of the tube (see Figure 1.3). Equilibrium in the vertical direction implies that the capillary height, hc, of the liquid column is:

[1.4]

Surface tension of water in standard conditions is equal to 0.075 N/m, therefore, in a capillary of 1 mm diameter the rise is of the order of 2 cm. If the diameter of the tube is instead 1 μm the capillary rise is 20 meters. In coarsely grained soils capillary rise is hence negligible and the soil over the ground-water table can be considered dry. Conversely, fine grained soils are saturated up to several tens of meters over the ground-water table.

Figure 1.3.Rise in a capillary tube

1.3. Soils as an aggregation of particles

As a first approximation, the structural arrangement of an elementary volume of soil can be schematized as in Figure 1.2. Solid particles occupy only a portion of the space relative to an element of soil. The remaining portion, called “volume of voids”, is occupied by a fluid, usually air and/or water.

The ratio of the volume occupied by voids, Vv, to the volume occupied by solids, Vs, is called void ratio, e:

[1.5]

Alternatively, porosity, n, is defined as the ratio of the volume of voids to the total volume, V, of a soil element:

[1.6]

It is clear that the higher the porosity, the easier it is for the grains to rearrange in a different configuration once this is perturbed by the action of external loads. On the other hand, a very dense soil has few degrees of freedom and hence needs a greater effort to change its initial configuration. Soil porosity is therefore one of the parameters largely influencing the soil mechanical behavior.

In order to define the range of soil porosities, an ideal material composed of rigid spheres of equal radius is considered. A simple cubic structure, in other words a configuration in which spheres are all disposed tidily one next to the other and every layer is disposed exactly as the one below, is characterized by a porosity equal to 0.476. This configuration is highly unstable. A small external perturbation is sufficient to reduce its porosity. Conversely, in a cubic tetrahedral configuration (spheres disposed at the vertex of a regular tetrahedral, in contact among them), porosity is much lower and equal to 0.259. In this case, the structure is very stable and an external perturbation will therefore cause a negligible rearrangement of the micro-structure with respect to the previous case. It is worth noting that if a closed portion of surface occupied by a set of particles and by the enclosed voids is isolated, an external perturbation will cause a decrease in volume in the case of loose sand, while it will cause an increase in volume in the case of dense sand (dilatancy).

The proposed model is only an example. Firstly, a soil is composed of particles of different sizes and non-rounded shapes. Moreover, smaller particles have a greater possibility of occupying a minor total volume, the volume of solids being equal. For instance, a sphere of radius R occupies a cube of radius 2R with a porosity of 0.476. On the other hand, eight spheres of radius R/2, and hence occupying the same volume of the spheres just considered, would fill the same cube only if disposed in the most unstable configuration previously described. However, the eight small spheres can dispose in several ways, for example in the tetrahedral configuration that is characterized by a much lower porosity. A sample of sand composed of several particle sizes will be characterized, in general, by a smaller porosity in comparison with a sample of sand of equal weight that is mono granular (composed of particles of the same size). To take into account the effect of grading in sands, it is more appropriate to refer, other than to porosity, to the relative density (density index), which is traditionally defined as:

[1.7]

where emax and emin are two void ratios, conventionally determined (refer to ASTM D2216-66), which define the loosest and the densest state for a criterion of sand.

Moreover, particles are not rigid. Calcareous and volcanic sands are composed of fragile grains that can crush under loading. As stated before on the effect of particle size, porosity will decrease not only as a consequence of grain rearrangement but also of the crushability of the particles themselves.

Finally, when particles develop cohesive bonds, configurations with very high void ratios are possible. For example, loess deposits and cohesive silts deposited by wind, can be characterized by porosities higher than 60% (e > 1.5). These configurations are stable only for tensional levels that are lower than the bond strength. For higher tensional levels, the bonds break and the soil assumes a much more compact configuration. The collapse of these kinds of soils usually causes big problems from an engineering viewpoint.

