Page 1

Pergamon

Int. J. Engng Sci. Vol. 35, No. 10/ll, pp. 1045-1063, 1997

~) 1997 Elsevier Science Limited. All rights reserved

Printed in Great Britain

0020-7225/97 $17.00 + 0.00 PII: S0020-7225(97)00024-4

A GENERAL NONLINEAR

CONSOLIDATION

MATHEMATICAL MODEL FOR SOIL

PROBLEMS

R. LANCELLOTFA

Dipartimento di Ingegneria Strutturale, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino,, 10129, Italy

L. PREZIOSI*

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino,, 10129, Italy

(Communicated by K. R. Rajagopai)

Abstract---~Ris paper presents a three-dimensional consolidation model, based on mixture theory.

Both the Euilerian and the Lagrangian formulations are given in one dimension for finite strain and

general material nonlinearity. Then the paper formulates the initial boundary value problems related

to several situations of relevant geotechnical engineering interest, such as consolidation between

draining and impervious boundaries subjected to stress and/or velocity conditions, consolidation under

own weight of a layer growing due to deposition of wet material, or to sedimentation of solid particles

in a quiescent fluid. © 1997 Elsevier Science Ltd.

1. INTRODUCTION

Since the publication of the pioneer article by Terzaghi [1] there has been a growing interest in

consolidation theory. This interest arises from both theoretical requirements, linked to the

analysis of porous media, and engineering applications, related to the prediction of settlement

rate, changes of soil properties with the evolution of its state, and, more recently, diffusion of

pollutants.

Terzaghi's consolidation theory was aimed at explaining one-dimensional consolidation

processes in a rather simplified formulation, in order to solve practical aspects. However, in his

original formulation [1] it was not clear which coordinates were being used, so that some

confusion arose on this point. Bjerrum et al. [2] first recognized that Terzaghi used a "reduced

coordinate" Z, which refers to the volume occupied by the solid substance. By using this

definition the real thickness dZ of a porous element is equal to (1 + e)dz, and the coefficient of

permeability used into Terzaghi's equations should be interpreted as a reduced coefficient,

linked to the usual one by the relation (1 + e)Kr = K.

By using this interpretation (see Ref. [3]) one may conclude that Terzaghi's original

formulation should be considered as a model in which the coefficients of hydraulic conductivity

and compressibility are constant, and self-weight is neglected, but not as an infinitesimal strain

theory. Only the formulation given in his later papers in terms of full thickness should be

considered as an infinitesimal strain theory.

Shortcomings of the theory were then outlined by Gibson et al. [3], who also provided a

comprehensive one-dimensional model. They removed the limitation of small strains and took

into account changes of compressibility and permeability during consolidation. A review has

been very recently written by de Boer [4].

Further models were developed in the mid-1960s by Davis and Raymond [5], Janbu [6] and

Mikasa [7]. The available solutions are compared in Refs [8, 9]. Finally, Cornetti and Battaglio

[10] have recently developed a general nonlinear model, providing a suitable solution technique

for well- and ill-posed problems.

* Author to whom all correspondence should be addressed.

1045

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1046 R. LANCELLOTTA and L. PREZ1OSI

The starting point of this paper is the theory of mixture which has been conceived and

developed a couple of decades ago to model the macroscopic behavior of complex systems in

which different constituents are mixed and interact at a microscopic level. This theory uses

classical continuum mechanics tools and is based on balance laws and conservation principles.

In its framework it can, in principle, model mixture of widely different origin, such as porous

media, suspensions, emulsions, foams, bubbly liquids, gas or solid mixtures, and so on. Basic

modeling and constitutive assumptions on several terms appearing in the resulting system of

partial differential equations particularize the theory to the specific field of application.

The theory has been presented by many authors. Among them, we recall for their review

style the papers by Atkin and Craine [11, 12], Bedford and Drumheller [13], Bowen [14, 15],

and MUller [16], the fundamental volume by Truesdell [17], and the recent one by Rajagopal

and Tao [18], where more references on the historical development of the theory can be

found.

This paper applies this approach to the deduction of a new fully nonlinear model aimed at

describing soil mechanics problems, characterized by finite deformation. The soil is assumed to

be saturated and fracture-free. No restriction applies on the stress-strain and permeability-void

ratio relations, but the fact that they must be one-to-one maps during compression.

