1 Introduction

Over the past few decades, numerous constitutive models for soils have been proposed, and more will be emerging in future [15, 36, 37]. When using a certain constitutive law in engineering design, one may be concerned with its calibration and validation.

1.1 Calibration of the parameters

A constitutive model normally involves a number of parameters. For example, the modified Cam-Clay (MCC) model includes five parameters related to the elastic and plastic behavior of soil [27,28,29, 32]. However, many of these parameters are derived from laboratory tests with particular stress paths that may not be representative for engineering problems. Moreover, the in situ stress history is often unknown, which hinders the use of laboratory tests.

One possible approach is to fit the constitutive model based on laboratory tests such as conventional triaxial and oedometer tests. In this case, some optimization methods can be used for model calibration. This fitting approach with optimization methods is not new. For instance, Pal et al. [23] used the genetic algorithm (GA) [10] to calibrate the parameters of the hierarchical single surface model. Sadoghi Yazdi et al. [31] adopted the particle swarm optimization (PSO) algorithm [11] to calibrate the parameters of the linear elastic-hardening plastic model with Drucker–Prager yield criterion. More applications can be found in references [1, 6, 16, 24, 54, 56].

1.2 Validation of the constitutive laws

The constitutive model should be validated by laboratory tests with various stress paths before engineering application. For example, Matsuoka et al. [20] developed a unified SMP model for clays and calibrated the parameters through conventional drained triaxial tests on normally consolidated Fujinomori clay (experimental data from Nakai and Matsuoka [22]). The calibrated parameters are then verified by independent test results, such as undrained triaxial compression and extension tests. More examples of model validation through independent test results can be found in the literature [4, 9, 12,13,14, 18, 21, 25, 26, 30, 34, 38, 39, 41, 44, 47, 50, 51].

Despite the aforementioned successes in the calibration and validation of constitutive laws, there still lacks standard methodologies. This manuscript presents an in-depth approach for calibrating and validating a constitutive model with conventional laboratory test results. Specifically, the unified hardening model for clays and sands (CSUH model) developed by Yao et al. [48] is considered. This constitutive model inherits the basic principles of the MCC model while extends its ability to model the mechanical behavior of both sands and clays. This manuscript reports the details of calibrating its eight basic parameters via consolidated drained triaxial compression (CD) tests. The calibrated parameters are further used to predict the consolidated undrained triaxial compression (CU) tests. This so-called “drained versus undrained” approach assures its universal applicability along various stress paths. The calibration solver and the related documentation and test cases can be found on the website: https://github.com/ChenZuyuIWHR/CSUH-BH.

In the remaining part of this manuscript, firstly, the theories of the CSUH model and its numerical implementation are detailed. Afterward, the particle swarm optimization algorithm based on multiple adaptive strategies (MAPSO) for model calibration is described. Finally, the model calibration and validation processes are presented, where the authors demonstrate the relative errors and determination coefficient of the predicted results.

2 The CSUH model

2.1 Model framework

It is well-known that state-dependent dilatancy and excess pore water pressure are two major challenges in constitutive modeling of soils [7, 8]. The original unified hardening (UH) model, proposed by Yao et al. [44,45,46] for overconsolidated clays based on the MCC model, is capable of characterizing these two issues by introducing a unified hardening law [50]. Because of this advantage, it is later extended for many kinds of soils, e.g., unsaturated soils [18] and structured soils [57], etc., and has gained popularity in many engineering practices [43, 52, 53, 55]. Recently, the original UH model is further extended for granular materials [48], which is the so-called CSUH model.

The CSUH model can degrade into the original UH model for overconsolidated clays and can further degrade into the MCC model for normally consolidated clays [48]. Therefore, the highlights of the CSUH model can be appreciated by comparing its basic features with the MCC model, as shown in Table 1. The parameters of the CSUH model as well as some intermediate variables are summarized in Table 2. In the following, a brief review of the CSUH model is outlined:

  1. 1.

