Some remarks on nonlinear consolidation models

Applied Mathematics Letters 18 (2005) 811–815 www.elsevier.com/locate/aml

Some remarks on nonlinear consolidation models✩ Ida Bonzania,∗, Renato Lancellottab a Department of Mathematics, Politecnico, Corso Duca degli Abruzzi 24, 10129-Torino, Italy b Department of Structural and Geotechnical Engineering, Politecnico, Corso Duca degli Abruzzi 24, 10129-Torino, Italy

Received 1 July 2004; accepted 1 August 2004

Abstract This paper deals with the modelling of nonlinear consolidation phenomena in a homogenous clay changing from an over to a normal consolidation regime. Specifically this paper develops a technical analysis of the model under the assumption of small deformations to derive a new class of models and to show how classical models known in the literature can be regarded as a particular case of the one dealt with in this paper. © 2004 Elsevier Ltd. All rights reserved. Keywords: Consolidation; Nonlinearity; Soil mechanics; Porous media

1. Introduction Mathematical models of consolidation phenomena have been developed starting from the classical consolidation theory by Terzaghi [1]. This theory is based on the assumption that consolidation of homogeneous clay layers occurs with coefficients of permeability and compressibility constant during the process. The various developments available in the literature, are documented in the review papers [2,3]. Properties of the consolidating clay may vary in space, due to both variations of soil type and stress history. As a consequence experimental results generally show nonhomogeneous and nonlinear behaviors which are not described by simple models. In particular some relevant phenomena can be described only if related to processes moving from an overconsolidated regime to a normally ✩ Under

the auspices of GNFM of the Istituto Nazionale di Alta Matematica.

∗ Corresponding author.

E-mail addresses: [email protected] (I. Bonzani), [email protected] (R. Lancellotta). 0893-9659/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.08.012

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consolidated one (i.e. in terms of plasticity theory, from an elastic domain to an elasto-plastic surface). A model to take into account the above phenomenon has been proposed in [4], where this type of behavior has been called a consolidation which changes type. Comparisons with experimental data have validated the above model which shows that change of type substantially modifies the pore pressure profiles. Various applications [5] and simulations obtained by application of the generalized collocation method [6] have shown that substantial quantitative differences can be obtained with respect to the description delivered by relatively simpler models. The above consideration motivates the contents of this paper which deals with an analysis of the model proposed in [5] to show how such a model generates, under suitable simplifications (mathematical and physical), some relatively simpler models available in the literature. The consolidation model is summarized in Section 2, while Section 3 deals with the above mentioned analysis. 2. Mathematical models for variable soil properties This section provides, in view of the analysis which will be developed in the next section, a concise description of the mathematical model proposed in [4]. The reader interested in additional information including comparisons with experimental data and simulations for various drainage conditions is referred to the already cited papers [4,5]. The physical–mechanical assumptions which generate the model are the following: Hypothesis 2.1. The state of the soil is described by the pore pressure and the void ratio. The independent variables are time and vertical coordinate. Therefore, analysis is developed in the one-dimensional case, i.e., the soil is laterally confined and the drainage can only occur in the vertical direction. The dependent variable is the pore pressure. Hypothesis 2.2. The soil is fully saturated and both phases, water and soil particles, behave as an incompressible medium. The self weight of the porous medium is ignored. Then, during the consolidation process the excess pore pressure can be evaluated by the difference between the effective normal stress at the end of the consolidation σ f and its current value σ  . In addition the total vertical stress σc generated by the external load is constant in time, i.e. σc = σ f − σ0 , where σ0 is the initial value of σ  . Hypothesis 2.3. Darcy’s law is applicable to the movement of the water through the soil. The constitutive laws linking the void ratio e to the effective stress σ  and the permeability coefficient k are the following:

 σ k e = e0 − Ic log10 = e0 + ck log10 , (2.1)  σ0 k0 where Ic is the compressibility index, ck is the permeability index, and e0 and k0 correspond to the initial value σ  = σ0 . The model is related to suitable dimensionless independent and dependent variables, as well as three dimensionless parameters. In detail, t is the time referred to T = H 2 /cv0 , where H is the drainage path and cv0 the initial consolidation coefficient; x is the vertical coordinate referred to H , with x ∈ [0, 1];

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u is the excess pore pressure divided by the vertical stress σc ; by Hypothesis 2.2 it follows that u ∈ [0, 1]; ε is the effective normal stress normalized by its initial value ε = σ  /σ0 ; it spans in the interval [1, 1 + σc /σ0 ]; while the characteristic parameters are η=

Ic , 1 + e0

λ=1−

Ic , ck

µ=

σ f − σ0 σ0

.

