## Contents

vi Contents

3 Effective stress 67

3.1 Effective stress equation and physical meaning of effective stress 68

3.1.1 Effective stress equation 68

3.1.2 Physical meaning of effective stress 68

3.2 Relationship between effective stress and shear strength/volumetric strain 69

3.2.1 Formula of volumetric strain of a soil 69

3.2.2 Shear strength formula of a soil 70

3.2.3 Effective stress and the constitutive relationship of a soil 72

3.3 Primary verification of the correlation between effective stress

and shear strength of unsaturated soils 72

3.4 Effective stress principle for unsaturated soils 86

3.4.1 Statement of effective stress principle 86

3.4.2 Simple applications of effective stress principle for unsaturated soils 87

4 Seepage equation of unsaturated soils 93

4.1 Seepage equation of saturated soil 93

4.1.1 Derivation of the formula of Darcy’s law for saturated soil 93

4.1.2 Seepage equation for saturated soils 96

4.2 Seepage equation for unsaturated soils 99

4.2.1 Soil-water interaction force of unsaturated soils 99

4.2.2 Motion equation and the coefficient of permeability for

unsaturated soils 102

4.2.3 Seepage equation for unsaturated soils 104

4.3 Formula of seepage force 106

4.4 The overflow condition of the gas in soils 112

4.4.1 The overflow condition of the closed air bubbles 112

4.4.2 The overflow condition for pore air while surface water is infiltrating 113

5 Discussion on some issues related to effective stress 117

5.1 Does Terzaghi’s effective stress equation need to be modified? 118

5.2 Is effective stress pseudo or real stress? 121

5.3 Effective stress and stress state variables of soils 122

5.4 Effective stress and soil skeleton stress 124

5.5 The effective stress of unsaturated soils 132

5.6 Should contractile skin be the fourth phase? 134

Units and symbols 139

References 143

Subject Index 147

Terzaghi (1925, 1936) proposed the concept of effective stress and the effective stress principle,

which established the basis of soil mechanics. However, Terzaghi neither provided

an accurate definition of effective stress nor explained its physical meaning. He only indicated

“All the measurable effects of a change of the stress, such as compression, distortion

and a change of the bearing resistance, are exclusively due to changes in the effective

stresses.” Therefore, researchers deduced the effective stress according to the test results of

shear strength and volumetric change and proposed many types of modification formulae for

effective stress equation. There were some other researchers who believed effective stress is

interparticle stress or soil skeleton stress. However, following their definition of interparticle

stress or soil skeleton stress, they could not obtain Terzaghi’s effective stress equation accurately.

On the other hand, Terzaghi’s effective stress equation is only applicable for saturated

soils, not for unsaturated soils. Bishop (1959) expanded this equation to unsaturated soils

and proposed the effective stress equation for both saturated and unsaturated soils. Similarly,

since the definition and physical meaning of effective stress were not clear, confusion was

encountered in the explanation of effective stress and determination of the coefficients in

effective stress equation.

This book aims to indicate the physical meaning of effective stress and illustrate that

effective stress is the soil skeleton stress, excluding the effect of pore fluid pressure. The

effective stress equation characterizes the equilibrium relationship of the inner forces in soil

mass and owns the uniform expression for saturated and unsaturated soils, which is permanently

established. It indicates the interparticle stress and soil skeleton stress, excluding pore

fluid pressure, are consistent with the effective stress and also elucidates why effective stress

governs the strength and deformation of soils.

To illustrate the physical meaning of effective stress and derive the effective stress equation,

we applied the methods of continuous medium mechanics to conduct internal force

analysis on the free bodies of the soil skeleton, pore water and pore air separately and deduced

the equilibrium differential equation for each phase of the soil. When the free body of the soil

skeleton is taken to conduct force equilibrium analysis, the stress on the surface of the free

body has two parts: one part is induced by pore fluid pressure, which only includes normal

stress, and the other part is due to all the other external forces (excluding pore fluid pressure),

which includes normal stress and shear stress. Comparing the equilibrium differential equation

of the soil skeleton to that of total stress, the relationship among total stress, pore fluid

pressure and soil skeleton stress due to the external forces (excluding pore fluid pressure) can

be obtained. It can be found that this relationship under full-saturated condition is Terzaghi’s

## Preface

effective stress equation. Consequently, we call the soil skeleton stress due to all the external

forces, excluding pore fluid pressure, effective stress; illustrate the reason why the effective

stress equation exists theoretically and on what condition the effective stress determines the

deformation and strength of a soil; and simultaneously indicate the effective stress equation

is valid, no matter whether for saturated soils or for unsaturated soils.

