A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes

applied sciences Article A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes Kazimierz Józefiak *,† , Artur Zbiciak † , Karol Brzezi ński † and Maciej Ma´slakowski † Faculty of Civil Engineering, Warsaw University of Technology, 16 Armii Ludowej Ave., 00-637 Warsaw, Poland; [email protected] (A.Z.); [email protected] (K.B.); [email protected] (M.M.) * Correspondence: k.jozefi[email protected] † These authors contributed equally to this work. Abstract: The paper presents classical and non-classical rheological schemes used to formulate constitutive models of the one-dimensional consolidation problem. The authors paid special attention to the secondary consolidation effects in organic soils as well as the soil over-consolidation phenomenon. The systems of partial differential equations were formulated for every model and solved numerically to obtain settlement curves. Selected numerical results were compared with standard oedometer laboratory test data carried out by the authors on organic soil samples. Additionally, plasticity phenomenon and non-classical rheological elements were included in order to take into account soil over-consolidation behaviour in the one-dimensional settlement model. A new way of formulating constitutive equations for the soil skeleton and predicting the relationship between the effective stress and strain or void ratio was presented. Rheological structures provide a flexible tool for creating complex constitutive relationships of soil. Citation: Jozefiak, K.; Zbiciak, A.; Keywords: secondary consolidation; settlement; rheology; plasticity; constitutive modelling; soil Brzezinski, K.; Maslakowski, M. A mechanics; kepes Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes. Appl. Sci. 2021, 11, 1980. 1. Introduction https://doi.org/10.3390/ Soil is a complex porous three-phase material in which many phenomena take place app11051980 simultaneously. Because of the complexity a lot of soil mechanics problems have to be solved numerically or by laboratory investigation [1]. For numerical calculations, Academic Editor: Ilhan Chang mainly using the finite element method (FEM), appropriate material models have to be implemented. As there are many kinds of soil types, differing in particle size fractions, Received: 2 February 2021 geological origin and the conditions of loading, there is no universal solution. Many Accepted: 20 February 2021 Published: 24 February 2021 commercial FEM software packages provide their own material models [2]. However, they also allow for extending the models as there is often a need to analyse some less typical soil Publisher’s Note: MDPI stays neutral mechanics problems or soil types more thoroughly. In case of porous media, constitutive with regard to jurisdictional claims in equations are generally formulated in the effective stress and applied for coupled pore published maps and institutional affil- pressure—stress numerical calculations of geotechnical problems [3]. iations. Consolidation and settlement calculations are very important aspects of geotechnical design. The settlement of soil can be divided into three parts: (i) Elastic response, (ii) primary consolidation settlement related to the drainage of water, (iii) secondary settlement, related to the creep of the soil skeleton. In most cases only the primary consolidation is considered as it causes the largest settlement for the majority of soils. However, for organic Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. or soft clay soils, the secondary compression can be also large and has to be taken into This article is an open access article account [4]. Consolidation of the soft soil layer is also important in the case of column distributed under the terms and supported embankments, where the load transferred to the column and geosynthetic layer conditions of the Creative Commons strongly depends on the settlement of the soft layer [5,6]. Attribution (CC BY) license (https:// Rheological schemes provide a flexible tool for creating complicated constitutive creativecommons.org/licenses/by/ relationships of various materials. By applying rheological schemes to model the behaviour 4.0/). of the soil skeleton and coupling them with the equation of pore water flow, it is possible Appl. Sci. 2021, 11, 1980. https://doi.org/10.3390/app11051980 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 1980 2 of 12 to describe the transient (time-dependant) response of soil taking into account secondary consolidation of viscoelastic soils [7–12]. In the study, a method of modelling the transient process of primary and secondary consolidation of soil based on Burgers rheological scheme is presented. The Burgers structure is commonly used to model the behaviour of asphalt mixtures [13,14]. Here, it will be applied for the soil skeleton in order to simulate the transient process of consolidation of an organic soil layer. It should be noted that the presented procedure