In particular, Section 2 will introduce the basic concepts of mixture theory which will be used

in the following sections and the specialization of the theory which can be obtained under

suitable assumptions on the inertial and viscous effects. These hypotheses are usually verified in

most geotechnical problems of interest, and enable the deduction of a three-dimensional

consolidation model.

The one-dimensional model in the general case of finite strain and material nonlinearity is

then studied in deeper detail in Sections 3 and 4, which deal, respectively, with the Eulerian and

the Lagrangian formulation. In Section 4 it is also outlined that several models suggested in the

literature are special cases of the one presented here and can be deduced from it under

restricting hypotheses.

Finally, Section 5 deals with the often overlooked problem of the formulation of the

boundary and interface conditions related to several practical problems of geotechnical

engineering interest. For instance, consolidation between draining and impervious boundaries

subjected to stress and/or velocity conditions, consolidation of a layer growing due to deposition

of wet material, or to sedimentation processes in a quiescent fluid above it. Particular attention

is paid to the nontrivial problem of the formulation of the proper boundary conditions and of

the evolution equations determining the positions of the interfaces which delimit the

consolidating layer.

2. THE THEORY OF MIXTURES

The theory of mixtures which is based on balance laws and conservation principles can be

applied to several scientific fields which have in common the fact that the system can be

schematized as a mixture of continua interacting at a microscopic scale.

In the isothermal case and in absence of chemical reactions or phase changes, the theory gives

rise to the following system of equations:

-- + v.(~a,.O -- o,

(2.1)

~t

- -

Ot

÷ v.(~,v,) = o,

(2.2)

Page 3

Nonlinear mathematical model for soil consolidation problems 1047

[ Ors )

p.,~bsk Ot + v.,'Vv~ = V"[]'s + p~bsg + m~. (2.3)

0vt )

pi~bl Ot + vl'Vvl

= V'-I]-i + pl~blg -- m~, (2.4)

where s stands for the solid phase and I for the liquid phase. Furthermore, referring to the p-

constituent:

(1) pp is the: "true" density--that is the density of the material which is used as the p-

constituent of the mixture, e.g. Pwater = 1000 kg/m 3.

(2) ~bp is the volume fraction, i.e. the volume occupied by the p-constituent over the total

volume. Therefore, assuming saturation, ~b~ + ~b~ = 1.

(3) Vp is the velocity.

(4) q]-~ is the partial stress tensor, which describes the behavior of the p-constituent when

the other constituent is co-present. The difference between the partial stress -~p and the

stress T e of the p-constituent taken separately, usually termed Reynolds stress, is well

emphasized by a comparison with the model obtained using the ensemble average

approach [19, 20].

(5) g is the gravitational acceleration.

(6) m~ is the momentum supply [21] (also named internal body force or interaction force

[14, 15]) and is related to the local interactions between the constituents across the

interface separating them. It appears in both the momentum equations (2.3) and (2.4)

with a different sign, because of its nature of an interaction (action-reaction) force

between constituents.

REMARK 2.1. Adding the continuity equations (2.1) and (2.2) gives

V.vc = 0,

where

(2.5)

is the so-called composite velocity.

v~ = ~sVs + ~lvt, (2.6)

In this way a divergence-free velocity field is identified. In handling the model it is very useful

to exploit this property using equation (2.5) in place of a continuity equation, either equations

(2.1) and (2.2).

If, instead, equations (2.1) and (2.2) are first multiplied by Ps and p~ and then added, the usual

balance law for the mixture as a whole is obtained

Opm

--

Ot

q-- V "(pml0m) ~-- 0, (2.7)

where

Pm = ps~bs + Plt~I,

is the so-called composite density, i.e. the density of the mixture, and

(2.8)

p,d~,v., + pl,;blvl

V m =

Pm

(2.9)

is the so-called mass average velocity, i.e. the velocity of the center of mass of the mixture.

