Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2023.40.0076 Acta Polytechnica CTU Proceedings 40:76–82, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague EXTRACTING GENERAL KNOWLEDGE OF MODEL PARAMETERS FOR CLAYS OUT OF NUMEROUS LABORATORY TESTS Veronika Pavelcováa, ∗, Tomáš Jandab a Czech Technical University in Prague, Faculty of Civil Engineering, Department of Geotechnics, Thákurova 7, 166 29 Prague, Czech Republic b Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7, 166 29 Prague, Czech Republic ∗ corresponding author: veronika.pavelcova@fsv.cvut.cz Abstract. This paper is concerned with the process of determining the material parameters of advanced constitutive models for clays. Provided results are applicable already in the early stages of geotechnical survey when no detailed data from laboratory tests of soils expected at construction site are known. A worldwide database of laboratory tests obtained during four year operation of web calibration application ExCalibre was used for this purpose. This application allows automatic calibration of three advanced constitutive soil models based on detailed data from triaxial and edometric tests. The paper describes the process of determining the confidence intervals of each material parameter of modified Cam-clay and hypoplastic clay models for low (CL) and high (CH) plasticity clay according to unified soil classification system (USCS) and tabulates their typical ranges. The purpose of presented work was to contribute to the use of advanced constitutive models among practical engineers. Keywords: Hypoplastic clay, modified Cam-clay, calibration, material parameters. 1. Introduction In engineering practice, it can be observed that ma- jority of geotechnical engineers limit themselves to the use of basic non-linear soil models of the Mohr- Coulomb or Drucker-Prager type when describing the interaction of the structure and the subsoil and mostly avoid critical state models when solving standard tasks. The general lack of knowledge of these ad- vanced material models among practical engineers may be one of the reasons. Also standard civil engi- neering studies and the most commercial geotechnical programs primarily promote only basic models. The second reason is that it is significantly more dif- ficult to determine the input parameters of advanced non-linear models in comparison to the basic linear models. While the inputs for the use of basic non- linear models are shear strength parameters that can be determined directly from the results of standard laboratory or field tests, and their ranges are even tab- ulated for each soil type, critical state models require prescription of parameters describing the behavior of the given material not only under isotropic compres- sion but also parameters characterizing the current state of the soil. A practical engineer often encounters a situation where, although he knows that there exist models which are capable to describe the behavior of modeled soil significantly more accurately than linear constitutive models, he is no longer able to determine the necessary parameters that govern the model. 2. Critical state models Both, the hypoplastic clay [1] and modified Cam- clay [1, 2], constitutive models are based on the theory of critical state. The main features of these models are similar but unlike the hypoplastic model the modified Cam-clay model distinguishes the elastic and plastic part of strain components and therefore allows the straightforward visualization of permanently strained plastic zones in FEM calculations. On the other hand, the model of hypoplastic clay can correctly represent the associated gradual reduction in stiffness upon activation of plastic behavior rather than a sudden drop as predicted by the modified Cam-clay model. 2.1. Hypoplastic clay The hypoplastic model for clay takes into account the following important properties of soils: • The stiffness depends on the current value of the mean effective stress σeffm and the direction of load- ing. • The strength depends on the value of the mean ef- fective stress σeffm and current density characterized by the void ratio e. The model distinguishes between material parame- ters that are constant for all possible states of a par- ticular soil and state variables that evolve during straining. In this sense the hypoplastic model is fun- damentally different from the Mohr-Coulomb model which employs distinct values of its material param- eters for the same soil in different density states. In 76 https://doi.org/10.14311/APP.2023.40.0076 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 40/2023 Extracting Knowledge of Model Parameters out of Laboratory Tests Parameter Name Unit Material parameters N Position of the normal con- solidation line [-] λ∗ Slope of the normal consoli- dation line [-] κ∗ Slope of the swelling line [-] φc Friction angle in the critical state [°] ν Parameter controlling the value of the shear modulus [-] State parameter e Void ratio [-] Table 1. Material parameters of hypoplastic clay model. Figure 1. Normal consolidation line and swelling line assumed in hypoplastic model for clay [1]. particular, the hypoplastic model for clay requires five material parameters and one state variable summa- rized in Table 1. 