    Basic framework. The CSUH model extends the isotropic normal compression line (NCL) of the MCC model to a more generalized form for sands. The new NCL becomes a curved line by introducing a new interception Z. In the CSUH model, the NCL of the MCC model is called the reference normal compression line (RNCL). The NCL of the CSUH model asymptotically approaches the RNCL when the parameter Z is equal to N. In addition, the CSUH model indicates that the current state of point B is not necessarily unloaded from an isotropic compression line, as proposed by the MCC, but may be from a more generalized anisotropic compression line (ACL). As illustrated in Figs (a) and (b) of Table 1, the ACL is defined by the parameters Z and χ.

  2. 2.

    Plastic potential. The MCC model adopts the associated flow rule, while the CSUH model adopts the non-associated flow rule. The parameter M in the plastic potential equation of the MCC model is replaced by Mc in the CSUH model.

  3. 3.

    Yield Function. The CSUH model adopts a drop-shaped yield function by introducing a critical state parameter χ. This parameter can adjust the vertical distance between the NCL and the critical state line (CSL), i.e., \(\Delta e_{{A_{2} F}}\) shown in Fig (c) of Table 1, leading to a feasible control of the dilatancy.

  4. 4.

    Hardening law. The hardening law of the CSUH model adopts a similar form as the unified hardening parameter in the UH model for overconsolidated clay. A new dilatancy parameter m is introduced in the expression of the characteristic state stress ratio Mc to control the rate of dilatancy. The larger m is, the earlier the current stress ratio η exceeds the current Mc, leading to volume dilation.

Table 1 Comparison between the CSUH and the MCC models
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Table 2 The basic parameters and intermediate variables of the CSUH model
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As shown in Table 2, the CSUH model involves eight basic material parameters, with the first five being the same with the MCC and the original UH models [44, 48]. The CSUH model can degrade to the original UH model by setting parameter Z = N and by setting parameters χ and m to be zero. Due to this feature, the CSUH model is able to uniformly decipher the mechanical behaviors of both clays and sands. For details regarding the formulation and prediction performance of the CSUH model, the reader may refer to the literature [48].

2.2 Numerical integration

The CSUH model is an incrementally elastoplastic model which requires step-by-step integrations for establishing the stress–strain and excess pore pressure relationships, as follows:

$$\left\{ {\Delta \varepsilon_{i} } \right\} = \left[ {C_{ij} } \right]\left\{ {\Delta \sigma^{\prime}_{j} } \right\},\,\,\,\,i = 1,2,3,\,\,\,\,j = 1,2,3$$
(1)

where \(\left[ {C_{ij} } \right]\) represents the elastoplastic matrix \(\left[ {C_{ij}^{{{\text{ep}}}} } \right]\) during loading process and elastic matrix \(\left[ {C_{ij}^{{\text{e}}} } \right]\) during unloading process, i.e., elastic process. The elastoplastic matrix \(\left[ {C_{ij}^{{{\text{ep}}}} } \right]\) is defined as

$$\left[ {C_{ij}^{{{\text{ep}}}} } \right] = \left[ {C_{ij}^{{\text{e}}} } \right] + \left[ {C_{ij}^{{\text{p}}} } \right]$$
(2)

with

$$\left[ {C_{ij}^{{\text{e}}} } \right] = \frac{1}{E}\left[ {\begin{array}{*{20}c} 1 & { - \nu } & { - \nu } \\ { - \nu } & 1 & { - \nu } \\ { - \nu } & { - \nu } & 1 \\ \end{array} } \right],$$
(3)

and

$$\left[ {C_{ij}^{p} } \right] = c_{p} \frac{{M_{c}^{4} - \eta^{4} }}{{M_{f}^{4} - \eta^{4} }}\frac{{\frac{\partial f}{{\partial \sigma^{\prime}_{j} }}}}{{\frac{\partial g}{{\partial p^{\prime}}}}}\frac{\partial g}{{\partial \sigma^{\prime}_{i} }},\,\,\,i = 1,2,3,\,\,\,\,j = 1,2,3$$
(4)

being the elastic and plastic matrixes, respectively. \(\left[ {C_{ij}^{e} } \right]\) can be obtained from the elastic modulus E varying with mean effective stress p′ and Poisson’s ratio ν. The plastic matrix \(\left[ {C_{ij}^{p} } \right]\) can then be obtained from intermediate variables listed in Table 2 and the partial derivative formulas in the literature [48].