can be applied for any viscoelastic rheological structures. This shows that the great potential of constitutive models developed in different research fields can be straightforwardly utilized to solve geotechnical problems. An example of such a problem is the modelling of secondary consolidation. The 1D constitutive equations derived using rheological schemes can be implemented as a part of the evolution law of the Cap yield surface of the Modified Drucker–Prager/Cap model available in commercial FEM codes. On the other hand, the behaviour of soil in the effective stress space (e − log s0 curve) can be modelled by using non-classical rheological schemes as will be demonstrated in this paper. To do this, the non-classical Kepes element [15] will be introduced in a rheological scheme of the soil skeleton in effective stress, and based on that scheme a system of algebraic-differential equations will be formulated. This system of equations has not been presented previously in the geotechnical literature. Moreover, it should be emphasised that the Kepes element has not been applied to the geotechnical problem either. A detailed description of the Kepes rheological model for both 1D and 3D cases can be found in the monograph by Zbiciak [16]. 2. One-Dimensional Consolidation Model The one-dimensional consolidation problem is the consolidation of an infinite satu- rated homogeneous clay layer, loaded by an infinite and uniform pressure. The infinite layer of soil undergoing consolidation can be drained on both sides (see Figure1) or only on one side and rests on a non-deformable base (e.g., rock). Only excess pore water pressure, which arises after application of the external load, is considered. q(t) z H Figure 1. One-dimensional consolidation problem. Despite the fact that one-dimensional formulation seems to be a simplistic approach it is often used for the prediction of the settlement of engineering structures, such as road embankments [17–19]. The simplest model of one-dimensional consolidation was presented by Karl Terza- ghi [20]. The partial differential equation of flow can be expressed as: k ¶2s0(z, t) ¶#(z, t) = 2 (1) gw ¶z ¶t 0 where k is the coefficient of permeability, gw is the unit weight of water, s is the effective stress function of depth and time, and # is strain function of depth and time. Appl. Sci. 2021, 11, 1980 3 of 12 Assuming the linear constitutive law for the soil’s skeleton, in which the strain function #(z, t) is related to the effective stress s0(z, t) by the following equation s0(z, t) #(z, t) = (2) M where M denotes the oedometer modulus, Equation (1) can be solved with appropriate initial and boundary conditions. For the two-side drainage (see Figure1) the boundary conditions are as follows s0(0, t) = q(t) s0(H, t) = q(t) (3) where q(t) is a function of the applied load and H is the thickness of the soil layer. The initial condition is s0(z, 0) = 0. Equation (1) can be solved both analytically and numerically [20]. In the paper [10] numerical solutions of the Terzaghi problem were demonstrated. Moreover, some more advanced rheological schemes modelling the soil skeleton were also taken into consideration and the results of numerical solutions were compared with known analytical formulas [8]. The simple linear relationship between the effective stress and strain (Equation (2)) does not take into account secondary consolidation to be observed in organic or soft clay soils. This can be shown by comparison of the model prediction with oedometer laboratory test data carried out on a remoulded sample of soil with about 7% organic matter content (see Figure2). Taking different values of parameters M and k, it was impossible to obtain a satisfying fit to the oedometer test results. 0.5 ] 0.4 mm Oedometer IL test results; Δσ= 100 kPa [ 0.3 organic soil sample;I om = 6,7% 0.2 M= 4.5 MPa,k= 10 -10 m/min Settlement M= 4.2 MPa,k= 10 -11 m/min 0.1 M= 1.5 MPa,k= 10 -12 m/min 0.0 0 20 000 40 000 60 000 80 000 Time[min] Figure 2. Terzaghi solution in comparison to test data. 3. Classical Rheological Schemes for Consolidation Various classical rheological structures, represented by connections of springs and dashpots, are widely used to model the rate-dependent behaviour of asphaltic materials (see [21–25]) and polymers [26]. Such schemes for modelling problems of soil mechanics were adopted, e.g., in papers [7,8]. Gibson and Lo [8] presented an analytical solution of one-dimensional consolidation in which the relationship between the effective stress and vertical strain was formulated according to the standard model. 3.1. Burgers Model for Soil Skeleton In this section, we introduce another rheological scheme not widely used in soil mechanics—the Burgers model (see Figure3). This model is commonly used to model the behaviour of asphalt mixtures [13,14]. Here, it will be applied for the soil skeleton to simulate the transient soil response during consolidation and to obtain the settlement change in time.

Load more