2.1.1. Parameters N and λ∗ Parameters N and λ∗ define the position and the slope of the isotropic normal compression line (NCL) in the ln(σeffm )×ln(1+e) space as illustrated in Figure 1. The normal compression line defines a theoretical response of the soil sample during isotropic compression test. Since the model assumes incompressible grains the resulting volumetric strain depends on the decrease of volume of pores only and thus on the void ratio. 2.1.2. Parameter κ∗ Parameter κ∗ defines the slope of the swelling line in the ln(σeffm ) × ln(1 + e) space as illustrated in Figure 1. Unlike the NCL which is fixed by parameter N the position of the swelling line depends on the previous compression stress. In addition, unlike simple elasto- plastic models, the slope of the unloading-reloading line is non-linear in hypoplasticity, so that the pa- Parameter Name Unit Material parameters e0 Position of the normal con- solidation line [-] λ Slope of the normal consol- idation line [-] κ Slope of the swelling line [-] Mcs Slope of the critical state line [-] ν Poisson’s ratio [-] State parameter pc The preconsolidation pres- sure [Pa] Table 2. Material parameters of modified Cam-clay model. rameter κ∗ should thus be calibrated by means of simulation of a laboratory test, rather than by di- rectly evaluating its slope. 2.1.3. Parameter φc Parameter φc is the critical angle of internal friction. The angle of repose can not be measured for fine- grained soils and the critical friction angle has to be determined from triaxial shear test. The critical friction angle correspond to the maximal mobilized friction angle for normally consolidated samples where no softening occurs during the triaxial shear test. Or the critical friction angle corresponds to the mobi- lized friction angle after the peak for overconsolidated samples. 2.1.4. Parameter ν Parameter ν controls the value of the shear modulus. The bulk and shear modulus are linked through the parameter ν in the isotropic normally consolidated state. During shear test the shear modulus evolves non-linearly and its magnitude is being controlled by the parameter ν and by the distance from the failure state. 2.2. Modified Cam-clay The modified Cam-clay takes into account the follow- ing important properties of soils: • The stiffness depends on the current value of the mean effective stress σeffm and the direction of load- ing. • The strength depends on the value of the mean effective stress σeffm a current density characterized by the void ratio e or by the preconsolidation pres- sure pc. The model introduces five material parameters that are constant for all possible states of a particular soil and one state variable that evolves during straining. All parameters are summarized in Table 2. 77 Veronika Pavelcová, Tomáš Janda Acta Polytechnica CTU Proceedings Figure 2. Normal consolidation line and swelling line assumed in modified Cam-clay model [2]. 2.2.1. Parameters e0 and λ Parameters e0 and λ define the position and the slope of the isotropic normal compression line (NCL) in the ln(σeffm ) × e space as illustrated in Figure 2. The normal compression line defines a theoretical response of the soil sample during isotropic compression test. Since the model assumes incompressible grains the resulting volumetric strain depends on the decrease of volume of pores only and thus on the void ratio. 2.2.2. Parameter κ Parameter κ defines the slope of the unload- ing/reloading line in the ln(σeffm )×e space as illustrated in Figure 2. Unlike the NCL which is fixed vertically by parameter e0 the position of the swelling (unload- ing/reloading) line depends on the maximal previous compression stress denoted as the preconsolidation pressure pc. 2.2.3. Parameter Mcs Parameter Mcs is the slope of the critical state line plotted in the σeffm × J space, see Figure 3. The value is computed from the critical state angle φc by: Mcs = 2 √ 3 sin φc 3 − sin φc . (1) 2.2.4. Parameter ν Parameter ν is the Poisson ratio known from linear elasticity. It controls the ratio of radial and axial elastic strains under uniaxial loading and consequently the ratio between the stress dependent bulk and shear modulus. 