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1048 R. LANCELLOTYA and L. PREZIOSI

Summing up the two momentum equations (2.3) and (2.4) and using equation (2.7) gives the

momentum equation for the mixture

where

vm

0t

)

Pm + Vm'VVm = V'Tm+ Ping, (2.10)

~Yrn : -l-s @ ~-' -- Ps(~s(Vs -- Vm) (~ (Vs -- Vm) -- PI~bl(Vl- Vm) ~ (Vl -- Vm), (2.11)

is the stress tensor of the mixture considered as a whole (see Refs [19, 20,22]) and the quantities

u s -- 1 1 ~ m and vt - Vm in equation (2.11) are the so-called diffusive velocities.

As before, it may be convenient to consider the momentum equation for the mixture

[equation (2.10)] in place of one of the momentum equations for the constituents (say, the one

referring to the solid constituent), essentially for two reasons:

1. It does not contain the interfacial term m,~.

2. It is possible to perform experiments on the mixture as a whole to determine the stress

constitutive equation for Tm. In fact, neither q]-~ nor q]-~ can be measured directly. On the

other hand, direct experiments on Tm, TI, and Ts, where Ts and T, are the "true" stresses

related to the constituents taken separately, can be done.

A basic equation introduced in all consolidation models is Darcy's law, which can be deduced

from the momentum equation for the liquid constituent equation (2.4) under the following

assumptions [15]:

1. Inertia of the liquid constituent is neglected.

2. The viscous forces for the liquid phase are neglected, so that the stress tensor reduces to

a scalar.

3. The interaction force depends linearly on Vs- v~ through an invertible matrix, which

depends on the deformation of the soil skeleton.

One then has

v, - v= =

--

(VP,- pIg), (2.12)

/z4,,

where /.~ is the liquid viscosity, K is the "effective permeability" tensor, P~ is the pore liquid

pressure,

1

E.= T (~:,r~- D)

is the Lagrangian (finite deformation) strain tensor for the soil skeleton, and I:= is its

deformation gradient.

Finally, if not only the inertia of the liquid constituent, but also that of the solid constituent is

neglected, then the momentum balance for the mixture equation (2.10) rewrites as a stress

equilibrium equation. The three-dimensional model can then be written as

o4,=

--

Ot

+ V.(~b=v 0 = O, (2.13)

V "vc = O, (2.14)

vj - v= = - - (VPI- pIg), (2.15)

Page 5

Nonlinear mathematical model for soil consolidation problems 1049

V.Tm + Ping = O. (2.16)

One of the advantages to using equations (2.13)-(2.16) is that one does not have the usual

difficulties associated with prescribing boundary conditions for problems formulated within the

full context of mixture theory. This difficulty is one of the main drawbacks when applying

mixture theory. ~he problem of how to specify stress boundary conditions in mixture theory has

been discussed in detail by Rajagopal and coworkers in Refs [18,23]. Further comparisons

between the results obtained using mixture models and Darcy's law can be found in Ref. [24] in

the case of an incompressible porous material.

3. THE EULERIAN FORMULATION IN ONE-DIMENSIONAL CONDITIONS

In this and the following sections it is assumed that both flow and strain take place along the

vertical direction z (g =- gez), and that the medium is isotropic in a horizontal plane, so that

consolidation occurs along a principal direction of the permeability tensor.

Recalling that the continuity equation of the solid constituent equation (2.1) can be written in

Lagrangian form as [see also equation (4.1)]

¢;

¢8 ' det U:s - (3.1)

where ~b~ is the volume ratio of the undeformed reference configuration and that in one

dimension the only nontrivial component of the Lagrangian strain tensor is

1 { ¢.2 __ )

Ezz= -~ \--~ 1 ,

(3.2)

then the dependence of the permeability tensor K on H:s is equivalent in one-dimensional

problems to that on Cs.

The same is not true for the stress tensor Tm which refers to the whole mixture. In this case

the possible dependence on the Lagrangian strain tensor involves not only the volume fraction,

as in equation (3.2), but also the fluid properties.

Equations (2.13)-(2.16) can then be written in one dimension as

-- + (¢sv~) = O, (3.3)

Ot Oz

0vc

--

az

= 0, (3.4)

where

K(4,s) ( oP,

/z(1-¢s) \ 0z + p' g

)

v,-v~= ' (3.5)

OPi

--

Oz

0o"

Oz

+ -- + Pmg=O, (3.6)

a' = ~rm-- PI (3.7)

Page 6

1050 R. LANCELLO'I"I'A and L. PREZIOSI

is the "effective stress" and trm is the total stress in the mixture (both of them are taken positive

in compression.)

