A general nonlinear mathematical model for soil consolidation problems

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

Pergamon

A GENERAL

~) 1997 Elsevier Science Limited. All rights reserved Printed in Great Britain PII: S0020-7225(97)00024-4 0020-7225/97 $17.00+ 0.00

NONLINEAR MATHEMATICAL MODEL CONSOLIDATION PROBLEMS

FOR

SOIL

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. I N T R O D U C T I O N

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

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: --

~t

- -

Ot

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

(2.1)

o,

(2.2)

÷ v.(~,v,)

=

Nonlinear mathematical model for soil consolidation problems

[ Ors

)

0vt

)

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

pi~bl Ot + vl'Vvl

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

1047

(2.3)

(2.4)

where s stands for the solid phase and I for the liquid phase. Furthermore, referring to the pconstituent: (1) pp is the: "true" density--that is the density of the material which is used as the pconstituent 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,

(2.5)

v~ = ~sVs + ~lvt,

(2.6)

where

is the so-called composite velocity. 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 q-- V "(pml0m) ~--0, Ot

(2.7)

Pm = ps~bs + Plt~I,

(2.8)

-where

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

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

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

(2.9)

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

Pm

vm ) 0t + Vm'VVm = V ' T m + Ping,

(2.10)

where ~Yrn : -l-s @ ~-' -- Ps(~s(Vs -- Vm) (~ (Vs -- Vm) -- P I ~ b l ( V l - 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 -- 11~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= =

-

-

( V P , - 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

vj - v= =

+ V.(~b=v0 = O,

(2.13)

V "vc = O,

(2.14)

- -

( V P I - pIg),

(2.15)

Nonlinear mathematicalmodel for soil consolidationproblems V.Tm + Ping = O.

1049 (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 E U L E R I A N F O R M U L A T I O N IN O N E - D I M E N S I O N A L C O N D I T I O N S 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)]

¢; d e t U:s -

¢8 '

(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, Ot Oz

(3.3)

0vc -= 0, az

(3.4)

K(4,s) ( oP, v,-v~=

/z(1-¢s)

\ 0z

) + p' g

'

(3.5)

OPi 0o" -+ -+ Pmg=O, Oz Oz

(3.6)

a ' = ~rm-- PI

(3.7)

where

1050

R. LANCELLO'I"I'A and L. P R E Z I O S I

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 v,-

vs -

m

1 - ~b.

,

(3.9)

where

-

-

+ (ps -

•

(3.10)

L a:

Equation (3.8) and (3.9) then explicitly determine

•b

s

v, = C(t) + - 1-4~ v, = C(t)

-

Q,

Q.

(3.11) (3.12)

which, in turn, can be substituted in the continuity equation (3.3) to give 0.~_____2_+ C(t) OZ--Ad~ _ Ot Oz

0 (qbsQ), Oz

(3.13)

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].

Nonlinear mathematical model for soil consolidation problems

1051

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

e-

~bl

1 - --

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)--~--+ ( 1 oz

+

20( e) ~z

+

Q )

= 0,

(3.18)

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

Q= K(e) ( 3o-' Ps-Pl - + - - g tz - Oz l+e

)

•

(3.19)

4. T H E L A G R A N G I A N F O R M U L A T I O N IN O N E - D I M E N S I O N A L C O N D I T I O N S 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,

(4.3)

which can be written using equation (4.2) as

~'~

~b I

+ -~-

[(~l(13i - l/s) ] = O.

(4.4)

1052

R. L A N C E L L O T r A 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

epl + Ps

--

OZ

+ - - g = O ,

l+e

Oz

or using equations (3.7) and (4.2), Oer'

+

OZ

OPi

epl + p,, + - g = O. OZ l+e*

(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 OZ + Plg--~-~ =0.

(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

l+e

(v,-

vs) =

k(e)

Ou

Pig

OZ

OZ

--,

Oz

(4.8)

or

v , - vs =

k(e)

l+e*

Ou

Pig

e

OZ

,

(4.9)

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

+ (1 + e*)

OQ OZ

= O,

(4.10)

where, recalling the relation between hydraulic conductivity and permeability,

k(e) Pig

K(e)

--,

#

(4.11)

Nonlinear mathematical model for soil consolidation problems

Q --

k(e) l+e* (OFT' - p~g l + e OZ

1053

Ps--P_________~l) +

1+

e* g

(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 Oe OQ + v~ - - + (1 + e) = O, Ot " Oz Oz

(4.13)

or recalling equation (3.12),

Oe Ot

--

+C(t)

Oe -~Z

-

Q

Oe -~Z

+(l+e)

oQ OZ

(4.14)

=0.

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

(1 + e)

- Q - ~ z = (1

+

e) 2

1

l +e

0Q

Oz

2 0

(l+e) 20z

e) ~z

~

Q

' (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)

i~e Oe l + e* ~ r Oe ] o---t + f(e) a---Z - p, g ~ [ g(e) ~ J,

(4.17)

equation (4.10) reduces to

1054

R. LANCELLO'I"rA and L. PREZIOSI

where

f(e)=

(,,) --~(-1

d r,,(e, ] ,

l+e* l+e

g(e) = k(e) - -

(l+e*)-d-~-eLl+ e

dE de

(4.18)

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. U n d e r these hypothesis equation (4.17) reduces to ae 0t

k

--

Pig

do" 02e (1 + e*) d--e- 0 z 2 '

(4.20)

or recalling equation (4.19) as Oe

02e -

Ot

cv - -

(4.21)

Oz 2 '

where

cv -

k(1 + e*) avplg

,

(4.22)

is the consolidation coefficient. D u e to the linearity b e t w e e n stress and strain, equation (4.19) can also be written in terms of the effective stress:

a(7' Ot

~20" - cv - -

Oz 2 "

(4.23)

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

Nonlinear mathematical model for soil consolidation problems

1055

a p p l i e d l o a d is t i m e i n d e p e n d e n t as well as s p a t i a l l y i n d e p e n d e n t . I f this is t h e case, then c3u

02u - cv - -

Ot

(4.24)

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 O [ k(e) do-' Oe ] +-(l+e*)--=0. Ot Oz p~g de Oz

--

(4.25)

If the constitutive relation have the nonlinear expressions O-*P

k = k* - -o -,r

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

(4.26)

then the consolidation coefficient c~ -

k(e)(1 + e*)

do"

Pig

de

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

-

plgCc

(4.27)

is constant and ,equation (4.25) reduces to the nonlinear formulation given by Davis and R a y m o n d [5]:

0o-' = c,

Ot

az 2

o-'

\ 0z

.

(4.28)

It is of interest to note that the substitution of equation (4.27) into equation (4.25) gives Oe 02e 0 t - cv -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 R a y m o n d [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-' '~

(4.30)

\ o-'/'

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~, which is the gas compression law in adiabatic processes. IJ[$ JS: IO/ll-r

(4.31)

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 ratiopermeability. 5. I N T E R F A C E AND BOUNDARY C O N D I T I O N S 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 - v~(zb(t),t) = C(t) - Q(zb(t),t). dt

(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)

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

(5.6)

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

dt

or the relative boundary condition

(5.7)

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

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) =

(5.8)

pm(Z,t)g d z.

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

~Zh*(0 ~ Prn(Z,t) dz = d Zh(t)

J

z..(t)

[(Ps - Pl)~Ps(Z,t) + Pl] d z Zh(I)

. pl[Zb'(t) . . zb(t)] . + (1

p' / ~ z'