2.2.5. Preconsolidation pressure pc The model has one state variable – the preconsolida- tion pressure pc. For an isotropic compression test the preconsolidation pressure pc corresponds to the maximal mean stress σeffm to which the soil has been subjected. The kinematics of the model is evident from Figure 2. Soil sample with an initial state a (it has void ratio ea and preconsolidation pressure pac ) is loaded isotropically up to the compressive mean stress σeffm = pac . The deformation during this loading is nonlinear but elastic, i.e. if the soil is unloaded from this point it returns along the κ-line back to the state with e = ea and no permanent volumetric strain occurs. If the soil is loaded beyond the point pac it moves along the λ-line. During this loading both the elastic and the plastic volumetric strain develop. When the soil is unloaded from the point σeffm = pbc it moves along the new κ-line defined by the new value of preconsolidation pressure σeffm = pbc. As evident from the initial and final (in unloaded state) values of the void ratio ea > eb the permanent volumetric strain developed during this loading sequence [1]. 3. Automated calibration of constitutive models Even though the development of more advanced con- stitutive soil model theories which take into account bounding surface plasticity, whether those based on the theory of plasticity or hypoplasticity [3, 4], or baro- desy [5] continues steadily forward, the development of automated calibration procedures stays behind. The calibration procedures are regarded as the inverse anal- ysis, since the parameters of a constitutive model are not known in advance but the reaction of the system is known. The goal of the calibration is to minimize the objective error function E(U ) which is defined as a function of a difference between the observation or experiment Uexp and simulation Unum. The most com- mon is the least square method. Nevertheless, more sophisticated dimensionless formulas can be used [6]. 3.1. ExCalibre As mentioned in the introduction, the Faculty of Civil Engineering of CTU in cooperation with the Faculty of Science of Charles University developed a ExCal- ibre web application [7] which performs automatic deterministic calibration of the hypoplastic model for sands [8], hypoplastic model for clays [9] and modified Cam-clay model [10]. From 2018, the application [1] is available free of charge for registered users who agree to provide the uploaded data for research and development purposes. A typical workflow of model calibration in ExCalibre is the following: (1.) The user downloads a laboratory protocol tem- plate which is prepared for the user in the form of an MS Excel file. The template is different for clays and sands. (2.) The user fills in the results of laboratory tests. A minimum of one consolidated isotropi- cally undrained (CIUP) triaxial test with pore pres- sure measurement for clay soils or one consolidated isotropically drained (CID) triaxial test for sandy 78 vol. 40/2023 Extracting Knowledge of Model Parameters out of Laboratory Tests Figure 3. Yield surface of the modified Cam-clay model [2]. soils and one standard oedometric (OED) test must be completed for successful calibration. In addition to those already mentioned, it is possible to fill in the classification according to USCS and index characteristics of the soil. These parameters are currently not used in calibration but they are stored for research purposes. (3.) The user uploads the laboratory protocol back to the web application. As a result, the applica- tion proposes optimal model parameters and graphs comparing the course of measured and simulated laboratory tests. These graphs serve to visually check the calibration of the model. (4.) It is possible to manually modify individual model parameters to adjust the calibration or to examine the influence of individual parameters on model predictions after the calibration. Large pool of available data collected worldwide pro- pose an effective basis for statistical processing, deter- mination of typical ranges of individual parameters and search for correlations between individual param- eters presented in this paper. 4. Data processing All data work was concentrated in universal scripts written in the Python programming language due to the repeatability of laboratory test data analysis. It is a higher-level programming language with dynamic support for data types and support for programming paradigms. The programming language [11] is freely available. The format conversion of laboratory protocols from Microsoft Excel Spreadsheet xlsx ExCalibre template to Comma-separated values csv was the first step of data processing. It was a necessary step due to the complexity of the xlsx file structure. xlsx file is a zip package that contains xml files representing in- dividual sheets of the spreadsheet which are always supported by rels files containing specifications for linking individual xml files together to form a com- plete file. Compared to complex xlsx file, csv is a simple file format that is intended for exchanging tabular data