In this way we can conclude that Terzaghi’s effective stress equation does not need to

be modified and that the effect of pore fluid pressure has to be excluded from internal force

analysis when using soil skeleton stress or interparticle stress to explain effective stress. The

perspective that effective stress should be deduced according to the test results of strength

and volumetric change is inappropriate. For unsaturated soils, we can determine the coefficients

in Bishop’s effective stress expression and give them explicit physical meanings.

On the other hand, the expression of Darcy’s law is obtained in this book based on the

equilibrium differential equation of pore water, which, in laminar seepage, is the condition

that the moving resistance is linearly proportional to the velocity of pore water. For unsaturated

soils, the expression of the relationship among coefficient of permeability, degree of

saturation and water potential is derived.

The key point of this book is the interpretation of effective stress, based on the equilibrium

differential equation of component phases. By carefully reading the book, it can be

found that the equilibrium differential equation, rather than effective stress, has fundamental

and foundational meaning for establishing the theoretical system of soil mechanics. The equilibrium

differential equation is essential for soil mechanics, which is the core content in this

book. The effective stress equation is a natural result of the equilibrium differential equation,

and the equation of Darcy’s law and the coefficient of permeability of unsaturated soils are

only specific applications of the equilibrium differential equation of pore fluid. The issues

discussed in the last chapter of this book are related to the equilibrium differential equation

or effective stress. These issues are important to soil mechanics, while most of them are controversial.

The discussion on these issues aims to eliminate these controversies and help the

readers better understand the role of equilibrium equation and master the concept of effective

stress. For simplicity, this book only provides the equilibrium differential equation of soil

mechanics in static condition, which are readily extended to dynamic condition.

Actually, it is very simple to derive the equilibrium differential equation by taking the

soil skeleton as the study subject and considering the effect of pore fluid pressure and external

forces separately to conduct internal force analysis. However, no one has done this before.

Biot (1941) obtained the consolidation equation by introducing the effective stress equation

into the total stress equilibrium equation, which actually reflects the equilibrium of the soil

skeleton. Zienkiewicz and Shiomi (1984) and the other researchers also derived the equilibrium

equation of the soil skeleton via inducing the effective stress equation into the total

stress equilibrium equation in the stress-strain computation of soils. Fredlund and Morgenstern

(1977) took the soil skeleton, pore water, pore air and contractile skin (surface tension

skin) as analysis subjects independently and deduced the equilibrium equation and the stress

state variable that controls the equilibrium of soil structure, whereas he did not directly derive

the equilibrium differential equation of the soil skeleton. In his research, the soil structure

included the soil skeleton and contractile skin; the equilibrium differential equation of the

soil skeleton was obtained by subtracting the equilibrium equation of pore water, pore air

and contractile skin from that of total stress. In the theory of mixtures, the soil skeleton,

pore water and pore air were treated as study subjects separately to derive their individual

equilibrium differential equation. However, in the internal force analysis of the soil skeleton,

it did not distinguish the two different equilibrium force systems of pore fluid pressure and

external forces. In its definition of soil skeleton stress, the effects of the external load and pore

fluid pressure are both included, and the physical meaning of effective stress is consequently

hidden.

Most of the content in this book is the reorganization and summary of previously published

research work by the authors. Before this, Shao (1996) proposed the interaction principle

for multiphase media, including (1) in the force and deformation analysis of soil mass,

as a multiphases medium, each phase of the soil skeleton, pore water and pore air should be

treated as an independent analysis subject; (2) the interaction between interphase forces can

be described by a pair of force and counterforce in the force analysis of each phase medium

and (3) the deformation and motion of the soil skeleton, pore water and pore air are only

determined by their own state variables, boundary condition and initial condition. He derived

Terzaghi’s effective stress equation for saturated soils and the effective stress equation for

unsaturated soils (Shao, 2000, 2011a, 2011b, 2012; Shao and Guo, 2014; Shao et al., 2014)

based on the interaction principle for multiphase media. He also discussed some issues (Shao

et al., 2014), such as whether the effective stress is real stress of soil mass, whether the effective

stress equation needs to be modified, the definition and formula of seepage force, the

stress state variable and effective stress equation of unsaturated soils, and whether the contractile

skin of unsaturated soils needs to be treated as an independent phase.

