A general nonlinear mathematical model for soil consolidation problems
ABSTRACT This paper presents a threedimensional consolidation model, based on mixture theory. Both the Eulerian 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.

Article: On a Powder Consolidation Problem
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ABSTRACT: RESUMEN RESUMEN The problem of the consolidation of an aerated fine powder under gravity is considered . The industrial relevance of the problem is discussed and a mathematical model is introduced . The mathematical structure is that of a coupled system for three unknowns , pressure , stress, and height of the powder in the ( axisymmetric ) bunker containing it . The system itself consists of a parabolic PDE, an ODE, and an integral equation determining a free boundary corresponding to the height of the powder . Existence and uniqueness of a solution is established . A numerical method based on a formulation of the semidiscretized problem as an index 1 DAE is proposed and implemented . The feasibility of the approach is illustrated by computational results .01/2001;  SourceAvailable from: psu.edu
Article: On a Powder Consolidation Problem.
[Show abstract] [Hide abstract]
ABSTRACT: . The problem of the consolidation of an aerated fine powder under gravity is considered. The industrial relevance of the problem is discussed and a mathematical model is introduced. The mathematical structure is that of a coupled system for three unknowns, pressure, stress and height of the powder in the (axisymmetric) bunker containing it. The system itself consists of a parabolic PDE, an ODE and an integral equation determining a free boundary corresponding to the height of the powder. Existence and uniqueness of a solution is established. A numerical method based on a formulation of the semidiscretized problem as an index 1 DAE is proposed and implemented. The feasilibility of the approach is illustrated by computational results. Key words. consolidation, multiphase, parabolic, free boundary, DAE, integral equation. AMS subject classifications. 35K55, 65L80, 65M06, 76S05 1. Introduction. One important factor determining the mechanical properties of fine powders is the possible pres...SIAM Journal of Applied Mathematics. 01/2001; 62:120.  SourceAvailable from: L. Fusi[Show abstract] [Hide abstract]
ABSTRACT: We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.Nonlinear Analysis Real World Applications 01/2006; 7:10481071. · 2.20 Impact Factor
Page 1
Pergamon
Int. J. Engng Sci. Vol. 35, No. 10/ll, pp. 10451063, 1997
~) 1997 Elsevier Science Limited. All rights reserved
Printed in Great Britain
00207225/97 $17.00 + 0.00 PII: S00207225(97)000244
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 threedimensional 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 onedimensional 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 selfweight 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 onedimensional 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 mid1960s 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 illposed problems.
* Author to whom all correspondence should be addressed.
1045
Page 2
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 fracturefree. No restriction applies on the stressstrain and permeabilityvoid
ratio relations, but the fact that they must be onetoone 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 threedimensional
consolidation model.
The onedimensional 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" densitythat 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 pconstituent 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 pconstituent when
the other constituent is copresent. The difference between the partial stress ~p and the
stress T e of the pconstituent 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 (actionreaction) force
between constituents.
REMARK 2.1. Adding the continuity equations (2.1) and (2.2) gives
V.vc = 0,
where
(2.5)
is the socalled composite velocity.
v~ = ~sVs + ~lvt, (2.6)
In this way a divergencefree 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 socalled composite density, i.e. the density of the mixture, and
(2.8)
p,d~,v., + pl,;blvl
V m =
Pm
(2.9)
is the socalled mass average velocity, i.e. the velocity of the center of mass of the mixture.
Page 4
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 : ls @ ~'  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 socalled 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 threedimensional 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 ONEDIMENSIONAL 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 onedimensional
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)
14~
v, = C(t)  Q.
which, in turn, can be substituted in the continuity equation (3.3) to give
(3.12)
0.~_____2_ + C(t) OZAd~ _
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 singlevalued relation between the volume ratio and the excess stress, equation
(3.14) takes the structure of a nonlinear convectiondiffusion equation.
In this case equation (3.14) can be written using the excess stress as a state variable as
where xlt = EJ 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 VoigtKelvin 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
PsPl
g
l+e
)
 
Oz
+ • (3.19)
4. THE LAGRANGIAN FORMULATION IN ONEDIMENSIONAL CONDITIONS
As has already been stated in the previous section, the material form of the continuity
conditions is given by
d
d7 (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 onedimensional 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 timevarying domain. However, as will be shown in
Section 5, problems with timedependent 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 ratioeffective stress, or the void ratiopermeability relationship, and for this reason
represent a general formulation. Anyway, in order to solve them, the above relationships must
be specified, as "singlevalued" functions. In particular, the fact that the relation is singlevalued
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
ot + f(e) aZ 
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 lefthand 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*) de 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 selfweight 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)
or
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/llr
Page 12
1056 R. LANCELLOTTA and L. PREZ1OSI
Furthermore, it must be observed that it is quite usual to use a void ratiopermeability
relationship similar to the one established for the void ratioeffective 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 wellformulated and wellposed 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 &b1. Hence, one can specialize the interface conditions formulated, for instance, by
MUller [16] to the onedimensional 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 righthand side of equation (5.29) is the mass of solid in the layer,
and therefore is constant, equation (5.26) simplifies to
d Zfree
di (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).