Equation (3.4) implies that the composite velocity vc does not depend on z, that is

~b,vs + (1 - ~b~)v, = C(t),

(3.8)

where, as will be shown in Section 5, the determination of C(t) has to be done on the basis of

the boundary conditions and is related to the value assumed by the composite velocity at the

boundaries.

Substituting equation (3.6) in equation (3.5) gives

a

m ,

v,- vs - 1 - ~b. (3.9)

where

-- + (ps - •

L a:

Equation (3.8) and (3.9) then explicitly determine

(3.10)

•b

s

v, = C(t) + - - Q,

(3.11)

1-4~

v, = C(t) - Q.

which, in turn, can be substituted in the continuity equation (3.3) to give

(3.12)

0.~_____2_ + C(t) OZ--Ad~ _

Ot

0 (qbsQ),

Oz

(3.13)

Oz

or

where the constitutive relation for the effective stress tr' is still to be specified.

If the mixture is assumed to behave like an elastic material, that is

tr' = E(4~,), (3.15)

where E is a single-valued relation between the volume ratio and the excess stress, equation

(3.14) takes the structure of a nonlinear convection-diffusion equation.

In this case equation (3.14) can be written using the excess stress as a state variable as

where xlt = E-J is the inverse of equation (3.15), and ~, = d~/dtr'.

Particularizations of the dependence of the permeability and stress on the volume ratio

(possibly together with other simplifying assumptions) give rise to several models presented in

the literature to describe soil consolidation [8, 9].

In reality, equation (3.15) is a simplification, since the soil skeleton and the liquid cannot

deform independently, but have to carry the load by joint deformation. This suggests that a

better model would treat the mixture at least as a Voigt-Kelvin solid [20, 25, 26].

Page 7

Nonlinear mathematical model for soil consolidation problems 1051

REMARK 3.1. Equation (3.13) rewrites in term of the void ratio

~bl 1

e - - -- 1, (3.17)

i.e. the volume occupied by the liquid constituent over the volume occupied by the solid

constituent, as

de

--

Ot

C(t)--~-- +

oz

20(

e) ~z

Q )

+ (1 + = 0, (3.18)

where Q, defined in equation (3.10), rewrites in terms of the void ratio as

Q= K(e) ( 3o-'

- -

tz

Ps-Pl

-g

l+e

)

- -

Oz

+- • (3.19)

4. THE LAGRANGIAN FORMULATION IN ONE-DIMENSIONAL CONDITIONS

As has already been stated in the previous section, the material form of the continuity

conditions is given by

d

d--7 (p~sdet~:~) = O, (4.1)

which in one dimension reduces to

Oz

OZ

49~

~b,

l+e

1 +e* '

(4.2)

where e* is the void ratio in the undeformed reference configuration.

As usual in continuum mechanics, the material coordinate z indicates the current position of

the particle that in the undeformed reference configuration was identified by the reference

coordinate Z.

The local form of the mass conservation relation of the liquid phase over a material volume

fixed on the solid phase assumes the form

0

-~7 P"~'-~- + -~- [p,4,,(,,,- ,,,)] = o,

which can be written using equation (4.2) as

(4.3)

~'~ ~b I + -~- [(~l(13i - l/s) ] = O.

(4.4)

Page 8

1052 R. LANCELLOTrA and L. PREZIOS1

In the Lagrangian formulation it is then more convenient to consider as a state variable the

void ratio e defined in equation (3.17) which allows one to rewrite equation (4.4) as

]

--

0t

+ (1 + e*) (vi- Us) = O. (4.5)

~

The equilibrium of the global porous element is given by

O0"rn 0 Z

OZ

epl + Ps

+ - - g = O ,

l+e

--

Oz

or using equations (3.7) and (4.2),

Oer'

OZ

OPi

OZ

epl + p,,

+ -

l+e*

+ - g = O. (4.6)

At this point, in consolidation theory it is common to introduce the excess pore pressure u,

namely the quantity in excess to the hydrostatic value, so that the equilibrium of the fluid phase

requires

OPi

OZ

Ou

OZ + Plg--~-~ =0.

Oz

(4.7)

Note that in this equation, the definition given to the excess pore pressure implicitly accounts

for any interaction effect.

