Schofield wroth 1968

Critical State

Soil Mechanics

Andrew Schofield and Peter Wroth Lecturers in Engineering at Cambridge University

Preface

This book is about the mechanical properties of saturated remoulded soil. It is written at the level of understanding of a final-year undergraduate student of civil engineering; it should also be of direct interest to post-graduate students and to practising civil engineers who are concerned with testing soil specimens or designing works that involve soil. Our purpose is to focus attention on the critical state concept and demonstrate what we believe to be its importance in a proper understanding of the mechanical behaviour of soils. We have tried to achieve this by means of various simple mechanical models that represent (with varying degrees of accuracy) the laboratory behaviour of remoulded soils. We have not written a standard text on soil mechanics, and, as a consequence, we have purposely not considered partly saturated, structured, anisotropic, sensitive, or stabilized soil. We have not discussed dynamic, seismic, or damping properties of soils; we have deliberately omitted such topics as the prediction of settlement based on Boussinesq’s functions for elastic stress distributions as they are not directly relevant to our purpose. The material presented in this book is largely drawn from the courses of lectures and associated laboratory classes that we offered to our final year civil engineering undergraduates and advanced students in 196 5/6 and 1966/7. Their courses also included material covered by standard textbooks such as Soil Mechanics in Engineering Practice by K. Terzaghi and R. B. Peck (Wiley 1948), Fundamentals of Soil Mechanics by D. W. Taylor (Wiley 1948) or Principles of Soil Mechanics by R. F. Scott (Addison-Wesley 1963). In order to create a proper background for the critical state concept we have felt it necessary to emphasize certain aspects of continuum mechanics related to stress and strain in chapter 2 and to develop the well-established theories of seepage and one-dimensional consolidation in chapters 3 and 4. We have discussed the theoretical treatment of these two topics only in relation to the routine experiments conducted in the laboratory by our students, where they obtained close experimental confirmation of the relevance of these theories to saturated remoulded soil samples. Modifications of these theories, application to field problems, three-dimensional consolidation, and consideration of secondary effects, etc., are beyond the scope of this book. In chapters 5 and 6, we develop two models for the yielding of soil as isotropic plastic materials. These models were given the names Granta-gravel and Cam-clay from that river that runs past our laboratory, which is called the Granta in its upper reaches and the Cam in its lower reaches. These names have the advantage that each relates to one specific artificial material with a certain distinct stress – strain character. Granta-gravel is an ideal rigid/plastic material leading directly to Cam-clay which is an ideal elastic/plastic material. It was not intended that Granta-gravel should be a model for the yielding of dense sand at some early stage of stressing before failure: at that stage, where Rowe’s concept of stress dilatancy offers a better interpretation of actual test data, the simple Granta-gravel model remains quite rigid. However, at peak stress, when Granta-gravel does yield, the model fits our purpose and it serves to introduce Taylor’s dilatancy calculation towards the end of chapter 5. Chapter 6 ends with a radical interpretation of the index tests that are widely used for soil classification, and chapter 7 includes a suggested computation of ‘triaxial’ test data that allows students to interpret much significant data which are neglected in normal methods of analysis. The remainder of chapter 7 and chapter 8 are devoted to testing the

Table of contents

Contents

Preface Glossary of Symbols

Glossary of symbols

The list given below is not exhaustive, but includes all the most important symbols used in this book. The number after each brief definition refers to the section in the book where the full definition may be found, and the initials (VVS) indicate that a symbol is one used by Sokolovski. The dot notation is defined in §2, whereby denotes a small change in the value of the parameter x. As a result of the sign convention adopted (§2) in which compressive stresses and strains are taken as positive, the following parameters only have a positive dot notation associated with a negative change of value (i.,

x&

+x&=−δx):v,l,r. The following letters are used as suffixes: f failure, r radial, 1 longitudinal; and as superfixes: r recoverable, p plastic (irrecoverable).

a Height of Coulomb’s wedge of soil 9. a Cross-sectional area of sample in axial-test 5. Half unconfined compression strength 6. Coefficient of consolidation 4.

d Diameter, displacement, depth 1, 2, 3. e Voids ratio 1. General function and its derivative 3. Height, and height of water in standpipe 3.

i Hydraulic gradient 3. k Maximum shear stress (Tresca) 2. k Coefficient of permeability 3. k Cohesion in eq. (8) (VVS) 8. l Length of sample in axial-test 5. l Constant in eq. (8) cf. λ in eq. (5) 8.