1.4. Interaction with pore water

Inter-granular voids can be partially or totally filled with water. The degree of saturation, Sr, is defined as the percent ratio of the volume occupied by water to the volume of voids:

[1.8]

The soil water content, w, is defined as the ratio of the mass of water within the sample, Ww, to the mass of the solid part, Ws, this being the dry weight of the considered sample,

[1.9]

Let γw be the unit weight of water and γs the unit weight of the material composing the grains. The water content is then linked to the degree of saturation and to the void ratio by the relationship:

[1.10]

where

[1.11]

The value of Gs does not greatly differ for the principal types of minerals composing the grains of a soil and usually ranges from 2.5 to 2.9. The Gs value of quartz is 2.65, of calcite is 2.71, with 2.7 the typical average value of clayey minerals.

It is evident that the unit weight of a volume of soil is different from the unit weight of the grains and depends on the water content. Let γ be the total unit weight of a certain sample of soil:

[1.12]

In particular, for a dry sample, the dry unit weight, γd, is equal to:

[1.13]

while for a saturated sample the total unit weight is equal to:

[1.14]

Finally, a sample submerged in water is subjected to an up-thrust that is equal to the weight of the volume of water displaced. The buoyant unit weight of a soil is hence equal to:

[1.15]

Notice that this result is equally achieved by considering the soil sample as composed of a unique material, or by considering the solid part, namely the single particles, independently of the fluid.

1.5. Transmission of the stress state in granular soil

Imagine applying a load to a volume of soil composed of rigid particles. The stress state will be transmitted from the boundaries within the sample through the contacts between the particles. To study this phenomenon, tests on discs made of CR-39 co-polymer can be run, and exploiting photo-elasticity techniques, the average stress and strain tensors can be determined.

Figure 1.4.Distribution of the stress state in a sample of granular material composed of disks of polymeric material (Drescher and De Josselin de Jong, 1972)

Exposed to rays of polarized light, the transparent “grains” develop isochromatic patterns that are a function of the intensity of the stress state. Figure 1.4 (Drescher and De Josselin de Jong, 1972) shows the chains of aligned contact points along which forces are transmitted. The thicker the black line in the figure, the greater the intensity of the transmitted forces. Analogously Figure 1.5, from Calvetti (1998), illustrates the results of a numerical simulation on a set of cylinders of circular section enlightening the same phenomenon.

Figure 1.5.Numerical simulation of the distribution of the stress state in a sample of granular material with the discrete element method (Calvetti, 1998)

Some grains are intensely stressed, others less and others do not even carry any load. The most interesting aspect observed, both in the real test and in the numerical simulation, is that chains are not stable. For a certain value of load, two grains, initially in contact, slide over one another, interrupting the contact. Hence, the column is no longer able to transmit the load. Grains that were previously loaded suddenly are unloaded, the load is immediately redistributed and a new chain is formed. If the number of grains is large enough, this intense internal assessment is not externally visible, that is to say the loads and displacements applied to the sample vary with continuity without any abrupt change in the internal structure.

Due to column instability, the fundamental parameter ruling the overall soil behavior is the resistance to sliding of the contacts, more than the stiffness of the grains themselves. This resistance is due principally to the friction among grains and partly to their cementation (often absent). The law governing the shear strength is known as Coulomb’s equation (even if it was firstly formulated by Amonton a century before). This law establishes that two rigid bodies in contact will slide one with respect to the other when the shear force at contact, T, reaches a proportion of the normal force at contact N:

[1.16]

where μ is a coefficient that depends only on the type of material and not on the dimensions of the bodies in contact.