It is then assumed that the excess pore pressure is related to the flow through Darcy's law

which rewrites in a Lagrangian framework as

e k(e) Ou OZ

--,

Oz

- -

l+e

(v,- vs) = (4.8)

Pig OZ

or

k(e) l+e*

Ou

v,- vs = , (4.9)

Pig e OZ

where k(e) is the hydraulic conductivity.

By substituting this latter equation into equation (4.5), using equations (4.6) and (4.7), and

taking into account the continuity equation (4.2), the Lagrangian finite strain formulation for

the one-dimensional consolidation problem assumes the form

Oe

--

Ot

OQ

OZ

+ (1 + e*) = O, (4.10)

where, recalling the relation between hydraulic conductivity and permeability,

k(e) K(e)

--,

#

(4.11)

Pig

Page 9

Nonlinear mathematical model for soil consolidation problems 1053

Q k(e) l+e* (OFT'

p~g

l+e

P s - - P _ _ _ _ _ _ _ _ _ ~ l )

e* g

1+

-- - -

OZ

+ (4.12)

is the Lagrangian expression of equation (3.19).

REMARK 4.1. Equation (4.10) is equivalent to equation (3.18), as it is expected to be. In fact,

recalling equation (4.2), the transformation of equation (4.10) in the Eulerian formulation gives

Oe

Ot

Oe OQ

Oz

+ v~ --

" Oz

+ (1 + e) = O, (4.13)

or recalling equation (3.12),

Oe

--

O t

Oe

-~Z

Oe

-~Z

oQ

O Z

+C(t) - Q +(l+e)

=0.

(4.14)

The last two terms in equation (4.14) can be written as

(1 + e) - Q -~z = (1 + e) 2 1 0 Q

Oz

2 0 Q

l +e

(l+e) 20z

e) ~z ~ '

(4.15)

which, when substituted back into equation (4.14), gives equation (3.18).

The Lagrangian formulation is often preferred to the Eulerian formulation since it does not

involve the C(t) term which appears, for instance, in equations (3.13) and (3.18). Furthermore,

equation (4.10) has the advantage that it has to be integrated on a fixed domain, while equation

(3.13) usually inw9lves integration over a time-varying domain. However, as will be shown in

Section 5, problems with time-dependent mass within the layer need be formulated from an

Eulerian viewpoint.

REMARK 4.2. Equations (3.13) and (4.10) have been derived without any assumption regarding

the void ratio-effective stress, or the void ratio-permeability relationship, and for this reason

represent a general formulation. Anyway, in order to solve them, the above relationships must

be specified, as "single-valued" functions. In particular, the fact that the relation is single-valued

means that one is focusing on compression problems obtained, for instance, imposing a load on

the soil layer. The release of the load may, in fact, in principle, lead to an expansion of the layer,

which certainly will not follow the same rule due to anelastic soil behavior.

In the following it will be shown that both linear and nonlinear infinitesimal formulations are

special cases of equation (3.13) or equation (4.10). Before doing that it is useful to observe that

if the mixture is assumed to behave as an elastic material,

tT' = - E (e), (4.16)

equation (4.10) reduces to

i~e

o---t + f(e) a---Z -

Oe l + e*

p, g

~ r

~

Oe ]

[ g(e) ~ J, (4.17)

Page 10

1054

where

R. LANCELLO'I"rA and L. PREZIOSI

(,,)

--~(-1

d r,,(e, ]

f(e)= (l+e*)-d-~-eLl+ e ,

l+e* dE

l+e

g(e) = k(e) - -

(4.18)

de

The advection term on the left-hand side drops if gravity is neglected.

4.1 Conventional linear infinitesimal strain theory

The conventional linear infinitesimal theory introduced by Terzaghi is based on four main

assumptions:

(1) Infinitesimal strains, namely the deformation of a consolidating layer is small compared

to its initial thickness, which implies that the Lagrangian coordinate Z can be replaced

by the Eulerian coordinate z, and as a consequence the volume (l+e) can be

approximated by (1 + e*).

(2) Linear elastic constitutive relation between void ratio and effective stress

de

d~'

- av (4.19)

where av is the compressibility index, strain and stress level independent.

(4) The hydraulic conductivity k is constant.