Coefficients of volume compressibility 4.

n Porosity 1. n Normal stress (VVS) 9. p Effective spherical pressure 5.

Equivalent pressure cf.

cu cvc

ff ', ,hh w

mvc,mvs

pe σ'e 5. Undrained critical state pressure 5. Critical state pressure on yield curve 6. Pressure corresponding to liquid limit 6. Pressure corresponding to plastic limit 6. Pressure corresponding to Ω point 6.

p*, q* Generalized stress parameters 8. p, q Uniformly distributed loading pressures (VVS) 9. Equivalent stress (VVS) 9.

q Axial-deviator stress 5. Undrained critical state value of q 5. Critical state value of q on yield curve 6.

r Radial coordinate 3. r

pu px pLL pPL pΩ

p'

qu qx

1 , r 2 Directions of planes of limiting stress ratio (VVS) 9. s Distance along a flowline 3. Parameters locating centres of Mohr’s circles (VVS) 9.

s, t Stresses in plane strain App. C

s+,s−

t Tangential stress (VVS) 9. t Time 1.

12 t Half-settlement time 4.

u Excess pore-pressure 3.

Total pore-pressure 1.

u Velocity of stream 1. v Velocity 1. Artificial velocity 3.

v

uw

va s Seepage velocity 3. v Specific volume 1. vx Critical state value of v on yield curve 6. Ordinates of κ-line and λ-line 6. Specific volume at liquid limit 6. Specific volume at plastic limit 6. Change of specific volume corresponding to plasticity index 6. Specific volume corresponding to Ω point 6.

w Water content 1. w Weight 1. x, y, z Cartesian coordinate axes 1. x

vκ,vλ vLL vPL ∆vPI vΩ

t Transformed coordinate 3. A,At Cross-sectional areas 3, 4. A,Am Fourier coefficients 4. AB,, B Pore-pressure coefficients 7. Compression indices 4, 6.

D Diameter 1. D

CcC', c

0 A total change of specific volume 5. E Young’s modulus 2.

E& Loading power 5. Potential functions (Mises) 2, 2, app. C

F Frictional force 9. G Shear modulus 2. G

F F,', F*

s Specific gravity 1. H Maximum drainage path 4. H Abscissa of Mohr-Coulomb lines (VVS) 8. K Bulk modulus 2. K, K 0 Coefficients of earth pressure 6. L Lateral earth pressure force 9. M Mach number 1. N Overcompression ratio 7. N Normal force 9. Vertical loads in consolidometer 4. Active and passive pressure forces 9, 9.

P,Pw,Ps PA,PP P& Probing power 5. Q Quantity of flow 3. R Force of resistance 1. T Tangential force 9. Tv Time factor 4.

∑'