Taking into account the above hypotheses, balance equations and constitutive laws give 
 
e ∂ k ∂u ∂   =   ∂t 1 + e ∂ x k˜ ∂ x    ε = 1 + µ(1 − u) e − e0 = −η log10 ε    1 + e0    k   = ελ−1 , k0

(2.2)

(2.3)

where k˜ = k0 ln 10/µη M . The first equation represents the mass balance equation combined by Darcy’s law, the second one, deduced by Hypothesis 2.2, plays the role of momentum balance of the porous medium stating the dependence of the effective normal stress on the excess pore pressure, i.e. σ f − σ  = σc u; finally the third and last equations translate in the new variables the constitutive laws of Eq. (2.1). From a mathematical point of view the above system is constituted by four equations depending on the four dimensionless unknown u, e, k/k0 , ε and on the three parameters introduced in Eq. (2.2). Technical calculations generate the following model: 
2  ∂u µη M ∂u ∂ 2u = , (2.4) h(u) 2 + p(u)  ∂t µη(u) − η (u)ε(u; µ) ln(ε(u : µ)) ∂x ∂x where

µ (2.5) p(u) = (1 − λ) h(u). ε A suitable function η = η(u − u c ) models the sharp variation of such parameter induced by the two different consolidation regimes:



(u−u c ) c) + η exp α (u−u exp −α M u 1−u

, η = η(u − u c ) = ηm (2.6) (u−u c ) c) + η exp α (u−u exp −α m u 1−u h(u) = (1 + e0 )[1 − η(u) log10 ε]2 ελ ,

with α a positive parameter to be identified. The slope of η with respect to u i.e. the term η (u; u c , ν, α) is obtained by derivation. Both parameters λ and µ are assumed, in [4], to be constant during the whole consolidation process. In fact, taking into account experimental results, in many practical applications the ratio Ic /ck may be regarded as constant, since the permeability index ck shows the same sharp variation of Ic in correspondence of the same critical value u c . Finally, if parameter η is considered constant during the

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whole consolidation process (i.e. the consolidation does not change type), the particular model obtained by Eq. (2.4) has been previously introduced and discussed in paper [7]. 3. Mathematical model in the case of small deformations This section deals with a technical analysis of the model briefly described in Section 2 with the aim of showing how introducing additional assumptions it can generate simpler models, however useful for the applications. The first step consists in the derivation of a one-dimensional and nonlinear model suitable to describe a consolidation process which crosses from an overconsolidated regime to a normally consolidated one, under the additional assumption that the total volume is almost constant in time, i.e. (1 + e) ∼ = (1 + e0 ), which can be regarded as a small deformation assumption. Then the mass balance equation can be written as follows:

  e ∂ k ∂u ∂ , = ∂t 1 + e0 ∂ x k˜ ∂ x

(3.1)

(3.2)

once the Darcy law has been taken into account. If compared with the first equation of System (2.3), one can observe that the void ratio e is now referred to the initial volume (1 + e0 ) and not to the actual volume (1 + e) as it should be in a general consolidation process. This approximation is now justified by the assumption of small deformations. In Eq. (3.2) the left hand side may be substituted by 1 ∂e , 1 + e0 ∂t

(3.3)

while the right hand side, taking into account the last equation of System (2.3), may be written as
 ∂ µη M λ−1 ∂u ε . (3.4) ∂ x ln 10 ∂x Before deriving, the function ε is substituted by 1 + µ(1 − u) as given by the second equation of System (2.3). Then the derivation with respect to the time of Eq. (3.3) is performed taking into account the third equation of System (2.3), with ε = 1 + µ(1 − u). Both η and u depend on the time, so the following equation is obtained by deriving Eq. (3.3) 1 ∂e η(u)µu ∂u = −η log10 [1 + µ(1 − u)] + . 1 + e0 ∂t ln 10[1 + µ(1 − u)] ∂t