(5) Gravitational effects are negligible.

Under these hypothesis equation (4.17) reduces to

ae k

0 t Pig

do" 02e

-- (1 + e*) d--e- 0 z 2 ' (4.20)

or recalling equation (4.19) as

O e 02e

- cv - - (4.21)

Ot Oz 2 '

where

k(1 + e*)

cv - , (4.22)

avplg

is the consolidation coefficient.

Due to the linearity between stress and strain, equation (4.19) can also be written in

terms of the effective stress:

a(7' ~20"

- cv - - (4.23)

Ot Oz 2 "

Note that the Terzaghi equation is usually formulated in terms of excess pore

pressure. Changes of effective stress are equal to changes of excess pore pressure if the

Page 11

Nonlinear mathematical model for soil consolidation problems 1055

applied load is time independent as well as spatially independent. If this is the case,

then

c3 u 02u

- cv -- (4.24)

Ot

OZ 2 "

4.2 Nonlinear in[initesimal strain theory

If self-weight is neglected, e* is constant, and the effective stress is a unique function of void

ratio, then equation (4.17) becomes

Oe

--

Ot

O [ k(e)

Oz

do-' Oe ]

de

+-- (l+e*)-- --

Oz

=0. (4.25)

p~g

If the constitutive relation have the nonlinear expressions

O-*P

k = k* --, e = e* - Cclog (o-'/o-*'), (4.26)

o-r

then the consolidation coefficient

k(e)(1 + e*) do"

Pig

k*o-*' (1 + e*)

plgCc

c~ - - (4.27)

de

is constant and ,equation (4.25) reduces to the nonlinear formulation given by Davis and

Raymond [5]:

0o-' = c,

Ot

o-' . (4.28)

az 2 \ 0z

It is of interest to note that the substitution of equation (4.27) into equation (4.25) gives

Oe

0 t - cv --

02e

OZ 2 '

(4.29)

that means that changes of void ratio must satisfy the diffusion equation.

Equation (4.28) is less restrictive than equation (4.23) or equation (4.24) as far as the

constitutive relations are concerned, so it can be regarded as a more general formulation of an

infinitesimal strain theory.

As already mentioned, equation (4.26) as used by Davis and Raymond [5] is the most popular

in geotechnical literature. Butterfield [27], however, discussed in details shortcomings related to

the use of this equation, so that at present it appears that a more consistent one should be the

material compression law suggested by the same author:

dV _

v

A( do-' '~

\ o-'/'

(4.30)

where V = 1 + e is the specific volume and ,l is an experimental constant.

It can be of some interest to note that, by putting a = 1/,L the previous equation can be also

written in the forra

o-' V ~ = o-'0Vc~, (4.31)

which is the gas compression law in adiabatic processes.

IJ[$ JS: IO/ll-r

Page 12

1056 R. LANCELLOTTA and L. PREZ1OSI

Furthermore, it must be observed that it is quite usual to use a void ratio-permeability

relationship similar to the one established for the void ratio-effective stress. In this respect it

remains to be proved that a relation similar to equation (4.30) also applies to void ratio-

permeability.

5. INTERFACE AND BOUNDARY CONDITIONS

5.1 General procedure

The evolution of the system can be obtained solving, by suitable methods [28], the initial

boundary value problem associated to the models proposed in the preceding sections, e.g.

equations (3.13), (3.16), (3.18) or (4.10). The structure of the equation is such that, besides

the initial condition, two boundary conditions, one on each boundary, are needed in order

to have a well-formulated and well-posed problem. In addition, in the Eulerian formulation

the quantity C(t) appearing, for instance, in equations (3.13), (3.16) and (3.18) and the evolution

equations for the moving boundaries still need to be determined.

The statement of the boundary conditions must be related to the analysis of the physics

of the system. This section will consider several practical situation of geotechnical interest.

Before entering the details, it has to be noted that several boundary conditions can be applied

at the extrema of the consolidating layer. In fact, the boundary may be draining (referred

to with the subscript "dr"), with the liquid that can flow through it with a negligible resistance,

or impervious (referred to with the subscript "imp"), which prevents the liquid to flow through.

It can be fixed, can move at a given velocity, or a load can be applied on it. In addition,

the boundary can be an interface with a pool of liquid which filtrates through the consolidating

layer due to gravity or to a hydraulic gradient.