• Chapter 1 Basic Concepts Table of Conversions for S Units
  • 1 Introduction
  • 1 Sedimentation and Sieving in Determination of Particle Sizes
  • 1 Index Tests
  • 1 Soil Classification
  • 1 Water Content and Density of Saturated Soil Specimen
  • 1 The Effective Stress Concept
  • 1 Some Effects that are ‘Mathematical’ rather than ‘Physical’
  • 1 The Critical State Concept
  • 1 Summary
• Chapter 2 Stresses, Strains, Elasticity, and Plasticity
  • 2 Introduction
  • 2 Stress
  • 2 Stress-increment
  • 2 Strain-increment
  • 2 Scalars, Vectors, and Tensors
  • 2 Spherical and Deviatoric Tensors
  • 2 Two Elastic Constants for an Isotropic Continuum
  • 2 Principal Stress Space
  • 2 Two Alternative Yield Functions
  • 2 The Plastic Potential Function and the Normality Condition
  • 2 Isotropic Hardening and the Stability Criterion
  • 2 Summary
• Chapter 3 Seepage
  • 3 Excess Pore-pressure
  • 3 Hydraulic Gradient
  • 3 Darcy’s Law
  • 3 Three-dimensional Seepage
  • 3 Two-dimensional Seepage
  • 3 Seepage Under a Long Sheet Pile Wall: an Extended Example
  • 3 Approximate Mathematical Solution for the Sheet Pile Wall
  • 3 Control of Seepage
• Chapter 4 One-dimensional Consolidation
  • 4 Spring Analogy
  • 4 Equilibrium States
  • 4 Rate of Settlement
  • 4 Approximate Solution for Consolidometer
  • 4 Exact Solution for Consolidometer
  • 4 The Consolidation Problem
• Chapter 5 Granta-gravel
  • 5 Introduction
  • 5 A Simple Axial-test System
  • 5 Probing
  • 5 Stability and Instability
  • 5 Stress, Stress-increment, and Strain-increment
  • 5 Power
  • 5 Power in Granta-gravel
  • 5 Responses to Probes which cause Yield
  • 5 Critical States
  • 5 Yielding of Granta-gravel
  • 5 Family of Yield Curves
  • 5 Hardening and Softening
  • 5 Comparison with Real Granular Materials
  • 5 Taylor’s Results on Ottawa Sand
  • 5 Undrained Tests
  • 5 Summary
• Chapter 6 Cam-clay and the Critical State Concept
  • 6 Introduction
  • 6 Power in Cam-clay
  • 6 Plastic Volume Change
  • 6 Critical States and Yielding of Cam-clay
  • 6 Yield Curves and Stable-state Boundary Surface
  • 6 Compression of Cam-clay
  • 6 Undrained Tests on Cam-clay
  • 6 The Critical State Model
  • 6 Plastic Compressibility and the Index Tests
  • 6 The Unconfined Compression Strength
  • 6 Summary
• Chapter 7 Interpretation of Data from Axial Tests on Saturated Clays
  • 7 One Real Axial-test Apparatus
  • 7 Test Procedure
  • 7 Data Processing and Presentation
  • 7 Interpretation of Data on the Plots of v versus ln p
  • 7 Applied Loading Planes
  • 7 Interpretation of Test Data in (p, v, q) Space
  • 7 Interpretation of Shear Strain Data
  • 7 Interpretation of Data of ε&and Derivation of Cam-clay Constants
  • 7 Rendulic’s Generalized Principle of Effective Stress
  • 7 Interpretation of Pore-pressure Changes
  • 7 Summary
• Chapter 8 Coulomb’s Failure Equation and the Choice of Strength Parameters
  • 8 Coulomb’s Failure Equation
  • 8 Hvorslev’s Experiments on the Strength of Clay at Failure
  • 8 Principal Stress Ratio in Soil About to Fail
  • 8 Data of States of Failure
  • 8 A Failure Mechanism and the Residual Strength on Sliding Surfaces
  • 8 Design Calculations
  • 8 An Example of an Immediate Problem of Limiting Equilibrium
  • 8 An Example of the Long-term Problem of Limiting Equilibrium
  • 8 Summary
  • φ,φm Angles of friction in Taylor’s shear tests 5.
  • φ+,φ− Inclinations of principal stress on either side of discontinuity (VVS) 9.
• ψ Streamline function 3.
• Γ Ordinate of critical state line 5. - ∆ Caquot’s angle (VVS) 9. - Λ Parameter relating swelling with compression 6.
• M Critical state frictional constant 5. - Major principal stress (VVS) 9.
• Ω Common point of idealized critical state lines 6.

1

Basic concepts

1 Introduction

This book is about conceptual models that represent the mechanical behaviour of saturated remoulded soil. Each model involves a set of mechanical properties and each can be manipulated by techniques of applied mathematics familiar to engineers. The models represent, more or less accurately, several technically important aspects of the mechanical behaviour of the soil-material. The soil-material is considered to be a homogeneous mechanical mixture of two phases: one phase represents the structure of solid particles in the soil aggregate and the other phase represents the fluid water in the pores or voids of the aggregate. It is more difficult to understand this soil-material than the mechanically simple perfectly elastic or plastic materials, so most of the book is concerned with the mechanical interaction of the phases and the stress – strain properties of the soil-material in bulk. Much of this work is of interest to workers in other fields, but as we are civil engineers we will take particular interest in the standard tests and calculations of soil mechanics and foundation

engineering.