Figure 1.6.Sliding of a brick on a rough plane

Tabor (1959) provided a simple explanation of [1.16]. For simplicity refer to Figure 1.6, where a brick on a rough plane is illustrated. The surface of the brick is rough and the contact area between the two bodies is small compared with the apparent area A. The normal strength σy, that can by transmitted from one body to the other through the contact points is limited. In order to satisfy vertical equilibrium the effective contact area has to grow to a certain value, Ac , such that

[1.17]

Under this stress state, the two surfaces develop a sort of cold welding. Suppose we now apply a shear force, T. If τy is the shear strength of the weld, sliding will take place at

[1.18]

Since τy and σy are constants that are characteristic of the material that composes the grains, it follows from Amonton’s law that

[1.19]

which is independent of the apparent contact area.

As the normal stresses at the contacts determine the resistance to sliding, it follows that the more a soil is confined by compression stresses the higher the shear stresses must be in order to induce significant strains.

Moreover, smaller or larger strains will take place in the sample according to the grains structure, more or less dense (the external load increment being equal). When a column collapses, a new one will form more easily in a dense sample than in a loose one. Also the shape of the particles is relevant. For instance, rounded particles rotate more easily and oppose less resistance to a structure rearrangement compared to elongated or elliptical ones. As a consequence, a sample composed of elongated particles will be more rigid and resistant than a sample of equal weight composed of rounded particles.

If the particle configuration is compact, that is if the material is in a dense state, and if the particles are rigid, a load increment will lead to a less compact configuration and will be accompanied by an increase in the overall volume of the sample. This phenomenon is referred to as dilatancy.

To visualize this phenomenon, consider Figure 1.7, in which two rough rigid plates are in contact. Assume the lower plate to be fixed. The action of a force T implies a relative displacement of the two plates with the consequent raise of the upper plate. Since voids increase, the overall volume between the two plates increases.

Note from the figure that in order for displacement to take place, the upper plate has to win the resistance due to the friction between the grains and also the one due to geometry. Geometry in its turn controls dilatancy. It is therefore evident that a strong relationship exists between dilatancy and strength.

Notice that grains cannot always be assumed to be rigid. If the sample is loaded by rigid plates that impose the uniformity of displacements at the boundaries as in Figure 1.7, the average stress state will be given by the force necessary to generate this displacement divided by the area of the load plate. However, the effective contact area between the grains is much smaller than the total area of the plates. This implies that the stress state at contacts is at least three orders of magnitude higher. Small average stresses can therefore generate very high stress states at the contacts among grains. In calcareous and volcanic sands this can lead to the breaking of grains. In contrast, in quartz sands this can occur only at very high average stresses (higher than 1 MPa), as for instance at the base of foundations piles.

Figure 1.7.The phenomenon of dilatancy

1.6. Transmission of the stress state in the presence of a fluid

In a saturated soil under hydrostatic conditions, the stress state of the pore fluid is equal to the hydrostatic pressure due to the weight of the fluid above. For porous enough materials such as soils, voids are continuous and water can seep through the pores driven by differences in the hydraulic head.

The pressure of water tends to separate the grains one from each other, facilitating their sliding. Imagine filling the void space between the brick and the rough plane, considered in section 1.5 (Figure 1.6), with a fluid under pressure; the fluid having a pressure equal to u.

According to vertical equilibrium, under the assumption of a contact area much smaller than the total one:

[1.20]

Hence,

[1.21]

and then sliding takes place at

[1.22]

The higher the pressure of the water, the lower the force T necessary for sliding to occur.

Let τ be the average shear stress on the plane of area A and σ the average normal stress on the same plane, equation [1.22] then gives

[1.23]

where σ′ is referred to as effective stress.

The overall deformability of a volume of soil is governed by the relative sliding among grains. It follows the soil mechanical behavior is ruled by the effective stresses and not by the total ones. Therefore, it will be assumed as a postulate that the soil mechanical behavior depends solely on the effective stress state and on its changes.

If soil is not fully saturated, water tends to concentrate around the contact areas (Figure 1.8), and its surface tension gives rise to adhesion among the grains pressed together. Thereafter, since shear strength is provided by the frictional strength generated by capillary pressure, the grains remain in contact, even in the absence of external compression loads. This strength, often referred to as apparent cohesion, vanishes if soil is submerged in water and thus the water pressure becomes positive.