This book discusses the definition of the “soil skeleton” and clarifies that the soil skeleton

should include a portion of bonding water in pores. Why a portion of bonding water should

be treated as a component of the soil skeleton is discussed in Section 1.4. There is a coefficient,

degree of saturation, before the term of matric suction in the effective stress equation

for unsaturated soils derived from the equilibrium differential equation. However, the stress

expression in the formula of the shear strength of unsaturated soils obtained by Vanapalli and

Fredlund (Vanapalli et al., 1996) via experiments has the same form as the effective stress

equation we derived, only the corresponding coefficient is effective degree of saturation (refer

to Section 3.2.2 in this book). This implied that the parameter for the degree of saturation in

the effective stress equation should be effective degree of saturation. This might not be a new

perspective, which, however, has neither been clarified in previous theoretical research of soil

mechanics nor been practically applied. Apparently, this perspective is crucial to the theoretical

research of soil mechanics, at least may change the definition of dry soils, and may affect

the stress-strain characteristic and the research on the constitutive relationship of fine particle

soils. It is not an easy-to-complete task to prove that the pore water corresponding to residual

water content should be treated as a component of the soil skeleton. The micro-scale research

on the structure of soils appears very important herein. Unfortunately, the current research

results are inadequate to verify this perspective (Section 1.4 in this book), while the available

references about this aspect are limited. The corresponding research can be conducted

in micro-scale and macro-scale. The micro-scale research includes finer and more accurate

tests on soil-water structure and interaction, molecular dynamic simulation and the research

of soil-water combined structural model. The macro-scale research can be preceded with the

characteristics of strength and deformation change of fine particle soils.

During the derivation of the equilibrium equation via taking the free body of the soil

skeleton to conduct the internal force analysis, pore fluid pressure and the effect of the other

external forces are separately considered for two reasons. First, when we take the free body of

the soil skeleton that does not include pore fluid to conduct internal force analysis, the forces

acting on the soil skeleton by pore fluid should appear, which includes the forces due to

equilibrated fluid pressure and non-equilibrated fluid pressure. The force due to equilibrated

fluid pressure is self-equilibrated with fluid pressure. It can and should be considered independently;

and the force due to non-equilibrated fluid pressure should be represented with a

couple of action and reaction forces, i.e., seepage forces. Second, different from the effect of

the other external forces, the uniform pore fluid pressure only induces volumetric change of

soil particles and contributes to the shear strength of soils on the contact surfaces between

particles (as shown in Section 3.2 in this book).

The definition and physical meaning of effective stress are found by separately considering

the effects of pore fluid pressure and the other external forces. This finding is helpful in

eliminating the always-existing controversy on effective stress and unifying the research on

saturated soil mechanics and unsaturated soil mechanics.

Compared to all known theories of soil mechanics, this book presents some other new

perspectives and results. Except for the definition and physical meaning of effective stress,

these new results also include (1) the development of Terzaghi’s effective stress principle

(Section 3.4); (2) the effective stress equation of unsaturated soils (including nearly saturated

soils containing absorbed air bubbles) (Section 3.1); (3) the correction of the formula

of seepage force (Section 4.3); (4) the preliminary verification of the relationship between

effective stress and shear strength and deformation of unsaturated soils (Section 3.3); (5) the

theoretical formula of the permeability coefficient of unsaturated soils; and (6) the escape

condition for occluded air bubbles in water (Section 4.4). As of yet, the test verification for

the effective stress of unsaturated soils has not been finished, and the verification for the theoretical

formula of the permeability coefficient of unsaturated soils and the escape condition

for occluded air bubbles in water is under way.

The research in this book assumes that the soil is homogeneous, ignoring the shear

strength of pore water (expect for the pore water that is treated as a component of the soil

skeleton) and the interaction force between pore water and pore air induced by the relative

flow. It should also be noted that the work reported here has been established based on the

following important postulations: (1) the strength of soil mass is that of the soil skeleton, and

(2) the deformation of soil mass is that of the soil skeleton. These two postulations are of

great significance for the research of soil mechanics.