Actually, boundaries can be considered as nondeformable porous materials with given

volume fraction. For instance, a solid impervious boundary is a "porous" material with volume

ratio &b----1. Hence, one can specialize the interface conditions formulated, for instance, by

MUller [16] to the one-dimensional case with negligible inertia we are dealing with.

If the boundary Zb(t) is fixed on the soil skeleton, as in most of the cases which will follow,

then one has that both the composite velocity vc and the stress of the mixture o" m have to

be continuous across the boundary

vc(zb(t),t) = &bVsb(t) + (1 -- ~bb)Vlb(t), (5.1)

O'm(Zb(t),t ) = O'b(/), (5.2)

where subscript "b" indicates a quantity referring to the boundary and ~b is taken positive

in compression.

Furthermore, the velocity of the boundary is equal to the velocity of the soil grains at the

boundary, which may be related to the volume ratio (or the stress) at the boundary through

equation (3.12):

dZb

dt

- v~(zb(t),t) = C(t) - Q(zb(t),t).

(5.3)

If the velocity vb(t) of the boundary is given, then equation (5.3) determines both the term

C(t) in equations (3.13) and (3.18):

C(t) = Oh(t) + O(zb(t),t),

(5.4)

and, of course, the evolution equations for the moving boundary

dZb - vb(t).

dt

(5.5)

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Nonlinear mathematical model for soil consolidation problems 1057

Furthermore, since C(t) is constant throughout the layer, going to the other boundary zb.

one can also determine either its velocity, and therefore the evolution equation for the

boundary,

dzw

dt

- vb(t) + Q(Zb(t),t) -- Q(zb.(t),t),

(5.6)

or the relative boundary condition

Q(zw(t),t) = vb(t) - Vb-(t) + Q(zb(t),t).

(5.7)

As far as the stress condition is concerned, it is useful to observe that the integration of

the stress equilibrium equation (3.6) over the consolidating layer [zb(t), Zb*(t)] relates the

stresses at the two boundaries:

~ z~.( t)

o" (Zb(t),t) - o" (Zb*(t),t) + P,(zb(t),t) - Pl(Zb*(t),t) = pm(Z,t)g d z.

d zh(t)

(5.8)

Using, then, the definition of composite density [equation (2.8)], one has

~Zh*(0 ~ z..(t)

Prn(Z,t) dz = [(Ps - Pl)~Ps(Z, t) + Pl] dz

d Zh(t) J Zh(I)

pl[Zb'(t)

.

zb(t)] + (1

.

p' /~ z'<'>

ps / Jzh(o

.. ps~bs(t,z) d z,

(5.9)

where the last integral is equal to the mass Ms(t) of the solid constituent in the layer, which

is constant if there is no flow of soil grains through the boundaries, as occurs in many practical

cases. In fact,

~Zh.(t) f Zh*(O)

Ms(t) = psdPs(t,z) dz = ps~s(O,z) dz = Mo.

(5.10)

d Zh(O J zh(O)

Substituting equations (5.9) and (5.10) back into equation (5.6), one can then write

/ \

or' (zb(t),t) - tr' (zb*(t),t) + P,(zb(t),t) - P,(zb.(t),t) = p,g[zw(t) - Zb(t)l + ( 1 - P___L ) Mo g. (5.11)

\

Ps /

Some sample cases will now be described in order to apply the theoretical results to specific

configurations of geotechnical engineering interest.

5.2 Imposed velocities on impervious boundary and stress on draining boundary

As a first practical example, assume that the layer is between a draining and an impervious

boundary. Since the composite velocity is continuous across the impervious boundary and is

constant across the layer, it has to be constantly equal to the value assumed on the impervious

boundary. Therefore,

vc(t,z) = C(t) = Oimp(t ) VZ, (5.12)

where Vimp(t ) is the velocity of the impervious boundary.

This means, for instance, that if the impervious boundary is fixed at the bottom of the layer

and the draining: top boundary Zdr(t) is pushed down, then, since the composite velocity is

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1058 R. LANCELLOTTA and L. PREZIOSI

also constant across Zdr(t), there will be a pool of still liquid forming over the draining boundary,

while in the bulk of the layer the liquid will move up as the solid consolidates down, so that

the composite velocity vanishes everywhere.