It is appropriate at the outset of this book to comment on present standard practice in soil engineering. Most engineers in practice make calculations and base their judgement on the model used two hundred years ago by C. A. Coulomb 1 in his classic analysis of the active and passive pressures of soil against a retaining wall. In that model soil-material (or rock) was considered to remain rigid until there was some surface through the body of soil-material on which the shear stress could overcome cohesion and internal friction, whereupon the soil-material would become divided into two rigid bodies that could slip relative to each other along that surface. Cohesion and internal friction are properties of that model, and in order to make calculations it is necessary for engineers to attribute specific numerical values to these properties in each specific body of soil. Soil is difficult to sample, it is seldom homogeneous and isotropic in practice, and engineers have to exercise a considerable measure of subjective judgement in attributing properties to soil. In attempts over the last half-century to make such judgements more objective, many research workers have tested specimens of saturated remoulded soil. To aid practising engineers the successive publications that have resulted from this continuing research effort have reported findings in terms of the standard conceptual model of Coulomb. For example, typical papers have included discussion about the ‘strain to mobilize full friction’ or ‘the effect of drainage conditions on apparent cohesion’. Much of this research is well understood by engineers, who make good the evident inadequacy of their standard conceptual model by recalling from their experience a variety of cases, in each of which a different interpretation has had to be given to the standard properties. Recently, various research workers have also been developing new conceptual models. In particular, at Cambridge over the past decade, the critical state concept (introduced in §1 and extensively discussed in and after chapter 5) has been worked into a variety of models which are now well developed and acceptable in the context of isotropic hardening elastic/plastic media. In our judgement a stage has now been reached at which engineers could benefit from use of the new conceptual models in practice. We wish to emphasize that much of what we are going to write is already incorporated by engineers in their present judgements. The new conceptual models incorporate both the standard Coulomb model and the variations which are commonly considered in practice: the words cohesion and friction, compressibility and consolidation, drained and undrained will be used here as in practice. What is new is the inter-relation of

3

tube is placed in position in a constant temperature bath. After certain specified periods a pipette is inserted to a depth in the tube, a few millilitres of the suspension are

withdrawn at that depth and are transferred into a drying bottle.

z= 100 mm

Let us suppose that originally a weight W of solid material was dispersed in 500 millilitres of water, and after time t at depth 100 mm a weight w of solid material is found in a volume V of suspension withdrawn by the pipette. If the suspension had been sampled immediately at then the weight w would have been WV/500 but, as time passes, w will have decreased below this initial value. If all particles were of a single size, with effective diameter D, and if we calculate a time

t= 0

() 1 2

18 G D

z t s w

D γ

μ −

= (1)

then before time tD the concentration of sediment at the sampling depth

would remain at its initial value WV/500: at time t

(z= 100 mm) D the particles initially at the surface of the tube would sink past the depth z= 100 mm,and thereafter, as is shown in Fig. 1, there

would be clear liquid at the depth z. It is evident that if any particles of one specified size are present at a depth z they are present in their original concentration; (this is rather like dropping a length of chain on the ground: as the links fall they should preserve their original spacing between centres, and they would only bunch up when they strike the heap of chain on the ground). Therefore, in the general case when we analyse a dispersion of various particle sizes, if we wish to know what fraction by weight of the particles are of diameter less than D, we must arrange to sample at the appropriate time tD. The ratio between the weight w withdrawn at that time, and the initial value

