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We recall first the effective stress and constitutive relationships as defined in Equation (1.16) of Chapter 1 which we repeat below.

(2.1a)

or

(2.1b)

This effective stress is conveniently used as it can be directly established from the total strains developed.

However, it should be remembered here that this stress definition was derived in Chapter 1 as a corollary of using the effective stress defined as below:

(2.2a)

or

(2.2b)

which is responsible for the major part of the deformation and certainly for failure.

In soils, the difference between the two effective stresses is negligible as α ≈ 1. However, for such materials as concrete or rock, the value of α in the first expression can be as low as 0.5 but experiments on tensile strength show that the second definition of effective stress is there much more closely applicable as shown by Leliavsky (1947), Serafim (1954), etc.

For soil mechanics problems, to which we will devote most of the examples, α = 1 will be assumed. Constitutive relationships will still, however, be written in the general form using an incremental definition

(2.3a)

(2.3b)

The vectorial notation used here follows that corresponding to stress components given in (1.1). We thus define the strains as

(2.4)

In the above, D is the “tangent matrix” and dε⁰ is the increment of the thermal or similar autogenous strain and of the grain compression . The latter is generally neglected in soil problems.

If large strains are encountered, this definition needs to be modified and we must write

(2.5)

where the last two terms account for simple rotation (via the definition in 2.6b) of the existing stress components and are known as the Zaremba (1903a, 1903b)–Jaumann (1905) stress changes. We omit here the corresponding vectorial notation as this is not easy to implement.

The large strain rotation components are small for small displacement computation and can be frequently neglected. Thus, in the derivations that follow, we shall do so – though their inclusion presents no additional computational difficulties.

The strain and rotation increments of the soil matrix can be determined in terms of displacement increments du_i as

(2.6a)

and

(2.6b)

The comma in the suffix denotes differentiation with respect to the appropriate coordinate specified. Thus

If the vectorial notation is used, as is often the case in the finite element analysis, the so‐called engineering strains are used in which (with the repeated index of ∂u_i,i not summed)

(2.7a)

or

However, the shear strain increments will be written as

(2.7b)

or

We shall usually write the process of strain computation using matrix notation as

(2.8)

where

(2.9)

And for two dimensions, the strain matrix is defined as:

(2.10)

with corresponding changes for three dimensions (as shown in Zienkiewicz et al. 2005).

We can now write the overall equilibrium or momentum balance relation for the soil–fluid “mixture” as

(2.11a)

or

(2.11b)

In the above, w_i (or w) is the average (Darcy) velocity of the percolating water.

The underlined terms in the above equation represent the fluid acceleration relative to the solid and the convective terms of this acceleration. This acceleration is generally small and we shall frequently omit it. In derivations of the above equation, we consider the solid skeleton and the fluid embraced by the usual control volume: dx ⋅ dy ⋅ dz.

Further, ρ_f is the density of the fluid, b is the body force per unit mass (generally gravity) vector, and ρ is the density of the total composite, i.e.

(2.12)

where ρ_S is the density of the solid particles and n is the porosity (i.e. the volume of pores in a unit volume of the soil).

The second equilibrium equation ensures the momentum balance of the fluid. If again we consider the same unit control volume as that assumed in deriving (2.11) (and we further assume that this moves with the solid phase), we can write

(2.13a)

or

(2.13b)

In the above, we consider only the balance of the fluid momentum and R represents the viscous drag forces which, assuming the Darcy seepage law, can be written as

(2.14a)

(2.14b)

Note that the underlined terms in (2.13) represent again the convective fluid acceleration and are generally small. Also note that, throughout this book, the permeability k is used with dimensions of [length]³·[time]/[mass] which is different from the usual soil mechanics convention k′ which has the dimension of velocity, i.e. [length]/[time]. Their values are related by where and g′ are the fluid density and gravitational acceleration at which the permeability is measured.

The final equation is one accounting for the mass balance of the flow. Here we balance the flow divergence w_ii by the augmented storage in the pores of a unit volume of soil occurring in time dt. This storage is composed of several components given below in order of importance:

1 the increased volume due to a change in strain, i.e.: δijdεij = dεii = mTdε

2 the additional volume stored by compression of void fluid due to fluid pressure increase: ndp/Kf

3 the additional volume stored by the compression of grains by the fluid pressure increase: (1 − n)dp/KSand

4 the change in volume of the solid phase due to a change in the intergranular effective contact stress .

Here K_T is the average bulk modulus of the solid skeleton and ε_ii the total volumetric strain.

Adding all the above contributions together with a source term and a second‐order term due to the change in fluid density in the process, we can finally write the flow conservation equation

(2.15)

This can be rewritten using the definition of α given in Equation (1.15b) as

(2.16a)

or in vectorial form

(2.16b)

where

(2.17)

In (2.16), the last two (underlined) terms are those corresponding to a change of density and rate of volume expansion of the solid in the case of thermal changes and are negligible in general. We shall omit them from further consideration here.

Equations (2.11), (2.13), and (2.16) together with appropriate constitutive relations specified in the manner of (2.3) define the behavior of the solid together with its pore pressure in both static and dynamic conditions. The unknown variables in this system are:

 The pressure of fluid (water), p ≡ pw

 The velocities of fluid flow wi or w

 The displacements of the solid matrix ui or u.

The boundary condition imposed on these variables will complete the problem. These boundary conditions are:

1 For the total momentum balance on the part of the boundary Γt, we specify the total traction ti(t) (or in terms of the total stress σij nj (σ⋅ G) with ni being the ith component of the normal at the boundary and G is the appropriate vectorial equivalence) while for Γu, the displacement ui(u), is given.

2 For the fluid phase, again the boundary is divided into two parts Γp on which the values of p are specified and Γw where the normal outflow wn is prescribed (for instance, a zero value for the normal outward velocity on an impermeable boundary).

Summarizing, for the overall assembly, we can thus write

(2.18)

and

Further

(2.19a)

and

(2.19b)

It is of interest to note, as shown by Zienkiewicz (1982), that some typical soil constants are implied in the formulation. For instance, we note from (2.16) that for undrained behavior, when w_i,i = 0, i.e. with no net outflow, we have (neglecting the last two terms which are of the second order).

or

and

or

If the pressure change dp is considered as a fraction of the mean total stress change m^T dσ/3 or dσ_ii/3, we obtain the so‐called B soil parameter (Skempton 1954) as

Using the assumption that the material is isotropic so that

where K_T is (as defined in equation (1.10), the bulk modulus of the solid phase and μ is once again Lamé’s constant. B has, of course, a value approaching unity for soil but can be considerably lower for concrete or rock. Further, for unsaturated soils, the value will be much lower (Terzaghi 1925; Lambe and Whitman 1969; Craig 1992).