In addition to equation (5.12), because of equations (5.1) and (5.3), one also has

Vimp(t) = Vs(Zimp(t),t) = Vl(Zimp(t),t),

or, because of equation (3.9),

(5.13)

Q(zimo(t),t) = 0. (5.14)

This boundary condition can be equivalently written as

(90 -r

OZ

--

(Zimp(t), t) + (Ps - Pl)(bs(Zimp(t),t)g = 0, (5.15)

which, if the solid skeleton is assumed to behave elastically [see equation (3.15)], is a Robin-

type condition:

drips ( qbs(Zimp(t)'t)) -~Z (Zimp(t)'t) + (Ps -- P,)Cs(Zimp(t),t)g = 0. (5.16)

to be imposed on the impervious boundary.

If a stress boundary condition ~r(t) is imposed on the draining boundary, then the liquid

pressure is continuous across the boundary and

or' (Zdr(t),t) = ~r(t), (5.17)

which provides the other boundary condition.

Though the velocity of the draining boundary Zdr(t) is not provided, it can be easily deduced

specializing equation (5.6) by keeping in mind equation (5.14):

dZdr

dt

- Vimp(t) -- Q(zdr(t),t),

(5.18)

while the evolution of the other boundary is obviously determined by

d Zimp = Vimp(t).

dt

(5.19)

Actually, in many, if not all, practical situations, the impervious boundary is fixed, which means

that Vimp(/) ----0 and Zimp(t)=Zo.

Besides equations (5.18) and (5.19), the Eulerian formulation also requires the determination

of C(t), which is given by equation (5.12).

In this problem, however, it is more convenient to consider the Lagrangian formulation

[equation (4.10)] joined with the Lagrangian version of equations (5.15) and (5.17):

a~r'

aZ

(Pl- P~)g

l+e*(Z~mp) '

(Zimp't)=

(5.20)

tr' (Zd,,t) = o'(t). (5.21)

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Nonlinear mathematical model for soil consolidation problems 1059

5.3 Consolidation of a draining layer

Assume now that both boundaries are draining. The bottom one is at rest, i.e. Zbot(t)=Z0,

while a stress condition is imposed on the top boundary Z,op(t):

tr'(Ztop(t),t) = tr(t). (5.22)

Since the bottom boundary is at rest, then from equation (5.4)

C(t) = O(zo,t),

and from equation (5.6)

(5.23)

dztop _ Q(zo,t) - Q(ztop(t),t).

dt

(5.24)

If it is assumed that the pore liquid pressure at the bottom is constantly maintained at the

value existing be, fore the load is applied, that is

e,(zo) = eatm + p,g(Ztop(O) - Zo),

(5.25)

then there may be some water filtrating through the top boundary, so that the liquid pressure

at the top of the layer is

P,(ztop(t),t) = Patm + pbg[z,ee(t) - Ztop(t)], (5.26)

where zfr~(t) is the position of the free surface of the water pool.

Substituting ectuations (5.22), (5.25) and (5.26) into equation (5.11) gives

/ \

0" (Zo,t)= o'(t) -b plg[Zfree(t) -- Ztop(0)]-1- ( 1 - P-_! }Mog.

Ps /

(5.27)

\

If the water above the layer is constantly withdrawn, then zfree(t)= Ztop(t), otherwise it has

to be computed through a mass balance of the liquid:

dp,(Z,t) dz + dz = ~bl(Zo,t)[vl(Zo,t) -- vs(Zo,t)] = Q(zo,t),

(5.28)

dt LJz,, J z,,,,

or

d Zfree

dt (t)= Q(zo,t) + -- --

1 d f z,,,,<,)

!

Ps d t j z,,

psdps( z,t) d z.

(5.29)

Since the integral on the right-hand side of equation (5.29) is the mass of solid in the layer,

and therefore is constant, equation (5.26) simplifies to

d Zfree

d------i-- (t) = Q(zo,t).

(5.30)

In conclusion, equations (5.22) and (5.27) with Zfree(t) either equal to Ztop(t), or detemined

by the integration of equation (5.30), are the boundary conditions to be used both in the

Lagrangian and t]he Eulerian formulation. This last formulation, however, also needs equations

(5.23) and (5.24).