between different systems. The data of the csv file are stored in rows just like in the default table while the individual items in the row are further separated by a predetermined character. The indi- vidual sheets of the input xlsx laboratory protocol form separate csv files which are, for organizational reasons, stored in a folder with a name corresponding to the original name of the xlsx file so that one folder represents one complete laboratory protocol. Written script cycles through the folders represent- ing individual laboratory protocols with csv files re- placing Microsoft Excel Spreadsheet and extracts char- acteristic information about the laboratory protocol using predefined classes. The extracted data is further inserted into a dictionary-type data structure. A dic- tionary is a data type sometimes called an associative array. It consists of key-value pairs. Each key-value pair maps a key to an associated value. In the algo- rithm described , the key corresponds to the name of a folder representing one laboratory protocol. Key values are assigned using classes so that one line of the dictionary represents the data of one laboratory protocol. The first value of the dictionary corresponds to the relative path to the file, assigned attributes are: USCS classification of soil, basic index characteristics such as specific gravity, Atterberg’s limits or angle of repose. It is followed by representation of particle size distribution and loading and quantity of performed oedometric and triaxial tests. The ExCalibre application contains a public labo- ratory test database so that these protocols can be used for testing the functionality of the calibration application by a new user. Knowing that, filtering of duplicate laboratory protocols was another step of data processing. A simple condition consisting of • USCS classification of soil, • the largest sieve and the corresponding passing in the grain size analysis, • the initial void ratio of the oedometric test of the naturally consolidated soil sample, • the initial void ratio of the oedometric test of the reconstituted soil sample 79 Veronika Pavelcová, Tomáš Janda Acta Polytechnica CTU Proceedings Hypoplastic clay Modified Cam-clay λ∗ κ∗ N ν φc λ κ e0 ν Mcs Count 7 7 7 7 7 7 7 7 7 7 Mean value 0.119 0.010 1.396 0.276 25.3 0.297 0.016 2.784 0.307 0.997 Standard deviation 0.082 0.003 0.659 0.062 2.0 0.365 0.012 2.521 0.065 0.088 Minimum value 0.063 0.006 0.872 0.170 22.8 0.104 0.010 1.237 0.220 0.888 25 % quantile 0.068 0.009 0.960 0.245 23.9 0.107 0.010 1.383 0.250 0.935 50 % quantile 0.094 0.010 1.173 0.280 25.6 0.155 0.010 1.752 0.340 1.008 75 % quantile 0.123 0.010 1.521 0.320 26.2 0.249 0.018 2.734 0.355 1.032 Maximum value 0.296 0.014 2.763 0.350 28.8 1.107 0.039 8.263 0.380 1.147 Table 3. Basic statistical description of calibrated model parameters for CH soil. was used for the determination and removal of dupli- cate laboratory protocols. The results of this class- driven filtering revealed that only 58 of the total 1 924 laboratory protocols are non-duplicated. The remain- ing protocols are duplicates of another 161 protocols in various frequencies. In total, the database would therefore contains 219 unique laboratory protocols. Considering USCS composition of non-duplicate pro- tocols only low (CL) and high (CH) plasticity clay soils were selected for further processing. All of the laboratory protocols of soils classified as CH or CL have been uploaded to the ExCalibre web applica- tion and material parameters for hypoplastic model for clay soils and modified Cam-clay were calibrated and added to the dictionary-like dataframe collect- ing protocol information. Thus the already existing dataframe was extended by 5 columns with material parameters of hypoplastic clay model (see Table 1) and another 5 columns for modified Cam-clay model material parameters (see Table 2). 5. Results The database contained only 7 laboratory protocols of soil classified as high-plasticity clay after removing all duplicates. All of them were successfully calibrated for both of constitutive models – hypoplastic clay and modified Cam-clay. The summarization of basic statistical description of calibrated model parameters is summed up in Table 3. Strong correlations between Atterberg’s limits (liquid limit plastic limit wL and plastic limit wP ) and slope of the NCL line of both calibrated models have been observed. Example of correlation between plastic limit and slope of the NCL assumed in hypoplastic model for clay can be seen in Figure 4 and the same trend applies to the position of the NCL. All considerable correlations are summarized in Table 4 but limited number of protocols has to be considered. Overall 72 laboratory protocols of soil classified as low-plasticity clay contained the database after remov- ing all duplicates of which it was possible