The numerical integration in a conventional triaxial test can be started with the input of the axial strain increments Δε1 and gives the output of the principal stress increments (\(\Delta \sigma^{\prime}_{1}\), \(\Delta \sigma^{\prime}_{2}\) and \(\Delta \sigma^{\prime}_{3}\)) and the lateral principal strain increments (Δε2 and Δε3). The boundary conditions and the controlling equations are listed in Table 3.

Table 3 Boundary conditions and the controlling equations of the CSUH model for modeling conventional triaxial tests
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The loading–unloading criterion is expressed by the following elastic trial function in which the principal stress increments are calculated using the elastic matrix.

$$\Delta f_{{{\text{trial}}}} = \frac{\partial f}{{\partial \sigma^{\prime}_{1} }}\Delta \sigma^{\prime}_{1} + \frac{\partial f}{{\partial \sigma^{\prime}_{2} }}\Delta \sigma^{\prime}_{2} + \frac{\partial f}{{\partial \sigma^{\prime}_{3} }}\Delta \sigma^{\prime}_{3}$$
(5)

The above criterion indicates that with a positive \(\Delta f_{{{\text{trial}}}}\), the numerical step is a loading process during which the principal stress increments and lateral principal strain increments need to be recalculated by the elastoplastic matrix; otherwise, the step is an unloading process. With the model parameters, boundary conditions, and controlling equations, the stress–strain and strain-pore pressure relationships in conventional triaxial tests can be calculated according to the flowchart in Fig. 1.

Fig. 1
figure 1

Flowchart of the iterative process of the CSUH model in triaxial tests

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3 The particle swarm optimization algorithm

3.1 Framework of the swarm population

The canonical particle swarm optimization (PSO) algorithm was originally developed by Kennedy and Eberhart to optimize nonlinear functions and train the neural network [11, 3]. To better balance the exploration (global search) and exploitation (local search) capabilities, Wei et al. [40] proposed a PSO variant based on multiple adaptive strategies (MAPSO), which has a promising performance on complicated problems. Thus, the MAPSO is adopted to calibrate parameters of constitutive model in this study. MAPSO divides all the exemplars, i.e., particles, in the canonical PSO algorithm into Ns swarms, and then different exemplars Xi can automatically select their learning exemplars (ALE) to improve their diversity.

As shown in Fig. 2, the exemplars are contained in matrices with the following definitions:

Fig. 2
figure 2

Candidate learning exemplars for different exemplars Xi in each generation

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Swarm: A swarm consists of three exemplars arranged in columns. Based on error value, the three exemplars in a swarm are categorized into elite Ei, mediocrity Mi, and inferior Ii, among which the elite exemplar has the least error.

Population matrix Γ: This matrix is organized by Ns swarms. It includes all the exemplars with number of N1 = 3 × Ns.

“Best so far” matrix B: This is a vector consisting of Ns elements out of the population matrix, and each represents the best elite exemplar of the swarm to date with the same column. The elements are denoted by B1, B2, …, BNs. The exemplar Bi is the first candidate learning exemplars of Ei, Mi and Ii.

Submatrices E, EM and EMI: These submatrices are defined for picking up the second candidate learning exemplars of Ei, Mi and Ii.

The optimization algorithm will proceed with N1 loops in each generation. Each generation updates all the exemplars in the population matrix that will be renewed according to the rules of arrangement that have elite, mediocre and inferior exemplars sorted and rearranged based on their error values. In addition, to rationally utilize the computational resource, the MAPSO algorithm adopts an adaptive adjustment for population size (APS) by deleting or adding exemplars according to the generation that the error value of the best exemplar remains unchanged.

3.2 The process of the MAPSO algorithm

The MAPSO algorithm proceeds mainly according to the following procedures:

Step 1: The initial exemplar Xi is given randomly in the search area R, and the initial velocity Vi is given randomly in the interval [− (Ru − Rl)/5, (Ru − Rl)/5] with Ru and Rl being the upper and lower limits of R. After filling up in the population matrix Γ by the N1 number of exemplars, their error values are calculated as: \(\Lambda_{i} = f({\mathbf{X}}_{i} ),\;i = 1,\;2, \ldots ,\;N_{1}\).