(3.5)

To derive Eq. (3.4), the dependence of the state variable u on the space x is considered. Then the following result is obtained: 

2 ∂ µη M µη ∂u ∂u M λ−1 λ−2 [1 + µ(1 − u)] = (λ − 1)[1 + µ(1 − u)] (−µ) ∂ x ln 10 ∂x ln 10 ∂x  ∂ 2u (3.6) + [1 + µ(1 − u)]λ−1 2 . ∂x

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Once both Eqs. (3.5) and (3.6) are inserted in Eq. (3.2) the resulting equation may be solved with respect to ∂u/∂t; at the end of such calculations the following evolution equation for the dependent variable u is obtained: 
2  2u µη M ∂ ∂u ∂u = ελ 2 + (1 − λ)ελ−1 µ , (3.7)  ∂t µη(u) − η (u)ε ln ε ∂x ∂x where ε(u; µ) = 1 + µ(1 − u). The above equation represents a new consolidation model for soil with change of type, when the total volume may be considered almost constant during the whole consolidation process. Now it can be shown how the above model contains as particular cases some specific models known in literature. For instance, under the assumption that η = η M = const, which means that the consolidation does not change type, the model proposed by Juarez-Badillo in [8] is obtained: 
2  ∂u ∂u ∂ 2u λ−1 = (1 + µ − µu) (1 + µ − µu) 2 + (1 − λ)µ . (3.8) ∂t ∂x ∂x If in addition compressibility and permeability indexes Ic and ck are assumed to vary in the same way (i.e. λ = 0) during the whole consolidation process one obtains the model by Davis and Raymond [9]:
2 ∂ 2u ∂u µ ∂u = 2+ . (3.9) ∂t 1 + µ − µu ∂ x ∂x Finally, if η = const, λ = µ = 0 the linear Terzaghi model [1] is mathematically deduced, ∂u ∂ 2u (3.10) = 2. ∂t ∂x A description of the above models reported in Eqs. (3.8)–(3.10) as well as a parameter sensitivity analysis is available in paper [10]. References [1] K. Terzaghi, Die berechnung der durchlassigkeitsziffer des tones aus dem verlauf der hydrodynamischen spannungserscheinungen, Sitzber. Akad. Wiss. Wien Math.-Nat. Kl. 132 (1923) 125–138. [2] R. De Boer, Highlights in the historical development of the porous media theory: Toward a consistent macroscopic theory, Appl. Mech. Rev 49 (1996) 201–262. [3] R. De Boer, Toward a consistent macroscopic theory, Appl. Mech. Rev 49 (2000) 201–262. [4] M. Battaglio, N. Bellomo, I. Bonzani, R. Lancellotta, Non-linear consolidation models of clay which change of type, Internat. J. Non-linear Mech. 38 (2003) 493–500. [5] M. Battaglio, I. Bonzani, D. Campolo, Nonlinear consolidation models of clay with time dependent drainage properties, Math. Comput. Modelling (2004) (in press). [6] N. Bellomo, Nonlinear models and problems in applied sciences: From differential quadrature to generalized collocation methods, Math. Comput. Modelling 26 (1997) 13–34. [7] P. Cornetti, M. Battaglio, Nonlinear consolidation of soil modeling and solution techniques, Math. Comput. Modelling 20 (1994) 1–12. [8] E. Juarez-Badillo, General theory of consolidation for clays, in: R.N. Yong, F.C. Townsend (Eds.), Consolidation of Soils: Testing and Evaluation, Amer. Soc. for Testing Materials, Philadelphia, 1986, pp. 137–153. [9] E.H. Davis, G.P. Raymond, A nonlinear theory of consolidation, Geotecnique 15 (1965) 161–173. [10] S. Arnod, M. Battaglio, N. Bellomo, Nonlinear models in consolidation theory: Parameter sensitivity analysis, Math. Comput. Modelling 24 (1996) 11–20.