Figure 1.8.Formation of water menisci around grain contacts in unsaturated soil

In fine grained soils, the role of porosity is played by water content. According to [1.10], in the case of a fully saturated soil

[1.24]

in which void ratio and water content are synonymous except for a multiplicative non dimensional factor.

Nevertheless, fine grained soils behave according to this relationship only in a limited range. In fact, for high water contents, fine grained soils will tend to behave more as a viscous fluid than as a solid, even though porous and saturated. The liquid limit, wL, is defined as the transition water content associated with the minimum shear strength under which it is not possible to define a continuous solid any longer. Obviously a similar level is arbitrary and therefore it is defined by a standard procedure.

The value of wL depends on the mineralogical characteristics of the clay and increases with the specific surface. If the water content of a soil sample decreases (for example under a compression load), the particles will tend to get closer and the strength to increase. Below a certain value, the water content becomes too low and the soil loses ductility (plasticity) characteristic of fine grained soils. Soil then becomes brittle and strength no longer increases with the decrease in water content, or at least grows at a lower rate. This limit of plasticity, wP, is arbitrarily determined by means of a procedure that is reported in ASTM standards. The difference between wL and wP is referred to as the plasticity index (PI). Liquid and plastic limits are called Atterberg limits from the name of the Swedish agronomist that conceived them in order to empirically derive soil properties. In fact, although in a qualitative manner, Atterberg limits, and in particular wL and PI, are related to some important mechanical characteristics; such as oedometric compressibility and short term shear strength. Moreover, since they can be determined by employing simple equipment and at an extremely cheap cost, they still have a relevant practical use. Indeed, fine grained soils classification is based on the Atterberg limits (plasticity chart).

1.7. From discrete to continuum

Theoretically, it would be possible, exploiting powerful computers, to follow the evolution of the single grains subjected to an assigned load path. However, such an approach is certainly unjustified from an economical viewpoint. Moreover, at present, the computational burden could turn out to be higher than the time necessary for the construction of the designed structure. Furthermore, to solve a boundary values problem, the knowledge of at least the geometry of the problem, namely dimensions and distribution of particles, is necessary. It is apparent that this level of detail of knowledge is not achievable. It is therefore necessary to adopt an alternative strategy to predict the behavior of an engineering structure, which does not necessarily imply knowledge of the stress state at each contact.

Even though soil can be interpreted as the assembly of independent particles, the fundamental idea at the basis of soil mechanics is that, with respect to the dimensions of the structures with which it interacts (e.g. foundations, diaphragm walls, tunnels, etc.), dry soil can be considered as a solid continuum. It is then reasonable to define at every point the stress and strain states. The stress state is defined by the stress tensor, σij, and is linked via equilibrium relationships to the external loads. Conversely, the strain state is defined by the strain tensor, εhk, and is related via compatibility relationships to the displacement field, Uh.

The major conceptual issue is concerned with the fact that soil is a porous medium whose behavior depends on the effective stress at contacts. The pressure of the pore fluid then has to be taken into account; therefore, the fundamental assumption of pore fluid as a continuum, in which it is possible to measure the pressure, u, at every point, is introduced. Soil is hence schematized as a medium composed of two continua occupying the same volume of space; a solid continuum, ruled by the laws of solid mechanics, and a fluid continuum, governed by the laws of fluid mechanics. This medium will be referred to as porous medium. The two continua are assumed to act in parallel. As far as equilibrium is concerned, the total stress, σij , will be divided into two portions; one called effective stress, σ′ij , acting on the solid continuum (skeleton), and the second one constituted by the pore water pressure, u, acting on the fluid. Since free water can only stand isotropic pressures, the effective stress tensor will be defined as:

[1.25]

where δij is the unit second order tensor (Kronecker delta). The effective stress σ′ij can then be seen, from a physical viewpoint, as the fraction of the total stress that grains exchange via their contacts (see section 1.6). Nevertheless, it is important to note that although σ′ij is linked to the stress at contacts, it is numerically very different. In fact, σij and consequently σ′ij are defined at each point of the continuum as the ratio of the resultant to the overall cross-sectional area, and not only to the fraction occupied by the solids or to the sum of the contact areas.