Step 2: Sort the swarms in the population matrix based on their values of Λ. The population matrix thus looks like Fig. 2, which arranges the exemplars in ascending order arranged by inferior, mediocre and elite elements.

Step 3: Update the candidate learning exemplars in the “best so far” vector by making Bi = Ei, i = 1, 2,…, Ns, which is the first generation of candidate learning exemplars that have relatively small values of Λ.

Step 4: Generate a new population matrix with

$${\mathbf{X}}^{t + 1} = {\mathbf{X}}^{t} + {\mathbf{V}}^{t + 1}$$
(6)

where \({\mathbf{V}}^{t + 1}\) is a velocity of an exemplar expressed as follows:

$${\mathbf{V}}^{t + 1} = w \cdot {\mathbf{V}}^{t} + r_{{1}} \cdot ({\mathbf{B}}^{t} - {\mathbf{X}}^{t} ) + r_{{2}} \cdot \left( {{\mathbf{S}}^{t} - {\mathbf{X}}^{t} } \right)$$
(7)

The superscripts t and t + 1 represent the numbers of this generation and the next generation, respectively. In this equation, the first candidate learning exemplar Bt is the elite exemplar belonging to the same swarm of Xt. As shown in Fig. 2, St is the second candidate learning exemplar randomly taken from submatrices E, EM, or EMI if Xt is attributed to elite, mediocrity and inferior, respectively. This process of automatically selecting two candidate learning exemplars for a given exemplar is called ALE. w represents an inertia weight denoting how much the previous velocity is preserved and is equal to 0.9–0.8 × t/tm, where tm represents the maximum numbers of generation; and r1 and r2 are two random numbers uniformly distributed in the interval [0, 1]. The operation process of updating each exemplar in the population matrix by two candidate learning exemplars can be represented by a vector operation diagram, as shown in Fig. 3.

Fig. 3
figure 3

The method of updating exemplars Xt+1 in the MAPSO algorithm

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Step 5: Repeat step 2 for sorting the new population matrix and Step 3 for renewing the “best so far” matrix.

Step 6: Delete or add exemplars according to APS.

Step 7: Repeat Step 4 for a new round of generation until the loop time t reaches a preset limit tm.

Step 8: Select the one with the least error from Bi as the optimal output.

3.3 The optimization statements

The optimization for calibrating the CSUH model proceeds using the following definitions:

Exemplar: A vector containing the basic parameters represented by X = (M, ν, κ, λ, N, Z, χ, m) for the CSUH model.

Search area: The MAPSO algorithm requires a reasonable area defined by R for searching the optimized parameters. The limits of the five parameters (M, ν, κ, λ, N) are suggested by the MCC model [5, 19, 32, 33, 35, 49]. For sands, the parameter Z is close to the maximum void ratio, approximately (0.4–0.7) times of N. For clays, Z is equal to parameter N. The parameters χ and m lie, respectively, in the ranges of [0, 1) and \(\left[ {0,\;{{(1 - \chi )} \mathord{\left/ {\vphantom {{(1 - \chi )} {\left[ {\left( {\lambda - \kappa } \right)\left( {1 + \chi } \right)} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\left( {\lambda - \kappa } \right)\left( {1 + \chi } \right)} \right]}}} \right)\) [48]. Table 4 gives the suggested upper and lower boundaries of the parameters for the CSUH model.

Table 4 The upper and lower boundaries of the parameters
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Individual error: The following two criteria are used for assessing the relative errors between the experimental data and model prediction [2].

Mean relative error

$$MRE = \frac{1}{{N_{2} }}\sum\limits_{i = 1}^{{N_{2} }} {\frac{{\left| {y_{e}^{i} (x_{i} ) - y_{m}^{i} (x_{i} )} \right|}}{{\left| {y_{e}^{i} (x_{i} )} \right|}}}$$
(8)