Fig. 1 Process of Sedimentation on Dispersed Specimen

WV/500, is the required fraction. The fraction is usually expressed as a percentage smaller than a certain size. The sizes are graded on a logarithmic scale. Values are usually found for D = 0, 0, 0, and 0 mm, and these data6-11 are plotted on a curve as shown in Fig. 1. A different technique 5 , sieving, is used to sort out the sizes of soil particles of more than 0 mm diameter. A sieve is made with wire cloth (a mesh of two sets of wires woven together at right angles to each other). The apertures in this wire cloth will pass particles that have an appropriate intermediate or short diameter, provided that the sieve is shaken sufficiently for the particles to have a chance of approaching the holes in the right way. The finer sieves are specified by numbers, and for each number there is a standard 12 nominal size in the wire cloth. The coarser sieves are specified by nominal apertures. Clearly this technique implies a slightly different definition of diameter from that of sedimentary analysis, but a continuous line must be drawn across the particle size distribution chart of Fig. 1: the assumption is that the two definitions are equivalent in

4

the region about 0 mm diameter where sieves are almost too fine and sedimentary settlement almost too fast.

Fig. 1 Particle Size Distribution Curves

Civil engineers then use the classification for grain size devised at the Massachusetts Institute of Technology which defines the following names: Boulders are particles coarser than 6 cm, or 60 mm diameter; Gravel contains particles between 60 mm and 2 mm diameter; Sand contains particles between 2 mm and 0 mm diameter; Silt contains particles between 0 mm and 0 mm diameter; Clay contains particles finer than 0 mm (called two microns, 2μ).

The boundaries between the particle sizes not only give almost equal spacing on the logarithmic scale for equivalent diameter in Fig. 1, but also correspond well with major changes in engineering properties. A variety of soils is displayed, including several that have been extensively tested with the results being discussed in detail in this text. For example, London clay has 43 per cent clay size, 51 per cent silt size and 6 per cent sand size.

1 Index Tests

The engineer relies chiefly on the mechanical grading of particle sizes in his description of soil but in addition, two index numbers are determined that describe the clayeyness of the finer fraction of soil. The soil is passed through a sieve, B. No. 36 or U. Standard No. 40, to remove coarse sand and gravel. In the first index test 5 the finer fraction is remoulded into a paste with additional water in a shallow cup. As water is added the structure of fine soil particles is remoulded into looser states and so the paste becomes progressively less stiff. Eventually the soil paste has taken up sufficient water that it has the consistency of a thick cream; and then a groove in the paste (see Fig. 1) will close with the sides of the groove flowing together when the bottom of the cup is given a succession of 25 blows on its base. The paste is then at the ‘lowest limit’ of a continuing

6

that the simple engineering classification does consider the most important mechanical attributes of soil. It is hard to appreciate the significance of the immense diversity of sizes of soil particles. It may be helpful to imagine a city scene in which men are spreading tarred road stone in the pavement in front of a fifteen-story city building. If this scene were reduced in scale by a factor of two hundred thousand then a man of 1 metres in height would be nine microns high – the size of a medium silt particle; the building would be a third of a millimetre high – the size of a medium sand grain; the road stones would be a tenth of a micron – the size of what are called ‘colloidal’ particles; the layer of tar would correspond to a thickness of several water molecules around the colloidal particles. Our eyes could either focus on the tarred stones in a small area of road surface, or view the grouping of adjacent buildings as a whole; we could not see at one glance all the objects in that imaginary scene. The diversity of sizes of soil particles means that a complete survey of their geometry in a soil specimen is not feasible. If we select a volume of 1 m 3 of soil, large enough to contain one of the largest particles (a boulder) then this volume could also contain of the order of 10 8 sand grains and of the order of 10 16 clay particles. A further problem in attempting such a geometrical or structural survey would be that the surface roughness of the large irregular particles would have to be defined with the same accuracy as the dimensions of the smallest particles. An undisturbed soil can have a distinctive fabric. The various soil-forming processes may cause an ordering of constituents with concentration in some parts, and the creation of channels or voids in other parts. Evidence of these extensively occurring processes can be obtained by a study of the microstructure of the soil, and this can be useful in site investigation. The engineer does need to know what extent of any soil deposit in the field is represented by each specimen in the laboratory. Studies of the soil-forming processes, of the morphology of land forms, of the geological record of the site will be reflected in the words used in the description of the site investigation, but not in the estimates of mechanical strength of the various soils themselves. In chapters 5 and 6 we consider macroscopically the mechanical strength of soil as a function of effective pressure and specific volume, without reference to any microscopic fabric. We will suggest that the major engineering attributes of real soils can be explained in terms of the mechanical properties of a homogeneous isotropic aggregate of soil particles and water. We show that the index properties are linked with the critical states of fully disordered soil, and we suggest that the critical state strengths form a proper basis of the stability of works currently designed by practising engineers. Suppose we have a soil with a measured peak strength which (a) could not be correlated with index properties, (b) was destroyed after mechanical disturbance of the soil fabric, and (c) could only be explained in terms of this fabric. If we wished to base a design on this peak strength, special care would be needed to ensure that the whole deposit did have this particular (unstable) property. In contrast, if we can base a design on the macroscopic properties of soil in the critical states, we shall be concerned with more stable properties and we shall be able to make use of the data of a normal soil survey such as the in situ water content and index properties. We gave the name clay to particles of less than 2 micron effective diameter. More properly the name clay should be reserved for clay minerals (kaolinite, montmorillonite, illite, etc.). Any substance when immersed in water will experience surface forces: when the substance is subdivided into small fragments the body forces diminish with the cube of size while surface forces diminish with the square of size, and when the fragments are less than 0 micron in size the substance is in ‘colloidal’ form where surface forces predominate. The hydrous-alumino-silicate clay minerals have a sheet-like molecular