Moreover, compatibility between the two continua requires the decrease in volume of the solid continuum to be equal to the volume of water expelled from that volume. In fact, both solid particles and water are assumed to be incompressible.

The importance of the effective stress tensor is related to a fundamental experimental result achieved by Terzaghi (1923). In fact, he noticed, by comparing the behavior of saturated soil samples subjected to stress states differing only in the value of the pore pressure, that strength and deformability were the same. He inferred then that the value of the pore water pressure alone does not influence the mechanical behavior of the soil and called it neutral pressure. Conversely, strength and deformability depend only on the effective stress, and are therefore called effective. Rendulic (1937) provided later exhaustive experimental evidence.

For unsaturated soils, a relevant role is played by water suction that can become very high (in absolute value). In the late 20th century, thanks to Alonso, Gens and Josa (1990), the complex interaction between water and solid skeleton has also been investigated for partially saturated soils.

The dependence of soil mechanical behavior on the effective stress implies the need to measure the neutral pressure at every point. Three situations can be distinguished: still ground-water table, stationary and transient flow.

Refer to Figure 1.9. For the sake of simplicity, the water table is assumed to correspond with the horizontal soil surface. The vertical stress at any arbitrary depth z can then be derived. For reasons of symmetry, shear stresses on the lateral surfaces of the prism illustrated in the figure are zero, hence, according to vertical equilibrium:

[1.26]

Moreover, equilibrium of the fluid phase implies that

[1.27]

and, thus, according to the definition of effective stress

[1.28]

Figure 1.9.Vertical stress on a soil element at depth z below the horizontal ground surface with still ground-water table at surface level

Consider now Figure 1.10. Imagine having to excavate to a depth H and lowering the water table to this level with a pumping system. According to Pascal’s principle of communicating vessels, water will flow from the reservoir towards the excavation to restore the initial level. Thus, in order to maintain the difference in water level, water must be continuously pumped from the excavation. After a certain period of time, the flow, Q, will become constant and stationary flow will be established. The value of the pressure at every point can then be derived via a continuity equation stating the balance between the mass of water entering and leaving the volume of soil.

Figure 1.10.Stationary seepage

Refer, finally, to the case in Figure 1.11. Imagine building an embankment composed of granular material on top of a saturated clay layer. The embankment will behave as an external load, q, on the soil. This load has to be equilibrated by an increase in the soil total stress, Δσv. To determine the magnitude of the portion of load transmitted to the solid skeleton and the portion transmitted to the water, it is necessary to consider that an increase in stress on the solid skeleton implies the rearrangement of its structure. This can occur only with a partial expulsion of water from the pores. In fact, both water and solid particles are considered incompressible at the ordinary stress levels. Due to the low permeability of soil, a certain amount of time is necessary in order for the soil to rearrange its structure.

Figure 1.11.Layer of clayey soil subjected to the load of a soil embankment

In order to understand this process, a simple analogy is presented (Figure 1.12), in which the spring represents the soil skeleton. Let P be the total load, Ps the load carried by the spring and Pw the load carried by the water.

Figure 1.12.Schematization of the problem in Figure 1.11. Ps and Pw change over time. Initial condition y=0, Ps=0

1.8. Stress and strain tensors

Before proceeding any further, it seems suitable to recall some definitions and proprieties concerning the stress and strain tensors.

The stress state at a point is expressed by a second order symmetric tensor, known as the Cauchy stress tensor, characterized by nine components, which can be written in matrix form as:

[1.29]

or with engineering notation as:

[1.30]

In which the diagonal entries represent the normal stresses acting orthogonally to the surface of normal xi, while the off-diagonal terms represent the shear components acting on the surface of normal xi and directed as xj.