Determination coefficient

$$R = 1 - \sqrt {\frac{{\sum\nolimits_{i = 1}^{{N_{2} }} {\left( {y_{e}^{i} (x_{i} ) - y_{m}^{i} (x_{i} )} \right)^{2} } }}{{\sum\nolimits_{i = 1}^{{N_{2} }} {\left( {y_{e}^{i} (x_{i} )} \right)^{2} } }}}$$
(9)

where \(y_{e}^{i} (x_{i} )\) and \(y_{m}^{i} (x_{i} )\) represent the ith experimental and model-predicted y with the same xi, respectively, as shown in Fig. 4. N2 is the total number of experimental points in a single curve, MRE represents the relative difference between the experimental data and model prediction, and R describes the analogy in the distribution tendency of the two datasets. In this study, we set MRE < 0.2 and R > 0.85 as the criteria for an acceptable prediction. In the parameter optimization of CSUH, y represents the variables q, p′, εv, u or e, while x represents ε1.

Fig. 4
figure 4

The individual error of a single curve fi

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It should be noted that the two criteria have a limitation when the values of \(y_{e}^{i} (x_{i} )\) are close to zero, making the denominators of Eqs. (8) and (9) close to zero as well. The errors calculated by them are therefore exaggerated. On this occasion, these criteria may not be applicable.

Total error: Suppose there are altogether N3 comparisons between the experimental data and model prediction, as shown in Fig. 4 (here N3 = 3), whose mean relative errors are MREj, j = 1, 2,…, N3. The total error Λ is defined as their average value.

$$\Lambda = \frac{1}{{N_{{3}} }}\sum\limits_{{j = {1}}}^{{N_{{3}} }} {{{MRE}}_{j} }$$
(10)

Obviously, a smaller total error indicates a better fitting exemplar. For N3 relationships yj = fj(X), j = 1, 2,…, N3, find the X associated with the minimum total error subject to the restriction that X is within the search area R, then the optimization statement is

$$\begin{gathered} \min \, \Lambda = f({\mathbf{X}}) \hfill \\ {\text{s.t.}}{\mathbf{X}} \in {\mathbf{R}} \hfill \\ \end{gathered}$$
(11)

In this study, the investigated relationships are (1) N3 number of axial strains versus the deviatoric stress. (2) N3 number of axial strains versus volumetric strain and (3) one curve depicting isotropic compression. N4 is the total number of individual errors in drained tests and is equal to 2 × N3 + 1.

4 Calibration and validation

4.1 Clay material

4.1.1 Calibration

The experimental results from drained and undrained conventional triaxial compression tests on saturated Lower Cromer Till (LCT) clay were collected from Pestana et al. [25]. The LCT clay in isotropic compression test is compressed to \(p^{\prime}_{{\text{c}}}\) and then unloaded with different over-consolidation ratios (OCRs). The pre-consolidation pressure \(p^{\prime}_{{\text{c}}}\) (also denoted by \(\sigma^{\prime}_{{\text{c}}}\)) equals 770 kPa, and the corresponding void ratio ec is 0.345. The isotropic compression test result is actually a part of CD tests related to the initial values of e0 and lnp0 of the individual triaxial compression tests associated with various confining pressures p0. In the consolidation stage, the isotropic compression test is carried out to obtain the initial values of e0 and lnp0 before shearing stage of the triaxial compression tests. Thus, the isotropic compression test result is actually a part of CD tests.

The experimental data from isotropic compression test and CD test are adopted for the MAPSO algorithm to calculate the basic parameters of the CSUH model. Here, the population dimension N1 = 75 in each generation. The best exemplar, selected from 80 × N1 random initial exemplars, is taken as the initial guess parameter. The set of parameters that leads to the minimum total error is taken as the optimization parameter. The change in the total error value in the optimization process is shown in Fig. 5. The initial guess and final optimization parameters are given in Table 5.

Fig. 5
figure 5

The generation t versus the total error value Λ during the parameter optimization of the LCT clay

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Table 5 Optimization parameters for LCT clay by MAPSO
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In this particular case, the researchers also performed isotropic compression and extension tests for the LCT clay, from which one may gain the actual basic parameters M = 1.2, κ = 0.009, λ = 0.06 and N = Z = 0.752, as shown in Fig. 6 [25, 48]. This thus gives us an opportunity to check the performance of the calibration method. The comparisons between the actual basic parameters and the optimized parameters show that all the parameters except for κ are in good agreement. The isotropic compression test is a part of the CD test, while the isotropic extension test is not. Therefore, the isotropic extension test is not used in the back analysis of parameters, which may be the reason why the optimized κ differs greatly from the actual κ.