to calibrate 56 for hypoplastic clay and 61 for modified Cam-clay model. The reason why not all of the protocols were successfully calibrated will be further examined. Basic statistical description of calibrated model parameters 25 30 35 40 Plastic limit wP [-] 0.10 0.15 0.20 0.25 0.30 Sl op e of th e no rm al c on so lid at io n lin e * [ -] Figure 4. Correlation between plastic limit and slope of the NCL assumed in hypoplastic model for clay. Hypoplastic clay Mod. Cam-clay λ∗ N ν φc λ e0 Mcs wL 0.92 0.95 0.41 0.91 0.94 0.40 wP 0.95 0.92 0.52 0.44 0.89 0.88 0.44 Table 4. Correlations between Atterberg’s limits of high-plasticity clay soil and model’s parameters. is also summarized and can be examined in Table 5. Examples of parameter’s distribution are shown in his- togram in Figure 5. In comparison to high-plasticity clay, only moderate correlations between Atterberg’s limits and model parameters have been observed for low-plasticity clay. In particular, higher soil mois- ture at Atterberg’s limits indicates higher parameter N with correlation coefficients about 0.41. The similar trend also applies to modified Cam-clay’s parameter e0 with correlation coefficients about 0.46 for the liq- uid limit and the 0.41 for plastic limit. Correlation 0.40 has been found between moisture at liquid limit and the slope of the NCL line λ. Another correlations seem insignificant for practical applications. 6. Conclusions This paper briefly explained each individual input parameters of both hypoplastic and modified Cam- 80 vol. 40/2023 Extracting Knowledge of Model Parameters out of Laboratory Tests Hypoplastic clay Modified Cam-clay λ∗ κ∗ N ν φc λ κ e0 ν Mcs Count 56 56 56 56 56 61 61 61 61 61 Mean value 0.067 0.008 0.948 0.217 29.1 0.107 0.010 1.388 0.270 1.120 Standard deviation 0.035 0.005 0.283 0.109 6.6 0.062 0.006 0.597 0.112 0.320 Minimum value 0.001 0.000 0.507 0.010 2.9 0.002 0.000 0.112 0.010 0.101 25 % quantile 0.042 0.005 0.725 0.155 25.0 0.059 0.008 0.949 0.200 0.984 50 % quantile 0.064 0.007 0.899 0.230 30.0 0.096 0.010 1.246 0.260 1.199 75 % quantile 0.090 0.010 1.170 0.290 33.3 0.149 0.011 1.797 0.390 1.274 Maximum value 0.178 0.025 1.667 0.400 43.6 0.284 0.036 3.075 0.400 1.790 Table 5. Basic statistical description of calibrated model parameters for CL soil. 0.00 0.05 0.10 0.15 Slope of the normal consolidation line * [-] 0 2 4 6 8 10 12 14 16 18 Fr eq ue nc y (a). Hypoplastic clay model. 0.00 0.05 0.10 0.15 0.20 0.25 Slope of the normal consolidation line [-] 0 2 4 6 8 10 12 14 16 Fr eq ue nc y (b). Modified Cam-clay model. Figure 5. Histograms of slopes of the NCL for hypoplastic clay and modified Cam-clay model. clay constitutive model and it summarized the issue of automatic parameter calibration. ExCalibre web calibration application, a practical tool for calibrat- ing these material parameters based on the results of standard laboratory tests, was introduced in Section 2 and Section 3. Then a method of how could existing data be used to obtain general knowledge of the prop- erties of these individual parameters was presented in Section 4 and finally, the paper summarized extracted basic statistical data of model parameters with respect to USCS classification in Section 5. The main aim of this research is to help with devel- opment of a robust automatic calibration procedure which will be applicable at any stage of geotechni- cal survey to promote the use of advanced models in geotechnical practice. The determination of the confidence intervals along with correlations between material parameters and index soil properties was only the first step of data analysis and other more complex studies will follow. Acknowledgements The support provided by the SGS project No. SGS22/030/OHK1/1T/11 and the GAČR project No. 22-12178S is gratefully acknowledged. References [1] ExCalibre – SoilModels Automatic Calibration. [2022-07-26]. https://soilmodels.com/excalibre-en/ [2] M. Šejnoha, J. Pruška, T. Janda, M. Brouček. Metoda konečných prvků v geomechanice: Teoretické základy a Inženýrské aplikace. České vysoké učení technické, 2015. ISBN 978–80–01–05743–8. [3] J. Jerman, D. Mašín. Hypoplastic and viscohypoplastic models for soft clays with strength anisotropy. 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[2022-07-26]. https://www.python.org 82 https://doi.org/10.1007/s11440-012-0199-y https://www.python.org Acta Polytechnica CTU Proceedings 40:76–82, 2023 1 Introduction 2 Critical state models 2.1 Hypoplastic clay 2.1.1 Parameters N and lambda* 2.1.2 Parameter kappa* 2.1.3 Parameter phi c 2.1.4 Parameter nu 2.2 Modified Cam-clay 2.2.1 Parameters e0 and lambda 2.2.2 Parameter kappa 2.2.3 Parameter Mcs 2.2.4 Parameter nu 2.2.5 Preconsolidation pressure pc 3 Automated calibration of constitutive models 3.1 ExCalibre 4 Data processing 5 Results 6 Conclusions Acknowledgements References