7

structure with electric charge on the surfaces and edges. As a consequence clay mineral particles have additional capacity for ion-exchange. Clearly, a full description of the clay/water/solute system would require detailed studies of a physical and chemical type described in a standard text, such as that by Grim. 14 However, the composite effects of these physico-chemical properties of remoulded clay are reflected to a large measure in the plasticity index. In §6 we show how variation of plasticity corresponds to variation in the critical states, and this approach can be developed as a possible explanation of phenomena such as the al sensitivity of leached post-glacial marine clay, observed by Bjerrum and Rosenqvist. 15 In effect, when we reaffirm the standard soil engineering practice of regarding the mechanical grading and index properties as the basis of soil classification, we are asserting that the influence of mineralogy, chemistry and origin of a soil on its mechanical is behaviour is adequately measured by these simple index tests.

1 Water Content and Density of Saturated Soil Specimen

If a soil specimen is heated to 105°C most of the water is driven off, although a little will still remain in and around the clay minerals. Heating to a higher temperature would drive off some more water, but we stop at this arbitrary standard temperature. It is then supposed that the remaining volume of soil particles with the small amount of water they still hold is in effect ‘solid’ material, whereas all the water that has been evaporated is ‘liquid’. This supposition makes a clear simplification of a complicated reality. Water at a greater distance from a clay particle has a higher energy and a lower density than water that has been adsorbed on the clay mineral surface. Water that wets a dry surface of a clay mineral particle emits ‘heat of wetting’ as the water molecules move in towards the surface; conversely drying requires heat transfer to remove water molecules off a wet surface. The engineering simplification bypasses this complicated problem of adsorption thermodynamics. Whatever remains after the sample has been dried at 105°C is called solid; the specific gravity (Gs) of this residue is found by experiment. Whatever evaporates when the sample is dried is called pore-water and it is assumed to have the specific gravity of pure water. From the weights of the sample before and after drying the water content is determined as the ratio:

. weightofsolids

weightofporewater watercontentw=

In this book we will attach particular significance to the volume of space v occupied by unit volume of solids: we will call v the specific volume of unit volume of solids. Existing soil mechanics texts use an alternative symbol e called ‘voids ratio’ which is the ratio between the volume of ‘voids’ or pore space and the volume of solids: v= 1 +e.

A further alternative symbol n called porosity is defined by n=(v− /)1 v=e/v. Figure

1(a) illustrates diagrammatically the unit volume of solids occupying a space v, and Fig. 1(b) shows separately the volumes and weights of the solids and the pore-water. In this book we will only consider fully saturated soil, with the space (v− 1 )full of pore-water.