Fig. 6
figure 6

Comparisons between the isotropic compression and extension tests of LCT clay [25] and CSUH simulations in e ~ lgp′ coordinates

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With the optimized parameters listed in Table 5, the mechanical response of the LCT clay can be predicted by the CSUH model. Figure 7 shows the comparison between experiments and model prediction in terms of axial strain ε1 versus deviatoric stress q and axial strain ε1 versus volumetric strain εv. The individual errors MRE and R of most prediction summarized in Table 6 meet the criteria MRE < 0.2 and R > 0.85. Only the prediction of the volumetric strain the values of relative for OCR = 10 slightly exceeds the criteria, while the absolute difference between experimental and predicted results is small. A comprehensive analysis shows that the prediction of volumetric strain is good. In general, the CSUH model gives an excellent prediction of the deviatoric stress and the volumetric strain of LCT clay in the drained test under different OCRs.

Fig. 7
figure 7

Comparisons between the CD test results of LCT clay [25] and CSUH simulations: a axial strain versus deviatoric stress and b axial strain versus volumetric strain

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Table 6 The MRE and R values of model prediction in CD tests on the LCT clay
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4.1.2 Validation

The stress–strain relations during the undrained tests are more easily affected by the dilatancy than that in drained tests. The volumetric strain is usually smaller than the deviatoric strain during the drained shearing. Thus, if the prediction of the volumetric strain caused by the dilatancy is inaccurate, the prediction deviation of the stress–strain relations in the drained stress path is not significant. Moreover, dilatancy does not affect the stress path and the residual strength in the drained conditions; thus, it has less influence on the stress–strain relations in the drained conditions.

However, in the undrained conditions, the volumetric strain increment is zero, and there is a coupling relationship between the recoverable and the unrecoverable increments. In an undrained shearing, the unrecoverable volumetric strain increment is affected by its deviatoric component through dilatancy. Then, the unrecoverable volumetric strain increment influences the recoverable volumetric strain increment through the coupling relationship, thereby affecting the mean effective stress. The dilatancy greatly affects the effective stress path, and thus, it can significantly affect the strength of soils under undrained conditions. A slight error in the description of dilatancy under the undrained conditions may result in a sizeable stress–strain relation deviation, and the phenomenon “one false step will make a great difference” will appear. Therefore, it is essential and critical to validate the constitutive model through undrained tests.

In what follows, we will validate the CSUH model by predicting the undrained mechanical response of LCT clay using the calibrated parameters listed in Table 5. The selected experiments are related to OCR values of 1, 1.25, 2, 4, and 10. Figure 8 shows the comparisons between the experiments and prediction in terms of ε1 versus \({{\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)} \mathord{\left/ {\vphantom {{\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)} {\left( {2\sigma^{\prime}_{{\text{c}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2\sigma^{\prime}_{{\text{c}}} } \right)}}\) and ε1 versus u.

Fig. 8
figure 8

Comparisons between the CU test results of LCT clay [25] and the CSUH predictions: a axial strain versus deviatoric stress and b axial strain versus excess pore pressure

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Similar to Table 6, we also calculate the relative errors MRE and R in the prediction of undrained triaxial tests on LCT clay. The total errors of the ε1 versus \({{\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)} \mathord{\left/ {\vphantom {{\left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right)} {\left( {2\sigma^{\prime}_{c} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2\sigma^{\prime}_{c} } \right)}}\) relationships for all cases of OCR values are good, meeting the criteria of MRE < 0.2 and R > 0.85 as shown in Table 7. With respect to the ε1 versus u relationship, we find that MRE and R cannot meet these two criteria for OCR = 4 and 10. A further inspection on Fig. 8b reveals that the values of u are very close to zero, making both MRE and R be exaggerated by using Eqs. (8) and (9). In general, the predictions on excess pore pressure also show good agreement with the experiments except for the case of OCR = 10 in which the values MRE and R are 0.708 and 0.453, respectively.