In engineering notation, normal stresses are expressed with the Greek letter σ and only one index, while shear stresses are indicated with the letter τ.

Any second order tensor can be expressed as the sum of an isotropic tensor and of a deviator tensor:

[1.31]

where

[1.32]

is the average isotropic pressure and σij the stress deviator. Einstein’s notation has been exploited in the expression of p. According to this convention when an index variable appears twice in a single term it implies summation. From now on, this convention will always be adopted in the text.

Similarly,

[1.33]

It is apparent, from the definition of effective stress, that

[1.34]

[1.35]

Only the isotropic portion of the stress tensor is influenced by the water pressure, while the effective deviator stresses coincide with the total stresses. Hence, from now on any specification will be omitted for the deviator term.

Similarly, the strain state at a point is expressed by a second order symmetric tensor as:

[1.36]

or with engineering notation as:

[1.37]

The diagonal entries represent the longitudinal strains along the direction xk , while the off-diagonal terms are equal to half the variation of initially orthogonal fibers. Also the strain tensor can be divided into an isotropic and a deviator part

[1.38]

where

[1.39]

is the volumetric strain. The tensor, εhk, referred to as the strain deviator, is linked to the change in shape of the volume element.

By definition, the change of reference frame (passing from a Cartesian coordinate system to another system) implies that the tensor aij of second order transforms into the tensor ahk, as follows:

[1.40]

where mhi contains the cosines of the angles between the axis xh of the new Cartesian frame and the axis xi of the old one. The components of the tensor ahk are different from those of aij. Nevertheless, some quantities, symmetric with respect to the indices and obtained as a combination of first, second and third order of the components aij, remain constant. In particular:

[1.41]

[1.42]

[1.43]

I1, I2 and I3 are referred to as the first, second and third invariants of the tensor aij, respectively.

It is immediately evident that p is equal to one third of the first invariant of the stress tensor, while εv is equal to the first invariant of the strain tensor.

Similarly, the invariants of the deviator part bij can be defined as

[1.44]

It is apparent that the first invariant of the deviator is zero, while

[1.45]

[1.46]

For every symmetric second order tensor, a reference frame, referred to as principal and for which all the entries aij with i ≠ j are zero, exists. The diagonal entries are called principal components and the invariants can be expressed as functions of these components, for example:

[1.47]

[1.48]

[1.49]

and hence

[1.50]

[1.51]

[1.52]

[1.53]

1.9. Bibliography

AASHO (1986) “Standard specifications for trasportation materials and methods of sampling and testing. vol.I”, Specifications, AASHO, Washington.

ASTM (1992) “Annual book of standards”, Soil and Rock, vol. 04.08, Philadelphia.

AGI (1963) Nomenclatura geotecnica delle terre, RIG, no.4, pp. 275–286

Alonso E.E., Gens A. & Josa A. (1990) “A constitutive model for partially saturated soils”, Géotechnique, vol. 40, no. 3, pp. 405–430.

ASTM D2216-05. Laboratory Determination of Moisture Content of Soil, American Society for Testing and Materials.

BS 1377 (1981) Methods of Testing for Soil for Civil Engineering Purposes, BSI, London

BS 5930 (1981) Code of Practice for Site Investigations, BSI, London.

Calvetti F. (1998) Micromeccanica dei materiali granulari. Tesi di Dottorato. Politecnico di Milano. Dipartimento di Ingegneria Strutturale.

Drescher A., De Josselin de Jong G. (1972) “Photoelastic verification of a mechanical model for the flow of a granular material”, J. of Mech. Phys. Solids, no. 20, pp. 337–351.

Rendulic L. (1937) Ein Grundgesetz der Tonmechanik und sein experimenteller Beweis. Der Bauingenieur, vol. 18, 459.

Tabor D. (1959) “Junction growth in metallic friction: the role of combined stresses and surface contamination”. Proc. Roy. Soc. Series, a 251, pp. 378–393.