9

simply as a space in which some properties such as strain energy or plastic power are continuously distributed So far the advances in solid state physics which have been accompanied by introduction of new materials and by new interpretations of the properties of known materials, have not led to a revival of Navier’s formulation of elasticity. There is a clear distinction between workers in continuum mechanics who base solutions of boundary value problems on equations into which they introduce certain material constants determined by experiment, and workers in solid-state physics who discuss the material constants per se. Words like specific volume, pressure, stress, strain are essential to a proper study of continuum mechanics. Once a sufficient set of these words is introduced all subsequent discussion is judged in the wider context of continuum mechanics, and no plea for special treatment of this or that material can be admitted. Compressed soil, and rolled steel, and nylon polymer, must have essentially equal status in continuum mechanics. The particular treatment of saturated soil as a two-phase continuum, while perfectly proper in the context of continuum mechanics is sufficiently unusual to need comment. We have envisaged a distribution of clean solid particles in mechanical contact with each other, with water wetting everything except the most minute areas of interparticle contact, and with water filling every space not occupied by solids. The water is considered to be an incompressible liquid in which the pore pressure may vary from place to place. Pore-water may flow through the structure of particles under the influence of excess pore-pressures: if the structure of particles remains rigid a steady flow problem of seepage will arise, and if the structure of particles alters to a different density of packing the transient flow problem of consolidation will occur. The stress concept is discussed in chapter 2, and the total stress component σ normal to any plane in the soil is divided into two parts; the pore-pressure uw and the effective stress component σ', which must be considered to be effectively carried

by the structure of soil particles. The pore-pressure uw can be detected experimentally if a porous-tipped tube is inserted in the soil. The total stress component can be estimated from knowledge of the external forces and the weight of the soil body. The effective stress component σ' is simply calculated as

σ' =σ−uw (1)

and our basic supposition is that the mechanical behaviour of the effective soil structure depends on all the components of the effective stress and is quite independent of uw.

Fig. 1 Saturated Soil as Two-phase Continuum

In Fig. 1 the different phases are shown diagrammatically: each phase is assumed to occupy continuously the entire space, somewhat in the same manner that two vapours sharing a space are assumed to exert their own partial pressures. In Fig. 1 a simple tank is shown containing a layer of saturated soil on which is superimposed a layer of lead shot. The pore-pressure at the indicated depth in the layer is uw = γw×hw, and this applies whether hw is a metre of water in a laboratory or several kilometres of water in an ocean

10

abyss. The lead shot is applying the effective stress which of controls the mechanical behaviour of the soil.

Fig. 1 Pore-pressure and Effective Stress

The introduction of this concept of effective stress by Terzaghi, and its subsequent generalization by Rendulic, was the essential first step in the development of a continuum theory of the mechanical behaviour of remoulded saturated soils.

1 Some Effects that are ‘Mathematical’ rather than ‘Physical’

Most texts on soil mechanics 17 refer to work of Prandtl which solved the problem of plane punch indentation into perfectly plastic material; the same texts also refer to work of Boussinesq which solved various problems of contact stresses in a perfectly elastic material. The unprepared reader may be surprised by the contrast between these solutions. In the elastic case stress and strain vary continuously, and every load or boundary displacement causes some disturbance – albeit a subtle one – everywhere in the material. In direct contrast, the plastic case of limiting stress distribution leads to regions of constant stress set abruptly against fans of varying stress to form a crude patchwork. We can begin to reconcile these differences when we realize that both types of solution must satisfy the same fundamental differential equations of equilibrium, but that the elastic equations are ‘elliptic’ in character 18 whereas the plastic equations are ‘hyperbolic’. A rather similar situation occurs in compressible flow of gases, where two strongly contrasting physical regimes are the mathematical consequence of a single differential equation that comes from a single rather simple physical model. The general differential equation of steady compressible flow in the (x, y) plane has the form 19

() 1 2 0

2 2

2 2 = ∂

∂ + ∂

∂ − x y

M

φ φ (1)

where φ is a potential function, and M (the Mach number) is the ratio u/vso of the flow velocity u and the sonic velocity vso corresponding to that flow. The model is simply one of a fluid with a limiting sonic velocity. To illustrate this, we have an example in Fig. 1(a) of wavefronts emanating at speed vso from a single fixed source of disturbance O in a fluid with uniform low-speed flow u<vso. After a time t, the centre of the disturbance has been carried a distance ut downstream,