Table 7 Evaluation of the individual errors MRE and R in the prediction of undrained triaxial tests on the LCT clay
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4.2 Granular material

4.2.1 Calibration

A further simulation is performed to examine the proposed method for calibrating and validating the CSUH model in predicting the mechanical behavior of granular materials. A saturated rockfill material from the Changhe dam (CHD) is selected for this investigation [17]. This material was thoroughly investigated by Liu et al. [17] through isotropic compression, CD, and CU tests with a high-pressure triaxial apparatus.

The model parameters for the rockfill material are calibrated using the MAPSO algorithm based on isotropic compression and CD tests. The initial and optimized results are listed in Table 8.

Table 8 Optimization parameters for the rockfill material from the Changhe dam
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The simulated results of the isotropic compression test and the drained triaxial tests on the rockfill material are, respectively, shown in Figs. 9 and 10. It can be seen that the CSUH model gives rise to excellent predictions of both tests. Most of the predictions agree well with the test results and meet the criteria with MRE < 0.2 and R > 0.85 as shown in Table 9, except for the prediction of volumetric change with the confining pressure of 400 kPa, in which the MRE and R is 0.45 and 0.47, respectively. These results suggest that the CSUH model can sufficiently describe dilatancy of rockfill materials.

Fig. 9
figure 9

Comparisons between the isotropic compression test results of the rockfill [17] and the CSUH simulations in the e ~ lgp′ coordinates (MRE = 0.003, R = 0.996)

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Fig. 10
figure 10

Comparisons between the CD test results of the rockfill [17] and the CSUH simulations: a axial strain versus deviatoric stress and b axial strain versus volumetric strain

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Table 9 Evaluation of the individual errors MRE and R in the prediction of drained triaxial tests on the rockfill
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4.2.2 Validation

The following content is related to the validation of the CSUH model with the calibrated parameters. Likewise, the parameters listed in Table 8 are used to predict the undrained behavior. The comparisons between experiments and model prediction are shown in Fig. 11.

Fig. 11
figure 11

Comparisons between the CU test results of the rockfill [17] and the CSUH predictions: a axial strain versus deviatoric stress and b axial strain versus excess pore pressure

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An inspection of Fig. 11 suggests that the numerical predictions of the relationships ε1 versus q and ε1 versus u in undrained triaxial tests agree well with the experimental results. Most of the predictions meet the pre-described criteria except for the tests with confining pressure of 400 kPa in which the values of MRE and R of the relationships of ε1 versus q are 0.261 and 0.757, respectively, and the values of MRE and R of the relationships of ε1 versus u are 0.685 and 0.490, respectively, refer to Table 10. This may be attributable to the fact that granular material may exhibit remarkable dilatancy under low level of confining pressure. In general, the CSUH model can reasonably describe the strain hardening, as well as the phenomenon that the excess pore water pressure first increases and then decreases in the undrained tests.

Table 10 Evaluation of the individual errors MRE and R in the prediction of undrained triaxial tests on the rockfill
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5 Conclusions

This paper presents a standard methodology for calibrating and validating constitutive models with conventional triaxial tests results. For this purpose, we consider a recently proposed constitutive model, the CSUH model, for clays and sands. The CSUH model is an upgrade in theory and innovative development of the modified Cam-Clay model. By introducing the compressive hardening parameter ps, the CSUH model can describe the compressive hardening characteristics of both sands and clays. In addition, the CSUH model adopts the same hardening law of the original UH mode, which is able to capture dilatancy and strain softening behaviors of overconsolidated soils. By combing all the features in a single framework, the CSUH model is able to sufficiently decipher the mechanical behaviors of various soils (e.g., clays, sands, and rockfill materials, etc.) in a uniform way, which is exactly what Wroth and Houlsby expected [42].

It is believed that verifying the CSUH model with the independent tests can increase the credibility of this research. With the help of the MAPSO algorithm, the parameters of the CSUH model can be calibrated through drained triaxial compression tests on clay and rockfill materials. The accuracy of the optimization is examined by two relative errors. Furthermore, the CSUH model with the calibrated parameters is validated by simulating undrained triaxial compression tests. The prediction shows an excellent agreement with the experimental results. In general, the results demonstrate that the CSUH model has a strong ability for predicting the mechanical behavior of various soils.